Computational Logistics (Lecture Notes in Computer Science)

Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris...

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Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany

6971

Jürgen W. Böse Hao Hu Carlos Jahn Xiaoning Shi Robert Stahlbock Stefan Voß (Eds.)

Computational Logistics Second International Conference, ICCL 2011 Hamburg, Germany, September 19-22, 2011 Proceedings

13

Volume Editors Jürgen W. Böse Carlos Jahn Hamburg University of Technology, Institute of Maritime Logistics Schwarzenbergstraße 95 (D), 21073 Hamburg, Germany E-mail: {juergen.boese; carlos.jahn}@tu-harburg.de Hao Hu Xiaoning Shi Shanghai Jiao Tong University School of Naval Architecture, Ocean and Civil Engineering Dongchuan Road 800, Shanghai 200240, China E-mail: {hhu; sxn}@sjtu.edu.cn Robert Stahlbock Stefan Voß University of Hamburg, Institute of Information Systems Von-Melle-Park 5, 20146 Hamburg, Germany E-mail: [email protected]; [email protected]

ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-24263-2 e-ISBN 978-3-642-24264-9 DOI 10.1007/978-3-642-24264-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011936536 CR Subject Classification (1998): F.2, D.2, I.2, C.2, F.1, I.6 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Computational logistics comprises the planning and implementation of large/complex logistics tasks using computations and advanced decision support. It is applied in various areas, such as the ﬂow and storage of goods or services as well as related information from their source to their destination. Typically, optimization models and algorithms are developed, veriﬁed and applied for planning and executing complex logistics tasks, e.g., ﬁnding the most eﬃcient scheduling/plan for the transport of passengers or goods. These models and algorithms are integrated with advanced computer technology to get satisfactory results in appropriate time even for large-scale problem instances and to provide interactivity, visualization etc. for a better understanding and problem solution. Furthermore, computational logistics involves the use of information systems and modern communication and information technology (IT) for the design, planning and control of logistics networks as well as the complex tasks within them. The International Conference on Computational Logistics (ICCL) provides an opportunity for researchers and practitioners in the ﬁeld of computational logistics to present their latest results and ﬁndings in a fruitful and open-minded environment. This volume of the Lecture Notes in Computer Science consists of selected papers presented at the Second International Conference on Computational Logistics, held at the University of Hamburg, Germany, during September 19–22. ICCL 2011 was the second of its kind, the ﬁrst was held in 2010 in Shanghai, China. ICCL 2011 was located in the most beautiful city in Germany, Hamburg. Hamburg is part of one of the most important logistics networks. Among others, the city features two airports within the premises of the city, one of the three biggest container ports in Europe belonging to the largest container ports worldwide. As a metropolitan region, numerous trade activities involving multitudes of companies take place every day providing great challenges for optimization upand downstream along the value chain. This makes Hamburg the perfect place to inspire new ideas in the ﬁeld of computational logistics as well as information and communication sciences. On top of this, the year 2011 also marked two speciﬁc events making Hamburg the perfect place for hosting this conference. First, Hamburg was awarded as the Green Capital of Europe 2011, making The Challenge of Sustainability a special focus of the conference. Moreover, the joint eﬀort between Shanghai, host of the ﬁrst ICCL, and Hamburg, being the host of this one, reveals the good collaboration between these two cities, marking the 25th anniversary of the partnership between the two cities.

VI

Preface

The contributions presented at the conference as well as the selected papers in these proceedings highlight recent developments in computational logistics. We grouped the contributions in three parts as follows: – Part I: Transport Services – Part II: Logistics Systems and Production – Part III: Maritime Shipping and Container Terminals Organizing a conference and publishing the proceedings is of course an endeavor involving many people in numerous activities. We ﬁrst want to thank all authors and presenters for their contributions. Moreover, we greatly appreciate the valuable help and cooperation from the members of the international Program Committee as well as the referees. September 2011

J¨ urgen W. B¨ ose Hao Hu Carlos Jahn Xiaoning Shi Robert Stahlbock Stefan Voß

Organization

Organizing Chairs Committee Hao Hu Stefan Voß

Shanghai Jiao Tong University, China University of Hamburg, Germany

Organizing Committee J¨ urgen W. B¨ ose Xiaoning Shi Robert Stahlbock

Hamburg University of Technology, Germany Shanghai Jiao Tong University, China, and University of Hamburg, Germany University of Hamburg, Germany, and FOM University of Applied Sciences, Essen/Hamburg, Germany

Program Committee and Referees Hannes Bade Michael G.H. Bell Johann Bergmann J¨ urgen W. B¨ ose Birgitt Brinkmann Reiner Buhl Marco Caserta Marielle Christiansen Teodor Gabriel Crainic Joachim R. Daduna Wolfgang Domschke Karl D¨ orner Rene Eisenberg Kjetil Fagerholt Jens Froese Hans-Otto G¨ unther Richard F. Hartl

Fraunhofer Center for Maritime Logistics and Services CML, Hamburg, Germany Imperial College London, UK Hamburg University of Technology, Germany Hamburg University of Technology, Germany Leuphana University L¨ uneburg, Germany Fraunhofer Center for Maritime Logistics and Services CML, Hamburg, Germany University of Hamburg, Germany Norwegian University of Science and Technology, Norway Universit´e du Qu´ebec `a Montr´eal, Canada Berlin School of Economics and Law, Germany TU Darmstadt, Germany Johannes Kepler University Linz, Austria HPC Hamburg Port Consulting, Germany Norwegian University of Science and Technology, Norway Jacobs University Bremen, Germany TU Berlin, Germany University of Vienna, Austria

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Organization

S¨ onke Hartmann Geir Hasle Sin C. Ho Dmitry Ivanov Carlos Jahn Rune Møller Jensen Achim Koberstein Herbert Kopfer Leo Kroon Gilbert Laporte Hong K. Lo Arne Løkketangen Frank Meisel Rudy Negenborn Julia Pahl Matthew Petering Warren B. Powell Robert Rauer Juan Jos´e Salazar Gonz´ alez Silvia Schwarze Hans-J¨ urgen Sebastian Xiaoning Shi Dongping Song Grazia Speranza Sven Spieckermann Robert Stahlbock Theodore Tsekeris Jens Wollenweber David Woodruﬀ

HSBA Hamburg School of Business Administration, Germany SINTEF Applied Mathematics Department of Optimisation, Norway Aarhus University, Denmark University of Hamburg, Germany and St. Petersburg University, Russia Hamburg University of Technology, Germany IT University of Copenhagen, Denmark University of Paderborn, Germany University of Bremen, Germany Erasmus University Rotterdam, The Netherlands HEC Montr´eal, Canada The Hong Kong University of Science and Technology, Hong Kong Molde College, Norway Martin Luther University Halle-Wittenberg, Germany Delft University of Technology, The Netherlands University of Hamburg, Germany K¨ uhne Logistics University, Germany Princeton University, USA Hamburg University of Technology, Germany University of La Laguna, Spain University of Hamburg, Germany RWTH Aachen, Germany Shanghai Jiao Tong University, China, and University of Hamburg, Germany University of Plymouth, UK Brescia University, Italy Simplan AG, Germany University of Hamburg, Germany Centre of Planning and Economic Research (KEPE), Greece Fraunhofer SCS, Germany University of California (Davis), USA

Table of Contents

Transport Services Combinatorial Auctions in Freight Logistics . . . . . . . . . . . . . . . . . . . . . . . . . Heiner Ackermann, Hendrik Ewe, Herbert Kopfer, and Karl-Heinz K¨ ufer

1

Shipper Decision Support for the Acceptance of Bids during the Procurement of Transport Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tobias Buer and Herbert Kopfer

18

Aspects of Information Management in Road Freight Transport . . . . . . . . Joachim R. Daduna

29

The Single-Stage Location-Routing Problem with Time Windows . . . . . . Halil Ibrahim G¨ und¨ uz

44

A Cross Entropy Multiagent Learning Algorithm for Solving Vehicle Routing Problems with Time Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tai-Yu Ma

59

Modelling the Synchronization of Transport Means in Logistics Service Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dorota Slawa Mankowska, Christian Bierwirth, and Frank Meisel

74

Optimization of Infectious Medical Waste Collection Using RFID . . . . . . Pamela C. Nolz, Nabil Absi, and Dominique Feillet

86

The Pickup and Delivery Problem with Cross-Docking Opportunity . . . . Hanne L. Petersen and Stefan Ropke

101

Application of an RFID-Based System for Construction Waste Transport: A Case in Shanghai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tingting Ruan and Hao Hu

114

Strategic and Operational Planning of Bike-Sharing Systems by Data Mining - A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Patrick Vogel and Dirk C. Mattfeld

127

Logistics Systems and Production Economic Impacts of the Alternative Reuse of Empty ISO Containers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Großkurth, Robert Stahlbock, and Stefan Voß

142

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Table of Contents

Optimal Issuing of Perishables with a Short Fixed Shelf Life . . . . . . . . . . . Ren´e Haijema

160

The Maximum Flow Problem with Minimum Lot Sizes . . . . . . . . . . . . . . . Dag Haugland, Mujahed Eleyat, and Magnus Lie Hetland

170

Multiobjective Evolutionary Algorithm for Redesigning Sales Territories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loecelia Ruvalcaba, Gabriel Correa, and Vittorio Zanella

183

Optimizing Complex Logistics Systems with Approximative Consideration of Short-Term Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tobias Winkelkotte

194

Maritime Shipping and Container Terminals Application of RFID Technology at the Entrance Gate of Container Terminals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lei Hu, Xiaoning Shi, Stefan Voß, and Weigang Zhang

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A Three-Level Hierarchical Workload Management Scheme for Yard Cranes in Container Terminals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shell Ying Huang, Xi Guo, and Mei Mei Lau

221

A Service-Oriented Model for the Yard Management Problem in Container Terminals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jian Gang Jin, Jin Xin Cao, Jiang Hang Chen, and Der-Horng Lee

233

Container Terminal Yard Operations – Simulation of a Side-Loaded Container Block Served by Triple Rail Mounted Gantry Cranes . . . . . . . . Jan Klaws, Robert Stahlbock, and Stefan Voß

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Randomized Algorithm with Tabu Search for Multi-Objective Optimization of Large Containership Stowage Plans . . . . . . . . . . . . . . . . . . Fan Liu, Malcolm Yoke Hean Low, Wen Jing Hsu, Shell Ying Huang, Min Zeng, and Cho Aye Win A Variable Neighborhood Search Heuristic for Tramp Ship Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fotini Malliappi, Julia A. Bennell, and Chris N. Potts Fast Generation of Near-Optimal Plans for Eco-Eﬃcient Stowage of Large Container Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dario Pacino, Alberto Delgado, Rune Møller Jensen, and Tom Bebbington Game Theoretical Aspects in Modeling and Analyzing the Shipping Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaoning Shi and Stefan Voß

256

273

286

302

Table of Contents

Scheduling Yard Cranes Considering Crane Interference . . . . . . . . . . . . . . . Ulf Speer, Gerlinde John, and Kathrin Fischer Solving the Resource Allocation Problem in a Multimodal Container Terminal as a Network Flow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elisabeth Zehendner, Nabil Absi, St´ephane Dauz`ere-P´er`es, and Dominique Feillet

XI

321

341

A Simulation Study for Evaluating a Slot Allocation Model for a Liner Shipping Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sebastian Zurheide and Kathrin Fischer

354

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371

Combinatorial Auctions in Freight Logistics Heiner Ackermann1 , Hendrik Ewe1, , Herbert Kopfer2, and Karl-Heinz K¨ ufer1 1 2

Fraunhofer ITWM, Kaiserslautern, Germany [email protected] Chair of Logistics, University Bremen, Germany

Abstract. Freight business is a huge market with strong competition. In many companies, planning and routing software has been introduced, and optimization potentials have been widely exploited. To further improve eﬃciency, especially the small and medium sized carriers have to cooperate beyond enterprise boundaries. A promising approach to exchange transportation requests between freight carriers is provided by combinatorial auctions and exchanges. They allow bundles of items to be traded, thereby allowing participants to express complex synergies. In this paper we discuss various goals for a combinatorial request exchange in freight logistics and provide the reasoning for our design decisions. All goals aim to improve usefulness in a practical environment of less-than-truckload (LTL) carriers. We provide experimental results for both generated and real-life data that show signiﬁcant savings and are often close to a heuristic solution for the global optimization problem. We study how bundling and restricting the number of submitted bids aﬀect the solution quality.

1

Introduction

Freight business has become a huge market, having a volume of 76.1 bn. A C alone in Germany in 2008, for example (all numbers taken from [8]). Competition is tough and freight carriers in recent years have introduced routing and planning software to improve eﬃciency within their companies. But the market is quite scattered, with many small and medium sized companies. Although average business sizes have increased recently, in Germany 60% of the surveyed 2600 businesses employ less than 50 people. As a consequence, it is crucial for further optimization to aim at eﬃcient cooperation between companies, as an additional approach for enhancing the competitiveness. But there are not only economic reasons, of course, that argue for a stronger cooperation: also from an ecological point of view it is desirable to decrease driving distances, and to especially avoid deadheads, i.e., parts of tours driven without any load. Currently, a freight carrier’s planning process is often organized as follows: In the course of the day, new requests for transportation are acquired, and the planned tours for the own ﬂeet of trucks are constantly updated. A request

Part of this work was funded by Stiftung Rheinland-Pfalz f¨ ur Innovation research grant 961 - 386261/864. Corresponding author.

J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 1–17, 2011. Springer-Verlag Berlin Heidelberg 2011

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H. Ackermann et al.

consists of a pickup and a delivery location, a pickup and a delivery date and possibly time windows, and a further description of the goods to be transported (like, e.g., the weight and the loading meters). The requests are typically scheduled a few days in advance, the very next day mostly being the earliest pickup date possible. Because of these short-term aspects, the problem has a certain online-ﬂavor to it, and the human dispatcher has to decide quickly, which requests to accept and which not, and how to set prices. For this reason he will sometimes accept something in the ﬁrst place, but then be able to acquire something ﬁtting even better just a couple of hours later. To be more ﬂexible in the planning process, carriers often have a (small, ﬁxed) set of subcontractors. All requests that were initially accepted for servicing, but in the end still not ﬁt in the scheduled tours, are forwarded to them. Using a subcontractor is typically more costly than including the request into a carrier’s own tour, because chances are high that the subcontractor can also not include an arbitrary request in any of his routes very well. At this point, cooperation with other freight carriers seems especially promising: If some of them can ﬁt undesirable requests into their routes, eﬃciency can increase substantially. Of course there are already various mechanisms and platforms for freight exchanges available on the Internet. Most of them are open to everybody and work more or less like a bulletin board. A big problem is quality of service. If a carrier oﬀers a request on these platforms, he does not really know who is going to acquire it and therefore will be responsible for executing his request. But most carriers would like to provide a constantly high level of service and quality to their customers, which they can then no longer guarantee. Personal conversation with a major German freight carrier conﬁrm that this is indeed a big concern and a serious reason for not using existing systems. Secondly, most of the systems do not account for the highly synergetic character of freight business. In many situations, carriers will only want to accept a certain request if they do also get a speciﬁc second one, which is complementary to the ﬁrst one in their respective situation. Bidding on both requests separately exposes them to the risk of getting just one. What they need to express instead is what we call a “bundle bid” on both items: They either want all requests in the bundle (for the stated price), or none of them. Similarly, one can easily imagine situations where just one arbitrary out of a number of requests is wanted. Bidding on one speciﬁc item may leave the carrier losing, whereas bidding on multiple requests involves the risk of winning more than one of them. The basic idea to solve these issues and to still exploit the synergies of collaboration is to create a closed computer system that enables carriers to trade requests by means of a combinatorial exchange. While this idea is generally straight-forward, the details of composing such a system are not. Especially the underlying mathematical model needs to be designed carefully to provide a suitable framework that addresses the carriers’ needs. We will discuss the requirements on such a system in detail in the next section, where we introduce our approach for a combinatorial exchange platform for collaborating freight carriers. We describe our use case and motivate our design decisions, also by contrasting

Combinatorial Auctions in Freight Logistics

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them with the mathematical models currently available in literature. In Section 3 we give a brief overview on challenges in proﬁt sharing, and in Section 4 we show how the users of the system are supported in their decisions by our system. The practical application of the presented model is studied in Section 5. We conclude in Section 6.

2

Modeling Combinatorial Exchanges

In this section we describe our main objectives when modeling the combinatorial exchange, and present the resulting design decisions. With all goals we aim to improve usefulness of the system in a practical freight carrier environment. Nevertheless, many ideas and results are transferable to similar settings. 2.1

Modeling Goals

Our design goals are directly derived from practical considerations and serve to increase acceptance for the system amongst freight carriers. One of their main concerns certainly is data privacy protection. Since many competing carriers participate in the collaboration platform, sensitive data may under no circumstances leak or be derived from other information published to the system. While the former is more of a technical challenge and requires a solid software design, the later must be faced by designing an auction that only requires minimal information from all participants. A second issue freight carriers are concerned about is losing their decision competence to a (black box) software they do not fully understand. The measures we propose to address these worries are, on one hand, leaving as much authority at the carriers’ side as possible. On the other hand, we aim to create an easy way to understand and transparent auctioning system, particularly on the part of the bidding language and the proﬁt sharing. Closely related to this is the predictability of the auction outcome. Of course the participants cannot anticipate whether their bids will win or not, but they must be enabled to express which scenarios they can imagine for their ﬂeet and which they can not. They must never be surprised by an auction outcome that contradicts their planning. Key to this is the design of a suitable bidding language and a proper allocation algorithm. The auction platform strongly depends on suﬃcient participation, especially as synergies amongst carriers are often superlinear, and bigger auctions promise larger beneﬁts. Therefore, it is a crucial point to provide eﬀective incentives, for example by oﬀering free participation and by designing a system that is guaranteed to only improve the carriers’ situations. One more goal is the smooth integration of the auction platform into the carriers’ existing planning processes without causing much overhead. On the technical side this involves building interfaces to access the company data, which will not be discussed here. On the organizational side it requires to establish the new process within the company. To overcome this barrier we carefully analyze the use case in the next section and derive our auction model from it.

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2.2

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Use Case

We would like to design an auction that allows freight carriers to frequently exchange requests for transportation, based on their current state of planning. A typical use case would be to hold an auction one to three times per day. As our goal is to integrate into the existing workﬂow, we do not want a complete rescheduling. Instead we would like to complement pre-planned tours by new requests to increase eﬃciency and improve proﬁt, and forward requests, which are currently candidates for subcontracting, to other carriers at a lower price. The requirements from practice have the nice side eﬀect that they also reduce complexity compared to a complete rescheduling of requests. Generally the auctioning system should require as little additional time and eﬀort from the dispatcher as possible. Therefore, we suggest strong decision support modules to enable (semi-)automation for the tedious and non-critical tasks. For the same reason we do not consider iterative auctions with multiple turns of oﬀering and bidding: Applying such a procedure several times per day seems unsuitable. Consequently, we end up with the almost classical chronology of an auction-based cooperative freight exchange: 1. Oﬀering phase, where requests are placed on the platform to have them fulﬁlled by other participants, 2. Bidding phase, where bids on the oﬀered requests are placed, 3. Assignment phase, where the Winner Determination Problem (WDP) is solved to determine the redistribution of requests amongst participants, 4. Proﬁt sharing phase, where the generated proﬁt is shared according to a previously selected proﬁt sharing scheme. Dividing up the user interaction into steps 1 and 2 is not critical, as long as participants do not regret decisions from phase one in phase two, when they receive additional information. The split of oﬀering and bidding into two separate steps is again also a means to reduce complexity of the system. It helps the users to focus on the current task, since they can work through both phases consecutively and independently from each other. Throughout the auction, all coalition partners participate both as an oﬀering and a bidding party. However, there is no constraint to both oﬀer and bid on bundles, every participant can just as well take one of the both roles. All stages are synchronized between carriers, such that each of them is always in the same stage. Oﬀering Phase. Participants may formulate both bids and oﬀers as bundles of requests, attached with a price. For an oﬀering bundle, one or more executing carriers are sought and the price indicates the maximum amount the oﬀering carrier is willing to pay, if all the requests in the bundle are executed. Oﬀering requests as a bundle instead of oﬀering them each individually not only allows to express synergies between requests (e.g., if they have similar origins and/or destinations), but also guarantees that either all or none of the request within the bundles are sold.

Combinatorial Auctions in Freight Logistics

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Bidding Phase. In the second phase, all requests that have been submitted to the exchange platform are revealed to the participants. To meet the data privacy requirement, only the very basic request information are given. Participants do neither know the oﬀering party nor the oﬀering price, and do not have to worry about potential constraints arising from bundle oﬀers. This is taken care of by the platform automatically. The carriers now check the tours in the preliminary transportation plan and decide, whether some of the oﬀered requests make a good supplement to the planned routes. They then submit bundles of requests to the platform and thereby express their willingness to fulﬁll a set of requests for a certain monetary reward. In contrast to oﬀering, the price set for a bidding bundle tells the minimum amount of money asked by the bidder to fulﬁll the complete set of requests contained in the bundle. We do not require any additional information from the carriers, be it valuations for bundles they are not bidding on, for subsets of their bundles, or for single requests. As a technical aspect, for bidding bundles we furthermore transmit a label identifying all other bundles this bundle conﬂicts with. This implements the XOR functionality described in Section 2.3. Assignment Phase. After everybody has placed their bids, we reassign requests to carriers to minimize total fulﬁllment costs. In Combinatorial Auction literature this is commonly referred to as the Winner Determination Problem (WDP). To simplify further treatment, we consider each oﬀering bundle also as a bidding bundle of the very same carrier on the same set of requests for the same price. That way we do not have to take care of the special case of requests not being sold. Solving the WDP requires solving a weighted Set Packing Problem (SPP), which is known to be computationally hard [9]. The result is a partition of the set of bidding bundles into winning and losing ones. This determines the actual transfer of requests from one company to another, and is the basic input data for the subsequent proﬁt sharing phase. For the bidding bundles we can easily relax the bundling by removing some requests while keeping the price the same. This is a valid modiﬁcation because no carrier will object driving less for the same money. This feature can solve overlaps of big bundles, which all contain some common request. In such a case, only one of the bundles is assigned the request, which can sometimes lead to better revenues. Mathematically this means solving a weighted Set Covering Problem (SCP) instead of a SPP. Proﬁt Sharing Phase. The total amount of proﬁt generated by the reassignment of requests among the participants of the auction can be simply determined as the sum of the oﬀering bundles prices minus the sum of the winning bidding bundles prices. The goal now is to distribute it in a way considered fair by all participants. Important ingredients for a proﬁt sharing scheme can be found in literature (see, e.g., [16]), but generally it is not possible to achieve all desired properties together. More details on this can be found in Section 3.

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Expressing Evaluations

Each carrier has to evaluate the savings when giving away certain requests, or determine the additional expenses when acquiring new requests from the auctioning platform. One way to do so is a before-after comparison: The carrier optimally allocates all requests to his own trucks and subcontractors as a baseline scenario and compares the costs to the new setting (with the new requests included or with the requests given away, respectively). The diﬀerence, called Δ-price, can be used by the carrier as the bundle price. Calculating the Δ-price is easy when oﬀering requests allotted for subcontracting. When giving them away, all tours of own trucks remain untouched and the only diﬀerence is the subcontracting costs, which can be typically just summed up for all oﬀered requests. Determining the Δ-price is a lot more complicated when integrating new requests into existing tours. One has to solve a Pickup and Delivery Problem (PDP) for potentially each possible bundle to be evaluated, whose number may be exponential in the number of oﬀered requests. Clearly, this is infeasible to solve optimally in general. But also the user himself is a limiting factor, since nobody is able to merely inspect 210 or even 220 bundles, even when they are generated automatically in the ﬁrst place. For these reasons we have to reduce complexity, especially of the bidding process, by a considerable amount. Our model tackles this problem by the following measures: Fixed initial tours to decouple oﬀering and bidding. We connect to a state of planning where a subset of the own requests is assigned to the own ﬂeet, and tours for the individual trucks have been planned. By ﬁxing this assignment of requests to trucks (but not the actual tours, of course), we avoid the complexity of redistribution. Instead we trust the assignment which has been planned by the dispatcher in a previous planning step, probably with the help of software tools and his rich experiences. Keeping the current assignment as stable as possible is, of course, also appreciated by the carriers, as it simpliﬁes integration into their organizational processes. The requests that are candidates for subcontracting can now be blindly oﬀered to the other carriers, using the assumed subcontracting costs as a price. With this strategy, the participant either forwards a request to another carrier at a lower price, or does not sell it and sticks to his initial subcontracting plans. In both cases there is no loss for the carrier compared to his status quo. This decouples oﬀering from the subsequent bidding activities. The user can focus on the single steps and does not have to oversee any complex dependencies. Independence of trucks among each other. As there is no rearrangement of requests between own tours in the course of the auction, there are also no interdependencies between vehicles. For any request, of course, one has to consider that in the end it may only be assigned to one truck. In our model, this is taken care of automatically by the auctioneering software. Hence the user can freely plan bundles for a certain truck without having to care about overlaps with other

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bundles already submitted in the context of a diﬀerent vehicle. In terms of the bidding language, bundles from diﬀerent trucks can be connected by OR-bids if they do not contain a common requests, and by XOR-bids otherwise. XOR bids within vehicles. When considering a set of requests that could potentially extend a truck’s tour, only very few of the requests can be evaluated independently of each other in terms of Δ-costs: In some cases integrating one request leads to exclusion of other requests because of weight, loading meters, or time constraints. In other cases the modiﬁcation of the route, caused by integrating a certain request, changes the Δ-costs of other requests. Therefore, the requests’ valuations are not simply additive, and connecting them by ORbids is not reasonable. In our model we use XOR-bids instead, such that at most one of the bids for each vehicle is realized. The user can simply imagine to build diﬀerent scenarios for the vehicle from which at most one is chosen. That way he can work on the creation of each single scenario (i.e., bundle) without having to worry about other scenarios he created before. Altogether we end up with an OR-of-XOR bidding language: For each vehicle we connect bundle oﬀers by XOR-bids, while the bundles of diﬀerent vehicles can be linked by OR, since they are independent of each other (provided the auctioneer ensures that each requests is just assigned once). We know from auction theory that this bidding language can express any user valuation, although some valuations may require an exponential number of bundles. But these worst-case examples (see [19], p.219), do not seem relevant for practical purposes in freight logistics. Furthermore, in our scenario we are concerned with rather big loads and long loading times, so we typically see not more than 2 to 5 requests per truck and the number of relevant bundles is limited. Nevertheless, we have to assume that the set of submitted bundles is not always complete, and there might be no submitted bundle for a set of requests that is yet relevant for the carrier. To easy that problem we use automated subbundling. This actually has already been described in Section 2.2 as a relaxation of the set packing constraints. Seen from the bidders point of view, every bid (S, v(S)) on a subset S of the available requests, with a price v(S), implies a set of bids with the same price on all subsets T of S, i.e. a set of bids {(T, v(S)) | T ⊂ S}. As stated before, the justiﬁcation for this is that the oﬀering carrier will not mind driving less requests for the same amount of money. This concept is often referred to as ’free disposal’ for regular auctions. The term does not quite ﬁt in reverse (procurement) auctions, as participants do not get more but less than they asked for. We propose the term ’free omission’ instead. A question yet to be answered is, of course, how to determine the bundles that are automatically reduced. We propose a randomized procedure that is balanced between participants in the long run. 2.4

Existing Approaches

Both the slightly older survey by de Vries and Vohra [28] and the more recent one by Abrache et al. [1] give a great overview on the current literature on

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combinatorial auctions. Also, there is the excellent book “Combinatorial auctions” [6] which covers a broad range of aspects, starting from the game theoretic fundamentals and ending at examples of practical applications. From a game-theoretic point of view, designing the rules for an auction falls into the ﬁeld of mechanism design. Here, one of the historical corner stones certainly is Vickrey’s paper from 1961 [27]. His ideas on auction theory where later generalized by Clarke [4], Groves [11]. Combinatorial Auctions where ﬁrst described by Rassenti [23] and quickly emerged into an active ﬁeld of research. Myerson and Satterthwaite introduce the dilemma of mechanism design and show that trade-oﬀs with respect to the desired properties of mechanisms are inevitable. We will come back to that in the section about proﬁt sharing. Closely related to the design of a combinatorial auction is the choice of the bidding language. The paper by Nisan [18] and the corresponding chapter in [6] provide a comprehensive overview. Many authors try to ﬁnd domain speciﬁc languages, that are simple enough to be understood by practitioners and yet able to express most meaningful valuations [3,26]. Conen et al. [5] try to narrow down the number of bids by iteratively asking the participants for their preferred bundles. This does not seems applicable to our multi-vehicle scenario, where bundles from diﬀerent trucks interdepend. Krajewska and Kopfer [14] propose an OR-bidding language. [7] and [10] propose a special kind of matrix-form bidding, which allows to solve the WDP in polynomial time. Generally, the choice of bidding languages needs to ﬁt the practical problem, because otherwise bundling is rarely used by participants [22,3]. This is of course a strong point for decision support during bundling. A couple of papers propose tools to deal with that: Hsieh [13] reveals multipliers of Lagrangian Relaxation to determine the most promising single items in a reverse iterative combinatorial auction. Lee et al. [15] propose a non-linear mixed integer formulation for the carrier’s bid generation problem and solve it by a decomposition method. Wang et al. [29] give a heuristic for generating bids, that is founded on the assumption that all bids are equally likely to win in the auction, which seems too restrictive for our scenario. There are few suggestion on how to actually share the proﬁt between participants of an exchange. In practical applications, using VCG-payments is mostly out of question, as this leads to a mechanism that is not budget-balanced. Parkes et al. [21] investigate how to modify VCG-payments to achieve budget-balance while keeping individual rationality and as much incentive compatibility as possible. They propose a threshold rule to prevent agents with large incentives from manipulation. Krajewska and Kopfer [14] propose to distribute the proﬁt between both the sellers and the buyers. They require all participants to price all single items that are for sale, to be able to distribute bundle proﬁt on a single item basis. Pan et al. [20] apply the well-known Shapley Value to distribute proﬁt in a spectrum auction. ComEx [12,24] is the integrated system we found most related to our approach. It features a combinatorial auction to exchange delivery services between proﬁt centers by means of an OR-of-XOR bidding language. Their model assumes

Combinatorial Auctions in Freight Logistics

9

a strong geographical relationship between customers and proﬁt centers, and tries to reassign those that lie in the boundary regions of two or more centers. Customers too far away from a proﬁt center are clustered into groups by a greedy heuristic and oﬀered to the other centers. In the bidding phase, sets of these clusters have to be integrated into the bidders’ existing tours. This seems a bit inﬂexible, as the clusters are built from the oﬀering party’s point of view and there is no possibility to break up clusters during bidding.

3

Profit Sharing

After having determined the sold oﬀers and the winning bundles in the WDP, the realized proﬁt now has to be distributed amongst the participants. Generally, we would like to relate the amount of money disbursed to a participant to the value he adds to the grand coalition. There are two possible ways to assess that: On a per bundle basis, or per participant. As it should not matter in terms of proﬁt share by which carrier a certain bundle is oﬀered (i.e., in which context), we distribute the proﬁt on a “neutral” per bundle basis which does not take into account the owner of the bundle at all. As a nice side eﬀect, this prevents pseudonymous bidding1 . 3.1

Game Theoretic Properties

We would like to frame certain properties for our proﬁt sharing scheme. As we already speak of “proﬁt sharing” and not just generally of “pricing”, we anticipate that we would like to set prices in a way that achieves so-called budget balance (BB): The total amount paid by the oﬀering carriers should exactly match the amount received by the winning bidders, since we would like to cover all carrier payments from within the platform. Secondly, we necessarily have to respect the given bundle prices: Oﬀering carriers should not pay more than they indicated, and bidders should receive at least as much as they asked for. Mostly this is referred to as individual rationality (IR), sometimes also as voluntary participation. Unfortunately, the famous theorem of Myerson and Satterthwaite [17] states that requiring BB and IR already prohibits any mechanism from being eﬃcient2 . Following Parkes et al. [21], we choose to implement a mechanism that is not incentive-compatible (IC) (so participants will have an incentive to misreport their valuations), but in exchange computes an optimal reallocation based on the reported bundle values. This is of great value in a practical implementation, since participants can retrace the decisions of the mechanism and the ex-post allocation can be easily justiﬁed.

1 2

For an overview on pseudonymous bidding in combinatorial auctions see the corresponding chapter in [6]. An eﬃcient mechanism maximizes total increase in value over all participants.

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Fairness

In addition to the two properties BB and IR, we would like to distribute proﬁt in a way that is considered “fair” by all participants. One ingredient is the so-called CORE property. It states that every subcoalition of agents is awarded at least as much proﬁt as they could have generated amongst themselves. Therefore it is a certain stability criterion for the chosen proﬁt distribution. The decision that is left to be made is, whether all submitted bundles or just the ones that actually contribute to the trade should be considered in the proﬁt sharing process. If we take into account all submitted bundles and not only sold oﬀers and successful bids, we introduce an extra level of competitiveness to the proﬁt sharing. A bundle’s value is then not only determined by how much money it saves the coalition, but also by how replaceable it is. As a result, oﬀering bundles (asks) are typically valued much higher than bids, as they are not replaceable at all. The diﬀerences may grow extreme in highly competitive markets, where any bid can be substituted by a diﬀerent bid with an only slightly higher price. As a second disadvantage, such a proﬁt sharing model would reward participants for solely oﬀering services to the other participants, which leaves the door wide open for manipulative bidding. We therefore think of winner determination and proﬁt sharing as two separate processes: First we determine the best allocation, afterwards we share the proﬁt amongst the trading parties. Consequently we should then drop all information about competition and rather try to determine a bundle’s value solely by its contribution to the total proﬁt. The three properties BB, IR, and CORE, applied to the set of trading bundles, set the framework for a class of proﬁt sharings that can be represented by a monetary ﬂow network. We refer to [2] for details on this and just propose to use the Shapley Value [25] at this point. For extremely large instances, the computational burden might be too large and an approximation scheme, like the one described in [2], must be used.

4

Decision Support

Participating in the exchange process is a complex task for all users. It is particularly challenging to select bundles of requests in the bidding phase, as there are exponentially many possible bundles for each of the user’s vehicles. For example, if just 20 requests were oﬀered to the exchange and one carrier has 5 trucks, he theoretically has to assess over 5 million diﬀerent bundles and select the most promising ones. We provide the user with a special decision support module called the bidding agent, to help solving this task. For each vehicle, a ranking of bundle proposals is generated. Each bundle is ranked by the diﬀerence of a “market price” and its Δ-price (cf. Section 2.3): The bigger this diﬀerence is, the more costs can probably be saved by integrating this request into the respective tour. To generate the bundle suggestions, our algorithm randomly tries to include the pickup and

Combinatorial Auctions in Freight Logistics

11

the delivery location of a new request into the existing route, while respecting the given side constraints. The most promising solutions are kept and the procedure is iterated. We do not only keep the “best” solutions in terms of the costs diﬀerence just explained above, but we also try to keep variety high: A bundle is allowed to be worse than another if it contains a diﬀerent set of requests. That way we broaden our bids and cover more of the oﬀered requests. Of course, the number of bundles generated by this procedure can sometimes still be too high to be evaluated by the user. In Section 5.1 we report on how only submitting a small fraction of the generated bids aﬀects total savings.

5

Results

We tested our implementation of the freight exchange on both real world data and generated scenarios. The real world data was obtained by a major German logistics company and represents an extract of one week of their business data. The company is organized as three proﬁt centers, which play the role of individual participants in the exchange. The artiﬁcial scenarios were generated by our own software tool and vary in some parameters to reﬂect diﬀerent use cases for the freight exchange. Some scenarios were designed to have similar characteristics as the practical data we obtained, others were intentionally created to deviate. This allows us to study for which real world scenarios our exchange model is especially suited. Table 1. The generated and the practical test data used in the experiments load/unload #requests #requests avrg. avrg. tour scenario #carriers #tours times per req. in tours subcontr. #req./tour duration gen5a gen5b gen5c gen5d gen5e gen5f gen5g gen5h gen15a gen15b gen15c gen15d day day day day day

1 2 3 4 5

10 10 10 10 10 30 30 30 3 3 3 10

50 50 50 50 50 150 150 150 45 45 45 150

1:00h 1:00h 1:00h 0:45h 0:30h 1:00h 0:30h 0:30h 0:30h 0:45h 1:00h 1:00h

98 120 116 134 147 378 389 579 149 132 149 401

100 100 100 100 100 300 300 300 90 90 90 300

1.96 2.40 2.32 2.68 2.94 2.52 2.59 3.86 3.31 2.93 2.42 2.67

9:13 10:03 9:59 10:06 9:38 10:49 10:27 11:36 10:17 10:32 10:14 11:06

3 3 3 3 3

55 49 51 56 43

1:00h 1:00h 1:00h 1:00h 1:00h

91 75 71 85 70

218 203 190 195 210

1.65 1.53 1.39 1.52 1.63

8:20 7:55 7:31 7:30 7:44

All problem instances, both practical and generated, are listed in Table 1 together with some characteristic indicators. Before we turn our attention to the actual results, we should shortly discuss how to measure improvements. We take a practical approach and measure the decrease in total costs (i.e., the sum of all tour costs and subcontracting costs for all participants) achieved by the redistribution of requests. Our model takes as input parameters the costs for

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the own ﬂeet and for the subcontractors. For the own ﬂeet, we account 100A C+ A C + 0.10 A C . Please A C , while for subcontractors we charge 100A C + 1.20 km 1.00 km t·km note that this measure is very manipulable, since increasing subcontracting costs would result in bigger savings. Nevertheless, we think we took a well-balanced and practical choice, and especially the comparison to the global optimization problem that we describe in the next paragraph provides a meaningful measure. Probably the most interesting question to answer is by how much our exchange model is able to improve the participants’ situations. This, of course, strongly depends on the input data: The better the planning already is, and the fewer the tours overlap geographically, the less the savings can actually be. To get a feeling for each scenario’s room for improvement, we would like to compare our solution to the global optimum, that is, to a complete re-scheduling of all requests to vehicles, across all participants. We will refer to this as the global optimization problem. Solving the global optimization problem includes solving a Pickup-andDelivery Problem, which is known to be computationally hard to solve optimally, so we can only try to ﬁnd a good solution using heuristics. We use the commercial product Ilog Dispatcher, which includes various tabu-search variants and other sophisticated heuristics to model and solve the global optimization problem. Unfortunately, some restrictions from practice seemed impossible to model with Ilog Dispatcher, so we were not able to calculate the global problem solution for the practical data. Table 2. Total savings, relative to the pre-auction situation, using bundling and submitting all bundles suggested by the decision support tool. In three cases, no solution could be found by the global heuristic.

scenario auction gen5a gen5b gen5c gen5d gen5e gen5f gen5g gen5h gen15a gen15b gen15c gen15d day day day day day

1 2 3 4 5

localOpt LO→aucglobal LO→auc(LO) tion→LO heurist. auctionExt tionExt→LO

11.1% 8.4% 7.4% 11.2% 16.0% 3.6% 5.1% 1.4% 8.9% 6.1% 2.1% 1.3%

5.2% 5.5% 6.2% 10.5% 10.7% 3.5% 2.5% 5.9% 5.9% 5.4% 9.5% 6.6%

10.9% 10.9% 12.7% 15.6% 20.4% 6.4% 6.4% 7.9% 18.6% 17.3% 10.7% 8.5%

20.5% 21.2% 23.0% 30.9% 38.9% 18.3% 29.0% 27.1% 18.2% -

19.0% 18.0% 16.8% 23.0% 29.4% 17.2% 16.8% 18.0% 17.1% 17.0% 13.6% 14.0%

18.6% 18.5% 19.1% 25.2% 27.6% 18.8% 15.7% 18.7% 22.5% 21.1% 16.5% 17.0%

1.5% 1.1% 3.1% 2.7% 1.6%

-

-

-

2.4% 2.5% 4.3% 2.8% 2.5%

-

To test our exchange model, we let all carriers of a scenario oﬀer all of their subcontracted requests, and then use the bidding agent described in Section 4 and submit all proposed bundles. The results are given in Table 2. Savings for the pure auction range from 1.1% to 16%, with tentatively larger savings for scenarios with more requests per truck. The results can be improved

Combinatorial Auctions in Freight Logistics

13

substantially with a “local” pre- and post-optimization: If every carrier tries to redistribute requests between trucks optimally both before and after the auction, savings increase to 6.4% to 20.4%. For this optimization we also use Ilog Dispatcher. Our model is very ﬂexible in the amount of requests that are oﬀered by the carriers: They can freely decide whether to only oﬀer selected requests that do not ﬁt in, to oﬀer all requests that are currently scheduled for subcontracting, or to even oﬀer requests that are currently pre-planned for tours of own vehicles. They can then simply bid on their own requests and either get them back (without worsening their situation), or sell them to some other carrier, participating in the generated surplus. To explore this aspect, we simulated the following scenario: For every truck with at least two scheduled requests, the carrier oﬀers the one request which ﬁts worst, i.e., for which the eﬀort of including it into the tour is highest compared to some “market price”. The results for this simulation can be found in the column “auctionExt” of Table 2. The savings increase to 13.6% to 29.4% for the generated scenarios, and even further when we combine“auctionExt”with the local pre- and post-optimization described above. The eﬀect is a lot smaller for the practical data in absolute numbers, but the increase is up to 60% relatively (for“day 1”from 1.5% savings up to 2.4% savings), which is still a considerable amount. One reason for smaller savings is that in the practical data around 55% of the trucks just have a single scheduled request and therefore do not oﬀer any additional request at all (compared to the regular auction). 5.1

Reduced Number of Submitted Bids

In a daily practical use, no user will like to blindly submit the bundles proposed by the bidding agent. At least he or she will verify the bundles and probably also set the price for each of them. This requires a very strict ﬁltering of the proposed bundles, such that not more than a handful per truck need to be inspected. Especially in the “auctionExt” scenarios, this means a drastic reduction. We tested by how much the total proﬁt decreases, if we only submit at most 3 bundles per truck. We simply chose those bundles that promised the largest total savings, compared to a given market price. The results are given in Figure 1. Despite the huge decrease in the number of submitted bids (by a factor of up to 30), the results are still very good and most of the proﬁt can still be realized in all scenarios. We also tested a more elaborate heuristics for the selection of the 3 bundles, which tries to avoid that all trucks submit bundles for the same requests, but results did not substantially change. 5.2

Set Covering vs. Set Partitioning

To verify the eﬀects of using the Set Covering Problem (SCP) formulation as described in Section 2.2, we have also solved the SPP as a comparison for selected instances. To achieve as much of an eﬀect as possible, we have chosen “auctionExt” scenarios with a reduced number of bids, as described in the previous

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30%

all bundles max 3 per truck

20% 10%

day 5

day 4

day 3

day 2

day 1

gen15c

gen15d

gen15b

gen5h

gen15a

gen5f

gen5g

gen5e

gen5d

gen5c

gen5a

gen5b

0%

10,000 5,000 0

Fig. 1. Top: Percental savings when submitting all bids or a maximum of 3 per truck in the “auctionExt” scenarios (numbers are with respect to pre-auction costs). Bottom: Reduction in the number of submitted bundles.

section. In these cases, typically a lot of large bundles are submitted and more overlaps occur than in regular scenarios. Nevertheless, the diﬀerences between using SCP and SPP were not very big and costs savings never exceeded 1% of the original costs. We still recommend using the SCP formulation, as in some cases it might save some money and we are not aware of any disadvantages. 5.3

Eﬀect of Bundling

One major advantage of a combinatorial auction is that users may express synergies between items by forming bundles. We tested for all of our scenarios, how much of the achieved savings can be attributed to bundling. We simply restricted the bundle size to 1 in the bidding agent for comparison. Especially for the “auctionExt” settings, a major amount of proﬁt is then lost, as it can only be realized by bundeling. The detailed results can be found in Figure 2. 30%

bundles singletons

20% 10%

day 5

day 4

day 3

day 2

day 1

gen15d

gen15c

gen15a

gen15b

gen5g

gen5h

gen5f

gen5e

gen5d

gen5c

gen5b

gen5a

0%

Fig. 2. Percental savings with singleton bids and with bundles bids in the “auctionExt” scenarios (numbers are with respect to pre-auction costs).

Combinatorial Auctions in Freight Logistics

6

15

Conclusion and Further Research

In this paper we presented a complete modeling proposal for a combinatorial auction for transportation requests. We showed up the crucial points when designing such a complex system and gave reasons for our decisions. Our four-stage approach aims at reducing complexity for the user while keeping expressiveness of the bidding language and predictability of the auction outcome. It is meant to be integrated into a freight carrier’s daily planning process. The experiments suggest that our design scales well, in the sense that oﬀering more requests promises higher monetary gains. A decrease of total execution costs of up to 27.6% for the generated scenarios and 4.3% for the real-life data shows signiﬁcant savings. Maybe the fact that in many cases the auction result is close to the solution of the global optimization problem found by Ilog Dispatcher is even more meaningful. The auction performs especially well for scenarios with an average of at least 2 or 3 requests per truck. This also explains the lower revenues on the practical data and some generated scenarios. Our experiments show that bundling can be indeed a major contributor to the savings achieved. Especially in the well-performing scenarios with more requests per truck, bundling more than doubles the savings. This strongly supports using a combinatorial auction instead of a simple bilateral trade mechanism. Formulating the Winner Determination Problem as a Set Covering Problem, on the other hand, barely improves the result. Nevertheless, this might be diﬀerent in scenarios where requests are located in a speciﬁc way geographically. With our generated scenarios and our test data, it seems as if all participants compose similar bundles, which prevents overlaps to occur. Reducing the number of bundles proposed by the decision support systems is a big topic for further research. While our achieved results are a good starting point, it is desirable to further identify promising bundles that are likely to win. One way to improve on that is to include information about the competitors into the selection process.

References 1. Abrache, J., Crainic, T., Gendreau, M., Rekik, M.: Combinatorial auctions. Annals of Operations Research 153(1), 131–164 (2007) 2. Ackermann, H., Ewe, H., K¨ ufer, K.H., Schr¨ oder, M.: Modeling proﬁt sharing in combinatorial exchanges by network ﬂows. Tech. Rep. 205, Fraunhofer Institute for Industrial Mathematics (2011), http://www.itwm.fraunhofer.de/en/press-and-media/ reports-of-the-itwm.html 3. An, N., Elmaghraby, W., Keskinocak, P.: Bidding strategies and their impact on revenues in combinatorial auctions. Journal of Revenue and Pricing Management 3(4), 337–357 (2005) 4. Clarke, E.H.: Multipart pricing of public goods. Publ. Choice 18, 19 (1971)

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5. Conen, W., Sandholm, T.: Preference elicitation in combinatorial auctions. In: EC 2001: Proceedings of the 3rd ACM conference on Electronic Commerce, pp. 256–259. ACM, New York (2001) 6. Cramton, P., Shoham, Y., Steinberg, R.: Combinatorial Auctions. MIT Press, Cambridge (2006) 7. Day, R.W.: Expressing preferences with price-vector agents in combinatorial auctions. Ph.D. thesis, University of Maryland (2004) 8. DSLV Deutscher Speditions- und Logistikverband e.V.: Zahlen, Daten, Fakten aus Spedition und Logistik (May 2010) 9. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (January 1979) 10. Goossens, D., Spieksma, F.: The matrix bid auction: micro-economic properties and expressiveness. Tech. Rep. urn:hdl:123456789/120980, Katholieke Universiteit Leuven (2006), http://ideas.repec.org/p/ner/leuven/urnhdl123456789-120980.html 11. Groves, T.: Incentives in teams. Econometrica 41(4), 617–631 (1973) 12. Gujo, O., Schwind, M., Vykoukal, J., Weiß, K., Stockheim, T., Wendt, O.: ComEx: Kombinatorische Auktionen zum innerbetrieblichen Austausch von Logistikdienstleistungen. Wirtschaftsinformatik (1), 201–218 (2007) 13. Hsieh, F.S.: Combinatorial reverse auction based on revelation of lagrangian multipliers. Decision Support Systems 48(2), 323–330 (2010) 14. Krajewska, M.A., Kopfer, H.: Collaborating freight forwarding enterprises. OR Spectrum 28(3), 301–317 (2006) 15. Lee, C.G., Kwon, R.H., Ma, Z.: A carrier’s optimal bid generation problem in combinatorial auctions for transportation procurement. Transportation Research Part E: Logistics and Transportation Review 43(2), 173–191 (2007) 16. Moulin, H.: Fair division and collective welfare. MIT Press, Cambr. (2003) 17. Myerson, R.B., Satterthwaite, M.A.: Eﬃcient mechanisms for bilateral trading. Journal of Economic Theory 29(2), 265–281 (1983) 18. Nisan, N.: Bidding and allocation in combinatorial auctions. ACM Conference on Electronic Commerce, 1–12 (2000) 19. Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, New York (2007) 20. Pan, M., Chen, F., Yin, X., Fang, Y.: Fair proﬁt allocation in the spectrum auction using the Shapley value. In: Proc. of the 28th IEEE Conf. GLOBECOM 2009, pp. 4028–4033. IEEE Press, USA (2009) 21. Parkes, D.C., Kalagnanam, J., Eso, M.: Achieving budget-balance with vickreybased payment schemes in combinatorial exchanges. Tech. rep., IBM Research Report RC 22218 (2001); updated (March 2002) 22. Plummer, C.L.: Bidder Response to Combinatorial Auctions in Truckload Procurement. Master’s thesis, Massachusetts Institute of Technology (2003) 23. Rassenti, S.: 0-1 Decision Problems with Multiple Resource Constraints: Algorithms and Applications. Ph.D. thesis, University of Arizona (1981) 24. Schwind, M., Gujo, O., Vykoukal, J.: A combinatorial intra-enterprise exchange for logistics services. Inf. Syst. E-Business Mngmnt. 7(4), 447–471 (2009) 25. Shapley, L.S.: A value for n-person games. In: Kuhn, H., Tucker, A. (eds.) Contributions to the Theory of Games II, Annals of Mathematical Studies, vol. 28, pp. 307–317. Princeton University Press, Princeton (1953)

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Shipper Decision Support for the Acceptance of Bids during the Procurement of Transport Services Tobias Buer and Herbert Kopfer University of Bremen, Chair of Logistics, Wilhelm-Herbst-Straße 5, 28359 Bremen, Germany {tobias.buer,kopfer}@uni-bremen.de http://www.logistik.uni-bremen.de

Abstract. Combinatorial reverse auctions can be used by shippers in order to procure transportation services from carriers. After carriers have submitted their bids, a shipper has to decide about the allocation of transport services to carriers, i. e., a shipper has to solve the winner determination problem of the auction. This paper focuses on a bicriteria winner determination problem in which a shipper has to select a subset of the set of bids and simultaneously decide about the desired trade-oﬀ between total transportation costs and the quality of the entire transportation services. To solve this bicriteria optimization problem a metaheuristic is developed that computes a set of non-dominated solutions based on the concepts of multi-start, large neighborhood search and a bicriteria branch-and-bound procedure. Compared to previous results in the literature, the proposed algorithm is able to improve the set of non-dominated solutions for 14 out of 30 benchmark instances.

1

Introduction and Literature Review

Shippers often tender their required transport services via reverse auctions. Freight carriers oﬀer a price for which they would agree to take over a certain transport contract. In this context, a transport contract is a framework contract with a duration of about one to three years. A transport contract deﬁnes, which volume and which kind of transport requests should be fulﬁlled at which level of service. Usually, in a single transport procurement auction, many heterogenous transport contracts are tendered and allocated to many carries. For these reasons, transport procurement auctions are often designed as combinatorial auctions [3,4,12]. In practice, those auctions often represent large-scale procurement events. Caplice and Sheﬃ report about combinatorial transport auctions from practice with an annual procurement volume of up to 700 million USD and up to 5,000 tendered transport contracts [4] . In a combinatorial auction, carriers are allowed to submit an arbitrary number of bundle bids. A bundle bid is a bid on any subset of the tendered transport contracts. Moreover, bundle bids are all-or-nothing bids, i. e., either a carrier is J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 18–28, 2011. c Springer-Verlag Berlin Heidelberg 2011

Shipper Decision Support for the Acceptance of Bids

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awarded all transport contracts which are combined in a bundle bid or none of them. Bundle bids oﬀer the advantage, that carriers can express their preferences for any combination of transport contracts, which are often non-additive [9,10,12]. Therefore, a carrier can combine those contracts which oﬀer high synergies and can be executed in a well-balanced tour, for example. Furthermore, with a bundle bid carriers avoid the risk of winning only some of the transport contracts and losing those which enable the construction of a cost eﬃcient tour. Generally, transport auctions proceed in three stages: pre-bidding stage, bidding stage, and post-bidding or allocation stage. This paper is concerned with a winner determination problem which has to be solved by the shipper within the allocation stage after all bundle bids have been submitted. The problem is called Bicriteria Winner Determination Problem based on the Set Covering Problem (2WDP-SC). In the 2WDP-SC, the shipper has to decide, which of the submitted bids he should accept while simultaneously minimizing the total procurement cost and maximizing the total service level of the transport contracts. The problem was introduced by [2] and is modeled as a bi-objective combinatorial optimization problem. The attainable transport quality is included as an additional objective function. This allows a shipper to ﬁnd out the realizable trade-oﬀ between transport cost an transport quality. In practice, shippers usually do not exploit their full potential for cost savings in order to improve service quality of the procured transport contracts [12]. Other winner determination problems of combinatorial auctions that try to integrate quality aspects in the decision making process are described in [3,4,5,6]. Primarily, these approaches try to integrate quality aspects as some kind of side constraint or they use penalty costs to disadvantage low quality carriers or bundle bids, respectively. None of these approaches uses an additional optimization criterion. However, a multi criteria approach could be beneﬁcial especially if the desired trade-oﬀ between transport costs and handling of the transport quality is a priori unknown to the shipper. Note that for a shipper, identifying the desired trade-oﬀ is one of the most challenging tasks in the procurement of transport contracts [4]. In the next section we present the studied bi-objective winner determination problem for combinatorial transport procurement auctions. Section 3 describes the developed solution approach which computes a set of non-dominated solutions based on the metaheuristic concepts of multi-start and large neighborhood search, as well as a bicriteria branch-and-bound procedure. The heuristic is evaluated in Section 4 by means of a benchmark test. Section 5 concludes the paper.

2

Bi-objective Winner Determination Problem

The bi-objective winner determination problem of a combinatorial transport procurement auction based on a set covering formulation (2WDP-SC) asks for the subset of submitted bundle bids which covers each transport contract at least once and simultaneously minimizes the sum of the prices of the bundle bids and maximizes the total transport service quality [1,2]. We are given a set of oﬀered transport contracts T , a set of bundle bids B, and a set of carriers C (m = |T |, n = |B|). The carrier that submitted

20

T. Buer and H. Kopfer

bundle bid b ∈ B is indicated by cb , cb ∈ C. To each bundle bid b ∈ B a price pb ∈ R+ is attributed. Let Am×n be a 0–1 matrix, the m rows represent transport contracts and the n columns represent bundle bids. If bid b ∈ B contains transport contract t ∈ T , then at,b = 1, otherwise at,b = 0. If at,b = 1, then we say b covers t. Furthermore, for all t ∈ T, c ∈ C the parameter qt,c ∈ N indicates at which level of service quality the transport contract t is executed by carrier c. A shipper prefers higher values of qt,c . In practice however, the range of values of qt,c that are really used will probably be rather limited. The decision maker is represented by the shipper. He has to decide, which of the bundle bids should be accepted as winning bids, i. e., a solution X is a subset of the submitted bundled bids B. The binary decision variable xb indicates, wether bundle bid b ∈ B is accepted as winning bid (xb = 1 ⇔ b ∈ X) or not (xb = 0 ⇔ b ∈ X). The 2WDP-SC is deﬁned by the expressions (1) – (4). pb xb , (1) min f1 (X) = b∈B

max f 2 (X) =

t∈T

s. t.

max{qt,cb at,b xb },

(2)

b∈B

at,b xb ≥ 1,

∀t ∈ T,

(3)

∀b ∈ B.

(4)

b∈B

xb ∈ {0, 1},

The ﬁrst objective function (1) minimizes the total procurement costs of the shipper. The second objective function (2) maximizes the total service quality of all procured transport contracts. Constraint (3) guarantees, that each transport contract is covered by at least one bundle bid. As a contract can be procured more than once but has to be executed only once, only the highest attainable service quality per transport contract is added in f 2 . Please note that using the strict equal sign in (3) would avoid this issue, however, in that case the total procurement cost would never be lower and probably even higher compared to the greater-or-equal variant. Finally, expression (4) ensures, that each bundle bid is an all-or-nothing bid, i. e., a bundle bid can be either accepted as winning bid or not. Partial acceptance of bundle bids is prohibited. The expressions (1), (3), and (4) deﬁne the well-known NP-hard set covering problem. For ease of presentation, the 2WDP-SC is treated as a pure minimization problem, i. e., we set f2 (X) := (−1)f 2 (X) and replace (2) by min f2 (X).

3

Hybrid Pareto Search Metaheuristic

The proposed solution approach is called Hybrid Pareto Neighborhood Search (HPNS). HPNS is consisting of a construction phase and an improvement phase. In the construction phase, a number of non-dominated solutions are iteratively constructed. In the improvement phase, non-dominated solutions are repeatedly destroyed and reconstructed. The reconstruction process is based on a greedy and as well on a branch-and-bound approach. That is, the reconstruction process is

Shipper Decision Support for the Acceptance of Bids

21

hybridized with the bi-objective Epsilon-Lookahead-Branch-and-Bound (LBB) that was introduced in [2]. As a consequence, a single destroyed solution usually leads to several new non-dominated solutions. We ﬁrst introduce some notation. Let k be the number of objective functions and let X 1 , X 2 be two feasible solutions. X 1 weakly dominates X 2 , written X 1 X 2 , if fi (X 1 ) ≤ fi (X 2 ), i = 1, . . . , k. X 1 dominates X 2 , written X 1 ≺ X 2 , if fi (X 1 ) ≤ fi (X 2 ), i = 1, . . . , k and fi (X 1 ) < fi (X 2 ) holds at least for one k. An approximation set is a set of feasible solutions which do not ≺-dominate each other. The approximation set which contains those feasible solutions which are not weakly dominated by any other feasible solution is called Pareto-optimal set. Finally, we introduce the operator which joins two approximation sets and removes all dominated solutions from the set union. Let A1 and A2 be two approximation sets, then A1 A2 = A1 ∪ A2 \ {X ∈ A1 ∪ A2 | ∃X ∈ A1 ∪ A2 ∧ X ≺ X} . 3.1

(5)

Multi-start Construction Phase

Algorithm 1 repeatedly constructs a number of feasible solutions until a termination criterion is met. The termination criterion is conﬁgured in Section 4.2. To construct a single feasible solution, the following steps are executed. Every bundle bid b is rated with respect to the current infeasible solution X by the two rating operators P (b, X) and Q(b, X) as introduced below. P (b, X) and Q(b, X) assign lower values to better bundle bids. If b neither contains a transport contract which is not covered by the current solution X, nor does b improve the objective function value of f2 , then both rating operators return the value ∞. Adding further bundle bids to a solution will not change the rating of bid b and therefore we do not need to rate b again. Rated bundle bids are managed by the so called Restricted Candidate List (CL) which is known from the metaheuristic Greedy Randomized Adaptive Search Procedure (GRASP, [8]). Here, the CL holds all non-dominated bundle bids according to the evaluation given by the two rating criteria P (b, X) and Q(b, X). Therefore, the size of the CL is implicitly given by the cardinality of the set of non-dominated bundle bids and is not controlled by a parameter. After every bundle bid has been rated, a bundle bid is drawn randomly from CL and added to the current solution X. These steps are repeated, until X becomes feasible. The bundle rating operators P (b, X) and Q(b, X) were also used in [1], note that P (b, X) was already introduced by Chv´ atal [7]. Lower values represent better bundle bids. The set of transport contracts which are covered by b is denoted by τ (b), i. e., τ (b) ⊆ T . Likewise, τ (X) denotes the set of transport contracts that are covered by solution X, i. e., τ (X) ⊆ T . The rating operators are deﬁned as follows: ∞ if | τ (b) \ τ (X) |= 0 , (6) P (b, X) = pb else. |τ (b)\τ (X)|

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T. Buer and H. Kopfer

⎧ ⎪ ⎨∞ f2 (X ∪ {b}) − f2 (X) Q(b, X) = ⎪ ⎩− | τ (b ) |

if f2 (X ∪ {b}) − f2 (X) = 0 , else.

(7)

b ∈X∪{b}

The rating operator P (b, X), deﬁned in (6), divides the price pb of bundle bid b by the number of transport contracts that are not yet covered by the current solution X. The rating operator Q(b, X), deﬁned in (7), considers the potential increase of the total service quality by adding bundle bid b. The increase in service quality is divided by the total number of covered transport contracts of the resulting solution X ∪ {b}. Hence, an increase in service quality which is simply achieved by unnecessarily covering transport contracts multiple times is penalized.

Algorithm 1. Construction phase Input: instance data, i. e., a set of bundle bids B A←∅ while termination criterion not met do X←∅ B ← B while X infeasible do CL ← ∅ foreach b ∈ B \ X do if P (b, X) = ∞ and Q(b, X) = ∞ then B ← B \ {b} // do not rate b in further iterations else CL ← CL {b} end end select b randomly from CL X ← X ∪ {b } end B ← B A ← A {X} end return approximation set A

3.2

Improvement Phase

After an approximation set A has been constructed during the multi-start construction phase the algorithm HPNS tries to improve each solution in A. As shown in Algorithm 2, a solution X is drawn randomly from A. The procedure is based on the ideas of large neighborhood search [11]. At ﬁrst, the procedure destroySolution uses a destroy rate of 20 percent to destroy a feasible solution X. For that purpose, for each bundle bid b in X a uniformly distributed random

Shipper Decision Support for the Acceptance of Bids

23

number between 1 and 100 is drawn. If the random number is less or equal to 20, then b is removed from X. During preliminary testing, static destroy rates between 15 and 40 percent were evaluated. Furthermore, from this interval different destroy rates were randomly applied to solutions during the improvement phase. In the end, no signiﬁcant improvements compared to a simple static destroy rate of 20 percent were observed. The procedure destroySolution returns the solution X d , which is most likely infeasible. In the second step, a greedy reconstruction strategy (cf. Algorithm 3) is applied to the infeasible solution X d . The evaluation of bundle bids within this strategy depends on the value of an iteration counter iter which is initialized at ﬁrst and incremented in each iteration of Algorithm 2. In Alg. 3, each bundle bid is rated by P (b, X d ), if iter is an even number, otherwise it is rated by Q(b, X d). The best rated bundle bid is added to X d . These steps are repeated until a certain cover quote is attained. The parameter cover quote is deﬁned as the number of transport contracts covered by solution X d divided by the total number of transport contracts m. After that, the greedy reconstruction heuristic terminates. The reconstructed but infeasible solution is called X r . To continue, an exact bi-objective branch-and-bound procedure is applied the solution X r . The used procedure is called Epsilon-Lookahead-Branch-andBound (LBB) and is described in [2]. LBB starts with X r as initial solution and exactly explores the decision space based on X r . By doing this, only those bundle bids covering contracts which are not covered by X r are considered during the branch-and-bound search. When LBB terminates, it returns an approximation set A . The solutions of the best approximation set found so far are joined with the solutions of A according to the operator (5). After that, the iteration counter iter is incremented and the steps are repeated again. As termination criterion for the improvement phase a time duration is used which is conﬁgured in Section 4.2.

4

Evaluation

The performance of the proposed heuristic HPNS is evaluated by means of a computational experiment with thirty instances of the 2WDP-SC. We focus on the quality of the obtained solutions and therefore introduce the hypervolume indicator in Section 4.1 as a quality measure for an approximation set. Numerical results are given in Section 4.2 and compared to the results computed by three other heuristics for the 2WDP-SC described in [1] and [2]. 4.1

The Hypervolume-Indicator as a Quality Measure for Approximation Sets

The comparison of heuristics for single objective optimization problems is often done by comparing the objective function values of the best found solutions. In multi objective optimization, the heuristics compute an approximation set. Even for a single run on a single instance we have to compare sets of solutions which

24

T. Buer and H. Kopfer

Algorithm 2. Improvement phase Input: approximation set A iter ← 0 while termination criterion not met do select a solution X from A X d ← destroySolution(X) X r ← greedyReconstructionHeuristic(X d , iter) A ← LBB(X r ) A ← A A iter ← iter + 1 end return approximation set A

Algorithm 3. greedyReconstructionHeuristic Input: infeasible solution X d , iteration counter iter γ ← |τ {X d }|/m while γ ≤ cover quote do g ∗ ← +∞ foreach b ∈ B \ X d do if iter is even then gb ← P (b, X d ) else gb ← Q(b, X d ) if gb < g ∗ then g ∗ ← gb b∗ ← b end end X d ← X d ∪ {b∗ } γ ← |τ {X d }|/m end return X d

introduces additional complexity. A discussion of challenges and suggestions for the comparison of approximation sets is given by Zitzler et al. [13]. To compare the computed approximation sets, we use the hypervolume indicator introduced by Zitzler and Thiele [15]. The hypervolume indicator IHV (A) assigns a positive real number to an approximation set A. Approximation sets with higher indicator values are considered to be better. To calculate the hypervolume of an approximation set A, a reference point r has to be established. The reference point has to be chosen in such a way that each objective function vector of a solution in A weakly dominates r (cf. Figure 1a). The hypervolume of the approximation set A is the volume of the set of points in the objective function space which are weakly dominated by the solutions in A and bounded by r. In Figure 1a the hypervolume of A is marked light grey and the hypervolume of another approximation set B is marked dark grey. Note that the dark grey area partly overlaps the light grey area. As Figure 1b illustrates, the

Shipper Decision Support for the Acceptance of Bids

25

hypervolume of an approximation set A increases, if a new solution is found which is non-dominated by any existing solution in A. In Figure 1b two nondominated solutions are added to the approximation set A, which results in an increase of IHV (A) by the shaded area.

f2

×

r

f2

×

r

A B

X1

X2

f1 (a) Approximation sets A and B with marked hypervolume

f1 1

2

(b) X and X increase IHV (A)

Fig. 1. Hypervolume indicator IHV 4.2

Numerical Results and Discussion

The computational experiments were executed on an Intel Pentium Dual Core CPU T3200 with 2.0 GHz. The algorithm HPNS has been implemented without parallelization in Java 1.6, so the algorithm uses only a single CPU core. For the study at hand, we use the benchmark instances introduced in [2] which are available in the electronic appendix of [1]. As in previous studies of the 2WDPSC, the reference point r was set to (f1 (B), 0) for each instance. That is, r is weakly dominated by every feasible solution. Furthermore, we normalize r and each objective function vector of a solution, cf. page 204 of [1]. This seems necessary, as for the used instances, the domains of both objective functions diﬀerentiate in some order of magnitude. The heuristic HPNS was parameterized as follows. The total runtime was set to three minutes. The termination criterion of the construction phase (cf. Algorithm 1) was set to 30 consecutive iterations without an improvement of the approximation set, i. e., no new non-dominated solution was constructed. Preliminary tests suggested, that this termination criterion outperforms a termination criterion based on the available runtime as used in [1]. The termination criterion of the improvement phase (cf. Algorithm 2) was set to a total runtime of three minutes minus the time required for the construction phase. The probability to remove a bundle bid was set to 20 percent. The cover quote as termination criterion for the greedy reconstruction heuristic was set to 98 percent. During preliminary testing, it has been observed that the heuristic HPNS is rather sensitive to changes of the parameter cover quote. Smaller values led to instance sizes which could sometimes not be solved by LBB in reasonable time, while higher cover quotes left very few decision possibilities.

26

T. Buer and H. Kopfer

Table 1. Comparison of the IHV values obtained by four heuristics IHV No.

|B|

|T |

|C|

ρ

HPNS

GSP R

GSQR

EA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

500

125

25

1000

125

25

250

25

25 50 75 25 50 75 25 50 75 25 50 75 25 50 75 25 50 75 25 50 75 25 50 75 25 50 75 25 50 75

0.9002 0.9108 0.9063 0.9435 0.9439 0.9604 0.9068 0.9061 0.9032 0.8832 0.8924 0.8983 0.9666 0.9722 0.9805 0.9536 0.9576 0.9557 0.9416 0.9448 0.9432 0.9064 0.9007 0.8968 0.8939 0.9001 0.8982 0.8605 0.8746 0.8902

0.8992 0.9114 0.9036 0.9513 0.9566 0.9587 0.9037 0.9038 0.9010 0.9064 0.8984 0.9004 0.9694 0.9807 0.9756 0.9552 0.9546 0.9531 0.9461 0.9573 0.9532 0.9067 0.8997 0.8962 0.8928 0.8979 0.8925 0.8893 0.8855 0.8942

0.8923 0.9053 0.8958 0.9498 0.9531 0.9544 0.8993 0.8972 0.8970 0.8995 0.8944 0.8962 0.9771 0.9789 0.9783 0.9534 0.9530 0.9528 0.9523 0.9534 0.9503 0.9046 0.8965 0.8949 0.8897 0.8917 0.8922 0.8883 0.8959 0.8927

0.8914 0.9038 0.8983 0.9479 0.9508 0.9535 0.9021 0.9001 0.8961 0.8927 0.8943 0.8937 0.9720 0.9786 0.9746 0.9531 0.9532 0.9510 0.9337 0.9530 0.9508 0.9022 0.8942 0.8944 0.8902 0.8947 0.8887 0.8708 0.8827 0.8939

0.9064 0.9197 0.0315

0.9051 0.9232 0.0306

0.8994 0.9210 0.0319

0.9011 0.9186 0.0321

50

2000

125

25

250

25

50

500

median mean standard deviation

25

50

100

Each of the thirty instances has been solved by HPNS exactly once. The obtained values of the hypervolume indicator IHV are given in Table 1. The size of each instance is characterized by columns two to ﬁve. At this, the value of ρ in column ﬁve indicates the degree of synergy between transport contracts. Higher synergies between transport contracts tend to result in bundle bids which cover a greater number of transport contracts. The results are compared to three other algorithms called GSP R , GSQR , and EA. The approaches GSP R and GSQR are both introduced in [1] and are based on a GRASP that applies Path Relinking

Shipper Decision Support for the Acceptance of Bids

27

as a post optimization procedure. The heuristic is an evolutionary algorithm (EA) based on the improved Strength Pareto Evolutionary Algorithm (SPEA2, [14,2]). In terms of the hypervolume indicator IHV , the two heuristics HPNS and GSP R perform best. For 14 instances, HPNS is able to compute new best-known approximation sets. Accordingly, HPNS computes the highest hypervolume for an instance 14 times. GSP R computes the highest hypervolume 12 times, GSQR achieves the best value 3 times, and the EA reaches the best value only once. Concerning the median IHV the heuristic HPNS outperforms the three other heuristics GSP R , GSP R , and EA, which underlines the competitiveness of HPNS. However, considering the IHV mean over all thirty instances, the heuristics HPNS ranks only third. The lower mean value of HPNS can be explained with a lower robustness of HPNS compared to GSP R (compare standard deviations in Table 1). In particular, for some instances HPNS obtains signiﬁcantly worse hypervolume values than GSP R , e. g., for the instances 4, 5, 10, 20, 21, 27, and 28.

5

Conclusion and Outlook

In this paper, we have proposed a new metaheuristic algorithm called HPNS to solve the bi-objective winner determination problem 2WDP-SC. The 2WDPSC has to be solved by a shipper during the procurement of transport services after all carriers have submitted their bundle bids. The developed algorithm HPNS combines the ideas of GRASP and large neighborhood search with an exact branch-and-bound procedure. The performance of the algorithm HPNS has been evaluated by means of thirty benchmark instances from the literature. To compare the solution quality to three other heuristics, the quality of the found non-dominated solutions has been measured by the hypervolume indicator. According to this measure, HPNS has been able to compute 14 new best values. Over all thirty instances, the obtained mean hypervolume of HPNS ranks third. However, HPNS obtains the best median hypervolume among all approaches. Future improvements of HPNS should deal with the question, how to better integrate the metaheuristic and the exact solution approach. The present approach seems to be critically aﬀected by the parameter cover quote which has a strong impact on the size of the decision space that is explored by branch-andbound. Furthermore, some preference information from the shipper could be collected in an interactive manner during search and used to guide the heuristic in order to improve decision support. Acknowledgments. This research was supported in part by the German Research Foundation (DFG) as part of the project ”Kooperative Rundreiseplanung bei rollierender Planung”.

References 1. Buer, T., Pankratz, G.: Grasp with hybrid path relinking for bi-objective winner determination in combinatorial transportation auctions. Business Research 3(2), 192–213 (2010)

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2. Buer, T., Pankratz, G.: Solving a bi-objective winner determination problem in a transportation procurement auction. Logistics Research 2(2), 65–78 (2010) 3. Caplice, C., Sheﬃ, Y.: Optimization-based procurement for transportation services. Journal of Business Logistics 24(2), 109–128 (2003) 4. Caplice, C., Sheﬃ, Y.: Combinatorial auctions for truckload transportation. In: Cramton, P., Shoaham, Y., Steinberg, R. (eds.) Combinatorial Auctions, pp. 539– 571. MIT Press, Cambridge (2006) 5. Catal´ an, J., Epstein, R., Guajardo, M., Yung, D., Mart´ınez, C.: Solving multiple scenarios in a combinatorial auction. Computers & Operations Research 36(10), 2752–2758 (2009) 6. Chen, R.L.Y., AhmadBeygi, S., Cohn, A., Beil, D.R., Sinha, A.: Solving truckload procurement auctions over an exponential number of bundles. Transportation Science 43(4), 493–510 (2009) 7. Chv´ atal, V.: A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4(3), 233–235 (1979) 8. Feo, T.A., Resende, M.G.: Greedy randomized adaptive search procedures. Journal of Global Optimization 6(2), 109–133 (1995) 9. Kopfer, H., Pankratz, G.: Das Groupage-Problem kooperierender Verkehrstr¨ ager. In: Kall, P., L¨ uthi, H.J. (eds.) Operations Research Proceedings 1998, pp. 453–462. Springer, Berlin (1999) 10. Pankratz, G.: Analyse kombinatorischer Auktionen f¨ ur ein Multi-Agentensystem zur L¨ osung des Groupage-Problems kooperierender Speditionen. In: Inderfurth, K., Schw¨ odiauer, G., Domschke, W., Juhnke, F., Kleinschmidt, P., W¨ ascher, G. (eds.) Operations Research Proceedings 1999, pp. 443–448. Springer, Berlin (2000) 11. Shaw, P.: Using constraint programming and local search methods to solve vehicle routing problems. In: Maher, M.J., Puget, J.-F. (eds.) CP 1998. LNCS, vol. 1520, pp. 417–431. Springer, Heidelberg (1998) 12. Sheﬃ, Y.: Combinatorial auctions in the procurement of transportation services. Interfaces 34(4), 245–252 (2004) 13. Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Transactons on Evolutionary Computation 7, 117–132 (2003) 14. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Giannakoglou, K., Tsahalis, D., Periaux, J., Papailiou, K., Fogarty, T. (eds.) Proceedings of the EUROGEN 2001 Conference, pp. 95–100. International Center for Numerical Methods in Engineering (CIMNE), Barcelona (2002) 15. Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach. IEEE Transactions on Evolutionary Computation 3(4), 257–271 (1999)

Aspects of Information Management in Road Freight Transport Joachim R. Daduna Berlin School of Economics and Law Badensche Straße 52, D - 10825 Berlin, Germany [email protected]

Abstract. For freight transport (and in particular road freight transport) the availability of a suitable (intra- and inter-organizational) information management is of decisive significance. Therefore, the technical (and organizational) framework as well as the essential instruments will be presented and analyzed. This will be followed by an examination of the operational planning and the operations of the road freight transport where economic factors influence the discussion as well as ecological aspects. It will be illustrated that an intelligent linkage between different elements of the information and communication technology and the automotive engineering as well as with the transport infrastructure leads to a significant improvement regarding the operational processes in road freight transport. This does not only refer to the costs of transport operations but also reflects positive ecological effects and improvements of transport security.

1 Information Management in Transport Logistics The (logistical) information management becomes an increasingly crucial element in the field of planning and controlling the production processes based on division of labor that occur in the view of (cross-company) supply chains. However, even in reference to such production structures the management of the resource information that is understood as an (independent) factor of production (see, e.g., [19]: 19pp) cannot be restricted to the range of one separate company. Next to the internal (incompany) information management there is also the need of an external (processoriented) information management (see, e.g., [7]). These represent a mandatory precondition for an efficient planning, controlling and monitoring of (increasingly globalized) provision of service processes, there is a need for finding an adequate structural and technical arrangement. At first it is to define the operational functions that need to be integrated whereat external specifications resulting from interoperational activities in cross-linked structures has to be considered as they might restrain the room for maneuvers of the company enormously. The information management is process-oriented in its basic approach and therefore a focus on activities regarding the operational and dispatching level in the intraand inter-organizational area occurs. As a result, according to the classical definition of the term logistics, we can assume a space- and time-oriented basic structure. These J.W. Böse et al. (Eds.): ICCL 2011, LNCS 6971, pp. 29–43, 2011. © Springer-Verlag Berlin Heidelberg 2011

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J.R. Daduna

include the functional activities of procurement and transformation as well as the allocation and distribution of information under the consideration of qualitative and quantitative user requirements, especially regarding cost aspects as an essential part to be considered. In particular, the last aspect includes a high potential for conflicts while on one hand the permanent system availability with a maximal dimensioning of technical systems implies a low default risk, but it is on the other hand associated to higher costs. However, interruptions in the flow of information can include extensive consequences for the processes and is under certain circumstances furthermore related to extra borrowing costs. Therefore the effort of providing information has to be in a reasonable relation to the situational benefit due to the fact that the profitability of the application of the systems has to be in the focus. Based on the underlying tasks three issues are being defined that should still not be observed as isolated from one another (see e.g. [7]), due to the fact that functional inter-relations and interdependencies exist. • Company-related or as with limited participants defined (operational) application systems, also including the field of operational planning. • Company-related or as with limited participants defined telematic systems. • Cross-company information and communication systems in the context of a computer-aided logistics management, i.a. for the management of logistical facilities. Another part of (macro-logistic) information management is constituted by different publicly-operated systems that can indirectly influence logistics processes. In general this refers to applications that primarily serve for the (collective) monitoring and control of traffic flows and road pricing systems. In the following the functional aspects of a capable information management in road freight transport will be observed not only in regard of (micro-)economic effects but also considering macro-economic and ecological aspects. First of all, the technical basics of transport monitoring and control will be analyzed. The next section includes operation-related applications that affect the road freight transport processes. An outlook of expected developments is given at the end of the remarks.

2 Technical Framework The technical framework analyzed here, generally cover the field of macro-logistical (i.e., externally provided) systems that have to be linked to the internal information and communication structures as well as the (technical) vehicle equipment. The focus of the observation lies on the (collective) recording and provision of traffic data, the (individual) detecting and visualization of positioning data and also the usage of the data within the navigation systems. Vehicle-to-infrastructure communication (V2I) and vehicle-to-vehicle communication (V2V) are being included here as well. In order to use the full range of the given opportunities a suitable information and communication system structure is required for data collecting, demand-related processing and user-oriented provision of necessary data. In the following sections some of the major (technical) basics and also possibilities of use of a capable information management will be presented.

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2.1 Traffic Data Detecting The detecting of data largely occurs automatically along (selected) routes (with a relevant traffic volume) whereat different technical systems are being applied. The objective of data collecting is an illustration of the actual traffic situation based on the elemental parameters of driving speed and traffic density. Besides the data about traffic flows further information (e.g. weather, road conditions) can be recorded that provides additional information about external (traffic volume-related) factors. The most important methods of data detection are the following (see e.g. [33]): • Stationary measuring points: Applied techniques are inductive loop vehicle detectors, passive infrared sensors, radar and microwave sensors, and optoelectronic sensor networks. • Recording and evaluation of video camera pictures at selected traffic junctions for traffic monitoring. • Evaluation of airborne traffic flow recordings associated with vehicle detection and traffic census: Data collection by satellites, aircrafts, helicopters and unmanned aerial vehicles (UAV) applying optoelectronic scanner as well as infrared- and radar systems. • Floating car data (FCD): Data collection on the basis of stationary beacon systems or in combination with satellite-based location techniques. • Fleet generated FCD: Data collection by recording the movements of a (sufficient) number of vehicles (GPS-captured taxi-fleets, etc.) in big cities and congested urban areas, e.g., in order to recognize traffic disruptions. • Acquisition of FCD-related data via mobile communication systems: Data recording by means of flows along a stated cell structure. • Extended floating car data (xFCD): Incorporating (technical) vehicle behavior and the state of activity of vehicle-internal systems. The data processing is determined on one hand by the formulation of forecasts concerning the actual processes and on the other hand by the development of recommendations for action or control instructions. The approach is based on (macroscopic) traffic flow models (see, e.g., [26]), that help to pursue three essential objectives, failure detection, completion of traffic data, and traffic jam length and dead time estimations. The captured results have to be provided to road users according to their individual needs, whereas it can take place in different ways. However, a situational timing regarding the design of the information flows is essential that also offers the possibility for users to react towards certain traffic situations in time. 2.2 Communication and Location A capable (logistic) information management needs sufficient communication structures for necessary intra- und inter-organizational cooperation as well as for the integration into macro-logistic systems and its application, respectively. In addition location systems are required which allow a sufficiently exact detecting of the movement of (mobile) objects (vehicles, load units, etc.) in traffic networks and logistics facilities. To visualize efficiently the location data, spatial views in the form of digital maps are needed based on geographic information systems (GIS). In the following these three complexes are presented with its essential structures.

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Communication systems To attain an integrated and high-productive information management a permanent available (and failure-free) communication infrastructure with corresponding interconnections is necessary, allowing real-time data flows if required. Only in this way the information flows connected with the operational processes can be handled efficiently. The required techniques are available, as, e.g., telephone networks, internet, wireless local area networks (WLAN), trunked radio networks, satellite communication networks as well as cellular networks. Which of these options will be applied, primarily depends on the (operational) requirements and the technical performance and suitability. Location systems In location processes the position of objects are determined (in the 2D- and 3Dspace). Here different process principles are underlying, which in part can be used in combined versions. These are the beacon-based location, the (vehicle on board) location based on route calculation by, e.g., deduced reckoning, map matching and pattern recognition (see, e.g., [9]: 369pp) as well as wave-based and mobile radio systems. If the use of vehicle on board location systems is not forced by specific requirements, the satellite-based (radio) location is in the foreground. This has its origin in the global positioning system (GPS) developed in USA for military applications (see, e.g., [9]: 177pp; [23]: 107pp). Nearly parallel in the former Soviet Union the Globalnya Navigatsionnaya Sputnikovaya Sistema (GLONASS) system was designed (see, e.g., [9]: 245pp; [23]: 255pp), whose importance is currently very low for applications in logistics. Independent from these two developments which had been characterized by the political situation at that time, also commercially operated systems are build as for example the EutelTRACS system in the European area (see, e.g., [23]: 100pp). Besides these existing location systems the Galileo project must be seen which has been initiated within a European Community (EC) program (see, e.g., [9]: 257pp) (availability expected in 2014). Due to the worldwide importance for monitoring and control of logistic processes in the following the GPS system is presented as well as its extended version, the differential GPS.

 Global Positioning System The basic concept of the GPS system is to attain an accurate location of objects in real-time, which will be available worldwide. The applications have been intended for (at first for military) land-based vehicles, vessels and aircrafts, In recent years the system becomes opened for private user, as, e.g., for monitoring logistic processes. The GPS system mainly consist of three segments (see [23]: 110pp): space segment, control segment and user segment (see Figure 1). • The space segment is formed by minimum 24 (and up to 30) satellites, which allow (in the height of 20.230 km) a widely worldwide complete coverage. • The control segment is formed by in all five facilities (see, e.g., [23]: 114pp) which different functions, a main control (and monitoring) station and four monitoring station, while three of them operate also as a ground antenna stations. • The user segment covers all occurring objects in this field (for example in logistics the deployed vehicles), which are located with a guaranteed accuracy. For determining a (more) accurate position a connection to at least four satellites are necessary, whereas these (in a global average) amount to 10 m with a availability of

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70% (see, e.g., [9]: 218). If a higher accuracy is required, improvements are made by system extensions, so, e.g., with the differential GPS (DGPS).

Fig. 1. Basic structure of the GPS system segments

 Differential Global Positioning System With the DGPS (see, e.g., [8]: 221p; [9]: 218pp) a higher accuracy is attained by applying locally fixed reference stations (see Figure 2). Based on a suitable terrestrial infrastructure it is currently possible to get an accuracy up to 1 cm for certain applications.

Fig. 2. Basic structure of the DGPS system

Visualization of detected data To operationalize the data attained from location procedures a visualizable transformation layer is necessarily, that means, only by transferring the detected data to such a layer they will be usable information. For such transformation standardized GIS

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systems are of great importance. In this context the standards of the United States National Digital Cartographic Data Standards Task Force (US DCDSTF) for spatial database systems are an important basis, which contains a defined number of spatially referenced data (see, e.g., [30]: 40pp). The (formal) structuring is based on four basic element (geo objects), nodes, lines, areas and volumes (see, e.g., [8]: 159pp), which are normally not statically determined but underlying changes over time. The stored data are available in functional differentiated layers (see, e.g., [16]), that means, thematic subsets are created, which can be structured based on different aspects, e.g., depending on the responsibility for data detecting and processing as well as user-oriented. An additional aspect is that only those data should be shown which are needed by a user. Based on these reflections it is appropriate to realize (referring to transport specific applications) a reduced (slim) GIS (see, e.g., [10]) or a GIS-T (see, e.g., [29]: 32pp), respectively, that based on publicly available GIS data (see, e.g., [16]). 2.3 Vehicle Navigation The application of navigation systems for an individual guidance (see, e.g., [17]: 144pp: [26]) offers the opportunity to quickly adapt transport operations to the current situation and to attain a reduction of errors. For the most part the framework therefore is a (satellite-based) tracking process as well as a sufficient precise electronic map. According to the available data it can be differentiated between two forms, the usage of static and dynamic (continuously updated) network information. Another aspect is the data supply, based on decentralized (vehicle-internal) and centralized (vehicleexternal) data bases (see, e.g., [17]: 145), whereas the following differentiation occurs: •

•

Onboard systems: In this type of system, the technical equipment necessary for routing processes and the network information is located in the (internal) device. As a result the available data is statically and does not reflect the current traffic conditions. Offboard systems: In this case the system intelligence is located in an external control center. From there the vehicles integrated into a navigation service are being provided with the necessary information based on continually updated (network) data.

The (technical) procedure regarding the navigation is generally based on a combination of satellite-based and vehicle-onboard location. This applies to the usage of GPS elements as well as to the integration of dead reckoning and the map matching techniques (see, e.g., [9]: 369pp). Further developments in the field of (3D-) visualization are based on, e.g., the usage of Google Earth and methods of the (3D-) geo visualization. 2.4 Vehicle-Related Technical Support and Automation On one hand this refers to advanced driver assistance systems (ADAS) that support the user in his activities (automated to a large extent) but without replacing him (not even partially), i.e., the freedom of individual decision-making is generally not limited but if only in a subarea. In this context it is related to autonomous systems, comfort systems, driver information systems and efficiency-enhancing systems (see, e.g.,

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[21]). On the other hand we find (fully) automated systems that do not require a driver from the technological point of view. From the perspective of logistics, the systems of interest are specifically those that contribute to improvements of the transport processes, help to reduce search efforts of the drivers, and enable an increase in capacity regarding the road network and in security (see, e.g., [27]). In the foreground are applications in the 3F-parameter space as well as in V2V and V2I communication (based on ad-hoc-networks) (see, e.g., [32, 34]). In the concept of platooning (see, e.g., [18, 22]) to reduce the (minimum) distances between the vehicles of a convoy (e.g. of larger commercial trucks), many of these techniques are being represented. The technical capabilities allow a defined number of vehicles (until a maximum of seven trucks) to form such a convoy (see, e.g., [14]: 8pp), whereat besides the improved utilization of the infrastructure fuel savings of about 10% for the following vehicles evolve. Using ADAS security aspects have to be considered first, also concerning physical and mentally discharge of the driver (see, e.g., [24]). Nevertheless, negative aspects are in the discussion (see, e.g., [1]) due to the fact of a de facto incapacitation of the driver whereas his active involvement is reduced to a simple function. A complete automation of vehicles in the road transport will not be feasible in the nearer future although technical requirements (see, e.g., [25]) are fulfilled and positive effects can be observed regarding transport and the economic efficiency. Substantial reasons in favor of a prohibition are legal restrictions that concern the questions of guilt and liability, especially in view of a mixed-mode traffic with both, manually- and automatically-driven vehicles. However, applications within closed network structures can be realized.

3 Support of Operations in Road Freight Transport The existence of suitable technical systems and (software-) tools and also a capable information management are of significant importance for an efficient operation of (road-) freight transport. Respectively, the focus lies on an extensive interconnectivity of the individual (operational) functions in order to guarantee a data and information flow free of any system-related media discontinuity. Figure 3 reflects a basic structure for the processes concerning the road freight transport. This structure underlines the fact that transport operations cannot be seen as (overall) isolated partial processes, they are rather virtually cross-linked with differing external and internal areas. Regarding the internal part of the process it includes a wide range of procedures starting with commercial data management but also referring to personnel and technical management. In this case information technology (IT) services do not necessarily need to be performed by in-company resources, it is also possible to use application service providing (ASP) solutions (see, e.g., [31]) and cloud or on-demand computing offers (see, e.g., [35]). In addition, there exists the possibility to utilize external linkages with integrated (IT) service providers and electronic platforms (e.g. for freight and storage exchange). Furthermore, in this context it needs to be accounted for the integration into (macro-logistical) information and monitoring systems. In the following sections the discussion will focus on three issues, fleet management (FM) and cargo tracking, applications of supply chain monitoring and control and the tracking and tracing of shipments.

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Efficiency analysis / Reporting

Personnel administration

Vehicle routing

Accounting data management

Inter-organisational level E-markets (Freight / truck capacities)

Shipment tracking & tracing

Order management Truck (On-board computer)

Truck fleet and garage management

Dispatching

Monitoring load units

Fleet management

Road transport operations

Navigation systems

Traffic mangement and information systems

Traffic management center

Road pricing systems

Vehicle location / communication Macrologistic Level

Fig. 3. Information technology for transport operations in road freight transport

3.1 Fleet Management and Load Unit Tracking The underlying approach of a FM concept is reflected by a continuous communication between a dispatcher in a (control) center and the drivers of the currently deployed vehicles and also the (continuous) detection of vehicle movements. The necessity of using such tools has been proven, overall caused by demands of the forwarders about a preferably time-specific navigation of logistical processes. Apart from this externally induced enhancement of the degree of service, there is also the need of improved instruments for planning as well as monitoring and control (see, e.g., [6]). The fundamentals of a capable FM can be divided into two elements: • •

Location and communication systems for a determination of current vehicle positions (tracking) and the transmission of the information to a control center. Transmission of (technical and operational) data or information between vehicles (or drivers) and control center.

The main (operational) objectives of FM are aimed at creating an efficient organization of operating processes, where not only scheduling and monitoring but also the provision of an extensive (and legally secured) documentation (tracing) are essential factors. This includes foremost an extensive (technical) recording of the actual vehicle operations, amongst others with date, driving and idle times, the current load factor as well as (vehicle-based and driver-based) operating data (see, e.g., [2]). By having such (generally updated) (tracking) data, the possibility exists to actively influence the processes, i.e., there is no limitation towards an (ex post-oriented) reaction, it is rather possible to act on a target-oriented and time-specific way, eventually also to undertake preventing measures. Here, the interface is found in form of the board computer that enables data (and voice) radio communication between the driver (and/or the

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on-board computer, respectively) and a control center but also handles the controlling of onboard equipment (e.g., chip card-, radio frequency identification (RFID)-, and barcode-readers) and also different sensors. Initial points for an intervention into the processes are not only (short-term) disturbances in the operational processes, but also include adaptations concerning the order situation that claim an adjustment of the available resources and respectively a (shortterm) rearrangement of processes (as, e.g., the vehicle schedules). Whereas the quality of the decision-making in such situations depends on the volume and suitability of the available information and moreover on the date of availability, the usage of fleet management systems (FMS) leads to distinct operational advantages. Moreover, the time-related processes can be simplified and accelerated as well as improved in terms of quality (for example by reducing transmission errors) when having a electronic linkage. The controlling of processes constitutes a controversial function in the FM as it the (vehicle-related) tracking process inevitable implies an extensive (technological-based) control of the employees affected (see, e.g., [1]). An important aspect thereby is the avoiding of time- and space-related deviations from the given timeframe, also under the consideration of preventing (vehicle) thefts. At this point it also needs to be underlined that a procedure of delivering a complete monitoring of the processes is indispensable in cases such as a (legally) prescribed documentation of transport operations. This is the case when operating hazardous material transports (see, e.g., [12]) where the route of the vehicles is defined by (external) authorities due to legal regulations (see, e.g., [28]). Also for frozen goods (EC-wide) regulations increasingly force a complete proof of correct transport procedures to ensure the compliance of the cool chain. With the introduction of digital tachographs (see, e.g., [4]: 60pp) a new era of the automated monitoring of driver and vehicle data is being initiated. It is the overall objective to ensure a forgery-proof data collection and storage and a direct (and simplified) access to control (for the responsible authorities). The data that is recorded with the help of digital tachographs contains essential information for dispatching that enables an integration of system structures of the FM and the working time management system. From the perspective of the affected driver personnel, the mentioned possibilities of an (automated) monitoring should not only be seen in a negative context. Positive effects have to be considered as well, such as the opportunity of a fast and targeted intervention in case of emergencies (e.g. traffic accident, attacks) possible due to current position reports, also in combination with the alerting and control of operations of the responsible security forces. Moreover the advantages arising from the advanced technical monitoring of the vehicles or single aggregates must be mentioned as they serve for the decisionmaking in case of (technical) malfunctions (or incidents) and create the possibility of a preventive and condition-related maintenance. The focus is a continuous documentation of the state of aggregates (motor, gear, etc.) whereas the measured data are transmitted, for instance to the responsible workshop management (in real-time or in case of need) with the help of telemetry systems (see, e.g., [37]: 281pp). Parallel to that, the cargo-related status recording needs to be recognized (for example supported by sensors for temperature measurement and data recording related to RFID-technologies (see, e.g., [36]) that are already mandatory for certain procedures

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(fresh goods and frozen products) (see, e.g., [11]). Apart from the (technical) monitoring this includes the (automated) documentation. In addition to the above described basic functions of FM, three arguments are of importance (see, e.g., [20]), continuous transmission of temperature data for a documentation (vehicle/trailer), establishing of limits for alert in case of illegitimate temperature deviations, and alert in case of a not scheduled opening of the cooling box doors. It is for several reasons useful to functionally integrate the resulting measurements into a FMS as the information structures and possibilities of intervention also allow for a short-term intervening into the processes, especially in order to prevent the occurrence of transport losses. Obvious operational advantages are a result of a more sophisticated usage of the available means of transport but also positive macro-economic and environmental aspects can be seen as supplementary advantages. These do also result from a reduction of transport-related ecological damage, thus every decrease of mileage in connection with logistical operations, also caused a decrease of the traffic volume and pollutant emissions. This is to say that the mentioned operational objectives (including efficiency and cost savings) are not contrary to the macro-economic and environmental aims, it can rather be understood as being complementary to each other. Nevertheless, the FMS faces limits regarding its monitoring possibilities that are often not included in the analysis. The detailed documentation of processes is for instance only applied to the deployed vehicles and processes related to the vehicles, hence weak points arise concerning the monitoring of load units, especially in the field of multi modal (road-rail) transport. An intermediate storage as in a road-rail transshipment terminal forces an interruption in the (vehicle-related) monitoring that is not desirable, also in view of theft problems. In addition, a statement about the position of a load unit can only be concluded from driver-related information whereat it has to be assumed that no mistakes occurred when the cargo was loaded. In contrast, if the transport object (e.g., container, swap body, etc.) is being equipped with an independent monitoring system (see, e.g., [3]: 131pp), it is possible then to determine continuously the locations. The monitoring that starts with the forwarder and ends at the client, is independent from the respective transport systems and can also be used within transshipment facilities (multi modal (road-rail) transshipment terminals, seaport container terminals, etc.). The significant restrictions that can occur due to radio shadows can be overcome by using (facility-related) RFIDinstallations. Besides the improved customer information, for example by providing frequently updated status information (position of the load unit, predicted delivery date, etc.), the security aspects has to be regarded that results from an (independent) load unit monitoring (see, e.g., [3]: 176pp). It is furthermore important to mention the insurancerelated considerations that occur as a (overall) continuous documentation of all processes of load unit handling (transportation, warehousing) generally reduce the risk of damage or loss. This relieves the option of enlarging the negotiation scope (significantly) when determining the insurance rate of the transport. Altogether, the advantages of FMS leverage the disadvantages even though a critical analysis is necessary regarding some aspects. In order to reach positive effects it is necessary to fully integrate it into the internal and external information management. Based on this more fields of application that contribute to the enhancement of the (logistical) service provision processes can be found.

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3.2 Applications of Supply Chain Controlling Demand-driven monitoring and control of vehicles regarding transport operations on the way to scheduled destinations work as an interesting FMS application (see, e.g., [13]). The basic idea of these considerations is to ensure a timely planned arrival of the vehicles in order to prevent congestions at delivery and/or pick-up facilities. Figure 4 shows the process structure of this kind of monitoring and control. Central plant supervision

Message (Departure forwarder)

Control position

Order confirmation Forwarder

Waiting position / Truck

Instruction to enter

Code input / Gate

Unloading (Stock receipt)

Exit gate

Order confirmation Dispatcher Truck movements Information flows

Fig. 4. Monitoring and control in delivery and/or pick-up operations

The basic structure here does not only relate to a Just-in-Time (JIT)-concept where intra-organizational restrictions of the capacities are of importance, it also relates to a shift of (time-based and cost-related) risks which result from disruptions in transport processes on the delivery level. 3.3 Shipment Tracking and Tracing In this concept individual shipments are the major objective. Therefore the specific supply chain from the forwarder to the customer is shown (see, e.g., [27]). Due to the linkage between information and material flow, the primary issue of internal information management arises. This includes the two function of tracking the shipment and tracing it, e.g., for to proof correct (logistics) services, in some cases carried out to verify the fulfillment of legal obligations (see, e.g., [5]). On the basis of the available in-company data, also the information of customers can be improved significantly in the framework of an external information management that allows customers to get access to their shipment data. At this point the Internet, e.g., can serve as a communication medium between customers and (logistics) service providers which are often used to carry out the transport operations. The necessary (technical) infrastructure for the introduction of shipment tracking and tracing for a service provider generally includes the detection devices outside and inside the vehicles as well as in stationary facilities (depot, hub, etc.) for the applied identification techniques (generally optoelectronic as well as electromagnetic and electronic systems) and also specific software systems. Furthermore components are used from existing communication structures (e.g. mobile telephony, internet). Nevertheless, this generates a mere punctual tracking of the position data rather than a continuous documentation as explained in Figure 5.

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Depot

Forwarder / Seller

Data flows

Reporting

Transport operations

Entry / Exit

Recording status data

Entry

HUB Exit

Client

Exit / Entry

Depot

Fig. 5. Schematized procedure of the shipment tracking and tracing

When analyzing major advantages and disadvantages of these concepts from the perspective of the service provider, especially the tracing aspect must be underlined. Consequently, by making use of punctual tracking it is possible to differentiate the responsibility besides the proof of the order execution. Furthermore the increase of the adherence to delivery dates within the delivery processes as well as the reduction of losses through a simplified tracking of the shipment is of high importance. Another positive effect evolves from the improved possibilities to control the drivers and warehousemen. Nevertheless, the investments in vehicle equipment and stationary facilities need to be considered and also higher IT-costs should be expected. Additional information about the reporting of a shipment that is done continuously and with a low expenditure represents an extensive advantage for the customer in planning and dispatching processes, especially regarding processes with a critical time frame. When considering the importance of such an interface for the customer and its relation to the service provider, it can be said that the investment expenditure and the operational costs remain relatively low compared to the additional benefit.

4 Conclusion and Outlook As it is revealed in the argumentation, complex structures form a fundamental part of a logistic information management in the field of road freight transport. This is not only determined by (in this area available) systems for computer-aided (operational) planning and decision making of logistic processes on the in-company (micrologistic) level but to a great extent on the macro-logistic level and the here existing information and communication infrastructure. If it is possible to shape the structures accordingly and to apply them, significant advantages can be generated. These arise from cost savings and more efficient processes in service provision and furthermore in different areas from positive effects on macro-economic and environmental aspects. Caused by different basic conditions, these effects are varying case-related, so that an exact quantification is generally not possible. Due to that fact the appearing effects will be mentioned in the following, also to give a short overview on possibilities (and limits) of a powerful information management in road freight transport.

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

In-company effects Besides the mentioned cost effects the (qualitative) enhancement concerning the service provision processes must be considered as they positively influence the competitiveness. The following aspects are of main interest in this case: • Reduction of the necessary driving distances through (dynamic) routing based on actual information of traffic conditions and through an improved guidance with the help of capable navigation systems. • Enhanced response and preventive actions taking in case of derivations from the planned data due to an improved state of information reached by operational data in real-time. • Advancement of transparency in in-company processes with corresponding advantages in planning and decision making situations. • Increase of security in road traffic as well as to external impacts and also in relation to a reduction of costs resulting from damage. Further positive effects evolve from the customer perspective in form of an improved transparency regarding the logistic procedures that lead to a higher certainty for planning. In addition, a more specific data recording and evaluation encourages an advanced documentation, so that appearing sources of error can be determined.  Macro-economic effects By reducing the driving distances the consumption of resources and the strain on infrastructure from road freight transport can be decreased. This refers to the construction (and expansion) as well as to the maintenance of the traffic infrastructure but also to the follow-up costs resulting, e.g., from traffic accidents. It becomes clear that an efficient information management can counteract the stated (negative) external effects of road freight transport what also creates positive monetary effects.  Environmental effects In context of the reduction of driving distances an explicit decrease of ecological damage is the result. Here, the long-term consequences on the macro climate and the short-term (local) influences, especially in highly densified agglomeration areas have to be considered. In this context it can be observed that efficient logistics in fact are environmentally-friendly per se, what was known much longer than all in the last years discussed Green Logistics-concepts. Next to these positive effects, there is also the need of analyzing the accruing investment and operational costs borne by public institutions as well as private companies that have to be examined by preparing a cost-benefit analysis (including monetarized advantages). Only under a positive result it would make sense to realize the discussed measurements. The innovative improvements made in the field of information and communication technologies that have been presented in the last decades (see, e.g., [15]) are accompanied by considerable cost savings regarding investment and operational costs that lead to a growing potential of customers. The increase of market penetration that is essential for the necessary technologies of a powerful information management enlarges the group of (potential) users and therefore the further possibilities for interorganizational cooperation.

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20. Krüger, M., Böckle, M.: Kühlketten lückenlos online überwachen - Intelligente Sendungsverfolgung schließt Lücke bei unternehmensübergreifenden Transportprozessen. In: Bullinger, H.-J., ten Hompel, M. (eds.) Internet der Dinge, pp. 273–280. Springer, Berlin (2007) 21. Küçükay, F., Haney, D.: Assistenzsysteme in der Fertigungstechnik. In: Gesamtzentrum für Verkehr Braunschweig (GZVB) (ed.) Automatisierungs- und Assistenzsysteme für Transportmittel, Braunschweig, pp. 123–143 (2004) 22. Kunze, R., Ramakers, R., Henning, K., Jeschke, S.: Organization and Operation of Electronically Coupled Truck Platoons on German Motorways. In: Xie, M., Xiong, Y., Xiong, C., Liu, H., Hu, Z. (eds.) ICIRA 2009. LNCS, vol. 5928, pp. 135–146. Springer, Heidelberg (2009) 23. Mansfeld, W.: Satellitenortung und Navigation. 2nd rev. and ext. edn. Vieweg, Braunschweig (2004) 24. Nakanishi, Y.J.: Vehicle and road automation. In: Nof, S.Y. (ed.) Springer handbook of automation, pp. 1165–1180. Springer, Dordrecht (2009) 25. Ollero, A., Castaño, Á.R.: Automation of mobility and navigation. In: Nof, S.Y. (ed.) Springer handbook of automation, pp. 279–294. Springer, Dordrecht (2009) 26. Papageorgiou, M., Ben-Akiva, M., Bottom, J., Bovy, P.H.L., Hoogendoorn, S.P., Hounsell, N.B., Kotsialos, A., McDonald, M.: IST and traffic management. In: Barnhart, C., Laporte, G. (eds.) Transportation. Handbooks in Operations Research and Management Science, vol. 14, pp. 715–777. North-Holland, Amsterdam (2007) 27. Placzek, T.S.: Potenziale der Verkehrstelematik zur Abbildung von Transportprozessen im Supply Chain Event Management. Logistik Management 6(4), 34–46 (2004) 28. Planas, E., Pastor, E., Presutto, F., Tixier, J.: Results of the MITRA project - Monitoring and intervention for the transportation of dangerous goods. Journal of Hazardous Material 152, 516–552 (2008) 29. Rodrigue, J.-P., Comtois, C., Slack, B.: The geography of transport systems. 2nd rev. and updated edn. Routledge, London (2009) 30. Schoch, M.: Verwendung feinräumiger geographischer Informationen in aggregierten Verkehrsprognosen. Nomos, Baden (2004) 31. Scholz-Reiter, B., Toonen, C., Windt, K.: Logistikdienstleistungen. In: Arnold, D., Isermann, H., Kuhn, A., Tempelemeier, H., Furmans, K. (eds.) Handbuch Logistik. 3rd rev. edn., pp. 581–607. Springer, Berlin (2008) 32. Sichitiu, M.L., Kihl, M.: Inter-vehicle communication systems - A survey. IEEE Communications Surveys & Tutorials 10(2), 88–105 (2008) 33. Sperb, H.: Sammlung und Aufbereitung von Verkehrsinformationen. In: Siegle, G., Thielmann, H. (eds.) Mobil mit digitalen Diensten, pp. 229–243. Hüthig, Bonn (2003) 34. Tacconi, D., Miorandi, D., Carreras, I., Chiti, F., Fantacci, R.: Using wirelesssensor networks to support intelligent transportation systems. Ad hoc Networks 8, 462–473 (2008) 35. Zhang, Q., Cheng, L., Boutaba, R.: Cloud computing - State-of-the-art and research challenges. Journal of Internet Services and Aplication 1(1), 7–18 (2010) 36. Zöller, S., Meyer, M., Steinmetz, R.: Drahtlose Sensornetze als Werkzeug zur Echtzeiterkennung und -verarbeitung von Events in der Supply Chain. In: Schönberger, R., Elbert, R. (eds.) Dimensionen der Logistik - Funktionen, Institutionen und Handlungsebenen, pp. 805–820. Gabler, Wiesbaden (2010) 37. Zogg, J.-M.: Telemetrie mit GSM/SMS und GPS-Einführung. Franzis´, Poing (2002)

The Single-Stage Location-Routing Problem with Time Windows Halil Ibrahim G¨ und¨ uz Deutsche Post Lehrstuhl f¨ ur Optimierung von Distributionsnetzwerken RWTH Aachen [email protected]

Abstract. The well-known capacitated facility location problem (CFLP) and the vehicle routing problem with time windows (VRPTW) have been studied intensively over the last decades. In most distribution systems, depot location and routing decisions are implemented independently. Low-quality solutions are obtained if a sequential method, e.g., locate depots ﬁrst and plan routes second, is used. In location-routing problems (LRP), location and routing are solved simultaneously. Here, our goal is to combine the CFLP and the VRPTW into a single-stage LRP with time windows (SSLRPTW), which covers more realistic aspects, especially time aspects, of many real problems. In order to eﬃciently solve the SSLRPTW for large-scale instances, a tabu search heuristic is proposed. This method outperforms the used sequential method.

1

Introduction

The logistics industry is one of the most important economic factors, and indicators and logistics costs often represent a large portion of company expenses. The bulk of logistic costs are comprised of location, transportation, and handling costs. To reduce them, depot location and vehicle routing are crucial choices. Both problems are usually tackled independently in order to reduce the complexity of the overall problem. In particular, aspects of future route plans are neglected in most distribution systems, and only approximated. This approximation needs, however, a priori knowledge of transport services, and research has shown that this strategy often leads to suboptimal solutions (Salhi and Rand [23]). The set of LRPs within the location theory combines location and route plan decision levels. In recent years, the attention given to the ﬁeld of locationrouting has increased and many of the published works deal with real problems. Location-routing has a wide ﬁeld of application. For example, rubber smallholders [17], goods transportation [20], military [14], evacuation [6], and the paper industry [8] to name just a few. Especially in postal logistics, a huge variety of LRP applications exists (see, e.g., [5], [13], and [26]). A very good overview of LRP is given by Nagy and Salhi [16]. In this paper we consider an SSLRPTW and capacity restrictions at both customers and depots. Time restrictions in LRP have hardly been considered yet. Jacobsen and Madsen [10] consider a latest delivery time at customers in a two J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 44–58, 2011. c Springer-Verlag Berlin Heidelberg 2011

The Single-Stage Location-Routing Problem with Time Windows

45

stage newspaper distribution network. Wasner and Z¨ apfel [26] mainly restrict the routes to maximum time duration in a parcel distribution network. A periodic LRP, where customers are visited frequently over a given multiperiod horizon, is tackled with a hybrid evolutionary algorithm by Prodhon [21]. Moreover, to our best knowledge, this is the ﬁrst time that time windows have been applied to LRP. In some real problems external truck companies perform the routing. In those cases route plans are implemented for a long period of time where time windows of customers must be respected. Thus, the routing is rather tactical than operational. We propose a tabu search algorithm with with add, drop, and swap moves. Tabu search was also applied by Albareda-Sambola, Diaz and Fernandez [2] to LRP restricted to one single route per capacitated open depot. Tuzun and Burke [25] proposed a tabu search approach for LRP with capacitated routes but uncapacitated depots. This paper is organized as follows. Section 2 introduces the required notation, deﬁnes the problem and proposes an integer linear optimization model. The location subproblem, solved as an integer linear program, and the arising vehicle routing problem with time windows, solved with construction and improvement heuristics, are presented in Section 3, followed by the explanation of the proposed tabu search heuristic in Section 4. Computational results are presented in Section 5. We close with some concluding remarks.

2

Problem Definition

The SSLRPTW studied in this paper is deﬁned on a weighted (not necessarily complete) directed graph G = (V, A, C, T ). The node set V consists of a subset D of m potential depot sites and a subset I = V \ D of n customers. C and T are weights, corresponding to the travelling costs cij and the travelling time tij (includes service time si at node i), associated with the set of arcs A linking any two nodes i and j. Each depot site has a capacity Qd , opening hours [opend , closed ], and opening costs Fd . Further, each customer has a demand qi and has to be served during the time window [ai , bi ]. For service purposes, a homogenous ﬂeet K of vehicle with capacity C is available and any subset can be placed at any depot site. The task is to determine the location of open depots, the assignment of the customers to open depots, and the vehicle routes serving the customers with minimum overall costs such that the following constraints hold: – Each customer is assigned exactly to one open depot and served by exactly one vehicle during its time window (a waiting time wi at customer i is allowed). – Each vehicle is used once at the most. – Each vehicle route begins and ends at the same open depot during the opening hours. – The vehicle load does not exceed the vehicle capacity. – The total demand of the customers assigned to an open depot does not exceed the depot capacity.

46

H. I. G¨ und¨ uz

To formulate the SSLRPTW, the following binary variables are necessary: 1 if depot d ∈ D is open yd = 0 otherwise zdi

xkij

=

1 if customer i ∈ I is assigned to depot d ∈ D 0 otherwise

⎧ ⎨ 1 if node j ∈ V is directly visited after node i ∈ V by vehicle k ∈ K ⎩ 0 otherwise

=

Further, we need the following time variables: Ti

:

arrival time at customer i ∈ I

wi

:

waiting time at customer i ∈ I

startkd

:

departure time of vehicle k ∈ K at depot d ∈ D

endkd

:

return time of vehicle k ∈ K at depot d ∈ D

The linear program of the SSLRPTW can be stated as follows: min zSSLRP T W = Fd · yd + cij · xkij d∈D

(1)

k∈K (i,j)∈A

subject to

qi · zdi ≤ Qd · yd ∀ d ∈ D

(2)

i∈I

j∈V

xkij = 1

∀j∈I

(3)

xkji = 0

∀ k ∈ K, i ∈ V

(4)

xkdj ≤ 1

∀k∈K

(5)

xkij ≥ 1

∀ S ⊆ I, 2 ≤ |S|

(6)

∀ d ∈ D, i ∈ I, k ∈ K

(7)

∀k∈K

(8)

∀i∈I

(9)

k∈K i∈V

xkij −

j∈V

d∈D j∈V

k∈K i∈S j∈V \S

s∈V

(xkds + xksi ) ≤ zdi + 1

qj xkij ≤ C

i∈V j∈I

Ti + wi ≥ ai

The Single-Stage Location-Routing Problem with Time Windows

T i ≤ bi startkd endkd

47

∀i∈I

(10)

≥ opend

∀ k ∈ K, d ∈ D

(11)

≤ closed

∀ k ∈ K, d ∈ D

(12)

− M (1 − xkij ) − (Tj − Ti − wi − tij ) ≤ 0

∀ i ∈ I, j ∈ I, k ∈ K

(13)

∀ i ∈ I, j ∈ I, k ∈ K

(14)

− tdj ) ≤ 0

∀ d ∈ D, j ∈ I, k ∈ K

(15)

M (1 − xkdj ) − (Tj − startkd − tdj ) ≥ 0

∀ d ∈ D, j ∈ I, k ∈ K

(16)

∀ i ∈ I, d ∈ D, k ∈ K

(17)

∀ i ∈ I, d ∈ D, k ∈ K

(18)

M (1 −

xkij )

− M (1 −

− (Tj − Ti − wi − tij ) ≥ 0

xkdj )

− (Tj −

startkd

− M (1 − xkid ) − (endkd − Ti − wi − tid ) ≤ 0 M (1 −

xkid )

−

(endkd

− Ti − wi − tid ) ≥ 0

xkij ∈ {0, 1} ∀ i, j ∈ V, k ∈ K

(19)

yd ∈ {0, 1} ∀ d ∈ D

(20)

zdi ∈ {0, 1} ∀ d ∈ D, i ∈ I

(21)

Ti , wi ∈ ZZ + ∀ i ∈ I

(22)

startkd ,

endkd

∈ ZZ + ∀ d ∈ D.

(23)

The objective function minimizes the sum of depot and transportation costs. Capacity constraints of the open depots and the used vehicles are satisﬁed through inequalities (2) and (8). Constraints (3) and (4) known as ‘degree constraints’, guarantee the uniqueness and continuity of a route performed by a vehicle. Each vehicle is used once at the most through constraints (5). Constraints (6) eliminate subtours and constraints (7) ensure that a customer is only served by a vehicle assigned to the same open depot. While constraints (9) and (10) imply that the arrival time (with additional waiting time) at a customer is within its time window, constraints (11) and (12) guarantee that each route begins and ends at a depot during its open hours. Inequalities (13)-(18) determine the arrival time on a route, the departure, and the return time of a route performed by a vehicle. If xkij = 1 holds, than inequalities (13) and (14) reduce to the equation Tj = Ti +wi +tij , otherwise to the relaxed inequality −∞ ≤ Tj −Ti −wi −tij ≤ ∞. The same holds for (15)-(16) and (17)-(18) for startkd and endkd instead of Ti and Tj , respectively. Finally, the integrality constraints (19)-(23) state the binary or integer nature of the decision variables. For the purpose of time variables description, we discretize the time horizon into time points and code them as integer values. This formulation includes O(|K| · |V |2 ) binary variables, O(max{|K · |D|, |I|}) integer variables, and O(2|V | − 2|D| ) constraints. Therefore, only very smallscale instances can be solved with commercial solvers. Obviously, the SSLRPTW is NP-hard, since it reduces to the well known vehicle routing problem with time windows when |D| = 1. Thus, the use of heuristics is essential for largescale SSLRPTW instances. We tackled the problem with a tabu search approach described in Section 4.

48

H. I. G¨ und¨ uz

3

Subproblems

The SSLRPTW comprises the location of depots, the allocation of customers, and the construction of routes. We decompose the overall problem into subproblems, described in this section and propose a solution method, which is a modular part of our tabu search approach. 3.1

Capacitated Facility Location Problem with Time Windows

The negligence of routes reduces the SSLRPTW to the CFLP, ﬁrst introduced by Balinski [3] with additional time windows (CFLPTW). We use the notation in Section 2 to deﬁne the problem. The task is to determine the location of open depots and the assignment of the customers to open depots with a minimum sum of ﬁxed costs and assignment costs, such that the following constraints hold: – Each customer is assigned exactly to one open depot (single-sourcing) and reachable before the end of its time window. – The earliest arrival time at a depot from each assigned customer is before its closing time. – The total demand of the customers assigned to an open depot does not exceed the depot capacity. The CFLPTW model is identical to the original CFLP if time restrictions are checked during a preprocessing phase. Therefore, we need to check whether – opend + tdj ≤ bj and – max{opend + tdj , aj } + tjd ≤ closed hold, otherwise we can ﬁx the assignment variable zdj := 0 or remove arcs (d, j) and (j, d) from G. The CFLPTW with single sourcing can be stated as follows: min zCF LP T W = Fd · yd + (cdj + cjd ) · zdj (24) d∈D

(d,j),(j,d)∈A

subject to i∈I

qi · zdi ≤ Qd · yd

∀d∈D

(25)

∀I ∈I

(26)

zdi ∈ {0, 1}

∀ d ∈ D, i ∈ I

(27)

yd ∈ {0, 1}

∀ d ∈ D.

(28)

zdi = 1

d∈D

The objective function (24) sums all ﬁxed and assignment costs. Assignment costs are deﬁned by travel costs from depot to customer and back. Inequalities (25) guarantee that a customer can only be assigned to an open depot and capacity constraints of the depots are fulﬁlled. Single-sourcing restrictions are

The Single-Stage Location-Routing Problem with Time Windows

49

satisﬁed through constraints (26). Finally, constraints (27) and (28) deﬁne the binary assignment variables and location variables, respectively. If open depots are speciﬁed, than only an allocation problem with time windows remains (APTW). Hence, constraints (25) and (26) remain only for the open depots. Large-scale instances with up to 1000 customers and 350 potential depots can be solved exactly with commercial solvers. 3.2

Vehicle Routing Problem with Time Windows (VRPTW)

The vehicle routing problem with time windows arises from SSLRPTW if depot locations and a feasible assignment of customers are known. Then, the task is to construct routes for each depot and its assigned customers with the following constraints: – Each customer is served by exactly one vehicle during its time window (waiting is allowed). – Each route begins and ends at the depot during the opening hours. – The vehicle load does not exceed the vehicle capacity. The goal is to minimize the overall transportation costs. We omit a description of a model and refer the reader to [24]. For further VRPTW literature we refer the reader to Br¨ aysy and Gendreau [4]. Instead, we brieﬂy give an overview of the used heuristic methods taken from [4]. First, we construct routes with the savings heuristic of Clarke and Wright [7]. At the beginning, each customer is served individually by a separate route. Combining two routes, serving customers i and j, results in a cost savings Sij = cid − cdj − cij . We select the arc (i, j) linking customers i and j with maximum positive Sij such that the combined route is time and capacitative feasible. Further, we restrict the linking to arcs retaining the traversing order of the previous two routes (see Figure 1). Reversing a route order from (d, h, ..., i, d) to (d, i, ..., hd ) leads in most cases to time infeasibility. The savings procedure is applied iteratively. We use local search methods to improve feasible solutions. The initial solution is obtained from the savings procedure. Arc-Exchange mechanisms are applied to ﬁnd neighboring solutions. All neighbors of the current solution are examined and compared to it. The most improving neighbor replaces the current solution and the search continues. Well-known are the 2-opt and 3-opt arc-exchange procedures of Lin [12]. The following arc-exchange moves are applied: 2-opt, 2-opt∗ , Or-opt, relocate, exchange, and cross-exchange.

Fig. 1. Savings operation retaining the traversing order

50

H. I. G¨ und¨ uz

The 2-opt tries to improve a route by replacing two node disjoint arcs by two other arcs. Figure 2 depicts that the orientation of the route cannot be preserved. 2-opt∗ is similar to 2-opt, but it combines two routes so that the last customers of a given route are introduced after the ﬁrst customers of the other route. Thus, the orientation of the routes is preserved.

Fig. 2. 2-opt operation

The Or-opt introduced by Or [18] is another arc-exchange and a special case of 3-opt preserving the orientation of the route. It replaces three disjoint arcs by new three arcs, as shown in Figure 3.

Fig. 3. Or-opt operation

The relocate operator moves a customer from one route to another, and the exchange operator simply swaps customers of diﬀerent routes. Thus, we are able to change the assignments if the relocate and exchange operation is applied to routes assigned to diﬀerent depots. The basic idea of cross-exchange is to remove two arcs (i − 1, i) and (u, u + 1) from a ﬁrst route. Further, two arcs (j − 1, j) and (v, v + 1) are removed from a second route. Then the segments (i, . . . , u) and (j, . . . , v) are swapped. This is done by adding new arcs (i − 1, j), (v, u + 1), (j − 1, i), and (u, v + 1), as illustrated in Figure 4. All feasibility checks of the described operators can be done in linear time by applying them and iterating through the new routes. Linear time checks are too expensive if the operators are called several thousand times. A more

Fig. 4. Cross-exchange operation

The Single-Stage Location-Routing Problem with Time Windows

51

eﬃcient way is to check in constant time by using the (inverse) resource extension function generalized to segments, as introduced by Irnich [9]. Algorithm 1 gives an overview of how the multi-depot VRPTW (MDVRPTW) is solved in this paper. Algorithm 1. MDVRPTW 1: for each depot and its assigned customers do 2: Construct routes with the savings method. 3: end for 4: for each depot and its routes do 5: Improve routes by arc-exchange operators in the following sequence: 2-opt, 2opt∗ , Or-opt, relocate, exchange, cross-exchange. 6: if an operator improves a solution then 7: Stop and update the route(s). Go to 4. 8: end if 9: end for 10: for each pair of depots and their routes do 11: Improve routes by arc-exchange operators in the following sequence: relocate, exchange . 12: if an operator improves a solution then 13: Stop and update the route(s). Go to 10. 14: end if 15: end for 16: if an operator in 11 improved the solution at least once then 17: Go to 4. 18: end if

4

Tabu Search Algorithm for the SSLRPTW

‘Locate ﬁrst and route second’ type heuristics are sequential methods for LRP, whereas the iterative methods (e.g., [19] and [22]) iterate between the location and routing phases until a stop criterion is met. Although, iterative methods produce better solutions than sequential methods, they still have drawbacks. Both subproblems are treated as equal, therefore, the neighborhood space cannot be searched intensively. A more suitable approach is a hierarchical classiﬁcation of the subproblems, where the location is the main problem and the routing is subordinated. Thus, we refer to the nested method approach proposed by Nagy and Salhi [15]. First, we deﬁne the neighborhood structure of the SSLRPTW by the moves add, drop, and swap, introduced by Kuehn and Hamburger [11]. ‘Add’ means opening a closed depot, ‘drop’ means closing an open depot, and ‘swap’ is a simultaneous add and drop move. The change of a location mostly inﬂuences a connected area of closely located depots with their associated customers and the changed depot and its customers. Thus, we restrict the three moves to a region and a catchment area of an open depot, where after each move an MDVRPTW is solved. In addition, we have to deﬁne a neighborhood relation between two

52

H. I. G¨ und¨ uz

depots to specify the term closely in this context. The following deﬁnitions are taken from [15]. Definition 1. Two depots d1 and d2 are neighbors, if and only if at least one customer i exists, such that d1 and d2 are the nearest two depots to customer i. Definition 2. The region R(d) of an open depot d consists of the depot d itself, its customers, neighbor depots and their associated customers. Definition 3. The catchment area CA(d) of an open depot d is the smallest rectangle comprises depot d and its customers. Obviously, the neighborhood relation, the region of depot d and its catchment area is easy to create. Moreover, the assignment of customers can be restricted to the nearest opened depots. Further, we can limit an add and swap move to the catchment area of an investigated depot d, such that closed depots too far from depot d need not be considered because they cannot inﬂuence each other. Algorithm 2. Tabu Search for the SSLRPTW 1: Solve the CFLPTW ﬁrst and then the arising MDVRPTW to generate an initial solution S for the SSLRPTW. Calculate total costs C(S). 2: for each depot d in S do 3: Calculate the Region R(d). 4: Drop opened d from R(d), if this drop move is not tabu. Solve the APTW ﬁrst and then the MDVRPTW for R(d) \ {d}. Calculate the costs C(R(d) \ {d}) of the drop move. 5: for each closed depot d in the catchment area CA(d) do 6: Add d to R(d), if this add move is not tabu. Solve the APTW ﬁrst and then Calculate the costs C(R(d) ∪ {d}) of the add the MDVRPTW for R(d) ∪ {d}. move. 7: Swap d with d in R(d), if this swap move is not tabu. Solve the APTW \ {d}. Calculate the costs ﬁrst and then the MDVRPTW for (R(d) ∪ {d}) C((R(d) ∪ {d}) \ {d}) of the swap move. 8: end for 9: end for 10: Implement the move with the largest improvement, update S and C(S). 11: if S is better than the best known solution Sbest , i.e., C(S) < C(Sbest ) then 12: Set Sbest = S. 13: end if 14: Repeat 2 to 13 until a suitable stop criterion is reached.

Algorithm 2 describes the proposed heuristic. The initial solution for the SSLRPTW in line 1 is obtained by solving the CFLPTW ﬁrst and the MDVRPTW second. The solution of the CFLPTW, a set of open depots and assigned customers, serves as input for Algorithm 1 (MDVRPTW). Locations and assignments outside the region R(d) are not aﬀected by the moves in line 4, 6, and 7. All open depots in a region are determined by the moves, so only an assignment problem (APTW) has to be solved in these lines. Then, Algorithm 1

The Single-Stage Location-Routing Problem with Time Windows

53

is applied to the assignment solution. Thus, the routing is nested in the location phase. Note that a drop or swap move can lead in infeasible APTW due to time and capacity infeasibility. Further, the largest improvement in line 10 can be negative and result in a non-improving solution. In modern metaheuristics, such as tabu search, simulated annealing and genetic algorithms, the acceptance of non-improving solutions produces high quality solutions. In general, the acceptance of non-improving allows us to climb out of a local optima solution and to explore a big variety of the solution space. To forbid the immediate return to a local optima, we use the tabu strategy. In our algorithm, the reverse move of the move in line 10 is made tabu for a given number of iteration from lines 2 to 13. Our stop criterion is met if a maximum number of iterations or of non-improving move-selections (line 10) is performed.

5

Computational Study

The proposed heuristic was coded in C++ and executed on an Intel Core 2.15 Ghz computer with 3.24 GB RAM. CPLEX1 9.1 was applied to solve the CFLPTW and APTW. Before the computational results, we give a brief overview of the generated instances. 5.1

Test Instances

Since benchmarks for the SSLRPTW are not available, we derived SSLRPTW instances from the extended VRPTW Solomon benchmark [1]. We used the class RC1 with 400 customers, in order to have instances with clustered and randomly distributed customers. The 10 instances of RC1 vary in the time windows of customers. Some of the time windows are tight and some are relaxed. In each instance, the service time, demand, and coordinates are given. To add further potential depots to the existing depot, we take the smallest rectangle including all customers and draw horizontal and vertical lines parallel to the sides of the rectangle with equal distance. The result is a grid and every grid point in the rectangle is a potential depot. Overall, the generated instances consist of 50 potential depots. Open hours of the potential depots are deﬁned as [mini∈I ai , maxi∈I bi ] and their service time is set to zero. As ﬁxed costs we used the set of 100, 500, 1,000, and 3,000 monetary units (MU) combined with depot capacity 250, 500, 1,500, 3,000 quantity units (QU). The demand of all customers is 7,127 QU and the minimum required number of open depots is given in Table 1. Overall, 160 instances were created and tested. The vehicle capacity is adopted from the Solomon instances. Further, the travel costs cij and the travel time match the Euclidean distance multiplied by factor 10. 5.2

Computational Results

First, note that the 10 instances with the combination of the mentioned ﬁxed costs and depot capacity are called test group in this section. We use this notation to illustrate average results instead of explicit results of each instance. In 1

IBM ILOG mathematical program engine.

54

H. I. G¨ und¨ uz Table 1. Minimum required number according to the capacity Qd [QU] 250 500 1,500 3,000 Required depots 29 15 5 3

Table 2 each line corresponds to a test group. Colums ‘ depots’ correspond to the average number of open depots and ‘costs’ to the average overall costs of the initial and best obtained solutions, respectively. Column Δ provides the average improvement of the initial solution by the proposed tabu search heuristic. ‘CPU’ is the average CPU time, given in hh:mm:ss, and ‘ iter.’ the average number iterations needed by the tabu search heuristic. Table 2. Computational results of the tabu search heuristic test group Fd Qd

initial solution depots costs

best solution depots costs

Δ

statistics iter. CPU

100 250 100 500 100 1,500 100 3,000

48.0 47.0 47.0 47.0

43,593.5 41,152.7 41,137.7 41,140.8

41.4 36.4 34.8 34.5

41,471.4 39,259.2 39,261.7 39,221.3

4.91 4.63 4.60 4.70

41.3 36.5 34.3 36.9

0:08:38 0:12:05 0:12:41 0:13:30

500 250 500 500 500 1,500 500 3,000

41.0 39.0 39.0 39.0

59,732.2 56,515.5 56,500.5 56,489.7

31.8 19.8 16.4 16.1

59,732.2 49,473.7 48,735.7 48,529.9

6.62 12.52 13.82 14.16

33.3 33.7 37.0 38.1

0:11:03 0:16:46 0:23:17 0:24:07

1,000 250 1,000 500 1,000 1,500 1,000 3,000

35.0 32.0 32.0 32.0

75,164.4 69,417.1 69,535.5 69,529.3

30.9 17.0 11.7 11.8

71,847.8 69,417.1 55,135.6 55,239.7

4.42 15.94 20.80 20.64

20.9 28.0 36.0 34.4

0:09:26 0:14:38 0:38:02 0:38:36

3,000 250 3,000 500 3,000 1,500 3,000 3,000

29.1 19.0 18.0 18.0

131,940.2 98,724.7 96,098.1 96,096.5

29.1 15.2 5.9 5.5

131,285.0 88,960.2 69,566.5 69,092.6

0.50 9.92 27.70 28.21

16.2 20.8 29.2 30.0

1:01:51 0:15:10 2:04:05 2:17:37

For all instances, the CFLPTW was solved to optimality with CPLEX. We observed that the number of open depots remains unchanged within a group except for one instance in the group with ﬁxed costs 3,000 MU and capacity 250 QU. This indicates that time windows do not aﬀect the location decision of the CFLPTW. Furthermore, the number of open depots exceeds the minimum required number (compare Table 1 and Table 2), except again for the group with ﬁxed costs 3,000 MU and capacity 250 QU. Actually, the domination of the assignment costs of the CFLPTW model are the reason. Therefore, depots are located near to the customers to reduce the assignment costs, such that time windows are not restrictive anymore. In instances with increasing ﬁxed costs, we have a decreasing domination of assignment costs and if in addition tight depot

The Single-Stage Location-Routing Problem with Time Windows

55

capacity is available, then the number of open depots is equal to the minimum required number. By the following MDVRPTW, a ﬁrst feasible solution for the SSLRPTW is detected fast. In general, the ﬁrst obtained solution is far from the best solution found by our proposed tabu search heuristic. The average relative improvement of the initial solution is located between 4% and 29% (Table 2), except for one test group.

40

35

30

25

0.75-quantile minimum

improvement 20 in %

maximum 0.25-quantile median

15

10

5

0 RC1

Fig. 5. Range of relative improvement

As seen in Figure 5 and Figure 6, the relative improvement varies between 0.1% and 33.44% and the number of open depots is reduced up to 26. For 50% of the instances, the relative improvement is between 4.65% and 17.7%. Further, in 50% of the instances the depot number reduction is between 8 and 20. Actually, the decrease of depots permits the exploitation of economies of scale in the routing phase. The relative improvements and depot reduction below the 0.25-quantile are achieved mostly for the test groups with depot capacity 250 and 500 QU, whereas the results above the 0.75-quantile are achieved mostly for the test groups with depot capacity 1,500 and 3,000 QU with ﬁxed costs 1,000 and 3,000 MU. Small improvements are achieved for instances with low depot capacity 250 MU. With increasing depot capacity and ﬁxed costs, large improvements are obtained. This is due to the fact, that the depot number cannot be reduced as in instances with high depot capacity. Reducing depot numbers with low ﬁxed costs does not have the same impact on the overall costs as with higher ﬁxed costs.

56

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Fig. 6. Range of depot number

The CPU time shows, that the instances can be solved in acceptable time. We observed, that the CFLPTW model is solved fast, expect to the test group with ﬁxed costs 3,000 MU and depot capacity 250 QU. Because of the high ﬁxed costs and the tight ﬁxed costs many feasible solutions with similarly objective exist, such that CPLEX needs many iterations to abort with an optimal solution. Further, the used CPU time portion for the APTW is relatively high for instances with tight depot capacity. In this cases an iteration or a gap limitation could be useful for the CFLPTW or APTW, respectively. In all other cases more than 80% of the CPU time is used by the MDVRPTW Algorithm 1 for thousands of VRPTW instances arising during the tabu search Algorithm 2. Especially for the test groups with a few open depots we have large VRPTW instances. Therefore, in test groups with high ﬁxed costs and large depot capacity more than 90% of the CPU time is spent during the routing phases. To speed up, the use of less arc-exchange operators or an iteration limitation can be useful.

6

Conclusion

In this paper, both location and routing problems are tackled together, leading to the SSLRPTW. This consists of location depots, assigning customers, and planning routes with consideration of time windows and capacity restrictions at depots and customers. A tabu search heuristic is proposed to solve largescale instances of the SSLRPTW. It is based on the add, drop, and swap moves

The Single-Stage Location-Routing Problem with Time Windows

57

of Kuehn and Hamburger [11]. An initial solution is obtained by a sequential method. First, an appropriate CFLPTW is modeled as an integer linear program and solved with CPLEX. Afterwards, an MDVRPTW is solved. Feasible routes are created with the savings method of Clarke and Wright and improved with arcexchange operations. The proposed tabu search is tested on 160 instances with diﬀerent combinations of ﬁxed costs and depot capacity, as well as varying time windows. The solutions obtained, indicate that the sequential approach considers the time aspects insuﬃciently and shows that the proposed tabu search is an appropriate and far better method to solve the SSLRPTW. Since there are no exact methods available, future research should include lower bound calculations to evaluate the quality of the solutions. Further, the development of the proposed and other heuristics should be included in order to produce a higher quality solution.

References 1. Extended VRPTW Solomon benchmark, http://www.fernuni-hagen.de/WINF/touren/inhalte/probinst.htm 2. Albareda-Sambola, M., Diaz, J.A., Fernandez, E.: A compact model and tight bounds for a combined location-routing problem. Computers & Operations Research 32(3), 407–428 (2005) 3. Balinski, M.L.: Integer programming methods, uses, computation. Management Science 12, 253–313 (1965) 4. Br¨ aysy, O., Gendreau, M.: Vehicle routing with time windows, Part I: Route construction and local search algorithms. Transportation Science 39(1), 119–139 (2005) 5. Bruns, A., Klose, A., St¨ ahly, P.: Restructuring of Swiss parcel delivery services. OR Spectrum 22, 285–302 (2000) 6. Chan, Y., Carter, W.B., Burns, M.D.: A multi-depot, multiple-vehicle, locationrouting problem with stochastically processed demands. Computers & Operations Research 28, 803–826 (2001) 7. Clarke, G., Wright, J.W.: Scheduling of vehicles from a central depot to a number of delivery points. Operations Research 12, 568–581 (1964) 8. Gunnarsson, H., R¨ onnqvist, M., Carlsson, D.: A combined terminal location and ship routing problem. Journal of the Operational Research Society 57(8), 928–938 (2006) 9. Irnich, I.: Resource extension functions: properties, inversion, and generalization to segments. OR Spectrum 30, 113–148 (2008) 10. Jacobsen, S.K., Madsen, O.B.G.: A comparative study of heuristics for a twolevel routing-location problem. European Journal of Operational Research 5, 378–387 (1980) 11. Kuehn, M.J., Hamburger, A.A.: A heuristic program for locating warehouses. Management Science 9, 643–666 (1963) 12. Lin, S.: Computer solutions of the traveling salesman problem. Bell System Technical Journal 44, 2245–2269 (1965) 13. Lischak, C.: Standortplanung f¨ ur einen privaten Paketdienstleister. Ph.D. thesis, RWTH Aachen University (2001) 14. Murty, K.G., Djang, P.A.: The U.S. army national guards mobile training simulators location. Operations Research 47(2), 175–183 (1999)

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15. Nagy, G., Salhi, S.: Nested heuristic methods for the location-routeing problem. Journal of the Operational Research Society 47(9), 1166–1174 (1996) 16. Nagy, G., Salhi, S.: Location-routing: Issues, models and methods. European Journal of Operational Research 177, 649–672 (2007) 17. Nambiar, J.M., Gelders, L.F., Van Wassenhove, L.N.: Plant location and vehicle routing in the Malaysian rubber smallholder sector: A case study. European Journal of Operational Research 38(1), 14–29 (1989) 18. Or, I.: Traveling salesman-type combinatorial problems and their relation to the logistics of regional blood banking. Ph.D. thesis, Northwestern University (1976) 19. Perl, J., Daskin, M.S.: A uniﬁed warehouse location-routing methodology. Journal of Business Logistics 5(1), 92–111 (1984) 20. Perl, J., Daskin, M.S.: A uniﬁed warehouse location-routing. Transport Research Part B 19B(5), 381–396 (1985) 21. Prodhon, C.: A hybrid evolutionary algorithm for periodic location-routing problem. European Journal of Operational Research 210, 204–212 (2011) 22. Salhi, S., Fraser, M.: An integrated heuristic approach for the combined location vehicle ﬂeet mix problem. Studies in Locational Analysis 8, 3–21 (1996) 23. Salhi, S., Rand, G.K.: The eﬀect of ignoring routes when location depots. European Journal of Operational Research 39, 150–156 (1989) 24. Toth, P., Vigo, D. (eds.): The vehicle routing problem. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2002) 25. Tuzun, D., Burke, L.: A two-phase tabu search approach to the location routing problem. European Journal of Operational Research 116, 87–99 (1999) 26. Wasner, M., Z¨ apfner, G.: An integrated multi-depot hub-location vehicle routing model for network planning of parcel service. Operations Research Letters 20, 403–419 (2004)

A Cross Entropy Multiagent Learning Algorithm for Solving Vehicle Routing Problems with Time Windows Tai-Yu Ma LET-ISH, 14, Avenue Berthelot F-69363 Lyon Cedex 07 [email protected]

Abstract. The vehicle routing problem with time windows (VRPTW) has been the subject of intensive study because of its importance in real applications. In this paper, we propose a cross entropy multiagent learning algorithm, which considers an optimum solution as a rare event to be learned. The routing policy is node-distributed, controlled by a set of parameterized probability distribution functions. Based on the performance of experienced tours of vehicle agents, these parameters are updated iteratively by minimizing Kullback-Leibler cross entropy in order to generate better solutions in next iterations. When applying the proposed algorithm on Solomon's 100-customer problem set, it shows outperforming results in comparison with the classical cross entropy approach. Moreover, this method needs only very small number of parameter settings. Its implementation is also relatively simple and flexible to solve other vehicle routing problems under various dynamic scenarios.

1 Introduction The vehicle routing problem with time windows (VRPTW) has been known as one of the NP-hard problems in combinatorial optimization. The problem consists of delivering goods to a set of customers, which must be visited within given time windows, at a minimum cost under available capacitated vehicle constraints. The standard VRPTW has been the subject of intensive study because of its importance in real applications, such as pickup and delivery problems in transportation of goods and fleet operation management. The exact and heuristic algorithms have been proposed and applied to Solomon's test instances and many real-life situations. Recent reviews in these solution techniques for the VRPTW can be found in [1, 2, 3]. The state of the art of exact methods has successfully solved most of Solomon's 100-customer benchmark instances. However, only very limited large instances have been solved to optimality [4]. Hence, an intensive effort has been engaged in proposing efficient heuristics for solving the VRPTW and related vehicle routing problems. The state of the art of heuristics includes: local search [5, 6], adaptive large neighborhood search [7] and evolutionary mechanism [8, 9]. Basically, these heuristics apply two main procedures to construct and improve the solutions: the route construction procedure and the route improving procedure. The former consists of generating a set of feasible routes based on deterministic or stochastic route-building heuristics. This procedure provides initial points in order to apply some improving J.W. Böse et al. (Eds.): ICCL 2011, LNCS 6971, pp. 59–73, 2011. © Springer-Verlag Berlin Heidelberg 2011

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local search algorithms for finding high-quality solutions on the neighborhoods. In general, the neighborhood search space is very large; one needs to develop efficient searching strategies to find local optima. To this end, the local search algorithms are widely used, which consist of replacing a subset of initial solutions such that better solutions can be found. Numerous variants based on the local search framework have been proposed such as iterative local search [10, 11], multistart local search [12, 13] and adaptive multistart local search [5]. Basically, these local search algorithms generate initial solutions by the route construction heuristics or some random mechanisms, and then apply the local search heuristics such as k-opt, i.e., replacing k edges on a route traveled by a vehicle with k edges not on this route, or swap move, exchanging the position of two nodes on the same route, to improve solution quality. To avoid being trapped into local optima, related perturbation or randomization procedures need to be applied. The performance of the local search heuristics depends not only on the interaction between the route construction and the improvement procedures but also on the strength of the perturbation or randomization of the two procedures. To this issue, an adaptive large neighborhood search method has been proposed [7]. The authors proposed a flexible framework aiming to adaptively choose a set of local search techniques. This method provided an auto-adjusted mechanism to intensify or diversify searching neighborhood according to the performance of the set of heuristics. However, it needs to implement a set of heuristics, which is more timeconsuming and complicated than single-heuristic-based algorithm. Different from the aforementioned heuristics, the agent-based distributed solution techniques have been proposed recently. Vokrinek et al. [17] proposed a vehicle routing problem solver based on multiagent framework. The solver is composed of three types of agents in order to collect demand (task agent), allocate demand (allocation agent) and find routes (vehicle agent). The route construction algorithm is based on greedy search heuristics. The experiments were conducted for capacitated vehicle routing problem. The average solution quality within 91.3% optimality was reported. Barbucha and Jedrzejowicz [16] developed a multiagent platform for simulating dynamic vehicle routing system, i.e., customer requests arrive when the vehicles are running. The system is composed of a set of agents with different functionalities for executing different tasks such as initialization, customer request generation and route assignments for requests. Simple insertion rules are applied for customer request assignment to the vehicles. Other applications based on the multiagent framework in logistics can be found in [18]. In summary, these studies have developed simulation tools based on the multiagent approach to static or dynamic vehicle routing problems. However, the solutions found by these multiagent approaches are still far from the optima. In this work, a multiagent learning algorithm based on cross entropy (CE) method [14] is proposed. We associate a set of routing probabilities with nodes (customers) on the network, iteratively guiding the vehicles to find optimal routes, which minimize total cost, and satisfying capacity and time window constraints. Based on the performance of the routes traveled by the vehicles in each iteration, the choice probability distributions of next outgoing nodes are iteratively updated. We consider the optimum solution as a rare event to be estimated based on the importance sampling theory. More precisely, we specify a random mechanism to generate feasible solutions (samples), controlled by a set of parameterized probability

A Cross Entropy Multiagent Learning Algorithm for Solving VRPTW

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distribution functions (pdf). Based on the performance of the samples, the parameters are updated iteratively by minimizing Kullback-Leibler cross-entropy in order to generate better solutions in next iterations. A set of vehicle-specific transition matrices is associated with the nodes of the network to construct subsequently a feasible route for each vehicle. As the capacity and time window constraints need to be satisfied during the route construction process, a sequential importance sampling technique is utilized by constructing the solution sequentially, conditional on vehicle’s capacity and customer’s time windows constraints. The stochastic route construction procedure is repeated until all customers are serviced. As the classical CE method may be trapped on the local optima at its early stages, some local search techniques are combined with the CE method to avoid this problem and to improve significantly the convergence speed of the classical CE method. The rest of this paper is organized as follows. In Section 2, we define the VRPTW problem and provide its mathematical formulation. Section 3 introduces the concept of the classical CE method and the proposed multiagent system for the VRPTW problem. Based on the performance of routes travelled by the vehicles, a hybrid scheme combining the agent-based CE algorithm and the local search procedure is proposed. This scheme enables local search performed only on a small subset of good solutions. Section 4 provides the computational results for Solomon's 100-customer VRPTW instances. Finally, the conclusions and future extensions are discussed.

2 Problem Formulation The mathematical formulation of the VRPTW problem can be stated as follows. Let G (V , E ) be a directed graph with a vertex set V and an arc set E. The vertex set is composed of one depot (node 0) and n customers 1 to n, denoted as V = {0,1,..., n} . The arc set is E = {(i, j ) i ≠ j , ∀i, j ∈ V } . Each customer is associated with an amount of

goods di to be delivered. Let M = {1,2,..., M } be a set of homogeneous/heterogeneous vehicles with M being the total number of vehicles. Each vehicle m has a fixed capacity qm . Each customer is associated with a time window [ai , bi ] , for which the customer cannot be serviced before ai and after bi . Let si be the arrival time for customer i. The service time for customer i is denoted as u i with u i > 0 , but no service time at the depot, i.e., u 0 = 0 . Let the depot be associated with a scheduling time window [a0 , b0 ] , for which any vehicle cannot depart the depot before a0 and return to it after b0 . Each arc (i, j ), ∀i ≠ j has an asymmetric travel time tij and an operation cost cij . The problem is to determine a set of routes, originating and terminating at the depot, such that total cost is minimal by satisfying: (1) each customer is serviced exactly once; (2) vehicle capacity cannot be violated; (3) all customers must be serviced within the service time windows. Let rm denote a route, starting and terminating at the depot, composed of a sequence of customers visited by the vehicle m. The k-th visited customer of vehicle m is denoted as rm (k ) . Let δ m = {0,1,2,..., nm , nm + 1} be a sequence

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of visiting order of the customers in rm with rm (0) = rm ( nm + 1) = 0 . We define an indicator yim as 1 if customer i is serviced by vehicle m, and 0 otherwise. The set of routes for all vehicles is denoted as r = r1 , r2 ,..., r M . We formulate the VRPTW

{

}

problem mathematically as follows: Min S (r ) =

 

(1)

≤ qm , ∀m ∈ M

(2)

= 1, ∀i ∈ V \ {0}

(3)

crm ( k −1) rm ( k ) m∈M k∈δm \ {0}

subject to

d

i∈rm \ {0}

y

i

im

m∈M

srm ( k −1) + u rm ( k −1) + trm ( k −1) rm ( k ) ≤ srm ( k ) , ∀k ∈ δ m \ {0}

(4)

ai ≤ si ≤ bi , ∀i ∈ V

(5)

yim ∈ {0, 1}, ∀i ∈ V \ {0}, ∀m ∈ M

(6)

The objective function Eq. (1) minimizes the total travel cost. The constraints (2) mean that each vehicle cannot load goods exceeding its capacity restriction. The constraints Eq. (3) ensure that each customer is serviced exactly once. The constraints Eq. (4) ensure the consistency of service time for next visiting customer. The time windows are imposed by Eq. (5). The constraints Eq. (6) are the integrality constraints. Note that the above formulation is route-based, convenient for constructing routes for each vehicle based on the multiagent framework.

3 Cross Entropy Learning Algorithm The main concept of the CE method is to associate the optimization problem with an estimation problem throughout the route-searching process. The search process is characterized by a set of density functions associated with the nodes of the network. These density functions are iteratively updated based on the minimization of the Kullback-Leibler distance (cross entropy) between density functions and optimal density functions. In the following, we present first the multiagent model for general vehicle routing problem simulation. Then we describe the cross entropy multiagent learning algorithm to solve the VRPTW problem. 3.1 Multiagent Model for Vehicle Routing Problems with Time Windows The multiagent simulation framework is very convenient for modeling and simulating the general vehicle routing problems since it captures the system behavior dynamics resulting from the interactions of supply and demand. Basically, the system is

A Cross Entropy Multiagent Learning Algorithm for Solving VRPTW

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composed of a set of heterogeneous agents with predefined behavior. The advantage of the multiagent approach resides on its flexibility in capturing complex interactions between different components of the system. The simulation of the VRPTW problem by the multiagent approach describes vehicle’s optimal route search process under its capacity and service time windows constraints. In current application, the system is specified as: • •

•

Customer agent: a set of customers with fixed demand is known a priori. Each customer agent needs to be visited exactly once. Vehicle agent: a set of vehicle agents with fixed capacity is available in the system. Vehicle agents depart from and return to the depot by picking up customer requests under its capacity and service time windows constraints. The violation of these constraints is not allowed. Environment (network): it is represented by a directed graph on which a set of vehicles operate on it. The network is characterized by a set of nodes representing the customers and a set of links associated with related characteristics. In static case, the travel time of link is fixed. In dynamic case, one can replace it with time-dependent travel cost function.

The present multiagent framework is convenient in dynamic situations with timedependent travel cost or stochastic customer demand. 3.2 Cross Entropy Learning Algorithm The main idea of the proposed cross entropy learning algorithm is that we specify a parameterized random mechanism for vehicle agents to generate feasible routes. Based on the performance of these "samples" (a sample is a feasible solution of the VRPTW), the parameters of the random mechanism are updated towards the optimal solutions. This stochastic search algorithm is originated from the importance sampling techniques aiming to increase the accuracy of rare-event probability estimation. Basically, the CE learning algorithm is composed of two steps [14, 19]: (1) generate a set of samples according to some stochastic mechanism; (2) update the parameters of stochastic mechanism based on the performance of the samples in order to generate better solutions at next iteration. As the classical CE method may be trapped in local optima, we propose a hybrid scheme by combining local search algorithms in order to overcome this problem and increase the solution quality. The random feasible route generation mechanism moves vehicle agents from current customer/depot to next unvisited customers, one at each step, based on Markov chain on graph G(V,E), starting and ending at the depot. A set of vehiclespecific transition matrix (a routing probability matrix), P={ p 1 , p 2 ,..., p M } are associated with each node of the graph to construct subsequently a feasible route for one vehicle agent. The stochastic route construction procedure is repeated until all customers are serviced. Different from the classical CE method, the idea of the proposed method is based on the sequential importance sampling techniques [21], i.e., to construct a solution sequentially, conditional on capacity, while time windows constraints and partial solutions are already constructed before the current state. Note

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that the proposed CE learning algorithm is quite intuitive. It needs only to rescale the transition probabilities at current visiting node by eliminating infeasible candidates (next infeasible not yet visiting customers) in order to generate a feasible route at each step. If there is no feasible candidate to visit, the vehicle agent returns to the depot and next vehicle starts its route construction. Let the transition matrix of vehicle m be p m = ( pijm ), ∀m ∈ M with pijm being the

transition probability for the vehicle m moving from node i to node j. Note that p m is a V × V matrix with the first column and row being the depot. We set pijm > 0 if i ≠ j , and 0 otherwise. Each vehicle constructs a route sequentially based on its transition matrix and a prohibition list (infeasible candidates) until current state. As mentioned above, we cast the original optimization problem to an estimation problem of rare event probability. To increase the sampling performance at each iteration, a sequence of new sampling densities, called importance sampling densities, need to be chosen. The optimal importance sampling density P* can be iteratively derived by minimizing the Kullback-Leibler cross-entropy distance [14, 19]. First, a criterion of rare event (an approximate to optimal solution) γ is associated with each iteration and updated according to the performance of independent and identically distributed (i.i.d.) samples. We associate γ with an indicator I {S (r ) ≤γ } being 1 if the performance of solution is better than γ , i.e., S (r ) ≤ γ , and 0 otherwise. Let f (r; P) denote the probability distribution function (pdf) of r (defined around Eq. (1)), parameterized by P. The probability of a global tour r is the multiplication of outgoing node choice probabilities in vehicle's route constructing process. Its logarithm can be written as: M ( r ) nm +1

  I {

ln f (r; P ) =

m =1

k =1 i , j

rm ∈R ij ( k )

m } ln pij ,

(7)

where M (r ) is the set of vehicles utilized for the solution r. R ij (k ) denotes the set of feasible routes such that the k-th transition (move) is from node i to node j. The optimal important sampling distribution based on the minimization of the CE distance can then be obtained as: P* = arg max E Pw−1 I {S (r ) ≤ γ} ln f (r; P) P

(8)

Note that the above equation derives the optimal important sampling pdf based on known Pw −1 . It is equivalent to the following optimization problem: E Pw −1 I {S ( r ) ≤ γ } ln f (r; P)

Max

(9)

subject to

p

m ij

+

j∈Λ ( i )

= 1, ∀i ∈ V , ∀m ∈ M

(10)

A Cross Entropy Multiagent Learning Algorithm for Solving VRPTW

65

pijm ≥ 0, ∀(i, j ) ∈ E , ∀m ∈ M

(11)

where Λ+ (i) denotes the successors of node i. By applying Karush-Kuhn-Tucker optimality conditions, the optimal solution of the above optimization problem can be obtained by differentiating with respect to pijm as: n m +1   I {rm ∈R ij ( k )} Ε Pw −1  I {S (r ) ≤ γ } k =1  

N



pijm =

n m +1   I {rm ∈R i ( k )}  Ε Pw −1  I {S (r ) ≤ γ} k =1  

≈



  I{



S (r ) ≤ γ}

n m +1

 I{

rm ∈R ij ( k )



}

 k =1  n m +1    I {S (r ) ≤ γ} I {rm ∈R i ( k )}   h =1  k =1 

h =1 N



(12)



where R ij (k ) is the set of routes, for which the k-th transition is from node i to node j. R i (k ) is the set of routes for which its k-th transition is starting from node i. N is the sample size. Note that the above updating rule states that pijm is updated based on the transition

proportion from node i to node j of the vehicle m, conditional on the global tour performance satisfying S (r ) ≤ γ . As the above important sampling pdf could generate some infeasible tours in the sequential route construction process, a rescaling procedure is conducted at each step such that next node to be visited is drawn from the feasible node set. The CE learning algorithm is stated as follows. Algorithm 1: The CE learning algorithm Step 1: Initialize P0 as a uniform pdf over the node set V. Order the endings of the time windows bi , ∀i ∈ V from the smallest to the biggest. Set the iteration index w = 1. Step 2: Generate N i.i.d. samples according to the feasible route generation algorithm (described later). Step 3: Order the solution performance S (r ) as a sequence S (r1 ) ≤ S (r2 ) ≤ ... ≤ S (rN ) . Apply the local search procedure (described later) for the samples in the λN best solutions. We set 0.1 ≤ λ ≤ 0.3 . Step 4: Order the solution performance after the local search. The ρ -quantile of the performances S (~r ) , i.e., ρN  th lowest cost ( x denotes the smallest integer greater than or equal to x), is denoted as a new “rare-event” criteria: γ w = S( ρN )

(13)

As for the value of ρ and the sample size N, it is recommended that as the sampling number increases, the ρ value decreases. We set ρ as N = c2 V

2

c1 V

and

with 1 ≤ c1 ≤ 10 and 0.5 ≤ c2 ≤ 1 , where the sample size N is smaller

than in the classic CE method, reducing considerably the computational times.

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Step 5: Calculate Pw by Eq. (12) and apply the smoothed updating rule as follows: Pw := αPw −1 + (1 − α )Pw ,

(14)

It is recommended that α takes a value in the range of 0.4 ≤ α ≤ 0.9 (see [19]). Step 6: To avoid being trapped into local optimum, the following dynamic parameter adjustment process is applied. If S (rw* −2 ) − S (rw* −1 ) / S (rw* −1 ) ≤ θ , where 0.02 ≤ θ ≤ 0.03 , multiply λ by k1 and ρ by k 2 with 1.5< k1 <3, 1< k 2 <2, where S (rw* −1 ) is the cheapest travel distance obtained in iteration w-1. If for some iteration w , the value of γw stabilizes, i.e., γ w = γ w −1 = ... = γ w − c , where c is a constant, or w = w max then stops; otherwise set w = w + 1 and go to step 2. Algorithm 2: Feasible route generation algorithm This procedure rescales the transition matrix p m (eq. 12) for each vehicle in a sequential way such that each customer (node) is visited exactly once and satisfies vehicle’s capacity and time windows constraints. The complexity of the rescaling procedure is O (n 2 ) . Step 1: Initialization. Set all customers as unvisited and vehicles depart from the depot. Set position index k=0 for each vehicle. Step 2: Rescaling choice probability of next visiting customers. For vehicle m currently located at its node i (customer), rescale the choice probability pijm for next visiting node j as follows. First, set the choice probability of infeasible nodes (not satisfying capacity and time windows constraints, eq. (2)-(4)) as 0 and then normalize choice probabilities over all feasible outgoing nodes j to sum up to 1, i.e., pijm :=

pijm

p

m ij ′

, where Λ is the set of feasible nodes for vehicle m at node i.

j ′∈Λ

Step 3: If next visiting node is the depot, stop for vehicle m and set m: = m + 1, otherwise repeat step 2. 3.3 Local Search Procedure To improve the solution quality for the samples in the ρ -quantile of the best solutions in main algorithm 1, we apply the local search procedure. The procedure contains two phases. First, a greedy local search is applied. Then a route-exchange local search procedure is applied. •

Greedy search: this procedure aims to reduce the number of utilized vehicles and exchanges some nodes to get the largest travel cost reduction. It contains two steps as follows:

A Cross Entropy Multiagent Learning Algorithm for Solving VRPTW

67

a. b.

•

Remove the route with the least customers, Insert all nodes of the removed route, one node at a time, at the cheapest position of other routes. Route exchange search: this procedure applies randomly one of the following two local search methods: c. Reverse move: reverse current ordering two nodes of the same route, d. K-Or-opt [15, 22]: remove k nodes of current route and insert them in another position of the same route so that a feasible set of routes is preserved. This local search is implemented in the order of 1-Or-opt, 2-Oropt and 3-Or-opt.

Note that one can also apply some more efficient local search techniques such as LKH local search [12] or sequential search procedures [15] to obtain an improved solution quality.

4 Computational Experiments The algorithm is tested on three datasets C1, R1 and RC1 of Solomon's 100-customer benchmark instances [20]. These datasets reflect different characteristics of customers’ positions, tight or loose time windows constraints and vehicles’ capacity. The algorithm is programmed in C++. The results are shown in Table 1. The best known or optimal solutions are listed in the left part of Table 1. The results show that the proposed algorithm finds nearoptimal solutions with an average error of 6.24% (N=10,000) in comparison with optimal solutions. As the reported results are based on only 1 to 3 tests, better solutions can be found if trying more tests and modifying related parameter settings. The comparison of running times and solution quality based on different sample size shows that the running time is basically proportional to the sample size. Figure 1 reports the outperforming results of the proposed hybrid scheme compared with the classical cross entropy method. Note that the hybrid scheme utilizes fewer samples ( 0.7 n 2 ) than the cross entropy method ( ≈ 10n 2 ) to achieve better solution quality. Figure 2 presents the average performance of different sample size on 12 instances of C1, R1 and RC1 classes. Figures 3-5 presents the impact of different parameter settings on solution quality. These tests are performed on the R102 dataset of Solomon's 100 customer instances. Figure 3 presents the smooth parameter α on the convergence speed. The test experiences suggest that α may be taken between 0.6 and 0.9 to obtain better solution quality. The impact of the size of the elite sample, parameterized by ρ , on solution quality is shown in Figure 4. It suggests a range of [0.05, 0.1] for ρ . If ρ is too small, say 0.01, the algorithm converges quickly to local optima. The influence of the sample size of local search on the solution quality is not very significant (Figure 5). However, more efficient local search techniques play an important role in increasing the performance of the proposed algorithm.

827.3 826.3 827.3 827.3 1,466.6 971.5 1,234.6 932.1 1,457.4 1,132.3 1,372.7 1,114.2

C101 C103 C105 C107 R102 R104 R106 R108 RC102 RC104 RC106 RC108

840.1 890.6 867.1 843.1 1,551.0 1,061.0 1,334.0 1,040.3 1,571.8 1,248.8 1,477.6 1,242.0

Distance1 1,105 1,479 1,366 1,407 1,325 1,876 1,292 1,557 1,285 1,721 1,311 1,427

2,509 3,359 3,104 3,196 3,010 4,263 2,936 3,537 2,918 3,909 2,978 3,242

Time2 (sec.) 898.7 896.3 869.3 841.1 1,539.1 1,077.8 1,323.1 1,044.2 1,584.4 1,226.6 1,465.6 1,241.2

Distance1

CE multiagent learning algorithm Time1 (sec.)

1.55% 7.78% 4.82% 1.91% 5.75% 8.04% 8.05% 8.26% 7.85% 9.98% 3.71% 8.97% 6.39%

Relative error (%)

N=7,000

8.63% 8.47% 5.08% 1.67% 4.94% 9.76% 7.17% 8.67% 8.71% 8.02% 2.87% 8.90% 6.24%

Relative error (%)

N=10,000

1,930 1,942 2,250 2,157 2,151 2,643 2,074 2,367 2,354 2,242 2,045 2,126

Time1 (sec.) 4,384 4,411 5,111 4,900 4,886 6,003 4,712 5,378 5,347 5,092 4,647 4,829

Time2 (sec.)

Remarks: 1. The parameter settings are D 0.7, U 0.05, O 0.2, T 0.02, k1 2.0, k 2 1.2 for all instances, except C103 with U 0.07, O 0.15 . 2. The above mentioned result is based on 1 to 3 runs for each instance. 3. Time1 is the total time executed on Dell Latitude E6400 with 2.53GHz and 3.48G memory. Time2 is based on the execution of the same program on Asus Intel Pentium M processor 740 with 512MB memory. The speedup ratio between the two laptops is about 2.27. 4. The best know solution is based on [23].

Average error

Optimum/ best known solution4 Distance

Instance

Table 1. Application of CE multiagent learning algorithm to Solomon's VRPTW 100 customer instances

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1400 CE + local search CE only

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Fig. 1. Performance of CE multiagent learning algorithm compared with the classical CE method (without applying local search and dynamic parameter adjustment of Algorithm 1) (Solomon’s 50-customer C101 instance, ρ =0.05, α =0.7, optimum= 362.4) 3500 N = 0.2n2 N = 0.7n2 N = n2

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Fig. 2. Influence of sample size on the average performance of the algorithm over 12 instances of C1, R1 and RC1 classes of Solomon’s 100-customer instances. The average optimum is 1,082.47.

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α = 0.4 α = 0.7 α = 0.9

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Fig. 3. Influence of the smooth parameter α on the algorithm performance (Solomon’s 100customer R102 instance, ρ = 0.05, λ = 0.2, θ = 0.02, k1 = 2.0, k 2 = 1.2 N=7000, optimum is 1,466.6) 2400

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Fig. 4. Influence of the elite sample size on the algorithm performance (Solomon’s 100customer R102 instance, N=7,000, α = 0.7, λ = 0.2, θ = 0.02, k1 = 2.0, k 2 = 1.2 )

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2400

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Fig. 5. Influence of the size of local search on the algorithm performance (Solomon’s 100customer R102 instance, N=7,000, ρ = 0.05, θ = 0.02, k1 = 2.0, k 2 = 1.2 )

5 Conclusions In this paper, a cross entropy learning algorithm is proposed for solving vehicle routing problems with time windows. The advantage of the proposed approach is that the optimal routing probability is iterative learned based on importance sampling and rare event simulation theory. By combing the local search procedure and introducing the dynamic parameter adjustment procedure, the proposed method can avoid being trapped in the local optimum and find quickly near-optimal solutions. The numerical tests show also that the hybridization of the local search technique and the CE method can improve efficiently the solution quality. As the proposed method is based on an adaptive learning procedure, it provides a general method for solving vehicle routing problems under stochastic environment. Moreover, the proposed method needs only very small number of parameters settings. Its implementation is also relatively simple and flexible for various vehicle routing problems. Currently, the numerical study based on some Solomon's 100-customer VRPTW instances is implemented. Future extension concerns the application of multiagent framework on general dynamic vehicle routing problems solving. It is also interesting to apply an appropriate data structure and more efficient implementation techniques for large scale instance (200 to 1,000 customers). Moreover, other efficient local search techniques can be implemented to improve local search performance. Acknowledgements. This research has benefited from a grant of ANR (the French Agency for Research) project, Mutualisation et Optimisation de la Distribution Urbaine de Marchandises (MODUM).

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References 1. Cordeau, J.F., Desaulniers, G., Desrosiers, J., Solomon, M.M., Soumis, F.: The VRP with time windows. In: Toth, P., Vigo, D. (eds.) The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications, pp. 157–194 (2002) 2. Braysy, O., Gendreau, M.: Vehicle routing problem with time windows. Part I: Route Construction and Local Search Algorithms. Transportation Science 39, 104–118 (2005) 3. Braysy, O., Gendreau, M.: Vehicle routing problem with time windows. Part II: Metaheuristics. Transportation Science 39, 119–139 (2005) 4. Golden, B., Raghavan, S., Wasil, E. (eds.): The vehicle routing problem, latest advances and new challenges. Operations Research/Computer Science Interfaces Series, vol. 43. Springer, Berlin (2008) 5. Ibaraki, T., Imahori, S., Kubo, M., Masuda, T., Uno, T., Yagiura, M.: Effective local search algorithms for routing and scheduling problems with general time window constraints. Transportation Science 39(2), 206–232 (2005) 6. Hashimoto, H., Ibaraki, T., Imahori, S., Yagiura, M.: The vehicle routing problem with exible time windows and traveling times. Discrete Applied Mathematics 154, 2271–2290 (2006) 7. Pisinger, D., Ropke, S.: A general heuristic for vehicle routing problems. Computers & Operations Research 34, 2403–2435 (2007) 8. Braysy, O., Dullaert, W., Gendreau, M.: Evolutionary algorithms for the vehicle routing problem with time windows. Journal of Heuristics 10, 587–611 (2004) 9. Homberger, J., Gehring, H.: A two-phase hybrid metaheuristic for the vehicle routing problem with time windows. European Journal of Operational Research 162, 220–238 (2005) 10. Martin, O., Otto, S.W., Felten, E.W.: Large-step Markov chains for the TSP incorporating local search heuristic. Operation Research Letters 11, 219–224 (1992) 11. Hashimoto, H., Yagiura, M., Ibaraki, T.: An iterated local search algorithm for the timedependent vehicle routing problem with time windows. Discrete Optimization 5, 434–456 (2008) 12. Helsgaun, K.: An effective implementation of the Lin-Kernighan traveling salesman heuristic. Datalogiske skrifter, Writings on Computer Science, no. 81. Roskilde University (1999) 13. Braysy, O., Hasle, G., Dullaert, W.: A multi-start local search algorithm for the vehicle routing problem with time windows. European Journal of Operational Research 159, 586–605 (2004) 14. Rubinstein, R.Y.: The Cross-Entropy Method for Combinatorial and Continuous Optimization. Methodology and Computing in Applied Probability 2, 127–190 (1999) 15. Irnich, S., Funke, B., Grunert, T.: Sequential search and its application to vehicle-routing problems. Computers & Operations Research 33, 2405–2429 (2006) 16. Barbucha, D., Jedrzejowicz, P.: Multi-agent platform for solving the dynamic vehicle routing problem. In: Proc.of 11th Int. IEEE Conf. on Intelligent Transportation Systems, pp. 517–522 (2008) 17. Vokrinek, J., Komenda, A., Pechoucek, M.: Agents Towards Vehicle Routing Problems. In: Proc. of 9th Int. Conf. on Autonomous Agents and Multiagent Systems, pp. 773–780 (2010) 18. Davidson, P., Henesey, L., Ramstedt, L., Tornquist, J., Wernstedt, F.: An analysis of agent-based approaches to transport logistics. Trans. Res. Part C 13, 255–271 (2005)

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19. De Boer, P.T., Kroese, D.P., Mannor, S., Rubinstein, R.Y.: A Tutorial on the CrossEntropy Method. Annals of Operations Research 134(1), 19–67 (2005) 20. Solomon, M.M.: Algorithms for the vehicle routing and scheduling problems with time window constraints. Operations Research 35, 254–265 (1987) 21. Rubinstein, R.Y., Kroese, D.K.: Simulation and the Monte Carlo Method. Wiley Series in Probability and Statistics (2008) 22. Croes, G.: A method for solving traveling-salesman problems. Operations Research 6, 791–812 (1958) 23. Jepsen, M., Petersen, B., Spoorendonk, S., Pisinger, D.: Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows. Operations Research 56(2), 497–511 (2008)

Modelling the Synchronization of Transport Means in Logistics Service Operations Dorota Slawa Mankowska, Christian Bierwirth, and Frank Meisel School of Economics and Business, Martin-Luther University Halle-Wittenberg, Germany, {dorota.mankowska,christian.bierwirth,frank.meisel}@wiwi.uni-halle.de

Abstract. Synchronization of vehicle operations plays an important role for logistics service providers. Examples of logistics businesses with synchronization requirements can be found, e. g., in swap trailer transportation and truck-meets-truck traﬃc. In this paper, we provide a classiﬁcation of the various types of synchronization requirements for transport means in logistics applications. For supporting the planning of logistics service operations, we formulate a new mixed-integer programming model for vehicle routing, where vehicle routes are synchronized at ﬁxed and given points. Computational results are provided for assessing the solvability of the model under various problem settings.

1

Introduction

Research on vehicle routing problems (VRPs) in service networks is a very active ﬁeld of research due to the variety of practical applications, see [1]. An important extension to the class of vehicle routing problems is the coupling of routes of two or more vehicles, as is required by companies who oﬀer mobile services with synchronized vehicle operations. So far, vehicle synchronization is hardly considered in the literature, see [2,3,4]. The contribution of this paper is the formulation and testing of a new model for vehicle routing with synchronization constraints. The vehicle routing problem with synchronization constraints is to ﬁnd vehicle routes for serving a set of customers, where some nodes must be visited by more than one vehicle. In our paper, we consider the case where some of the customers require to be served by two vehicles. This problem occurs, e. g., if a customer must be served by operators with diﬀerent skills. For each customer there may also exist a time window, in which the service must take place. The objective is to minimize the total distance traveled by all vehicles. The problem is a combination of diﬀerent vehicle routing problems like the VRPTW (see e. g. [5]), where each customer must be served within a time windows and the HVRP (see, e. g., [6]), where a heterogeneous vehicle ﬂeet is considered. Some further vehicle routing problems are related to the problem considered here. For example, in the truck and trailer problem, truck and trailer operations are planned in such a way that the trucks have to relocate the trailers on their J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 74–85, 2011. c Springer-Verlag Berlin Heidelberg 2011

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routes, see, e. g., [7]. However, the trailer is a non-autonomous transport mean that is immobile without a truck. Hence, there is no such thing like planning individual routes for both of the transport means and, thus, synchronization of the routes is not part of the problem. In the vehicle routing problem with crossdocking, there is a precedence relation such that vehicles that pick up goods from customers must return to the cross-dock before vehicles can start from the cross-dock for distributing the goods to other customers, see, e. g., [8]. In this problem, vehicles meet at just one point (the cross-dock) but, diﬀerent to the vehicle routing problem with synchronization constraints, services at customer locations are not synchronized. Finally, synchronization of vehicles in a dynamic context is considered by [9], where focus is put on how to delay already planned services, when new customer requests become known. The outline of the paper is as follows. In Section 2 we explain the diﬀerent types of synchronization that may be required between mobile servers. In Section 3 we propose a mixed-integer programming model for the routing of two vehicles with synchronization at a set of ﬁxed and given points. We provide numerical results for randomly generated test instances in Section 4. The paper is concluded in Section 5.

2

Classification of Synchronization Types

Synchronization of vehicles means to couple the routes of two or more vehicles. Depending on the ﬁeld of application, vehicle routes must be synchronized in time and/or space leading to the four synchronization types shown in Table 1. Table 1. Synchronization types Temporal synchronization Spatial synchronization

ﬁxed points variable points

simultaneous

with precedence

Type I

Type II

Type III

Type IV

The spatial dimension of synchronization deﬁnes the locations where vehicle synchronization can take place. In Types I and II, the vehicles have to meet at ﬁxed and given synchronization points, whereas in Types III and IV, synchronization points are an outcome of the planning. The temporal dimension deﬁnes the order in which vehicles must visit a synchronization point. In Types I and III, synchronization points must be served simultaneously by vehicles, whereas in Types II and IV, a precedence relation must be respected for the vehicles. Synchronization of Type I, where vehicles must meet at their routes at ﬁxed points, occurs when load must be transshipped among vehicles at certain preset

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Fig. 1. Fixed point synchronization of Type I (left) and Type II (right)

locations. In synchronization of Type II, the vehicles must visit the same predeﬁned customer set, but merely a precedence relation is speciﬁed for the arrival of the vehicles. An example is swap trailer transportation, where one vehicle drops an empty trailer at a customer and, at a later point in time, another vehicle picks up the loaded trailer. In Type III, the vehicles have to meet simultaneously at synchronization points that are selected in the planning. This type occurs, for example, if vehicles have to meet for exchanging load, where the meeting location can be chosen ﬂexibly within the routes. This mode of operation is also called truck-meets-truck traﬃc, see [10]. If vehicles can load and unload their cargo themselves, the only requirement for load exchange is that the vehicle that provides cargo arrives at a synchronization point before the vehicle that receives this cargo. Such a case is covered by the synchronization of Type IV (precedence at variable synchronization points). Figure 1 illustrates the two types for ﬁxed point synchronization at the example of two vehicles. The space-time diagrams show the positions of the vehicles in the course of time. At the begin of the service process, both vehicles are located in the depot. Figure 1 (left) shows a ﬁxed point synchronization with simultaneous service (Type I). A single synchronization point is preset at customer 3. For enabling its simultaneous service, vehicle 1 must wait until vehicle 2 arrives at this customer. In contrast, Figure 1 (right) shows the case of less restrictive synchronization, where a precedence relation exists for the vehicle arrivals at customer 3. Here, vehicle 1 must arrive at customer 3 before vehicle 2, but it can leave without waiting for vehicle 2. As a result, vehicle 1 returns to the depot earlier. A further issue in synchronizing vehicles arises if only a subset of customer locations can be accessed by a vehicle. Such a limitation is observed for example in urban regions, where some customers may have deﬁcient infrastructure for being accessed by large vehicles. In such a situation, the set of variable synchronization points for the spatial synchronization in Types III and IV is restricted to those locations that can be accessed by all vehicles. The possible cases for the synchronization of two vehicles are shown in Figure 2. In this ﬁgure, C1 (respectively C2 ) refers to the set of customers that can be visited by vehicle 1 (respectively vehicle 2). Then, C1 ∩ C2 is the set of possible synchronization points. Following situations are possible:

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(i) The sets of serviceable customers for both vehicles are equal (C1 = C2 ). The selection of synchronization points is not restricted. (ii) The set of serviceable customers for one vehicle is a strict subset of the set of serviceable customers for the other vehicle (C1 ⊂ C2 ∨ C2 ⊂ C1 ). Synchronization can take place only at customers from C1 (or from C2 ). (iii) There exist customers, which have to be served by vehicle 1 and there exist customers, which have to be served by vehicle 2. Here, each vehicle has its own set of customers to be served and synchronization takes place at the intersection of C1 and C2 .

Fig. 2. Possible relations between sets of serviceable customers for two vehicles

3

Mixed Integer Programming Model for Two Vehicles

In the following, we propose a model for synchronizing two vehicles under synchronization of Type I and Type II. Given are a set of customers C and a ﬂeet V = {v1 , v2 } consisting of two heterogeneous vehicles. The vehicles are initially located at a depot, which refers to node 0 in the network. C¯ = C ∪ {0} is the set of all nodes in the network. The distance between any pair of nodes i, j ∈ C¯ is given by dij . We assume that the network is complete, and that the traveling time from node i to node j corresponds to the distance dij . Service duration piv denotes the time needed to serve customer i ∈ C by vehicle v ∈ V . Here, the heterogeneity of the ﬂeet is expressed by the diﬀerent service times to represent the various equipment of vehicles or the individual skills of the service operators. For each customer i, a time window [ei , li ] is given, in which the service must be started. The accessibility of customers by vehicles is modeled using a binary matrix [aiv ], where aiv = 1, if vehicle v ∈ V can visit customer i ∈ C, 0 otherwise. Spatial synchronization points are deﬁned by S ⊆ C, i. e., as a subset of customers who need service of both vehicles. Clearly, for each synchronization point i ∈ S it must hold that aiv1 = aiv2 = 1. For the temporal synchronization, a min max minimal time gap δiv and a maximal time gap δiv between the service start 1 v2 1 v2 min max = δiv = times of vehicle v1 and vehicle v2 at customer i ∈ S are given. If δiv 1 v2 1 v2 0, the vehicles have to serve the customers simultaneously (synchronization of min Type I). If δiv > 0, vehicle v1 must start the service at customer i at least 1 v2 min max ≥ δiv1 v2 time units before v2 (synchronization of Type II). If, additionally, δiv 1 v2 min max δiv1 v2 > 0 is set, vehicle v2 must not start the service later than δiv1 v2 time units

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after the service start time of v1 . A precedence constraint where vehicle 2 can start the service of customer i only after vehicle 1 ﬁnished the service can be min max expressed by setting δiv = piv1 and δiv = ∞. Precedence constraints for a 1 v2 1 v2 reverse vehicle order (service of v2 before service of v1 ), are modeled similarly. The routing of the vehicles is modeled by binary decision variables xijv , which ¯ 0 otherwise. take value 1 if vehicle v ∈ V moves directly from i ∈ C¯ to j ∈ C, The temporal variable tiv ≥ 0 denotes the service start time of vehicle v ∈ V ¯ The vehicle routing problem with synchronization constraints is at node i ∈ C. modeled as follows. The objective (1) is to minimize the total distance traveled by vehicles. dij · xijv (1) min → Z = ¯ j∈C ¯ v∈V i∈C

Constraints (2) guarantee that the route of each vehicle starts in the depot and ends in the depot. x0jv = xj0v = 1 ∀v ∈ V (2) j∈C

j∈C

Constraints (3) ensure that every customer who does not ask for a combined service of both vehicles is visited by only one of the two vehicles. xijv = 1 ∀i ∈ C\S (3) ¯ v∈V j∈C

Constraints (4) represent ﬂow balancing constraints, ensuring that vehicle v leaves customer i if it visits the customer. xjiv = xikv ∀i ∈ C, v ∈ V (4) ¯ j∈C

¯ k∈C

Constraints (5) determine the service start times of vehicles at customers, taking into account the service durations piv and the travel times dij . Here, M refers to a suﬃciently large positive value, which is set to li for deriving a tight model formulation. The constraints furthermore avoid cycles in the vehicle routes. ¯ j ∈ C, ¯ v∈V tiv + (piv + dij )xijv ≤ tjv + M (1 − xijv ) ∀i ∈ C,

(5)

Constraints (6) let the service of customers start within their individual time windows. xijv ≤ tiv ≤ li xijv ∀i ∈ C, ∀v ∈ V (6) ei ¯ j∈C

¯ j∈C

Synchronization of services for customers in set S is modeled in Constraints (7)(9). Constraints (7) model the spatial synchronization by ensuring that each synchronization point is visited by every vehicle v ∈ V . xijv = 1 ∀i ∈ S, ∀v ∈ V (7) ¯ j∈C

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The temporal synchronization of vehicles is modeled in (8) and (9). The minimal time gap between the arrivals of the two vehicles is guaranteed by (8). min tiv1 + δiv ≤ tiv2 1 v2

∀i ∈ S

(8)

The maximal time gap between the vehicle arrivals is guaranteed by (9). max tiv1 + δiv ≥ tiv2 1 v2

∀i ∈ S

(9)

Constraints (10) deﬁne the domains of the binary decision variables xijv . Here, aiv · ajv = 1 if and only if vehicle v can access both nodes i and j, 0 otherwise. This additional domain restriction ensures feasibility of routes, by avoiding trips to customers which are inaccessible for vehicle v. xijv ∈ {0, aiv · ajv }

¯ ∀v ∈ V ∀i, j ∈ C,

(10)

The domains of service start times are deﬁned in (11). tiv ≥ 0

¯ ∀v ∈ V ∀i ∈ C,

Fig. 3. Synchronization of Type I (left) and Type II (right)

(11)

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Example: Figure 3 (left) shows the solution to the example instance if simulmin max taneous synchronizations is required at the customers, i. e. δiv = δiv =0 1 v2 1 v2 is preset ∀i ∈ S. The total route length of this solution is Z = 279.4. Figure 3 (right) shows the solution where precedence relations are given for synchronization points, i. e., vehicle 1 must arrive at a synchronization point before vehicle 2 but it does not have to wait for the second vehicle. This solution has a total route length of Z = 274.2, which is a saving of 1.9 % compared to the solution in Figure 3 (left). This example illustrates that less restrictive synchronization with Type II can lead to solutions of lower cost than synchronization of Type I. To get further insight into the chronology of the vehicle operations, Figure 4 shows space-time diagrams of the two solutions generated for the example instance. In Figure 4 (top), it can be seen that the service of customers 2, 4, and 6 is performed simultaneously by the two vehicles as is required under synchronization of Type I. In contrast, synchronization with precedence (Type II) allows for a non-simultaneous service as is observed for customers 2, 4, and 6 in Figure 4 (bottom). Here, waiting times of vehicles occur only due to customer time windows but not for synchronizing the vehicles. The space-time diagrams reveal that the two solutions diﬀer signiﬁcantly in the vehicle routes and in the service start times at the customers. This example shows the sensitivity of the

Fig. 4. Space-time diagrams for synchronization with simultaneous service (top) and precedence requirements (bottom)

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described problem, in which even small changes of the synchronization requirements have a large impact on the solution structure and the objective function value. It illustrates the importance of carefully incorporating the particular type of synchronization found in a speciﬁc practical application into the planning of vehicle service operations.

4

Numerical Experiments

We conduct computational experiments for assessing the solvability of the synchronization model by means of a standard MIP solver under varied problem size, synchronization requirements, and customer time windows. For the tests, we generate three sets of instances, namely A, B, and C. Set A contains ten small sized instances with ten nodes each, i. e., one depot and nine customers. Set B contains ten medium sized instances with 15 nodes each, i. e., one depot and 14 customers. Set C contains ten large sized instances with 20 nodes each, i. e., one depot and 19 customers. The nodes are randomly located in an area 50 × 50. The distances between nodes as well as the travel times are Euclidean. Two vehicles are available for serving customers. We assume that each vehicle can visit each customer, i. e., aiv = 1 ∀i ∈ C, v ∈ V . The service durations piv for customers i ∈ C by vehicle v ∈ V are drawn from the interval [1, 2, . . . , 7]. Customer time windows are generated as explained in the respective tests. Synchronization requirements are as follows. For every small, medium, and large instance three, four, and ﬁve synchronization points are randomly selected min max = δiv = 0 to enforce from the set of customers, respectively. We set δiv 1 v2 1 v2 synchronization of Type I. For testing synchronization of Type II, this setting is later modiﬁed. We use ILOG CPLEX 12.2 with a runtime limit of one hour per instance on a 2.80 GHz Intel Core Duo. Table 2. Numerical results for set A with varied customer time windows Time windows S

M

Z

1 2 3 4 5 6 7 8 9 10

290.8 303.2 242.3 302.1 324.6 330.3 279.7 243.2 294.0 288.0

0 0 0 0 0 0 0 0 0 0

00:01 00:01 00:01 00:02 00:01 00:02 00:01 00:04 00:01 00:02

266.9 289.6 214.6 270.7 298.9 279.4 272.9 243.2 271.1 247.7

0 0 0 0 0 0 0 0 0 0

00:03 00:06 00:01 00:18 00:04 00:04 00:01 00:04 00:04 00:05

261.8 282.6 214.6 238.8 281.7 266.4 272.9 223.7 263.5 247.7

0 0 0 0 0 0 0 0 0 0

00:08 00:05 00:01 00:03 00:07 00:03 00:01 00:02 00:56 00:32

261.8 282.6 214.6 238.8 281.7 226.4 272.9 223.7 263.5 247.7

0 0 0 0 0 0 0 0 0 0

00:38 00:16 00:01 00:04 00:38 00:06 00:03 00:04 00:22 00:07

average 289.8

0

00:02 265.5

0

00:05 255.4

0

00:12 251.4

0

00:14

gap CPU

Z

N

Instance

gap CPU

Z

L gap CPU

Z

gap CPU

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For the ﬁrst test, we solve instance set A under varied time window lengths for the customers. More precisely, we solve the instances under small (S) time windows (length = 70 time units), medium (M) time windows (140 time units), large (L) time windows (200 time units), and no time windows (N). The computational results are shown in Table 2. The table presents for each instance and each time window setting the objective function value Z of the best solution found within the runtime limit, the gap [in %] of the best solution against the lower bound delivered by CPLEX, and the computation time CPU [mm:ss] required for solving instances. The 0-gaps in this table reveal that all instances are solved to optimality independently of a particular time window setting. Actually, all instances are solved within less than one minute of computation time. Still, we observe a growth in the average computation time when turning from small to large and to no time windows. Contrasting runtimes, the average objective values decrease when time windows are enlarged. This ﬁnding is expected as more ﬂexible time windows open up potential for shortening the total route lengths of vehicles. In the second test, we solve instance set A under a varied number of synchronization points. Here, two, three, four, and ﬁve synchronization points are selected randomly for each instance. Time windows are set to medium size (M) as described for the ﬁrst test. The computational results for each instance and each number of synchronization points are shown in Table 3. Again, all instances are solved to optimality independent of the particular number of synchronization points. The average computation time varies in the number of synchronization points, but a clear trend cannot be identiﬁed. The average objective function values increase signiﬁcantly when turning from two to more synchronization points. For |S| = 5, average route lengths are about 21.4 % larger than for |S| = 2. This ﬁnding is explained by a additional travel eﬀort that is required for the vehicles Table 3. Numerical results for set A with varied number of synchronization points Number of synchronization points |S| 2

3

Z

1 2 3 4 5 6 7 8 9 10

257.7 258.2 220.4 242.4 281.5 237.0 257.0 207.2 243.9 245.2

0 0 0 0 0 0 0 0 0 0

00:06 00:04 00:02 00:04 00:10 00:06 00:02 00:02 00:07 00:08

266.9 289.6 214.6 270.7 298.9 279.4 272.9 243.2 271.1 247.7

0 0 0 0 0 0 0 0 0 0

00:03 00:06 00:01 00:18 00:04 00:04 00:01 00:04 00:04 00:05

280.4 291.5 254.7 253.6 324.9 301.5 285.7 240.9 279.3 259.5

0 0 0 0 0 0 0 0 0 0

00:07 00:04 00:03 00:05 00:10 00:03 00:01 00:01 00:58 00:03

291.0 353.4 256.3 277.6 339.0 311.2 299.4 254.6 285.8 305.5

0 0 0 0 0 0 0 0 0 0

00:03 00:06 00:02 00:07 00:20 00:03 00:01 00:01 00:11 00:01

average 245.0

0

00:05 265.5

0

00:05 277.2

0

00:10 297.4

0

00:05

gap CPU

Z

5

Instance

gap CPU

Z

4 gap CPU

Z

gap CPU

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83

when synchronized service is requested by a larger number of customers. For determining the amount of cost that follows from synchronization requirements, we furthermore solved all ten instances without any synchronization points (S = ∅). In this case the problem reduces to the VRPTW. We observe from this computation, that average cost without synchronization are 167.8 (detailed results are omitted here). Hence, we observe that already a few synchronization requirements (|S| = 2) increase cost by about 46 %. We conclude that synchronization has a strong impact on solution quality which emphasises the need to consider them in the vehicle operations planning. Next, we solve instance set A under varied minimal and maximal time gaps to analyze the sensitivity of solutions under diﬀerent synchronization requirements. For this purpose, we solve the instances under simultaneous synchromin max min = δiv = 0, and under precedence requirements with δiv = nization δiv 1 v2 1 v2 1 v2 max min max min max 0, δiv1 v2 = 50, with δiv1 v2 = 0, δiv1 v2 = 100, and with δiv1 v2 = 50, δiv1 v2 = 100. The computational results are shown in Table 4. All instances are solved to optimality again. The objective function values are aﬀected diﬀerently from the varied synchronization requirements. Taking the solutions with simultanemin max = δiv = 0) as a reference, the less restrictive case with ous service (δiv 1 v2 1 v2 min max δiv1 v2 = 0, δiv1 v2 = 50 enables improved solutions for six instances. The improvements range from a relative route length saving of 0.2 % (instance 1) to min max = 0, δiv = 100, 4.5 % (instance 6). For the even less restrictive case with δiv 1 v2 1 v2 four further improvements are observed, with a maximal improvement of 4.1 % min max = 50, δiv = 100, synchronization requirements (instance 9). In case δiv 1 v2 1 v2 min max impact solution quality diﬀerently compared with δiv = δiv = 0. Three 1 v2 1 v2 instances show a larger objective function value, three instances show the same solution quality, and four instances have shorter total route lengths. Improvements are explained by the ﬂexibility that results from the non-synchronous Table 4. Numerical results for set A with varied minimal and maximal time gaps min max Time gaps (δiv , δiv ) 1 v2 1 v2

(0,0)

(0,50)

Z

1 2 3 4 5 6 7 8 9 10

266.9 289.6 214.6 270.7 298.9 279.4 272.9 243.2 271.1 247.7

0 0 0 0 0 0 0 0 0 0

00:03 00:06 00:01 00:18 00:04 00:04 00:01 00:04 00:04 00:05

266.4 286.8 214.6 266.9 298.9 266.7 272.9 235.6 269.0 247.7

0 0 0 0 0 0 0 0 0 0

00:05 00:08 00:01 00:08 00:05 00:03 00:01 00:05 00:05 00:30

266.4 282.6 214.6 263.7 298.9 266.7 272.9 233.1 258.1 247.7

0 0 0 0 0 0 0 0 0 0

00:04 00:05 00:01 00:08 00:06 00:05 00:01 00:04 00:04 00.38

305.1 282.6 214.6 268.1 311.4 301.2 272.9 241.4 263.5 247.7

0 0 0 0 0 0 0 0 0 0

00:03 00:04 00:01 00:04 00:01 00:05 00:01 00:02 00:01 00:06

average 265.5

0

00:05 262.6

0

00:07 260.5

0

00:08 270.8

0

00:03

gap CPU

Z

(50,100)

Instance

gap CPU

Z

(0,100) gap CPU

Z

gap CPU

84

D.S. Mankowska, C. Bierwirth, and F. Meisel Table 5. Comparison of small, medium, and large instances (sets A, B and C) set A

set B

set C

Instance

Z

gap

CPU

Z

gap

CPU

Z

gap

CPU

1 2 3 4 5 6 7 8 9 10

261.8 282.6 214.6 238.8 281.7 226.4 272.9 223.7 263.5 247.7

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

00:54 00:20 00:02 00:08 00:39 00:33 00:05 00:07 02:44 00:44

344.4 317.1 283.1 232.8 303.4 300.4 304.9 234.9 315.1 282.4

0.0 0.0 0.0 0.0 9.1 0.0 0.0 0.0 0.0 0.0

11:26 14:47 04:27 01:22 60:00 04:02 01:05 00:19 14:02 22:58

308.6 335.5 318.3 311.1 317.9 313.9 324.4 282.7 324.6 328.8

3.7 4.8 5.0 0.8 13.5 0.0 3.0 0.0 12.1 8.3

60:00 60:00 60:00 60:00 20:05 60:00 60:00 08:12 60:00 60:00

average

251.4

0.0

00:14

291.9

0.9

13:27

316.6

5.1

50:50

min arrival requirements. Longer route lengths are explained by the non-zero δiv 1 v2 parameters which enforce a temporal distance between the vehicle arrivals. In a ﬁnal test series, we evaluate the impact of the instance size on the solvability of the problem. We solve all three instance sets A, B, and C for the case with no customer time windows. The computational results are shown in Table 5. It can be seen that already adding ﬁve customers to the problem (instance set B) leads to much longer runtimes than for set A. The average runtime of CPLEX grows to 13 minutes per instance. When turning to set C, only two instances are solved to optimality. CPLEX ﬁnds integer feasible solutions also for the remaining instances, but gaps of up to 13.5 % are observed. Although a deterioration of solution quality is expected for CPLEX, it is surprising how quickly the solver reaches its limits. The non-optimal solution quality for instances with 20 nodes together with the very long computation times are considered inadequate for a practical application. These results obviously call for the development of more powerful solution methods for routing vehicles under synchronization requirements.

5

Concluding Remarks

We have proposed a model for vehicle routing with synchronization constraints. The considered types of synchronization are simultaneous synchronization and precedence synchronization at ﬁxed points. From computational tests, it has been shown that increasing synchronization requirements lead to a strong increase in vehicle route lengths. This is because the travel eﬀort increases, if more customers require service by both vehicles. From a managerial perspective, these results emphasize the need of carefully planning vehicle operations, if synchronized services play an important role in a practical application. Further computational tests show that the model can be solved to optimality by

Modelling the Synchronization of Transport Means

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standard solvers when instances with up to 15 nodes are considered. Already for larger instances with 20 nodes, the ILOG solver runs into its boundaries and produces solutions with gaps up to 13.5 % within one hour of computation time. This result indicates that problem-speciﬁc solution methods and heuristics must be developed for solving larger instances of the vehicle routing problem with synchronization constraints. Since the presented model considers synchronization of only two vehicles, a further important issue of future research is to formulate a model that can cope with vehicle ﬂeets of larger size. Other issues of future research are to extend the model towards synchronization at variable points, to incorporate vehicle capacities, and to address alternative objectives like the minimization of the maximum route duration over all vehicles. Acknowledgments. This research is funded by the Deutsche Forschungsgemeinschaft (DFG) under project reference B02110263.

References 1. Golden, B., Raghavan, S., Wasil, E.A. (eds.): The Vehicle Routing Problem: Latest Advances and New Challenges. Operations Research, Computer Science Interfaces Series. Springer, New York (2008) 2. Drexl, M.: Synchronization in Vehicle Routing–A Survey of VRPs with Multiple Synchronization Constraints. Technical Report LM-2011-02, Chair of Logistics Management, Gutenberg School of Management and Economics, Johannes Gutenberg University, Mainz (2011) 3. Bredstr¨ om, D., R¨ onnqvist, M.: Combined vehicle routing and scheduling with temporal precedence and synchronization constraints. European Journal of Operational Research 191, 19–31 (2008) 4. Del Pia, A., Filippi, C.: A variable neighborhood descent algorithm for a real waste collection problem with mobile depots. International Transactions in Operational Research 13, 125–141 (2006) 5. Cordeau, J.-F., Desaulniers, G., Desrosiers, J., Solomon, M.M., Soumis, F.: VRP with Time Windows. In: Toth, P., Vigo, T. (eds.) The Vehicle Routing Problem, pp. 57–188. Society for Industrial and Applied Mathematics, Philadelphia (2002) 6. Choi, E., Tcha, D.-W.: A Column Generation Approach to the Heterogeneous Fleet Vehicle Routing Problem. Computers and Operations Research 34, 2080–2095 (2007) 7. Scheurer, S.: A tabu search heuristic for the truck and trailer routing problem. Computers & Operations Research 33, 894–909 (2006) 8. Wen, M., Larsen, J., Clause, J., Cordeau, J.-F., Laporte, G.: Vehicle routing with cross-docking. Journal of the Operational Research Society 60, 1708–1718 (2009) 9. Roesseau, L.-M., Gendreau, M., Pesant, G.: The Synchronized Vehicle Dispatching Problem. Technical Report CRT-2003-11, Conference paper, Odysseus 2003, Centre de Recherche sur les Transports, Universit´e de Montr´eal, Canada (2003) 10. Weise, T., Podlich, A., Reinhard, K., Gorldt, C., Geihs, K.: Evolutionary Freight Transportation Planning. In: Giacobini, M., Brabazon, A., Cagnoni, S., Di Caro, G.A., Ek´ art, A., Esparcia-Alc´ azar, A.I., Farooq, M., Fink, A., Machado, P., et al. (eds.) EvoWorkshops 2009. LNCS, vol. 5484, pp. 768–777. Springer, Heidelberg (2009)

Optimization of Infectious Medical Waste Collection Using RFID Pamela C. Nolz, Nabil Absi, and Dominique Feillet Ecole des Mines de Saint-Etienne, CMP Georges Charpak, F-13541 Gardanne, France {nolz,absi,feillet}@emse.fr

Abstract. In this paper we consider the collection of infectious medical waste, produced by patients in self-treatment and stored at pharmacies. The problem is formulated as a collector-managed inventory routing problem, encompassing stochastic aspects, which are common in such problems. Social objectives, speciﬁcally the satisfaction of pharmacists and the local authority, as well as the minimization of public health risks, are considered for the real-world motivated inventory routing problem. To optimize the planning process for a predeﬁned time horizon, we take advantage of radio frequency identiﬁcation technologies. We design a tabu search based algorithm to optimize the determination of visit dates and corresponding vehicle routes. The suggested approach is tested on real-world data from the region of Provence-Alpes-Cˆ ote d’Azur, in France. The results for diﬀerent waste collection scenarios are analyzed and compared in order to evaluate the performance of the proposed solution method.

1

Introduction

The quantity of waste material disposed of by home health and medical care services, as well as by self-attending patients is increasing [20]. Therefore, the treatment of infectious waste, such as contaminated needles or syringes, is of great importance. Infectious waste, especially sharp objects, need to be handled with great care in order to prevent the spread of diseases. The unsafe disposal of health-care waste poses public health risks, considering that health workers, waste handlers or pedestrians might come into contact with contaminated waste if it is dumped into areas without restricted access. In the south-east region of France, the disposal of medical waste accumulated by patients in self-treatment, is commonly organized as follows. At pharmacies, patients receive empty boxes which are dedicated to the safe storage of their medical waste. Full boxes are brought back by the patients to the pharmacies, where they are stored. When a certain storage capacity at a pharmacy is reached, a pharmacist makes a call to the local authority, which organizes the pick-up of the waste material. Full boxes are then collected by a vehicle and deposited at collection centers in order to be disposed of. An external company periodically visits and depletes collection centers. Currently, a pharmacist makes a call J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 86–100, 2011. c Springer-Verlag Berlin Heidelberg 2011

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to the local authority when the maximal storage capacity (a container for full boxes) at the pharmacy is reached. The pharmacy is visited one or two days after giving a call to the local authority. Consequently, the number of boxes at a pharmacy is only known by the collector when a call is received, and the planning of the collection tours is made on a daily basis. The current system has three main inconveniences: First, transportation costs are high because tours are only optimized day per day. Second, the service at pharmacies is bad, not only for pharmacists but also for patients. If a container is full, waiting to be emptied by the collector, patients cannot enter their boxes and have to disturb the pharmacist, causing an unpleasant situation for both of them. Third, the collector cannot plan his/her tours in advance. In this paper, we envisage the improvement of the planning process by integrating Radio Frequency Identiﬁcation (RFID) technology. Being equipped with RFID tags, boxes can automatically be registered when entered into a container at a pharmacy. This allows the collector to control the stock at pharmacies, because the exact number of boxes can be accessed at any time. Pharmacies can be visited according to a schedule determined by the local authority for the whole time horizon, depending on the number of boxes and the distances between the pharmacies. The number of boxes arriving at a pharmacy is stochastic and is therefore not known exactly a priori. However, by considering the number of boxes stored at the pharmacies at the beginning of the time horizon, the waste collection tours can be planned for a predeﬁned period. Two conﬂicting aspects have to be considered when determining visit dates for pharmacies. On the one hand, pharmacies should not be visited too late, as their storage capacity will be exceeded. On the other hand, pharmacies should not be visited too early (when only few boxes are stored), as each visit interrupts the working process of the pharmacists. Possessing information on when collections of waste material will be carried out during the whole planning period, pharmacists can adjust their time schedule and better anticipate their daily activities. The ﬂexibility in determining the visit dates of pharmacies and the corresponding tours, makes the simultaneous optimization of collection and routing possible. However, the whole process becomes less reactive, as the planning is based on information available at the beginning of the time horizon. In order to represent both interest groups, the local authority as well as pharmacists, the total cost is composed of ﬁxed cost when a tour is carried out, transportation cost and inventory cost at the pharmacies. This study is part of a project called ‘Trace-de-TIC’, funded by the French Environment and Energy Management Agency (ADEME). The agency ﬁnances and develops research and technological innovation with the aim of protecting the environment and turning the concept of sustainable development into a concrete reality. Our project contains several innovative and new aspects. First of all, the problem formulation combines characteristics of inventory routing and stochasticity in an applied research study. Secondly, we consider social objectives for the collection of infectious medical waste. To be more speciﬁc, we address four

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performance indicators, namely minimization of public health risks, satisfaction of pharmacists, satisfaction of the local authority and total distance of collection tours. In order to ameliorate the time planning for both, the pharmacists and the local authority, we provide a mathematical formulation of the waste collection problem described above, addressed as a stochastic Inventory Routing Problem (IRP), by exploiting the beneﬁts of RFID technology. The IRP deals with the distribution (or collection) of a product to several facilities over a given planning horizon [3]. The proposed solution approach can be applied in two diﬀerent ways in order to handle the stochastic IRP. On the one hand it can be embedded in a rolling horizon framework, where the same problem is solved every day based on the newly available data. In this way, transportation costs can be improved. On the other hand, the planning can be done once for the whole time horizon, considering the initial inventory. This allows the pharmacists and the collector to plan and coordinate their activities for the whole time horizon. The remainder of the paper is organized as follows. In the next section, a literature review is presented. In Section 3, the problem is described in detail. In Section 4, the solution procedure is introduced and explained. Computational experiments are presented in Section 5. Some concluding remarks and perspectives are provided in Section 6.

2

Literature Review

The infectious medical waste collection problem under investigation contains characteristics of inventory routing and stochasticity. It is furthermore embedded in an applied research project. This is why we partition the literature review into three sections: Inventory routing problem, Real-world waste collection problems, and Stochastic vehicle routing problem. 2.1

Inventory Routing Problem

Bertazzi et al. [3] present an overview of IRPs, regarding diﬀerent formulations and several extensions. Federgruen and Simchi-Levi [10] provide a mathematical formulation of the single-depot single-period IRP. Campbell et al. [4] study the IRP as well as the stochastic IRP, where the future demand of a customer is uncertain. Stockout cost at a customer is modeled with a penalty function that contains ﬁxed and variable components. Campbell and Savelsbergh [6] present a two-phase approach for an IRP. In the ﬁrst phase a delivery schedule is determined by solving an integer programming model and in the second phase a set of delivery routes is constructed by applying heuristics. Yu et al. [28] study an IRP with split deliveries, allowing the delivery to each customer in each period to be performed by multiple vehicles. Le Blanc et al. [17] present a collector managed IRP for reverse logistics, where the collection company is responsible for the inventory to be picked up and recycled. Solyali et al. [25] present a robust IRP with uncertain demands. The authors determine the delivery quantities and

Optimization of Infectious Medical Waste Collection Using RFID

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delivery times, as well as the vehicle routes, and they allow backlogging of demand at customers. They propose mixed integer programming formulations of the problem and its deterministic variant. 2.2

Real-World Waste Collection Problems

Baptista et al. [2] present a Periodic Vehicle Routing Problem (PVRP) case study considering the collection of recycling paper containers in Portugal. The number of visits to containers within a period is included in the problem formulation as a decision variable. Nuortio et al. [22] address a stochastic PVRP with time windows for the collection of municipal solid waste in Eastern Finland, which is solved with a metaheuristic. Kulcar [15] presents a case study on waste collection in Brussels, evaluating several means of collection, such as vehicle, rail and canal. Mourao and Almeida [21] examine a capacitated arc routing problem for a refuse collection VRP. They present lower bounding techniques and a heuristic in order to solve a real-world problem for the collection of household waste in Lisbon. Teixeira et al. [26] provide a case study on the collection of recyclable urban waste. Periodic vehicle routes are planned for the collection of three diﬀerent waste types, regarding a relatively long time horizon. Tung and Pinnoi [27] study a vehicle routing and scheduling problem for waste collection in Hannoi, Vietnam. The problem is divided into several phases, where waste has to be picked up and delivered at diﬀerent stages, while time windows have to be respected. Campbell et al. [5] present a case study for a real-world IRP. They develop a two-phase solution approach. In the ﬁrst phase they determine the customers to be served on a speciﬁc day and the amount to be delivered, and in the second phase they determine the delivery routes and schedules. Shih and Chang [24] develop a computer system for the collection of infectious hospital waste. Their solution approach for the considered PVRP consists of two phases. Firstly, the authors develop vehicle routes for all customers without considering allowable day combinations. In the second phase, they apply a mixed integer programming method to assign the routes to days while respecting all constraints. 2.3

Stochastic Vehicle Routing Problem

Gendreau et al. [11] provide a review paper on stochastic VRPs, referring to stochastic customers, stochastic travel times and stochastic demands. Most of the papers which can be considered relevant for our waste collection problem focus on the solution of a stochastic program with recourse. In the ﬁrst stage, an a priori solution is determined, where the realizations of the random variables are not known. In the second stage a recourse or corrective action can be taken, if for example the vehicle capacity is exceeded. The aim is to determine a solution of minimum expected total cost. The same authors develop a tabu search heuristic called Tabustoch [12] for the VRP with stochastic demands and customers. Laporte and Louveaux [16] present a two-index formulation of the SVRP and explain the integer L-shaped Method for continuous stochastic programming,

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which is based on branch and cut. Hjorring and Holt [14] use the L-shaped method to solve a single-VRP. A relaxed IP is solved and additional constraints (optimality cuts) are added for a tighter approximation of route failure cost. Rei et al. [23] propose a hybrid heuristic combining Monte Carlo sampling and local branching to solve a single VRP with stochastic demands. Mendoza et al. [19] propose a memetic algorithm for the solution of a multi-compartment VRP with stochastic demands. The authors apply genetic operators in combination with local search procedures in order to solve the stochastic program with recourse. Ak and Erera [1] present a paired-vehicle recourse strategy, considering the coordination of vehicles in pairs. A tabu search heuristic is applied to realworld instances from Istanbul, Turkey. Christiansen and Lysgaard [7] introduce a branch-and-price algorithm for the capacitated VRP with stochastic demands. The authors formulate a set partitioning problem and develop a dynamic programming approach for the solution of the column generation subproblem.

3

Problem Description

We formulate the problem as a Stochastic Collector Managed Inventory Routing Problem (SCMIRP). Two decisions have to be made: (1) when to visit each pharmacy, (2) how to route the vehicle to minimize costs. One vehicle is located at the depot, where it starts and ends each of its tours, which is at most one tour per day. We consider a planning period of several weeks, where the inventory level at the end of the time horizon can be diﬀerent from the starting level. The total cost is composed of transportation cost, ﬁxed vehicle cost associated with a tour, and inventory cost. Inventory cost represents two components, determined by the diﬀerence between a predeﬁned level (number of boxes) at a pharmacy and the number of boxes actually picked up by the vehicle. Unitary excess inventory cost incurs when a predeﬁned maximum storage capacity (a container for full boxes) at a pharmacy is reached. A penalty cost at a pharmacy incurs, if the number of boxes at the pharmacy is lower than a predeﬁned threshold inventory level. Note that for a pharmacist a visit is perturbing as long as a minimum storage level has not been reached, as he/she has to stop the working process in order to hand out only few boxes. The problem can then be presented as a two-stage stochastic optimization problem with recourse. In the ﬁrst stage, decisions have to be made on the visit dates for each pharmacy and the tours by which the vehicle visits the pharmacies. The actual number of boxes at a pharmacy becomes known, when the pharmacy is visited. Therefore, the number of boxes to be collected at each pharmacy is determined in the second stage. Note that for our model, it makes no diﬀerence whether the number of boxes at the pharmacies becomes known at once (e.g. assessed via RFID technology), or gradually, going from one pharmacy to the next. This particularity is explained by the fact that in the second stage the whole inventory at each pharmacy is collected. Let G be a complete graph where V = {0, 1, ..., n} is the vertex set and A is the arc set. One vehicle is located at the depot, represented by vertex 0. A

Optimization of Infectious Medical Waste Collection Using RFID

91

distance or travel time matrix satisfying the triangle inequality is deﬁned on A. Vertices i = 1, ..., n represent the pharmacies, where boxes for infectious medical waste are collected. cij denotes the distance or travel time between vertices i and denotes j. cf is the ﬁxed vehicle cost, that incurs by carrying out a tour. cthrd i the penalty cost incurring by a visit to pharmacy i, if the threshold inventory level Iithrd at pharmacy i is not reached. cmax is the unitary excess inventory cost i incurring by a visit at pharmacy i, if the maximal accepted inventory level Iimax at pharmacy i is reached. t ∈ 1, ..., T expresses the set of discrete time instants. Ii0 is the initial inventory at pharmacy i. ξit denotes the number of boxes deposited at pharmacy i by unit of time as a random variable. The variable Iit denotes the inventory at pharmacy i at time t. vit is the number of boxes exceeding the maximal accepted inventory at time t. zit is the number of boxes picked up at pharmacy i at time t. wit denotes the binary variable which is equal to 1 if pharmacy i is visited at time t, and 0 otherwise. xtij is equal to 1 if pharmacy j is visited immediately after pharmacy i at time t, and 0 otherwise. The binary variable yit is equal to 1 if Iithrd is not reached when visiting pharmacy i at time t, and 0 otherwise. The variables xtij and wit determine the ﬁrst-stage decision, while the variables zit determine the second-stage decision. Using this notation, the following stochastic inventory routing problem can be formulated. First stage: min

n n T

cij xtij +

i=0 j=0 t=1

n T

cf xt0j + E(Q(x, w, ξ))

s.t.

(1)

j=1 t=1 n

xt0j ≤ 1 ∀t = 1, ..., T

(2)

j=1 n

xtip =

i=0, i=p n

n

xtpj

∀p = 0, ..., n; t = 1, ..., T

(3)

j=0, p=j

xtij = wjt

∀j = 0, ..., n; t = 1, ..., T

(4)

xtij = wit

∀i = 0, ..., n; t = 1, ..., T

(5)

i=0 n j=0 T

wit ≥ 1

∀i = 0, ..., n

(6)

t=1

xtij ≤ |S| − 1

∀S ⊆ {1, ..., n} ; t = 1, ..., T

(7)

i,j∈S

xtij ∈ {0, 1} ∀i = 0, ..., n; j = 0, ..., n; t = 1, ..., T

(8)

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P.C. Nolz, N. Absi, and D. Feillet

wit ∈ {0, 1} ∀i = 1, ..., n; t = 1, ..., T

(9)

Second stage: Q(x, w, ξ) = min

n T

cmax vit + i

i=1 t=1

Iit = Ii0 +

t

n T i=1 t=1

ξis −

s=1

t−1

zis

cthrd yit + i

n i=1

cmax IiT +1 i

∀i = 1, ..., n; t = 1, ..., T

s.t.

(10)

(11)

s=1

IiT +1 ≥ IiT − ziT − Iimax

∀i = 1, ..., n

(12)

zit ≤ Iit + M (1 − wit )

∀i = 1, ..., n; t = 1, ..., T

(13)

zit ≥ Iit − M (1 − wit )

∀i = 1, ..., n; t = 1, ..., T

(14)

zit ≤ M (wit ) ∀i = 1, ..., n; t = 1, ..., T vit ≥ zit − Iimax zit ≥ Iithrd (wit − yit )

∀i = 1, ..., n; t = 1, ..., T ∀i = 1, ..., n; t = 1, ..., T

(15) (16) (17)

Iit ≥ 0 ∀i = 1, ..., n; t = 1, ..., T + 1

(18)

zit ≥ 0 ∀i = 1, ..., n; t = 1, ..., T

(19)

vit ≥ 0

(20)

∀i = 1, ..., n; t = 1, ..., T

yit ∈ {0, 1} ∀i = 1, ..., n; t = 1, ..., T

(21)

The objective function of the first-stage problem is given by (1). It expresses the minimization of the total cost, which comprises the transportation cost, the ﬁxed cost and the expected inventory cost. The expectation is taken with respect to the distribution of the vector ξ of random variables ξit . The actual inventory cost Q(x, w, ξ) results as the solution of the second-stage problem (10) - (21). In the terminology of stochastic programming, Q(x, w, ξ) is called the recourse function. It expresses the expected cost of the decision to be made after the unknown parameter values become known. Constraints (2) state that at most one tour can be carried out per discrete time instant. Constraints (3) ensure that if the vehicle visits a pharmacy at a discrete time instant, it also leaves the pharmacy at the same time instant.

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Constraints (4) and (5) guarantee that a pharmacy is only visited when an arc leads to that pharmacy. Constraints (6) guarantee that each pharmacy is visited at least once during the whole time horizon. Constraints (7) are the subtour elimination constraints. Constraints (8) and (9) deﬁne the variables of the firststage problem. In the second-stage problem, equation (10) expresses the objective function as the sum of unitary excess inventory cost and penalty cost. Constraints (11) and (12) deﬁne the level of inventory of pharmacy i at time instant t. Constraints (13) and (14) ensure that the whole inventory is collected, when pharmacy i is visited. Constraints (15) force the number of collected boxes to be zero, if a pharmacy is not visited. Constraints (16) determine the number of boxes that exceed the maximal accepted inventory at pharmacy i. Constraints (17) deﬁne yit as equal to 1 if the number of boxes picked up at pharmacy i at time t is lower than the threshold inventory, and 0 otherwise. Constraints (18) to (21) deﬁne the variables of the second-stage problem.

4

Solution Approach

Due to the complexity of the problem, an exact solution method cannot be envisaged for realistic problem instances. Therefore, we propose a heuristic method in order to solve the SCMIRP for infectious medical waste. The solution approach is based on a tabu search (TS) algorithm [13] and a collection pattern based operator is developed to handle stochasticity. An initial solution of our inventory routing problem for infectious medical waste collection is generated by randomly assigning collection days to pharmacies. For each day, a Traveling Salesman Problem (TSP) is solved with the Lin-Kernighan heuristic [18]. The TSP consists of ﬁnding the shortest route, starting from a given city, visiting each of a speciﬁed group of cities, and then returning to the original point of departure [8]. The Lin-Kernighan heuristic is a generalization of 2-opt and 3-opt, swapping pairs of sub-tours in order to create a new tour. The neighborhood for our TS is inspired by periodic TSPs and is based on collection patterns, a collection pattern being a set of visit dates for a pharmacy. The neighbor of a solution is obtained by exchanging the collection pattern for a pharmacy i and determining the corresponding vehicle routes. The routing costs are re-optimized with the Lin-Kernighan heuristic when evaluating a pattern. The best collection pattern is determined for each pharmacy depending on the current solution. The best solution out of the neighborhood replaces the current solution by changing the collection pattern for a pharmacy i, leading to the best improvement, and by updating the vehicle routes. Figure 1 illustrates the calculation of the best neighbor, which is performed by solving a shortest path problem of the following structure. Each node of the graph represents a discrete time instant. Additionally, a ﬁctitious start node and a ﬁctitious end node complete the graph. Visiting a node means that the corresponding pharmacy i is visited at time instant (day) t. An arc leads from each node to all of its successors.

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Fig. 1. Shortest path problem for pharmacy i

The distance (cost) for the shortest path problem is composed of the routing cost diﬀerence Δt , which incurs when the corresponding node in the graph is visited, and the inventory cost dtt i , associated with the length of a traversed arc. For each discrete time instant, a TSP is solved with the Lin-Kernighan heuristic, assuming that pharmacy i is included in the tour as well as assuming that pharmacy i is not included. The diﬀerence Δt between the two TSP solution costs is associated with the node representing the time instant t. The inventory cost dtt i represents the expected penalty and excess inventory costs. In the described waste collection problem, the number of boxes disposed at a pharmacy cannot be known exactly a priori. Therefore, the number of boxes disposed at pharmacy i by unit of time is expressed by the random variable ξit . as the estimated cost of being We calculate the estimated inventory cost dtt i penalized either because the threshold inventory level is not reached or because the maximal accepted inventory level is exceeded. Given a pattern with the corresponding visit dates for pharmacy i, the estimated inventory costs can be calculated as follows.

dtt i

=

cmax i

∞ I=Iimax +1

P

(Iitt

= I) × (I −

Iimax )

+

cthrd i

Iithrd

P (Iitt = I) (22)

I=0

Iitt

=

t

ξir

(23)

r=t+1

Equation (22) represents the inventory cost, which is composed of the unitary excess inventory cost and the penalty cost at pharmacy i. In order to take account of the stochastic data, we calculate the probability of reaching a certain inventory

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level when pharmacy i is visited. Equation (23) determines the inventory at pharmacy i at time t , if a collection is done at time t. The inventory cost dst i , starting before the ﬁrst period of time, additionally takes account of the initial inventory Ii0 :

Iist

=

Ii0

+

t

ξir

(24)

r=t+1

For the cost dte i , leading to the ﬁctitious end node, only unitary excess inventory cost is considered, but not penalty cost. Note that dte i is the cost of leaving boxes exceeding the maximal accepted inventory at pharmacy i at the end of the time horizon. The shortest path is found by applying the algorithm of Dijkstra [9]. If the best pattern found, is the same as the current pattern, we determine the second shortest path in the graph. Note that the best pattern found remains the same, if for the current solution the pattern of every pharmacy is optimal. The second shortest path in the graph is found by forbidding once each arc contained in the shortest path, starting from the ﬁctitious end node. The determination of the second shortest path allows the algorithm to escape from a local optimum. The pharmacy for which the collection pattern is changed, is temporarily declared tabu. A solution containing tabu characteristics is accepted as incumbent solution, if it is better than the best known solution. We use a ﬁxed-length tabu list and a ﬁxed tabu duration.

5 5.1

Computational Experiments Test Instances

We tested the proposed solution approach on a real-world instance from the region of Provence-Alpes-Cˆ ote d’Azur. In this region, 30 pharmacies participate in the project ﬁnanced by ADEME, with the aim of improving the collection of infectious medical waste. Information on the number of boxes arriving at each pharmacy and the speciﬁcation of the threshold inventory level and the maximal accepted inventory level is gathered from the participating pharmacies. Each of the pharmacies is equipped with the required RFID technology. For the investigation, we assume that 1 vehicle is located at the base, where each collection tour of infectious medical waste starts and ends. The planning is made a priori for a time period of 22 days, each discrete time instant representing 1 day. 5.2

Numerical Results

We assume that full boxes of infectious medical waste arrive independently at pharmacies according to a Poisson distribution with mean λti . The estimated inventory costs for pharmacy i are then calculated as follows from equations (22) and (23).

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λ=

t

λri

(25)

r=t+1 dtt i

=

cmax i

∞

e

Iitt =Iimax +1

= cmax (λ−Iimax +(Iimax −λ) i

thrd

Ii tt Iitt λIi tt max thrd −λ λ (I − I ) + c e i i Iitt ! i Iitt ! tt

−λ

(26)

Ii =0

Iimax −1

Iitt =0

thrd

tt

e−λ

λIi

Iitt !

+Iimax e−λ

Ii max Iitt λIi thrd −λ λ e )+ c i Iimax ! Iitt ! tt Ii

=0

The evaluation of our solution approach is performed by comparing the current scenario to the RFID scenario for ﬁve diﬀerent problem settings. The current scenario is a realization of the data in real time. It is assumed, that each pharmacy makes a phone call to the local authority when reaching 85 % of the maximal accepted inventory. Boxes at a pharmacy are collected the day after having called the local authority. Table 1 contains the results of the comparison between the current scenario and the RFID scenario. It presents total cost, inventory cost, routing cost, number of routes, number of visits when the number of boxes is smaller than the threshold inventory level, number of visits when the number of boxes is higher than the maximal accepted inventory level, and the average number of boxes exceeding the maximal accepted inventory level. Considering the basic setting, it can easily be observed, that the total costs for the current scenario are twice as high as the costs for the RFID scenario. The advantage of the RFID scenario is that the collector can ﬂexibly combine pharmacies into tours in order to decrease total cost. This is performed by a trade-oﬀ between routing cost and inventory cost. For example, it might be preferable to visit a pharmacy which is close-by, even if the threshold inventory level has not been reached. Note that penalty cost incurs when a pharmacy is visited too early (with respect to the threshold inventory) and unitary excess inventory cost incurs when a pharmacy is visited too late (with respect to the maximal accepted inventory). In the current scenario, the local authority is not able to prevent visits above the excess inventory level, because a visit is only carried out when a pharmacist makes a call. For this reason, inventory cost is very high. The number of routes carried out in the current scenario is twice the number of routes in the RFID scenario. This is due to the fact that in the current scenario a tour is carried out whenever the local authority receives a call, even if only one pharmacy has to be visited. As the routing of the vehicle can only be optimized on a daily basis, routing cost is elevated as well. The same holds for the variable arrival rate setting, where the number of boxes arriving each day at the pharmacies varies over the time horizon.

total cost current scenario 479 227 RFID scenario 205 663 Variable arrival rate total cost current scenario 496 806 RFID scenario 224 085 High arrival rate total cost current scenario 1 153 590 RFID scenario 458 758 Low inventory cost total cost current scenario 354 127 RFID scenario 159 686 High inventory cost total cost current scenario 1 035 230 RFID scenario 287 198

Basic setting

inventory cost routing cost # routes # threshold visits # excess visits # excess boxes 695 000 340 227 20 0 73 2 30 000 257 198 12 3 3 1

inventory cost routing cost # routes # threshold visits # excess visits # excess boxes 13 900 340 227 20 0 73 1.9 14 400 145 286 8 10 30 4.2

inventory cost routing cost # routes # threshold visits # excess visits # excess boxes 794 000 359 588 20 0 155 5.1 83 000 375 758 19 16 32 2.1

inventory cost routing cost # routes # threshold visits # excess visits # excess boxes 155 000 341 806 20 0 72 2.2 31 000 193 085 10 9 19 1.2

inventory cost routing cost # routes # threshold visits # excess visits # excess boxes 139 000 340 227 20 0 73 1.9 11 000 194 663 10 2 7 1.2

Table 1. Current scenario vs. RFID scenario

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High arrival rate means that twice the number of boxes considered in the basic setting arrive at the pharmacies. The average number of boxes exceeding the maximal accepted inventory level is very high for the current scenario, given that a pharmacy is only visited after a call is made at an inventory level of about 85 % of the maximal accepted inventory. By carrying out almost twice the number of tours as in the basic setting, in the RFID scenario the high arrival rate of boxes can be handled. For low inventory cost, excess inventory cost and penalty cost used in the basic setting, are divided by a factor of 10. For high inventory cost these values are multiplied by a factor of 5. Consequently, in the RFID scenario, the number of visits underneath the threshold inventory level or above the maximal accepted inventory level is high when inventory cost is low. When inventory cost is high, on the contrary, visits underneath the threshold inventory level or above the maximal accepted inventory level are avoided. Fewer tours are carried out in the low inventory cost setting, and the number of excess boxes is rather high, as this does not heavily inﬂuence total cost. In the high inventory cost setting, on the contrary, more routes are carried out and the number of visits below the threshold inventory level and above the maximal accepted inventory level are kept small in order to achieve low total cost for the RFID scenario.

6

Conclusion and Perspectives

We presented a solution approach to optimize the determination of visit dates and the corresponding vehicle routes for infectious medical waste collection. We consider social objectives, such as the satisfaction of pharmacists and local authority and the minimization of public health risks. Experimental results show, that by integrating an RFID technology, the planning process can be ameliorated for both, local authority and pharmacists. The RFID scenario allows a decrease in total cost by 50 % compared to the current scenario, as inventory and routing costs can be kept low when optimizing visit dates a priori. The present solution approach is to be embedded into a decision support system. Together with the decision makers (pharmacists, local authority) the weight for the inventory costs and the lower and upper inventory levels can be determined. These values can be chosen independently for each pharmacy, in order to reﬂect the inconvenience caused by a visit or excess of inventory. Future research concentrates on the development of a sampling method, where a ﬁxed sample of random scenarios are optimized as realizations of the stochastic problem. This allows a better evaluation of the quality of solutions provided by our tabu search algorithm. Acknowledgments. The present research work was partly ﬁnanced by the French Environment and Energy Management Agency (ADEME) by grant #10 02C 0003. The authors gratefully acknowledge the support of this institution.

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The Pickup and Delivery Problem with Cross-Docking Opportunity Hanne L. Petersen and Stefan Ropke Technical University of Denmark, Department of Transport Bygningstorvet 115, 2800 Kgs. Lyngby {hlp,sr}@transport.dtu.dk

Abstract. In this paper, we consider the pickup and delivery problem with cross-docking opportunity (PDPCD). The problem arises from an industry application, and includes pickup requests, delivery requests, and pickup-and-delivery requests. Each pickup-and-delivery request can be served either as direct delivery by one truck, or by being picked up and transported to the cross-dock by one vehicle, and subsequently delivered at its ﬁnal destination by another vehicle. Handling times at customers sites and terminal are given. A typical daily instance includes 500–1,000 requests. We solve the problem using a Large Neighborhood Search (LNS) approach.

1

Introduction

The problem to be considered in the following is a pickup and delivery problem, where a number of requests must be served. Each request has given time windows, and the aim is to construct a set of vehicle routes, such that all requests are served using the available ﬂeet of vehicles, in the cheapest possible way. The cost of a given solution depends on driven distance, total time spent, required handling operations, etc. In the present application, the PDPCD, a cross-dock is available to help carry out the required operations. A cross-dock allows for transfer of goods between vehicles, and means that one request can be served by two vehicles in combination, with one vehicle performing the pickup and the transportation of the goods to the cross-dock, and the other vehicle transporting the goods from the cross-dock to its ﬁnal destination. A major advantage of using a cross-docking setup is that it allows for consolidation of goods, and thereby more eﬃcient use of vehicles. The application behind this paper originates with Alex Andersen Ølund A/S (AAØ), a major Danish transporter of ﬂowers, with activities reaching most of Northern and Mid-Europe. Their national distribution of ﬂowers, which is being considered in this paper covers some 500–1,000 daily requests, to be served by a ﬂeet of 170 trucks. The distribution includes pickup of ﬂowers at gardeners’ greenhouses, and delivery at ﬂorists and supermarkets. The greenhouses are typically located in the western part of Denmark, while the deliveries are all over the country, with J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 101–113, 2011. c Springer-Verlag Berlin Heidelberg 2011

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a higher concentration in the more densely populated areas in the east. The locations of the pickup and delivery points of AAØ in Denmark can be seen from Figure 1. This opens opportunities for use of cross-docking, where the ﬂowers are brought from the greenhouses to the cross-dock, and repacking takes place before the ﬂowers are sent on to their ﬁnal destinations.

Fig. 1. Distribution of customer locations in Denmark

The problem we study, denoted the the pickup and delivery problem with crossdocking opportunity (PDPCD), can be described as follows: We are given a set of transportation requests R = {1, . . . , n}. Each request i ∈ R involves an amount (di ) of small ﬂower containers to be transported from an origin to a destination. The origin and destination belonging to request i are numbered i and n + i, respectively. The set of all locations considered is denoted L = {0, 1, . . . , 2n} where location 0 is the cross-dock. A homogeneous ﬂeet K = {1, . . . , k} of vehicles is available for serving the requests. Each vehicle can contain up to 43 ﬂower containers. All vehicle routes start and end at the cross-dock and each route can visit the cross-dock several times to load and unload. Each location i ∈ L is associated with a time window [ai , bi ] within which the location must be visited. We note that requests often share locations (i.e., a site requiring deliveries from several companies) in this case the physical location will be associated with as many locations in L as there are requests visiting the physical location. Each request can either be transported from origin to destination using a single vehicle or it can be picked up by one vehicle, transported to the cross-dock, transferred

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to a second vehicle and delivered to the destination by that vehicle. We note that some requests either start or end their journey at the cross dock. The time needed for unloading (loading) at the cross-dock is dependent on the amount unloaded (loaded) and is described by the staircase function shown in Figure 2. Loading and unloading times are cumulative so the vehicle can spend up to 100 minutes at the cross-dock to do a complete reload. It is assumed that the goods unloaded from a vehicle are available for loading into a new vehicle when all goods that require unloading have been removed from the ﬁrst vehicle. Similarly, a vehicle cannot start loading until all goods are ready. Service times at pickup and delivery points are also determined using the staircase function from Figure2.

Service time (minutes)

50 40 30 20 10 0

0

5

10 15 20 25 30 35 40 45 50 Number of containers

Fig. 2. Service time as function of number of containers loaded/unloaded

Capacity limitations at the cross-dock are not considered: there are no restrictions on the amount of goods stored temporarily at the cross-dock or or on the number of vehicles that can be operating at the cross-dock simultaneously. This simpliﬁcation is justiﬁed by the fact that the cross-dock at AAØ is well dimensioned and that the company currently do not experience that the cross-dock forms a bottleneck in the operations. The objective of the problem is to minimize transportation cost. The costs are split into ﬁve elements: 1) a daily cost for using each vehicle, 2) a cost per minute for using each vehicle, 3) a cost per km driven, 4) a cost per container handled at the cross-dock, 5) a road toll for crossing the bridge between Funen and Zealand (the two major islands on Figure 1). When calculating travel time and distances road network distances are used. As there is an hourly rate for using each vehicle, it is in our interest to limit the duration of each route. This means that the departure time from the cross-dock should be scheduled instead of simply leaving when the cross-dock opens.

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5

1 2

6

1 2

3

4

4 7

5

6

7

3

Fig. 3. A sample routing plan using a cross-dock (the center square). The numbers identify individual requests – a circle denotes pickup and a rhombus denotes delivery. The dotted lines indicate morning trips, dashed lines afternoon trips, and full line evening trip. The symbols inside the cross-dock indicate that two orders are partial, and require pickup or delivery at the cross-dock.

Figure 3 shows an example of how vehicle routes could be constructed using a cross-dock. In Figure 3 the routes can be covered by 2 vehicles, and the diﬀerent types of lines indicate the order in which the routes must be operated. A crossdock contains one or more areas that are laid out for short-term storage of goods, from the time when it is oﬀ-loaded from an incoming vehicle, and until the reload and departure of the delivery vehicle, but in general it cannot be regarded as a storage opportunity, and as a rule of thumb, the cross-dock should be empty at the end of the working day. This underlines the function of the cross-dock as a tool that can be used in transportation planning, but not a long-term storage facility. Finally, the real-life application leading to this study is in the business of transporting fresh plants and ﬂowers, which have limited shelf life, further encouraging short storage times. The use of cross-docks in distribution leads to several interesting planning problems, not just from a vehicle routing perspective, but also regarding the layout and operations of the cross-dock facility itself. This includes assignment of trucks to doors, assignment of the storage areas, handling of goods inside the cross-dock, location of cross-docks, and many other problems. Some examples of derived problems can be found in [4]. However, in this paper we will focus exclusively on the associated routing problems. The literature on routing problems with cross-docking (or transfer) opportunity is scarce. The papers that study a problem most closely related to ours are [10] and [7]. [10] study a problem with multiple cross-docks (called transshipment points in the paper) and propose a heuristic for solving the problem. [7] study the

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problem with a single cross-dock, develop a mathematical model and a branchand-cut algorithm for solving the problem. Instances with up to six requests and two vehicles are solved to optimality. [8] study a dial-a-ride problem (pickup and delivery problem focused on passenger transportation, usually including service level constraints, see e.g., [6]) in which passengers can be dropped oﬀ and picked up at certain transfer points. The passenger is transported between the transfer points by means of a ﬁxed route service (e.g., a bus or tramway). A mathematical model is developed and the authors propose several ideas for strengthening the modeling. The solution approach is tested on an instance with 4 requests and 3 transfer points. The instance is solved to optimality in roughly 4,000 seconds. [9] and [21] study the PDPCD where all requests have to be cross-docked. Both propose mathematical models and metaheuristics for solving the problem. The literature on pickup and delivery problems is quite rich, it has been surveyed recently in [3,5,13]. The basic pickup and delivery problem most closely related to ours is the pickup and delivery problem with time windows. Recent successful heuristics for the problem are described in [18,11,12]. The most powerful exact methods for the problem are based on the set partitioning model [17,2]. The exact methods have solved some instances with up to 500 requests to optimality but unsolved instances with 75 requests also exits.

2

Solution Method

We solve the PDPCD using the metaheuristic LNS proposed in [19]. In an LNS metaheuristic the neighborhood is deﬁned implicitly by a destroy and a repair method. A destroy method destructs part of the current solution while a repair method rebuilds the destroyed solution. The destroy method typically contains an element of stochasticity such that diﬀerent parts of the solution are destroyed in every invocation of the method. The neighborhood of a solution is then deﬁned as the set of solutions that can be reached by ﬁrst applying the destroy method and then the repair method. The destroy method will typically destroy a relatively large part of the solution. This together with the many ways of repairing the solution mean that the neighborhood of a single solution contains a large number of solutions. A concept closely related to LNS is that of Very Large Scale Neighborhood Search (VLSN) deﬁned in [1]. While the LNS is a heuristic framework, VLSN is the family of heuristics that search neighborhoods whose sizes grow exponentially as a function of the problem size, or neighborhoods that simply are too large to be searched explicitly in practice, according to [1]. The LNS is one example of a VLSN heuristic. We use a variant of LNS called adaptive LNS (ALNS) proposed in [18]. In an ALNS several destroy and repair methods are deﬁned and a method for choosing between the diﬀerent destroy and repair methods based on their performance is employed.

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Pseudo-code for the ALNS is shown in Algorithm 1. Three variables are maintained by the algorithm. The variable xb is the best solution observed during the search, x is the current solution and xt is a temporary solution that can be discarded or promoted to the status of current solution. In line 2 the global best solution is initialized. In line 4 a destroy and a repair method is chosen based on performance, this is described in detail in [18]. In line 5 the heuristic ﬁrst applies the chosen destroy heuristic and then the chosen repair heuristic to obtain a new solution xt . More speciﬁcally, d(x) returns a copy of x that is partly destroyed. Applying r(·) to the partly destroyed solution repairs it, that is, it returns a feasible solution built from the destroyed one. In line 6 the new solution is evaluated, and the heuristic determines whether this solution should become the new current solution (line 7) or whether it should be rejected. Line 9 checks whether the new solution is better than the best known solution, here c(x) denotes the objective value of solution x. The best solution is updated in line 10 if necessary. In line 12 the termination condition is checked and in line 13 the best solution found is returned. We will describe the accept and stop criteria used in the current implementation in more detail in Section 2.1. Algorithm 1. Adaptive large neighborhood search 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

2.1

input: a feasible solution x xb = x; repeat select a destroy method d and a repair method r; xt = r(d(x)); if accept(xt , x) then x = xt ; end if if c(xt ) < c(xb ) then xb = xt ; end if until stop criterion is met return xb

Acceptance and Stop Criteria

In our implementation we use a simulated annealing criterion in the accept function in line 6 of Algorithm 1. The temporary solution xt is always accepted if t c(xt ) ≤ c(x), and accepted with probability e−(c(x )−c(x))/T if c(x) < c(xt ). Here T > 0 is the current temperature. The temperature is initialized at T0 > 0 and is gradually decreased during the search. The idea is to allow deteriorating solutions to be accepted with high probability in the beginning of the search and only accept a few or no deteriorating solutions towards the end. We use either (a) elapsed time or (b) iteration count as stopping criterion. The choice of stopping criterion impacts the temperature updating mechanism. If criterion (b)

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is used the T is updated by the formula T = αT in each iteration. α is selected such that the temperature reach Tend in the ﬁnal iteration. If criterion (a) is used then we set T = αη T0 where η is the elapsed time and α is selected such that αηend T0 = Tend, where ηend is the target time. The start and end temperatures are chosen using the method described in [14]. 2.2

Parallel Large Neighborhood Search

We use a parallel version of the ALNS by employing the software framework described in [16]. This allows us to take full advantage of current computers that often contain four or more computation units (cores). The idea of the parallelization is that the current (x) and best (xb ) solutions are shared among all processing units while lines 4 to 11 of Algorithm 1 are performed in parallel, with each processing unit working on a local version of the temporary solution (xt ). When choosing to accept or reject xt in line 6 the local xt is compared to the shared x and we update the shared x in line 7 if necessary. Similarly, the shared global best solution can be updated in lines 9–11. 2.3

Construction Heuristic

The initial solution is constructed by ﬁrst making the cross-dock-or-not decision for each request, and then constructing the routes for the resulting set of pseudorequests. A pseudo-request can consist of either a pickup operation, a delivery operation, or a pair of corresponding pickup and delivery operations. If it is decided at the initial stage that a request must be cross-docked, the request will be assigned a transfer limit time, and split into two pseudo-requests. These two pseudo-requests will then be treated separately during the route construction stage. Initially the cross-docking decision, whether to cross-dock or not, is made for each pickup-and-delivery request. If the time windows for a given request, in combination with necessary minimum driving and handling times, make crossdocking infeasible for the request, it is served as direct delivery. If the time gap between the pickup and delivery time windows is larger than some parameter, an+i − bi > λt , the request is cross-docked. The intuition behind this criterion is that if there is a long time (e.g., 8 hours) between earliest delivery and latest pickup time then it is unlikely that the request should be served by a single route, because that route would have a very long duration. If neither of these two conditions apply, we consider the direct distance between the pickup and delivery locations, and the distance when using the cross-dock. The request will be cross-docked if the ratio between these two distances is above some given d direct threshold, which is a parameter, dist distCD > λ . The routes can now be constructed from the resulting set of pseudo-requests by a greedy insertion. During this, we ﬁrst insert all pseudo-requests which contain two operations, and then inserting the remaining pseudo-requests, in order to insert the more complex pseudo-requests ﬁrst.

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Destroy Operators

The algorithm applies two diﬀerent destroy operators: one which removes visits randomly from the solution, and one which removes entire trips. The RandomDestroy operator takes two parameters, gmin and gmax , indicating the lower and upper bounds on the number of visits to remove at each iteration. At each invocation of the operator the number of visits to remove is determined as a random number in the given interval. The TripDestroy operator takes one parameter, hmin , which is the minimum number of requests to remove at each iteration. It will randomly select a trip to remove, and remove all requests contained in this trip from the solution. This will be repeated until the desired minimum number of requests has been reached. The tests reported in this paper used values gmin = 5, gmax = 30, hmin = 10. 2.5

Repair Operator

The algorithm applies a single repair operator, which consecutively inserts the removed requests by using a regret measure (see e.g., [15]), thereby giving the heuristic an opportunity to detect operations that could become expensive to insert at a later stage. Letting Δfi1 denote the cost of inserting operation i in the cheapest possible way, and Δfi2 the cost of inserting it in the second-cheapest way, the regret cost of inserting the operation is Δfi2 − Δfi1 , and all requests can then be inserted in order of decreasing regret cost. Whenever a request is considered for insertion, it is considered both for direct delivery and cross-docked handling. 2.6

Further Algorithm Features

In this section we propose two features of our heuristic that distinguish it from a straightforward ALNS implementation. Pre-sort at insertion. In order to reduce the calculation time spent on unfavorable insertion operations, the insertion procedure of our algorithm uses a pre-sort function to reduce the number of routes to check for feasible insertion. For each route the simple cost of increased distance of insertion is calculated, and the routes are ranked according to this cost, and only the ρ routes with lowest insertion cost are checked, with ρ as a parameter (ρ = 10 was used in the computational experiments). More speciﬁcally, we consider the ρ best routes, starting with the best feasible route. Thus if any routes at the top of this ranking are infeasible for insertion, they do not count towards the ρ routes, whereas any infeasible route after the ﬁrst feasible is counted. The feature has a positive impact on the quality of the solution that can be reached within a ﬁxed run-time. We note that the pre-sorting has some similarities to the granular tabu search proposed by [20].

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Return to best solution. In order to intensify the search, and allow each thread to beneﬁt from the eﬀort of the others, we allow each thread, at any iteration, to replace its current solution, with the globally best solution seen so far, with a given probability p, which is a parameter. We provide computational evidence for the impact of this parameter in Section 3.3.

3

Computational Results

The algorithm has been tested on data that have been provided by the company AAØ. The algorithm was implemented in C++ and the tests were carried out on a computer equipped with two Intel Xeon E6430 2.66GHz processors for a total of 8 computation cores. All 8 cores were utilized in the tests reported below. 3.1

Industry Instances

In this section, we present the results obtained on ﬁve instances, reﬂecting the operation on ﬁve diﬀerent days, in the spring of 2009, as presented in Table 1. The instances have been solved with a running time of 90 minutes, and the reported results are averages over 10 runs. Table 1. Results obtained on real-life instances (averages over 10 runs) Instance aaoe20 aaoe21 aaoe22 aaoe23 aaoe24

n

part. req.

loc.

init. sol.

avg. sol.

best sol.

avg. gap

avg. iter.

cd-%

726 784 982 883 585

190 194 251 192 179

444 549 601 628 404

1,102,420 1,142,510 1,336,710 1,209,620 998,277

509,163 646,806 784,713 652,980 585,771

495,233 630,317 777,371 631,840 573,821

2.8% 2.6% 0.9% 3.3% 2.1%

40,825 26,598 20,494 18,572 43,444

66% 74% 77% 75% 64%

The table shows the number of requests in each instance, the number of partial requests (requests which have either their pickup or delivery location at the crossdock), and the total number of unique locations. Then it lists the value of the initial solution, the average solution value over 10 computations, the value of the best solution seen, the percentage gap between the best and average solutions, the average number of iterations (step 4 to 11 of Algorithm 1) completed within the allotted 90 minutes, and ﬁnally the number of requests that are cross-docked in the best solution, as a percentage of the number of “full” (non-partial) requests. The table shows a considerable variation in the number of iterations that can be completed within the given 90 minutes, mostly corresponding to the varying

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size of the instances. For each instance, the diﬀerence between the best and average solution value is around 1%–3%. The results show that many requests are serviced via the cross-dock. This is partially due to the geographical structure of the problem, and, as we mentioned earlier, is not considered problematic for this case study. In the next section we further investigate how results are distributed when running multiple tests. In Table 2 we show the split of the solution cost between the diﬀerent components, ﬁxed vehicle cost, km cost, road toll, cross-dock handling cost, and hourly vehicle cost. Table 2. Split of solution cost Distance 35.10% Time cost 27.87% Vehicle daily cost 16.68% Road tolls 13.68% Cross-dock handling 6.68%

3.2

Variation of Results

In order to examine the degree of variation of the solution quality and thereby the robustness of the heuristic, a series of identical test runs was performed. Figure 4 shows the distribution of the objective values that were obtained in 50 algorithm runs. The objective values ranged between 538,555 and 598,843 with an average of 560,314 and a standard deviation of 11,101. The results show that there is a certain variation in the obtained objective values, resulting from the stochastic nature of the algorithm. The ﬁgure is based on results obtained using an early version of the algorithm, hence the objective values are generally higher than the results reported earlier, but we believe the pattern is still representative. In Figure 5 the solution value is shown as a function of the iteration count. We see a typical metaheuristic behavior where the largest improvements take place in the beginning of the search. 3.3

Return-to-Best

As described in Section 2.6, each thread of the parallel LNS will occasionally be allowed to update its current solution, with the global best solution. The probability with which this happens is a parameter, and Table 3 shows the objective values of the ﬁnal solution, when using diﬀerent values for this parameter.

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12

10

8

6

4

2

0 530000

540000

550000

560000

570000

580000

590000

600000

Fig. 4. Distribution of objective values obtained over 50 identical runs on instance aaoe20

Fig. 5. The development of the solution value (y-axis) over the course of 40,000 iterations (x-axis) of the LNS algorithm, corresponding to 1.5 hours running time

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Avg. obj. val.

% diﬀ.

0 0.1 0.3 0.5

519025 493588 504348 503779

0% −4.9% −2.8% −2.9%

As can be seen, the application of this feature can lead to considerable improvements of the objective value of up to 5%. It would be interesting to study if the same feature would beneﬁt other ALNS heuristics as well or if it is particularly well-suited for the problem at hand.

4

Conclusion

In this paper we have described a pickup and delivery problem with cross-docking opportunity, as encountered by a major Danish transportation company. We have implemented a parallel adaptive large neighborhood search algorithm to solve the problem, and here we have presented some initial results of applying the algorithm to some real-life problem instances, with 500–1,000 requests to be served each day. The results are promising. We have also presented some of the additional features of our algorithm, and examined the impact of letting each thread of the parallel algorithm return to the best known solution, as a successful intensifying measure. Acknowledgments. The work was sponsored by the Danish Agency for Science, Technology and Innovation (project ”Intelligent Freight Transport Systems”).

References ¨ Ergun, J., Punnen, A.: A survey of very large-scale neigh1. Ahuja, R., Ergun, O., borhood search techniques. Discrete Applied Mathematics 123, 75–102 (2002) 2. Baldacci, R., Bartolini, E., Mingozzi, A.: An exact algorithm for the pickup and delivery problem with time windows. Operations Research 59, 414–426 (2011) 3. Berbeglia, G., Cordeau, J.F., Gribkovskaia, I., Laporte, G.: Static pickup and delivery problems: a classiﬁcation scheme and survey. Top 15(1), 1–31 (2007) 4. Boysen, N., Fliedner, M.: Cross dock scheduling: Classiﬁcation, literature review and research agenda. Omega 38, 413–422 (2010) 5. Cordeau, J.F., Laporte, G., Ropke, S.: Recent models and algorithms for one-to-one pickup and delivery problems. In: Golden, B., Raghavan, R., Wasil, E. (eds.) The Vehicle Routing Problem: Latest Advances and New Challenges, pp. 327–357. Springer, Heidelberg (2008) 6. Cordeau, J., Laporte, G.: The dial-a-ride problem: Variants, modelling issues and algorithms. 4OR 1, 89–101 (2003)

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7. Cortes, C., Matamala, M., Contardo, C.: The pickup and delivery problem with transfers: Formulation and a branch-and-cut solution method. European Journal of Operational Research 200(3), 711–724 (2010) 8. H¨ all, C., Anderson, H., Lundgren, J., V¨ arbrand, P.: The integrated dial-a-ride problem. Public Transport 1(1), 39–54 (2009) 9. Lee, Y., Jung, H., Lee, K.: Vehicle routing scheduling for cross-docking in the supply chain. Computers & Industrial Engineering 51, 247–256 (2006) 10. Mitrovi´c-Mini´c, S., Laporte, G.: The pickup and delivery problem with time windows and transshipment. INFOR 44(3), 217–227 (2006) 11. Nagata, Y., Kobayashi, S.: Guided ejection search for the pickup and delivery problem with time windows. In: Cowling, P., Merz, P. (eds.) EvoCOP 2010. LNCS, vol. 6022, pp. 202–213. Springer, Heidelberg (2010) 12. Nagata, Y., Kobayashi, S.: A memetic algorithm for the pickup and delivery problem with time windows using selective route exchange crossover. In: Schaefer, R., Cotta, C., Kolodziej, J., Rudolph, G. (eds.) PPSN XI. LNCS, vol. 6238, pp. 536–545. Springer, Heidelberg (2010) 13. Parragh, S., Doerner, K., Hartl, R.: A survey on pickup and delivery problems. part II: Transportation between pickup and delivery locations. Journal f¨ ur Betriebswirtschaft 58(2), 81–117 (2008) 14. Pisinger, D., Ropke, S.: A general heuristic for vehicle routing problems. Computers & Operations Research 34(8), 2403–2435 (2007) 15. Potvin, J.Y., Rousseau, J.M.: A parallel route building algorithm for the vehicle routing and scheduling problem with time windows. European Journal of Operational Research 66, 331–340 (1993) 16. Ropke, S.: Parallel large neighborhood search - a software framework. In: MIC 2009, VIII Metaheuristics International Conference CDROM (2009) 17. Ropke, S., Cordeau, J.F.: Branch-and-cut-and-price for the pickup and delivery problem with time windows. Transportation Science 43(3), 267–286 (2009) 18. Ropke, S., Pisinger, D.: An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows. Transportation Science 40(4), 455–472 (2006) 19. Shaw, P.: Using constraint programming and local search methods to solve vehicle routing problems. In: Maher, M.J., Puget, J.-F. (eds.) CP 1998. LNCS, vol. 1520, pp. 417–431. Springer, Heidelberg (1998) 20. Toth, P., Vigo, D.: The granular tabu search and its application to the vehiclerouting problem. INFORMS Journal on Computing 15(4), 333–346 (2003) 21. Wen, M., Larsen, J., Clausen, J., Cordeau, J.F., Laporte, G.: Vehicle routing with cross-docking. Journal of the Operational Research Society 60, 1708–1718 (2009)

Application of an RFID-Based System for Construction Waste Transport: A Case in Shanghai Tingting Ruan1 and Hao Hu2 1

2

Sino-US Global Logistics Institute [email protected] School of Naval Architecture, Ocean and Civil Engineering Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, PRC, 200240 [email protected]

Abstract. Eﬀective management of construction waste imposes a big challenge to cities with large volume of construction activities. Shanghai had to handle these problems in preparing the 2010 World Expo such as overloading of the trucks, traﬃc safety on the road, environment pollution, and heavy manpower needed to supervise the transport process. This paper presents our eﬀort of applying Radio Frequency Identiﬁcation (RFID) technology to reengineer the construction waste transport. Details of the new RFID-based system are introduced with an in-depth comparison with the traditional system. A Fuzzy Comprehensive Evaluation (FCE) method is used to validate the eﬀectiveness of the new system. Case studies in Shanghai show that the RFID-based system helps to better monitor the transportation procedure and to avoid the existing problems signiﬁcantly. On the other hand, it shows that stakeholders such as project owners and transport contractors are not always satisﬁed with the governments’ strong position in the new system, which might prevent the wide application of RFID technology in construction waste transport.

1

Introduction

Construction waste is usually deﬁned as spoil, disposable materials and other wastes generated in the process of new construction, reconstruction, extension, demolition and decoration. Eﬀective management of construction waste imposes a big challenge to cities with large volume of construction activities. New buildings in China in recent years sum up to 2 billion square meters in total. Construction waste comprises 30 % to 40 % of the total urban waste. The construction of a 10, 000m2 building creates 500 to 600 tons of waste, while the demolition of a 10, 000m2 building creates 7,000 to 12,000 tons of waste according to industrial data [14]. The amount of construction waste was 6 million tons in November, 2010 in Shanghai, comparing to the 3.9 million tons per month from May to October when most construction sites were closed due to the World Expo. J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 114–126, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Therefore, the collection and transportation of construction waste become a very important issue. However, the problems of transporting construction waste have existed for a long time. For example, overloaded vehicles cause traﬃc accidents. About 18,000 cases of traﬃc violations were reported in Shanghai from November of 2009 to January of 2010 [6]. Late payment of fare by project owners is a popular complaint by the waste transport contractors. Additionally, the supervision of construction vehicles is relevant to ﬁve government authorities, and that frequently brings diﬃculty of eﬀective communication and data sharing [7]. RFID is an area of automatic identiﬁcation that is gaining momentum and is considered by some to emerge as one of the most pervasive computing technologies in history [9]. It is a proven technology that has been in use since the 1970s. RFID is becoming increasingly prevalent with the decrease of the cost. RFID is widely used ranging from inventory management to environment sensors, pebbles tracking and healthcare service [2,10,12]. Applications include proximity cards, automated toll-payment transponders, payment tokens, ignition keys of many millions of automobiles as a theft-deterrent, checking-in and out for parking space management, and books tracking in libraries [5]. Some researchers have proposed to apply RFID to construction waste management. In 2007, Chowdhury et al. indicated the use of RFID would not only bring down waste management costs, but also facilitate automating and streamlining waste identiﬁcation and waste management systems [3]. In 2009, RFID associated intelligent systems were used to track vehicle position in solid waste monitoring and management [1]. The authors were involved in the research and application of the RFID-based system to the 2010 World Expo in Shanghai. This paper will report the practice in Shanghai by describing a case in Zhabei District of Shanghai. In particular, the evaluation of the RFID system will be presented in detail as studies in this regard are scarce. The rest of this paper is organized as follows. Section 2 presents the traditional process of construction wastes transport and analyzes the existing problems. Section 3 describes the RFID-based system applied in construction waste management. We present the current process after process reengineering and solved problems in Section 4. The FCE method is used to assess the RFID system in Section 5. Finally, Section 6 discusses the current applications in Shanghai and concludes with some remarks.

2 2.1

Traditional Construction Waste Transport and Its Problems Transportation of Construction Wastes

Figure1 depicts the typical process of transportation of construction waste from constructions sites to unloading places. Vehicles, which are deployed by transport contractors, load the construction waste in construction sites and transport and unload them in the designated sites. However, some disadvantages exist in this commonly used system. For example, trucks unload construction waste in diﬀerent places for convenience. Project owners pay charges to a transport company

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according to the calculated quantity of construction waste in design stage rather than the actual transport quantity. Construction waste that is not unloaded in the designated place is also paid.

Fig. 1. Transportation of construction wastes

2.2

Discussions of Existing Problems

We investigated several construction projects in Shanghai. A project in Zhabei District, one of the downtown districts in Shanghai, is chosen as case study in this paper. The transport company has transported construction waste for nearly half a year for the project. The unloading place is in Baoshan District, one of the suburbs in Shanghai, and it is nearly 30 kilometers from the construction site. The transportation of one cubic meter of construction waste costs approximately 0.6 dollar per kilometer. We summarize several problems of this traditional transport system as follows: Threat to traﬃc safety and environment. In order to enhance earnings, drivers tend to drive at a high speed and for a long time, which is likely to cause serious traﬃc accidents. Also they might dump construction waste in an undesignated place for convenience or for more work load, which destroys the neatness of the city or even pollutes the environment. In addition, the lid of an overloaded vehicle cannot be closed, and therefore dust and waste in the vehicle can easily fall onto the road in the city. Late payment of fare by project clients. Late payment of fare by clients is a popular complaint by the waste transportation contractors. Transport contractors are paid behind schedule by project clients or main construction contractors, which may bring ﬁnancial diﬃculties to transportation contractors.

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Unregulated transport business. The entry barrier of the transportation industry is very low, bringing a very ﬁerce competition. Transport companies can make proﬁt by doing some irregular and illegal things, such as overload, speeding, undesignated dumping, modiﬁcation of trucks, and so on. Ineﬃcient management by the governments. It is diﬃcult for governments to monitor construction waste transport because relevant government authorities are not involved in the traditional process. Furthermore, one construction waste truck is regulated by ﬁve government authorities, i. e., Traﬃc Police, Transportation and Port Authority, Municipal Greening & City Appearance Administration Authority, Urban Management & Administration Execution Bureau, and Urban and Rural Construction and Transportation Committee. This complicated structure always brings diﬃculty in obtaining real-time information and data sharing.

3 3.1

Applying RFID Technology to Construction Waste Transport RFID Technology

RFID is an automatic identiﬁcation technology, where the information is stored on a device called tag or transponder and can be retrieved by another device called reader using electromagnetic coupling as the way to exchange data. RFID can be divided into two categories: active and passive. Active tags require a power source, while passive ones do not require batteries [13]. An RFID-based system is usually the integration of RFID technology, Global Positioning System (GPS) technology, Geographical Information System (GIS), video intelligent recognition system, and other related technologies. 3.2

Applying RFID to Construction Waste Transport

As shown in Figure 2, RFID antenna and RFID tags are installed on the windshield glass in the truck with e-tamper function. Tag readers are portable, and transport companies rent them from government and put them in construction sites and unloading places. RFID tag readers are also used by government employees to inspect the transportation of construction wastes. The 2.45 GHz active RFID tags and tag readers are used with an eﬀective range of about 80 meters and more than 5 years’ battery life (see Figure 3). Long reading range and high reading speed enable tag readers to identify objects with high accuracy. Each construction waste truck installs an RFID tag and GPS system. Controllers, sensors and video probes are installed in dumping sites. In the supervision center, GIS map, video intelligent recognition system and data exchange system are used to locate and track trucks. The RFID tag records information of transport contractors, vehicle identiﬁcation number, driver’s information, name and location of construction site and dumping sites, the desired route, departure and arrival times. The most important data is the weight of construction waste in the construction site and as well as the dumping site.

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Fig. 2. Trucks with RFID tags

Fig. 3. RFID tag and tag reader

When the truck arrives at the dumping site, it is detected by the RFID reader, and information is sent out to the supervision center. The whole process can be automatically ﬁnished and requires no manual actions by the driver. The RFIDbased system can also eliminate trucks’ overload which is a serious problem in old transport mode. GPS is an important component of the RFID-based system. GPS is a spacebased global navigation satellite system that provides location and time information under almost all weather conditions and at all times and anywhere on or near the Earth when and where there is an unobstructed line of sight to four or more GPS satellites. In the system, GPS is applied to trace and monitor the vehicle, in particular the speed and route.

4

Process Reengineering and New Measures

Shanghai municipal governments launched many new infrastructure projects and implemented lots of new technologies in preparing the 2010 World Expo. The new RFID-based system was adopted by the governments as a kind of high technology and an important measure to better manage construction sites many of which are related to the Word Expo. With the adoption of RFID technology, the traditional construction waste transport system introduced in Section 2.1

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119

was restructured which can be shown by three ﬂows, i. e., logistics, fund and information ﬂows. 4.1

The New RFID-Based System

In the new system, which is shown in Figure 4, a third-party bank account and a supervision center are set up by governments. In terms of logistics ﬂow, there are two kinds of changes. One is that drivers have no need to get oﬀ the trucks because an RFID reader automatically collects data when trucks pass by the installed RFID readers. The other is that drivers have to drive along the designated routes. In terms of fund ﬂow, project owners have to remit enough money into the third-party account in advance. The transport contractors will receive money from the third-party account only after the data collected in dumping sites show that the construction wastes has been unloaded in designated places. The new payment procedure helps to regulate both project owners and transport contractors. In terms of information ﬂow, the supervision center manages the whole system, collects and processes relevant data. In short, the new system enhances signiﬁcantly the management of construction waste transport, especially the monitoring of trucks.

Fig. 4. The RFID-based system

4.2

Advantages of the New System

Advantages of the new RFID-based system can be summarized as below. Improving traﬃc safety and environmental impact. The supervision center with RFID and GPS technologies monitor the weight, route and speed of the trucks, which can prevent problems of overload and speeding. In addition, governments set regulations for the drivers, and trucks are required to install safety protection devices, such as voice automatic reminders.

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Allowing no late payment of fare by clients. The establishment of the thirdparty bank account and regulation of new process does not allow defaults. Industry becomes well-managed. The barrier to entry transportation industry is increasing due to the adoption of the RFID-based system and rules made by the governments. There are only one or two transport contractors in each district of Shanghai approved and appointed by the district government. Transport contractors apply for qualiﬁcation to install an RFID-based system and rent tag readers from the government. Furthermore, drivers are qualiﬁed only if they have at least 3 to 5 years experience in truck driving and obtain required certiﬁcations. Trucks also need to accept regular inspections. Improving government management eﬃciency. Monitoring trucks is not diﬃcult anymore due to information synchronization in the supervision center, with sharing real-time information with all related departments. The condition of each truck can be obtained by portable tag readers, and the ﬂow of fund is clearly shown in a bank account.

5

Evaluation of the RFID-Based System

The FCE method is used to validate the eﬀectiveness of the RFID-based system in this study. 5.1

Fuzzy Comprehensive Evaluation Method

The method of FCE has been applied to many real world problems. Fuzzy factors can be expressed by quantized form after math transformation. It applies to the evaluation situations which are diﬃcult to be quantiﬁed by precise numbers [11]. A fuzzy subset is a group of semantics that describe uncertain things [8]. The approach can be modeled as Figure5. Basic arithmetic steps of the FCE method are as follows: Step 1. According to the basic principle of FCE method, evaluation conclusion can be divided into m evaluation grades, and the evaluation conclusion set is deﬁned as V = {y1 , y2 , . . . , ym }. Step 2. Build evaluation index set U reﬂecting all indicators from various aspects: U = {U1 , U2 , . . . , Us }, where Ui ∩ Uj = ∅, i = j. Let each evaluation sub-indicator set be Ui = {Ui1 , Ui2 , . . . , Uini }, where i = 1, 2, . . . , s; n = s n . i i=1 Step 3. The fuzzy relationship matrix from Ui to V is R = (rij,k )ni ∗m , where i = 1, 2, . . . , s; j = 1, 2, . . . , ni ; k = 1, 2, . . . , m, and rij,k ∈ [0, 1] denotes the membership degree of sub-indicator Uij against evaluation conclusion grade yk . Bi denotes single indicator evaluation of Ui treated as one single indicator, namely single evaluation matrix of U = {U1 , U2 , . . . , Us } is: R = (B1 , B2 , . . . , Bs )T .

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Fig. 5. The comprehensive evaluation model

Step n4.j The distribution of weight of Ui is Ai = {ai1 , ai2 , . . . , aini }, where j=1 aij = 1. Each Ui , as one part of U , reﬂects a certain attribute of U . The weight distribution in U is A = {a1 , a2 , . . . , as }, where ai =

nj

s

j=1 aij nj j=1 aij

.

i=1

The ﬁrst-level FCE is calculated by: Bi = Ai · Ri = (bi1 , bi2 , . . . , bim ).

(1)

With A and R, we get the second-level evaluation: B = A · R = (b1 , b2 , . . . , bs ).

(2)

The ﬁnal FCE gained from above steps reﬂects diﬀerent levels of various indicators in the evaluation set. The evaluation grade is judged by the maximum principle. The ith evaluation conclusion yi of evaluation conclusion set V is the ﬁnal evaluation grade, which is denoted by bi = max(b1 , b2 , . . . , bs ).

(3)

Here, we use the Delphi method to determine value of R and weight value of Ai . The Delphi method is a survey or interview method in which the expert panelists’ knowledge and presumptions on an issue or development process under study are collected in an interactive process. Delphi is especially useful when the phenomenon under study is complex or when the topic is somehow diﬃcult to deﬁne [4]. Construction waste management involves some factors which are diﬃcult to be quantitatively evaluated by speciﬁc criteria, and people in diﬀerent positions have diﬀerent opinions on one thing. Therefore, we combine diﬀerent experts’ opinions on membership degree together to determine values of R and A by using the Delphi method. 5.2

Establishment of the Evaluation Indicators

After doing some investigations, analysis and summaries of construction waste transportation in Shanghai, we establish a kind of two level assessment indices system, which is shown in Table 1.

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Evaluation index U

Evaluation Sub-indicators Ui U11 Transport eﬃciency of vehicles driving from construction site to dumping site U12 Eﬃciency of passing through the gate of unloading place

U1 Transport eﬃciency

U2 Environmental impact

U3 Traﬃc safety impact U4 Industry inﬂuence

U5 Cost impact

5.3

U21 Decrease of phenomenon that construction wastes are not unloaded in designated place U22 Decrease of dust pollution to road in city U31 Reduction in the amount of overloaded trucks U32 Reduction of speeding phenomenon U41 Diﬃculties of the barrier to entry transportation industry U42 Diﬃculties of government to monitor U51 U52 U53 U54

Improvement in preventing defaults Cost in transport contractor Expense of real estate developer Expense of government

Fuzzy Relationship Matrix

We invited ten experts, who come from government departments, project owners and transport contractors, to take part in the measure and evaluation of current process applied with an RFID-based system. The evaluation conclusion is divided into 5 grades (much better, better, no change, worse, much worse than before) that respectively correspond to scores 5, 4, 3, 2, 1. The ﬁgures in Table 2 indicate the scores graded by experts. Since the grades for each Uij given by experts are added up to 10, grade should be divided by 10 ﬁrst to satisfy rij,k ∈ [0, 1] . Then, the fuzzy relationship matrix of Uij , where i = 1, 2, . . . , 5, and j = 1, 2, . . . , ni , can be established as below.

0.6 0.8

0.2 0.0

0.1 0.0

0.0 0.0

0.7 0.0

0.3 0.7

0.0 0.3

0.0 0.0

0.0 0.0

0.3 R3 = 0.6

0.7 0.4

0.0 0.0

0.0 0.0

0.0 0.0

0.5 0.4

0.3 0.0

0.0 0.3

0.0 0.0

0.1 R1 = 0.2 R2 =

0.2 R4 = 0.3

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Table 2. New process evaluation index table in construction waste transportation Evaluation index U

Sub-indicators Ui

5

4

3

2

1

U1

U11 U12

1 2

6 8

2 0

1 0

0 0

U2

U21 U22

7 0

3 7

0 3

0 0

0 0

U3

U31 U32

3 6

7 4

0 0

0 0

0 0

U4

U41 U42

2 3

5 4

3 0

0 3

0 0

U5

U51 U52 U53 U54

3 0 0 0

3 0 0 6

4 0 0 3

0 7 6 1

0 3 4 0

⎡

0.3 ⎢0.0 R5 = ⎢ ⎣0.0 0.0

0.3 0.0 0.0 0.6

0.4 0.0 0.0 0.3

0.0 0.7 0.6 0.1

⎤ 0.0 0.3⎥ ⎥ 0.4⎦ 0.0

The value in a matrix denotes the membership degree of sub-indicator Uij against evaluation conclusion grade yk , where k = 1, 2, . . . , 5. 5.4

Obtaining Weights of Indicators

Here, we get the weight value of Ai by using the Delphi method. After surveying some experts in transportation industry of construction waste, we do some analysis and obtain weight values of the fuzzy synthesis system, which are shown in Table 3. From Table 3, we can get the distribution weight of Ui (i = 1, 2, . . . , 5): A1 = (0.50, 0.50), A2 = (0.52, 0.48), A3 = (0.50, 0.50), A4 = (0.49, 0.51), A5 = (0.21, 0.29, 0.29, 0.21). The weight distribution of the evaluation index set U is A = (0.14, 0.16, 0.21, 0.17, 0.32). 5.5

Comprehensive Evaluation

First-level comprehensive evaluation of sub-indicators set B1 − B5 . With a fuzzy operator, the ﬁrst-level fuzzy evaluation matrix can be obtained by the following formula: 0.1 0.6 0.2 0.1 0.0 = (0.15, 0.70, 0.10, 0.05, 0.00) B1 = A1 · R1 = (0.50, 0.50) · 0.2 0.8 0.0 0.0 0.0 Similarly, B2 = A2 · R2 = (0.364, 0.492, 0, 0, 0), B3 = A3 · R3 = (0.45, 0.55, 0, 0, 0),

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0.14

U11 U12

0.50 0.50

U2

0.16

U21 U22

0.52 0.48

U3

0.21

U31 U32

0.50 0.50

U4

0.17

U41 U42

0.49 0.51

0.32

U51 U52 U53 U54

0.21 0.29 0.29 0.21

U5

B4 = A4 · R4 = (0.251, 0.449, 0.147, 0.153, 0), and B5 = A5 · R5 = (0.063, 0.189, 0.147, 0.398, 0.203). Second-level comprehensive evaluation of evaluation index set B. After obtaining the ﬁrst-level fuzzy evaluation matrix, the second-level fuzzy evaluation matrix can be obtained through the same principle: B = A · R = A · (B1 , B2 , B3 , B4 , B5 )T = (b1 , b2 , b3 , b4 , b5 ) = (0.236, 0.430, 0.109, 0.160, 0.065). 5.6

Analysis of the Results

B is evaluated according to the maximum principle by max(b1 , b2 , b3 , b4 , b5 ) = b2 . Because the second index value is the largest in the evaluation results, the grade of an RFID-based system in construction waste transport belongs to the second grade. The sum of b1 and b2 resulting in 66.6 % means that the current process is better than before. From the feedback from the experts, we found that an RFID-based system perform excellent in the following three aspects: eﬃciency of passing through the gate of unloading place U12 , decrease of phenomenon that construction wastes are not unloaded in designated place B21 , and reduction of speeding phenomenon B32 .

6

Concluding Remarks

After the investigation of the application of RFID technology in construction waste transport in Shanghai and evaluation of its eﬀectiveness by using the FCE method, we ﬁnd that the RFID-based system helps to improve traﬃc safety and environmental protection, allow no late payment of fare by project clients,

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to establish a well-managed industry and improve the government management eﬃciency. Citizens in Shanghai beneﬁt from it eventually. A practically useful technology might not be generally acceptable in real world. Though evaluation of an RFID-based system is satisﬁed, it has not yet achieved widespread acceptance in Shanghai. Stakeholders such as project owners and transport contractors are not always satisﬁed with governments’ strong position in the new system. The main problem is the increase of cost. According to experts’ estimates, although the expense associated with governments’ supervision decreases, the expense of project owners increases nearly 200 %, and the cost of the transport contractors enhances about 25%. The new system beneﬁts the government most. At ﬁrst stage, the government has to pay a large amount of money to purchase equipments and software, but in the long run, labor cost and management expenses will decrease as well as complaints from citizens. Nevertheless, project owners have to pay money into a third-party bank account in advance, and this increase the capital cost. Transport contractors are forced to install RFID tags and rent tag readers, and thus they make less proﬁt. Moreover, the price of construction waste transportation is set by government, so they have no choice but to accept the new price. The industry is slightly monopolized by government, and it is not a sustainable mode for construction waste transport industry to widely accept the RFID-based system. To some extent, increase of cost prevents the widespread of RFID technology in construction waste management. Another problem is that the RFID-based process requires government stuﬀ to have a more standardized inspection on construction waste transport, and therefore inspection procedures are more complicated and sometimes tedious. For example, government employees need to carry portable tag readers to inspect the situation in construction sites and designated places. Therefore, better automation is needed to encourage the application of RFID technology in construction waste management. This study reports a case of applying RFID technology to the construction waste transport industry and evaluates its performance. More empirical and conceptual studies will be conducted in future in further reﬁning and validating this RFID-based system.

References 1. Arebey, M., Hannan, M., Basri, H., Abdullah, H.: Solid waste monitoring and management using RFID, GIS and GSM. In: IEEE Student Conference on Research and Development (SCOReD), pp. 37–40 (2009) 2. Benelli, G., Pozzebon, A., Raguseo, G., Bertoni, D., Sarti, G.: An RFID based system for the underwater tracking of pebbles on artiﬁcial coarse beaches. In: Proc. of the Third International Conference on Sensor Technologies and Applications, SENSORCOMM 2009, pp. 294–299 (2009) 3. Chowdhury, B., Chowdhury, M.: RFID-based real-time smart waste management system. In: Proc. of Australasian Telecommunication Networks and Applications Conference (ATNAC 2007), pp. 175–180 (2007)

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4. Laakso, K., Rubin, A., Linturi, H.: Delphi method analysis: The role of regulation in the mobile operator business in Finland. In: Proc. of PICMET 2010: Technology Management for Global Economic Growth 2010, pp. 1–7 (2010) 5. Mohamed, N., Garoot, A.H., Hazza, Z.M.: A case study on impacts of RFID adoption in tree inventory management. In: Proc. of 2nd IEEE International Conference on Computer Science and Information Technology (ICCSIT 2009), pp. 624–628 (2009) 6. n.a.: http://news.sina.com.cn/c/2010-12-07/162718441719s.shtml, last call (July 15, 2011) 7. n.a.: http://news.sina.com.cn/c/2011-01-18/133421833803.shtml, last call (July 15, 2011) 8. Ren, X., Zhang, G.: Research on decision support for Six Sigma project selection based on fuzzy evaluation. In: Proc. of the 4th International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM 2008), pp. 1–11 (2008) 9. Roberts, C.: Radio frequency identiﬁcation (RFID). Computers & Security 25(1), 18–26 (2006) 10. Sarac, A., Absi, N., Dauz`ere-P´er`es, S.: A literature review on the impact of RFID technologies on supply chain management. International Journal of Production Economics 128(1), 77–95 (2010); Integrating the Global Supply Chain 11. Sun, C., Ding, W.: Special skill evaluation in mechanical operation ﬁeld based on fuzzy evaluation method. In: 2010 International Conference on E-Health Networking, Digital Ecosystems and Technologies (EDT), vol. 2, pp. 538–541 (2010) 12. Tzeng, S.F., Chen, W.H., Pai, F.Y.: Evaluating the business value of RFID: Evidence from ﬁve case studies. International Journal of Production Economics 112(2), 601–613 (2008) 13. Want, R.: An introduction to RFID technology. IEEE Pervasive Computing 5(1), 25–33 (2006) 14. Zhou, W., Chen, J., Lu, H.: Status and countermeasures of domestic construction waste resources. Architecture Technology 40, 741–744 (2009)

Strategic and Operational Planning of Bike-Sharing Systems by Data Mining – A Case Study Patrick Vogel and Dirk C. Mattfeld University of Braunschweig, Decision Support Group, Muehlenpfordtstrasse 23, 38106 Braunschweig, Germany {p.vogel,d.mattfeld}@tu-bs.de

Abstract. Bike-sharing is a new form of sustainable urban public mobility. A common issue observed in bike-sharing systems is imbalances in the distribution of bikes. There are two logistical measures alleviating imbalances: strategic network design and operational repositioning of bikes. IT-systems record data from Bike Sharing Systems (BSS) operation that are suitable for supporting these logistical tasks. A case study shows how Data Mining applied to operational data offers insight into typical usage patterns of bike-sharing systems and is used to forecast bike demand with the aim of supporting and improving strategic and operational planning.

1 Introduction Bikes gain attention in urban public mobility. They allow sustainable transportation and “provide the missing link between existing points of public transportation and desired destinations” [1]. Bike-sharing is a short-term bicycle rental service for innercity transportation providing bikes at unattended stations. Recently, BSS have rapidly emerged in major cities all over the world. Bike-sharing providers face a challenging logistical task. They have to ensure high bike availability in order to satisfy customers. Short rental times and one-way use lead to highly dynamic spatial and temporal customer movements causing imbalances in the distribution of bikes. Mainly, there are two logistic measures alleviating imbalances: strategic location planning of bike stations and operational repositioning of bikes. Data Mining (DM) applied to operational BSS data gains insight into the complex causes of imbalances. According to Hand et al. [2] DM is “the analysis of (often large) observational data sets to find unsuspected relationships and to summarize the data in novel ways that are both understandable and useful to the data owner”. In future works, the obtained findings can be used for building sophisticated planning models from the field of Operation Research (OR). In Section 2 measures alleviating bike imbalances in BSS are presented. A DM framework for supporting strategic or operational planning of BSS is introduced. In a subsequent case study, two years of operational ride data from Vienna’s BSS J.W. Böse et al. (Eds.): ICCL 2011, LNCS 6971, pp. 127–141, 2011. © Springer-Verlag Berlin Heidelberg 2011

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“Citybike Wien” are analyzed according to the DM decision support framework. Typical usage patterns are determined and bike demand is forecasted which can support strategic and operational decisions.

2 Logistic Challenges in Strategic and Operational Planning of BSS BSS offer environment-friendly, individual urban mobility. Increase in bike use and enhancement of public transport options as well as reduction of congestion and pollution are primary goals of BSS [3]. The major drawback of bikes is that usage strongly depends on weather and topography. Typically, a network of bike stations is spread over the city providing free bikes or free boxes, respectively. Automated bike rentals are possible due to information and communication technology that also store cyclists’ rental activities for billing and control purposes. The most common type of provider are advertising companies [4]. Rights from municipalities to advertise on street furniture are received in exchange for providing a BSS. In order to keep rental times short, the first minutes of a ride are often for free. After that, fees increase linearly or progressively in certain time steps. These incentives let customers quickly return bikes, making them available for others to reach a high bike turnover. Short rental times and one-way use lead to imbalances in the spatial distribution of bikes at stations over time. That is why ensuring the availability of free bikes and boxes is logistically challenging and crucial for the usage and acceptance of BSS. The tendering for Arlington’s BSS indicates the relevance of balanced bike distribution. Here, the provider has to ensure that stations are not full of bicycles for more than 60 minutes during daytime and 180 minutes during nighttime [5]. There are two logistic measures alleviating these imbalances: • Operational (short-term) provider based repositioning of bikes. Here, staff uses vehicles to relocate bikes from full to empty stations. Costs for reposition of a bike in the “Vélib” amount to 3 $ [4]. In order to meet the highly fluctuant bike demand, a sophisticated management of the repositioning fleet is self-evident. Decisions to be made comprise when to relocate what amount of bikes from which full to which empty station. • In contrast to short-term reaction on bike shortage at stations, fluctuations can be absorbed by a suitable strategic (long-term) network design. Precisely because network design is expensive and cannot be changed on a short-term basis, sophisticated decisions about the number, size and location of stations are obligatory. That is why fluctuations in operational bike activities have to be anticipated in strategic planning in order to make some repositioning effort become superfluous. Ignoring these operational activities can lead to suboptimal location decisions [6].

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3 Decision Support Framework for Operational and Strategic Planning of BSS In recent years, some work has been done regarding planning and control as well as data analysis of BSS. However, only few scientific literature in this field is published. In the following, a brief overview of BSS articles dealing with DM and OR is given: • Froehlich et al. [7] analyze data comprising the number of available bikes at stations from Barcelona’s BSS. According to the number of available bikes in the course of day stations are grouped by cluster analysis. Visualization of the clustered stations show spatial dependencies, e.g., uphill stations tend be empty. • Kaltenbrunner et al. [8] also deal with data from Barcelona’s BSS. Bike usage patterns are determined that are similar to Froehlich et al. [7]. A time series model is presented that predicts the number of available bikes and boxes at stations. • Borgnat et al. [9] analyze the dynamics of bike movements in Lyon’s BSS. With the help of DM, temporal and spatial bike usage is examined. Time series analysis is applied for modeling and predicting daily and hourly rentals. Furthermore spatial patterns are examined by clustering bike flows between stations. Spatial and temporal dependencies exist between stations of clusters interchanging many bicycles. • Lin and Yang [10] present a mathematical decision model from the field of OR to determine an adequate number and location of bike stations considering bike availability at stations. Network structure of bike paths between the stations and travel paths for users between their origin and destination are output of the model. The model is tested with artificial bike demand data. These recent works either deal with mining of bike-sharing data or building decision models without incorporating real world BSS behavior. For example, Lin and Yang [10] only consider the availability of bikes and not the availability of boxes. They also admit that hourly rental fluctuations are ignored in their model. OR models for real world Bike-sharing applications should capture stochastic demand and dynamic planning periods. These models have an exceptional computational effort. With the help of DM, sophisticated OR models can be built by providing information about the system structure which is used to determine decision attributes and suitable aggregated data [11]. In this paper we present a decision support framework for strategic and operational planning of BSS. The framework comprises DM and OR. Here, we focus on DM to gain insight into the complex bike operations. Cluster analysis and time series analysis are applied to operational data in order to examine typical activity patterns and to model and forecast bike demand. The decision support framework is depicted in Figure 1. DM is often associated with Knowledge Discovery in Databases (KDD). The process oriented KDD embeds DM framed by data preprocessing and verification [12]. DM is distinguished from common statistics due to the secondary analysis of large amounts of operational data that has not intentionally been collected for this purpose. DM tasks are for instance descriptive modeling like clustering data and predictive modeling like regression and time series analysis [2]. Findings from DM

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are incorporated in the corresponding OR models on the different planning levels. The output of the OR models are used to modify the system’s appearance. Implemented decisions do not only affect the corresponding but also underlying planning levels. Strategic planning Data Mining: cluster analysis

Operations Research: network design

Operational planning Data Mining: time series analysis cluster analysis

Operation Research: fleet management

System operation and data collection (internal) ride data

(external) geographical / weather data

Fig. 1. Decision support framework for strategic and operational planning of BSS

In detail, the framework consists of the following components: • System operation and data collection: Ride data is collected from the system operation. Also external geographical and weather data is incorporated in order to support the data analysis. • Operational planning: With the help of time series analysis [13] temporal rental patterns are modeled. A time series model is moreover used to forecast daily and hourly rentals under given weather circumstances. Hence the provider is able to react on forecasted fluctuations with an adequate repositioning. Cluster analysis [14] is applied to group stations according to their pickup and return patterns. Geographical visualization gives the BSS operator a better understanding of spatial dependencies of the stations’ activity. With help of OR, cost efficient transportation requests and routing for the repositioning fleet are generated. • Strategic planning: From a strategic planning point of view it is necessary to at least understand bike activities at stations on the operational level. Cluster analysis is also capable of supporting strategic planning. Concerning network design in BSS, we are looking for spatial relations between bike usage at stations and location of stations. If there is a relation, activity patterns for potential stations can be anticipated according to their location factors by assigning the activity patterns from existing stations with the same location factors. In a following location decision model an adequate number, location and size of stations have to be determined whereas the anticipated operational activities and bike availability have to be considered.

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With the help of the decision support framework, operational data is analyzed for understanding usage patterns and forecasting demand. DM is the first step to give BSS planners insight into the system’s behavior in order to build sophisticated decision models. Also generated data can serve as input for these models. This can improve the quality of strategic and operational OR models.

4 Case Study: Forecast Bike Demand and Understanding BSS Behavior by Data Mining In the following case study, DM is used to understand and model causes for bike imbalances. Time series analysis is applied for supporting operational repositioning decisions. Operational ride data is transformed into models which describe temporal rental patterns and forecast bike demand on the basis of weather circumstances. In addition, spatio-temporal dependencies of bike pickups and returns at stations for operational and strategic decision support are examined. This is achieved by cluster analysis that yields groups of stations showing similar activity patterns. The analyzed data comprises two years of operational ride data from Vienna’s BSS “Citybike Wien” (http://citybikewien.at) which is operated by “Gewista Werbegesellschaft mbH”. Because bike usage is inheritably defined by weather circumstances, weather data which covers the same time span is also incorporated. Rapidminer Community Edition from Rapid-I GmbH (http://rapid-i.com) is used for data analysis. Results are visualized with the geographical information system Google Earth (http://earth.google.com). 4.1 Preprocessing A preprocessing phase is necessary in order to make the data suitable for DM. Ride and weather data are gathered. Also aggregation and normalization of data assures a solid basis for the subsequent data analysis. Ride Data The provided data set includes ride information in form of pickup station and timestamp as well as return station and timestamp (see Table 1). Almost 760,000 rides were recorded in the years 2008 and 2009. Trip durations are calculated by subtracting pickup from return timestamps. The system comprises 54 stations at the beginning of 2008. Six new stations were erected in 2008 and one in 2009. Geographical coordinates and the number of bike boxes are available for every station. Also the total number of bikes in the BSS is provided. Table 1. Example ride data set Pickup station

Pickup timestamp

Return station

Return timestamp

1074

2008-01-01 13:56:07

1052

2008-01-01 14:12:57

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In the preprocessing phase, data objects that show the following characteristics are removed: • Rides where bikes are reported as defect or stolen. • Rides with negative trip durations are excluded. This can happen if an error occurs at the bike box while returning a bike. • Rides that last less than 60 seconds which start and end at the same station. This indicates that a bike is immediately returned after being picked up. An actual ride does not take place. • Rides that start or end at test stations. The number of data sets declines to approximately 750,000 after removing the affected data sets. This still extensive amount of operational data has to be aggregated, because an inspection of single rides does not lead to a general impression of bike activities. A suitable timescale of the aggregation has to be chosen in order to detect temporal patterns in bike activity. The tradeoff is the following: the smaller the time windows, the larger the fluctuations, whereas larger time windows might smooth relevant observations. Citybike Wien grants that the first 60 minutes of a ride a free of charge, which is the case for more than 90 % of all trips. Average trip duration is 29 minutes, whereas the median trip duration is 15 minutes. Exploratory analysis yields that bike activity is highly fluctuant with respect to time of day, day of week and type of day. That is why ride data for both years are aggregated for 24 time windows per day of the week, leading to 168 values. Common approaches of traffic systems analysis correspond to these time windows [15]. 10000 9000 8000 7000

number of rides

6000 5000 4000 3000 2000 1000 0 0 6 Mo

12 18

0 6 Tu

12 18

0 6 We

12 18

0 6 Th

12 18

0 Fr

6

12 18

0 6 Sa

12 18

0 6 Su

12 18

time of week total

1st quarter

2nd quarter

3rd quarter

4th quarter

Fig. 2. Temporal ride patterns in the course of week

The total number of rented bikes in the course of week is depicted in Figure 2. Patterns for working days and weekends can be observed. Working days show three peaks. The subway stops service between 0-1 a.m. which might cause the night peak. Due to commuters there is a morning peak followed by another peak in the late

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afternoon hours. Here, commuting and leisure activities overlap. There are only two peaks on weekends. The night peak is more distinct, whereas the morning peak is absent indicating that the Citybike is predominantly used for leisure activities at weekends. In addition the number of rented bikes in the course of week aggregated per quarter is depicted. The shape of the curve is similar for every quarter, but the amplitude is different. This indicates that the proportion of hourly rides is constant throughout all seasons. For instance, 2.2 % of daily rides account for the hour between 8 and 9 a.m. for every Monday. Weather Data Bike usage is inheritably defined by weather circumstances. However, recent studies disagree about which weather effects have certain influence on bike usage [16–20]. In four of the five studies rainfall has a high impact on bike usage. Regarding temperature Brandenburg et al. [18] and Snizek [16] assume a correlation between bike usage and short term temperature variation whereas Girod [17] finds a seasonal temperature effect. Brandenburg et al. [18] also presume that cloudiness and sunshine are relevant factors. Therefore relevant weather factors from these six factors have to be determined and incorporated in a bike usage prediction model. Weather data is provided by the Austrian Institute for Meteorology and Geo Dynamics “Zentralanstalt für Meteorologie und Geodynamik” (www.zamg.ac.at). Sixteen weather factors were recorded at five weather stations that lie within a radius of 10 kilometers around Vienna’s city center. Geographical coordinates of all stations lie at hand. Due to the preliminary examination of weather impact on bike usage, six factors are selected from the data set (see Table 2). Factors temperature, rainfall, wind, humidity and sunshine are recorded every ten minutes at all five stations. Cloudiness in measured every hour at one of the five stations. Table 2. Weather factors weather temperature (tl) wind (ff) humidity (rf) sunshine (so) rainfall (rr) cloudiness (N)

description 10-min average of temperature 10-min average of wind speed 10-min average of humidity sunshine duration in 10-min 10-min sum of rainfall 1-h average of cloudiness

value range/unit 1/10 degrees Celsius (1°C=10) 1/10 ms-1 (1ms-1 = 10) 0-100 % 0-600 seconds 1/10 mm (1mm = 10) 10 steps of cloudiness (0-9)

According to the findings of the ride data preprocessing, weather factors are also aggregated on an hourly basis: • The hourly average for the 10-minute factors temperature, wind, humidity, sunshine and rainfall is generated for every weather station. Since those average values differ slightly at the five stations, the median of the averages is built. Analysis of the aggregated data shows that at least one value for every hour of the two years is recorded by all weather stations.

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• The factor cloudiness is recorded every hour. Values are missing in 20 % of the hours in two years. If a value is missing, it is replaced by the next recorded previous or subsequent value. • Dependencies of weather factors are analyzed with the help of a correlation matrix. The outcome indicates that humidity and sunshine are highly correlated with temperature. Since weather studies indicate that temperature has a high impact on bike usage, humidity and sunshine are not considered in the following analysis. 4.2 Forecasting Bike Demand Using Time Series Analysis As shown in the previous section, the number of rented bikes is highly fluctuant with respect to the day of week and hour of day. The fact that rentals are not independent of time points out that time series analysis [13] is suitable for modeling the number of rentals in a BSS. Also time series models are capable of modeling seasonal weather influences and short term fluctuations. In order to explore correlation between rides and weather, both ride and weather data are combined. Findings from an exploratory data analysis indicate that rain has a negative effect on daily and hourly rentals. Starting rain provokes sudden drops in rentals numbers whereas stopping rain causes an increase. Furthermore, the average hourly fluctuations in the weekly course are quite constant. This leads to a time series model for predicting the number of hourly rentals with two components. A cyclic component expresses the average number of rented bikes in terms of a seasonal ) times the fraction of a specific hour of a weekday amplitude for a given day ( ( ). The second component expresses the fluctuations for a given hour of a specific day ( ). (1) Daily Forecast Linear regression [21] is used to determine the correlation between rides and external factors explaining the daily bike usage ( ). The following factors are examined: • These daily aggregated weather factors are evaluated: average temperature T(d) [1/10 °C], rainfall in sum of rainfall SR(d) [1/10 mm] or binominal BR(d) [rain=1, no rain=0], average wind speed W(d) [1/10 mm/s], average cloudiness C(d) [1-9]. • Since there is an increase in rides between year 2008 and 2009, the effect of the system growth SG(d) is modeled by either the number of available bikes in the system or the number of bike boxes or the number of stations for a given day. In order to test the significance of the regression factors, the data is split up into training data (year 2008) and test data (year 2009). Minimizing the sum of squares is used to determine the factors. The output is the following regression: (2)

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The coefficient of determination for the model with coefficients according to Table 3 is 0.874. This points out how well the model is likely to model bike rentals under given weather circumstances. Four weather factors remain in the model that show an expected behavior (see Table 3). Temperature is the most important factor and increasing temperature leads to higher ride numbers and conversely. Cloudiness and sum of rainfall have almost the same negative effect on rides numbers, whereas sum of rainfall outperforms the binomial rain factor. Wind has the smallest effect of all weather factors. The factor for system growth is not part of the model, because negative coefficients were computed. A growing systems leading to less rides simply does not make sense. Weather factors might superimpose the impact of system growth. Table 3. Values for the daily component of the time series model factor temperature ( ) rainfall ( ) cloudiness ( ) wind ( ) constant ( )

coefficient 5.637 -2.140 -35.053 -3.824 660.708

standard deviation 0.155 0.274 3.997 0.825

standard coefficient ß 0.789 -0.169 -0.190 -0.100

36.414 -7.805 -8.770 -4.637

Hourly Fluctuations Although the coefficient of determination for daily ride modeling is very satisfying, there is need for modeling fluctuations of a specific hour . After erasing the daily component from aggregated rides numbers, hourly fluctuations remain. These hourly ride numbers depend on weather as well as ride numbers from previous hourly observations. In order to determine which previous hours, also called lag, are significant, the autocorrelation function (ACF) and partial autocorrelation function (PACF) [22] are analyzed. The ACF decays exponentially towards zero and the PACF dies out after two lags, pointing out that an auto-regressive (AR) process with the order of two is suitable for modeling hourly fluctuations [22]. The same set of weather factors for testing daily weather impact are used for determining hourly weather impact. It turns out that temperature ( ) and rain or no rain ( ) are the only relevant weather factors for hourly fluctuations with 0.584. ACF and PACF analysis of the AR(2) residuals still show autocorrelation for the 24th lag. That is why the hour of the day before ( ) is incorporated in the AR model, leading to the following model for hourly fluctuations with 0.603: (3) Coefficients are computed according to Table 4. Previously observed ride numbers have a greater effect on ride numbers than weather. The temperature impact on hourly ride numbers is less significant than starting rain.

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factor temperature ( ) rain or no rain ( ) ( ) ( ) ( ) constant ( )

coefficcient 0.0 003 -4.1 149 593 0.5 099 0.0 0.1 148 -0.0 063

standard deviation 0.002 0.477 0.007 0.007 0.007

standard coefficient ß 0.015 -0.063 0.593 0.099 0.148

2.0556 -8.7000 82.2997 13.6888 20.4998

Combining the cyclic component c and hourly fluctuations (eq. 1) leads to 0.918. The absolute averag ge prediction error for years 2008 and 2009 is 8.249 biikes per hour with a standard deviation d of 8.754 bikes, indicating how well the houurly rentals are captured by thee proposed time series model. One-step hourly forecassted ride numbers for a random mly selected week in 2009 are depicted in Figure 3. T The cyclic component predictss system wide ride numbers quite well. In particuular forecasted night hours show w almost no difference to recorded rides. During daytiime hours, the cyclic componeent is adjusted by hourly fluctuations. Both componeents capture the temporal rentalss patterns in a good manor.

Fig. 3. Recorded R and forecasted hourly ride numbers

4.3 Typical Behavior of BSS B Determined by Cluster Analysis In order to support BSS planners in operational and strategic decisions, insights iinto spatio-temporal dependenccies of bike activities are given. According to our assumption, bike pickups and a returns depend on the stations’ locations. That is w why the system wide temporal usage u patterns are broken down into station related pickkups and returns activities. The chosen timespan of one hour remains, but pickups and returns are aggregated for every e station whereas working days and weekend days are

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considered separately. Furthermore pickups and returns for every hour are divided by the total daily number of pickup or returns. This normalization is necessary in order to compare the activity at stations. This leads to 48 attributes describing the normalized hourly pickup and return activity for every station serving as input for the cluster analysis. Stations with only a few pickups or returns are excluded from the analysis, because these stations could show non-typical return and pickup patterns which distort the clustering. Cluster analysis determines groups of stations where hourly pickup and return activity within a group is similar to each other and different from stations in other groups [14]. Three different cluster algorithms for a varying number of clusters are applied: k-means (KM) [14], Expectation Maximization (EM) [14] and sequential Information-Bottleneck (sIB) [23]. One of the most well-known partitional cluster algorithms is KM. Partitional cluster algorithms seem to be a promising approach due to the high dimensionality of the considered data [24]. An initial partitioning with k clusters is created whereas the number of clusters has to be chosen beforehand. Objects are relocated from one cluster to another by minimizing the distances of objects within clusters and maximizing the distance of objects in different clusters. The EM algorithm is an extension of the KM paradigm. On basis of a weight that represents the probability of membership, each object is assigned to a cluster. sIB is an agglomerative clustering method capable of dealing with high dimensional data. The outcome of the three cluster algorithms is evaluated with different validation indices that determine the cohesion and separation of clusters. The Davies-BouldinIndex [25], Dunn-Index [26] and Silhouette-Index [14] determine the cohesion and separation of clusters according to the distance between data sets among each other and their clusters. The elbow criterion is applied for determining the desired number of clusters. Striking changes in the indices‘ values, called elbow, indicate an appropriate clustering [14]. High values for Dunn and Silhouette-Index indicate a proper clustering whereas the opposite holds for the Davies-Bouldin-Index. EM, KM and sIB are applied for k varying from two to ten. Results of the three indices for workings are depicted in Table 5. Table 5. Cluster validation results for different cluster algorithms Number of clusters k EM: Davies-Bouldin EM: Dunn EM: Silhouette sIB: Davies-Bouldin sIB: Dunn sIB: Silhouette KM: Davies-Bouldin KM: Dunn KM: Silhouette

2 1.33 0.23 0.30 1.34 0.26 0.30 1.35 0.26 0.30

3 1.66 0.24 0.21 1.66 0.24 0.21 1.70 0.24 0.21

4 1.57 0.26 0.22 1.62 0.26 0.23 1.58 0.24 0.22

5 1.51 0.27 0.24 1.59 0.26 0.21 1.54 0.24 0.22

6 1.63 0.25 0.20 1.84 0.20 0.21 1.69 0.24 0.19

7 1.54 0.24 0.20 1.73 0.29 0.21 1.65 0.24 0.19

8 1.56 0.29 0.23 1.62 0.32 0.22 1.60 0.26 0.19

9 1.64 0.29 0.22 1.60 0.29 0.26 1.76 0.26 0.20

10 1.39 0.29 0.25 1.62 0.29 0.28 1.51 0.23 0.19

According to the elbow criterion EM yields a proper clustering for 5 clusters indicated by a drop in the Davies-Bouldin-Index and peaks for Dunn und Silhouette. sIB and KM also yield the best clustering for k = 5. The EM clustering is chosen because it outperforms sIB and KM. A further inspection of the cluster results is

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necessary, because cluster validation indices only determine the cohesion and separation of clusters. Results of the cluster analysis are temporally and spatially validated in the following. The centroid, representing the average hourly fraction of daily pickups or returns, of every cluster is used for a temporal validation. The output of the cluster analysis is five different patterns of the temporal pickup and return activity at stations depicted in Figure 4. For a better understanding, clusters are labeled according to their daily pickup and return patterns. At commuter I stations at average 8 % of the total daily pickups occur between 8 and 9 a.m., for example. Pickups in the daily course

12%

Returns in the daily course

fraction of daily amount

10% 8% 6% 4% 2% 0% 0

2

4

6

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10 12 14 hour of day

16

18

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22

0

2

4

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10 12 14 hour of day

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Fig. 4. Activity patterns for working days in the daily course

• The commuter I cluster consists of 13 stations. This cluster shows a high pickup activity in the morning whereas return activities dominate in the evening hours. • 16 stations are assigned to cluster commuter II. These stations show a higher return activity in the morning compared to pickup activity (6 % vs. 2 %). In the evening, pickup activity is slightly higher than return activity. • The 12 stations of the leisure cluster show high evening and nighttime activity. • The tourist cluster is the smallest cluster with only 3 stations. This cluster’s daytime pickup and return activities dominate the other clusters’ activity. • The biggest cluster with 19 stations is labeled AVG because it reflects average pickup and return activities. When averaging the activity of all stations, the outcome is very similar to the pattern of this cluster. The clusters’ geographical distribution is visualized in Figure 5 in order to examine spatial reasons for the determined activity patterns. This gives planners a better understanding of the systems behavior leading to a better repositioning and location planning. Exploratory analysis of the clusters’ surroundings leads to the following findings:

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• The stations of cluster commuter I are located at the periphery with residential buildings. The commuter II stations are located around the city center that offers a high number of working places compared to other districts [27]. This leads to the assumption that workers use the Citybike for commuting from residential areas to working areas. Also tourist attractions are found around the city center causing high pickup and return activity in the evening. • Framed by commuter II station lays the tourist cluster station 1021. Citybike hands out tourist cards next to this station. The other two tourist stations are located near the famous tourist attractions Prater carnival in the northeast and castle Schoenbrunn in the southwest. This endorses that the high daily and low nightly activity is caused by tourist usage. • Stations of the night active leisure cluster tend to be located close to the city center, but at the edge of the BSS network. For example, popular night clubs and bars lay in the immediate vicinity of station 1041 and 1042 causing high nightly activity. • Although AVG stations are spread over the network, they still occur in small groups. Overlapping spatial factors might be a reason for this.

Fig. 5. Geographical distribution of clusters (aerial view of the city of Vienna)

The presented findings from the spatio-temporal analysis show that reasons for certain pickup and return activities at bike stations are complex and diverse. The presumption that activity patterns and the station’s location are correlated seems promising according to first results from the data analysis. A further investigation of the stations surroundings is needed to fully support the assumption.

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5 Conclusion BSS enhance inner-city public transport options. Ensuring high bike availability is crucial for the acceptance of such systems. Due to one-way use and short rental times imbalances in the spatial distribution of bikes occur. Logistic measures alleviating imbalances are sophisticated strategic network design and operational repositioning of bike with the help of OR models. A decision support framework is presented to gain insights into the complex bike activity by DM. This leads to better understanding about BSS structure and increases OR effectiveness. For supporting operational planning, time series analysis is used to determine factors influencing seasonal and short-term bike usage. Moreover, the model is capable of forecasting the system wide ride numbers. Clustering of stations according to their activity shows that stations with similar temporal activity patterns are geographically connected. With respect to repositioning fleet management, synergies can be obtained by serving areas rather than single stations that have similar bike imbalances. An exploratory analysis of the clustered stations’ surroundings indicates that activity patterns depend on the station’s location. This knowledge can be used for a sophisticated strategic planning of bike stations where operational bike fluctuations are anticipated.This leads to the conclusion that DM is a good approach to gain insight into the complex bike activities in BSS. Focus of further research lies in building sophisticated OR models with the help of the gained knowledge. Acknowledgements. We thank „Gewista Werbegesellschaft m.b.H.“ for providing data from their project “Citybike Wien” and discussing related issues and results. We also thank „Zentralanstalt für Meteorologie und Geodynamik“ for providing the weather data.

References 1. Midgley, P.: The Role of Smart Bike-sharing Systems in Urban Mobility. Journeys, 23–31 (May 2009) 2. Hand, D.J., Mannila, H., Smyth, P.: Principles of Data Mining. MIT Press, Cambridge (2001) 3. Shaheen, S., Guzman, S., Zhang, H.: Bikesharing in Europe, the Americas, and Asia: Past, Present and Future. Paper presented at the 89th Annual Meeting of the Transportation Research Board (2010) 4. DeMaio, P.J.: Bike-sharing: History, Impacts, Models of Provision, and Future. Journal of Public Transportation 12(4), 41–56 (2009) 5. Zahory, M.N.: Request for Proposals for the operation of the Arlington Bike-sharing Program (2009) 6. Salhi, S., Rand, G.K.: The effect of ignoring routes when locating depots. European Journal of Operational Research 39(2), 150–156 (1989) 7. Froehlich, J., Neumann, J., Oliver, N. (eds.): Sensing and Predicting the Pulse of the City through Shared Bicycling (2008)

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8. Kaltenbrunner, A., Meza, R., Grivolla, J., Codina, J., Banchs, R.: Urban cycles and mobility patterns: Exploring and predicting trends in a bicycle-based public transport system. Pervasive and Mobile Computing 6(4), 455–466 (2010) 9. Borgnat, P., Robardet, C., Rouquier, J.-B., Abry, P., Flandrin, P., Fleury, E.: Shared Bicycles in a City: A Signal Processing and Data Analysis Perspective (2010) 10. Lin, J.-R., Yang, T.-H.: Strategic design of public bicycle sharing systems with service level constraints. Transportation Research Part E: Logistics and Transportation Review 47(2), 284–294 (2011) 11. Meisel, S., Mattfeld, D.: Synergies of Operations Research and Data Mining. European Journal of Operational Research 206(1), 1–10 (2010) 12. Fayyad, U.M.: Advances in Knowledge Discovery and Data Mining. AAAI Press, MIT Press, Menlo Park, Calif (1996) 13. Box, G.E.P., Jenkins, G.M., Reinsel, G.C.: Time series analysis. Forecasting and control, 4th edn. Wiley, Hoboken (2008) 14. Tan, P.-N., Steinbach, M., Kumar, V.: Introduction to data mining. Pearson international Edition. Pearson, Boston (2006) 15. Pinkofsky, L.: Typisierung von Ganglinien der Verkehrsstärke und ihre Eignung zur Modellierung der Verkehrsfrage. Shaker, Aachen (2005) 16. Snizek + Partner Verkehrsplanung: Radverkehrserhebung Wien (2009), http://www.snizek.at/radverkehr/berichte/1254_Radverkehrserh ebungen-Bericht-August-2009.pdf, (June 30, 2011) 17. Girod, B.: Eigenschaften des Fahrradverkehrs. Analyse des Fahrradverkehrs aufgrund der Mikrozensus 2000 und Thurgau 2003 Verkehrsbefragungen. Institut für Verkehrsplanung und Transportsysteme, Eidgenössische Technische Hochschule Zürich, Zürich (2005) 18. Brandenburg, C., Matzarakis, A., Arnberger, A.: Weather and cycling—a first approach to the effects of weather conditions on cycling. Met. Apps 14(1), 61–67 (2007) 19. Haustein, S., Hunecke, M., Manz, W.: Verkehrsmittelnutzung unter Einfluss von Wetterlage und -empfindlichkeit. Internationales Verkehrswesen 59(9), 392–396 (2007) 20. Nankervis, M.: The effect of weather and climate on bicycle commuting. Transportation Research Part A: Policy and Practice 33(6), 417–431 (1999) 21. Backhaus, K.: Multivariate Analysemethoden. Eine anwendungsorientierte Einführung, mit ...6 Tabellen, 11th edn. Springer, Berlin (2006) 22. Schlittgen, R.: Angewandte Zeitreihenanalyse. Oldenbourg, München (2001) 23. Slonim, N., Friedman, N., Tishby, N.: Unsupervised document classification using sequential information maximization. In: Proceedings of the 25th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, pp. 129–136. ACM Press, New York (2002) 24. Berkin, P.: A Survey of Clustering Data Mining Techniques. In: Kogan, J., Nicholas, C., Teboulle, M. (eds.) Grouping Multidimensional Data. Recent Advances in Clustering, pp. 25–71. Springer, Heidelberg (2006) 25. Jain, A.K., Dubes, R.C.: Algorithms for clustering data. Prentice Hall Advanced Reference Series. Prentice-Hall, Englewood Cliffs (1988) 26. Abonyi, J., Feil, B.: Cluster Analysis for Data Mining and System Identification. Birkhäuser Verlag AG, Basel (2007) 27. Vienna Statistical Yearbook (2009), http://www.wien.gv.at/statistik/pdf/bezirksportraets09.pdf, (April 4, 2011)

Economic Impacts of the Alternative Reuse of Empty ISO Containers Peter Großkurth1, Robert Stahlbock2,3 , and Stefan Voß2 1

Student at the University of Hamburg and Hamburg University of Applied Sciences, Hamburg, Germany [email protected] 2 Institute of Information Systems, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany [email protected], [email protected] 3 FOM University of Applied Sciences, Essen/Hamburg, Germany

Abstract. With hundreds of thousands of ISO containers rejected from the world’s container ﬂeet every year due to age or empty container accumulation problems, the question of their potential reuse for other purposes arises. Moreover, beside the ecological impacts the economic impacts of the diﬀerent reuses are of great interest as such recycling activities may provide cost and energy saving opportunities. Next to providing some background information about containers and empty container management, this paper investigates the scale of containers that enter the secondary market, diﬀerent ways in which they can be put to new use and several concepts of economic impacts resulting from the repurposing activities. Furthermore, it suggests promising research opportunities in particular with respect to the use of information systems and modern communication and information technology (IT) for planning and control of container ﬂows and alternative reuse of containers.

1

Introduction

Besides its primary function as transportation equipment, an ISO shipping container has considerable potential for other purposes. Built to withstand heavy loads during the transport on land and sea, a container is a valuable object in regard to its structural design and the material it is composed of. This suggests its reuse for other, alternative purposes than the employment in the cycle of transportation, a matter which gains importance when facing some of the issues of empty container management. The questions arise as to which other functions empty ISO containers can be reused for and which ecological as well as economic impacts alternative reuses result in. This paper attempts to investigate diﬀerent ways in which empty containers can be repurposed and their economic impacts. For this purpose the remainder of the paper is structured as follows. Section 2 provides some important background information, by stating some facts about the importance and prominence of containers in the world’s transportation J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 142–159, 2011. c Springer-Verlag Berlin Heidelberg 2011

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network and about technical aspects of ISO containers. Section 3 is focused on the issue of reusing containers. It describes the ﬁeld of empty container management and highlights the processes which lead to containers leaving the transportation network. In Section 4, diﬀerent ways of reusing containers are presented, along with some examples. Furthermore, research opportunities within the ﬁeld of computational logistics are brieﬂy proposed. Diﬀerent concepts of the economic impacts of reusing empty containers are analyzed in Section 5. Finally, Section 6 concludes the paper.

2

ISO Containers

Since the ﬁrst successful introduction of an organized employment of containers for the transportation of goods by American entrepreneur Malcom McLean in the 1950s, containerization has changed the world. Following the standardization process in America and Europe in the 1960s, global transportation infrastructure adopted the container as its key element for the intermodal transportation of general cargo as it leads to a safer, quicker and cheaper ﬂow of goods. Ports and ships were built to handle containers, transportation geography changed with some ports replacing others as regional transportation hubs, many goods became available globally and production structures shifted towards horizontal integration [17]. Nowadays, worldwide more than 90 % of all general cargo is transported in shipping containers (see, e. g., [14, p. 3] or [34]). Figure 1 depicts global trade ﬁgures indicating that especially container freight is assumed to

Fig. 1. World GDP, world trade, seaborne trade, and container turnover – Data retrieved from [3,32,33,34,36,38,39]

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considerably increase over time. The global container ﬂeet exceeds 25 million TEU (twenty-foot-equivalent-unit), as is shown in Figure 2. Even though the ﬂeet capacity shrunk due to the global recession in 2008/2009, the ﬂeet is expected to reach the size of 30 million TEU by 2012 [37]. As will be discussed later, the size of the ﬂeet leads to a considerable amount of containers being sorted out of the transportation network each year, making the issue of their reuse an important matter.

Fig. 2. Global container ﬂeet 1990-2009 – Data retrieved from [37]

There are diﬀerent types of ISO containers. Besides the standard dry container, which appears most commonly in the lengths of 20 and 40 feet, there are containers for diﬀerent freight and loading types. Among these are open tops, hard tops, bulk containers, tank containers, ﬂat racks and refrigerated containers (reefers). As diﬀerent as these container types may appear, all share the ability to be transported in the intermodal system. However, with a share of over 84 % of all ISO containers [2], the standard containers are most widely used and therefore also play the essential part when it comes to discussing their reuse. There are several physical properties which prompt such a reuse. Yet also other types of containers which share the essential physical properties of standard containers can be reused for other purposes. Firstly, most containers are made of weatherproof steel, often referred to as Corten steel, which, however, is simply a brand name. Compared to other steel weatherproof steel is not overly expensive while maintaining good machining properties. Its major advantage lies in the fact that after initial corrosion, a

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patina is created through alloying elements in the steel which under most conditions protects it against further corrosion [8]. This property does not only beneﬁt the transport of containers on land and at sea. It is also advantageous for the possible reuse of a container. Secondly, the 13.67 m2 surface area within a 20 foot container compares to that of a small room, while its height of 2.33 m allows a human to move uprightly inside of it. As trivial as this might seem, it enables a container to be used as an actual room, thereby multiplying its possible functions. A third feature that also suggests an advantage to the reuse of a container are its structural qualities. 20 foot and 40 foot containers weigh 2.3 and 4 tons with payloads of 21.7 and 26.5 tons, respectively. With the requirement of being able to carry at least six other fully loaded ones, it is apparent that a container is a very strong and robust object [10].

3

Empty Container Management

The issue of the reuse of empty containers is attributed to the ﬁeld of empty container management. Empty container management deals with diﬀerent geographical levels and major players involved with the question of what to do with empty containers within the intermodal transportation network. Its main focus is to identify the problems and consequences of the movement of empty containers and to work on strategies, optimization methods and possible player decisions, with the aim to ﬁnd solutions for the problems [1]. The key players are the owners of containers, namely ocean carriers and the container leasing industry. In 2009, ocean carriers owned 59 % and leasing companies 41 % of the global container ﬂeet (see, e. g., [31, p. 56] and [34]). Yet other parties, such as container producers, depot owners, ports, shippers, politicians and even the general population are involved in the matter. On the global level, the problems with empty containers are caused mainly by the fundamental and chronic trade imbalances between regions of the world, such as North America and Asia [31, p. 53]. In some regions a high demand for empty containers arise, while other regions hold a major surplus of them. On one hand, this may lead to vast and costly repositioning activities for empties, while on the other hand it may lead to large accumulations of empties in speciﬁc regions. Generally, an increase in volume leads to even larger volume imbalances, i. e., large export ﬂows from Asia to North America and Europe with smaller import ﬂows to Asia (see Figure 3). The full scope of empty container management is too complex and detailed to be reproduced in this paper (see, e. g., [12,18,30]), yet some numbers point to its signiﬁcance in the world of transportation: it is estimated that the global share of the movements of empties exceeds 20 % of all container movements on sea [22, p. 10] and 40 % on land (estimation by Drewry Shipping Consultants, cited in [15, p. 224]), with annual costs of 17 to 20 billion US$ [31, p. 54].

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Fig. 3. Traﬃc of main global containerized cargo routes 2009 (million TEU) – Data retrieved from [34, p.19]

While most aspects of empty container management deal with containers that remain within the transportation network, the issue of containers which leave it is also of importance. There are two reasons for the possible rejection of containers from the cycle of transport. One is that the formerly mentioned trade imbalances and lack of repositioning activities can lead to large accumulations of empty containers. As will be discussed, such accumulations can cause a number of problems, with the only solution lying in the removal of the accumulation. This can lead to containers being sold out of the network [1]. The second reason lies in the fact that every container has a lifespan in the transportation cycle, usually 12 years, after which it will be sorted out of the ﬂeet and sold [2, p. 6]. Leasing companies usually sort out containers after conducting an investment calculation, where the sales proceeds of a container exceed its future revenue minus the cost of repair. Annually, about 5 % of the global container ﬂeet are replaced by new containers. Since 2004 more than one million TEU are rejected from the transportation network every year, in each of the years 2007, 2008 and 2009 the ﬁgure was 1.35 million TEU [37]. With such ﬁgures it becomes obvious that the availability and supply of containers for other purposes is rather an enduring, growing and, therefore, signiﬁcant matter than a singular or sporadic observation.

4

Alternative Reuse of Empty Containers

While optimization and information systems have been considerably used to improve some aspects of empty container management quite a few issues seem to have been neglected. Applications of modern decision support systems aim

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at, e. g., forecasting supply and demand of containers in particular areas or regions (i. e., forecasting surplus and deﬁcit and, therefore, the need of container repositioning activities) or improving the utilization of equipment. However, it would be helpful to further investigate to which extent a technology such as RFID (Radio Frequency Identiﬁcation) or GPS (Global Positioning System) can support decisions in the ﬁeld of alternative reuse of empty containers. This may also apply to other, more simple methods of tracking containers, such as capturing interface data. For example, it is expected that improvements in tracking of containers result in a better visibility of containers as well as of their status and condition. More accurate data on containers’ availability result in an improved database for planning purposes, better planning results and decisions, and ﬁnally cost savings. Container data can be fed into an information system, and the system can be used to plan time and location of a container’s exit of the transportation network and its alternative reuse. The identiﬁcation of requirements for successful IT applications (data availability, data quality, data ﬂow, etc.) in the area of empty container management, in particular considering alternative reuse of empty containers, is a fruitful research topic within the ﬁeld of computational logistics. However, this paper is not focused on those concrete computational aspects, but it is intended for providing an overview on economic impacts of alternative container reuse as a basis for fruitful research in that area in the future. With many diﬀerent ways in which empty containers can be put to a new purpose, this paper focuses on the following three, which appear to be the most common ones and the ones with the largest possible economic impact. 4.1

One-Way Transport Equipment

One possibility in which shipping containers can be temporarily reused is to employ them as one-way transport equipment to areas with high cost of repositioning. This use appears to be a step in the existence of many containers that were sorted out due to their age [2, p. 6]. After these containers were rejected from the ﬂeets of ocean carriers and leasing ﬁrms, they are sold to buyers wishing to ship goods to areas in which the repositioning of containers would be very eﬀortful and expensive. Such are areas which are far away from ports or which have poor transportation infrastructure, this being the case for many developing countries (see, e. g., [26]). Upon arrival in their destination, after serving as transport equipment for a ﬁnal time, these containers are then intended to ﬁnd another, more permanent function. This use of an old container for a one-way transport qualiﬁes as an alternative reuse as there is a clear diﬀerence to the common use of a container as a permanent transport equipment which would require repositioning after its unloading. This reuse and its resulting economic impacts may particularly beneﬁt from detailed research in the ﬁeld of computational logistics.

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Unmodiﬁed Reuse

The easiest way to give a sorted-out container a new and permanent purpose is to use it without any changes or modiﬁcations. Using them as storage facilities plays an essential role in this concept [2, p. 6], as the features of a lockable and weatherproof space can be fully utilized without need for structural alteration. The dimension in which containers are reused for storage ranges from the employment of single containers, as are often seen on the yards of companies or building sites, to a professional scale where large numbers of stationary containers are rented out to customers in the need of storage space. Other unmodiﬁed uses may include the creation of barriers on a compound, without the loss of the storage ability, or the application in art sculptures and installations, such as in the work of Belgian artist L. Deleu [27, p. 134]. 4.3

Reuse as Building Material

The most impressive alternative reuse for used containers is their application in architecture. Container architecture has got a lot of recognition in recent years and also includes the use of pre-made containers and container-like modules [27, p. 20], which, however, are of no concern here. Before the numerous ways in which sorted-out containers can be applied in buildings and the various functions of the created buildings are discussed it is important to explore the actual suitability of containers as building materials. Generally, the reuse as building material is subject to the condition of the containers. Especially containers which are sorted out due to their age are likely to show signs of damage. Yet remarkably, only a very small share of containers (about 1 in 1,000) is immediately scrapped for steal due to heavily damaged conditions. Many damages appear to be of a minor nature and can easily be taken care of in a potential reuse as building material. For example, perforated wall sections can either be replaced by simple welding procedures or used strategically to create openings for doors or windows.[28]. The formerly mentioned strong structure of containers often exceeds the structural requirements for building materials [16, p. 24]. Yet if alterations are made to the structure, such as the removal of wall and roof segments or even complete sides, these requirements may not be met anymore. However, by reinforcing the structure with adequate components, the structural suitability can be restored [27, p. 47]. An essential part is played by the heat insulation, especially as containers act as strong heat conductors through their metal structure. Insulation can be applied on the inside and outside, with inner insulation taking away some of the already scarce ceiling height, while outer insulation requires extra weatherprooﬁng at higher expense [28, p. 89]. Other building factors include sound insulation, ventilation, electrical grounding and the type of fundament, which is

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often simpler for container structures than for traditional building types. In total, containers have to meet the same building requirements as any other material, so that their applicability is solely limited by the building designer [28, p. 75]. The easiest way to use a container in architecture is to use the actual container as a building, where only the formerly mentioned requirements have to be met. Yet there are more sophisticated ways to employ containers in buildings. One is to use containers as modules in composite buildings where the containers may also act as load-carrying elements, besides their purpose to create enclosed space. The sole function of a load carrying structure is also possible [27, p. 16]. The manner of creating enclosed space with containers is also very variable. In some buildings the employed containers are hardly altered and merely stacked on top of each other, while in others they can be heavily modiﬁed, intricately combined and innovatively set together to create sophisticated ﬂoor layouts and building structures. In many buildings, containers are also combined with other building materials. The functions of the container buildings in developed countries are as numerous as traditional buildings. A categorization by [27, p. 11] distinguishes between public, domestic and commercial buildings. Among others, public container buildings include educational buildings such as schools and nursery schools and cultural buildings such as museums, galleries and event-locations. Many public container buildings are temporary structures. Domestic container buildings include many private houses in various shapes and sizes and numerous mobile buildings, which are often collapsible and only made of one or few containers. Dormitory projects, many of a charitable nature, are also often realized, as are domestic auxiliary buildings such as garages, saunas or pool-houses. Commercial container buildings are predominantly oﬃce and retail structures, yet there are also numerous examples of container caf´es, restaurants and bars. While container architecture seems to be attributed to developed countries, especially due to its often modern and innovative character, container buildings also play a role in developing countries. In fact they can frequently be found, especially in port-remote settings, where repositioning costs are high, as was pointed out. While most container structures in developing countries are of a basic appearance without sophisticated modiﬁcations, an innovative and creative approach to the buildings can often be witnessed. This may be due to less strict building requirements in developing countries or lack of their enforcing. This results in containers being used in all kinds of structures, including production facilities where they may serve multiple purposes. Also, container structures often ﬁnd widespread use in development aid projects. Figure 4 and Figure 5 show examples of public, domestic and commercial structures in both developed and developing countries.

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(a) Nomadic Museum, New York, USA; [4]

(b) Beach House Redondo, USA; [7]

(c) Freitag Store, Zurich, Switzerland; [23] Fig. 4. Examples of container structures in developed countries

Economic Impacts of the Alternative Reuse of Empty ISO Containers

(a) Oﬃce, Mwanza, Tanzania; Photo by P. Großkurth

(b) Room in a Shipyard, Mwanza, Tanzania; Photo by P. Großkurth

(c) Restaurant, Mwanza, Tanzania; Photo by P. Großkurth Fig. 5. Examples of container structures in developing countries

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Economic Impacts

One can identify various impacts of the alternative reuse of rejected containers, which are eclectic with regard to the respective group of interest. Yet it has to be said that predominantly the impacts can be described and analyzed on a conceptual level rather than a monetary level. This is due to the complexity of the subject, as well as the lack of suitable monetary data. 5.1

Impacts in Empty Container Management

As the question of container reuse is attributed to empty container management, it is reasonable to ﬁrst investigate the impacts in this ﬁeld. Looking at containers rejected due to their age, the fact that a used container is a reusable object still possessing worth makes the sale of such a container an important action alternative for the players involved. An asset which is eﬀectively depreciated still generates a monetary value, which at closer look is of a considerable size as the sales prices for used containers are relatively high and stable when comparing them with prizes for new containers. For example, in 2008 a new 20 foot container cost 2.200 e while a 12 year old 20 foot container sold for 1.100 e [2]. With the price of an old container being half the price of a new container, the monetary depreciation of the asset is just 50 % over 12 years and upon sale much of the initial purchase price can be regained. From an investment point of view this is very favorable as high liquidation proceeds result in a higher net present value of a capital asset. The economic advantage of the sale is also apparent when comparison with another alternative is made: scrapping for steal. While this is in principle an ecological and economical positive process, with almost half of today’s steel production consisting of recycled material [29], the proceeds generated in the sale of an old container to scrap yards are signiﬁcantly less than the sale into the secondary market [1]. When looking at empty container accumulations in areas of high surplus of empty containers, the decision to sell containers to the secondary market is the only alternative which does not result in high costs, as opposed to repositioning or long-term storage. This alternative can only be taken into account when the demand for empties in other regions can be satisﬁed without containers from the speciﬁc surplus region, which can be a dynamic problem depending on the market situation. The temporary storage of some empty containers is unavoidable and necessary in order to react to ﬂuctuating demand levels [1, p. 13], but from an economical perspective the permanent storage of a large number of containers is very disadvantageous as the empty containers represent unutilized assets. Especially in times of low prices for new containers (e. g., in 2001 and 2002) the habit evolves that empties pile up in places after just one or few trips to avoid repositioning costs while new containers are purchased for the transportation before adding to the pile-up [21]. For example, in the early years of the last decade, the prices for newly produced containers were rather low, ranging around US$ 1,300 for a 20 foot container, with costs of repositioning at a comparative scale, e. g., it

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cost leasing ﬁrms US$ 1,200 to reposition an empty container from the US-East Coast to Asia. While such settings prompt the accumulation of empty containers in some areas, an increase in container prices may on the other hand lead to high repositioning activities. This was, e. g., the case in 2004 and the following years, when higher prices for steel resulted in costs of new containers of over US$ 2,000 and repositioning became the economically more reasonable option (see, e. g., [1, p. 60], [31, p. 57]). On short term the lack of repositioning may be advantageous for the players, yet through the lack of utilization there is an enormous waste of material and building costs [6]. The employment of these unused containers for other purposes presents an economically better solution, as compared to disuse. Empty container accumulation can be regarded as a waste of capacities, but in addition several other problems are caused by empty container accumulation. Storage of containers in port regions takes up a lot of highly valued land, which could otherwise be used for better purposes [1, p. 3]. Piled up containers in port areas can interfere with the ports productivity. High stacks of empty containers present a safety hazard to both humans and assets, as well as being an aesthetical impairment and causing noise pollution [1, p. 64]. In particular, these eﬀects are negative when the accumulation is close to a residential area, even resulting in signiﬁcant depreciation of real estate (see, e. g., [19]). The listed problems and their economic consequences can only be solved by reducing the number of stored containers; therefore, a container sold out of the network makes a contribution to dealing with these problems. Another important economic impact in empty container management results from the reuse of rejected containers as one-way transportation equipment. Especially many developing countries have areas with high repositioning costs, due to a number of factors. Usually developing countries have passive trade balances, resulting in few goods to be shipped back and a high number of empty transports. Transportation infrastructure is inferior in many places. Railroad networks are underdeveloped and inland water transportation is nonexistent in many places, making trucks the primary means of transportation. With many roads in bad conditions and corruption adding to the costs, inland transportation is burdensome and very expensive. For example, the costs for the transport of a 20 foot container from Mombasa to Nairobi in Kenya in 2010 were 9.16 US$/km [26]. Comparing this with costs of moving a container, e. g., from Germany to China via train at 0.4 e/km [24] reveals the enormous diﬀerence in cost levels. Therefore, the fact that a one-way container does not have to be repositioned from such an area with high repositioning costs leads to substantial savings.

5.2

Impacts through Use as Building Material

While some people ﬁnd container buildings not appealing, others are convinced that they have become an important part of architecture. An assessment of container buildings based on aesthetic perception is not possible, yet an economic assessment is when comparison with traditional build types is made.

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Used containers already provide the body shell of a building, costing 60 e/m3 [27, p. 9]. Traditional building shell costs ﬂuctuate between time and region, ranging from 101 to 127 e/m3 in Germany in recent years [25,13]. It is apparent that the cost level for container body shells is well beneath that of traditional ones. The same pattern can be observed regarding the total building costs: Californian container architect P. DeMaria states his container buildings at US$ 135 per square foot are 35 % less expensive than traditional buildings in his region [5], which is even exceeded by container architect A. Kalkin, whose container houses at US$ 73 to US$ 90 per square foot cost less than half of the prices of houses in his region in the Northwest of the United States [11]. With only £ 40 per square foot, Urban Space Management’s container buildings in London cost a third of the £ 120 per square foot of traditional buildings [35]. In total, the cost savings of using containers as building material appear to be signiﬁcant. But building with containers does not only result in cost savings but also in savings of building time. Firstly, container structures usually do not require any excavation [16, p. 14] as fundaments are mostly made of simple concrete panels [27, p. 16]. This does not only lessen the environmental stress of the construction but also saves a lot of time. Secondly, time is saved as the body shell already exists through the pre-existing structure of the containers. This leads to further time savings as the containers can undergo any structural changes in advance of their assembly into a composite building made up building modules. Furthermore, the container modules typically do not loose their ability to be transported in an intermodal way, guaranteeing a quick and cheap transport to the building site. According to architect David Cross, container structures may have a 40 % quicker building time than traditional construction [9]. Economically speaking, a quick building time has several advantages. The sooner a building is completed, the sooner it can be put to use and start beneﬁting people. Building capacities are bound for a shorter time, resulting in cost savings, while disturbances and pollution from the building sites are reduced [16]. Generally, short planning and implementation times lead to reduction of investment costs [27, p. 18]. Positive economic as well as positive environmental impacts are also gained through the fact that many container buildings are collapsible and can be moved from one site to another without leaving any building residues. When such a building is not needed in one location anymore it does not have to be forfeited or demolished at high cost, thereby also not resulting in any pollution. The space it stood on can be put to a new use without the cost of clearing up. Often enough, the intermodal system can be used for the transportation of the building. At the new site, no high costs or material expenses have to be made, as the building already exists. The strong structure of containers can cause this cycle to be repeated often. A building structure with individual container components also allows for ﬂexibility during construction, making short-notice reactions to changing building requirements possible [16, p. 14]. Compared to the scrapping for steel, the reuse of containers for buildings is very energy saving and thereby also cost saving.

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Creating new steel from a 40 foot container requires around 8,000 kilowatt-hours of energy, while using it in a building structure requires only 400 kilowatt-hours, which is 5 % [11]. The reduced energy consumption results in monetary savings but also in a smaller environmental strain. The formerly described weatherproof steel which most containers are made of allows for reduced maintenance eﬀorts and costs, while not straining the environment with corrosion. Overall, it can be concluded that using containers as building material has several environmental advantages and results in high ﬂexibility. Despite all the listed positive aspects of container buildings, a few disadvantages have to be mentioned, which can also be of an economic nature. The enclosed space provided by containers sets limits to the realizable sizes of rooms in a building. Heat insulation may yet further reduce the available space, thin insulation comes at greater expense. In order to build large rooms, signiﬁcant structural changes have to be made to the container elements. Container buildings lacking good heat insulation can only be used temporarily, such as in event architecture [27, p. 78]. From an energetic point of view, containers have thermal bridges in their corners, which does not allow for energy-eﬃcient buildings [20]. This is very disadvantageous particularly in times of rising energy costs. Stacking containers leads to unfortunate doublings of container walls. In combination with the often perceived lack of esthetical appearance, these negative factors cause many people to dislike container architecture. 5.3

Impacts through Image Eﬀect

Some conceptual economic impacts can be identiﬁed which are based on the appearance of repurposed containers as environmentally responsible items of recycling, with their image as icons of globalization and transportation also playing a role. In recent years, environmental awareness has become an integral part in the mentality of many people. This can be contributed to the thread of global warming, as increasing demands for energy whilst facing the ebbing of fossil fuels. With this background, technical innovations and systems which are gentle in the consumption of energy and resources and thereby economically and environmentally eﬃcient gain increasing recognition and approval within large parts of the world population. Using old and rejected containers for other purposes represents such a system. As described, the prolonged use of containers in building projects preserves energy and other materials, while at the same time reducing its carbon footprint. Also, the interior of container buildings is often designed with environmentfriendly materials [16, p. 15]. Buildings made of reused containers therefore contribute to the protection of resources and the saving of energy. Existing container buildings also act as role models, as with the increasing media attention for container architecture and the rising awareness of its advantages, this contribution may even grow in the future.

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The images of used containers and container buildings can also be used as tools of marketing. Besides the described ecological image, an used container has an image of mobility and globalization, having traveled thousands of miles during its life cycle [27, p. 10]. Companies can utilize these images in order to create positive associations with their products or philosophies. This works well, as container buildings capture the attention of viewers through their unusual appearances, which are often cognitively surprising and physically intensive. The resulting emotional link that customers create between the images of the container buildings and brands or products is beneﬁciary for the company. An example can be seen with the Freitag Flagship Store in Zurich, Switzerland, shown in Figure 4(c). The Freitag company sells bags made from recycled material and utilizes the image of old containers to communicate its environmental philosophy. 5.4

Impacts in Developing Countries

The repurposing of used containers has to be viewed diﬀerently for developing countries. With many such countries containing areas with high cost of repositioning empties there is a considerable amount of former one-way containers available. Due to the high demand for building material, containers usually ﬁnd a new purpose quickly. It can be witnessed that the use of containers as buildings appears very natural and many-sided, as they provide good alternatives for many purposes and some building obstacles which exist in the developed world do not need to be overcome. Such are less strict building laws and a reduced need for insulation due to the often favorable climates. For many companies containers are essential facilities. But they also possess importance as simple housing means for many people. In comparison to local building materials containers can be of a superior nature with regard to quality, although usually being more expensive. As developing countries are often situated in regions of natural disasters, containers can play an important role in the aftermath of a catastrophe, serving as emergency housing. This purpose attributes them with important economical and humanitarian features. Due to their availability, rather low price and good transportability containers are often utilized by organizations of development aid, which ﬁnd many purposes for them, especially in remote areas.

6

Conclusion

This paper identiﬁed several approaches and concepts of economic impacts resulting from the alternative reuse of empty ISO containers. The impacts can be viewed from two perspectives which also take into account the diﬀerent groups of interest. On one hand, the indirect perspective looks at the impacts following the rejection of containers from the cycle of transportation due to age or empty container accumulation, where a role is played by the fact that a potential reuse prompts the rejection. On the other hand, the direct perspective regards the impacts following the actual repurposing of the containers. Together, the sorting-out and reuse form a sequence of steps in the life cycle of a container

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and it is remarkable that the impacts of these steps are in principle of a positive nature, regardless of perspective, as high cost savings are involved. Also from an environmental point of view the impacts are thoroughly positive. In this paper, most of the impacts could only be described conceptually, due to the lack of information and in particular of monetary data allowing for a more detailed analysis. The little data at hand, e. g., for the cost and time saving potential of container buildings, suggest that closer investigation of the subject will reveal results of a positive nature. Therefore, more detailed information would not only allow a deeper insight into the matter but would likely lead to an intensiﬁcation of the impacts, as positive information would further advertise the alternative reuse of containers. Advanced information about the cost saving potential in the reuse of containers as one-way transports in areas of high repositioning costs would aid empty container management through a more target oriented application, further reducing the empties movements and related costs. In the same way, further information about the economic and humanitarian value of the reuse of containers in developing countries could lead to a systematic allocation as a form of development aid. In conclusion, a profound scientiﬁc investigation of the economic impacts of the alternative reuse of empty ISO containers seems fruitful. Furthermore, detailed research in the ﬁeld of computational logistics is promising with respect to improvements of container reuse possibilities and processes and their positive economic impacts.

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25. Oberste Bauaufsichtsbeh¨ orde Saarland: Bekanntmachung der Obersten Bauaufsicht u ¨ ber die Festsetzung der durchschnittlichen Rohbauraummeterpreise (April 14, 2008), http://www.vpi-saarland.de/download/saar_rohbaukosten_2008.pdf 26. Rodrigue, J.P., Comtois, C., Slack, B.: The Geography of Transport Systems. Routledge, New York (2009) 27. Slawik, H., Bergmann, J., Buchmeier, M., Tinney, S. (eds.): Container Atlas – Handbuch der Container Architektur. Die Gestalten Verlag, Berlin (2010) 28. Smith, J.D.: Shipping containers as building components. Ph.D. thesis, Department of the Built Environment, University Brighton, UK (2006), http://www.cityzen.biz/containerresearch.pdf 29. Stahl-Informations-Zentrum im Stahl-Zentrum, D¨ usseldorf: Stahl-Recycling (2010), http://www.stahl-info.de/Stahl-Recycling/stahlrecycling.asp 30. Stahlbock, R., Voß, S.: Improving empty container logistics – can it avoid a collapse in container transportation? In: Kroon, L., Li, T., Zuidwijk, R. (eds.) Liber Amicorum in Memoriam Jo van Nunen, pp. 217–224. DINALOG Dutch Institute for Advanced Logistics, Rotterdam School of Management, Erasmus University, Rotterdam (2010) 31. Theofanis, S., Boile, M.: Empty marine container logistics: facts, issues and management strategies. Geo Journal 74(1), 51–65 (2009) 32. United Nations Conference on Trade and Development (UNCTAD) – Secretariat: Review of maritime transport (2008), http://www.unctad.org/Templates/Download.asp?docid=9248&lang=1 &intItemID=4398 33. United Nations Conference on Trade and Development (UNCTAD) – Secretariat: Review of maritime transport (2009), http://www.unctad.org/Templates/Download.asp?docid=12455& lang=1&intItemID=5248 34. United Nations Conference on Trade and Development (UNCTAD) – Secretariat: Review of maritime transport (2010), http://www.unctad.org/Templates/Download.asp?docid=14218& lang=1&intItemID=5746 35. Vidal, J.: Boxing clever, (January 14, 2004), http://www.guardian.co.uk/society/2004/jan/14/environment.environment 36. Volk, B.: Growth factors in container shipping (2002), http://maritimebusiness.amc.edu.au/papers/AMC3_GRO.pdf 37. World Shipping Council: Global container ﬂeet (2010), http://www.worldshipping.org/about-the-industry/ containers/global-container-fleet; (data based upon Containerisation International, Market Analysis,World Container Census 2010) 38. World Trade Organization: World Trade Report 2009. World Trade Organization (2009), http://www.wto.org/english/res_e/booksp_e/anrep_e/ world_trade_report09_e.pdf 39. World Trade Organization: World Trade Report 2010. World Trade Organization (2010), http://www.wto.org/english/res_e/booksp_e/anrep_e/ world_trade_report10_e.pdf

Optimal Issuing of Perishables with a Short Fixed Shelf Life Ren´e Haijema Operations Research and Logistics group, Wageningen University, The Netherlands

Abstract. The management of inventories of perishable products with a short maximal shelf life takes a good issuing policy next to a good ordering policy. Ordering policies of non-perishables are well studied in literature and implemented in Automated Store Ordering (ASO) systems and Computer Assisted Ordering (CAO) systems. These ordering policies are stock-level dependent and do not take the product ages into account. As a consequence when applied to perishable products they do not anticipate expected future outdating of products and thus result in unnecessary outdating and shortages. Improved ordering policies are proposed in the literature but hardly implemented in ASO and CAO systems as these are designed for the management of non-perishables. Without changing the order policies of such systems one may reduce outdating and shortages by issuing the products in a sophisticated way. We present therefore a stochastic model that is solved by stochastic dynamic programming.

1

Introduction

In the last (two) decades Automated Store Ordering (ASO) and Computer Assisted Ordering (CAO) systems have become more and more in use to improve the eﬃciency and eﬀectiveness of the inventory management at retailers [9,11]. Using point of sales data the stock levels in the store are updated automatically, and either an order is placed automatically by an ASO system or an order quantity is proposed by a CAO system. In the latter case the order is to be processed further by a decision maker who is responsible for the ordering process. ASO systems are mainly found by supermarkets and retailers that oﬀer a great variety of products. In settings with a smaller product line a CAO system can be used. Mechanisms to set a (proposed) order quantity are ordering policies that take as input the actual stock level (as tracked by the system) and any information on future demand (based on historical information on demand realizations and predictions based on promotions). In case stock replenishment happens daily often traditional ordering policies like (R, S) and (R, nQ) policy are considered. As most ASO and CAO systems are designed for the inventory management of non-perishables, these systems do not acknowledge the aging of products in stock and do not record the outdating of products. In recent work like [11,4,2,1,3] new J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 160–169, 2011. c Springer-Verlag Berlin Heidelberg 2011

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heuristics and ordering policies are proposed to improve the inventory management of perishables through such systems. So far most ASO and CAO system still rely on stock level dependent ordering policies that result in the unnecessary outdating of perishable products in stores. In earlier work we focussed on ordering policies for blood banks (see [10] and previous papers [5]) and have developed a CAO decision support tool. In this paper we focus on the issuing policy at hospitals: blood products are issued internally to diﬀerent departments (e.g. Oncology, Hematology, General Surgery). Compared to blood banks hospitals tend to face much higher percentages of shortage and outdating as they operate at a much smaller scale and as such do not beneﬁt from economies of scale. We show that hospitals can operate more effectively and eﬃciently by carefully selecting which products to issue for meeting the demand (without changing any ASO or CAO system in use). This paper aims to ﬁll part of the literature gap reported in the overview by Kareasmen and colleagues in [6] and [7]. Since one of the ﬁrst theses on perishable inventory management ([13] and [12]) the controlled issuance of perishable products is understudied while there is a great potential for savings. In current literature one either assumes that products are taken from stock according to a simple rule like FIFO or LIFO or some mixture, or according to the customer’s preference for a product of a speciﬁc age. Often the manager of a stock point has some way to control this process: e.g. by a keeping the forward and reserve inventories separated, or by having control of issuance themselves (like at blood banks and hospitals). This paper seems to be one of the ﬁrst papers that present a method to numerically compute an optimal issuing policy. Outline. In the next section we present the practical problem of inventory management of blood platelet concentrates at hospitals. In Section 3 the issuing problem is formulated as a Markov decision problem (MDP) that can be solved by Stochastic Dynamic Programming (SDP). Simulation results for the optimal issuing policy and alternative issuing policies are reported in Section 4. Finally in Section 5 follows a discussion and a conclusion.

2

Issuing BPCs at a Hospital

Blood platelets are of live saving importance. As small particles in the bloodstream platelets repair damaged blood vessels. Platelets deteriorate rapidly in quality even inside the blood stream, but most people’s platelets production at the bone marrow is suﬃcient to retain a safe level. However, after a major bleeding caused by a trauma or a surgery, a patient may temporarily have a lack of platelets. This lack needs to be resolved by a transfusion of donated blood platelets to boost the patients platelet level. Patients who get radiated or who receive chemotherapy suﬀer from a platelet function disorder and lack a suﬃcient production of blood platelets at the bone marrow. These patient need therefore be transfused regularly with platelets. The demand for platelets on a particular day is (highly) uncertain. Therefore blood banks and hospitals keep these life saving products in stock.

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In [4], [2] and [10] the relevant logistical processes involved with making blood platelet concentrates (BPCs) are discussed. Here we brieﬂy revisit this discussion and focus on the processes relevant for the ordering and issuing of BPCs at hospitals. Hospitals order BPCs at a blood bank in principal once a day. An order placed at the end of the day or early in the morning, say before 8am arrives within a few hours, say at the latest at noon. In case of a shortage an emergency delivery is arranged by the blood bank, which supplies about 30 hospitals. Many Dutch hospitals are relatively small or medium sized in terms of BPC usage and to keep outdating at an acceptable level stock levels are kept low. In practice it appears that the emergency deliveries happen quite frequent. When the people of the transport and distribution department are busy, the transports from a blood bank to a hospital is outsourced to certiﬁed express service providers and eventually by certiﬁed cabs. A reduction of shortages is desired as it reduces the hassle of arranging emergency transport, save money, and reduces the stress involved with shortages as lives may be at risk. As the demand for platelets is virtually stationary (but maybe periodic) hospitals may apply an order-up-to S rule with ﬁxed levels for each weekday (S1 , ..., S5 , S6 = 0, S7 = 0). S6 and S7 are zero as during the weekends blood banks accept no regular orders from the hospitals, since the production capacity is low as is the number of voluntary donors. To balance shortages and outdating at hospitals a clever issuing rule can be constructed by formulating the issuing of platelets as a Markov decision problem (MDP). The pharmacy of a hospital issues platelets throughout the day upon request by medicines according to their operating schedule and by emergency operations. To simplify the discussion of the MDP model we split the day into two parts as if issuing happens only at 8am and at 12am. At 8am hospitals assign and distribute the required number of BPCs to the diﬀerent departments (Hematology, Oncology, General Surgery) at which BPCs transfusions happen. This process is repeated at 12am, when demanded volumes for the afternoon are known to the inventory manager at the hospitals pharmacy. For practical use one may consider more than two issuing epochs. The solution of the MDP model described in the next section prescribes how many products from each age category to issue to meet the demand.

3

MDP Model

The problem of issuing BPCs is formulated as a discrete time MDP. The MDP has 2 decision epochs per day (8am and noon), which we label e =am and e =pm. The MDP is periodic with period 14 (=7 days), as regular transhipment of BPCs from a blood bank to the hospital are not scheduled for the weekends. As we explain below both the state and the action spaces are discrete and ﬁnite, hence numerical computation of an optimal issuing policy is possible. 3.1

State

At the start of an epoch e ∈ {am, pm} on a weekday d ∈ {1, ..., 7} ={Monday, ..., Sunday}, the total demand k till the next epoch is known as well as the number

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of BPCs in stock x = (x1 , ..., xm ). The stock consists of m age categories, where xr is the number of BPCs with a residual shelf life of r days. So the state of the Markov decision problem is the tuple i = (d, e, k, x). We assume that the demand over an epoch is ﬁnite, i.e. the demand will never be for more than K BPCs (0 ≤ k ≤ K). Then the state space I is ﬁnite as the elements of x are limited by the replenishment policy with ﬁxed order-up-to levels (S1 , ..., S5 , S6 = 0, S7 = 0) for each day of the week. 3.2

Action and Action Space

At each epoch a decision is to be taken on the number of BPCs to issue from each of the m age categories. This decision is represented as a vector a = (a1 , ..., am ). The action space A is ﬁnite as the number of BPCs to issue of a speciﬁc age category r cannot exceed the number of BPCs available and must be non-negative: ar ∈ {0, 1, . . . , xr }

∀r = 1, ..., m

(1)

and it does not make sense to issue more than needed, hence: m

ar ≤ k.

(2)

r=1

As in practice, we require to meet all demand when enough BPCs are in stock. is thus recorded as a shortage. The Any discrepancy between m r=1 ar and k m number of BPCs short is: sho(k, a) = k − r=1 ar . Any shortage is considered as lost demand: in practice either the required BPCs are delivered by an external process through a (more expensive) emergency delivery or the surgery is postponed to another day. 3.3

State Transition and Transition Probabilities

For the transition from state i = (d, e, k, x) to a next state j = (d , e , k , y), we consider two cases: 1. e = am – d = d and e = pm for all r = 1, . . . , m − 1: some BPCs are issued; – yr = xr − ar m ym = (Sd − r=1 xr )+ : the replenishment order is delivered at noon. 2. e = pm – d = mod(d, 7) + 1 and e = am – yr = xr+1 − ar+1 for all r = 1, . . . , m − 1: products left do age by 1 day; ym = 0 no replenishment, so no new BPCs available. The only stochastic component in the transition from state i to state j is the demand that is to be met in the next epoch: i.e. the value of k which follows dem a probability distribution Pd,e . Hence the probability Pi,j (a) of being the next epoch in state j, when taking action a in state i is thus: dem (k ) Pi,j (a) = Pd,e

(3)

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The stock state transition from x to y is deterministically set by the issuing decision a and, when e = am, by the replenishment quantity set by the orderup-to Sd policy. During a transition outdating happens only at the end of the day. The number of outdated BPCs is thus out(d, e, x, a) = (x1 − a1 ) · I(e = pm) (where I(b) = 1, if boolean expression b is true, and zero if b is false). 3.4

Direct Costs

A so-called direct cost function C(d, e, k, x, a) accounts for the cost over a single epoch starting in state (d, e, k, x) and taking decision a. The cost function includes linear holding costs, and costs to penalize the occurrence of shortages and outdating. We deﬁne the following direct cost function C(d, e, k, x, a) = chd,e

m

xr + csd,e · sho(k, a) + cod,e · out(e, x, a)

(4)

r=1

with coeﬃcients per weekday d and epoch e: chd,e holding costs per BPC in stock at the start of epoch e, csd,e shortage costs per demanded BPC that cannot be met from stock, cod,e outdating costs per BPC that expires at the end of day d. 3.5

Minimal Expected Costs per Week

The optimal policy that minimizes the expected costs per week can be obtained from solving the relative value vector or bias terms v and the average costs per epoch gd,e from the set of Bellman equations [8]: ⎞ ⎛ ∀i = (d, e, k, x) ∈ I : v(i) + gd,e = min ⎝C(i, a) + Pij (a) · v(j)⎠ , (5) a∈A

j∈J (i,a)

where j = (d , e , k , y) in which y depends on a amongst other, and J (i, a) I is the set of states j for which Pi,j (a) > 0. Equation 5 represent a system of |I| equations in |I| + 14 unknowns. Hence there is no unique solution (v, g1,am , ..., g7,pm ), and one may ﬁx one value v(i) for each periodic class (d, e). The other values are relative to these values. Hence the vector v is called a relative value vector. gde . The expected costs per week under the optimal policy is g = d,e

3.6

Value Iteration

To solve Equation 5, we apply an approximation scheme based on value iteration. To allow for an eﬃcient storage of the transition probabilities, we show how the computations can be done eﬀectively using the demand probabilities directly. K dem Pde (k ) · Vn−1 (d , e , k , y) (6) Vn (d, e, k, x) = min C(d, e, k, x, a) + a∈A

k =0

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Vn (d, e, k, x) is the (cumulative) expected cost over an horizon of n consecutive epochs when taking optimal decisions over the entire horizon starting in state (d, e, k, x). The horizon is split into two parts: the ﬁrst epoch and the next n − 1 epochs. By setting the vector V0 to zero, one iteratively computes the elements of V1 , V2 , etc. Clearly lim Vn (d, e, k, x) is unbounded, but the diﬀerences Vn − Vn−1 are n→∞ bounded but not converging. This is because the state space I consists of 14 subspaces, called periodic classes, set by the time related dimensions d and e. The diﬀerence Vn (d, e, k, x) − Vn−14 (d, e, k, x) does converge (to the long-run average cost per week g. The optimal issuing decision in state (d, e, k, x) is the vector a that minimizes Equation 6. For technical details, such as a proof of convergence we refer to [8]. The value iteration algorithm can be implemented eﬃciently by making use of the periodicity, nevertheless computation times can easily become excessively large when the number of states is very large, see [2]. For the case of issuing BPCs at small and medium sized hospitals the time needed to compute an optimal policy is within seconds up to a minute (depending on the implementation and the CPU). In Equation 6, one may consider in iteration n all possible actions a to ﬁnd out which one is minimizing the expected costs. Instead of full enumeration of all values of the action vector, one may solve the minimization problem by dynamic programming. That way problems with larger action spaces can be solved more eﬃciently. For cases with a larger state space, for which doing all the computations takes too long, one may use aggregation techniques to reduce the number of states: e.g. by modeling replenishments and demands to happen in batches of products rather than individual product units the state and action space is reduced signiﬁcantly. The MDP model for determining optimal issuance decisions can then act as a fair approximation, as in such cases replenishments commonly happens in batches (set by the supplier).

4

Numerical Case

We illustrate the model by applying it to a case of a medium sized hospital. Main objective is to compare the optimal issuing policy with existing static issuing rules. The comparison is done by dynamic Monte Carlo simulation under the same assumptions as made in the MDP model such that any diﬀerence in the simulated results can be related to diﬀerent issuing decisions. (Alternatively one could compute the results by Markov chain analysis.) The input data of the model and the respective results are discussed in the next subsections. 4.1

Case Data

We consider a medium sized hospital, where on average four BPCs per day are transfused. The blood bank delivers BPCs with a maximum shelf life of m = 4 days. Note that the eﬀective shelf life is 3.5 days at the hospital, as products are

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delivered at noon. We model the demand during an epoch as a Poisson process with means µd,e as reported in Table 1. To ease the analysis of the results we have assumed that the demand is stationary and aperiodic and that the mean demand is equally spread over the morning and the afternoon. Note that this implies, as in practice, the demand to continue during weekends whereas no orders are placed and no replenishments are received. The next-to-last line of the table shows the expected demand µL+R to cover d till the next replenishment. The order-up-to levels should be set high enough to cover this demand. A rule of thumb for setting the replenishment levels S

d when the demand till the next replenishment is uncertain is Sd = µL+R + 2 µL+R . d d

On Mondays, Tuesdays, Wednesdays, and Thursdays µL+R =µ d √d,am + µd,pm + µd+1,am = 6, and the order-up-to levels Sd is set to 11 (≈ 6 + 2 6) as reported on the last line of the table. Similarly the order-up-to level on Fridays is set to = 14. S5 = 21, as µL+R 5 Table 1. Means of Poisson demand and order-up-to levels Sd Weekday d = Mon Tue Wed Thu Fri Sat Sun µd,e=am : mean morning demand 2 2 µd,e=pm : mean afternoon demand µL+R : mean till next replenishment 6 d 11 Sd : order-up-to levels

2 2 6 11

2 2 6 11

2 2 2 2 2 2 6 14 10 11 21 0

2 2 6 0

To have a ﬁnite number of transitions to consider in equation 6, the Poisson distribution with mean 2 is truncated at a maximum of 7 (K = 7): that is the demand for 7 or more BPCs is modeled as demand for 7 BPCs. In this case the unit holding cost is negligible in relation to the procurement cost of BPCs of about 150 euro. In line with previous studies [5]shortages are penalized by setting the shortage cost csd,e ﬁve times as high as the outdating costs csd,e of 150 euro per BPC: cod,e = 150 and csd,e = 750 for all d and e. The outdating costs represents the unit costs that could be saved when an outdated BPC would not have been ordered. The shortage costs include a (big) penalty as lives may be at risk and the costs to arrange an emergency delivery from the blood bank or to postpone a surgery. 4.2

Case Results

By value iteration we have determined an optimal issuing policy, which we compare against simple rules like FIFO, LIFO and the combination of LIFO in the morning and FIFO in the afternoon. The latter issuing rule is denoted as LIFO-FIFO. To evaluate the rules one may apply value iteration to solve the underlying Markov chains. However, we have chosen to simulate the rules to get more detailed information at once. To get accurate results, each rule is simulated in 10,000 runs of 54 weeks including 2 weeks for warming up. The results are reported in Table 2.

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Table 2. Simulation results for optimal issuing versus FIFO, LIFO and LIFO-FIFO Optimal FIFO LIFO LIFO-FIFO Weekly costs % above optimum Outdating Shortages

1063 1539 2350 +45% +121% 16% 16% 25% 1.0% 3.1% 3.8%

1415 +33% 17% 2.4%

The trade-oﬀ to make is between outdating and shortages. From literature one would expect that the optimal policy closely resembles FIFO, as issuing the oldest products ﬁrst reduces the outdating. From Table 2, we learn the optimal issuing policy greatly saves on compared to FIFO issuance, while the outdating ﬁgures are is the same. The average costs per week of the optimal issuing policy is 1063 euros per week. Under LIFO these costs are twice as large. FIFO appears to be 45 % above the optimal cost level. Surprisingly, under LIFO-FIFO the average costs are closest to the minimal average costs per week, but still 33 % above optimal. The explanation for the large diﬀerences is found in a better timing of outdating by the optimal issuing policy. Investigating the detailed actions for some states we observe that, for example, in some states it is optimal to let products expire on Sundays by issuing younger BPCs. The eﬀect of this enforced or planned expiration is that a higher and apparently better order volume is derived on Monday via the ﬁxed order-up-to level. Consequently there will be less shortages on Tuesdays, as is supported by Table 3: 0.9 BPCs under FIFO and 0.2 BPCs under cost-optimal issuance. Annually the number of shortages is 52 − 15.6 = 36.4 BPCs lower under the optimal issuing policy. Insights: The issuing policy does not so much aﬀect the outdating percentage, but through a better timing of outdating shortages are reduced dramatically. Table 3. Optimal issuing versus FIFO: simulation results per weekday

Weekday

number of BPCs outdating shortage Optimal FIFO Optimal FIFO

Saturday Sunday Monday Tuesday Wednesday Thursday Friday

0.7 1.8 2.6 0.0 0.0 0.0 0.2

0.1 0.1 4.9 0.0 0.0 0.0 0.2

0.0 0.0 0.1 0.2 0.0 0.0 0.0

0.0 0.0 0.1 0.9 0.0 0.0 0.0

Weekly Annual

5.3 150

5.3 150

0.3 15.6

1 52

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Hence an ordering rule that causes too many shortages under FIFO or LIFO can show a good performance under a dynamic issuing rule. By issuing the BPCs in a cost-optimal way shortages are reduced. This enables stock managers to reduce outdating by lowering the order-up-to levels somewhat while keeping shortages at an acceptable level. Hence the optimal issuing policy indirectly enables to reduce outdating next to shortages.

5

Discussion and Conclusions

We have shown that an optimal issuing policy for perishables with a short shelf life can be computed by formulating and solving a Markov Decision Problem. Such an optimal policy may perform much better than existing policies like FIFO and LIFO issuance. In a numerical example of a hospital that issues twice per day BPCs to its departments (but get replenishments on weekdays only) it appears that FIFO issuance implies more than three times as many shortages than optimal issuance. From a computational point of view we have discussed how to reduce the computation time when dealing with large state and action spaces. We expect that one may obtain similar results for shops that sell fresh food, like dairy products, fruit and vegetables, as these products have a short life time of just a couple of days while replenishments may not happen on all weekdays and shops may be open seven days a week. The issuance in such shops is than controlled by a careful allocation of products to the forward and reserve inventories. These control decisions require a DSS tool that can be used next to an CAO or ASO system and requires no change of other inventory management processes. The reduction in shortages is signiﬁcant which may give some room for reducing the outdating by lowering the stock levels a bit. Thus optimal issuance provide beneﬁts at the economic and environmental level as well as increases the customer service level. Acknowledgments. The author thanks Michal Bilinski for coding the SDP algorithm and his co-supervisor Mark R. Kramer.

References 1. Broekmeulen, R.A.C.M., van Donselaar, K.H.: A heuristic to manage perishable inventory with batch ordering, positive lead-times, and time-varying demand. Computers & Operations Research 36(11), 3013–3018 (2009) 2. Haijema, R.: Solving large structured markov decision problems for perishable inventory management and traﬃc control, Ph.D. thesis, Univeristy of Amsterdam - Tinbergen Institute - Amsterdam School of Economics, 12 (2008) 3. Haijema, R.: A new class of stock level dependent ordering policies for perishables with a short maximum shelf life. International Journal of Production Economics (2011); doi:10.1016/j.ijpe.2011.05.021

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4. Haijema, R., van der Wal, J., van Dijk, N.M.: Blood platelet production: Optimization by dynamic programming and simulation. Computers and Operations Research 34(3), 760–779 (2007) 5. Haijema, R., van Dijk, N.M., van der Wal, J., Smit Sibinga, C.: Blood platelet production with breaks: Optimization by SDP and Simulation. International Journal of Production Economics 121, 467–473 (2009) 6. Karaesmen, I., Scheller-Wolf, A., Deniz, B.: Planning production and inventories in the extended enterprise. In: Managing Perishable and Aging Inventories: Review and Future Research Directions. International Series in Operations Research & Management Science. ch. 15, vol. 151, pp. 393–436. Springer, Heidelberg (2011) 7. Pierskalla, W.P.: Operations research and health care, a handbook of methods and applications. In: Supply Chain Management of Blood Banks, pp. 104–145. Kluwer Academic Publishers, New York (2004) 8. Puterman, M.L.: Markov decision processes: Discrete stochastic dynamic programming. Wiley Series in Probability and Mathematical Statistics (1994) 9. van der Vorst, J.G.A.J., Beulens, A.J.M., de Wit, W., van Beek, P.: Supply chain management in food chain: Improving performance by reducing uncertainty. Int. Trans. Op. Res. 5(6), 487–499 (1998) 10. van Dijk, N.M., Haijema, R., van der Wal, J., Smit Sibinga, C.: Blood platelet production: a formal approach for practical optimization. Transfusion 49(3), 411–420 (2009) 11. van Donselaar, K., van Woensel, T., Broekmeulen, R., Fransoo, J.: Inventory control of perishables in supermarkets. International Journal of Production Economics 104, 462–472 (2006) 12. van Zyl, G.J.J.: Inventory control for perishable commodities, unpublished Ph.D. dissertation, University of North Carolina (1964) 13. Veinott Jr., A.F.: Optimal ordering, issuing, and disposal of inventory with known demand, unpublished Ph.D. dissertation, Columbia University (1960)

The Maximum Flow Problem with Minimum Lot Sizes Dag Haugland1 , Mujahed Eleyat2 , and Magnus Lie Hetland3 1

3

Department of Informatics, University of Bergen, Bergen, Norway 2 Miriam AS, Halden, Norway Department of Computer and Information Science, Norwegian University of Science and Technology, Trondheim, Norway

Abstract. In many transportation systems, the shipment quantities are subject to minimum lot sizes in addition to regular capacity constraints. That is, either the quantity must be zero, or it must be between the two bounds. In this work, we consider a directed graph, where a minimum lot size and a ﬂow capacity are deﬁned for each arc, and study the problem of maximizing the ﬂow from a given source to a given terminal. We prove that the problem is NP-hard. Based on a straightforward mixed integer programming formulation, we develop a Lagrangean relaxation technique, and demonstrate how this can provide strong bounds on the maximum ﬂow. For fast computation of near-optimal solutions, we develop a heuristic that departs from the zero solution and gradually augments the set of ﬂow-carrying (open) arcs. The set of open arcs does not necessarily constitute a feasible solution. We point out how feasibility can be checked quickly by solving regular maximum ﬂow problems in an extended network, and how the solutions to these subproblems can be productive in augmenting the set of open arcs. Finally, we present results from preliminary computational experiments with the construction heuristic.

1

Introduction

In transportation systems, as well as in production and manufacturing, some operations might be eﬀective only when the processed quantity lies above a given threshold. Reasons for such restrictions are of diverse nature. Operations might require lots to be large in order to be cost eﬀective, the products appear only in batches of a minimum size, or underlying mechanical and chemical processes require a minimum level of operation. Operations involving setup costs are somehow related to those involving minimum lot sizes. Their resemblance is reﬂected by the corresponding optimization models, which in both cases imply the introduction of binary variables representing the decision of whether to activate the operation. An obvious distinction is that negative eﬀects of a yes-decision in the case of setup costs are conﬁned to the objective function, whereas in the case of minimum lot sizes, also feasibility is aﬀected. As a consequence, solution approaches that work well in the former case may be non-trivial to translate to the latter. J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 170–182, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Production planning models respecting lot size decisions and constraints appear abundantly in the operations research literature. Most of these are extensions of the classic capacitated lot sizing problem (CLSP), of which several surveys [1,5,7] are found. The CLSP amounts to optimize the lot sizes over a multi-period horizon, when, in addition to holding and production costs, a setup cost is incurred in each period of production. In its simplest form, the CLSP model does not include lower bounds on the lot sizes. As suggested, e.g., in the model by Voß and Woodruﬀ [11], minimum lot sizes may replace or come in addition to setup costs, and the model updates are straightforward. Network ﬂow models for optimizing transportation decisions are also often extended by binary variables in order to reﬂect setup costs. The fixed charge network flow problem [4,9] is a direct extension of the minimum cost ﬂow problem, where a ﬁxed cost for utilizing the arc is added to the ﬂow proportional term. Another extension of the minimum cost ﬂow problem is the minimum cost circulation problem [10], in which lower ﬂow bounds are deﬁned on the arcs. Contrary to the ﬁxed charge network ﬂow problem, the circulation problem is modeled as a linear program. The minimum lot sizes are considered as hard constraints in the sense that it is not an option to put the ﬂow to zero in case it is infeasible or suboptimal to respect the bound. Flow models acknowledging this option do not seem to be well studied in the scientiﬁc literature. However, for the reasons given above, we believe that such models have relevance. For instance, the Miriam Regina system [8] for testing production availability and deliverability in the process industry and in energy applications, incorporates a ﬂow model where processes can be operated in a semi-continuous range (either they are turned oﬀ or they are assigned a capacity above a non-zero bound). In this article, we study network ﬂow optimization subject to minimum lot size constraints. For reasons of simplicity, and consistent with the approach taken in the Miriam Regina system, we choose ﬂow maximization as the underlying model. The purpose of the work is to suggest eﬃcient computational methods that, despite the proven intractability of the problem, are able to produce nearoptimal solutions with modest computational eﬀort. The rest of the article is organized as follows. In Section 2, we introduce some nomenclature and mathematical notation, and deﬁne our network ﬂow model in rigorous terms. Section 3 provides a proof of NP-hardness of the problem. In Section 4, we formulate the problem as a mixed integer programming problem, and we show how Lagrangean relaxation can be applied in order to compute upper bounds on the maximum ﬂow. We also give a fast method for translating the (possibly infeasible) relaxed solutions into feasible ones. In Section 5, we develop a construction heuristic based on an augmenting path approach. Computational results with this heuristic are given in Section 6.

2

Problem Definition

Let G = (N, A) be a directed graph with node set N and arc set A, and nonnegative integer vectors and u of lower and upper ﬂow bounds, respectively.

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We assume that G contains a unique source s ∈ N and a unique sink t ∈ N , and that A contains a circulation arc (t, s) with ts = 0 and uts = ∞ from the sink to the source. We consider the problem of maximizing the ﬂow from the source through the network to the sink, such that the ﬂow at each arc is either zero or between the two bounds. This is equivalent to maximizing the ﬂow recycled along arc (t, s). By deﬁning the set of circulations in G as ⎧ ⎫ ⎨ ⎬ F (G) = x ∈ IRA xij − xji = 0 ∀i ∈ N , + : ⎩ ⎭ j:(i,j)∈A

j:(j,i)∈A

the problem is expressed as: max

xts , x ∈ F (G), xij ∈ {0} ∪ [ij , uij ] (i, j) ∈ A.

(1) (2) (3)

Henceforth, we say that arc (i, j) is closed if xij = 0, and open otherwise. We ¯ = A \ X be the set of closed arcs. let X ⊆ A be the set of open arcs, and let X Let GX denote the subgraph with node set N and arc set X. We say that X is feasible if there exists some ﬂow vector x ∈ F (G) satisfying ij ≤ xij ≤ uij ¯ Let z(X) denote the maximum for all (i, j) ∈ X, and xij = 0 for all (i, j) ∈ X. value xts can take under these conditions, and let z(X) = −∞ if X is infeasible. Observe that the empty set is feasible with z(∅) = 0.

3

Computational Complexity

Proposition 1. Problem (1)–(3) is NP-hard. Proof. The proof is by a polynomial reduction from the Subset Sum problem, which is known to be NP-complete [2, p. 951]. Given a ﬁnite set {a0 , a1 , . . . , an } of positive integers, the problem is to decide whether there exists a subset S ⊆ {1, . . . , n} such that i∈S ai = a0 . Deﬁne the digraph with node set N = {s, t, v0 , . . . vn } and arc set A = {(s, v1 ), . . . , (s, vn ), (v1 , v0 ), . . . , (vn , v0 ), (v0 , t)}, and let the ﬂow bounds be v0 t = uv0 t = a0 and svi = vi v0 = usvi = uvi v0 = ai (i = 1, . . . , n). Hence, if there does exist a subset S as requested, then an optimal solution to (1)–(3) is to send a0 units from s to t via nodes vi , i ∈ S. Otherwise, the only feasible solution is x = 0.

4

Integer Programming Model and Lagrangean Relaxation

By deﬁning the decision vector y ∈ {0, 1}A, where yij = 1 if xij ∈ [ij , uij ] and yij = 0 if xij = 0, we arrive at the mixed integer linear program max xts ,

(4)

The Maximum Flow Problem with Minimum Lot Sizes

x ∈ F (G),

173

(5)

xij − ij yij ≥ 0, (i, j) ∈ A,

(6)

xij − uij yij ≤ 0, (i, j) ∈ A. y ∈ {0, 1}A.

(7) (8)

Deﬁne F (G, u) = {x ∈ F (G) : xij ≤ uij ∀(i, j) ∈ A}, and the maximum ﬂow out of node i ∈ N when the lower ﬂow bounds are relaxed as ⎧ ⎫ ⎨ ⎬ Mi = max xij : x ∈ F (G, u) . ⎩ ⎭ j:(i,j)∈A

Then any feasible solution to (4)–(8) satisﬁes j:(j,i)∈A ji yji ≤ Mi , stating that a set of arcs entering node i cannot be opened if the sum of their lower ﬂow bounds exceeds the maximum ﬂow out of the node. A Let λ ∈ IRA + and μ ∈ IR+ , where λts = μts = 0, and consider the Lagrangean relaxation of (4)–(8): (9) L(λ, μ) = max (i,j)∈A (hij xij + dij yij ) ,

x ∈ F (G, u),

j:(j,i)∈A ji yji ≤ Mi ,

y ∈ {0, 1}A.

(10) i ∈ N,

(11) (12)

1, (i, j) = (t, s) and dij = μij uij − λij ij . That is, we λij − μij , (i, j) = (t, s), have applied the Lagrangean multipliers λij and μij to (6) and (7), respectively. To strengthen the relaxation, the upper ﬂow bounds are kept in (10) and the valid inequalities (11) are added. Consequently, we arrive at a minimum cost ﬂow problem in the x-variables. In the y-variables, the problem is decomposed into a set of |N | knapsack problems, and is hence solvable in pseudopolynomial time. where hij =

4.1

A Heuristic for Finding Feasible Solutions

Since (9)–(12), the optimal solution of which is denoted (xL , y L ), is a capacitated minimum cost ﬂow problem in the x-variables, the arcs (i, j) ∈ A : 0 < xL ij < uij deﬁne no cycles in G. Let this arc set be contained in some AT ⊆ A where (N, AT ) is a tree (arc directions disregarded). That is, ¯ we have xL ij ∈ {0, uij } for all (i, j) ∈ AT = A \ AT . With the purpose of increasing the number of feasible arc ﬂows, each iteration of the following heuristic sends ﬂow along a cycle consisting of arcs in AT and exactly one arc, k, in A¯T . As a result, arc k replaces some cycle arc (i, j) ∈ AT , which in its turn is assigned ﬂow equal to either 0, ij or uij . Throughout the heuristic, we thus have that all non-tree arcs have either zero ﬂow or ﬂow on either of their bounds. Initially, none of them have ﬂow on their lower bound.

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The tree arcs are assigned ﬂow values between zero and the upper bound. If no arc has ﬂow strictly between zero and the lower bound, a feasible solution is found, and the heuristic terminates. For all k ∈ A¯T , let Ck denote the arc set of the unique cycle consisting of arc k and arcs in AT . We accept to let k replace some tree arc if and only if, for some number f , both the following conditions hold when f units of ﬂow are sent along Ck : 1. All currently feasible arc ﬂows remain feasible. 2. At least one currently infeasible arc ﬂow becomes feasible. An arc in Ck is said to be a forward (backward) arc if it is (not) directed consistently with k. To remove ambiguity, we assume that the f ﬂow units are sent in the forward direction. That is, all forward arcs in Ck have their ﬂow values increased (decreased) if f > 0 (f < 0), and the converse is true for the backward arcs. If no arc meeting the conditions can be found, the heuristic terminates and fails to ﬁnd a feasible solution. Proposition 2. The set of f -values satisfying condition 1 consists of a closed interval and at most |Ck | distinct values. Proof. Consider a traversal of the cycle starting by arc k = (i0 , i1 ), and let i0 , . . . , i|Ck |−1 denote the nodes hence visited. Let i|Ck | = i0 , and let Fm denote the set of f -values that make the ﬂow on all arcs on the path (i0 , . . . , im ) feasible (m = 0, . . . , |Ck |). Assume Fm = [lbm , ubm ] ∪ Pm , m < |Ck |, where |Pm | ≤ m, and observe that the assumption holds for m = 0 (let lb0 = −∞, ub0 = +∞). If km = (im , im+1 ) is a forward arc then the set Sm of f -values rendering its ﬂow value feasible is Sm = Im ∪ {pm }, where ⎧ xkm = 0 ⎨ ([km , ukm ] , 0) , (Im , pm ) = ([0, ukm − km ] , −km ) , xkm = km ⎩ ([km − ukm , 0] , −ukm ) , xkm = ukm . Otherwise, we have Sm = Im ∪ {pm }, where ⎧ xkm = 0 ⎨ ([−ukm , −km ] , 0) , (Im , pm ) = ([km − ukm , 0] , km ) , xkm = km ⎩ ([0, ukm − km ] , ukm ) , xkm = ukm . This yields Fm+1 = Fm ∩ Sm = ([lbm , ubm ] ∩ Im ) ∪ Pm+1 , where Pm+1 consists of all values in Pm contained in Sm and also pm if pm ∈ Fm . It follows that Fm+1 is composed of an interval and at most m + 1 singular values, and the result follows by induction.

The proposition covers the special case where f = 0 is the only value satisfying the condition, in which case the interval in question is empty. The proof of Proposition 2 is constructive in giving an algorithm for computing the feasible ﬂow assignments to Ck . Once we have found that condition

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1 is met by f ∈ [lb, ub] ∪ {p1 , . . . , p|Ck | }, we check whether any of the values lb, ub, p1, . . . , p|Ck | satisﬁes the second condition. Among all such values, we select the one maximizing xts if (t, s) ∈ Ck . Otherwise, a value maximizing the number of feasible arc ﬂows is chosen. It is straightforward to see that by this choice of f -value, at least one arc in Ck will be assigned zero ﬂow or ﬂow on one of its bounds, and is hence qualiﬁed to leave AT . The above procedure is repeated for all k ∈ A¯T in an arbitrary order, and interrupted when a feasible ﬂow is found or when the two conditions cannot be met. 4.2

Variable Fixing

Combining a feasible ﬂow and the upper bound L(λ, μ) on the maximum ﬂow may help to ﬁx the value of certain y-variables. Assume the heuristic above or any other method has identiﬁed some ﬂow vector xH , and assume there

1 feasible

exists 1 1 , where y some other feasible solution x , y = 1 for some k ∈ A. Let x0 , y 0 k

1 1 yk0 = 0. Since x0 , y 0 is feasible in (9)– be identical to x , y , except that

0 1 = (i,j)∈A hij x1ij + dij yij − (12), we have L(λ, μ) ≥ (i,j)∈A hij x0ij + dij yij then L(λ, μ)+μ u − μk uk +λk k ≥ x1ts −μk uk +λk k . It follows that if x1ts > xH k k ts H λk k > xH . Hence, if μ u − λ ≤ x − L(λ, μ), opening arc k cannot yield k k k k ts ts solutions that are superior to xH , and consequently, we ﬁx yk = 0. Since x = 0 is feasible, we obtain as a special case that all arcs k for which μk uk − λk k ≤ A −L(λ, μ) can be closed. This holds regardless of how λ ∈ IRA + and μ ∈ IR+ are chosen, and the tighter the upper bound L(λ, μ) the more variables can be ﬁxed. 4.3

The Lagrangean Dual

In order to have the tightest possible upper bound on the maximum ﬂow, and thereby to be able to ﬁx a maximum number of variables, it is desirable to solve the Lagrangean dual problem minλ,μ∈IRA L(λ, μ), e.g. by the popular subgradient + (dual descent) algorithm. Since the relaxed problem in the y-variables does not have integrality property, L(λ, μ) has a potential to dominate the bound the minimum bound minλ,μ∈IRA + provided by the LP-relaxation of (4)–(8). It is easily seen that the latter is simply the max ﬂow in G when the lower ﬂow bounds are neglected. Therefore, the improved bound comes for the cost of solving |N | knapsack problems in each iteration of the subgradient algorithm, in place of solving a single maximum ﬂow problem. The size of any knapsack problem is however proportional to the indegree of the corresponding node, and for sparse graphs this represents a modest computational cost.

5

Construction Heuristic

In this section, we give a heuristic method that starts with X empty, and then gradually extends X as long as extensions can be found. This will produce a

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sequence of arc sets, some of which will be feasible while others are infeasible. Whenever X is feasible, we attempt to extend it such that the max ﬂow is improved. Otherwise, we look for extensions that reduce the constraint violations. The best feasible X hence encountered is output from the algorithm. This algorithmic idea is indicated by algorithm 1. Algorithm 1. Idea(G,,u,(t, s)) X ← {(t, s)} // The initial set of open arcs consists uniquely of the circulation arc z ∗ ← 0, X ∗ ← X // best solution ever repeat Check whether X is feasible if feasibility check positive then Let x be a ﬂow allocation in GX maximizing xts if z(X) > z ∗ then z ∗ ← z(X), X ∗ ← X // best solution ever S ← extension of X suggested to increase xts else S ← extension of X suggested to reduce constraint violation X ←X ∪S until S = ∅ return X ∗

5.1

Checking Feasibility

Finding a feasible arc set X is not a trivial problem. Neither is it trivial to ﬁnd a good extension S of an already feasible X, and we may encounter that X ∪ S is inferior to X and even infeasible. Checking whether any arc set is feasible, is however accomplished by solving a standard max-ﬂow instance in an auxiliary network deﬁned as follows [6, Section 10.2]: Let GX = (N , A ) be a digraph with node set N = N ∪ {s , t } and arc set A = X ∪ {(t , s )} ∪ {(s , i) : i ∈ N } ∪ {(i, t ) : i ∈ N }. That is, we add to GX an auxiliary source s and an auxiliary sink t and a circulation arc between them. We also add arcs from s to each node in N , and from each node in N , we add an arc to t . With each arc (i, j) in the extended digraph, we associate a ﬂow capacity (an upper ﬂow bound) cij , but we do not deﬁne lower ﬂow bounds in this network (or we can assume they are all 0). We let cij = uij − ij for all (i, j) ∈ X and ct s = ∞. For the new arcs joining the auxiliary source with other nodes, the capacities are deﬁned as cs i = j:(j,i)∈X ji for all i ∈ N . Likewise, capacities on arcs entering t are deﬁned as cit = j:(i,j)∈X ij for all i ∈ N . This means that the capacity of arc (s , i) becomes the sum of lower bounds on arcs in GX entering node i, while the capacity of arc (i, t ) is the sum of lower bounds on arcs in GX leaving node i. Observe that cs i = cit = ij . i∈N

i∈N

(i,j)∈X

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Proposition 3. If x ∈ Z A is a flow allocation in GX such that xt s is maxi mized, then the arc set X is feasible if and only if xt s = (i,j)∈X ij .

Proof. See Theorem 10.2.1 in [6].

It follows that feasibility of X can be checked by solving a standard (that is, without lower ﬂow bounds) max-ﬂow problem in GX . If the maximum ﬂow equals the total capacity of the arcs leaving s (and thereby also the total capacity of the arcs entering t ), then X is feasible. Otherwise, X is infeasible. We solve the standard max-ﬂow problem in GX by an augmenting-path algorithm. In each iteration, ﬂow is sent along a path from s to t in the residual network of GX , and the algorithm terminates when no such path exists. 5.2

Allocating Flow to a Set of Feasible Arcs

Assume we have computed x by the augmenting-path algorithm, and veriﬁed that the condition in Proposition 3 holds. We conclude that X is feasible. Actually, the ﬂow satisfying (2)–(3) is given as xij = xij + ij for all (i, j) ∈ X. This is however not necessarily the ﬂow allocation that yields z(X) = xts . To maximize xts , we go on searching for augmenting paths in the residual network of GX , but now the paths go from s to t. All arcs incident to s or t are ignored, because the ﬂow here must be unchanged. The maximum ﬂow in GX is ﬁnally found by adding ij to xij for all (i, j) ∈ X. 5.3

Extending a Feasible Solution

If x is the feasible ﬂow allocation to GX that maximizes xts , then the residual graph of GX has no path from s to t. Hence, an extension of X should produce such a path in order to open for more ﬂow from s to t. We therefore consider the residual of the entire network G, where also currently closed arcs are included, and search for a ﬂow-augmenting path using one of the criteria explained in ¯ becomes the desired Section 5.5. If such a path is found, its intersection with X extension S. 5.4

Extending an Infeasible Solution

Assume now that we have veriﬁed that x does not satisfy the condition in 3. We conclude that X is infeasible, and we start the search for an extension of X that hopefully reestablishes feasibility. It is easily seen that the suggested transformation xij = xij + ij for all (i, j) ∈ X produces a solution violating the ﬂow conservation constraints (2) for at least two nodes, whereas the ﬂow bounds (3) are respected. More precisely, we can identify (at least) one node i+ with excess entering ﬂow, and (at least) one − node i+ and i− such that of entering ﬂow. That is, we can ﬁnd i with shortage j:(j,i)∈X xji+ − j:(i,j)∈X xi+ j > 0 and j:(j,i)∈X xji− − j:(i,j)∈X xi− j < 0. The idea is now to extend X in such a way that ﬂow can be sent from i+ to ¯ for arcs that create new paths joining the two i− . To this end, we search in X unbalanced nodes.

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Let G be the digraph consisting of all nodes and arcs in G and GX , i.e., ¯ are added. Deﬁne the capacities cij = an extension of GX where all (i, j) ∈ X ¯ uij − ij also for all (i, j) ∈ X. Assign the ﬂow x (see Section 5.1) to the network G , and ﬁnd a path P = (NP , AP ) with node set NP and arc set AP from i+ to i− in the residual network G (x ). If there is no such path, we conclude that ¯ denote the set of new arcs X cannot be extended. Otherwise, let S = AP ∩ X along the path. In general, G (x ) may contain several ﬂow-augmenting paths from i+ to i− . Care must be shown when selecting P , because once S is accepted as extension of X, the new arcs will never leave again. A necessary condition for feasibility of X ∪ S is that the arcs (i, j) ∈ S introduce a ﬂow-augmenting path from i+ to i− when added to the residual network GX (x ). This is however not a suﬃcient condition, because the lower bounds of the new arcs may provoke new infeasibilities. Taking the lower bounds into account when introducing new arcs seems to be hard to accomplish in rigorous terms, and will therefore be dealt with heuristically as explained in Section 5.5. It is not guaranteed that X ∪ S is feasible. This will be checked in the next iteration of algorithm 1, which makes a call to the procedure discussed in Section 5.1. 5.5

Finding Paths in Residual Graphs

We have developed three methods for ﬁnding paths from s to t to extend a feasible solution, and from i+ to i− to extend an infeasible solution. Their computational burdens are modest as they can be accomplished by a simple rewriting of Dijkstra’s shortest-path algorithm. The ideal path is one where the smallest upper bound is large and the largest lower bound is small. Our path ﬁnding methods vary in the way they are adapted to this observation. Maximizing the largest lower ﬂow bound: In the ﬁrst method, we simply ﬁnd a path P = (NP , AP ) for which max(i,j)∈AP ij is minimized. Excess ﬂow (see Section 5.4) and upper bounds do not aﬀect the choice of path. The method is implemented by replacing the summation of path and arc lengths (lower ﬂow bounds) in Dijkstra’s algorithm by a maximum operation over the two arguments. Large diﬀerence between the smallest upper and the largest lower bounds: In the second method, the ambition is to ﬁnd a path P with min(i,j)∈AP uij −max(i,j)∈AP ij large, while also taking excess ﬂow into account. Let ei denote the excess ﬂow at node i, and for any i ∈ NP , let Li and Ui denote, respectively, the minimum and maximum amounts of ﬂow node i has to receive if all arcs in AP are opened. Then, for all (i, j) ∈ AP , we have (let Ls = −∞, Us = +∞) Lj = max {Li + ei , ij } and Uj = min {Ui + ei , uij }. As an approach to make Ui+ − Li+ (or Ut − Lt ) large, we apply these recursive formulae in a heuristic inspired by Dijkstra as shown in algorithm 2.

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Algorithm 2. LargeCapacityDiﬀerence(i+,i− ) di ← −∞, Li ← 0, Ui ← ∞, ∀i ∈ N di+ ← 0, NP ← {i+ }, AP ← ∅, i ← i+ repeat for ∀j ∈ N \ NP : (i, j) ∈ A do L ← max{Li + ei , ij }, U ← min{Ui + ei , uij } if U − L > dj then Lj ← L , Uj ← U , dj ← U − L , pj ← pi // Best known path to j goes via i Find i ∈ arg max {dj : j ∈ N \ NP } NP ← NP ∪ {i}, AP ← AP ∪ {(pi , i)} until i− ∈ NP return (NP , AP )

A hybrid method: Finally, we suggest a method combining the ideas of the two ﬁrst methods. Consider the if-statement of algorithm 2. Determining whether a new best path to j has been found, is in the hybrid method based on a comparison either between U − L and dj or between L and Lj . When |L − Lj | is small in comparison to |U − L − (Uj − Lj )|, then the decision is made as in algorithm 2, otherwise we compare L to Lj . In our implementation, we have applied |L − Lj | < 3|U − L − (Uj − Lj )| as the criterion for using the rule of algorithm 2.

6

Computational Experiments

We have evaluated the construction heuristic in Section 5 by applying it to a set of input graphs. The instances were obtained by ﬁrst generating four base graphs GW 6, GW 7, GW and GL, using the RMFGEN-generator of Goldfarb and Grigoriadis [3], which takes four parameter values a, b, c1 , and c2 as input. The actual parameter values and graph sizes are shown in Table 1. However, the generator was modiﬁed so that the capacities of the in-frame arcs are also generated randomly in the range [c2 , c2 a2 ]. We generated six instances of each graph that diﬀer by the percentage of arcs that have nonzero lower ﬂow bounds. For example, GL–40 is generated such that 40% of the arcs have nonzero lower ﬂow bounds. These arcs were selected randomly by drawing from the entire arc set A, and for each of them, ij was randomly generated in the range [uij /4, uij ]. In Table 2, we give the optimal ﬂow, produced by supplying the mixed integer programming formulation of Section 4 to CPLEX, the upper bound given by the LP-relaxation of the same model, and the ﬂow obtained by the three variants (see Section 5.5) of the heuristic. The variants are in the table denoted 1, 2, and 3, consistent with the order in which they are introduced in Section 5.5. As seen from the table, the LP-bound is for all four base graphs tight when ˙ of the arcs have lower ﬂow bounds. This means that in such no more than 50% instances, the minimum lot size constraints do not aﬀect the maximum ﬂow. For more constrained instances, the maximum ﬂow drops below the bound, and

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Table 1. Base graphs

Graph

a

GW 6 GW 7 GW GL

4 5 5 15

Parameters b c1 7 8 15 5

2,000 1,500 2,000 2,000

c2

|N |

|A|

10,000 6,000 10,000 10,000

112 200 375 1,125

432 815 1,550 5,100

Table 2. Optimal ﬂow, LP-bound and ﬂow produced by the heuristic

Path ﬁnding method 1 2

Instance

Optimal

LP-bound

GW 6–30 GW 6–40 GW 6–50 GW 6–60 GW 6–70

80,868 80,868 80,868 80,073 80,073

80,868 80,868 80,868 80,868 80,868

54,703 80,868 52,451 4,590 0

80,868 59,121 59,121 6,454 0

80,868 59,121 59,121 4,590 0

GW 7–30 GW 7–40 GW 7–50 GW 7–60 GW 7–70

79,286 79,286 79,286 79,286 79,286

79,286 79,286 79,286 79,286 79,286

79,286 79,286 79,286 76,167 2,752

79,286 79,286 69,903 51,343 20,660

79,286 79,286 79,286 79,286 2,752

GW –30 GW –40 GW –50 GW –60 GW –70

130,587 130,587 130,587 96,971 77,587

130,587 130,587 130,587 130,587 130,587

130,587 130,587 34,714 12,255 0

130,587 130,587 34,714 17,394 0

130,587 130,587 20,563 0 0

1,338,057 1,338,057 1,338,057 1,335,045 1,335,045

1,338,057 1,338,057 1,338,057 1,338,057 1,338,057

1,338,057 1,338,057 1,046,717 0 0

1,338,057 1,338,057 3111 0 0

1,338,057 1,338,057 123,895 0 0

GL–30 GL–40 GL–50 GL–60 GL–70

3

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the construction heuristic struggles to ﬁnd other solutions than the trivial zero solution. When the proportion of arcs with a minimum lot size is small, the heuristic seems to have a fair chance to ﬁnd good solutions. The results are inconclusive in the choice of variant of the heuristic. We believe that the heuristic’s lack of success in highly constrained instances is due to its greedy nature: When the arcs along some path are opened, excess ﬂow will be induced by their lower ﬂow bounds. Whether it in later iterations will be possible to open paths to account for this excess ﬂow seems diﬃcult, and if judged incorrectly, the heuristic will be left with excessive ﬂow for which there is insuﬃcient capacity. A more sophisticated procedure for arc selection could improve the construction heuristic. With many arcs subject to minimum lot size constraints, it becomes unlikely to ﬁnd long paths where the largest lower ﬂow bound is smaller than the smallest upper bound. Making a feasible ﬂow allocation then requires that the ﬂow is split at one or more nodes. Consequently, the heuristic should investigate more general subgraphs than paths when considering new arcs to be opened.

7

Conclusions

We have introduced the maximum ﬂow problem in directed graphs with minimum lot size constraints on the arcs, imposing either zero ﬂow or ﬂow between the lower and upper capacity. The disjunctive nature of the new constraints makes the problem NP-hard. Based on a mixed integer programming formulation, we have shown how the problem can be approached by Lagrangean relaxation. This approach involves a method for computing strong upper bounds on the maximum ﬂow, and a method for ﬁxing binary variables based on the upper bounds. We have also suggested a construction heuristic for the problem, and presented results from some computational experiments. This work will be followed up by experimental evaluation of the Lagrangean relaxation technique. Approaches for strengthening the construction heuristic will also be developed and tested numerically.

References 1. Bahl, H., Ritzman, L., Gupta, J.: Determining lot sizes and resource requirements: A review. Operations Research 35(3), 329–345 (1988) 2. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001) 3. Goldfarb, D., Grigoriadis, M.: A computational comparison of the dinic and network simplex methods for maximum ﬂow. Annals of Operations Research 78, 83–123 (1988) 4. Hirsch, W., Dantzig, G.: The ﬁxed charge problem. Naval Research Logistics Quarterly 15, 413–424 (1968) 5. Jans, R., Degraeve, Z.: Modeling industrial lot sizing problems: a review. International Journal of Production Research 46(6), 1619–1643 (1968)

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6. Jungnickel, D.: Graphs, Networks and Algorithms. Springer, Heidelberg (2008) 7. Karimi, B., Ghomi, S.F., Wilson, J.: The capacitated lot sizing problem: a review of models and algorithms. Omega 31, 365–378 (2003) 8. http://www.miriam.as 9. Nahapetyan, A., Pardalos, P.: Adaptive dynamic cost updating procedure for solving ﬁxed charge network ﬂow problems. Computational Optimization and Applications 39, 37–50 (2008) 10. Tardos, E.: A strongly polynomial minimum cost circulation algorithm. Combinatorica 5, 247–255 (1985) 11. Voß, S., Woodruﬀ, D.: Connecting mrp, MRP II and ERP supply chain production planning via optimization models. In: Greenberg, H. (ed.) Tutorials on Emerging Methodologies and Applications in Operations Research, p. 8.1–8.30. Springer, Heidelberg (2005)

Multiobjective Evolutionary Algorithm for Redesigning Sales Territories Loecelia Ruvalcaba1 , Gabriel Correa1, and Vittorio Zanella2 1

Universidad Aut´ onoma de Aguascalientes, Depto. de Sistemas de Informaci´ on, Av. Universidad, 940, Ciudad Universitaria, 20130 Aguascalientes, Ags., M´exico 2 UPAEP, 21 Sur 1103, Colonia Santiago, 72160, Puebla, Puebla, M´exico {lgruvalcaba,jgcorrea}@correo.uaa.mx, [email protected]

Abstract. Redesigning sales territories is a strategic activity that seeks to improve customer’s service level, sales costs and the quality’s life of the salesmen to gain a competitive advantage in the market. In this paper we propose a multiobjective evolutionary algorithm for redesigning sales territories inspired by a company dedicated to sell products along Mexico. One objective seeks to minimize new turnover variation against the current ones of the salesmen. The other objective looks at compacting territories through minimizing the sum of the distance traveled of its salesmen. Each territory is restricted to a maximum workload and the conservation of the residence places of the salesmen in new territorial conﬁgurations. Through an evolutionary algorithm we seek to solve large instances that have not been solved by an exact method.

1

Introduction

The high degree of competitiveness in the markets forces companies to focus on strategic activities. These activities diﬀerentiate them from its competitors. However, it requires that these activities are of practical implementation and do not generate excessive costs. Therefore, a major strategic activity aimed at improving the level of customer service, sales costs and the quality of life of the sales force, is redesigning sales territories. Redesigning sales territories consists of grouping small geographic areas called sales coverage units (SCU) in larger geographic areas called territories, in such a way that the latter are acceptable according to managerially relevant criteria [23]. Depending on the context, the managerial criteria may be motivated economically (e.g., minimal/maximal/average of current or potential sales, workload, number of customers or travel times) or have a population base (number of inhabitants, voter population), besides spatial constraints (contiguity and compactness) [11]. The particularity of the managerial criteria of the companies make it diﬃcult to generalize the problem. Additionally, although it is occasionally possible to reduce the managerial criteria of the companies to a single objective, they are often diﬃcult to deﬁne in these terms, since reality asks for multiple objectives. Therefore, deﬁning multiple objectives gives a better idea of the scope of the J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 183–193, 2011. c Springer-Verlag Berlin Heidelberg 2011

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problem. Designing sales territories is an application area that can be viewed as a districting problem and has been classiﬁed as NP-hard combinatorial optimization problem [18]. The consideration of multiple objectives for districting problems was proposed ﬁrst by Zoltners [22]. Guo et al. [10] developed and applied a multiobjective computational tool for devising optimal zoning called the Multi-Objective Zoning and AggRegation Tool (MOZART). MOZART is a graphical user interface that treats the problem of zone design as a problem of partitioning a mathematical graph that integrates Geographic Information System (GIS). Ricca [12] suggests a set of optimality criteria that ﬁnd good territorial aggregations and adopts the Old Bachelor Acceptance Heuristic to identify them. To understand the trade-oﬀ between these criteria, the author introduces appropriate optimality indexes and minimizes their weighted combinations with several diﬀerent sets of weights. Wei and Chai [19] present a multiobjective hybrid metaheuristic approach for GISbased spatial zoning. They propose a hybrid metaheuristic that integrates tabu search and scatter search method for generating non-dominated alternatives. Tavares-Pereira et al. [18] propose an evolutionary algorithm with local search for a multiobjective public service districting problem. The algorithm was applied to a real-world problem on Paris region public transportation. Ricca and Simeone [13] consider a multicriteria political districting problem they transform into a single objective model using a convex combination. They compare the behavior of the local search metaheuristics: descent, tabu search, simulated annealing, and old bachelor acceptance. The case application is performed with Italian political districting [13]. Gentile and Tiddi [8] propose a clustering algorithm for multiobjective automatic design of traﬃc zones. The method applies to a real case with the aim of analyzing the sensitivity of the result to the parameters of the heuristic and to the weights of the problem objectives. Correa et al. [5] propose a biobjective model for redesigning sales territories. The model looks for minimizing the total sum of the distances and the sales volumes variation for each salesman with respect to the current situation. They solve the model using the -constraint method to obtain the true eﬃcient set, and a heuristic method to obtain the approximate eﬃcient set. Both eﬃcient sets are compared to determine the quality of solutions obtained by the heuristic method. Salazar-Aguilar et al. [15] propose a GRASP (Greedy Randomized Adaptive Search Procedure) framework for a commercial territory design problem. They empirically developed two general schemes with two strategies for each one. They evaluated and compared these strategies with the Nondominated Sorting Genetic Algorithm (NSGA-II). Their test instances are based on real data from a beverage distribution ﬁrm in Mexico. The use of multiobjective evolutionary algorithms (MOEA) to solve problems has been driven primarily by the nature based on the population of the evolutionary algorithm (EA), which allows the generation of various elements of the optimal set of Pareto solutions (PF, Pareto’s Front) in a single run. Moreover, the complexity of multiobjective optimization problems (e.g., very large search spaces, uncertainty, noise, and disjointed Pareto curves), may avoid the use (or

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185

application) of traditional solutions of techniques multiobjective [4]. There are previous works that use EA to the design of territories with single or multiple objectives. Among works with a single objective [2] is included who propose a new approach to mono-objective zone design relying on the use of point data for electoral districting. They use a genetic algorithm (GA) and the process revolves around the positioning of the points that are used as the centers for the new regions to be built. Ba¸c˜ao et al. [1] presented and formalized mathematically an electoral districting problem. They used diﬀerent encoding for the problem, suited to GA optimization together with diﬀerent objective functions. A practical real world example for the Portuguese government is given and tests performed to evaluate the eﬀectiveness of the GA approach. Takashi [17] presents a method that adapts a basic GA to the zone designing problem where the objective function is to minimize the total distance to schools. He obtained the eﬀectiveness of a proposed GA when compared with other zone design methods. He solved a study case of Suita city, Osaka Prefecture. Xiao [20] describes a framework that uniﬁes the design and implementation of EAs for diﬀerent types of geographic optimization problems. A graph representation is used for deﬁning spatial structure. The framework is applied to a political redistricting problem. Some works for multiobjective design of territories include [3] who present a new technique based on MOEA to a large class of land management problems. They construct curves that illustrate the trade-oﬀs among various services. Their results illustrate that the MOEA-based approach can produce results equal to or signiﬁcantly more diverse than conventional integer programming techniques. Tavares-Pereira et al. [18] propose a MOEA for a public service districting problem [22]. Datta et al. [6] present a MOEA for a graph partitioning problem that involves the eﬀective partitioning of a graph into several disjoint subgraphs/zones, under multiple objectives and constraints. The developed MOEA is a modiﬁed form of NSGA-II. They applied this MOEA to four graphs randomly generated for partitioning them by optimizing three common objectives under ﬁve general constraints. In this paper we use an EA for redesigning sales territories for a real company that sells its products along Mexico, extending [5]. This company looks for two objectives in its new territorial conﬁguration. The ﬁrst seeks to minimize new turnover variation with respect to the current situation by each salesman and second, seeks to compact territories through the minimization of total distance traveled by its salesmen. In addition, the company established as constraints a maximum workload, the conservation of its sales force and the use of residence places of the salesmen as centers in new territorial conﬁgurations.

2

Problem Description

The case study in this paper is based on a real application from a company that distributes its products along Mexico. The company works with a system of compensation based on commission. The salesmen have built or inherited their customer portfolio and therefore its turnover through time without any

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control by the company management. There is no uniformity in the turnovers or workloads. It also presents territorial overlaps that result in an increase in service costs. Each territory has a particular geography, subject to the connectivity provided by the main communication routes (roads and state highways). Additionally the customer concentration is subject to the geography of the territory itself. To control its sales force, the ﬁrm divided the Mexican Republic into regions. For each region there are n salesmen. For each salesman the following is known: – A customer portfolio that represents the sum or amount of sales which is a base for the commission paid to each salesman. – Each salesman visits personally each customer. So, each customer has an attention time predeﬁned or known. – From the above condition it appears that every salesman has to travel a certain distance from his residence town to the customer locality. These distances are associated with variable travel costs. These costs must be reduced. – Salesmen currently do not have a work area (territory sales) deﬁned which leads to the existence of several customers who live in the same location but are served by diﬀerent salesmen, and in some cases, where a salesman has customers in the residence town of another salesman. In summary, there is no control of the salesmen assignment by the company. The objective pursued is to redesign the conﬁguration of the current regions considering two objectives deﬁned by the management of the company. The ﬁrst objective seeks to minimize the turnover variation in the new and existing portfolios from the salesmen so that they retain their income. The second objective looks for compact territories through minimizing the sum of distance traveled per salesman. The above is subject to a maximum load of working hours and indivisible SCUs that can only be served by a salesman. It is important mentioning that for this problem each population is considered as a SCU. Therefore, each SCU represents the sum of characteristics of all individual customers existing in the population that the SCU represents. It has also been established that a residence SCU must be considered as territorial center, so those residence SCUs with more than one salesman will be divided and each resident salesman retains his sales share.

3

Mixed-Integer Programming Model

Let C be a set of customers or SCUs demanding a sales volume mi ∈ R+ . They require wi ∈ R+ hours of attention per period. They are attended by a number of vendors V and they are at a minimum distance dij ∈ R+ from the salesmen. Each salesman has a current turnover T nowi ∈ R+ , which is obtained by adding the proportion of the demand xij of SCU i assigned to that salesman. Each salesman lives in a SCU k. In the redesigning of the new territories, we seek that new turnover T newi ∈ R+ has a minimum diﬀerence with respect to the current

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turnover T nowi . It was also established that the workload of the salesmen must be less than or equal to a maximum workload Cmax ∈ R+ and each SCU should be treated by a single vendor xij ∈ {0, 1} except those SCUs where more than one salesman lives. In this case, it has been determined that each salesman must retain its current share of sales aik ∈ R+ in the SCU. The problem described is modeled as a mixed-integer programming model as follows. Min Min

i∈V

i∈V

|1 −

j∈C dij xij 1 T nowi T newi |

(1) (2)

subject to

mj xij = T newi ∀i ∈ V

j∈C

(3)

xij = 1 ∀j ∈ C

(4)

wj xij ≤ Cmax ∀i ∈ V

(5)

i∈V

j∈C

xik = 1 ∀i ∈ V ∀k ∈ K

(6)

xij ∈ {0, 1} ∀i ∈ V ∀j ∈ C

(7)

Objective (1) aims to minimize the total sum of distances covered by the salesmen; (2) seeks to minimize the sum of the absolute diﬀerences between the new and current sales volume of every salesmen. The new sales volumes are obtained in accordance with the customers allocated to each salesman (Equation (3)). Equation (4) forces that each SCU is served by a salesman. Equation (5) restricts the workload of each salesman to the maximum workload Cmax . Equation (6) seeks to maintain the current ratio of sales in the residence SCU of the salesmen, while (7) ensures that the SCUs not being a place of residence will be attended in its entirety by only one salesman. Note that the model may be modiﬁed in various ways each having a diﬀerent conceptual character. Moreover, it can only be implemented in its entirety with appropriate adaptation. For instance, in the model we explicitly assume that mi and wi are proportional. Cmax is an upper limit set by the company (which might even be a requirement given by law as is the case in Mexico). In the current way of formulating Equation (6) we have maintained consistency with the binary character of the x-values. Though, this requires the use of ﬁctitious SCUs to represent diﬀerent proportions of sales for each salesman. Alternatively, one may deﬁne appropriate values aik and relate xij -values to those aik -values. If aik -values are predeﬁned appropriately, one might use a constraint as follows: xij = aik ∀i ∈ V ∀j ∈ C ∀k ∈ K

(8)

188

4

L. Ruvalcaba, G. Correa, and V. Zanella

Evolutionary Algorithm for Designing Territories

EAs imitate the basic principles of the evolution theory and apply genetic operators like mutation, crossover, and selection to a sequence of alleles. The sequence of alleles is the equivalent of a chromosome in nature and is constructed by a representation that assigns a string of symbols to every possible solution of the problem [14,21].

Fig. 1. Representation of an individual

The genetic operators’ features considered in this work are described as follows: – Although the mathematical model suggests the use of an individual with a multidimensional chromosome (i.e., number of SCUs and number of salesmen) their representation is simpliﬁed in the EA. Thus, an individual has a onedimensional chromosome. The chromosome is composed of as many genes as the number of SCUs of the problem, i.e., |C| elements. Each locus is associated with a SCU number and the value of the alleles is associated with the number of salesmen j. This last number indicates the salesman that caters to the SCU. Thus, the value of j must be within 1 and |V | (Figure 1). – The process of selection usually prefers better individuals to drive the search into good regions of the search space. So, in this work the selection of an individual for crossover and mutation operations is performed according to its proportional ﬁtness, i.e., according to roulette-wheel selection. To ensure that the best individuals are selected, a technique commonly known as “aggregation” is applied, because it combines all the objectives of the problem into one [4]. So the probability Pr (hi ) of selecting hi is obtained from the following equation Pr (hi ) =

wg /F itness(Objgh ) i i∈C Objghi

(9)

g∈G

where hi represents each of the individuals in the population, wg ≥ 0 are the weight coeﬃcients representing the relative importance given to each objec tive within the problem, where also g∈G wg = 1 and F itness(Objghi ) = 1/Objghi . For a more dispersed PF, the weighting coeﬃcients are assigned randomly in each generation.

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– The new population is obtained by means of two techniques. The ﬁrst consists in a survival operation based on a vector evaluation GA (VEGA), which was proposed by Schaﬀer; cf. [4]. VEGA is basically a simple GA with a modiﬁed selection mechanism. In each generation a proportion of the population (50%, in this case) is generated by making a selection according to an objective function at a time (i.e., 25% of the population for each objective). Second part of the population is obtained through a one-point crossover. The one-point crossover generates two individuals, whose parents are obtained randomly according roulette-wheel selection (Equation (9)). One-point crossover is random. The ﬁrst part of the chromosome is obtained from one parent and the rest from another. To diversify the search and prevent from premature convergence a mutation operator is used. We have established that mutation occurs every three generations and that 25% of the individuals in a population are mutated. For mutation, the individuals are select randomly. Two types of mutation operators are applied. The ﬁrst consists of a random reallocation of SCUs. Here the chosen SCU is allocated to a diﬀerent salesman of the current one. The second type of mutation consists of exchanging two SCUs. For this, two SCUs allocated to two diﬀerent salesmen are randomly selected and are exchanged between them. – We established convergence as termination condition. To try preventing convergence to a local minimum, we decided that this condition is present when there is a certain number of successive generations with similar average ﬁtness. In case of a local minimum, the new population will consist of only 20% of individuals obtained from VEGA and the remaining from crossover. Additionally we mutate up to 40% of the population. We suppose convergence to a global minimum when the average ﬁtness of the latest number of local minima is similar.

5

Comparison Metrics

It is known that in this case several PFs are obtained with the EA, we compare these in pairs S1 and S2 through the error ratio (ER) and the generational distance (GD); see [4] for background on these metrics. Here, ER indicates the percentage of S1 solutions, which are not members of S2 . Mathematically this is represented in equation (10). ER =

n

ei /n

(10)

i=1

where n is the number of vectors in S1 ; ei = 0 if vector i is a member of S2 and ei = 1, otherwise. On the other hand, GD is a way of estimating the distance between the sets S1 and S2 . Mathematically this is deﬁned as follows. n 2 i=1 di (11) ER = n

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L. Ruvalcaba, G. Correa, and V. Zanella Table 1. Characteristics of test instances

Instance number Number of salesmen Number of SCUs Number of diﬀerent |V | |C| residence SCUs |K| 1 48 4 3 2 31 4 2 3 93 6 6 4 618 42 29 Table 2. Feature summary of PFs obtained by an exact method Instance number

∗

1 2 3 4 : Results

Time required Range of values Number of non-dominated solutions Hours:Minutes 1 20:15 [6,915 ... 13,133] 30 0:05 [5,106 ... 5,892] 6 1:58 [9,246 ... 9,996] 149 840:38 * * were not obtained.

where n is the number of non dominated vectors in S1 and di is the Euclidian distance (measured in objective function space) between each of them and the closest member of S2 . Note, that under optimal behavior of the algorithm both ER and GD are equal to 0, since all vectors in S1 belong to S2 .

6

Experimentation and Computational Results

To simplify sales management, the company (described in Section 2) divided Mexico into several regions. We reuse this territorial division in order to reduce the complexity of the problem. Three regions and the entire instance are used to test the scope of the EA. We used network distances in this work. The point to point distances are obtained from a (non-complete) graph. In this graph the edge weights were calculated from the distance between primary and secondary highways associated to SCUs and maps of Mexico’s States obtained from the SCT (Communications and Transportation Secretary) [16] and maps of Mexico’s Highways [9]. We obtain the network distances by partial implementation of Floyd’s algorithm [7], omitting the peer route calculation. Table 1 shows the characteristics of the test instances used in experimentation. Correa et al. [5] previously obtained the solutions for an exact method. They solved the problem with the -constraint method using GAMS 21.0/CPLEX 8.0. Table 2 shows a summary of times, value range and non-dominated solutions values which are obtained by an exact method for each instance. On the other hand, we implemented the EA in Free Pascal for Windows Version 2.2.2. Both methods were run on a HP Compaq laptop nx6125, processor AMD Semptron 2800+ at 1.60 GHz and 2 GB in RAM memory. Ten experiments per instance were carried out. We used diﬀerent population sizes for each instance. Table 3 shows the average features summary of the PFs obtained by

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Table 3. The average features summary of the PFs obtained by the EA proposed Instance Time required Average ranges Average ranges Number of number Hours:Minutes of turnover of total distance non-dominated Min Max Min Max solutions 1 1:03.06 0.2428 0.6041 6,915 15,556.5 33 2 0:00.13 0.0187 0.3145 5,106 5,892.0 6 3 0:27.07 0.0003 1.9490 9,246 19,340.0 66 4 17:05.80 13.8087 3.6007 75,467 612,086.6 200 Table 4. Mean, standard deviation and coeﬃcient of variation by instance Instance Mean Standard number ER GD ER 1 2.08E-01 1.69E-03 1.34E-02 2 0.00E+00 8.84E-05 0.00E+00 3 2.92E-01 2.58E-03 1.39E-01 4 8.02E-01 1.66E-01 7.29E-03

deviation Coeﬃcient GD ER 2.31E-03 6.42E-02 7.58E-05 0.00E+00 3.15E-03 4.76E-01 1.83E-01 9.09E-03

of variation GD 1.37E+00 8.58E-01 1.22E+00 1.11E+00

the proposed EA. Table 4 presents the mean, standard deviation and coeﬃcient of variation obtained from comparison metrics per instance.

7

Conclusions and Discussion

In this paper we have presented a biobjective evolutionary algorithm applied to redesigning sales territories. The model considers the conservation of the sales force and existing customers. As a ﬁrst objective, we seek to minimize the new turnover variation with respect to the current ones to hold the income of the salesmen by the concept of commissions per sales, while the second objective seeks to minimize the total distance covered in the new territories. The places of residence of the salesmen are territorial centers in the new conﬁgurations and when more than one vendor resides in the same locality, it retains its current share of sales. In addition, the workload required to care for the new territories should be less or equal than 160 hours per month. The problem considered is NP-hard, so it was resolved by using an EA. We considered a diﬀerent population size for each instance. The initial population is semi-random. For the generation of new populations, we use the VEGA modiﬁed selection mechanism and a system of crossover between individuals, whose roulette-wheel selection is obtained from an aggregation function that integrates the two objectives through random weights ranging from one generation to another. Moreover, each three generations a 25% of individuals is mutated to give more variety to the population. The termination condition is convergence-based. Through the EA, we solve large instances that could not be resolved through the exact method. The EA tends to minimize the average values of the objectives over time. A feasibility analysis will be necessary for the implementation of the solutions obtained by the exact method versus the EA, because this will

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provide the basis for the improvement of the results obtained or to take some additional considerations into the model. However, it is up to the decision maker to select those conﬁgurations that better adapted themselves to his needs. Basing on this evaluation it is determined about whether it is required to readjust the parameters of the EA or if a more robust implementation of multiobjective evolutionary strategies is required, thus obtaining better solutions for large instances, and reduce the values of ER for these. By now, it is clear that the conﬁgurations obtained through the EA improve the current conﬁguration since they ensure that only one salesman attends the non residence SCUs. This condition avoids the duplicity of travel costs. In addition, the determination of routes and scheduling for the SCUs is desirable, as the intention is to have a suitable integration of travel costs and times. The latter point will allow us to evaluate in a more direct way the impact of the new territorial conﬁgurations and their implantation. Additionally, since the PF shows a higher reduction for total distance than for turnover it will be necessary to test the eﬀect of normalization of the objectives in this ﬁeld. Acknowledgement. We appreciate the support of CONACYT and Autonomous University of Aguascalientes during this research and for the presentation of this work.

References 1. Ba¸ca ˜o, F., Lobo, V., Painho, M.: Applying genetic algorithms to zone design. Soft Computing 9, 341–348 (2006) 2. Ba¸ca ˜o, F., Painho, M.: A point approach to zone design. In: 5th AGILE Conference on Geographic Information Science, Palma, Balearic Islands, Spain (2002) 3. Bennett, D., Xiao, N., Armstrong, M.: Exploring the geographic consequences of public policies using evolutionary algorithms. Annals of the Association of American Geographers 94, 827–847 (2004) 4. Coello, C., Lamont, G., Van Veldhuizen, D.: Evolutionary Algorithms for Solving Multi-Objective Problems. Springer, Berlin (2007) 5. Correa, J., Ruvalcaba, L., Olivares-Benitez, E., Aguilar, J., Macias, J.: Biobjective model for redesign sales territories. In: 15th Annual International Conference on Industrial Engineering: Theory, Applications and Practice (IJIE), Mexico (2010) 6. Datta, D., Figueira, J., Fonseca, C. M., Fernando, T.P.: Graph partitioning through a multi-objective evolutionary algorithm: A preliminary study. In: Genetic and Evolutionary Computation Conference (GECCO 2008), Atlanta, GA, pp. 625–632 (2008) 7. Floyd, R.W.: Algorithm 97: Shortest path. Communications of the ACM 5, 345 (1962) 8. Gentile, G., Tiddi, D.: Clustering methods for the automatic design of traﬃc zones. In: SIDT International Conference, Milan, Italy (2009) 9. Roji, G.: By Mexico’s highways (2010) 10. Guo, J., Trinidad, G., Smith, N.: MOZART: a multi-objective zoning and aggregation tool. In: Proceedings of the Philippine Computing Science Congress (PCSC), pp. 197–201 (2000)

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11. Kalcsics, J., Nickel, S., Schr¨ oder, M.: Towards a uniﬁed territory design approach - applications: Algorithms and GIS integration. Fraunhofer ITWM, Kaiserslautern (2005) 12. Ricca, F.: A multicriteria districting heuristic for the aggregation of zones and its use in computing origin-destination matrices. INFOR 42(1), 61–77 (2004) 13. Ricca, F., Simeone, B.: Local search algorithms for political districting. European Journal of Operational Research 189, 1409–1426 (2008) 14. Rothlauf, F.: Representations for Genetic and Evolutionary Algorithms, 2nd edn. Springer, Berlin (2006) 15. Salazar-Aguilar, M.A., R´ıos-Mercado, R.Z., Gonz´ alez-Velarde, J.L.: GRASP strategies for a bi-objective commercial territory design problem. Journal of Heuristics (2011); doi:10.1007/s10732-011-9160-8 16. Secretar´ıa de Comunicaciones y Transportes / Communications and Transportation Secretary, http://www.sct.gob.mx/carreteras/, last call (July 15, 2011) 17. Takashi, K.: Designing elementary school districts using a genetic algorithm: Case study of Suita City, Osaka. Geographical Review of Japan 79(4), 154–171 (2006) 18. Tavares-Pereira, F., Figueira, J., Mousseau, V., Roy, B.: Multiple criteria districting problems. The public transportation network pricing system of the Paris region. Annals of Operations Research 154, 69–92 (2007) 19. Wei, B.C., Chai, W.Y.: A multiobjective hybrid metaheuristic approach for GISbased spatial zone model. Journal of Mathematical Modelling and Algorithms 3, 245–261 (2006) 20. Xiao, N.: A uniﬁed conceptual framework for geographical optimization using evolutionary algorithms. Annals of the Association of American Geographers 98, 795–817 (2008) 21. Z¨ apfel, G., Braune, R., B¨ ogl, M.: Metaheuristic Search Concepts: A Tutorial with applications to Production and Logistics. Springer, Berlin (2010) 22. Zoltners, A.: A uniﬁed approach to sales territory alignment. In: Sales Management: New Developments from Behavioral and Decision Model Research, pp. 360–376. Marketing Science Institute, Cambridge (1979) 23. Zoltners, A., Sinha, P.: Sales territory alignment: A review and model. Management Science 29, 1237–1256 (1983)

Optimizing Complex Logistics Systems with Approximative Consideration of Short-Term Costs Tobias Winkelkotte Deutsche Post Chair of Optimization of Distribution Networks RWTH Aachen University

Abstract. This paper provides an approach which optimizes complex logistics networks strategically, considering the total operational and tactical costs. Real-world-sized instances of models which consider the costs of all time-horizons exactly are very complex and very diﬃcult to solve. Beside this it does not make much sense to completely plan operational details (that will never be executed later on). So, we formulate a model which uses an appropriate approximation of short-term costs. This makes the model and the solution process much easier. But the model turns out to be very complex, nevertheless. So we introduce a heuristic to solve it and to gain a satisfactory solution.

1

Introduction

Strategic planning of distribution networks means considering logistics processes in each time horizon: There are strategic decisions, e.g., on location of facilities and their capacities, but there are also tactical and operational processes which have to be taken into account. Such processes could be the allocation of resources, the planning of distribution tours, the assignment of drivers to vehicles etc. The problem now is, that at the time when strategic planning takes place the decision maker does not have any concrete information about operational details. If there are, for example, vehicle routes to plan, at the time of facility location one does not know which customers have to be served on which day, where the speciﬁc customers will be located, and which demand they will have. So the question is how to consider processes one does not know anything speciﬁc about. A common approach to solve such problems is to solve a vehicle routing problem (or whatever the operational problem is) with a set of exemplary customers and demands. But this method has some problems. Apart of that it is somehow artiﬁcial one has a high eﬀort to plan tours that will never be performed in reality – and this only to calculate costs of tours, that may, above that, not be realistic. So in this paper we are going to follow a diﬀerent approach: Using only the available information (that is: a customer and demand distribution) we will approximate tactical and operational costs to gain appropriate cost information to use in the strategic problem. This requires an extension of the facility location J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 194–208, 2011. c Springer-Verlag Berlin Heidelberg 2011

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problem which meets the requirements of the situation in which we do not have information about speciﬁc customer locations. 1.1

The Problem to Solve

In this paper we are going to adapt the approach with which a lot of realistic distribution networks might be planned: In order to locate facilities, managers look at a map of the distribution area and choose locations for a number of facilities. Then they draw border lines around each facility to deﬁne the regions for which each facility is responsible. The result is a partition of the total area into smaller regions, and the located facilities have to organize the distribution process within the region they are in charge for. The decision where to draw a border line depends on the demand density, but not on potential customer locations. Since this method uses only visual information it might not lead to an optimal solution. So, in this paper we want to adapt this methodology in order to ﬁnd optimal locations of facilities and an optimal partition of the area into regions. On the ﬁrst sight this requires the formulation of a capacitated facility location problem (CFLP): Assuming a ﬁnite set of potential facilities, one decides on whether to open each of these facilities and which customers to assign to them. In the present case we will not assign actual customers, but small geographical objects, which in the following we are going to call “districts”, where the (unknown) customers are living in. The only available information about these districts (beside their sizes) are the customer and demand distributions. When assigning such geographical areas to facilities there will be arising a problem: In the “planning-by-hand” approach described above one will always get connected areas to allocate to each facility, which is a desired feature of distribution areas. But when we plan this by solving a mixed-integer linear program, the solution most probably will not have this property. So we need to add a new set of constraint to the CFLP which guarantees the connectivity of the regions. Unfortunately these connectivity constraints will, as we see later, make the program very very complex. It will not be possible to solve it exactly for realworld-sized problem instances. This makes it necessary to implement heuristic search methods to at least be able to ﬁnd a near-optimal solution. 1.2

Structure

This paper is structured as follows: In Section 2 we will provide a mixed-integer, but in general non-linear program, with which we are able to model planning situations as described above. Starting from the well-known CFLP, some extensions will be introduced, and we will brieﬂy demonstrate how to construct an appropriate objective function which calculates short-term costs approximatively. We also use a linearized version of this model to calculate an optimal solution, which will be, however, only possible for small instances. So in Section 3 we are going to give an overview of a specialized heuristic search method

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which ﬁnds near-optimal solutions of the non-linearized model and which can even be applied to big instances.

2

The Model

The main goal of the problem described above is the location of facilities. Therefore, the starting point of the model to formulate comes from the location theory. A good overview of such models gives, e.g., Klose [7]. For our problem we choose a CFLP with single sourcing (CFLP-SS) and with capacity stages. Winkelkotte [9], Section 2.1.1 justiﬁes this decision. In the next subsection we ﬁrst introduce the CFLP-SS, and after that we continue with transforming this model into another one that describes the problem from Section 1. 2.1

The Capacitated Facility Location Problem

With the CFLP-SS, a set of customers, I, who have certain demands, wi (i ∈ I), have to be served from the facilities to be located. These facilities have to be chosen from the set of potential facilities, P. A facility, p ∈ P, can be opened n with diﬀerent capacities, which are given by the (ﬁnite) set Kp = {Kp1 , . . . , Kp p }. The costs of running facility p ∈ P with capacity k ∈ Kp are given by fpk , and the costs of allocating customer i ∈ I to facility p ∈ P are given by cpi . With these assumptions we deﬁne the following decision variables: 1, if facility p ∈ P is working with capacity k ∈ Kp ypk := 0, otherwise 1, if customer i ∈ I is allocated to facility p ∈ P zpi := 0, otherwise With this we formulate the CFLP-SS as follows: min cpi zpi + fpk ypk p∈P,i∈I

s.t.:

(1a)

p∈P k∈Kp

zpi = 1 ∀i ∈ I

(1b)

i∈I

ypk = 1 ∀p ∈ P

k∈Kp

i∈I

wi zpi ≤

Kpk ypk ∀p ∈ P

(1c) (1d)

k∈Kp

ypk ∈ {0, 1} ∀k ∈ Kp , p ∈ P

(1e)

zpi ≥ 0 ∀i ∈ I, p ∈ P

(1f)

Equation (1a) is the objective function, which minimizes the total costs of location and allocation. With (1b) we can ensure that each customers is allocated once and only once to any facility, whereas (1c) guarantees that all the facilities

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will be open on one and only capacity stage (note that capacity 0 is also deﬁned as a capacity stage). Constraint (1d) makes sure that the sum of demands of the customers who are allocated to the same facility does not exceed its capacity. The non-negativity constraints, (1e) and (1f) make all decision variables being 0 or 1, the latter therefore containing the single-sourcing property. This model has now to be applied to our special case, where the customers are geographical units and where the customers allocated to the same facility have to satisfy a connectivity condition. 2.2

Additional Constraints

Before formulating constraints to ensure connectivity of regions we have to stipulate an appropriate structure on which the deﬁnitions base. At ﬁrst we are dealing with geographic objects, i.e., areas, so that we can not decide for each point in the plane where it should be allocated to. The CFLP-SS is a discrete program, so we also do need a discrete structure. As mentioned above we partition the whole area into small pieces which we call districts. This partition has to have certain properties: – The districts must not overlap each other. – The union of all districts have to cover the total area. In the following we will only take partitions into account which fulﬁll these conditions. In the next step we deﬁne the so-called connectivity graph, which is an undirected unweighted graph, G = (V, E), where the set of vertices, V , contains exactly one vertex for each district, and the set of edges, E, is deﬁned by: E := {(vi , vj ) ∈ V × V : the corresponding districts of the vertices vi and vj do have a common border}. Per deﬁnition, G is a connected graph, which will be described with its adjacency matrix, A := (aij )ni,j=1 (n being the number of districts). A subset of districts is called connected, if the corresponding subgraph of G is a connected graph. A connected set of districts, i.e., a connected subgraph of G, will be called a region. In what follows we can use the discrete structure of the connectivity graph to describe connectedness and regions. Obviously, the partition of the total area is not unique: there is an inﬁnite number of possibilities to conduct a partition. When performing the partition one should take care that the districts are neither to big nor to small. If they are to big, it might not be possible to ﬁnd an optimal set of regions. If they are to small, the numerical eﬀort to solve the problem will be to big without really being able to improve the solution compared to a partition with bigger districts. The size of the districts of a partition surely depends on the context. We suggest to do it rather pragmatically: Use a partition for which it is easy to gain data, e.g., administrative districts which are already well-deﬁned.

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Anyway, we assume the partition to be given and appropriate. The question now is: Given a facility and a set of allocated districts – how can we decide whether or not the districts are connected? The only information we have for this decision are the values of the decision variables of model (1). It is easy to see, that a region is connected, if and only if:

aij ≥ 1

(2)

i∈S,j∈V \S

for any subset of districts, S ⊆ V, S ∈ {∅, V }. But this condition is obviously not enough, because S should be the result of the optimization and can not be assumed before. So we have to manipulate the index sets on the left-hand side of the subequation and the constant on its right-hand side. Without going into further details, we end up at the following constraint, which guarantees connectivity for all regions:

aij zpj ≥ 1 +

i∈S1 ∪S2 , j∈S1 ∪S2

zpi − |S1 ∪ S2 |

i∈S1∪S2

(3)

∀S1 , S2 V mit S1 ∩ S2 = ∅, aij = 0 for i ∈ S1 , j ∈ S2 , p ∈ P

This condition has to be satisﬁed for all p ∈ P to make the solution feasible. The complete proof of this constraint can be found in Winkelkotte [9]. After having made sure that every region is connected, we would like to add another new constraint: If the regions are connected it seems to make sense that the facility, which is responsible for the region, is also located within it. Since the districts cover the total area, and under the assumption, that the potential facilities are also situated within it, then for each facility there must be a district which it belongs to. We state this by the following parameter: πpi :=

1, if facility p ∈ P is located within district i ∈ V 0, otherwise

Then, a facility lays in its region, if the following constraint is satisﬁed:

zpi ≥ πpi

k∈Kp

ypk ∀i ∈ V, p ∈ P

(4)

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Considering all this we end at the following model, which we call CFLP with Connectivity Constraints: min cpi zpi + fpk ypk (5a) p∈P i∈V

N.B.:

p∈P k∈Kp

zpi = 1 ∀i ∈ V

p∈P

wi zpi ≤

i∈V

(5b)

Kpk ypk ∀p ∈ P

(5c)

k∈Kp

ypk = 1 ∀p ∈ P

k∈Kp

zpi ≥ πpi

(5d)

ypk ∀i ∈ V, p ∈ P

k∈Kp

aij zpj ≥ 1 +

i∈S1 ∪S2 , j∈S1 ∪S2

(5e)

zpi − |S1 ∪ S2 |

i∈S1∪S2

(5f)

˜ i ∈ S1 , j ∈ S2 , p ∈ P ∀S1 , S2 V mit S1 ∩ S2 = ∅, aij = 0 fAr ypk , zpi ∈ {0, 1} ∀i ∈ V, k ∈ Kp , p ∈ P

(5g)

The objective, (5a), is, for the moment, the same as in model 1, as well as the constraints (5b), (5c), (5d), and (5g). Constraint (5e) is the constraint that ensures the facility to be in the allocated region, and (5f) is the connectivity constraint introduced above. The problem with this model is, that the number of connectivity constraints is exponentially big. One can show, that it is equal to O(3|V | ) (see Winkelkotte [9], Section 2.2). That makes it impossible to solve realistic instances of the model in a reasonable time, which we will see in Section 2.4. Therefore, in Section 3 we are going to introduce a heuristic solution method that is able to ﬁnd near-optimal solutions in a relatively short time. But at ﬁrst we have to further extend the model: We did not talk about how to realistically calculate tactical and operational costs, which is the main idea of the approach. This will be done in the next subsection. 2.3

Objective Function

Subject of this paragraph is the formulation of an objective function which approximates the tactical and operational logistics costs. Such a cost function will always describe the processes of a speciﬁc logistics network, so it is not possible to provide a general function to use as an objective. But this is the strength of the model and of the algorithm provided in Section 3: It is possible to strategically optimize every distribution network, independent of its special operational details. One “only” has to analyze the short-term processes and ﬁnd a cost function which estimates these costs appropriately. Such analysis is necessary for the model but, however, every logistics provider should analyze his processes in order to be able to run the network economically.

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Hence, in this paper, we can not provide a general model. But we are going to describe how such a logistics cost function could look like. We assume the following usual usual: The logistics network consists of a set of facilities, from where the distribution processes start (these facilities are to be found with our model). The distribution tours, starting from the facilities, have to serve each customer of his demand. We assume the total demand to be much bigger than the capacity of a truck, such that a lot of tours have to be conducted. Let us assume for the moment that we have only one facility with a precisely deﬁned distribution area. How does the logistics cost function look like in this case? The ideas of the following analysis are based on Daganzo [3] and [4]. Starting point of the analysis is the customer distribution: We do not know where the single customers are living at, but we do know how dense they are distributed over the area. The customer density will be given by δ (customers per unit of area). We also need to know the customers demands, which in average we assume to be equal λ (items per customer). With some further information about the distribution area – let A be its size and r¯ the average distance from the facility to any point within the area – and about some “technical” details – mainly the truck capacity, C – we can estimate the distances to overcome. We use the cost rates cs , cv , and cd for the costs of one stop, of handling costs of a single item and the costs of overcoming one distance unit, respectively, to formulate the following cost function of distribution: X + 0,57N δ −1/2 )cd , (6) C where N := Aδ is the number of customers, and X := N λ is the total demand. This example is quite simple, of course. But we do not want to go into more details, because this gives no new insights. But it is possible to incorporate much more details into such a cost function. It is, e.g., possible to calculate costs of a two-stage distribution network: Trucks ﬁrstly transport the items to transshipment points, where the items a loaded onto other (smaller) trucks, which travel to the customers to serve them. A respective cost function contains a independent variable for the number of transshipment points, and the costs of such a system result from the cost minimal number of transshipment points. A more detailed analysis of this case can be found in Winkelkotte [9]. Campbell [1] provides a model to integrate inventory costs. For deeper insights into the theory of logistics systems analysis the interested reader is referred to Daganzo [5]. Now, in model (5), neither the facility nor the assigned area are ﬁx – these are the results we want to calculate with the model. Hence, the cost function depends on which districts are allocated to a certain facility, so C := C(zpi ). The cost function (6) has to be applied to the union of these districts, each of which has diﬀerent values for all parameters of C. In order to only use the simple cost function (6) we calculate weighted averages of all values and insert them into the cost function. Finally the resulting objective function to be used in model (5) can, e.g., be written as: C({zqi : q = p}) + fpk ypk (7) min C := Xcv + ncs + (2¯ r

i∈V p∈P

p∈P k∈Kp

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In general, this function is not linear. We do not even have any general information about the mathematical structure of the function. So it is impossible to use an exact solver to ﬁnd a solution for the problem as it is now. However, in the next subsection we calculate a linearized model. But the results are in some ways not satisfactory, which will be explained later. 2.4

Exact Solution

In this paragraph we summarize the solutions we received by solving model (5) with CPLEX. Obviously it is not possible to solve the model with objective function (7), but it is possible to linearize it. Therefore, we deﬁne cpi to be the approximated logistics costs of serving district i from facility p. We can calculate these costs before starting CPLEX and receive total costs for each region by just adding the cost factors up, like it is shown in Eq. (5a). However, another problem arose when trying to solve the model: Due to the very large number of constraints (the model does have O(3|V | ) connectivity constraints) it is not only impossible to solve it, but CPLEX is also not able to just represent it as a solvable mathematical program. So we used a row-generating method, which ﬁrstly solves the program without any connectivity constraints. Then the algorithm investigates whether there are violated connectivity constraints and, if there are any, will add these to the program to resolve it. With this method it was possible to solve most of the small instances, but for the medium-sized and big instances it is be possible to ﬁnd a solution. The absolute objective function values are not of interest here, because they say nothing about the performance of the algorithm or about the solution. We only use them to evaluate the heuristically calculated solutions, which we receive later on. By this time, only the computing times might be interesting, which are shown in Figure 1. This graphic shows that there are a lot of instances which can be solved to optimality within a short time, but there are also some instances (which are very small) for which CPLEX needs more than an hour. Before drawing out attention to the heuristic method the exact solution (if is are one) should be evaluated qualitatively. At ﬁrst we see that only small instances can be solved. This is a problem of course, since the small instances do not have a realistic size. However, it might be possible to even solve some bigger instances, if the total area is decomposed into several smaller ones or something like that. But there is another, more serious problem: CPLEX is only able to solve the model with a surrogate objective function. That means, that it does not only solves a model, but also a simplified model of reality. So one must ask the question whether this still describes the real-world processes, which is essential for stressable solutions. Beside this it seems useless to search for exact solutions with a huge eﬀort while the model uses approximations to calculate the resulting costs. But for heuristic methods, it does not matter what the objective function, or even the constraints, look like. Such methods can be applied to a model of any structure, and this will be done in the next section.

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Fig. 1. Distribution of CPLEX computing times

3

Heuristic

As we have seen in the previous section it is not possible – or at least it takes a long time – to solve model (5) exactly. So in this section we present an approach to ﬁnd a near-optimal solution heuristically. It should be mentioned, that it is not the aim of a heuristic to ﬁnd the exact optimal solution. Of course, we want to ﬁnd as best a solution as possible. But, using heuristical methods, the lack of solution quality can be justiﬁed by several reasons: – At ﬁrst there is, of course, the trade-oﬀ of solution quality and computing time. Facility location problems are, admittedly, strategic problems for which the decision can take a longer time, but it is also necessary to run scenario analyses, where a lot of instances should be solved. So it is also important, that the solving times are not too long. – All optimization models are what they are called: models! They are not an exact description of reality, so an exact solution of a model is not necessarily an exact solution of the real-world problem. Actually, it might be far away from it. – Input data is never exact, especially operational data in strategic problems. So why calculate exact solutions based on non-exact data? In our case we even use approximations of the resulting costs. To sum it up, we decide upon using a heuristic. Our starting point are local search procedures which are well-known from ordinary facility location problems. The best-known local search methods in facility location are Add and Drop, which, starting from a feasible solution, open or close facilities and reallocate the customers. Since we have facilities which can be utilized on diﬀerent capacity stages we split both of these methods into two:

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– Add: The capacity of a facility can be enlarged to the next stage, but opening a closed facility is not possible – Open: Facilities, which are closed in the current solution, can be opened on any capacity stage – Drop: A facility’s capacity will be reduced by one stage – Close: A Facility, which is currently open on any stage, will be closed completely Furthermore we have to consider the additional constraints which stipulate the connectivity of regions, i.e., we can not freely choose which set of districts we want to allocate to a facility. We will explain the idea of ensuring connectivity exemplarily with the Add heuristic in Section 3.1. Considering connectivity makes the mentioned heuristic not very eﬀective, because only very small changes can be accomplished. The reason is that regions can only be modiﬁed at their borders. For bigger changes of the solution we introduce the following two heuristics which can aggregate two facilities or split one into two: – Centralize: Two facilities which regions border to each other will be closed, and another one will be opened instead. All the districts which have been allocated to the ﬁrst two will be assigned to the new facility. It is also possible that the new facility coincides with one of the old ones. So, two very small facility can be combined to a reasonable-sized one. The advantage of this operator is that we do not have to check whether the new region is connected. – Partition: This is the reverse operator to “Centralize”. One facility will be closed and its districts will be assigned to two others which were closed before. If the region has become very big, this operator makes two smaller ones from it. After repeatedly conducting these operators, it often happens that the facility is positioned very unfavorably within its region. To ﬁnd better positions, we introduce a seventh operator: – Switch: All districts that are assigned to the same facility will be removed from it and assigned to another one which was closed before. The new facility has to be positioned inside the region (according to constraint (5e)), so we do not have to check a lot of candidates. These seven local search operators will be applied simultaneously, in order to ﬁnd a local minimum very fast. The procedure is stirred by a metaheuristic. We use tabu search here, because this is a metaheuristic that is based on local search, and a lot of authors have reported about good experiences with tabu search applied to facility location problems, e.g., Crainic et al. [2] or Voß [8]. Since tabu search is a well-known method we do not want to explain it here. A good introduction of the general idea and details of diﬀerent concepts can be found in Glover and Laguna [6].

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3.1

Local Search

In this section we are going to illustrate how well-known local search heuristics are adapted such that the resulting solution meets the requirements of model (5). The idea is mainly the same for all the seven procedures, so we only explain it exemplarily for the Add method. Details of the other methods can be found in Winkelkotte [9]. When enlarging a facility with the Add heuristic it is not possible just to reallocate the customers (i.e., the districts) say by solving an allocation problem. In general the resulting allocation will not form connected regions. So we have to use another strategy. In the following we assume that the facility to enlarge is already open and that there is a region (i.e., as set of connected districts) assigned to it. Then the corresponding neighborhood of the current solution is given by: ˜p,k+1 = 1, ADD(y, z) := {(˜ y, z˜) : ∃!ˆ p :ypk ˆ =1 ⇒y ˆ Kpk y˜pk ≤ Kpk ypk ∀p ∈ P \ {ˆ p}}, k∈Kp

(8)

k∈Kp

whereas this deﬁnition only refers to the properties of the facilities. Additional constraints, like they are given in Model (5) have to be fulﬁlled. The general philosophy of enlarging the region is to only look at the neighboring districts. Only these are of interest to be added to the region, because only these have a direct connection to, and the connectivity constraint continues to be satisﬁed for this region. If there are several neighboring districts we have to choose some of them, which we do by solving a knapsack problem. This is subsumed in the following algorithm: 1. Deﬁne R := {i ∈ I : zpi ˆ = 1}, DR := i∈R wi , K := current capacity stage, K := next capacity stage. 2. Determine the set of neighboring districts which do not contain an open facility, and for which the it is cheaper to serve them from pˆ than from the current facility: cpj zpj }, N := {j ∈ I \ R : ∃i ∈ R : aij = 1, ypk = 0∀p ∈ P, k ∈ Kp , cpi ˆ < p∈P

3. 4. 5. 6. 7.

and the total demand of these districts, DN := j∈N wj If N = ∅: Go to 6. If DR + DN ≤ K: Set R := R ∪ N , DR := DR + DN . Go to 2. Otherwise: Choose the best subset of N by solving a knapsack problem. If DR > K: Go to 10. Determine the set of neighboring districts, which do not contain an open facility: N := {j ∈ I \ R : ∃i ∈ R : aij = 1, ypk = 0∀p ∈ P, k ∈ Kp } with a total demand of DN := j∈N wj .

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8. If DR + DN < K: Set R := R ∪ N , DR := DR + DN . Go to 6. 9. Otherwise: Choose the best subset of N by solving a knapsack problem. 10. Remove all j ∈ R from their original facilities. STOP. The application of this algorithm on a simpliﬁed case is depicted in Figure 2. After applying this algorithm the region assigned to pˆ will be connected. But by removing districts from other regions these might be cut into pieces. So it is necessary to check the connectivity of the regions from which districts have been removed.

Fig. 2. Add procedure (simpliﬁed): (1) Connectivity graph of the starting situation with three regions (black, dark gray, light gray). The black region is to enlarge. (2) Investigation of all regions that can be reached from the facility. (3) Determination of the districts which border to the black region. (4) Demand of neighboring districts does not exceed the facility’s demand. All the districts are added to the region. Again the neighboring regions are determined. (5) Capacity is not big enough to add all the districts. Choose the best subset of the districts and add them to the region. The other regions go back to their original facilities. (6) Also the last district’s demand is too big for the capacity. It remains at its former facility.

3.2

Computational Results

We have conducted some hundreds of computational tests to ﬁnd parameters of the tabu search algorithm which lead to the best solutions. The numerical results of these test are irrelevant at this place. The important message is: It is possible to solve instances of any size, even with thousands of districts and hundreds of potential facilities. The computing times are reasonable. Of course they are as longer as bigger the instances are, but at least it was possible to get any result. The small instances which have also been solved exactly (see Section 2.4) could be solved within seconds.

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Fig. 3. Comparison of the optima with the heuristically determined solutions (using the linearized objective function)

Fig. 4. Diﬀerences between solutions with the linearized and the not-linearized model

But much more important than the computing times is the quality of the solutions. A comparison is only possible for the small instances for which we could ﬁnd the exact solution. The deviations are smaller than 5% for most of the instances, and smaller than 10% for all instances, as it is shown in Figure 3. Much more important is the comparison of the solutions of the linearized version of model (5) and of its non-linearized one. This shows how much of solution quality will be lost when linearizing a model. Figure 4 shows by which

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factor the linearized solution is bigger than the non-linearized one. One can easily see, that there are instances with deviations of almost 50%. Admittedly that does not show whether the solutions (and not the objective function values) diﬀer from each other, but if a company wants to decide whether or not to execute the plan, it might lead to wrong decisions.

4

Conclusion

In this paper we have provided a model which plans facilities with their capacities and assigns distribution areas to them. For this planning a mixed-integer optimization model has be formulated, that is based upon the well-known facility location problem, but contains some additional constraints to guarantee the connectedness of the regions. The reason for formulating this model was to be able to calculate tactical and operational costs. Such cost are relevant for strategic planning, but can not be calculated in advance, because there is no appropriate information available. Hence we use analytical methods which gain average costs approximatively. This can be done with relatively simple formulas, but which are, unfortunately, nonlinear. Because of the high complexity of the model and its non-linear objective function it was not possible to solve real-world-sized instances exactly. Therefore, we implemented some heuristic search methods, also based on well-known procedures in facility location theory, to ﬁnd local optimal solutions. In order to not get trapped in local minima, we use tabu search as a metaheuristic. Computational test have shown that the results obtained with this algorithm are quite good compared to exact optima (so far it was possible to ﬁnd them), and that these results can be found in a relatively short computing time. To sum it up, we can say that it is possible to optimize strategic facility location problems considering short-term costs completely, but approximatively. One has a bigger eﬀort to analyze the logistics system, but one should always prefer to solve as much of a model by thinking as possible instead of using computing power to plan details one does not need to know. With this philosophy one is also able to learn something about the system, which is not possible if one only uses a computer.

References [1] Campbell, J.F.: Designing logistics systems by analyzing transportation, inventory and terminal cost trade-oﬀs. Journal of Business Logistics 11, 159–179 (1990) [2] Crainic, T., Gendreau, M., Soriano, P., Toulouse, M.: A tabu search procedure for multicommodity location/allocation with balancing requirements. Annals of Operations Research 41(4), 359–383 (1993) [3] Daganzo, C.F.: The length of tours in zones of diﬀerent shapes. Transportation Research B 18B, 135–145 (1984)

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[4] Daganzo, C.F.: The distance traveled to visit N points with a maximum of C stops per vehicle: an analytic model and application. Transportation Science 18, 131–150 (1984) [5] Daganzo, C.F.: Logistics Systems Analysis, 4th edn. Springer, Heidelberg (2005) [6] Glover, F., Laguna, M.: Tabu Search. Kluver, London (1997) [7] Klose, A.: Standortplanung in distributiven Systemen – Modelle, Methoden. Anwendungen, Physica Heidelberg (2001) [8] Voß, S.: Observing logical interdependencies in tabu search: Methods and results. In: Rayward-Smith, V., Osman, I., Reeves, C., Smith, G. (eds.) Modern Heuristic Search Methods, pp. 41–59. Wiley, Chichester (1996) [9] Winkelkotte, T.: Strategische Optimierung von Distributionsnetzwerken – Ein Optimierungsmodell und heuristische L¨ osungsverfahren zur Planung von Standorten und Absatzgebieten mit approximativer Ber¨ ucksichtigung der taktischen und operativen Logistikprozesse. Shaker (2011)

Application of RFID Technology at the Entrance Gate of Container Terminals Lei Hu1 , Xiaoning Shi1,2 , Stefan Voß2 , and Weigang Zhang1 1

Department of International Shipping and Logistics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China hulei [email protected], [email protected], [email protected] 2 University of Hamburg, Institute of Information Systems, Von-Melle-Park 5, 20146 Hamburg, Germany [email protected], [email protected]

Abstract. Radio Frequency Identiﬁcation refers to using transponders or tags aﬃliated with an object for the purpose of identiﬁcation and tracking by means of radio waves. This paper focuses on container port operations, emphasizing on the current status of these operations and its business bottlenecks. Based on that, we discuss related solutions for improving eﬃciency from the perspective of orderly balance and seamless connection in diﬀerent operational processes at the entrance gate of container terminals.

1

Introduction

With the gradual promotion of the concept of the Internet of Things (IoT), its related technologies are expected to have impact on the operational processes of any kinds of logistics, and further promote their eﬃciency and eﬀectiveness. As one of the technologies that enable the implementation of the IoT, Radio Frequency Identification (RFID) is becoming increasingly important and it is used in production, manufacturing as well as supply chain management. Many RFID applications seem to focus on closed-loop scenarios devised to solve particular problems in real business cases where alternative solutions are not feasible [10]. RFID tools play an important role in supporting assembly lines, medical, logistics, and supply chain management processes. RFID tools can identify, categorize, and manage the ﬂow of goods and information throughout the supply chain. Moreover, RFID brings greater visibility to business processes, e.g., in supermarkets, customs authorities, etc. In an ideal world, it can ensure the necessary data transfer to reach optimal supply chain conditions. Innovation management and process re-engineering of container terminals may refer to the analysis and redesign of workﬂows of port operations [18]. Related re-engineering can be used in ports to lower costs and increase quality of service. Information technology may be seen as key enabler for a radical change in ports and terminals. To which extent RFID is an enabler for related change still needs to be investigated. Transportation companies around the world value J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 209–220, 2011. c Springer-Verlag Berlin Heidelberg 2011

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RFID technology due to its impact on the business value and eﬃciency. Since RFID technology is mature, we can use this technology in the access control systems of container terminals. In this way, we may decrease the workload in the gate of the container terminal and improve the eﬃciency in receiving the containers. Regarding yard management, shipping and freight and distribution centers are some areas where RFID tracking technology is used. We ﬁrst sketch the functioning of container terminal operations in Section 2. In Section 3, the working principles of RFID are introduced. In Section 4 we discuss options to re-engineer selected port operations, together with a discussion on the application of RFID in the container entrance gate for road trucks. The latter section is moderately interleaved with a related literature review. Based on a small case, in Section 5, we reach conclusions on potential beneﬁt of applying the RFID technology as well as summarize some relevant further research topics.

2

Container Terminal Operations

Operation processes at a container terminal can be divided into import and export operation processes. In this paper we focus on the export processes and business bottlenecks. Seaport container terminals may be seen as open systems of material ﬂow with a large variation in size, function, and layout. Basically, they are very similar in structure and related sub-systems (see Figure 1). The waterside (ship operation or berthing area) is equipped with quay cranes for loading and unloading of vessels. Import and export containers are stocked in a yard which usually is divided into a number of blocks. Special stacking areas may be reserved, e.g., for reefer containers, which need electrical supply for cooling, or for storing hazardous goods. Separate areas may also be used for empty containers. Some terminals employ sheds for stuﬃng and stripping containers or for additional logistics services. The truck and train operation area links the terminal to the hinterland and outside transportation systems. The chain of operations for export containers can be described as follows (see Figure 2 and [15]). After arrival at the terminal by truck or train, the container is identiﬁed and registered with its major data (e.g., content, destination, outbound vessel, shipping line), picked up by internal transportation equipment, and distributed to one of the storage blocks in the yard. The respective storage location is given by row, bay, and tier within the block and is assigned in real time upon arrival of the container at the terminal. To store a container at the yard block, speciﬁc cranes or lifting vehicles are used (e.g., rail mounted gantry cranes RMG). Finally, after the arrival of the designated vessel, the container is transported from the yard block to the berth where quay cranes load the containers onto the vessel at pre-deﬁned stacking positions. The operations to handle an import container are performed in the reverse order. Scheduling the huge number of concurrent operations with all the diﬀerent types of transportation and handling equipment involves extremely complex planning problems. Following the discussion in the extensive surveys of [16,14], in view of the everchanging terminal conditions and the limited predictability of future events and their timing, they often must be solved in an online fashion or even in real time.

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Truck and Train Operation Area Hinterland Operation

Empty Stock Yard Import/Export Stock

Yard Moves Sheds

Quayside Operation Ship Operation Area

Fig. 1. Operation areas of a seaport container terminal and ﬂow of transports [16, p. 6]

Quayside

Landside

Stack with RMG

Quay Crane

Vehicles

Vehicles Trucks, Train

Vessel

Fig. 2. Transporting and handling procedures of a container [16, p. 13]

Consider a speciﬁc example of the container transport chain. When it comes to exporting of containers, the operation processes are as follows. 1. The shipper ﬁrst consolidates his cargoes in containers. 2. After the commission of the shipper or its freight forwarder, the trucking ﬁrm will transport the containers to the yard of the container terminal. 3. Before going into the container yard, the truck has to ﬁrst check-in at the container entrance gate. During this process, the gate house workers will check the EIR (Equipment Interchange Receipt) and related information of this container, such as the container size, its reference number, etc. 4. After the container has been checked, it will be unloaded, e.g., with a gantry crane, and placed on the yard. 5. When a vessel comes, the port uses related equipment to discharge and load the containers to the vessel.

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The procedure of importing the containers is similar to that of exporting containers. When it comes to the transshipment of the containers, the operation processes are shown as follows (including domestic as well as international transshipment containers). When the terminal unloads the transshipment containers, it needs the manifest of the transshipment container or the related EDI information, and the transshipment containers should be transported to a speciﬁed transshipment zone. Exporting transshipment containers need the notice certiﬁed by the customer. Then the terminal can load the transshipment containers. Export containers are usually delivered to the ports two or three days before the arrival of related vessels. But at that time the export manifests collected by the shipping agency need not be complete. Still at many ports worldwide only by manually completing data can the terminal collect the containers as there are incomplete (or non-correct) export lists. As there is only one piece of export manifest for one vessel, the terminal gates might even record the information of the containers manually only by referring to the documents, i.e., EIR and packing lists, and after this the containers can pass the entrance gates. It is obvious that this kind of process not only wastes time, but also increases the possibility of including errors [19]. Suppose we record the data completely when containers pass the gates, this will be at the expense of at least 30 to 60 extra seconds. If a vessel carries 2000 containers on average, then this procedure might take 10 to 20 hours to complete the process of moving the containers to the yards or container freight stations, possibly leading to a lower access rate of the gates in the terminal. To improve the recognition accuracy, a container number recognition system needs to identify the reference number at least twice. Finally, the results are based on the integrated complementary information of the container surfaces. Since containers are transferred around the world, the identiﬁcation number of the containers might be damaged or even disappear. This issue also limits the development of eﬃcient operations of the container terminal.

3

Technical Aspects of RFID Systems

RFID belongs to Automatic Identiﬁcation (Auto-ID) technologies. This family of technologies includes the famous bar code system, optical character readers and some biometric technologies (like retinal scans). Auto-ID technologies have proved to reduce time and working resources needed and to increase data accuracy. Despite their practical value, the fact that a person is needed to manually scan items is itself a constraint. It is exactly this part where RFID revolutionizes Auto-ID technologies as mentioned in [17]. An RFID system consists of three parts: a scanning antenna, a transceiver with a decoder to interpret the data, and a transponder, the RFID tag, which has been programmed with information. A typical RFID tag consists of a microchip attached to a radio antenna mounted on a substrate. The chip can store as much as 2 kilobytes of data [17] and beyond. For example, information about a product or shipment-date of manufacture, destination and sell-by data can be stored in a tag. Tags can be passive, active or semi-active. An active tag contains some types of power source

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Tag/Transponder

Reader/Antenna

Computer

Fig. 3. Working principles of RFID

on the tag, whereas the passive tags rely on the radio signal sent by the reader for power. Semi-active tags use a battery to run the microchip’s circuitry but not to communicate with the reader. The reader emits radio waves in ranges of anywhere from one inch to 100 feet or more, depending upon its power output and radio frequency used. The data is interchanged with the monitoring computer for processing after the reader decodes the data encoded in the tag’s integrated processing. The working process is shown in Figure 3. Regarding fundamentals and applications of RFID, Finkenzeller [3] is of value as an introductory reference. Next we review RFID literature with a speciﬁc focus on container terminals and logistics and provide a speciﬁc application scenario.

4

Application of RFID to Improve the Container Terminal Operation

The intelligent management of container terminals mainly consists of the passive RFID tag with an UHF band between 860-960MHZ, an RFID reader, the communication system, a common data management system and the car software (onboard unit). The passive RFID tag can contain the container information, the cargo information and the information of the logistic chain. The passive RFID tags are attached to the lintel of the container. The container gate house or entrance gate as well as all handling equipment components (e.g., reach stacker, straddle carrier, quay cranes, RMG) are all attached with RFID readers [24]. When a container which is ﬁxed with the passive RFID tags passes any handling or yard equipment, all the information of the container may be checked by the RFID readers in this equipment. The information can be used in the modern management of the container terminal. There are quite a few more or less conceptual papers on the analysis of RFID technology implementation in container terminals; see, e.g., [1,2,9,13,21,26]. Harder and Voß [6] provide basis and applicable understanding of cost-beneﬁt analysis on RFID application to the shipping industry. RFID usage may envisage several beneﬁts including better “information visibility” allowing for improved tracking and tracing options etc. Another example is better container port security. If sensors are installed inside containers to monitor changes in light, humidity or pressure, indication could be given that someone had interfered with it. A successful test on this issue for Yantian International Container Terminals is mentioned in [22].

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Some researchers also observe ways to implementing automatic handling systems in China’s container yards (see, e.g., [23]). In addition, [11] describe the research and development of an RFID prototype system that integrates with mobile commerce (m-commerce) in a container depot to enhance its daily operations and support its location management. Wang et al. [20] introduce the application of RFID to Shanghai Port as a case study together with the analysis of future trends of such application to container related transportation. From the perspective of logistics services, in most cases challenges come from demand sides [7]. Above mentioned technology can be applied not only on cargo transportation but also health care logistics. For example, [27] did a comprehensive observation on framework and evaluation of health care related processes with RFID technology implementation. An example where RFID has successfully been applied to a real-world case in postal logistics is found, e.g., in [12]. The paper deals with the re-engineering and design optimization of a warehouse for package storage operations occurring for a courier express company in Italy. The study considers the use of RFID tags to facilitate identifying items in a package delivery service facility. 4.1

A Specific Application Scenario

The main business of container terminal enterprises is divided into domestic and foreign container trade by providing ships with loading and unloading operations. Other than that, programs are available related to scheduling, billing, clearance after checking, etc. The operation is more complex for exporting containers compared with importing containers, since we must rely on the reliabilities of the information of the containers to load the container on board exactly, including the name of the vessel, the voyage, and the reference numbers of the bill of lading, etc. However, that information is mainly collected by the workers who are assigned to posts for speciﬁc container gates. If someone wants to transport an export container to a terminal, the terminal needs to check the paper document with the manual data of the actual container. Only if they match, the terminal can accept the container for further operations. Obviously this kind of process dramatically increases the workload of the workers who receives containers, reduces the speed of the traﬃc ﬂow and prolonges the stopping time of the trucks. Moreover, when several vessels arrive at the same time, the chance of error will be larger. Currently, when the container is transported through the container gate house it needs to manually check the container size, the container number, the seals, etc. State-of-the-art identiﬁcation technology in many terminals is a kind of practical method based on OCR (Optical Character Recognition) technology. But as time goes by, the number sticking on containers may become fuzzy, so it becomes not so tangible for the gate house workers to identify containers. The antenna of the RFID on the container gate house is shown in Figure 4. The working principles of the RFID reader of the container gate house are sketched in Figure 5. Related reading regarding the processes at entry gates is provided in [15,4].

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Antenna

Fig. 4. Antenna located at the gate of a container terminal

RFID Antenna

RFID Tag

RFID Reader

Computer

Fig. 5. Working principles of RFID readers

When the container ﬁxed with passive RFID tags goes through the entrance gate, the loadmeter senses there are containers passing. At the same time the loadmeter sends signals to the RFID reader, after that the RFID reader begins to work. It reads the related information of the container recording by the passive RFID tags through locating the RFID tags on the container, thus it can automatically identify the container number. The Container Terminal Management System records the information automatically by the RFID reader. After the information is checked by the information center of the container terminal, the container is allowed to leave the port, at the same time the information center sends the time that the container took in the entrance gate and related information to the Common Data Management Platform. This process needs no human intervention. To compare the eﬃciency of the entrance gate before

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using the RFID system with the one after using the RFID system, we take the entrance gate of the Shanghai Waigaoqiao Terminal as example with the time as the main index to determine whether the RFID system is eﬃcient. Preliminary investigation of the related gates shows the data provided in Table 1. 4.2

Truck Handling at the Terminal Entrance – Conceptual Aspects

New technology is creating opportunities for an entirely new wave of re-engineering eﬀorts. In [5], re-engineering as a role model was implemented in the back oﬃce, the factory, and the warehouse. Later on, it has been applied to the front oﬃce and the revenue-producing side of the business: product development, sales, and marketing. In case a port operator is regarded as an enterprise and the port operation process is viewed as workﬂow within enterprises, re-engineering would most probably have to occur as long as either new demand is increasing or new information technology is about to be applied. From its inception, re-engineering has been a close partner of information technology. Technology enables the processes that are the essence of re-engineering to be redesigned. The container gate house service system is a typical queuing system, the major processes are trucks passing through the gate. The object in this system is a container truck. According to a large number of internal and external statistical data, most of the container gate house service processes can be considered as a M/Ek/S model [25]. The arrival process of trucks follows a Poisson distribution Pn = P (n) =

λn −λ e , n!

n = 1, 2, 3, ...

(1)

with n being the number of daily arriving trucks, λ the daily average number of arriving trucks, and P (n) denoting the probability of n trucks arriving in a day. After arrival, the passing time of the trucks follows a k-Erlang distribution fk (t) =

μk(μkt)k−1 −μkt e , (k − 1)!

t>0

(2)

with μ being the number of trucks served in a single gate every day. In the container gate house service system, service station number is the number of gates owned by the port, we designate it by S. Set the strength factor of system loading according to the M/Ek/S queuing model: s−1 −1 1 λ 1 1 λ S n ( ) + ( ) P0 = n! μ S! 1 − ρ μ n=0

(3)

Application of RFID Technology at the Entrance Gate

Pn = P (n) =

1 λ n n! ( μ ) P0 , 1 λ n S!S n−s ( μ ) P0

(n < S) (n ≥ S)

217

(4)

where P0 is the probability of no trucks arriving in a day, namely the probability that every gate of the port is idle. The main performance indicators in a port services system include: 1. The average number of trucks waiting in the container gate house: Lq =

∞

(n − S)Pn =

n=S+1

(S × ρ)S × ρ P0 S!(1 − ρ)2

(5)

2. The average number of arriving container trucks: LS =

∞

nPn = Lq + S × ρ

(6)

n=0

3. The average residence time of trucks: WS = LS /λ 4. The average waiting time of the trucks: Wq = Lq /λ In the following we consider Daxie Terminal, Ningbo (China), as an example. Table 2 provides details on a sample of 20 trucks between 2:30 and 3:30 PM, May 11th 2011 to calculate related performance indicators. Each truck is transporting one container passing through the gate without RFID technology. Moreover, on May 11th the total number of container trucks entering the port was 1904 TEU. Daxie in Ningbo has eight gates in total, among which ﬁve are entrance gates and three are for exits, i.e., S = 5. We may denote by tw the waiting time for each truck when it tends to enter into the terminal. tp is the processing time for manually processing the EIR documents as there is no RFID. ts = tw + tp . Based on these numbers we may perform the following calculations. Based on 1904 TEU we have λ = 79.33/h, ρ = 0.2314, P0 = 0.17. Moreover, Lq = 0.0002, LS = 1.157, WS = 226.83s, and Wq = 0.04s. From these calculations, we can see that the average waiting time Wq is nearly 0, reﬂecting that the container gates are enough for the Daxie container ports. WS = 226.83s indicates that the average service time of one truck is less than four minutes. On one hand, if we use RFID technology in the container gates, then the passing time tp will be decreased, thus the average waiting time can decrease. On the other hand, if the time of the trucks passing through the container gate house decreased, then the container terminal can decrease the number of the gates. For the present data we may conclude the phenomenon of queueing is not frequent. But when building a new container terminal, if the port authorities consider using RFID technology to design the container gate house, one might build a smaller number of gates.

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Table 2. Time of container trucks passing through the gate Container No. tw /second tp /second ts /second YMLU4956169 90 40 130 INKU6102043 89 152 241 SPNU2879389 78 72 150 YMLU5035019 124 46 170 DFSH6277779 144 57 201 CCLU9233314 87 67 154 YMLU8171209 123 69 192 HJCU4140364 131 71 202 GLDU7464425 56 68 124 CBHU8312882 88 63 151 GCSU6008340 87 49 136 BMOU4783317 75 110 185 GLDU0879225 54 56 110 KMTU7326329 94 68 162 UACU3371086 102 258 360 OOLU8858380 121 75 196 TGHU9799940 197 87 284 GLDU7213760 262 79 341 HUCU1072820 184 92 276 HJCU3205776 88 68 156 average passing time E(t) 113.7 82.35 196.05 Variance D(t) 2602.43 2308.13 4955.84

5

Conclusions and Further Research

In this paper, we discussed the current status of container port operations as well as the application of RFID in the container gate house and the container yard. For the discussion of the application of RFID in the entrance gate, we can conclude that by using an RFID system we can decrease the passing time of the container trucks. More speciﬁcally, from the discussion of Section 4, we can conclude that if we use the RFID technology in the container gate house, then we can improve the eﬃciency of the turnover of the container trucks, thus decrease the waiting of the trucks and also decrease the parking area of the container trucks. In this sense, we can expand the area of the storage yard. Due to the limited real world data that has been obtained, the potential re-engineering within the container yard has not been discussed in this paper. However, that would be one of research topics of our interest in the near future. Besides, in the application of RFID in the container yard we have not considered other conveyance when transporting containers from the yard to the terminal apron. This could be another topic for further research. As real business cases on RFID in the shipping industry are still quite rare, related ﬁeld studies would be of great beneﬁt. This might help to close the credibility gap regarding RFID mentioned in [8].

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A Three-Level Hierarchical Workload Management Scheme for Yard Cranes in Container Terminals Shell Ying Huang, Xi Guo, and Mei Mei Lau School of Computer Engineering Nanyang Technological University Block N4, Nanyang Avenue, Singapore 639798 {assyhuang,guox0006}@ntu.edu.sg

Abstract. We propose a three-level, hierarchical scheme for yard crane (YC) workload management in container terminals. Level 1 distributes YCs among different rows in the storage yard at suitable times based on predicted future workload. This is done a few times during a shift of 8 hours. Level 2 dispatches YCs to work in various non-overlapping working zones in each row for the time window in between two rounds of YC re-distributions at Level 1. Level 3 determines the serving sequences of vehicle jobs for an YC in a working zone over a period of time (e.g., a sub-planning window). The algorithms for levels 2 and 3 have been published elsewhere. This paper proposes the proportional distribution and the uniform distribution strategies for YC deployment at level 1. We compare the performance of the three-level hierarchical scheme in terms of average job waiting times and the average number of overflow jobs at the end of each planning window under the two distribution strategies.

1 Introduction Container terminals, which serve as hubs of container transshipment, are crucial nodes and play an important role in the marine transportation network. The demand on high quality services from container terminals includes efficiency and reliability in container handling which in turn requires the terminal to utilize its resources efficiently. Figure 1 shows a large part of a typical layout of a conventional container terminal. In such terminals, the storage yard is often divided into several tens of yard blocks in a number of rows parallel to the quay. Each yard block may have more than 30 slots (yard bays) of containers stored in length. Vehicles travel along lanes to transfer containers between quayside and yard side. When multiple vessels are loading and unloading, vehicles may arrive at different slot locations of a yard block for storing and retrieving containers. Ex-terminal trucks may also arrive through terminal gates to unload export containers or to load import containers. YCs are the interface between vehicles and container stacks in the storage yard. One most important performance target of container terminal operation is to minimize vessel turn-around time. It means the YCs need to serve vehicle jobs as efficiently as possible to reduce the delay of vehicles at the yard side in order to continuously feed the quay sides to support quay crane (QC) operations. The main objective at the yard side is then translated to minimizing the average vehicle job waiting time for YCs service. J.W. Böse et al. (Eds.): ICCL 2011, LNCS 6971, pp. 221–232, 2011. © Springer-Verlag Berlin Heidelberg 2011

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Fig. 1. Layout and container flows in a typical terminal (BLK: block)

The YC workload distribution in the yard is uneven and changes dynamically over time. To match the changing workload distribution, YCs need to move from time to time. The time taken by an YC to move from one job location to another is referred to as gantry time. YC gantry times, together with YC service times, contribute to vehicle waiting times and may become a significant part of a YC’s busy time. The movements of a rubber tired gantry (RTG) Crane include: linear gantry and cross gantry. Cross gantries are movements from one block to another in a different row, e.g., from BLK 2 to BLK 6 in Figure 1. An YC doing cross gantry has to make two 90° turns which take much longer time than linear gantry and may delay the vehicle movements by blocking the lanes. So too many YC movements will lower YC productivity but too few movements will deprive some vehicles from prompt service in some parts of the yard. After equipment ordering at the beginning of a shift, a common practice is to initially assign YCs to various yard blocks. Then a re-distribution of YCs among the yard blocks is done from one planning window to another to match the dynamically changing workload. Each planning window may be one hour, two hours or even four hours. Workload of a block is commonly estimated by the number of arriving jobs expected in a planning window or a quantity proportional to the number. This practice may result in some blocks having no YCs in charge for the entire planning window. This will cause very long vehicle waiting times in these blocks. It may also deploy more than one YC to some blocks. This means carefully synchronized YC operations are needed to avoid YC clashes. Sometimes when YCs block each other long waiting times are unavoidable. Since the distribution of YCs does not change within each planning window, there is no way to dynamically respond to changes in workload distribution in the storage yard. If the length of the planning window is long, e.g., one hour or longer, this is very inflexible. On the other hand, if the length of the planning window is short, like 15 minutes or half an hour, it may cause frequent cross gantry movements of YCs with substantial loss of productive working time.

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Therefore, we propose a hierarchical scheme for YC operation management which is organized into three levels as shown in Table 1. Suppose a suitable number of YCs has been assigned to work for the current shift. Table 1. Hierarchical YC operations management

Level 1

Level 2

Level 3

Distribution of YCs to different Rows for each planning window in a shift Time Partition of a planning window into subplanning windows Space Partition of each row into YC working zones for each sub-planning window YC dispatching in individual working zones

Level 1 distributes YCs among different rows at suitable times based on predicted future workload. This is done a few times during a shift of 8 hours. Level 2, in between two rounds of YC re-distributions at level 1, is the workload partition for each row in two dimensions: time and space. The partition in the time dimension is to divide a planning window into a number of sub-planning windows of variable lengths. The space partition is for each sub-planning window. It divides a row of yard blocks into a number of non-overlapping zones, not necessarily in units of yard blocks. This means the space partition changes from one sub-planning window to another in response to the dynamically changing workload distributions. Level 3 determines the serving sequences of vehicle jobs for an YC in a service zone over a sub-planning window. We propose to use estimated average job waiting time as metrics for workload partitioning and YC dispatching. The best partitioning and dispatching plans should yield the minimum average vehicle job waiting time calculated from all the jobs in the planning window. Minimizing vehicle waiting time means minimizing the delay vehicles experience in the storage yard which will reduce the waiting times of the QCs for vehicles. This leads to smooth and continuous operations of the QCs and reduction of the vessel turnaround time, one of the most important performance indicators of a container terminal. The hierarchical scheme aims to minimize overall average vehicle job waiting time through flexible time and space partitioning of workload by re-distributions of YCs to cope with dynamic workload distribution over time at the yard side. A modified A* search algorithm and a modified backtracking algorithm [3, 7, 8] are designed to find the optimal dispatching solution for an YC in a zone over a period of time at level 3. The modified algorithms are able to find the optimal solution from over 2.4x1018 possible dispatching sequences within seconds. Algorithms for space partition and time partition in level 2 are proposed in Guo and Huang [5, 4, 6]. Even though the outline of the hierarchical scheme has been presented in our earlier papers [8, 6], this is the first time we put the 3 levels together to manage the YCs in the whole yard. In this paper, we also propose and compare two strategies for level 1 to distribute YCs to a number of rows of yard blocks. The objectives of the

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level 1 method are to minimize the average job waiting times and the number of overflow jobs at the end of each planning window for level 1. In the rest of the paper, we discuss the related work in Section 2. We present the hierarchical YC operation management system in Section 3. The overall performance of the hierarchical scheme and the two strategies of YC distribution at level 1 are evaluated by simulation experiments in Section 4. Finally, Section 5 concludes the paper.

2 Related Work The problems of scheduling and dispatching resources in container terminals have been widely studied in recent years [18, 20, 21]. On the topic of YC management, two main problems are: (i) deciding job servicing sequence for an YC which we refers to as YC dispatching problem in this paper; (ii) allocating YCs to different parts of the yard which we refers to as YC deployment problem in this paper. Kim and Kim [12] studied single YC dispatching for loading with a given load plan and a given bay plan. A Mixed Integer Programming (MIP) model is proposed to minimize the total gantry time. Later, Kim and Kim [13] and Kim et al. [11] extended the study of this problem by comparing exact optimization, a beam search heuristic and a genetic algorithm (GA). For large problems, the MIP has limited applicability due to the excessive computational times, while heuristics cannot guarantee optimal solutions. In addition, the assumption of having dedicated YCs just to support vessel loading operations in these works is not always the best for terminals with many berths and more yard blocks than cranes. Several works studied the dispatching problem with 2 YCs. Due to the problem complexity, commonly MIP models were employed to formulate the problems while heuristic methods to find near-optimal solutions. Jung and Kim [10] considered 2 YCs working in one shared zone for loadings with a GA and a simulated annealing (SA) algorithm to minimize the make-span. Solutions with possible YC interference were considered not feasible in the MIP model. Lee et al. [14] considered 2 YCs working in 2 non-overlapping zones with a SA algorithm to minimize the make-span. Cao et al. [1] considered double-rail-mounted gantry (DRMG) crane systems where two YCs can pass through each other along a row of blocks with a combined greedy and SA algorithm to minimize the loading time. Stahlbock and Voß [19] also studied DRMG operations to maximize productivity. Experiments showed the proposed SA algorithm dominates the priority rule-based heuristics for high workload. Zhang et al. [22] studied YCs deployment among yard blocks. It is formulated as a MIP model to minimize total unfinished workload at the end of each planning period and is solved by a modified Lagrangian relaxation method. However, only one transfer per YC is allowed in the 4 hour planning period, which may not be enough to match the changing workload distribution. Cheung et al. [2] also studied the problem with a Lagrangian decomposition solution and a successive piecewise-linear approximation approach. Workload in these two papers is based on the number of container moves (equivalent to the number of YC jobs), which at best is a coarse estimation of the YC workload. Ng [16] considered the problem of multiple YCs sharing a single bi-directional lane and modeled it as an IP to minimize total job completion time. The workload for

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each YC was estimated by a simple greedy heuristic. However, safety constraints of YC separation of a minimal distance were not considered. Petering et al. [17] claimed that YCs could only schedule for at most 1.5 container jobs to avoid deadlocks when YCs are in charge of overlapping zones. This reinforces our belief that nonoverlapping YC zones should be formed in our YC dispatching schemes to achieve high performance while maintaining safety constraints and avoiding deadlocks in real-time settings. A least cost heuristic (LCH) was proposed by Linn and Zhang [15] to decide on distributing YCs from blocks with surplus capacity to ones with YC shortage at the beginning of each planning window. They tested a storage yard with 5 rows each of which has two yards blocks. The number of YCs is equal to the number of yard blocks. The amount of overflow work produced by LCH was shown to be 3 % to 6 % higher than the solution by the MIP model. However, safety constraint of separating two YCs by a minimal distance was not considered when two YCs are working in one yard block. This may lead to YC clashes. It may also lead to the streams of vehicles to be served by two neighboring YCs (RTGs) blocking each other when the two YCs are working side by side. As a result, the performance in real operations would not be as good as what the original paper expects. In a larger storage yard, LCH may also cause severe starvation: some jobs may need to wait for several planning windows before receiving YC services. This happens when the number of jobs in the block is not judged to be high enough a cost for an YC to come.

3 The Three-Level Hierarchical YC Workload Management Scheme With the expected berthing and un-berthing times of vessels, the terminal operational management system will schedule QCs to unload and load the vessels. It can be computed from these QC schedules the number of QCs that will be working and for how long before this number changes. Even though there may be many uncertainties in terminal operations, it is a reasonable assumption that it is known whether there will be a vessel berthing or un-berthing in the next hour. It follows that we can assume that the knowledge of which QCs are working in the next hour is accurate. If there is a change in which QCs are working, it is known how they will change and when they will change with the next hour. Then according to the QC schedule for the next hour, the number of jobs that will arrive in the storage yard, their locations and arrival times can be estimated. This is because containers to be unloaded from vessels are allocated yard storage locations before vessel arrivals. Containers to be loaded onto vessels and their locations are also known before vessel arrivals. So our problem is to make decisions on how to deploy YCs in the yard and in what order the YCs should handle the jobs. The hierarchical scheme aims to achieve the best possible balance of workloads among YCs through flexible time and space partitioning of workload to cope with dynamically changing workload distributions. By restricting cross gantry movements to the decision points of level 1, we try to keep the costly cross gantry movements to a minimum. At the same time we maintain the responsiveness to the dynamic changes in workload distribution with the effective time and space partition of workload at

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level 2. For the completeness of the presentation and clarity in what support the levels 2 and 3 give to level 1, we will briefly describe the level 2 and 3 algorithms first. Then we present the two YC distribution policies for level 1. 3.1 Level 3: Computing an Optimal Job Sequence Level 3 generates the optimal YC job sequence for each working zone. A modified A* search algorithm by Guo and Huang [3] is employed to find the optimal dispatching solution for an YC in a zone over a planning horizon. It involves simulating YC operations following various possible dispatching sequences and uses an admissible heuristic to prune the computation of a partial sequence if it is deemed not a better solution. The dispatching sequence with the smallest vehicle waiting time as predicted by the dispatching algorithm will be the optimal sequence for one working zone. We also incorporate the admissible heuristics into a second backtracking algorithm which uses a prioritized search order to accelerate the computation [8, 7]. The algorithm is able to find the optimal solution from over 2.4x1018 possible dispatching sequences within seconds. 3.2 Level 2: Time and Space Partition of Workload for YC Deployment in a Row Level 2 dispatches YCs to work in various non-overlapping working zones in each row for the time window in between two rounds of YC re-distributions at level 1. Time partitioning algorithm and space partition algorithm are used for deploying YCs to handle the changing job arrival patterns in a row of yard blocks. The main differences between our approach and most of the other YC deployment methods in literature are: (1) Average vehicle job waiting time instead of the number of jobs is used to balance YC workload and to evaluate the quality of a partition; (2) YC working zone assignment is not in units of yard blocks and our space partition algorithm generates more flexible divisions of the workload from all blocks in a row; (3) Within the time window, YC deployment plan changes and YC deployment frequency is not fixed but is decided by our time partition algorithm with the objective of minimizing average vehicle waiting times. YC workload distribution at the row level involves two parts: the space partitioning and the time partitioning. The space partitioning aims to partition a row into a number of non-overlapping zones for individual YCs in a given planning window. Various possible space partition plans for the row are generated. The level 3 algorithm in our scheme is used to compute the optimal job sequence for the zones in each plan. The predicted total job waiting time of a partition plan of the row is the sum of the total job waiting time for each of the zones. The space partition algorithm will choose the partition plan that returns the shortest total waiting time as the one to be used for assigning working zones to YCs. The average waiting time for all the jobs in the row is the total waiting time divided by the number of jobs. So the partition plan with the shortest total waiting time is the plan with the minimum average waiting time. The details of the space partition algorithm are published in [5]. The time partitioning aims to achieve dynamic workload partition by partitioning the time between two rounds of YC re-distribution at level 1 into a number of

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sub-planning time windows. Thus, the space partition for different sub-planning time windows may be different. Given the current planning window T and the number of YCs, M in the row, the algorithm tries to divide T into 2 sub-windows, T1 and T2, of equal planning time. If any of the sub-windows has a number of jobs smaller than M, then T should not be divided. In this case, in T1 or T2, each YC may have less than one job on average and this would lead to idling YCs. When both T1 and T2 have at least M jobs, the best partition is either no partition (keep window T) or the best partition for T1 followed by the best partition for T2, whichever gives the lower total job waiting time. The resulting time partition will be a series of planning time windows of variable time lengths which returns the smallest total job waiting time of all combinations. The time partition algorithm evaluates the performance of a planning window by calling the space partition algorithm to get the best space partition for that time window. The details of the time partition algorithm are published in [4, 6]. M = total number of YCs; Rem = M; For each row R in the yard Ni = 

; //   returns the integral part of a real number

Rem = Rem – Ni; If Rem > 0 Sort the rows in descending order of (

);

Add 1 YC to each row in this order and update Ni until Rem = 0; Fig. 2. Proportional Distribution Algorithm

For each row i starting from the first row to the last row If (Ni < the current number of YCs in the row) // this row has surplus YCs Move its surplus YCs to a row in need of YCs starting from Row 1 Fig. 3. Deciding on YC movements between rows

3.3 Level 1: Deploying YCs among Different Rows The objective of the level 1 module is to deploy YCs among different rows to offer high service standard in the whole storage yard. This means job waiting times and the number of overflow jobs at the end of the planning windows should be minimized. Since dynamic balancing of workload among YCs in each row in terms of job waiting times is done at level 2 for each planning window, we investigate the effectiveness of two strategies for level 1. The first one is to allocate a number of YCs to each row in proportion to the number of jobs the row expects in the planning window. The advantage of this strategy is the more jobs a row expects in a planning window, the more YCs will work to meet the demand on YC service. The second strategy is to allocate approximately equal number of YCs to each row. The advantage of this strategy is that YCs stay in their rows and spend all their time in productively serving vehicle jobs.

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The algorithm for the first strategy is given in Figure 2. It first computes the number of YCs needed in each row R as the integer smaller than or equal to where M is the total number of YCs. If there are remaining YCs they will be allocated to rows in descending order of the fractional part of . After deciding on the number of YCs each row should have for the next planning window, the next decision is on the movement of YCs based on the number of YCs each row has in the current planning window. It should minimize the total distance of YC moves in the redistribution of them among the rows. The algorithm is given in Figure 3. The algorithm is guaranteed to produce the minimum YC moves because no 2 YCs will cross each other going from their source row to destination row. The second strategy uniformly deploys an approximately equal number of YCs across the rows. The difference in the number of YCs between rows will be at most one when the total number of YCs is not a multiple of the number of rows (assuming the number of blocks in each row is the same). If the number of blocks in each row is not the same, the number of YCs in this strategy will be proportional to the number of blocks (or the number of container stacks). YCs do not move between planning windows so all YCs are available to work from the beginning of each planning window. With level 1’s deployment of YCs to different rows in the yard for each planning window, our level 2 time partition algorithm will divide the planning window into suitable sub-windows for each row. For each sub-window of each row, the space partition algorithm will partition the row into non-overlapping zones. One YC will be deployed to one zone. Our level 3 algorithm computes the optimal job sequence for each YC.

4 Performance Evaluation 4.1 Experimental Design The terminal in our experiments has a layout similar to Figure 1 with 5 yard blocks in each row and 4 rows in the yard. Each block has 36 yard slots (bays). When an YC executes a cross gantry move, we assume that it takes 5 minutes to make a 90 degree turn and travel time from one row to the next row is 5 minutes. Therefore it takes 15 minutes for an YC to move from one row to a neighboring row and each additional row to cross needs 5 more minutes. As reshuffling operation of containers in the yard is commonly done in the lull periods of yard operations, we assume containers to be retrieved are already on the top of the slots and containers to be stored in yard will be placed on top of their slot locations. The YC mean process time is thus assumed to be 120s for each container job (loaded or empty), same as in Jung and Kim [10]. The usage of constant YC process time is also seen in Lee et al. [14]. The simulation model is developed under Visual C++ compiler. It is not the focus of this paper to study how to predict vehicle arrivals by real time data driven simulation or other methods. So vehicle inter-arrival times (IAT) follows an exponential distribution for a mixture of storing/retrieval jobs for multiple vessels.

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As shown in Table 2, four sets of job arrival scenarios are tested with 16 and 20 YCs in the yard respectively. Sets A and C represent quite static workload scenarios in the yard. For the whole 24 hours, A and C have vehicle jobs arriving with an exponential IAT with mean of 45 and 30 seconds respectively. The job locations are uniformly distributed among the yard blocks. Set B is a scenario with dynamically changing workload both in time and space. Within a 24 hour period, there are a number of sub-windows with different mean job arrival rates to the entire storage yard. This presents different total workloads to the entire terminal yard due to the varying number of QCs working caused by vessel arrivals and departures. So for each sub-window1 in Set B, the mean job IAT is a value generated from uniform distribution U(40, 50). The length of each sub-window1 is a value generated from U(120, 360). Table 2. Tested workload scenario Set A B C D

Time Mean IAT = 45seconds Sub-win1 U(120,360)minutes with mean IAT = U(40, 50)s Mean IAT = 30seconds Sub-win1 U(120,360) minutes with mean IAT = U(25, 35)s

Space Mean IAT = 45seconds Sub-win2 U(30,90) minutes Diff. probability to each blk Mean IAT = 30seconds Sub-win2 U(30,90) minutes Diff. probability to each blk

Within each sub-window1, there are smaller time windows (sub-window2) where the space distribution of jobs changes dynamically. This represents the situation where even with the same total workload terminal-wide, container job distribution may vary greatly across different yard blocks. This is because unloaded containers will be carried to different parts of the yard depending on their respective second carriers. Similarly, containers to be loaded onto a vessel may also come from various parts of the yards in specific orders matching the vessel’s port of call in the voyage. In different sub-window2s, job distribution to different blocks will be different. The length of each sub-window2 is a random value from U(30, 90). Within each subwindow2, a random number xi from U[0, 9] is generated for each block. The probability of a job going to block i in the sub-window2 is equal to xi/∑ x . Scenario Set D demonstrates similar patterns as Set B, but with a higher total workload. We simulate a daily operation of 24 hours. Each deployment period for Level 1 is one hour. Ten independents runs are carried out for each instance of scenarios A to D with a total of 16 and 20 YCs, respectively. 4.2 Result Analysis First we compare the average job waiting times under the two strategies of allocating YCs to various rows in the yard. In Table 3, each row represents the average job waiting times for one scenario as indicated by the first letter which is followed by the number of YCs in the yard. For example, A16 means scenario A with 16 YCs. The results for the two strategies, proportional distribution and uniform distribution are listed in the respective columns.

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A16 A20 B16 B20 C16 C20 D16 D20

Proportional 14.700270 9.277399 23.616490 19.298040 29.926870 17.086090 39.657200 29.632620

Uniform 11.682070 7.465943 20.035830 15.934280 24.934880 14.423840 37.106970 26.220830

The results show that our three-level hierarchical scheme for YC management performs satisfactorily. The average job waiting times for 16 YCs working in the yard of 20 blocks are below 40 seconds in all scenarios, including the scenario where there is on average a job arriving to the yard every 30 seconds. This number reduces to less than 30 seconds when there are 20 YCs. In general, when the workload distribution is static, as represented by scenarios A and C, the average job waiting times are shorter than the dynamic situations represented by scenarios B and D. Under the variations of workload distributions tested, the uniform strategy performs better than the proportional strategy by several seconds in not only the static but also the dynamic workload distributions. This means the loss in YC productivity when YCs move from one row to another outweighed the benefits of having proportional number of YCs to the number of jobs. The result also shows that our dynamic time and space partition algorithms at Level 2 is able to take charge of all the jobs in a row despite the changes in workload distributions. Table 4 shows the average number of overflow jobs at the end of the one hour planning windows in our experiments. Table 4. Average number of overflow jobs under the two strategies

A16 A20 B16 B20 C16 C20 D16 D20

Proportional 0.200000 0.145833 0.583333 0.491667 0.791667 0.479167 1.016667 0.783333

Uniform 0.187500 0.133333 0.475000 0.437500 0.779167 0.491667 1.083333 0.745833

An overflow job is one that arrives earlier than 2 minutes before the end of the hour but the job is still waiting for service at the end of the hour. It can be seen that the three-level hierarchical scheme is able to finish almost all the jobs that arrive within the planning window. It should be noted that in scenarios A and B, there is one

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job arriving in the yard every 45 seconds on average. For example, in scenario A, there are on average 810 jobs per hour in the yard of 20 blocks. In scenarios C and D there is one job arriving in the yard every 30 seconds on average. For example, there are 1,220 jobs on average in the yard. Between the proportional strategy and the uniform strategy, there is no significant difference.

5 Conclusions and Future Work A three-level hierarchical workload management scheme for YCs in container terminals is presented in this paper. YCs are allocated to different rows at level 1 at the beginning of each hour. One strategy is proportional distribution based on the number of jobs each row expects in each hour. The other strategy is uniform distribution of YCs which removes the need for time consuming YC cross gantry. Level 2 carries out time partition to divide the one hour window into a number of variable length sub-windows and then space partition to divide the row into a number of zones, not necessarily at the boundary of yard blocks. Level 3 computes the optimal job sequence for YCs. Our experimental results show that even under the dynamic changes in workload both in time and space, the hierarchical scheme is able to keep the average job waiting time below one minute. The scheme is able to finish almost all jobs at the end of each hour. Compared with the proportional distribution scheme, the uniform distribution of YCs among different rows of yard blocks produces slightly shorter average job waiting time with no significant difference in the number of overflow jobs under the tested scenarios. This is not to conclude that YC cross gantry will make performance deteriorate in all situations. There may be extreme workload distributions which makes proportional distribution or a more intelligent form of it a better strategy. Our future work includes the comparison of this scheme with the LCH scheme [15] and an improved LCH scheme [9]. In the same direction of the work, a scheme which adopts some of the ideas of the LCH into our three-level hierarchical scheme may increase the flexibility of YC deployment method to handle workload distribution which may vary a lot. Another direction of work is an integrated approach where allocation of containers in the yard helps to balance the YC workload in real operations.

References 1. Cao, Z., Lee, D.H., Meng, Q.: Deployment strategies of double-rail-mounted gantry crane systems for loading outbound containers in container terminals. International Journal of Production Economics 115, 221–228 (2008) 2. Chung, R.K., Li, C.L., Lin, W.: Interblock crane deployment in container terminals. Transportation Science 36, 79–93 (2002) 3. Guo, X., Huang, S.: Performing A* search for yard crane dispatching in container terminals. In: Proc. of the Intern. Conf. on Tools with Artificial Intelligence, vol. 1, pp. 263–267 (2008) 4. Guo, X., Huang, S.: Deciding on planning windows by partitioning time for yard crane management in container terminals. In: 1st Intern. Conf. on Computational Logistics (ICCL), Shanghai (2010)

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5. Guo, X., Huang, S.: A two stage yard crane workload partitioning and job sequencing algorithm for container terminals. In: Proc. of 25th Symposium on Applied Computing (2010) 6. Guo, X., Huang, S.: Dynamic space and time partitioning for yard crane workload management in container terminals. Transportation Science (accepted 2011) 7. Guo, X., Huang, S., Hsu, W., Low, M.: Simulation based hybrid algorithm for yard crane dispatching in container terminals. In: Proc. of the Winter Simulation Conference (December 2009) 8. Guo, X., Huang, S., Hsu, W., Low, M.: Dynamic yard crane dispatching in container terminals with predicted vehicle arrival information. Advanced Engineering Informatics 25(3), 472—484 (2011) 9. Huang, S., Guo, X.: An improved least cost heuristic for dynamic yard crane deployment in container terminals. In: 7th Annual IEEE Conference on Automation Science and Engineering (2011) 10. Jung, S., Kim, K.: Load scheduling for multiple quay cranes in port container terminals. Journal of Intelligent Manufacturing 17(4), 479–492 (2006) 11. Kim, K.H., Kang, J.S., Ryu, K.R.: A beam search algorithm for the load sequencing of outbound containers in port container terminals. OR Spectrum 26, 93–116 (2004) 12. Kim, K.H., Kim, K.Y.: An optimal routing algorithm for a transfer crane in port container terminals. Transportation Science 33, 17–33 (1999) 13. Kim, K.Y., Kim, K.H.: Heuristic algorithms for routing yard-side equipment for minimizing loading times in container terminals. Naval Research Logistics 50, 498–514 (2003) 14. Lee, D.H., Cao, Z., Meng, Q.: Scheduling of two-transtainer systems for loading outbound containers in port container terminals with simulated annealing algorithm. International Journal of Production Economics 107, 115–124 (2007) 15. Linn, R., Zhang, C.: A heuristic for dynamic yard crane deployment in a container terminal. IIE Transactions 35(2), 161–174 (2003) 16. Ng, W.: Crane scheduling in container yards with inter-crane interference. European Journal of Operational Research 164, 64–78 (2005) 17. Petering, M.E., Wu, Y., Li, W., Goh, M., de Souza, R.: Development and simulation analysis of real-time yard crane control systems for seaport container transshipment terminals. OR Spectrum 31, 801–835 (2009); doi:10.1007/s00291-008-0142-7 18. Stahlbock, R., Voß, S.: Operations research at container terminals - a literature update. OR Spectrum 30, 1–52 (2008) 19. Stahlbock, R., Voß, S.: Efficiency consideration for sequencing and scheduling of doublerail-mounted gantry cranes at maritime container terminals. International Journal of Shipping and Transport Logistics 2(1), 95–123 (2010) 20. Steenken, D., Voß, S., Stahlbock, R.: Container terminal operations and operations research – a classification and literature review. OR Spectrum 26, 3–49 (2004) 21. Vis, I.F.A., de Koster, R.: Transshipment of containers at a container terminal: An overview. European Journal of Operational Research 147, 1–16 (2003) 22. Zhang, C., Wan, Y.w., Liu, J., Linn, R.J.: Dynamic crane deployment in container storage yards. Transportation Research-B 36, 537–555 (2002)

A Service-Oriented Model for the Yard Management Problem in Container Terminals Jian Gang Jin1 , Jin Xin Cao2 , Jiang Hang Chen1 , and Der-Horng Lee1, 1

Department of Civil and Environmental Engineering National University of Singapore, Singapore {jin jiangang09,chen jianghang07,dhl}@nus.edu.sg 2 Department of Transportation Engineering Inner Mongolia University, Hohhot, China [email protected]

Abstract. This study proposes a novel service-oriented model for the yard management problem in container terminals. Two decisions of the yard management problem are integrated instead of being solved hierarchically: storage allocation for containers that arrive in near future and yard crane (YC) deployment for assigning yard cranes over the entire storage yard. A concept of YC assignment proﬁle is proposed in order to take into account the requirement of YC work patterns. The problem is formulated as a mixed integer linear program with the objective of minimizing total delayed workload and YC movement penalty as well as the total container moving distance. A rolling horizon approach (RHA) is developed to obtain good solutions in an eﬃcient manner.

1

Introduction

A container terminal is an interface between maritime and land transport, providing intermodal services. Considering that container terminals are highly concentrated with containers, the handling eﬃciency of a container terminal aﬀects the eﬃciency of the whole container shipping network. A container terminal can be roughly divided into two parts: quayside and landside. Quayside operation is directly open to container vessels which requires fast container loading and discharging, while the landside oﬀers a temporary storage area for containers. For some container ports especially in Asia, a common issue that occurs during growth periods is the problem of land scarcity resulting in a highly concentrated storage situation within the storage yard. The storage yard management problem is very complicated in that it involves two highly interdependent decisions: (1) Storage space allocation decision which is to determine where and how much space should be allocated for incoming containers; (2) YC deployment decision which is to dynamically deploy YCs in the yard blocks according to the workload distribution among the entire yard.

Corresponding author.

J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 233–242, 2011. c Springer-Verlag Berlin Heidelberg 2011

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vessel 1

vessel 2

vessel 3

B01

B06

B11

B02

B07

B12

B03

B08

B13

B04

B09

B14

B05

B10

B15

quayside

storage yard

inland customers

C1

container transshipment

C2

container import/export

C3

yard crane movement

Fig. 1. A schematic view of the storage yard management (B: block, C: customer)

Figure 1 shows a schematic view of the container movements in and out of the storage yard. Transshipment containers are unloaded from the quayside and moved back to the quayside after temporary storage in the yard. However, import/export containers have both quayside operations and interaction with inland customers. Containers are placed at certain yard blocks according to operational objectives, e.g., evenly distributed workload and short container moving distance. The yard management also involves the decision of the YC deployment and movement in the storage yard. The challenge of the yard management problem lies in the interdependence of the two decisions. On one hand, the storage allocation plan determines the distribution of YCs’ workload over the entire yard and aﬀects YC deployment decisions. On the other hand, workload delay in some yard blocks due to inappropriate YC deployment would make the blocks not fully accessible for storing incoming containers in the near future. In the existing literature, storage allocation is well-studied like [3,7,4,5]. Storage locations should be determined in such a way that retrieval operations afterwards can be conducted eﬃciently. With a determined storage allocation plan, the information of container moves in all blocks can be available for planning YC deployment. YCs need to be deployed according to the distribution of container moves in the entire yard. The literature on YC deployment problem is very scarce [8,1,6]. The deﬁciency of the existing literature is that storage allocation and YC deployment are considered separately and this makes the interaction between the two problems unexplored. It is necessary to integrate the two problems to achieve a higher eﬃciency for storage yard management. This research is based on the following two motivations: (1) The storage allocation decision and YC deployment decision should be integrated to achieve a better eﬃciency instead of being considered by a hierarchical approach.

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(2) The YC work pattern is container type dependent (e.g., for an import container group, the unloading operation requires an intensive YCs’ work within a short time while the delivery operation spans a long time and needs less YCs). The requirement of YC work patterns should be taken into account for YC deployment. This paper contributes to the literature by proposing a mixed integer programming model for the yard management problem. This model is capable of achieving a better operational eﬃciency by integrating two types of decisions of yard management problem, storage allocation and YC deployment, instead of solving them separately as in most existing literature. The model also features a serviced-oriented YC deployment strategy taking into account the requirement of YC work patterns. The remainder of the paper is organized as follows. The service-oriented model of the problem is described in Section 2. Section 3 introduces an RHA which is designed to solve the problem in an eﬃcient manner. The computational experiments are presented in Section 4. Finally, we conclude this paper in Section 5.

2

The Service-Oriented Model

This section presents the proposed service-oriented model for the yard management problem. The modeling concept is illustrated in Section 2.1, followed by the mathematical formulation in Section 2.2. 2.1

Model Description

The service-oriented model has two important concepts: Service: the demand of receiving and retrieving work that YCs should provide within a block at a working shift, denoted by (i, t) where i is the yard block and t represents the working shift. YC assignment profile: for each service, a set of YC assignment proﬁles is deﬁned each of which represents a YC assignment plan for the service. Note that the concept of the YC assignment proﬁle is a main feature of the service-oriented model and allows incorporating manual decisions from experience. When deﬁning the proﬁle set, the requirement of YC work patterns should be taken into account. For example, a service of loading operation should be conducted as fast as possible, only those proﬁles in favor of loading operation should be put into the set of feasible YC assignment proﬁles. So, by introducing the concept of the YC assignment proﬁle, we can conduct a service-oriented YC deployment plan and combine manual decisions with computer optimization. Figure 2 is an illustrative example where a working shift is further divided into three time steps. This example shows three proﬁles for a service with a workload of four crane time steps. One crane time step is a workload that requires one YC work for one time step. Proﬁle 1 assigns YC1 to the service for the entire

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working shift and brings YC2 at time step 3. At the end of the working shift, no workload is left. For proﬁle 2 in which only YC1 works for three time steps, partial workload is delayed and is carried over to the next working shift. Proﬁle 3 assigns two YCs at the ﬁrst two time steps, which is an assignment plan in favor of fast speed processing. The workload is ﬁnished at the end of the second time step. A working shift s=1

s=2

s=3

Profile 1:

YC1

YC1

YC1 & YC2

Profile 2:

YC1

YC1

Profile 3:

YC1 & YC2

YC1

YC1 & YC2

Fig. 2. An illustrative example of the concept of YC assignment proﬁle

A similar modeling concept can be found in Giallombardo et al. [2]. They use quay crane proﬁles to handle the assignment of quay cranes to containerships. As highlighted by the authors, the concept of quay crane proﬁles can capture real-world issues, and works well to represent the control that the terminal has on several aspects of quay crane assignment during the optimization process. In this paper we apply the concept of crane proﬁles to the storage yard management problem. The diﬀerence is that [2] study the quayside operations including berth allocation and quay crane assignment while our work focuses on the storage yard management issues (storage allocation for containers and YC deployment). 2.2

Model Formulation

The yard management problem is modeled over a given time horizon which is divided into a series of working shifts. The problem should (1) determine the storage locations for incoming container groups in such a way that the total container moving distance can be minimized, and (2) choose YC assignment proﬁles for each service with an objective of minimizing the total workload delay. We deﬁne the following sets and parameters. Sets: – – – – – –

K: the set of container groups Φ: the set of YCs Ω: the set of yard blocks T : the set of working shifts, T = {1, 2, · · · , t¯} S: the set of time steps within a working shift P (i, t): the set of feasible YC assignment proﬁles for service (i, t), ∀i ∈ Ω, t ∈ T

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Parameters: s – δpj (i, t): ∈ {0, 1}. 1, if YC j is used at time step s in proﬁle p; and 0, otherwise, ∀j ∈ Φ, s ∈ S, p ∈ P (i, t) – γij : ∈ {0, 1}. 1, if block i is the favorable block of YC j; 0, otherwise, ∀i ∈ Ω, j ∈ Φ – vj : the penalty of YC j for leaving its favorable block, ∀j ∈ Φ – cp : the YC processing capacity of proﬁle p ∈ P (i, t) – ak : the incoming working shift of container group k, ak ∈ T – bk : the outgoing working shift of container group k, bk ∈ T – ok : the origin (berthing position or terminal gate) of container group k ∈ K – dk : the destination (berthing position or terminal gate) of container group k∈K – qk : the storage space requirement of container group k ∈ K – lim : the container moving distance between block i ∈ Ω and berthing position/terminal gate m – ri : the delayed workload from the last planning horizon at block i ∈ Ω – Qi : the storage capacity of block i ∈ Ω – αkt : ∈ {0, 1}. 1, if t = ak ; and 0, otherwise, ∀k ∈ K, t ∈ T – βkt : ∈ {0, 1}. 1, if t = bk ; and 0, otherwise, ∀k ∈ K, t ∈ T – λ1 : the unit cost of container movement – λ2 : the unit cost of YC workload delay and moving penalty

Note that we associate each YC with a favorable block by introducing parameter γij , as a YC deployment system that restricts crane movement yields a higher quay crane work rate than a system that allows greater mobility [6]. In other words, the parameters γij deﬁne a YC deployment template. We also deﬁne K + (t) = {k|k ∈ K, ak ≤ t} as the set of container groups that arrive at the terminal at or before working shift t, and K − (t) = {k|k ∈ K, bk < t} as the set of container groups that leave the terminal before working shift t. Decision variables: xki : ∈ {0, 1}. 1, if Group k is put into Block i for temporary storage; and 0, otherwise. ∀i ∈ Ω, k ∈ K p yit : ∈ {0, 1}. 1, if YC assignment proﬁle p is adopted for service (i, t); and 0, otherwise. ∀i ∈ Ω, t ∈ T, p ∈ P (i, t) wit : the scheduled workload for Service (i, t).∀i ∈ Ω, t ∈ T uit : the delayed workload for Service (i, t).∀i ∈ Ω, t ∈ T With the above notation, the total container moving distance can be determined by the storage allocation decisions as follows: A=

k∈K i∈Ω

(liok + lidk )qk xki

(1)

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The YC workload delay and the penalty for leaving their corresponding favorable blocks can be calculated by the YC deployment decisions as follows: s p B= uit + vj (1 − γij )δpj yit (2) i∈Ω t∈T

j∈Φ i∈Ω t∈T s∈S p∈P (i,t)

Objective function: min {λ1 A + λ2 B}

(3)

The objective function consists of the above two parts which are converted to monetary cost by two parameters λ1 and λ2 . Constraints:

ri +

i∈Ω

qk xki −

k∈K + (t)

wit =

xki = 1

k∈K

∀k ∈ K

(4)

qk xki ≤ Qi

∀i ∈ Ω, t ∈ T

(5)

k∈K − (t)

qk αkt xki +

qk βkt xki

∀i ∈ Ω, t ∈ T

(6)

k∈K p yit =1

∀i ∈ Ω, t ∈ T

(7)

p cp yit

∀i ∈ Ω, t = 1

(8)

p cp yit

∀i ∈ Ω, t ∈ T \{1}

(9)

p∈P (i,t)

uit ≥ ri + wit −

p∈P (i,t)

uit ≥ ui(t−1) + wit −

p∈P (i,t)

s p δpj yit ≤ 1

∀j ∈ Φ, t ∈ T, s ∈ S

(10)

i∈Ω p∈P (i,t)

xki ∈ {0, 1} ∀i ∈ Ω, k ∈ K

(11)

p ∈ {0, 1} ∀i ∈ Ω, t ∈ T, p ∈ P (i, t) yit

(12)

wit ≥ 0 ∀i ∈ Ω, t ∈ T

(13)

uit ≥ 0 ∀i ∈ Ω, t ∈ T

(14)

Constraints (4) ensure that each container group should be allocated to a certain block for temporary storage. Constraints (5) deal with the storage capacity requirement. k∈K + (t) qk xki represents the total amount of containers that have arrived in Block i by Time Period t while k∈K − (t) qk xki is the total amount of containers that have left Block i before Time Period t. So the left-hand-side

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of Constraints (5) is the storage status of Block i at Time Period t, and it should not exceed the storage capacity. The scheduled workload wit for each service is deﬁned by Constraints (6). Constraints (7) on proﬁle assignment guarantee that only one proﬁle is used for each service (i, t). The delayed work(9). Note that the exact deﬁnition load uit is deﬁned by constraints (8) and p of uit should be uit = max{0, ri + wit − p∈P (i,t) cp yit } ∀i ∈ Ω, t = 1 and p uit = max{0, ui(t−1) + wit − p∈P (i,t) cp yit } ∀i ∈ Ω, t ∈ T \{1}. However, constraints (8) and (9) are equivalent to them as uit is non-negative and the objective function is to be minimized. For each time step, a certain YC should be used by at most one service as ensured by Constraints (10). Finally, constraints (11)-(14) specify the domain for the decision variables.

3

Solution Approach

In this section, we apply the RHA, which is commonly used in real-world container port operations, to obtain good solutions within a reasonable computational time. The RHA is a simple heuristic which decomposes the original problem into a series of smaller ones that can be solved eﬃciently by a commercial solver like CPLEX. As illustrated in Figure 3, planning is conducted within a ﬁxed horizon in which only immediate future is included and the horizon runs continually. Planning on time period t starts with updating the delayed workload and the storage status of the previous period t − 1. Such information is used as inputs for planning at the current time period. In order to consider the impact of current decision over future planning, the optimization for the current period t also includes τ time periods in the future, that is {t, t + 1, · · · , t + τ }. After solving the problem of current time period t, the delayed workload and the storage status for the current time period t can be obtained and passed on to the next period t + 1 as input information. The horizon runs continually until the planning horizon covers the last working shift. So, by rolling the planning horizon, we decompose the original problem into a series of smaller ones. planning horizon of period t xxx

t

t 1

xxx

t W

t W 1

xxx

planning horizon of period t 1

Fig. 3. The rolling planning horizon

Note that the length of a planning horizon τ is a user-deﬁned parameter. For our problem, a short planning horizon would include less container groups and less working shifts. Thus less computational eﬀort is needed to determine the storage locations for containers and YC assignment within the horizon. However, a long planning horizon would favor predictive power about the future and result

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in solutions closer to optimal one. The deﬁciency of setting a large value for τ is that long computational time is needed to obtain the optimal solution for each planning horizon. For solving the optimization problem on each planning horizon, we set a computational time limit tmax . If the problem cannot be solved to optimum by the time limit, the best solution that has been found by tmax is adopted. Because of the time limit, the performance of the RHA does not always get better as τ increases since the optimization problem on each planning horizon with a larger τ becomes harder to be solved to optimum. As a result, a sensitivity analysis should be conducted for τ to reach a compromise for the computational eﬀort and solution quality. In addition to high computational eﬃciency, another advantage of the proposed RHA is that it is able to capture dynamic changes of the container terminal operations. As the terminal operations are highly dynamic, perfect information about future events (e.g., vessel arrival and container delivery) is not always available. Hence, the optimization model should be rerun once the information is updated. As the whole planning problem is divided into a series of smaller sub-problems by the RHA, only the sub-problems aﬀected by the updated information need to be rerun. Thus, the computational cost of updating the aﬀected sub-problems is much lower than that of rerunning the whole optimization problem.

4

Numerical Experiments

In this section, computational experiments are conducted to test the performance of the proposed service-oriented model and the RHA. The solution approach is coded in C++ and CPLEX 12.1 is used as the underlying optimization solver, while the service-oriented formulation is coded in ILOG IDE 6.3 which also invokes CPLEX 12.1 solver. All computational experiments are conducted on a PC with 3GHz CPU and 4GB RAM. In the numerical experiments, a working shift is divided into three time steps. We set the working capacity of a YC for one time step to 100 and the storage capacity of a block to 1,500. Each YC is assigned a favorable yard block.The YC movement penalty vj is proportional to the geographical distance between its favorable block and other blocks. YC assignment proﬁles are generated in which YCs are allowed to travel only within the neighborhood of their favorable blocks. Regarding the RHA, we set τ = 2 and tmax = 10sec. Three sets of instances are randomly generated each of which has ﬁve instances with the same parameters as shown in Table 1. The arriving schedule ak is uniformly distributed within the whole planning horizon and the duration-of-stay (bk − ak ) in the storage yard is set to uniformly distributed within [1, 5]. The origins ok and destinations dk of container groups are randomly assigned to the berths at the quayside and the terminal gate. The storage space requirement of a container group qk is uniformly distributed in [50, 200]. Note that the instance scale of Set 3 is comparable to that of real-world operations.

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Table 1. Predetermined parameters for three instance sets

Set 1 Set 2 Set 3

|Ω|

|Φ|

|T |

|K|

4 8 12

4 8 12

8 12 16

35 100 180

Table 2 presents the computational results of all the test instances. The second and third columns report the optimal solutions or best solutions that have been found by the time limit 10,800 seconds and the CPU time consumed. As CPLEX can only handle smaller scale instances, we report the lower bounds returned by CPLEX in the fourth column denoted by LB in case that no optimal solution is found by the time limit. The results and computational times of RHA are shown in column 5 and 6. The last two columns report the gap between the result of RHA and CPLEX and LB, respectively. GAP 1 compares the results of RHA and CPLEX. As can be seen, near-optimal solutions or better solutions than those found by CPLEX can be obtained by RHA. Besides, RHA is more eﬃcient than CPLEX as much less time is consumed. GAP 2 shows the upper bound of the gap between the RHA solutions and the optimal ones. The results also indicate Table 2. Computational results for three instance sets GAP 1a GAP 2b

CPLEX

LB

RHA

Result (1) Time (sec)

(2)

Result (3) Time (sec)

(%)

19,821.70 67,669.90 33,373.40 14,079.30 13,893.80

1.23 1.25 0.86 15.91 6.01

0.88 0.79 1.08 2.19 0.89

(%)

Set Set Set Set Set

1-1 1-2 1-3 1-4 1-5

19,649.35 67,139.10 33,015.20 13,778.00 13,771.75

8.73 3.13 2.55 840.33 369.19

Set Set Set Set Set

2-1 2-2 2-3 2-4 2-5

70,120.00 44,930.30 63,656.55 21,651.00 56,951.60

> 10, 800 > 10, 800 > 10, 800 > 10, 800 > 10, 800

70,044.85 44,617.55 63,160.55 20,653.35 56,426.35

70,521.70 45,876.60 64,329.15 22,348.90 57,715.75

42.24 58.59 53.84 41.47 35.96

0.57 2.11 1.06 3.22 1.34

0.68 2.82 1.85 8.21 2.29

Set Set Set Set Set

3-1 3-2 3-3 3-4 3-5

95,178.25 71,875.95 79,158.80 70,989.95 115,137.00

> 10, 800 > 10, 800 > 10, 800 > 10, 800 > 10, 800

94,486.05 70,418.20 76,016.65 70,713.60 107,852.20

95,143.75 72,895.25 77,300.50 72,046.55 109,965.15

103.41 109.54 104.43 96.61 111.71

-0.04 1.42 -2.35 1.49 -4.49

0.70 3.52 1.69 1.88 1.96

Average

0.68

2.56

a b

: GAP 1 = [(3) − (1)]/(1) × 100 : GAP 2 = [(3) − (2)]/(2) × 100

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the good performance of RHA. In conclusion, RHA is a good heuristic approach that balances the computational eﬀorts and solution quality.

5

Conclusion

This paper has proposed a novel service-oriented model for the yard management problem in port container terminals. Unlike previous literature in which two sub-problems of the yard management problem, storage allocation and YC deployment, are solved hierarchically, we integrate the two decision problems to achieve a better storage yard management plan. Besides, the service-oriented model features a concept of YC assignment proﬁle which takes into account the requirement of YC work patterns. The problem is formulated as a mixed integer program and an RHA is applied to obtain near-optimal solutions eﬀectively. Computational experiments have shown that the solution approach can balance the computational eﬀort and solution quality.

References 1. Cheung, R.K., Li, C.L., Lin, W.Q.: Interblock crane deployment in container terminals. Transportation Science 36(1), 79–93 (2002) 2. Giallombardo, G., Moccia, L., Salani, M., Vacca, I.: Modeling and solving the tactical berth allocation problem. Transportation Research Part B: Methodological 44(2), 232–245 (2010) 3. Kim, K.H., Kim, H.B.: The optimal sizing of the storage space and handling facilities for import containers. Transportation Research Part B: Methodological 36(9), 821–835 (2002) 4. Lee, L.H., Chew, E.P., Tan, K.C., Han, Y.B.: An optimization model for storage yard management in transshipment hubs. OR Spectrum 28(4), 539–561 (2006) 5. Moccia, L., Astorino, A.: The group allocation problem in a transshipment container terminal. In: Proceedings of World Conference on Transport Research. University of California, Berkeley (2007) 6. Petering, M.E.H., Murty, K.G.: Eﬀect of block length and yard crane deployment systems on overall performance at a seaport container transshipment terminal. Computers & Operations Research 36(5), 1711–1725 (2009) 7. Zhang, C., Liu, J.Y., Wan, Y.W., Murty, K.G., Linn, R.J.: Storage space allocation in container terminals. Transportation Research Part B: Methodological 37(10), 883–903 (2003) 8. Zhang, C., Wan, Y.W., Liu, J.Y., Linn, R.J.: Dynamic crane deployment in container storage yards. Transportation Research Part B: Methodological 36(6), 537–555 (2002)

Container Terminal Yard Operations – Simulation of a Side-Loaded Container Block Served by Triple Rail Mounted Gantry Cranes Jan Klaws1, Robert Stahlbock2,3 , and Stefan Voß2 1

Student at the University of Hamburg and Hamburg University of Applied Sciences, Hamburg, Germany [email protected] 2 Institute of Information Systems, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany [email protected], [email protected] 3 FOM University of Applied Sciences, Essen/Hamburg, Germany

Abstract. When thinking about automated operations, ports and terminal operators worldwide are investigating new technology to increase the eﬃciency and throughput of their terminals. Recently, double and triple rail mounted gantry crane solutions have been considered and are already in use or currently implemented. While in a latest conﬁguration, a pair of twin cranes running on the same rails in addition to a large cross-over crane running on its own rails is used, it seems not yet clear to which extent using three cranes in one block or stack is proﬁtable even if cross-over cranes are not considered. This paper investigates the possible productivity increase for a side-loaded block layout when two or three rail mounted gantry cranes are used.

1

Introduction

In a world when the time between initial developments to introducing the ﬁnal product to the customer is shrinking continuously, the necessity of a fully functional logistic system is a basic requirement. Today over 60 % of the world’s deep-sea cargo is being placed in containers and transported by ship to harbors all around the world. Because of the increase in traﬃc, the terminals are under strong pressure to match the rising volume with the right amount of capacity. Therefore, the port is playing a key role in connecting water traﬃc with inland transportation, and the target is to minimize and utilize time most eﬃciently. Ports located near major cities usually have problems with the limited amount of storage and thus need to use their available storage space in the most eﬃcient way. The key to matching the increase in traﬃc with a ﬁxed amount of storage is the use of highly automated cargo handling equipments for in-yard transportation, stacking and storing. These technical tools can help to increase the throughput and decrease the ship turnaround time at the terminal. Nowadays these desired improvements are realized through the use of various technical J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 243–255, 2011. c Springer-Verlag Berlin Heidelberg 2011

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equipments such as automated guided vehicles (AGVs), automated rail mounted gantry (RMG), automated rubber tired gantry (RTG) and automated double rail mounted gantry cranes (DRMG). Nevertheless, ports and terminal operators are always looking for new technology to increase the eﬃciency and throughput of the terminal. As an example, Container Terminal Burchardkai (CTB) in Hamburg, Germany, for the ﬁrst time is going to implement a triple rail mounted gantry crane (TRMG) solution in 2012 [18,8,7]. In a TRMG conﬁguration, a pair of twin cranes running on the same rails in addition to a large cross-over crane running on its own rails is used. However, one has to ask and seek new answers to several questions before executing new plans. Does adding a crane to existing systems automatically lead to more productivity? And if it does, what is the extent of the increase in productivity? Furthermore, one would also have to test whether adding a crane might in fact lower productivity because of the cranes interfering with each other while operating. To provide appropriate answers to these questions there have been several attempts; see the literature review below. Most studies on latest technology as it can be found, e. g., in Hamburg assume that the investigated stacks or blocks in the container yard are operated with end loading. This means that pickup and delivery of the containers takes place in an area which is located at each end of the block (water/landside). Related study results can be applied in container terminals which already use end loaded stacks or when their existing layout can be modiﬁed. Terminals which are using side loaded stacks and which cannot modify their layout due to various reasons might be interested in a side loaded TRMG solution. Note that latest technology developments not only assume front and end loading but also side loading. For instance, the latest automated container system of ZPMC [14] utilizes an automated grid system on the waterside while accessing the blocks/stacks by side loading. The goal of our paper is to provide a simulation study comparing DRMG and TRMG solutions with side loading where overtaking of the cranes is not permitted. To the best of our knowledge this is the ﬁrst attempt to investigate a related side loaded TRMG block layout. After a short literature review we discuss gantry crane processes in Section 3 and the simulation in Section 4.

2

Literature

Container terminal operations have been the topic of almost uncountable research eﬀorts in recent years. A very comprehensive survey is provided and updated in [17,15]. A general exposition of various related problems can also be found in a recent handbook [1]. Scheduling single yard cranes and RTGs is discussed in quite a few papers but the handling of DRMGs is already an issue with much less consideration due to its innovative nature. A simulation approach for operations of an automated terminal is presented in [11] comparing the following concepts: – AGV/single RMG, – conveyance system/single RMG,

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– overhead grid rail/AGV, – rack/AGV in terms of performance and costs. The results show that automation could improve the performance of conventional terminals substantially at considerably lower cost. It turns out that the AGV/RMG-system is the most cost-eﬀective one. In [2] deployment strategies of DRMG systems for loading outbound containers in traditional yard truck based terminals (without AGVs) are proposed. The authors present an integer programming model and a greedy heuristic, a simulated annealing approach, and a combined DRMG scheduling heuristic for solving the problem. In [13] a comparison of single RMG, twin RMG, and DRMG is presented. Diﬀerent stacking alternatives are evaluated by means of simulation in terms of throughput, ﬂexibility, complexity, operational costs, and investment costs. In [12] twin RMG, DRMG, and TRMG are compared. Overall results show that the twin RMG is the best performing one when storage capacity is taken into account. Per stack module, the triple RMG is the most productive one. The outline is similar to what is available in the studies by [16] for DRMG and [4] for TRMG, where front and end loading is assumed. Most recently, [16] describe an approach for scheduling DRMGs and [4] for scheduling TRMGs in an automated container storage block with asynchronous hand over at the transfer areas at both block front ends. Side loading can be found in many container terminals world wide, but also at intermodal transshipment terminals; see, e. g., [6]. A simulation study on operation rules for automated container yards is presented in [9]. In particular, the authors investigate container stacking rules as well as operational rules of DRMGs in an AGV based system with AGVs driving in an extra lane of the large crane beside the block. They aim at testing two sequencing rules (ﬁrst-in-ﬁrst-out rule and minimum empty travel distance rule) and two crane dispatching rules with and without diﬀerentiating the two cranes based upon their capabilities.

3

Gantry Crane Processes in a Storage Block

A simpliﬁed terminal layout consists of ﬁve basic elements: – One or more quays. – Several cranes, loading and unloading vessels docked at a speciﬁc quay. – Transport means like AGVs/straddle carriers/tractor trailers which deliver the containers from the cranes to the storage block. – Diﬀerent kinds of gantry cranes, which handle the container storage inside the yard and load them onto trucks. – Trucks and trains ensuring the connection from the harbor to the hinterland transportation and therefore handle the transportation to the ﬁnal customer. Details on the structure and handling equipment of a terminal are provided, e. g., in [17]. A basic overview of operations and the basic ﬂow of transports at a seaport container terminal can be found in Figure 1.

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Truck and Train Operation Area Hinterland Operation

Empty Stock Yard Import/Export Stock

Yard Moves Sheds

Quayside Operation Ship Operation Area Fig. 1. Operation areas of a seaport container terminal and ﬂow of transports [17]

Because of the complex interactions that are caused by the above described layout this paper focuses only on the operations that are performed by the gantry cranes serving the storage block. All activities concerning unloading the vessel itself and transport of containers from the vessel to the yard and from the yard to the trucks which handle the inland transportation will not be considered. 3.1

Storage Block System and Layout

First a basic overview seems appropriate about the container block that is located between the waterside (quay) and the landside (inland transportation). A container block consists of many containers placed next and on top of each other. Usually the block has three predeﬁned dimensions in width, length and height, depending on the space that is available at the terminal and the restrictions by the equipment operating at the block. Subsequently when these dimensions are mentioned they refer to bays and rows in x- and y-dimension and tiers in z-dimension. In everyday business the block may consist of various types of containers, e. g., standard 20 ft and 40 ft containers, reefers which need electrical connection, dangerous goods and oversized (height/width) containers. To simplify matters this paper deals only with 20 ft and 40 ft standard containers (which seems in line with the fact that usually other containers like reefers or oversized containers are handled in speciﬁc areas). At one side of the block a two lane road is located right along the full length of the block. Here AGVs will stop right at the bay location of the eﬀected container to either deliver or pick up a container (this also relates to settings in intermodal (truck-train) terminals [6]). The two lane road is assumed so that transport means such as AGVs are able to pass each other while waiting for pick up or delivery of a container. The lane located closer to the block will be the lane for pickup and delivery, the lane located

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to the left will be the passing lane. Without the ability to pass each other there would have been a mass backup in traﬃc if an AGV would have stopped right at the ﬁrst bay of the block, blocking the access to the remaining 36 bays. The layout is shown in Figure 2, modifying expositions from [10,16]. Note that this description is not restricted to AGVs and RMGs as handling equipment could easily accommodate, e. g., tractor trailer units rather than AGVs. Furthermore, the situation at the CTB which served as basis for [4] is slightly diﬀerent as water access is from more than one side of the terminal; see, e. g., [8,7].

Fig. 2. Layout of a storage block served by a TRMG (conﬁguration with three cranes of the same height, running on the same rail)

3.2

Diﬀerent Options to Serve the Storage Block

The container block above can be operated through diﬀerent types of technical equipment, yet always depending on the overall applied terminal layout. This paper will focus on two possible ways how to handle in-yard transportation and storage. In the ﬁrst case the block is served by a DRMG system, meaning that two cranes with the same height are running on the same rail. Second the block is served by a TRMG system. In this TRMG conﬁguration, three cranes all with the same height are running on the same rail. In both setups the cranes cannot pass each other due to the fact that they all travel on the same rail tracks. To be able to provide a scientiﬁcally based answer to the initial hypothesis, many diﬀerent conditions, which may aﬀect the productivity, need to be considered. This paper concentrates on comparing the two diﬀerent systems with identical initial situation and identical boundary conditions.

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Block Fragmentation and Container Placement Inside the Block

Dividing a block can be done in various ways. One option is to split up the block in sections, one only for export containers and one only for import containers. Another possibility is to have no sections at all, i. e., import and export containers will be mixed while stored in the block. This simulation will use the mixed block layout to prevent dead lock situations. The container placement inside the block is based on a weighted search. A weighted search is a search through the entire block the container is assigned to. Every possible position for a container inside the block gets a score based on diﬀerent values for that position. Finally the container is placed at the position with the highest score. The weighted search used in our simulation focuses on three diﬀerent parameters (number of containers in a row, stacking trucks in transit to the bay and unstacking trucks in transit to the bay) and is always applied to the same block. The ﬁrst parameter leads to the result that containers will be placed in rows with the least number of containers already in that row. The two other parameters will prevent AGVs from parking in the same spot to stay as close as possible to the real environment. A constant value will be assigned to all three parameters and depending on the value of the constant the search is weighted. For all simulation runs all three parameters had the same constant value of -0.5 meaning that all parameters had the same impact. To illustrate two diﬀerent positions in the block will be picked at random and the weighted search will be applied; see Table 1 for an example. Due to the result of the weighted search the container will be placed at position 2 because the ﬁnal score is higher (-0.5) than the score for position 1 (-2). Table 1. Position scores in a block

Location (Bay/Row)

Position 1

Position 2

30/3

24/8

Number of containers already placed at this position

3

1

Number of AGVs already parked at the bay

1

0

Final score

3.4

3×(−0.5)+1× 1×(−0.5)+0× (−0.5) = −2 (−0.5) = −0.5

Restacking Problem

The restacking problem occurs when a container that needs to be moved is blocked by others on top of it (for a survey on related relocation and restacking problems see [3]). Considering the layout of the block in Section 3.1, the speciﬁc container in question can either be blocked by just one container or by a maximum of three containers on top of it. To get to the container that needs to be moved next seems obvious. One of the cranes just has to move the blocking containers. The diﬃculty concerning those situations is how to move the container in the least amount of time and with the smallest number of container movements. Because of the complexity of a restacking problem, a fully independent simulation study could be performed focusing on the time that is needed

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for each restacking operation. Therefore, this simulation will not diﬀerentiate between diﬀerent restacking algorithms and just use one algorithm based on the predetermined container placement strategy for the storage block. If a restacking operation needs to be performed the crane will look within the aﬀected bay for the row with the least amount of containers already in it. After the row is located the container is automatically placed there. If there is no free spot at all, the crane will switch the bay and the algorithm will start over again until the operation is ﬁnished. Choosing the row with the least amount of containers could also help to prevent new restacking operations. But to provide a scientiﬁc answer to the statement another simulation study using some other restacking algorithm needs to be performed. 3.5

Processes within the Block System

This paper diﬀerentiates between two kinds of diﬀerent jobs that can be executed by the cranes. The movement of the containers itself concerning the two jobs is the same. The only diﬀerence is that export containers have the vessel as ﬁnal destination and import containers have the yard as ﬁnal destination. The only diﬀerence between the two systems is that in the DRMG setup only two cranes will perform the stacking and unstacking operations in the block compared to three cranes in the TRMG setup. Possible container movements are: 1. Import containers will be unloaded by one of the three quay cranes, loaded to the AGVs which will ensure the transport of the containers from the vessel to the yard where they will be picked up from the RMGs and stored in the block using the weighted search explained above. 2. Export containers will be picked up by the RMGs inside the block, loaded to the AGVs which will ensure the transport from the yard to the vessel where they will be picked up from one of the three quay cranes and loaded into the vessel. Movements from the yard to the gate or to trucks which handle the hinterland transportation will not be considered in this simulation study. 3.6

Crane Rules

This section concentrates on the interferences of the cranes while operating. In order for the cranes to work without any kind of disturbances (e. g., collisions) the cranes need to follow some basic rules as follows (these rules or restrictions will apply for both systems DRMG and TRMG): – While operating, a safety distance between the cranes must be respected at all points in time. – The velocity of the cranes is considered constant. No acceleration and breaking time will be taken into account. – The length and the width of the block itself are predeﬁned. – A new job can only be started by a crane if the preceding job is already ﬁnished.

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– A loaded crane always has the right of way. – If two loaded cranes hinder each other, the crane with the earlier due date of its container has the right of way. – Before starting the simulation the block is assumed to be ﬁlled (either based on real data or) randomly with import and export containers using the container placement strategy described in Section 3.3 (in this way it would be easy to replace the blockﬁlling by real data). – A crane is performing a movement only in one direction at a time (Manhattan metric). – The simulation is based on a pre-deﬁned number of containers that will pass through the system. On which side of the system the containers will enter is based on the service applied to the vessel; see below.

4

Simulation Scenario Design

In this section we describe the settings of our simulation together with related results. As simulator we use the simulation software Flexsim CT 3.0 [5]. All experiments are run on a personal computer with 2 GB RAM and an AMD Turion X2 CPU, 2.2 GHz. 4.1

Simulation Setup

As basic setting, the following parameters are used. The block consists of 10 rows, 37 bays and 4 tiers counting up to 370 positions for 20 ft containers and 185 for just 40 ft containers. The overall simulation setup consists of four basic elements: 1. 2. 3. 4.

A speciﬁc quay (500 m long) for the vessel. Three quay cranes which will unload and load the vessel. AGVs which will transport the containers between the vessel and the yard. Two or three RMGs serving the block depending on which system is applied.

In our study we assume that all RMGs operating at the block regardless of the system they are part of (DRMG/TRMG) are of the same height and will all travel on the same rail tracks which means they are not able to pass each other. To avoid a collision between the RMGs a safety distance of at least two bays is implemented and respected at all points in time. The AGVs will travel along the side of the block and will stop right next to the bay location of the container where they will be either loaded or unloaded by one of the RMGs. The RMGs are not tied to a speciﬁc area of the block (bays/rows) and could (in theory) serve the whole block but are always restricted in their movements because of the safety distance. Therefore, there will be a dynamical distribution of the RMGs on the entire block. The total numbers of import and export containers are displayed in the hatch proﬁle, which is a table of operations that are to be executed on the hatch by a single quay crane. Operations are executed in order, commencing with the ﬁrst

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Fig. 3. Hatch proﬁle

row and proceeding to each subsequent row until the full table has been executed. The hatch proﬁle used for all simulation runs can be found in Figure 3. A service which consists of multiple hatches is always assigned to a vessel and describes how many import and export containers in total are loaded to or discharged from the vessel. The service used in all simulation runs consists of the hatch described above multiplied by three, meaning that a total of 240 containers will be discharged and 270 containers will be loaded onto the vessel. The 270 export containers will be assigned to the block using the weighted search while starting the simulation. By using this method there will be no required warm up time or a previous simulation run necessary to have the storage block ﬁlled with containers. Details on the service can be found in Figure 4.

Fig. 4. Service table

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The applied service results in a frequent traﬃc of AGVs between the quay cranes and the RMGs serving the yard. Because the transport of the containers between the quay and the yard is not the focus of this paper we assume that there are enough AGVs to prevent a bottleneck situation caused by the AGVs. Moreover, we assume no deadlock situations due to too many AGVs operating in the same area. RMGs and quay cranes therefore do not have to wait on AGVs and are just limited to their own performance. The RMGs serving the block will place import containers inside the block using the weighted search explained above. If export containers that are scheduled to move next are blocked a restacking operation needs to be performed. In this case the RMG looks for the next free spot inside the same bay and the blocking container will be moved there. The last part of this subsection provides a brief overview of how the simulations were performed. For all simulation runs just one vessel with the above mentioned service is processed. This counts up to 510 moves that need to be completed before the vessel can exit the berth. Entering and leaving the berth is always bonded with a certain amount of time that is needed to tie down the vessel at the quay and to untie the vessel before leaving the quay. For both operations a total of 75 minutes are considered into the calculation. To tie down the vessel 45 minutes are considered before the quay cranes can actually start unloading the vessel. After completing to load the vessel it takes another 30 minutes for the vessel to prepare to leave the quay. The simulation system is conﬁgured so that computation times are 60 times faster compared to the real world. 4.2

Results

For the results presented next, twenty simulation runs for each system (DRMG and TRMG) were performed. After some initial simulation runs it became obvious that the traﬃc was always concentrated at one part of the storage block. To avoid the concentration to just a small section of the block and cause a traﬃc jam by the waiting AGVs, the block was separated into three sections: two sections consisting of 12 bays and 10 rows and one section consisting of 13 bays and 10 rows, counting up to a total of 37 bays with 10 rows. The three sections were combined to an import/export area. A container coming from the vessel was assigned at random to one of the three sections of the area and the placement of the containers was still done using the weighted search. Furthermore the RMGs were still just limited in movements because of the safety distance due to the fact that they all were assigned to the import/export area and not just to a speciﬁc section of the block. Using this setup helped a lot to spread traﬃc throughout the whole block and to stay closer to the real environment. We also assumed that AGVs are able to recognize when an operation is nearly completed so they can begin to work on the next operation. Without this ability the AGVs would have waited in the yard next to their last discharge. Because this is ineﬃcient the value ”Look ahead” in the hatch proﬁle was set to one. The overall gain in productivity between both systems was quite signiﬁcant. By looking at the average time at berth for a vessel served by a DRMG system

Simulation of a Side-Loaded Container Block Served by TRMGCs

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Fig. 5. Average time at berth for a vessel whose storage block is served by either a TRMG or a DRMG system

and the average time at berth for a vessel served by a TRMG system a signiﬁcant increase was observed. The average time at berth for a vessel served by a DRMG system was 8.34 hours compared to an average of 7.17 hours for a vessel served by a TRMG system. The data shows that a TRMG system always performed better than the DRMG system in all twenty simulation runs; see Figure 5. This means an average reduction of more than 14 % of the time at berth for a vessel. Finally, we discuss some results to investigate the sensitivity of the above results. First we assume that the containers are no longer spread over the whole block but that they have been stacked more densely over each other. That is, rather than distributing containers over the whole block we assume that Table 2. Sensitivity analysis results for DRMG and TRMG systems Simulation run Segregated bays Segregated bays Empty no. DRMG TRMG bays/positions Berth time Berth time DRMG (in h) (in h) Berth time (in h)

Empty bays/positions TRMG Berth time (in h)

1 2 3 4 5 6 7 8 9 10

7.79 8.01 7.79 7.89 7.92 8.28 8.05 7.97 7.92 8.02

6.99 7.05 7.01 6.91 6.81 7.16 6.76 6.94 7.02 6.56

8.30 8.38 8.20 8.17 8.49 8.28 8.26 8.46 8.35 8.37

7.63 7.18 7.22 7.21 7.23 7.09 7.21 7.21 7.11 7.37

Mean

7.96

6.92

8.33

7.25

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containers are stored in segregated bays (see columns 2 and 3 of Table 2), i. e., bays holding containers which are all the same type. Another test assumes a setting similar to the weighted search described above but with a simpliﬁcation such that we are searching for empty bays and positions ﬁrst (see columns 4 and 5 of Table 2). Overall we can see that the productivity gain of a TRMG system over a DRMG system is similar no matter which of the modiﬁed search rules is applied. Additional tests would be necessary to recommend speciﬁc weights but this was beyond the scope of the reported simulation results.

5

Conclusion

In this paper we have studied a speciﬁc TRMG system in comparison with a related DRMG system. Compared to more advanced systems where one crane may overtake one or two others, this system may be more easily implemented in already existing terminals. Looking at the simulation results it can be clearly admitted that a block served by a TRMG system is more productive than a block served by a DRMG system. A third crane therefore leads to more productivity in the simulated scenarios. Note, however, that the achieved results are only valid for a block setup where the RMGs are side loaded. Further research in the performed simulation could focus on dual cycling mode which means, AGVs dropping oﬀ an import container at the block will automatically be loaded with an export container. This reduces empty travels of the AGVs between the yard and the quay and therefore contributes to even better productivity. Other ideas could focus on changing the algorithm on how to place containers in the yard or changing the rules for restacking operations. Various parameters can be change in the simulation setup and they will probably all have a diﬀerent eﬀect on the productivity of a terminal. Most interesting could be the comparison of a side loaded and an end-loaded stack served by a TRMG system. Here not only the time at berth for a vessel is important but also a possible gain in space using the end-loaded stack plays an important role when the overall ship traﬃc develops as predicted. Looking at the research in container terminal setups using simulation software like Flexsim CT 3.0 one has to admit that the development has just begun to pick up pace and will support more and more when deciding which layout or which system (TRMG/DRMG) for a container yard is more productive. This will help to choose the best possible solution before even starting to implement any changes.

References 1. B¨ ose, J.W. (ed.): Handbook of Terminal Planning. Operations Research, Computer Science Interfaces Series, vol. 49. Springer, New York (2011) 2. Cao, Z., Lee, D.H., Meng, Q.: Deployment strategies of double-rail-mounted gantry crane systems for loading outbound containers in container terminals. International Journal of Production Economics 115, 221–228 (2008) 3. Caserta, M., Schwarze, S., Voß, S.: Container rehandling at maritime container terminals. In: B¨ ose [1], pp. 247–269

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4. Dorndorf, U., Schneider, F.: Scheduling automated triple cross-over stacking cranes in a container yard. OR Spectrum 32, 617–632 (2010) 5. Flexsim Software Products: Flexsim CT Simulation Software User Guide, Version 3.0 (2010) 6. Gutenschwager, K., B¨ ose, J., Voß, S.: Eﬃziente Prozesse im Kombinierten Verkehr – Ein neuer L¨ osungsansatz zur Disposition von Portalkr¨ anen. Logistik Management 5(1), 62–73 (2003) 7. Hamburger Hafen und Logistik AG: Expansion plans in ﬁlm: Burchardkai times two, http://www.hhla.de/Expansion-plans.105.0.html?&L=1 last check of URL: (May 15, 2011) 8. Hamburger Hafen und Logistik AG: Technical Data Burchardkai, http://www. hhla.de/TECHNICAL-DATA.113.0.html?&L=1 last check of URL: (May 15, 2011) 9. Kim, K.H., Wang, S.J., Park, Y.M., Yang, C.H., Bae, J.W.: A simulation study on operation rules for automated container yards. In: Proc. of the 7th Annual Intern. Conf. on Industrial Engineering, Busan, Korea, pp. 250–253 (2002) 10. Koch, T.: Automatik-Portalkrane im CTA-Containerlager. Hebezeuge und F¨ ordermittel 44, 632–636 (2004) 11. Liu, C.I., Ioannou, P.: A comparison of diﬀerent AGV dispatching rules in an automated container terminal. In: Cheu, R.L., Srinivasan, D., Lee, D.H. (eds.) Proc. of IEEE 5th Intern. Conf. on Intelligent Transportation Systems, pp. 880–885. IEEE, Piscataway (2002) 12. Saanen, Y.: Automated container handling, http://www.freight-int.com/ categories/automated-container-handling/automated-container-handling. asp last check of URL: (May 15, 2011) 13. Saanen, Y.A., Valkengoed, M.V.: Comparison of three automated stacking alternatives by means of simulation. In: Kuhl, M.E., Steiger, N.M., Armstrong, F.B., Joines, J.A. (eds.) Proc. of the 2005 Winter Simulation Conf (WSC 2005), Orlando, FL, USA, December 4-7, pp. 1567–1576. ACM, New York (2005) 14. Shanghai Zhenhua Port Machinery Co. Ltd: Automated container system, http:// www.zpmc.com/showroom/index.html last check of URL:(May 15, 2011) 15. Stahlbock, R., Voß, S.: Operations research at container terminals - a literature update. OR Spectrum 30, 1–52 (2008) 16. Stahlbock, R., Voß, S.: Eﬃciency consideration for sequencing and scheduling of double-rail-mounted gantry cranes at maritime container terminals. International Journal of Shipping and Transport Logistics 2(1), 95–123 (2010) 17. Steenken, D., Voß, S., Stahlbock, R.: Container terminal operations and operations research – a classiﬁcation and literature review. OR Spectrum 26, 3–49 (2004) 18. ThyssenKrupp GfT Gleistechnik GmbH: Crane track girders. Brochure, http:// www.tkgftgleistechnik.de/download.html?eID=dam_frontend_push&docID=833 last check of URL: (July 07, 2011), (August 2007)

Randomized Algorithm with Tabu Search for Multi-Objective Optimization of Large Containership Stowage Plans Fan Liu, Malcolm Yoke Hean Low, Wen Jing Hsu, Shell Ying Huang, Min Zeng, and Cho Aye Win School of Computer Engineering, Nanyang Technological University #50 Nanyang Avenue, Singapore 639798 {liufan,yhlow,hsu,assyhuang,zengmin,wcaye}@ntu.edu.sg

Abstract. This paper describes a randomized algorithm with Tabu Search (TS) for multi-objective optimization of large containership stowage plans. The algorithm applies a randomized block-based container allocation approach to obtain a Pareto set of stowage plans from a set of initial solutions in the ﬁrst stage, and uses TS to carry out multi-objective optimization on the Pareto set of stowage plans in the second stage. Finally, a group of non-dominated solutions is generated based on objectives such as the number of re-handles, the completion time of the longest crane, the number of stacks that exceed the weight limit, the number of idle slots, horizontal moment diﬀerence and cross moment diﬀerence. Experimental results based on real data show that the proposed algorithm is able to obtain better stowage plans compared with human planners.

1

Introduction

Nowadays, approximately 90 % of the commodity around the world are containerized and transported by containerships [4]. The stowage planning of containers onboard the ships is one of the most critical problems in shipping industry and it aﬀects the shipping lines’ operating cost signiﬁcantly. Existing stowage planning process is mainly carried out by human planners. The quality of the stowage plans prepared by human planners largely depends on their experience and intuition about the characteristics of containers and stowage requirements. However, to become a professional stowage planner, it takes years of training onboard containerships. Thus, there is a shortage of experienced planners in the shipping industry. Furthermore, the ,market of maritime transportation has been growing dramatically over the past 20 years. This requires larger containerships to cater to the increased demands from the customers. The largest containership available now has reached the size of 15,000 TEUs (Twenty-Foot Equivalent Unit) [9]. Therefore, shipping lines are facing increasing challenges in generating eﬃcient and feasible stowage plans for large containerships. In order to help the shipping industry cope with this increasing challenge, our research work aims to develop a fully automated stowage plan generation system J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 256–272, 2011. c Springer-Verlag Berlin Heidelberg 2011

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for large containerships. The detailed architecture of the system can be found in our previous published work [8]. In this paper, we proposed a randomized algorithm with TS for multi-objective optimization of large containership stowage plans in the automated stowage plan generation system. The remainder of the paper is organized as follows. Section 2 reviews some related literatures in the ﬁeld of stowage planning. In Section 3, we present deﬁnitions of some key objectives in the stowage planning problem. Our proposed optimization algorithm in the automated stowage plan generation system is described in Section 4. In Section 5, case studies and experimental results are presented. Finally, Section 6 concludes this paper and outlines some future work.

2 2.1

Literature Review Previews Research

The Master Bay Plan Problem (MBPP) [1], which refers to the problem of deciding the stowage conﬁguration of containers on a containership, has been studied by shipping companies and researchers since 1970. Initially the researchers tried to use traditional method to solve this problem. They mainly focused on establishing a set of 0-1 linear programming formulation which can translate the stowage planning problem including all the constraints and requirements into a mathematical model. If the linear programming formulations are well deﬁned, theoretically an optimal solution can be obtained. However, the search space of the established mathematical model depends on the capacity of the ship, volume of containers and the complexity of the operational constraints. Even for a medium size containership, e.g., a 2,000 TEUs vessel, the problem becomes a non-trivial one due to the large number of variables and inequalities needed for the formulations. Moreover, the stowage planning problem has been proven to be NP-complete in [2,3]. It is very hard or even impossible to guarantee an optimal solution in a reasonable execution time for a real size commercial containership. On the other hand, the ship building industries have been growing dramatically over the years. Ships with new structures are coming out every year. It is sometimes diﬃcult to establish a set of linear programming formulations to express the stowage planning problem due to the nature of the special structure of the ships. Thus, researchers have been trying to develop heuristics to provide feasible and eﬃcient solutions. A brief review of some recent research works follows. The work by Ambrosino and Sciomachen in 2004 [1] described a strategy for solving the stowage planning problem. They proposed a heuristic approach before performing a 0-1 linear programming model. The idea is to take a set of heuristic preprocessing of containers and pre-stowing procedures to allow the relaxation of some constraints of the original 0-1 linear programming model, and ﬁnally reduce the search space of the model. Although this strategy reduced the processing time to a reasonable level, about 20 minutes for one plan, some assumptions applied in the preprocessing stage made the problem unrealistic. For example, usually a containership has to visit several ports in one voyage. At

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each port some containers onboard have to be discharged and new containers have to be loaded. But in [1], they only considered the loading problem at the ﬁrst port. Crane intensity, which stands for the utilization of quay cranes used to load and unload containers, was also not considered. Wilson and Roach [10,11] developed a methodology for generating computerized stowage plan. In their approach, instead of establishing a 0-1 linear programming model, they proposed a two-step stowage planning procedure which consists of a strategic level and a tactical level. First in the strategic level, a branch-and-bound algorithm was developed to assign generalized containers in each bay of a vessel. In the second tactical step, TS was applied to match each individual container into a proper location of one bay. Their approach is capable of obtaining a feasible solution but optimality is not guaranteed. Also, a single objective function is used for the optimization purpose. Kang and Kim [7] developed a method to divide the stowage planning problem into two sub-problems, one for allocating container groups into the cargo holds and one for determining the detailed allocation pattern of containers within each cargo hold. The ﬁrst sub-problem was solved by a greedy search algorithm, while the second sub-problem was solved by a tree search method. A single objective function was developed to evaluate the solutions. These two sub-problems were solved iteratively until the objective value cannot be improved anymore. However, in this research they simpliﬁed the problem by only allowing 40Ft container to be stowed onto the ship. Because their approach still involved linear programming, the computation time was also relatively long compared with Wilson and Roach’s approach [10,11]. The stowage planning problem is inherently a multi-objective optimization problem where some of the objectives can be conﬂicting. In multi-objective optimization, the goal is to look for a set of solutions which represents trade-oﬀs with respect to the constraints. But all the research work mentioned above simpliﬁed this requirement. They designed a single objective function which associated a weight to each objective factor of the problem and obtained a single value to evaluate the solution. Using such methodology, a solution with optimum value in term of a single objective function may not be the optimal in term of multiobjective optimization. Xiao et al. [12] proposed a block stowage algorithm to take the distribution of workload among quay cranes into consideration. This algorithm tried to optimize two important objectives, namely the number of re-handles and the quay cranes’ utilization at the same time. By setting tolerance factor to diﬀerent levels, solutions with a low number of re-handles and a high quay crane utilization could be obtained. Although their work did not yet consider the issue of ship stability, their proposed framework contained a planned module to solve the ship stability problem. In their follow-up work [8], besides the existing block stowage algorithm, Liu et al. proposed a block selection algorithm to further consider the quay crane utilization and ship stability problem. In this block selection algorithm, they proposed two diﬀerent approaches to cater to the requirements for quay crane

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utilization and ship stability, namely the workload-based approach and the distancebased approach. The former was designed to obtain solutions with better quay crane utilization, while the latter was aimed at achieving better container weight distribution along the ship. The experimental results showed that the block selection algorithm exhibited good performance in terms of quality of the solutions and execution time. However, as the two proposed approaches were mutually exclusive, the quay crane utilization and the even distribution of containers weight may not be optimized at the same time. In this paper, we describe a randomized algorithm with TS for multi-objective optimization of large containership stowage plans based on [12,8]. 2.2

Tabu Search

The initial idea of TS was introduced by Glover in 1977 [5], and in 1993 the principal method of TS was proposed by Glover, Taillard and de Werra in their research work [6]. They explained that a global optimum of an optimization problem could be obtained by making a sequence of moves from one feasible solution of an optimization problem. Each move from one solution to another is the best available, but some moves are deﬁned as tabu or illegal at certain iterations to avoid cycling and getting stuck to the local optimum. However, in some circumstance a tabu move may be desirable and an aspiration criterion is needed to allow the move to be made. Since its introduction, this mechanism has been successfully used to solve optimization problems such as job scheduling, layout optimization, traveling salesman problem and vehicle routing problem. In this paper, we applied TS to solve the stowage planning optimization problem for large containerships.

3 3.1

The Stowage Planning Problem Problem Description

The stowage planning problem is to generate a stowage plan for a given containership at a port which will go on a given voyage. A containership contains a number of bays with bay index increasing from bow to stern. There are two kinds of bays, 40 foot and 20 foot bays, respectively. A 40 foot bay consists of two contiguous 20 foot bays. Each bay is formed by a number of rows across the width of the ship. Each row is a vertical stack of stowage locations from bottom to top. Bays are usually divided by hatches into two sections, on deck and under deck. Detailed introduction of containership structure can be found in [8]. The voyage of the containership is a sequence of the ports this ship will visit to unload and load containers. The stowage planner will also be given a list of containers that are to be loaded at the current port and to be unloaded at the various downstream ports along the voyage. There are mainly two types of containers deﬁned by the physical size. The dimension of a 20 foot container is 8 feet wide, 8 feet 9 inches high and 20 feet long. A 40 foot container is with the same width and height as a

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20 foot container, but twice the length of it. Besides grouping the containers by their sizes, containers can be grouped based on their characteristics into general containers, reefers, high cubes and hazardous containers. For detailed information of the diﬀerent types of containers, one can refer to [8]. A stowage plan needs to be generated which optimizes on a number of objectives. 3.2

Objectives

The following objectives are considered. • Number of re-handles Containers are stowed in vertical stacks on a containership, which is a LIFO (last in, ﬁrst out) system. When a container is discharged from the ship, all the containers above it in the same stack must be discharged ﬁrst. In some cases, if the container is located under a hatch, in order to access this container, all the other containers above the same hatch must be discharged as well. Re-handle happens when a container is supposed to remain on board at one port, but for some reasons, either discharging other containers or stowing in new containers under it, this container has to be discharged ﬁrst and loaded back again. This kind of operations brings additional quay crane movements for containers, which results in extra cost in both time and money. Therefore, reducing the number of re-handles is one of the key objectives in our research work. It can be described as, min

[oh (ruh + ah ) + (1 − oh )rah ]

(1)

h∈H

H denotes the set of all the hatches. oh denotes whether hatch h is opened. ruh denotes the number of re-handled containers under hatch h. ah denotes the number of containers not being unloaded at the current port on hatch h. rah denotes the number of re-handled containers on hatch h.

oh = min(1,

x(p−1)ijkc

¯p i∈Bh ,j∈Rh ,k∈UTij ,c∈C

+

|x(p−1)ijkc − xpijkc |)

(2)

ˆp i∈Bh ,j∈Rh ,k∈UTij ,c∈C

Bh denotes the set of bays contain hatch h. Rh denotes the set of rows contained in hatch h. U Tij denotes the set of tiers under deck in bay i row j. C¯p denotes the set of containers being unloaded at port p. xpijkc is a variable indicating whether a container c is stowed in bay i row j tier k at port p. Cˆp denotes the set of containers on board the ship both before unloading and after loading at port p.

Randomized Algorithm with TS for Containership Stowage Planning

ah =

x(p−1)ijkc

261

(3)

ˆp i∈Bh ,j∈Rh ,k∈ATij ,c∈C

ATij denotes the set of tiers on deck in bay i row j. rah = {c ∈ Cˆp |∃i ∈ Bh , ∃j ∈ Rh , ∃k ∈ ATij , x(p−1)ijmc2 x(p−1)ijkc = 1, +

¯p m
max(0, x(p−1)ijmc2 − xpijmc2 ) ≥ 1}

(4)

ˆp −f rah m
f rah denotes the set of forced re-handled containers on hatch h. f rah = {c ∈ Cˆp |∃i ∈ Bh , ∃j ∈ Rh , ∃k ∈ ATij , x(p−1)ijmc2 ≥ 1} x(p−1)ijkc = 1,

(5)

¯p m
ruh = {c ∈ Cˆp |∃i ∈ Bh , ∃j ∈ Rh , ∃k ∈ U Tij , x(p−1)ijmc2 x(p−1)ijkc = 1, +

¯p m
max(0, x(p−1)ijmc2 − xpijmc2 ) ≥ 1}

(6)

ˆp −f ruh m
f ruh denotes the set of forced re-handled containers under hatch h. f ruh = {c ∈ Cˆp |∃i ∈ Bh , ∃j ∈ Rh , ∃k ∈ U Tij , x(p−1)ijmc2 ≥ 1} x(p−1)ijkc = 1,

(7)

¯p m
• Completion time of the longest quay crane Quay cranes are used to discharge and load containers for the ships. At every port, a containership may be served by a given number (usually 3-5) of quay cranes. Each crane will be assigned to discharge and load containers in several bays. Due to the physical size of quay cranes, they are not able to operate simultaneously on adjacent bays. If two quay cranes are working too close to one another, one of them must stop and wait until the other crane moves to a bay with enough safety distance. The waiting time of a crane is called ”idle time”. The ideal situation is for all the quay cranes to start and ﬁnish their workload at the same time with minimum idle time. However, it is usually not possible to achieve this in the

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real world. Thus the completion time of the longest quay crane decides the utilization of the quay cranes. The aim is to minimize the completion time of the longest quay crane. It can be described as, min(max{Ti−complete = Ti−worktime + Ti−idletime + Ti−movetime |i ∈ Cn }) (8) Ti−complete denotes the completion time of crane i. Ti−worktime denotes the time crane i spends on loading and unloading. Ti−idletime denotes the time crane i spends on idling. Ti−movetime denotes the time crane i spends on moving from one bay to another. Cn denotes the set of cranes. • Number of stacks that exceed weight limit Due to the structure of the containership, there is a limit for the total weight of each stack both on and under deck. This limit varies for diﬀerent bays and rows. The total weight of all containers in one stack cannot exceed the maximum weight limit of it. Therefore, reducing the number of stacks that exceed their weight limit to 0 is the one of the objectives in stowage planning problem. It can be described as, Sij ) (9) min( i∈B,j∈Ri

Sij is the variable indicating whether the stack in bay i row j exceeds the weight limit or not, 1 for exceeds the weight limit, otherwise 0. • Number of idle slots An idle slot represents an unusable empty stowage location on ship. It occurs in three diﬀerent situations. First, a stack cannot be completely ﬁlled due to the weight of containers in that stack which has reached the stack weight limit. Second, if a stack under deck stows at least one high-cube container, the top slot must be left empty otherwise the hatch cover cannot be closed. Third, for the empty slots under a hatch with containers stowed on top, in order to access these empty slots, extra re-handles are needed. A good stowage plan should maximize the use of all the stowage locations on a ship and hence minimize the number of idle slots. • Horizontal moment diﬀerence The diﬀerence of weight distribution across the horizontal dimension of the ship can result in a heel angle. A large heel angle will have safety implication when the ship is sailing. Thus in a good stowage plan, the moment caused by the weight of containers on the starboard side (right) of the ship, including all the locations on deck and under deck, must be as close as possible to the moment caused by the weight of containers on the port side (left) of the ship. It can be described as, min(|

wc × xpijkc × lijk

ˆp i∈B,j∈LRi ,k∈Tij ,c∈Cp ∪C

−

ˆp i∈B,j∈RRi ,k∈Tij ,c∈Cp ∪C

wc × xpijkc × lijk |)

(10)

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263

LRi denotes the set of rows in bay i on the left side of the ship. RRi denotes the set of rows in bay i on the right side of the ship. Tij =U Tij ∪ ATij . wc denotes the weight of the container c. lijk denotes the distance from row j of bay i to the longitudinal center line of the ship. • Cross moment diﬀerence The diﬀerence in weight distribution of containers across the longitudinal dimension of the ship causes a trim. For safety and fuel eﬃciency considerations, a target trim is desired for sailing. In this paper, we set the target trim to zero. Hence, one of the objectives in stowage planning is to make sure that the moment caused by the weight of containers on the bow part (front) of the ship, including all the locations on deck and under deck, must be as close as possible to the moment caused by the weight of containers on the stern part (back) of the ship. It can be described as, min(|

wc × xpijkc × lijk

ˆp i∈AB,j∈Ri ,k∈Tij ,c∈Cp ∪C

−

wc × xpijkc × lijk |)

(11)

ˆp i∈F B,j∈Ri ,k∈Tij ,c∈Cp ∪C

AB denotes the set of aft bays of the ship. FB denotes the set of forward bays of the ship. lijk denotes the distance from bay i to the transversal center line of the ship.

4

Randomized Algorithm with Tabu Search for Multi-objective Optimization of Large Containership Stowage Plans

This section describes the proposed randomized algorithm with TS for multiobjective optimization of large containership stowage plans. This algorithm is a follow-up work based on the idea of ”Block Stowage” proposed in [12] and ”Block Assignment Strategy” proposed in [8]. To help understanding, we present the block stowage heuristics below brieﬂy: Block Stowage Heuristics Step 1 Divide all the stowage locations on the ship into several ”blocks” based on their relative positions to the hatches. A ”block” is deﬁned as a group of locations which are under or on the same hatch covers. Step 2 Divide the containers in the loading list into diﬀerent groups according to their physical sizes and ports of destination. Step 3 Select a group of containers in a reverse order of the ports of destination, which means the group of containers going to the furthest port is selected ﬁrst, and a block according to a set of heuristic rules. Then assign the containers in that group to the stowage locations in the selected block. Step 4 Terminate the procedure if all the groups of containers have been planned, else continue to Step 3.

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Xiao et al. [12] and Liu et al. [8] described in their previous works that the quay cranes utilization and ship stability can be inﬂuenced through applying diﬀerent heuristic rules in Step 3. However, due to the complex nature of this problem, no single heuristic is found to be adequate for obtaining a good stowage plan for multi-objectives optimization purpose. To overcome the drawback of such deterministic block selection approach, in this paper we propose a randomized two-stage multi-objective optimization algorithm with TS, in which not only one but a large number of stowage plans are generated in a ”randomized” block selection way in the ﬁrst stage, and then a small number of stowage plans with better characteristics such as quay crane utilization, number of idle slots and number of stacks that exceed weight limit are selected for further optimization in the second stage. TS is implemented in the second stage to optimize each stowage plan in terms of the number of stacks that exceed weight limit, horizontal moment diﬀerence and cross moment diﬀerence. The detailed description of the randomized two-stage multi-objective optimization algorithm with TS for the large containership stowage planning problem is given in the following subsections. 4.1

Initial Stowage Patterns Generation Stage

In this stage, the aim is not to generate a stowage plan which is ready for use by the decision maker, but to obtain some good stowage patterns for further optimization. Step 3 is a critical step in ”Block Stowage”. If a deterministic block selection heuristic rule is applied in this step, only one stowage pattern can be obtained through this procedure. Some other potential stowage patterns with better objective values are neglected. Thus, we propose a randomized approach in the block selection step. The idea is that we generate a large number of stowage patterns in a ”randomized” way instead of generating a single stowage pattern in a deterministic way. ”Randomized” here does not mean totally random. Following the basic heuristics proposed by Xiao et al. [12], the blocks selected should still satisfy the requirement of minimizing the number of re-handles. Sometimes there is more than one block that satisﬁes the block selection requirement. In this situation a decision must be made on which block to choose. We deﬁne this situation as a ”decision point”. Randomized block selection only happens at each ”decision point”. Choosing diﬀerent blocks at diﬀerent ”decision points” can result in totally diﬀerent stowage patterns. This is because the consequence of a decision can propagate to a later ”decision point”, and the number of combinations of different decisions can be quite large. For example, if there are 6 ”decision points” with 3 choices for each, this results in 3×6=729 diﬀerent combinations. In order to speed up the process of generating such a large amount of stowage patterns, only physical size constraints, which includes constraints such as 20 foot containers must be stowed in 20 foot locations, 40 foot containers must be stowed in 40 foot locations, and no 20 foot containers are allowed to be stowed on top of 40 foot ones, are considered when assigning each individual container to

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a speciﬁed stowage location regardless of the weight of each container. Because at this stage only stowage pattern, which determines the number of re-handles, completion time of the longest crane and the number of idle slots, is important. All the other objectives relating to the weight of containers will be optimized in the next stage. After a desired number of stowage patterns are generated, Pareto sorting is performed on the stowage patterns based on three objectives: number of rehandles, completion time of the longest crane and number of idle slots. Then a Pareto set of stowage plans are obtained for further optimization in the next stage. 4.2

Solutions Optimization by Tabu Search

A ”move” is one of the important elements in TS, as the idea of TS is to explore the search space of feasible solutions by making a series of moves and ﬁnally reach an optimal solution. In our proposed optimization algorithm, a ”move” in this problem is deﬁned as a swap of two containers with the same size and port of destination. Merely moving one container from its original stowage location to another new empty slot on ship or swapping two containers with diﬀerent sizes or ports of destination is not a legal move in our algorithm. This is because only a swap of two containers with the same size and port of destination can maintain the stowage pattern obtained from the ﬁrst stage of the algorithm, which is already designed to optimize the number of re-handles, the completion time of the longest quay crane and the number of idle slots. All the other illegal moves will change the stowage pattern obtained from the ﬁrst stage. There are two possible outcomes from a sequence of illegal moves: 1. Result in a stowage pattern with more re-handles, longer completion time of the longest quay crane or more idle slots; 2. Result in another stowage pattern which has already been discovered in the ﬁrst stage. Another reason for not allowing the move of a single container to an empty location or swapping two containers with diﬀerent sizes or ports of destination is that it will signiﬁcantly increase the size of the search space of feasible solutions. Since these kinds of ”moves” will not improve the quality of stowage patterns, but only increase the computation complexity, it is necessary to forbid these moves. Tabu move is another critical element in TS. A move is deﬁned as tabu when its reverse move has just been made recently. If a move is regarded as tabu in one iteration, it cannot be used as a legal move to lead to another solution in the neighborhood even if the solution is the best one available in that iteration. This mechanism prevents the exploration process from being stuck in a local optimum, and allows some degradation moves to be made in order to explore new neighborhood which may contain global optimum. In our proposed algorithm, a

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tabu move is not only a reverse of a move which has just been made recently, which is not suﬃcient to allow the exploration process to escape from the local optimum. For example, there are four containers Ca , Cb , Cc , Cd stowed in location La , Lb , Lc , Ld respectively. Then given the following swaps: Ca swaps with Cb , Cc swaps with Cd , Ca swaps with Cd , Cb swaps with Cc , Ca swaps with Cc , and Cb swaps with Cd . After the sequence of moves, the four containers all return to their original locations. However, during the sequence of moves no reverse move occurs. Thus, we have redeﬁned the ”move” and ”reverse move” for this problem. Each ”move” consists of two half moves, each half contains the identity of the container and its newly assigned location. For instance, a ”move” of Ca in La swapping with Cb in Lb is presented in the form of [(Ca , Lb ),(Cb , La )] , the ”reverse move” is a move to let two half moves (Ca , La ) and (Cb , Lb ) occur simultaneously. A tabu list of size Tmax is needed to record the Tmax moves made most recently, which prevents cycles of length less than or equal to Tmax from occurring in the search procedure. In our proposed algorithm a basic unit in the tabu list is a half move of a ”reverse move” associated with the ”move” performed recently. Therefore, a tabu list actually contains 2Tmax half moves. If both the two half moves of one ”move” are found in the tabu list, the move is considered tabu and cannot be performed. This design prevents the above mentioned cycling case from happening. However, sometimes it is desirable to let a ”move” marked as tabu to be carried out. Under this condition, an aspiration mechanism is needed to overrule the tabu status of the ”move”, and allow it to happen. In this research we have deﬁned an aspiration criterion to allow a ”tabu move” to be made. The aspiration criterion is such that if the ”move” can lead to a solution which dominates the best global solution found so far, then it can be made regardless of its tabu status. In this research we applied the TS for multi-objective optimization. Let A and B be two solutions for stowage planning. Objectives vectors F(A)={f1 (A), . . ., fk (A)} and F(B)={f1 (B), . . ., fk (B)} are obtained using solution A and solution B. Thus solution A is considered to dominate solution B (also written as A B) if and only if ∀ i ∈ {1, . . ., k}: fi (A) ≤ fi (B) ∨ ∃j ∈ {1, . . ., k}:fj (A) < fj (B). It means that each objective value of A is no greater than the corresponding value of B and there exists at least one objective value of A which is strictly less than that of B. The objectives being optimized in this stage are the number of stacks that exceed weight limit, horizontal moment diﬀerence and cross moment diﬀerence. The TS algorithm in this research also requires a termination condition. Otherwise this search procedure will run forever. Each time a solution that dominates the best global solution found so far is obtained, the best global solution will be updated with it. If the number of iterations performed since the best global solution last updated is greater than a pre-determined maximum number of iterations, the procedure terminates and the best global solution is regarded as the optimal or sub-optimal solution.

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The proposed TS algorithm is applied to each solution in the Pareto set obtained in the ﬁrst stage. Pareto ranking will be applied to the set of solutions after TS again. The pseudo code of TS is presented in Figure1. In this algorithm, K is a pre-determined value used as the stopping condition.

Fig. 1. Pseudo code of the TS algorithm

5

Case Study

In our study a containership with a capacity of 5,436 TEUs is used in the tests. Six test cases are conducted to evaluate the performance of the proposed randomized algorithm with TS. These test cases are constructed using real containers data provided by a shipping company. The number of ports for the containers in the loading list ranges from 4 to 6. The loading volume in the loading list ranges from 900 to 1,400 TEUs. The containers onboard take up 70% to 85% of the containership capacity. In these test cases, 500 randomized initial stowage plans are obtained in the ﬁrst stage. Then the top n (n=max [20, number of non-dominated solutions] ) are passed to the second stage. In the TS stage, the termination condition K is set to 100, the size of tabu list Tmax is set to 100.

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The algorithm is implemented using the JAVA language in this research work. All these experiments are carried out on an Intel Core2 PC with 2.66GHz CPU and 2 GB RAM. To evaluate the performance of the proposed algorithm, the stowage plans generated by an experienced human planner and by applying the basic block stowage heuristics which we developed in our previous works [12,8] are provided for comparison. The proposed algorithm shows very good performance in terms of the computation eﬃciency and solutions quality for all 6 test cases. Two representative test cases are presented below. 5.1

Real World Case Study

In the ﬁrst experiment, we used a set of real world data from a shipping company. The ship sails from port A to port B, and the stowage planning process is done at port B. Table 1 shows the statistics of containers load list at port B. Ports C to G are downstream ports ordered according to alphabetical order. Table 1. Containers load list at port B for experiment 1 C

D

E

F

20 ft

number of containers average weight (ton) standard deviation of weight (ton)

Port

143 23.17 5.90

18 16.37 5.73

0 -

109 21.22 3.54

G 0 -

40 ft

number of containers average weight (ton) standard deviation of weight (ton)

166 18.47 7.53

73 15.85 7.16

69 4 0

39 26.67 4.81

158 4 0

It takes 361 seconds to obtain 500 initial solutions. Table 2 shows 8 nondominated solutions among the 500 initial solutions. It takes another 1,969 seconds to apply TS to the top 20 solutions. Finally, 10 non-dominated solutions are obtained. Table 2. Objective values of non-dominated solutions obtained after the ﬁrst stage for experiment 1 Solution ID

Completion time of longest crane (min)

Number of stacks exceeding weight limit

Number of idle slots

Horizontal moment diﬀerence (ton × m)

Cross moment diﬀerence (ton × m)

182 369 358 333 446 445 126 196

1,262 1,266 1,274 1,274 1,314 1,371 1,375 1,376

23 34 45 8 17 19 38 27

350 290 315 276 282 236 242 233

513.0 495.2 1,832.8 333.1 1,369.3 2,812.3 1,742.3 1,509.4

76,891.2 34,994.4 76,032.0 11,538.2 52,269.8 75,833.6 47,010.6 7,917.0

Table 3 shows the objective values of the 10 non-dominated stowage plans generated by the stowage planning system. The stowage plans generated by an

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experienced human planner and by applying the basic block stowage heuristics are also shown in the last two rows of the table. The values related to weight are calculated based on a ship conﬁguration with zero ballast. The number of stacks that exceed the weight limit, the number of re-handles and the horizontal moment diﬀerence (ton×m) are not listed in the table as the plans generated by both the stowage planning system and human planner have no stack that exceeds the weight limit. For the number of re-handles, the 10 solutions generated by the stowage planning system has no re-handles, while the plan provided by the human planner has 11 re-handles at the current port. For the horizontal moment diﬀerence, all the 10 plans generated by the stowage planning system are able to obtain an even horizontal moment, while the plan provided by the human planner has a horizontal moment diﬀerence of 5,736.4 ton × m. Solutions with IDs 369, 446, 332, 126, 196 and 262 dominated the solution generated by the human planner for all objectives. Solutions with IDs 14, 369, 332, 126, 196 and 262 dominated the solution generated by applying the basic block stowage heuristics for all objectives. Table 3. Objective values of plans generated by the stowage planning system after the second stage and human planner at port B for experiment 1 Solution ID

14 182 369 358 446 445 332 126 196 262 Human planner Basic block stowage

5.2

Completion time of longest crane (min)

Number of idle slots

Cross moment diﬀerence (ton × m)

1,262 1,262 1,266 1,274 1,314 1,371 1,373 1,375 1,376 1,412 1,567 1,428

332 324 282 278 270 220 226 218 214 214 288 376

20,352.4 64,269.8 18,483.4 60,902.6 33,863.6 61,655.0 0.0 28,460.0 0.0 1,237.2 39,347.6 30,365.2

Syntactic Case Study

Because the average container weight in the ﬁrst experiment which involves real data is not very heavy, it is relative easy to keep all the stacks within the weight limit. Thus, in the second experiment we obtained a set of real stowage planning data from a shipping company and only modiﬁed the weight of these containers, and expect to observe some stacks to exceed the weight limit after applying TS. We modiﬁed the weight of containers by dividing the containers into groups according to their sizes and destination ports. Four diﬀerent weights of 10 tons, 20 tons, 30 tons and 40 tons were assigned to containers in each group with equal proportion. The ship sails from port C to port D, and the stowage planning process is done at port D. Table 4 shows the statistics of containers load list at port D. Ports E to H are downstream ports.

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F

G

H

20 ft

number of containers average weight (ton) standard deviation of weight (ton)

Port

52 25 11.18

99 24.85 11.13

207 24.93 11.16

96 25 11.18

40 ft

number of containers average weight (ton) standard deviation of weight (ton)

245 25.06 11.20

6 25 9.57

37 25.41 11.29

-

It takes 285 seconds to obtain 500 initial solutions. Table 5 shows the 7 nondominated solutions among the 500 initial solutions. It takes 1,830 seconds to apply TS to the top 20 solutions. Finally, 12 non-dominated solutions are obtained. Table 5. Objective values of non-dominated solutions obtained after the ﬁrst stage for experiment 2 Solution ID

Completion time of longest crane (min)

Number of stacks exceeding weight limit

Number of idle slots

Horizontal moment diﬀerence (ton × m)

Cross moment diﬀerence (ton × m)

471 430 245 340 21 383 107

950 951 960 996 1,056 1,074 1,134

77 77 69 55 65 63 61

315 309 297 295 293 283 281

4,447.5 5,147.5 5,887.5 3,617.5 4,427.5 4,637.5 4,327.5

8,175.7 32,965.7 20,154.3 11,834.3 17,635.7 2,795.7 8,105.7

Table 6 shows the objective values of the 12 non-dominated stowage plans generated by the stowage planning system. The number of re-handles is not listed in the table as all the plans have no re-handle at the current port. The stowage plan generated by applying the basic block stowage heuristics is shown in the last row of the table. The human planner did not plan for the modiﬁed test case, so the stowage plan from human planner is not available in this case. It can be observed from the table that solutions with IDs 471, 340, 370 dominated the solution generated by applying the basic block stowage heuristics for all objectives. From these experiments we can see that the proposed algorithm is able to generate multiple (10 in Experiment 1 and 12 in Experiment 2 based on a setting of 500 randomized initial solutions being generated in the ﬁrst stage) non-dominated solutions in a reasonable time period (less than 40 minutes). Compared with the previous research work and the human planners (it usually takes 2 hours to generate one stowage plan by a human planner), the proposed algorithm is much faster. 60 % of the non-dominated solutions dominated the plan provided by the human planner and also the stowage plan generated by applying basic block stowage heuristics in the ﬁrst experiment. 25 % of the non-dominated solutions dominated the plan generated by applying basic block stowage heuristics in the second experiment.

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Table 6. Objective values of plans generated by the stowage planning system after the second stage for experiment 2 Solution ID

471 430 99 245 470 340 370 383 169 172 448 107 Basic block stowage

Completion time of longest crane (min)

Number of stacks exceeding weight limit

Number of idle slots

Horizontal moment diﬀerence (ton × m)

Cross moment diﬀerence (ton × m)

950 951 960 960 962 996 1,012 1,074 1,082 1,097 1,106 1,134 1,076

48 36 0 48 33 43 34 58 31 56 55 34 50

287 270 240 277 264 283 270 278 264 272 276 256 287

-0.1 0.0 -0.1 -0.3 0.0 -0.1 0.0 -0.1 0.0 0.7 -0.1 0.1 0.1

4.3 15,145.7 39,755.7 5.7 575.7 5.7 5.7 4.3 3,945.7 4.3 4.3 5.7 265.7

Also, compared with the one only stowage plan generated by applying basic block stowage heuristics, several (usually 10 based on a setting of 500 randomized initial solutions being generated in the ﬁrst stage) non-dominated solutions are obtained by the Randomized block stowage algorithm with TS. This provides more options that emphasize on diﬀerent objectives for the shipping lines in the real world. The decision maker can choose one stowage plan from the nondominated solutions (around 10 based on a setting of 500 initial solutions) as their ﬁnal stowage guide giving more importance to one or a few objectives that they concern most.

6

Conclusion and Future Work

In this paper, we proposed a randomized algorithm with TS for multi-objective optimization of large containership stowage plans. This algorithm exhibits very good performance in terms of plan quality and computational eﬃciency compared to the plans generated by human planners. By applying the algorithm proposed in this paper, we are able to generate multiple stowage plans while trying to optimize on diﬀerent objectives. This could help the shipping lines make better decisions in the real world. In our future work, we will try to reﬁne this algorithm to enable it to explore more possible solutions in a shorter execution time by employing parallel computing technology.

References 1. Ambrosino, D., Sciomachen, A., Tanfani, E.: Stowing a Containership: the Master Bay Plan Problem. Transportation Research 38, 81–99 (2004) 2. Avriel, M., Penn, M., Shpirer, N., Witteboon, S.: Stowage planning for container ships to reduce the number of shifts. Annals of Operation Research 76, 55–71 (1998)

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3. Avriel, M., Penn, M., Shpirer, N.: Containership stowage problem: complexity and connection capabilities. Discrete Applied Mathematics 103, 271–279 (2000) 4. Ebeling, C.E.: Evolution of a Box. American Heritage of Invention and Technology 23(4), 8–9 (2009) 5. Glover, F.: Heuristics for Integer Programming Using Surrogate Constraints. Decision Science 8, 156–166 (1977) 6. Glover, F., Taillard, E., de Werra, D.: A User Guide to Tabu Search. Annals of Operations Research 41, 3–28 (1993) 7. Kang, J.-G., Kim, Y.-D.: Stowage Planning in Maritime Container Transportation. Journal of the Operational Research Society 53, 415–426 (2002) 8. Liu, F., Low, M.Y.H., Huang, S.Y., Hsu, W.J., Zeng, M., Win, C.A.: Stowage Planning of Large Containership with tradeoﬀ between Crane Workload Balance and Ship Stability. In: Proceedings of the 2010 IAENG International Conference on Industrial Engineering, pp. 1537–1543 (2010) 9. http://www.maerskline.com/link/?page=brochure&path=/ our services/vessels, accessed on (April 25, 2011) 10. Wilson, I.D., Roach, P.A.: Principles of combinatorial optimization applied to container-ship stowage planning. Journal of Heuristics 5, 403–418 (1999) 11. Wilson, I.D., Roach, P.A.: Container stowage planning: a methodology for generating computerized solutions. Journal of Operational Research Society 51, 1248–1255 (2000) 12. Xiao, X., Low, M.Y.H., Liu, F., Huang, S.Y., Hsu, W.J., Li, Z.: An Eﬃcient BlockBased Heuristic Method for Stowage Planning of Large Containerships with Crane Split Consideration. In: Proceedings of the International Conference on Harbour, Maritime & Multimodal Logistics Modelling and Simulation, pp. 93–99 (2009)

A Variable Neighborhood Search Heuristic for Tramp Ship Scheduling Fotini Malliappi, Julia A. Bennell, and Chris N. Potts University of Southampton, Highﬁeld, Southampton, SO17 1BJ, United Kingdom http://www.soton.ac.uk

Abstract. This paper considers a classical ship scheduling problem in which the routing and scheduling of a heterogeneous ﬂeet of ships with time windows for pick-ups and deliveries at multiple ports is required. Assuming ﬁxed ship speeds, the problem of maximising proﬁt is addressed. A variable neighborhood search metaheuristic is proposed for this problem. A computational evaluation compares this variable neighborhood search procedure with multi-start local search and a previous tabu search approach. Computational results show that variable neighborhood search provides both the best-quality solutions and the fastest computation time compared to multi-start and tabu search. keywords: scheduling, maritime, transport, variable neighborhood search.

1

Introduction

Much international trade is carried out by sea [4]. The current costs of sea transport versus other modes of transportation make this type of shipping desirable, especially when large quantities are to be transported. The 2008 report by UNCTAD [16] points out that a tremendous increase in seaborne trade can be observed in those years. Therefore, demand for maritime transport has grown along with increases in the world’s gross domestic product. This growth is mainly attributed to developing countries and transition economies. The report also points out that, despite the global credit crunch, the world economy and trade did not respond to the recession and continued to grow until 2007. The most recent report by UNCTAD [17] states that 2008 marks the end of a “super cycle” as estimates show a signiﬁcant drop in world seaborne trade volumes (4.5%). Therefore, in line with observations made up until 2007, demand for maritime transport followed the global gross domestic product drop, which in turn aﬀected shipping. Ship scheduling is usually part of the operational and tactical planning levels of the global shipping industry. According to literature reviews in maritime transportation [4,5,14,15], ship scheduling problems can be further divided into three modes of operation; industrial, tramp and liner shipping. In industrial shipping, the cargo owner also controls the ships. In tramp shipping, the ships follow the available cargoes and only operate when full or when economically J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 273–285, 2011. c Springer-Verlag Berlin Heidelberg 2011

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viable. This type of operation is analogous to a taxi company where taxis respond to customers and do not operate if no customer has booked. Finally, in liner shipping, operations are based on a published timetable which forces them to operate on a ﬁxed route and to ﬁxed times irrespective of whether there is suﬃcient cargo to enable a proﬁt to be made on the route. This mode of operation resembles the way a bus company operates; the bus will be driven along a route on the published timetable whether or not it is full [5]. Industrial shipping operators are typically aiming to transport all cargoes at a minimum cost; thus, optimisation of both schedules and capacity is desirable. Industrial operators also try to ensure transportation of all of their cargoes primarily by using their own ﬂeet of ships, but also using charters as and when needed. In tramp shipping, the operators’ main objective is to maximise their proﬁt from both “Contracts of Aﬀreightment (COA) [1]” and optional cargoes. A COA is an agreement to transport a speciﬁed cargo quantity between speciﬁed ports within a given time frame. Finally, in liner shipping, the objective is to optimise the published schedules in order to become more attractive to potential customers because the suitability of a schedule to a customer has a strong inﬂuence on the demand [5,14]. Finally, the three modes of operation in seaborne shipping are not mutually exclusive because ships can transfer from one mode to another and operators can operate ships in diﬀerent modes. In this paper, we solve a routing and scheduling problem faced by many tramp shipping companies. A tramp shipping operator has a set of contract cargoes that it is committed to carry, but also has the option of transporting optional spot cargoes. These optional spot cargoes usually oﬀer the opportunity of increasing the revenue, and therefore tramp operators will often accept these cargoes if it is proﬁtable enough to do so. The contract cargoes are long term agreements between the shipping companies and cargo owners. Every cargo that the tramp operator has agreed to carry must be picked up from its loading port and delivered at its unloading port within their pre-speciﬁed time windows. In general, the cargo quantities and ship capacities are such that multiple cargoes are allowed on board. Finally, the goal of the shipping company is to develop proﬁtable routes and schedules for the contract cargoes as well as choosing spot cargoes that will help maximise the total proﬁt. The main contribution of this paper is the development of an eﬃcient variable neighborhood search (VNS) algorithm for a tramp ship scheduling problem. In a computational evaluation, we compare the VNS heuristic with multi-start local search and a tabu search heuristic presented in the literature, both in terms of solution quality and computational time. The organisation of this paper is as follows. Section 2 summarises relevant literature for tramp ship scheduling problems and Section 3 gives a brief description to the problem. A description of the proposed solution method follows in Section 4. Section 5 contains our computational study where the VNS is compared with the multi start local search of [1] and tabu search described by [11]. Finally, Section 6 provides some conclusions.

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2

275

Literature Review on Ship Scheduling

The literature on ship scheduling addresses research considering all three modes of operation. The majority of published papers deal with problems arising from the industrial shipping sector, although there is some recent increase in interest in the tramp sector. Only a small number of papers tackle problems in liner shipping. A variety of problems have arisen over the years in line with economic developments. The diﬀerent problem variations discussed in the literature mainly tackle one or both of the two most important aspects of any ship scheduling problem; robustness and ﬂexibility. A comprehensive but not exhaustive classiﬁcation of ship scheduling problems is given by Ronen [14] in the ﬁrst review of ship routing and scheduling problems. As mentioned above, the literature on tramp shipping problems is quite sparse and only a few papers tackle such problems. The reason for lack of research interest in this shipping sector is attributed to the historic existence of a large number of small tramp shipping companies operating in the market. However, more recently, increased demand and the tendency of larger companies to outsource the shipping of their cargoes has led to the growth of small companies and a corresponding increase in research interest. A typical ship scheduling problem tackling the pickups and deliveries of bulk cargoes in the tramp sector is presented in a recent paper by Brønmo et al. [1]. In this problem, cargoes are required to be transported from their origin to their destination port within a speciﬁed time window. In addition, the simultaneous transportation of multiple cargoes is allowed when constraints for ship and cargoes permit such operation. A classical ship scheduling problem containing both the assignment and routing of cargoes is solved using a multi-start local search heuristic. The eﬀectiveness of the proposed heuristic is tested against a benchmark set partitioning approach to the problem with schedules generated a priori and given as an input to the set partitioning problem. Computational results show that the multi-start local search heuristic generates optimal or nearoptimal solutions within a reasonable computational time. In a follow-up to the study of Brønmo et al [1], Korsvik et al. [11] solve the same classical ship scheduling problem using tabu search. Speciﬁcally, their proposed tabu search algorithm employs a relaxation mechanism in which infeasible solutions with respect to time window and capacity constraints are allowed during the search. This facilitates ﬂexibility in the exploration of diﬀerent areas of the solution space that might not be possible when infeasible solutions are not considered during the search. This relaxation mechanism proves particularly useful for tightly constrained cases because navigation to solutions with diﬀerent characteristics would otherwise be diﬃcult. Moreover, a diversiﬁcation procedure in the form of a penalty that is added to the objective function before the evaluation of a new solution is also introduced as another device for ensuring an adequate navigation of the solution space. Other features include a periodic route re-optimisation as well as a neighborhood reduction technique in order to reduce the computational time. A ﬁnal intensiﬁcation phase is performed using the same ﬁve neighborhoods employed in the multi-start local search approach

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of Brønmo et al. Computational results are presented to compare the proposed tabu search algorithm and the multi-start local search heuristic of Brønmo et al. The tabu search method performs better, especially when considering tightly constrained instances in terms of capacity and time window constraints. The above two studies are the only ones using local search based heuristics to solve the same tramp ship scheduling problem considered in this paper. Results from both are used in our computational evaluation in this paper. Extended versions of the problem, such as allowing ﬂexible cargo sizes, and the use of diﬀerent solution methods are considered by [2,3]. In parallel to research on optimisation techniques for ship scheduling problems, decision support systems have also been developed to help ship schedulers. Examples include a system designed by Kim and Lee [10] for ship scheduling of bulk trade, and a system known as “TurboRouter” proposed by Fagerholt [6].

3

Problem Description

We consider the same classical ship scheduling problem as in the study of Brønmo et al. [1]. The tramp ship scheduling problem consists of a set of n cargoes that have to be picked up and delivered. A cargo j (j=1,. . . ,n) has an associated quantity qj . Further, cargo j requires pick-up operation jp at a given port where it is loaded onto the ship, and a delivery operation jd at another port where it is unloaded. There are two types of cargoes; contract cargoes and optional spot cargoes. The contract cargoes are those that the tramp shipping company is committed to carrying and the optional spot cargoes are only serviced if the company ﬁnds it proﬁtable to do so. The number of available ships is denoted by m. Certain ships cannot visit some ports nor carry some of the cargoes. A ship can be at a port or at a point at sea when the planning period begins and at the end of the planning period the ship is at the last planned unloading port [1]. The capacity of a ship is denoted by Q, and ships can carry multiple cargoes simultaneously irrespective of other cargoes that are already on board the ship provided that the sum of all cargo quantities qj on board at any time does not exceed Q. Transporting cargo j returns a proﬁt Pj . Each ship starts at an origin 0 and then travel between a sequence of ports. The travel times for each leg of a ship’s route are computed from a distance matrix dh,k , where h and k are indices of the origin and the ports, and a ﬁxed speed of 20 knots. In addition, each operation jp and jd is associated with a time window [ejp , ljp ] and [ejd , ljd ], respectively, for the start of service (loading or unloading). The tramp ship scheduling problem is a proﬁt maximisation problem such that: i. operations jp and jd belong to the same ship route and the port for jp is visited before the port for jd ; ii. all contract cargoes are transported exactly once; iii. all optional spot cargoes are transported at most once; iv. the load of a ship at any time does not exceed the ship’s capacity Q;

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v. the service for operations jp and jd begins in the intervals [ejp , ljp ] and [ejd , ljd ], respectively; vi. the proﬁt P is maximised.

4

Variable Neighborhood Search

The complexity of our problem is such that metaheuristic methods provide the most suitable solution methodology. In this paper, we propose a VNS algorithm, and evaluate its performance computationally against the multi-start local search heuristic of Brønmo et al. [1] as well as the tabu search algorithm implemented by Korsvik et al. [11]. For multi-start local search and tabu search, we refer to the original papers for a detailed description of the algorithms. Below, we describe our VNS algorithm. VNS is a relatively recent metaheuristic which was ﬁrst introduced by Mladenovic and Hansen in 1997 [13]. It is based upon the idea of systematically changing the neighborhood within the search, and this simple idea makes it widely applicable to a huge number of problems. A major advantage of the method is that it needs very few parameters to be set. In addition, VNS diﬀers from other metaheuristics in the sense that it does not follow a trajectory. Instead, it explores increasingly distant neighborhoods of the current best solution and allows the move to another solution if and only if an improvement has been identiﬁed [9]. Therefore, many good properties of the current best solution are preserved when moving from one solution to another [8,9]. Following a move from one best solution to another, a local search routine is applied in order to obtain a local optimum. The procedure is then repeated, thereby producing increasingly good solutions. To our knowledge, research on local search based heuristics for the tramp ship scheduling problem is limited. Only two heuristics, multi-start local search and tabu search are previously applied in solving the tramp ship scheduling problem, and these are used as benchmarks in our study. Section 4.1 explains how our initial solution is generated. Some useful neighborhoods are introduced in Sect. 4.2. The initial solution is then put through the shaking phase of the algorithm, as described in Section 4.3. Then the resulting (usually worse) solution enters an intensiﬁcation phase in which a variable neighborhood descent is used to search for an improvement. An overview of the complete VNS heuristic is given in Section 4.4. 4.1

Initial Solution

The construction of a high-quality initial solution is very important for the VNS heuristic because it tends to result in superior ﬁnal solutions than when a lowerquality initial solution is used. Our algorithm uses an insertion heuristic to create an initial solution, details of which are provided below. In the insertion heuristic, we ﬁrst create a list of cargoes that are sequenced on non-increasing order of size. The rationale behind this is that the most desirable cargoes are scheduled ﬁrst in order to ensure a high starting proﬁt. The cargoes

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are then selected one-by-one from the top of the sorted list to be assigned to available ships. In the case that a cargo is infeasible in respect to all ship routes, this is entered into an infeasible cargo list. Insertion Heuristic 1. Make a list of unassigned cargoes. 2. Sort the cargoes in non-increasing order of the quantities to be carried. 3. Until the list is empty repeat; (a) Select the ﬁrst cargo j from the sorted list and a ship u. (b) Assign cargo j to ship u by ﬁnding the most proﬁtable position for jp and jd in the current schedule. (c) If cargo j is feasible when inserted into the route of ship u, remove it from the list. (d) If cargo j is infeasible when inserted onto the route of ship u and u is not the highest indexed ship, increase u by 1 and return to Step 3b. Otherwise, when cargo j is not feasible after attempting to insert it onto each of the ships, remove j from the list of unassigned cargoes and insert j onto a dummy ship. 4.2

Neighborhoods

We now introduce the following ﬁve neighborhoods that form a key component of our variable neighborhood local search. These neighborhoods are used in the quick local search and the extended local search of the multi-start local search method proposed by Brønmo et al. [1]. 1-resequence: A cargo j is removed from a ship u and then jp and jd are reinserted into the route of ship u in the best possible positions. 2-resequence: A cargo i and a cargo j are removed from a ship u and then ip , id and jp , jd are reinserted into the route of the same ship u in the best possible positions. Therefore, this is the same neighborhood as the 1-resequence described above but involving two cargoes. re-assign: A cargo j is removed from a ship u and then jp and jd are inserted into the route of another ship v in the best possible positions. 2-interchange: A cargo i is removed from a ship u and a cargo j is removed from ship v. Cargoes i and j are then inserted into the routes ship v and ship u, respectively (using the best possible positions for ip , id , jp and jd ). 3-interchange: A cargo i is removed from a ship u, a cargo j is removed from ship v and a cargo k is removed from a ship w. Cargoes i, j and k are then inserted into the routes of ship v, w and u, respectively (using the best possible positions for ip , id , jp , jd , kp and kd ). It is important to note that insertions are performed sequentially. Speciﬁcally, the best possible position is ﬁrst determined for the pickup cargo and considering

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that this is now ﬁxed, the best possible position is found for the delivery cargo. This has a signiﬁcant impact on computing times as it reduces the maximum number of possible exchanges from O(n2 ) to O(n), where n is the number of cargoes in a ship’s route. 4.3

Shaking

The shaking phase of a VNS algorithm helps in escaping the valley of local optimum and involves the use of six neighborhoods. A random solution is selected from each shaking neighborhood when the search does not produce any further improvement. A move is made to the ﬁrst solution found in the neighborhood irrespective of whether it is improving or deteriorating. Three of the six neighborhoods used for shaking are 1-resequence, re-assign, and 2-interchange as deﬁned in Section 4.2. The other three are each s-ship delete neighborhoods for s = 1, s = 2 and s = m that make use of a delete and recreate principle. Associated with the s-ship delete neighborhood is a parameter δ that speciﬁes the percentage of cargoes that are deleted from a selected ship. s-Ship Delete Neighborhood Repeat the following steps s times. Randomly select a ship u. Delete δ% of the cargoes from ship u. Temporarily remove the deleted cargoes from the problem. Calculate current proﬁt P . (a) Randomly select a cargo j that has not been previously selected from the dummy ship. (b) Select a ship v that has not already been selected (c) Assign cargo j to ship v by ﬁnding the most proﬁtable positions for jp and jd in the current schedule. (d) If the cargo j is feasible, remove it from the dummy ship. (e) If cargo j is not feasibly inserted on ship v, then select the next ship as the new ship v and restart from Step 4b. In the case that cargo j is not feasible after attempting an insertion on all ships, restart from Step 4a. 5. Calculate the new proﬁt P . 6. If the new proﬁt P > P , then accept the assignment, insert any temporarily deleted cargoes onto the dummy ship, and exit the neighborhood. 7. If P ≤ P restart from Step 4a

1. 2. 3. 4.

4.4

Variable Neighborhood Search Overview

The outline of the VNS heuristic is as follows. Variable Neighborhood Search Step I Set the shaking neighborhoods to be: N1 is 1-ship delete; N2 is 2-ship delete; N3 is m-ship-delete; N4 is 1-resequence; N5 is re-assign; and N6 is 2-interchange. Set smax = 6.

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¯1 is 1Step II Set the variable neighborhood descent neighborhoods to be: N ¯ ¯ ¯ resequence; N2 is re-assign; N3 is 2-interchange; N4 is 2-resequence; ¯5 is 3-interchange. Set lmax = 6. and N Step III Find an initial solution x using the Insertion Heuristic. Step IV Repeat until no further improvements can be made to the proﬁt (1) Set s = 1 (2) Repeat until s = smax (a) Shaking - generate solution x at random from the s’th neighborhood Ns (x) of x (b) Perform local search by variable neighborhood descent: (b1) Set l = 1 (b2) Repeat until l = lmax ¯l (x ). Find the best neighbor x of x in N Move or not. If f (x ) < f (x ) set x = x and l =1; otherwise, set l = l + 1 (c) Move or not. If this local optimum is better than the current best set x = x and continue the search with the shaking neighborhood s = 1; otherwise, set s = s + 1

4.5

Multi-start Local Search

Multi-start local search is designed to search for global optimal solutions by restarting the local search in order to overcome local optima and diversify their search [7]. The search is normally restarted from a diﬀerent solution once a region has been explored. This is especially useful in constrained scheduling problems where neighborhoods that maintain feasibility are scarce but the construction of diﬀerent solutions can be done relatively easily. Multi-start local search was the ﬁrst metaheuristic used to solve the tramp ship scheduling problem returning optimal or near-optimal solutions to real-life instances of the problem within a reasonable computational time [2]. Bronmo et al [2] apply a two-phase procedure in which a number of diﬀerent initial solutions are generated and then improved by local search. Speciﬁcally, the initial solutions are constructed by an insertion procedure that uses both random and deterministic elements. Then, in contrast to the general framework in which every initial solution is improved, a selection of the best initial solutions are improved by a local search heuristic which in this case is split into two phases; a quick and an extended local search. The quick local search improves a selection of best initial solutions, while the extended local search improves a selection of those best solutions found by applying quick local search. 4.6

Tabu Search

Tabu search is a metaheuristic that aims to proceed from a local optimum by allowing non-improving moves. A memory structure in the form of a tabu list prevents the same sequence of solutions being revisited. An eﬀective tabu search

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procedure usually needs to apply intensiﬁcation and diversiﬁcation strategies in addition to the basic form of tabu search. This enables the further exploration of the search space as well as focusing on the more desirable areas through the use intensiﬁcation that increases the possibilities of obtaining a better solution. In the implementation of tabu search by Korsvik et al. [11] initial solutions are constructed by ﬁrst sorting cargoes in non-decreasing order of the start of the time window for picking up cargo. Then each cargo (pickup and delivery) is assigned to the route of a randomly selected ship ensuring that all constraints are satisﬁed apart from those involving capacity and time windows can be violated. It is important to note that allowing infeasible solutions is an important feature of the algorithm. Allowing infeasibility permits greater exploration of the solution space and proves to be useful with tightly constraint instances. Tabu search iterations are then initiated and these include the tabu search relocate neighborhood, a relaxation mechanism that allows infeasible solutions in the search, a continuous diversiﬁcation strategy which penalises nonimproving moves, and whenever a new best solution is found intra-route exchanges.

5

Computational Study

The tramp ship scheduling problem is similar to the multi-vehicle pickup and delivery with time windows (m-PDPTW) problem. Therefore, due to the lack of real and benchmark cases for the tramp ship scheduling problem, our computational study uses modiﬁed benchmark data originating from the PDPTW which were taken from the 100-customer test instances of Li and Lim [12]. This dataset was used for all three methods implemented; VNS, multi-start local search and tabu search. 5.1

Instance Descriptions

The data set includes 57 instances which are then divided into three groups; Cclustered customers, R- uniformly distributed customers and RC - combination of randomly placed and clustered customers. Also, the three groups mentioned are further divided into two groups according to the time windows used; 1- for narrow time windows and 2- for wide time windows. It is important to note that the benchmark data provided by [12] were modiﬁed to ﬁt the speciﬁcations of the tramp ship scheduling problem. More speciﬁcally, two more columns were added in order to specify the initial proﬁt gained from carrying each cargo as well as to specify whether a cargo is a COA or a spot one. The input proﬁt column was made equal to the quantity needed to be carried as it is expected to be a reasonable assumption. It is important to note that the proﬁt calculation in the objective function of the problem also depends on fuel consumption which in turn takes into account the distance travelled. The cargo type column contains binary numbers with 1 being a COA and 0 being a spot cargo.

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Heuristic Parameter Settings

Table 1 shows how the VNS heuristic results vary when diﬀerent percentages of deleted cargoes are used in the shaking phase as well as diﬀerent combinations of shaking neighborhoods. The gap between initial and ﬁnal solution returns the highest increase for the most proﬁtable combination and speciﬁcally combination 7 in Table 1 at 40%. It is important to point out that initial solutions are the same across all combinations and percentages used. The neighborhood without a forbidden (tabu) list returns one of the worst results. This is due to not preventing cycling. Speciﬁcally when cargoes were removed from the list and not put into a memory list that would enable the algorithm to remember them, they would then be re-inserted into the same positions thus resulting in reverting back to the original solution without allowing new moves to take place. The infeasible list used in the neighborhood would put them back in which would also result in not having the perturbation initially aimed for by implementing the VNS procedure. Hence, we have chosen to use the combination of one, two and all ships with tabu list in the delete and re-create group of shaking neighborhoods with the delete percentage set to 40%. Table 1. VNS shaking neighborhoods comparison No. Description 1. 2. 3. 4. 5. 6. 7.

1 ship without forbidden list 1 ship with forbidden list 2 ships with forbidden list 1 and 2 ships with forbidden list All ships with forbidden list 1 and all ships with forbidden list 1, 2 and all ships with forbidden list

10%

20%

30%

40%

50%

60%

70%

80%

90%

9.23

9.33

9.57

9.64

9.75

9.77

9.83

9.74

9.78

9.23

9.43

9.72

9.84

9.82

9.72

9.80

9.66

9.84

9.25

9.37

9.62

9.89

9.72

9.67

9.65

9.62

9.76

9.22

9.47

9.81 10.01

9.91

9.89 10.02

9.88

9.94

9.24

9.44

9.67

9.54

9.51

9.66

9.65

9.49

9.71 10.16 10.35 10.35 10.24 10.22 10.14 10.13

9.54

9.54

9.77 10.01 10.43 10.87 10.67 10.67 10.56 10.39 10.42

Table 2 presents the diﬀerent parameters used in the two settings implemented for VNS. The percentage of deleted cargoes in the shaking neighborhoods was set to 40 for both settings as it was found to be the most beneﬁcial. Moreover, a diﬀerent number of neighborhoods was used in the implementation of VNS settings 1 and 2. The values 1 or 0 indicate which neighborhood was used in each VNS setting with 1 being true and 0 being false. Table 3 presents the multi-start local search parameters. Settings 2 and 3 are identical of the ones used by Brønmo et al. [1]. Finally, Table 4 shows the only

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Table 2. Variable neighborhood search - parameters Setting 1 Setting 2 Shaking neighborhoods Percentage of deleted cargoes

40

40

VND neighborhood 1-resequence Reassign 2-interchange 2-resequence 3-interchange

1 1 1 0 0

1 1 1 1 1

Table 3. Multi-start local search - parameters Setting 1 Setting 2 Setting 3 No. of initial solutions generated 100 Percentage of initial solutions generated randomly random No. of solutions improved by: Quick local search 70 No. of solutions improved by: Extended local search 30

100 15 6 1

1,000 15 14 4

Table 4. Tabu search - parameters Setting 1 Setting 2 No. of iterations set as stopping criterion

1,000

10,000

diﬀerence in the tabu search’s parameters which only involves the amount of iterations used as the method’s stopping criterion. 5.3

Computational Results

All three methods were implemented with diﬀerent parameters in order to obtain the best possible results. Since all methods make use of random numbers, average results were obtained after making 10 runs for each instance. All methods were coded in C and run on a PC with an Intel Core 2 Duo, 3GHz processor and 3 GB of RAM under windows XP. Table 5 shows the average percentage increase in proﬁt when VNS is compared against the two benchmark algorithms. Results shown for all the diﬀerent heuristic parameters used are always of positive sign, i.e., both variable neighborhood settings used are returning higher quality solutions against both multi-start and tabu search and their associated variations. In addition, the trend is consistent in both VNS settings with the worst performer against both being multi-start setting 2 and the best performer being multi-start setting 1.

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Table 6 presents the average run times for all heuristics and settings. The shortest average run times are returned by multi start setting 1 but at the same time this algorithm and setting scores the lowest against both settings of VNS. Both VNS settings return higher quality solutions in fast computational times. It is therefore clear that both VNS settings oﬀer the most potential to the tramp ship scheduling problem as they both return the highest average proﬁt with very fast average run times. Table 5. All methods comparisons - % proﬁt Method

% diﬀerence % diﬀerence VNS setting 1 VNS setting 2

Multi-start setting 1 Multi-start setting 2 Multi-start setting 3 Tabu search setting 1 Tabu search setting 2

3.30 11.14 5.47 4.77 4.61

4.77 12.78 7.02 6.20 6.03

Table 6. All methods - average run times in seconds Method Multi-Start setting 1 Multi-Start setting 2 Multi-Start setting 3 VNS Setting 1 VNS Setting 2 Tabu Setting 1 Tabu Setting 2

6

Average Time (s) 16 0.5 5 4 36 23 69

Concluding Remarks

A variable neighborhood search algorithm is presented for solving tramp ship scheduling problems. A multi-start local search algorithm of Brønmo et al. [1] and a tabu search of Korsvik et al. [11] were used as benchmarks. In the VNS heuristic an initial solution is generated by using a quantity sort procedure. A shaking phase is then used to escape the initial local optimum through the use of diﬀerent neighborhoods involving inter and intra route operators as well as a destruction procedure. The new often worse solution is then improved with a collection of neighborhoods that exist in variable neighborhood descend. The computational study shows that on average VNS oﬀers the most potential as is both fast and returns the highest average proﬁt amongst all three heuristics and settings investigated. The balance of quick response time and quality of solutions that the VNS algorithm returns is very important in quickly deciding whether a spot cargo is accepted or rejected which is a crucial element in the proﬁtability of a tramp shipping operator’s business.

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References 1. Brønmo, G., Christiansen, M., Fagerholt, K., Nygreen, B.: A multi-start local search heuristic for ship scheduling - a computational study. Computers & Operations Research 34(3), 900–917 (2007) 2. Brønmo, G., Christiansen, M., Nygreen, B.: Ship routing and scheduling with ﬂexible cargo sizes. Journal of the Operational Research Society 58(9), 1167–1177 (2007) 3. Brønmo, G., Nygreen, B., Lysgaard, J.: Column generation approaches to ship scheduling with ﬂexible cargo sizes. European Journal of Operational Research 200(1), 139–150 (2010) 4. Christiansen, M., Fagerholt, K., Nygreen, B., Ronen, D.: Maritime transportation. In: Barnhart, C., Laporte, G. (eds.) Transportation, Handbooks in Operations Research and Management Science, vol. 14, pp. 189–284. Elsevier, Amsterdam (2007) 5. Christiansen, M., Fagerholt, K., Ronen, D.: Ship routing and scheduling: Status and perspectives. Transportation Science 38(1), 1–18 (2004) 6. Fagerholt, K.: A computer-based decision support system for vessel ﬂeet scheduling – experience and future research. Decision Support Systems 37, 35–47 (2004) 7. Glover, F., Kochenberger, G.: Handbook of Metaheuristics. Kluwer, Dordrecht (2003) 8. Hansen, P., Mladenovic, N.: Variable neighborhood search: Principles and applications. European Journal of Operational Research 130(3), 449–467 (2001) 9. Hansen, P., Mladenovic, N.: Variable neighborhood search. In: Glover, F., Kochenberger, G. (eds.) Handbook of Metaheuristics, pp. 145–184. Kluwer, Boston (2003) 10. Kim, S., Lee, K.: An optimization-based decision support system for ship scheduling. Computers & Industrial Engineering 33(3-4), 689–692 (1997) 11. Korsvik, J.E., Fagerholt, K., Laporte, G.: A tabu search heuristic for ship routing and scheduling. Journal of the Operational Research Society 61(4), 594–603 (2010) 12. Li, H., Lim, A.: A metaheuristic for the pickup and delivery problem with time windows. In: IEEE International Conference on Tools with Artiﬁcial Intelligence, pp. 160–167 (2001) 13. Mladenovic, N., Hansen, P.: Variable neighborhood search. Computers & Operations Research 24(11), 1097–1100 (1997) 14. Ronen, D.: Cargo ships routing and scheduling: Survey of models and problems. European Journal of Operational Research 12(2), 119–126 (1983) 15. Ronen, D.: Ship scheduling: The last decade. European Journal of Operational Research 71(3), 325–333 (1993) 16. UNCTAD secretariat: Review of maritime transport. United Nations Conference on trade and development, United Nations, New York, Geneva (2008), http://www.unctad.org/en/docs/rmt2008_en.pdf 17. UNCTAD secretariat: Review of maritime transport. United Nations conference on trade and development, United Nations, New York, Geneva (2010), http://www.unctad.org/en/docs/rmt2010_en.pdf

Fast Generation of Near-Optimal Plans for Eco-Eﬃcient Stowage of Large Container Vessels Dario Pacino1 , Alberto Delgado1 , Rune Møller Jensen1 , and Tom Bebbington2 1

2

IT-University of Copenhagen, Denmark {dpacino,alde, rmj}@itu.dk Maersk Line Operations, Global Stowage Production, Singapore [email protected]

Abstract. Eco-eﬃcient stowage plans that are both competitive and sustainable have become a priority for the shipping industry. Stowage planning is NP-hard and is a challenging optimization problem in practice. We propose a new 2-phase approach that generates near-optimal stowage plans and fulﬁlls industrial time and quality requirements. Our approach combines an integer programming model for assigning groups of containers to storage areas of the vessel over multiple ports, and a constraint programming and local search procedure for stowing individual containers.

1

Introduction

Cost-eﬃciency and sustainability are not opposed objectives for the stowage plans generated daily by liner shipping companies. A stowage plan assigns containers to load in a port to vessel slots. An eco-eﬃcient stowage plan aims at achieving a minimum port stay. In this way, port fees are minimized and the vessel maximizes its time at sea, which can be used to reduce its speed and thus bunker costs and CO2 emissions. Eco-eﬃcient stowage plans, however, are hard to produce in practice. First, they are made under time pressure by human stowage coordinators just hours before the vessel calls the port. Second, deep-sea vessels are large and often require thousands of container moves in a port. Third, complex interactions between lowlevel stacking rules and high-level stress limits and stability requirements make it diﬃcult to minimize the makespan of cranes and, at the same time, avoid that containers block each other (overstowage). Finally, according to our industrial partner, stowage planning optimization algorithms must be fast. Runtimes of more than ten minutes are impractical, since stowage coordinators possibly need to run several forecast scenarios. This paper introduces a new stowage planning optimization approach that, similar to the currently most successful approaches (e.g, [26,18,1]), decomposes the problem hierarchically as depicted in Figure 1. First the multi-port master planning phase decides how many containers of each type to stow in a set of storage areas of the vessel using an integer programming (IP) model. Based on this distribution, a complete stowage plan is generated in the Slot Planning J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 286–301, 2011. c Springer-Verlag Berlin Heidelberg 2011

Fast Generation of Near-Optimal Eco-Eﬃcient Stowage Plans

Loadlists Vessel Data Port Data

Muti−Port Master Planning

Muti−Port Master Plans

Current Port

Slot

Master Plan

Planning

287

Stowage Plan

Fig. 1. Hierarchical decomposition of stowage planning into master and slot planning

phase where individual containers are stowed by a combination of constraint programming (CP) and local search (LS) models implemented in [8,21]. To avoid negative impact of the ﬁnal stowage plan in future ports, the multi-port master planning phase optimizes simultaneously master plans for the current port and a number of downstream ports. We evaluated our approach experimentally on 20 real instances provided by our industrial partner. Despite the NP-hardness of multi-port master planning and the previous lack of success solving the problem optimally for large vessels over multiple ports (e.g., [26,18,1]), 11 of the 20 instances can be solved optimally in less than 10 minutes using a standard IP solver. More interestingly, the computation time can often be reduced several orders of magnitude if we relax the problem by dropping the integrality constraint on the decision variables in a mixed integer programming (MIP) model, without aﬀecting solution quality. A 2-phase approach for stowage planning, combining our MIP approach for multi-port master planning with our slot planning algorithms, can generate complete stowage plans in less than 330 seconds for 16 of the 20 instances. This runtime is well within the time bound of 10 minutes assessed by our industrial partner to be necessary for supporting stowage coordination in practice. The remainder of the paper is organized as follows. Section 2 describes the problem. Section 3 introduces related work. Section 4 presents our 2-phase approach. Section 5 presents the experiments and Section 6 provides a summary.

2

Background and Problem Statement

ISO containers transported on container ships are normally 8’ wide, 8’6” high, and either 20’, 40’, or 45’ long. Due to the lack of support points, it is not possible to stack a 20’ container on a 40’ or 45’ container. High cube containers are 9’6” high and pallet wide containers are slightly wider and can only be placed side-by-side in certain patterns. Refrigerated containers (reefers) must be placed near power plugs. Containers with dangerous goods (IMO containers) must be placed according to a complex set of separation rules. The capacity of a container ship is given in Twenty-foot Equivalent Units (TEU). As shown in Figure 2, the cargo space of a vessel is divided into sections called bays and each bay is divided into an on deck and a below deck part by a number of hatch covers, which are ﬂat, leak-proof structures. A container is overstowing and causing extra crane moves if it is discharged later than a container stowed below it, either in the same stack or under a hatch cover (hatchoverstowage). Each sub-section of a bay consists of a row of container stacks divided into slots that can hold a 20’ ISO container. Figure 3 (a) and (b) show the container slots of a bay and stack, respectively. Stacks have max height and

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3 7

5

7 4

9

4

8 10

7

4

Line of sight

Lashing bridge

Bay

Hatch cover

Station

Waterline

Fig. 2. The arrangement of bays in a small container vessel. The vertical arrows show an example of the resulting forces acting on the ship sections between calculation points (stations). Single crane work hours for adjacent bays are shown at the top.

Bow

Stacks On Deck 1

2

3

4

5

6

8

7

Stack Fore

Aft

Metacentric height

3

Tiers

2

On Deck 1

4

4

θ

40’ High−Cube

4

Tiers

3

M

1

2

3

G

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Below Deck 2

Gravity force

40’ Reefer

Z Buoyancy force

1 1

Stern

2

3

4

5

6

7

20’ Reefer

20’

Stacks Below Deck

(a)

(b)

(c)

Fig. 3. (a) A vessel bay seen from behind (b) A side view of a stack of containers; as depicted, power plugs are usually situated at bottom slots (c) Transverse stability

weight limits. Below deck, cell guides secure containers transversely. Containers on deck are secured by lashing rods and twist locks with limited strength. Thus, container weights must normally decrease upwards in stacks on deck. Moreover, lashing rods of 20’ stacks must be accessible and stack heights must be under the vessel’s minimum line of sight. 45’ containers can normally only be stowed over the lashing bridge on deck. A container ship must sail at even keel and have suﬃcient transverse stability. Figure 3(c) shows a cross section of a ship. For small inclination angles, the volume of the emerged and immersed water wedges (shaded areas) and thus the distance GZ are approximately proportional with the angle such that the buoyancy force intersects the center line in a ﬁxed position called the metacenter, M [24]. For an inclination angle θ, the ship’s uprighting force is proportional to GZ = GM sin θ. GM is called the metacentric height and the center of gravity G must be on the center line and result in suﬃcient GM for the ship to be stable. Maximum and minimum draft restrictions apply due to port depths, working height of cranes, and the propeller. The trim is the diﬀerence between the aft and fore draft and must be kept within a given span. For a station position p, the shear force is the sum of the resulting vertical forces on vessel sections (see Figure 2) acting aft of p, and the bending moment is the sum of these forces

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times the horizontal distance to them from p. Both of these stresses must be within limits. The vessel also has transverse bending moment (torsion) limits. The meta center, draft, trim, and the buoyancy of each section of the vessel can be derived from its hydrostatic tables given its displacement and longitudinal center of gravity. A container ship transports containers between ports on a ﬁxed cyclic route. It is the liner shippers and not the port terminals that are in charge of producing stowage plans. A stowage plan assigns the containers to load in a terminal to slots on the vessel and it is often sent to the terminal shortly before calling it. The terminal guarantees a certain productivity in terms of the number and eﬃciency of assigned quay cranes, but otherwise the liner shipper has no control over its operations. Typically the terminal substitutes detailed container information for container types in order to optimize the load sequence [16]. In this work we assume the role of the liner shipper and focus on the generation of typed stowage plans for the current port. The plan is then completed by the terminal during the load sequencing, where the types are substituted with concrete containers. The main objective of stowage planning is to minimize the port stay by reducing the total number of moves through overstowage minimization and distributing these moves evenly over the available quay cranes. Since quay cranes are too wide to work on two adjacent bays, a good lower bound for the makespan of quay cranes is the maximum work time of a single crane over pairs of adjacent bays (10 hours in the example shown in Figure 2). Stowage planning should take cargo in future ports into account (e.g., to ensure enough reefer capacity). But since cargo forecasts are inaccurate, the plan must be robust. Achieving this includes arranging containers with same port of discharge (POD) in vertical stacks rather than horizontal layers, freeing up as much bottom space as possible for unexpected reefer and long-haul containers, and to avoid mixing containers with diﬀerent POD under deck to minimize the risk of hatch-overstows. It is impractical to study large optimization models that include all details of stowage planning. On the other hand, all major aspects of the problem must be modeled for the results to be valuable. For container types this includes 20’, 40’, and reefer containers because short containers cannot be stowed over long ones which can force overstowage, and reefer slots are scarce and placed at the bottom which may force understowage. In addition, since stability, trim, draft and stress moment limits should not fully be ignored, some weight classes of containers must be introduced. It is also important to take containers onboard the vessel when arriving the current port and containers to load in future ports into account. Finally since port stay minimization is essential, a realistic estimate of quay crane makespan and overstowage must be included.

3

Literature Survey

The number of publications on stowage planning has grown substantially within the last few years. Contributions can be divided into two main categories: singlephase and multi-phase approaches. Multi-phase approaches decompose the problem hierarchically. They are currently the most successful in terms of model

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accuracy and scalability. The earliest work can be traced back to a 3-phase heuristic [25], but the ﬁrst contribution that models several major aspects of the problem is a 2-phase approach that solves a master planning phase for multiple ports with a branch-and-bound algorithm and uses tabu search for bay planning [26]. Other approaches that use a similar decomposition include solving multi-port master planning with an iterative improvement approach based on the transportation simplex method [18] and a bin-packing heuristic [28]. 3-phase approaches include combinations of constructive heuristics, 0/1 IP, and metaheuristics (e.g., [1]) and heuristics combined with LS (e.g., [27]). Multi-phase approaches developed by the industry include a multi-stage placement heuristic using a number of lower-bounds [15] and a combination of sequential LP for master planning and a hierarchy of IPs for slot planning [14]. Except this last deployed industrial system which provides input data to our experiments, none of the previous approaches include all the major aspects of stowage planning mentioned at the end of Section 2. Single-phase approaches represent the stowage planning problem (or parts of it) in a monolithic optimization model. The earliest work can be traced back to Aslidis [4],1 who makes a thorough study of overstow minimization algorithms for single bays. The remaining work is categorized according to solution approach. IP approaches include a very accurate but intractable model of the complete stowage problem [6] and simple models [3,13,19] with limited scalability. CP approaches include an early simple model of the complete problem [2] and the combination of CP [8] and LS [21] used in this paper to generate near-optimal slot plans fast and robustly. A number of metaheuristics have been suggested: genetic algorithms (e.g.,[7,10]), simulated annealing [11], and placement heuristics (e.g., [5]). These approaches use simple models and often only focus on one aspect such as overstowage minimization. Additional single-phase approaches include simulation (e.g., [23]), expert systems (e.g., [9]), 3D-packing (e.g, [22]), and casebased methods (e.g., [20]). Also these approaches lack representative problem models.

4

Solution Approach

We propose the 2-phase approach shown in Figure 1. The ﬁrst phase, multi-port master planning, distributes container types to sub-sections of bays for the entire route of the vessel. This phase deals with high-level constraints, modeling weight classes of standard 20’ and 40’ containers, and reefer containers. More precisely, we model GM stability, trim and draft limits, weight distribution and shear forces, and optimize hatch-overstowage and crane makespan. IMO, palletwide, and high-cube containers are not modeled. The two former are often stowed in speciﬁc areas, while the latter can be modeled with specialized capacity constraints. We also do not model bending moments and ballast water which are the focus of our future work. The input to multi-port master planning is industrial data from a currently deployed stowage planning optimization tool [14], and it 1

However, some unpublished work from the 1970s using simulation has been cited [6].

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includes: 1) vessel data with ship layout and stress limits, 2) current port loadlist, and future port loadlists based on historical data, 3) port data with water depths, crane heights, and crane productivity. The input is adjusted such that for example weight and height constants fulﬁll quay crane height requirements and line of sight constraints. For that reason, the constants used in the models often change from port to port. The master plan for the ﬁrst port (the one we are making the stowage plan for) is used as input for the second phase, slot planning, to assign the containers of the types deﬁned in the master plan to concrete slots. In slot planning, all major stacking rules apply: containers must form stacks, 20’ containers cannot be stowed on top of 40’ containers, reefer containers can only be stowed in reefer slots, stack maximum height and weight limits must be fulﬁlled and cell capacity must be observed. Containers are assigned with the aim of minimizing overstowage, clustering containers with the same POD and freeing stack and reefer slots for robustness. 4.1

Multi-port Master Planning

The multi-port master planning phase assigns types of containers to sub-sections of bays (locations). Locations are either above or under a hatch cover and are used as a tool to model hatch-overstowage. Figure 3(a) shows four locations within a bay. Outer locations are symmetrically split (such as location 2 and 4 in Figure 3(a)) to ease transverse stability calculations. For 20’ and 40’ containers, we consider a set of four mutually exclusive container types T = {L, H, RL, RH}, respectively light and heavy containers and light and heavy reefer containers. To produce a robust plan, our model takes into account the current and a set of downstream ports P . We deﬁne transports TR as the set of pairs p1 , p2 where p1 , p2 ∈ P are the loading and discharging port of a container type. We deﬁne 40τ two sets of decision variables x20τ tl and xtl representing respectively the amount of 20’ and 40’ containers of type τ ∈ T to be stowed in location l ∈ L, where L is the set of all locations, during transport t ∈ TR. Although the weight typing of containers used in the model might seem too simplistic, one has to take into account that the average weights, Wt20τ and Wt40τ , for each type are calculated at transport level, making the classiﬁcation much more reﬁned. Following are the constraints of the proposed IP model: t∈TRON p

+ 40τ x20τ ≤ Cpl tl + 2xtl

τ ∈T

20τ R xtl + 2x40τ ≤ Cpl tl

∀p ∈ P, l ∈ L (1) ∀p ∈ P, l ∈ L (2)

τ ∈{RL,RH} t∈TRON p

α xατ tl ≤ Cpl

∀p ∈ P, l ∈ L, α ∈ {20, 40} (3)

τ ∈T t∈TRON p

l∈L

ατ xατ tl = LD t

∀α ∈ {20, 40}, τ ∈ T, t ∈ TR (4)

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t∈TRON p

G−ρ p ≤

τ ∈T α∈{20,40}

l∈L

− ≤ Sps

Gρl

t l∈LAf s

+ Wtατ xατ tl ≤ Wpl

+ρ Wtατ xατ tl ≤ Gp

∀p ∈ P, l ∈ L (5) ∀p ∈ P, ρ ∈ {L, V } (6)

τ ∈T α∈{20,40}

t∈TR ON p

t∈TRON p

+ Wtατ xατ tl ≤ Sps

∀p ∈ P, s ∈ S (7)

τ ∈T α∈{20,40}

For each port p ∈ P , constraints (1) and (2) deﬁne the capacity restrictions of + ), and each location l ∈ L, respectively for the total number of TEU allowed (Cpl ON R the number of reefer containers that can be stowed (Cpl ), where TR p is the set of all the transports on the vessel at departure from port p. Similarly constraint 20 40 ) and 40’ (Cpl ) that can be stowed in a (3) restricts the number of 20’ (Cpl location. Constraint (4) forces the loading of all containers in the loadlists. The and LD 40τ amount of containers to load is given by the constants LD 20τ t t . The average weight of each container type under a speciﬁc transport Wtατ is used in + constraint (5) to limit the load of containers to the max weight allowance Wpl reﬂecting draft limits and vessel capacity. Stability constraints can be calculated w.r.t. the center of gravity of the ship using the hydrostatic data table, and thus be satisﬁed by limiting its position using constraint (6). For each port −ρ p ∈ P , the center of gravity limits, G+ρ p and Gp , have been calculated, where ρ ∈ {L, V } represent the longitudinal and vertical components reﬂecting trim, GM , and draft limits. Due to the symmetrical deﬁnition of outer locations, we assume cargo to be equally stowed on each side of the ship thus making irrelevant the calculations of the transverse component (Gρl indicate the components of the center of gravity for each location l ∈ L). Given a set of stations s ∈ S (calculation points as shown in Figure 2), constraint (7) calculates the downward t is the set of locations force created by the cargo aft of each station s, where LAf s + − aft of station s, and Sps and Sps are the maximum and minimum shear limits at station s ∈ S for port p ∈ P . Since the weight of cargo is constant, buoyancy is included in the calculations behind the constant limits. In the master planning phase, overstowage minimization focuses on hatchoverstowage. This is modeled by a number of binary variables δpl ∈ {0, 1}, indicating the presence of containers to load or unload at port p ∈ P under on deck locations LO . This is accomplished with the following constraint i∈LU l

⎛

⎞ D 40τ ⎠ ⎝Rpi x20τ + ≤ M δpl ti + xti t∈TRA p

∀p ∈ P, l ∈ LO

(8)

τ ∈T

A O where LU l is the set of locations under l ∈ L , TR p is the set of transports D that are either loaded or unloaded in port p, and Rpi is the containers already on board the vessel when arriving at the ﬁrst port (hereafter referred to as the release) that are discharge from location i in port p. The indicator variable can

Fast Generation of Near-Optimal Eco-Eﬃcient Stowage Plans

now be used to calculate hatch-overstowage as follows: OV 40τ O Rpl + x20τ − M (1 − δpl ) ≤ ypl tl + xtl t∈TROV p

∀p ∈ P, l ∈ LO

293

(9)

τ ∈T

O which, given the set of transports Constraint (9) deﬁnes the cost variable ypl OV OV TR p (and release containers Rpl ) that overstow containers to load or unload at port p ∈ P , counts the overstowing containers. In constraints (8-11) we make use of BigM constants M tightened to the upper bounds of the constraints. Due to the fact that slot planning constraints force 20’ containers to be stowed under 40’ ones, there is the possibility of forcing the introduction of overstowage within locations. To alleviate this problem, we estimate the potential overstowage between 20’ and 40’ containers within each location. D20 + x20τ ∀p ∈ P, l ∈ L (10) Rpl tl ≤ M φpl OV 40 Rpl +

τ ∈T t∈TRA p P x40τ tl − M (1 − φpl ) ≤ ypl

∀p ∈ P, l ∈ L

(11)

t∈TROV τ ∈T

Constraint (10) introduces a new set of Boolean variables φpl for each port p ∈ P and location l ∈ L, indicating the presence of 20’ containers to load or unload. P in constraint (11). This These indicator variables deﬁne the cost variable ypl variable holds the number of potential overstows between 40’ and 20’ containers D20 OV 40 and Rpl are the number of containers in the release within a location. Rpl discharged and potentially overstowing in port p, respectively. Optimization of crane utilization is modeled as the minimization of the makespan of cranes. We calculate a lower bound of the makespan of cranes ypT in port p as demonstrated in Figure 2 with the following constraint: 40τ x20τ ≤ ypT ∀b ∈ B, p ∈ P (12) C T ime tl + 2xtl l∈Lb τ ∈T t∈TRA p

where B is the set of adjacent bays, Lb is the locations of b ∈ B, and C T ime is the average crane time needed to load or unload a container. Capacity constraints over reefer containers do not ensure feasibility of a location in the slot planning phase due to stacking rules. The following constraint 20τ τ R R Fpl ≤ ypl ∀p ∈ P, l ∈ L (13) xtl + 2x40τ − Cpl tl τ ∈T t∈TRON p

alleviate this issue by reducing the maximum capacity of reefer containers within + τ τ R , where Fpl = Cpl /Cpl for all non-reefer a location by a proportional factor Fpl containers and a factor 1 for all reefer containers. The reduction is then captured R . in the cost variable ypl The model minimizes a weighted sum of the cost variables T T O P R C O ypl + + C P ypl + C R ypl C yp . (14) p∈P l∈L

p∈P

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The weights (C O , C P , C R , C T ) have been derived from the deployed system of our industrial partner and thus reﬂect a reﬁned adjustment to the economy of stowage planning and the preferences of stowage coordinators. Complexity of Multi-Port Master Planning. The hatch overstow problem (HOP) is NP-complete [17]. This problem models the same hatch-overstow objective as multi-port master planning. Given the optimization version of the HOP, we can reduce it to the multi-port master planning problem by having only one crane and using the reefer capacities to disallow containers below deck. Thus, multi-port master planning is NP-hard. 4.2

Slot Planning

The master plan for the ﬁrst port, which is the port to generate a stowage plan for, becomes an input to the slot planning phase. In this way information is passed in a top-down fashion from the master planning to the slot planning phase. There is currently no information ﬂowing in the other direction. The input deﬁnes the number of containers of each type to stow in each location of the vessel, and by generating a slot plan for each location, a type-based stowage plan is created. Each slot plan is an independent sub-problem. It must decide which container type (if any container at all) to stow in each slot of the location. The container types here correspond to those deﬁned in the multi-port master planning phase (Section 4.1) and must satisfy the constraints and optimize the objectives mentioned in Section 4 for slot planning. We direct the reader to [8] for a more detailed presentation of the model. The slot planning phase is solved using a combination of the CP and LS algorithms described in [8,21]. The CP algorithm is initially run with a time limit of one second.2 Optimal or near-optimal solutions are often found within this time frame. In some cases, however, the CP approach is not able to ﬁnd solutions. When this happens, the LS algorithm is run and the resulting solution is used. Such situations originate in two cases: 1) the problem is too complex for CP to be solved in one second (only few cases), and/or 2) the problem is infeasible. The latter case happens due to the abstraction used in the multi-port master planning phase. For instance, when the total weight of containers assigned to a location is within limits, but due to stack weight limits and the arrangement of already onboard containers it is not possible to stow all containers in the location. Since the loadlist is not strict, and removing a few containers from the locations has small impact on the stability of the vessel, the LS algorithm has been modiﬁed to roll out containers that cannot be stowed.

5

Computational Results

To evaluate our approach, we use 20 instances from a stowage planning optimization tool [14] deployed by our industrial partner. Table 1 gives an overview 2

This corresponds to an upper bound of about 100 seconds for a typical large vessel when solving all the slot planning sub-problems sequentially.

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Table 1. Problem instances overview. Columns under Vessel indicate ship dependent data: Cap. is the maximum nominal capacity of the ship and Loc. is the total number of locations. Notice that given the same number of locations, diﬀerent vessels can have diﬀerent capacities. Columns under Route show information about the route, Ports indicates the number of calls in the route, Util. and Weight are the maximum utilization during the voyage in terms of TEU and weight. Moves is the total number of crane moves on the route. The Encoding columns present the number of Boolean variables (Bools) needed by the multi-port master planning phase after preprocessing, while Transp. is the total number of active transports. Instances Characteristics ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Vessel Cap. (TEU) 7,490 9,618 4,755 9,160 7,344 5,044 6,717 4,478 5,052 4,755 9,118 4,456 4,478 8,490 5,047 6,545 9,984 2,584 9,118 9,160

Loc. 90 100 67 100 71 76 65 36 70 67 80 61 36 100 71 80 87 40 80 100

Ports 8 4 4 16 10 14 7 9 9 5 11 6 10 6 9 6 4 5 11 10

Util (%) 49 82 55 87 69 81 80 78 74 70 90 41 75 81 64 10 69 28 93 79

Route Weight (%) 30 56 49 54 62 76 66 59 42 72 61 35 54 58 57 9 33 11 30 62

Moves 6,162 5,306 2,276 9,610 8,736 8,562 5,462 1,934 5,272 3,384 14,290 4,660 6,790 6,482 8,208 834 3,284 882 12,182 9,196

Encoding Transp. Bools 17 250 19 180 18 118 54 623 76 618 69 750 56 238 20 57 83 577 27 157 109 850 31 315 81 352 11 334 88 585 10 234 12 143 9 118 25 728 89 960

of the characteristics of the instances. The instances are real stowage problems that coordinators have solved using the deployed tool and thus have very high data quality. All experiments were run on a Linux machine with two Six Core AMD Opteron processors at 2.0 Ghz and 32 GB of memory. Multi-Port master and slot planning models were implemented in C++ and used respectively CPLEX 12.2, and Gecode 3.5 [12] libraries. Due to the non-deterministic nature of the LS algorithm, results of slot planning are reported in average over 10 runs of the algorithm. 5.1

Multi-port Master Planning Experiments

Given that no previous approaches have solved the master planning problem to optimality, we did not expect our IP model to be eﬃcient. The results, however, did rise attention. Table 2 presents the results of the IP model, and two experiments where the algorithm is stopped at a 2 and 5% gap from the LP relaxation. A 5% approximation is acceptable since forecasted loadlist data is imprecise. The results are quite interesting for two reasons. First, the IP model was able to calculate optimal solutions for 13 of the 20 instances within a 5

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Table 2. Multi-Port Master Planning with IP. The ﬁrst column is the instance number. The next columns present grouped results of three runs of the model: the ﬁrst for optimality and the others ending respectively at 2 and 5% gap from the LP relaxation. Column Obj is the optimal value, and column Gap the distance to optimality w.r.t. Obj. Times are reported in Time, while time to generate a complete stowage plan is shown in column Total which includes the runtime of the slot planning phase. Instances that could not be solved within 5 hours are marked with timeout. The bold face shows results obtained within 10 min. IP Results ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Optimal Obj. Time 5 (10 ) (sec.) 10.99 6.15 18.73 7.41 3.77 8.83 28.87 385.68 20.64 5,988.48 timeout 30.22 558.65 77.52 2.24 6.71 2,476.37 6.85 19.30 timeout 2.89 24.71 timeout 11.53 151.76 23.90 1.78 4.62 1.32 timeout timeout timeout

Gap (%) 0.18 1.30 0.00 0.14 2.11 0.17 1.31 0.11 0.01 0.00 0.00 0.00 0.00 -

2% Gap Time (sec.) 6.15 2.52 8.49 229.08 1,390.92 timeout 39.06 1.12 1,033.35 16.38 timeout 7.06 4,658.59 151.76 timeout 1.78 1.32 0.46 timeout 11,319.07

Gap (%) 4.47 1.30 3.81 3.63 2.11 5.21 1.31 0.11 4.29 0.00 0.00 0.00 0.00 -

5% Gap Time (sec.) 2.98 2.52 5.19 96.65 1,390.92 timeout 10.00 1.12 1,033.35 6.90 timeout 7.06 2,488.22 151.76 timeout 1.78 1.32 0.46 timeout 7,272.95

Total (sec.) 7.94 16.15 10.13 110.28 1,396.18 timeout 15.26 5.2 1,034.01 17.95 timeout 15.01 2,497.41 162.49 timeout 3.24 15.30 1.36 timeout 7,282.80

hours limit. Second, taking the results with a 5% gap, it is possible to generate a complete stowage plan within 10 minutes for 12 of the instances, suggesting the need for further research on IP models. As expected, the objective value is dominated by the overstowage objectives (9) and (11). MIP relaxation. To tackle the weaknesses of the IP model, we propose an MIP model where we relax the integrality constraints over the decision variables 40τ x20τ tl and xtl . We experimentally evaluate the MIP relaxation by comparing its results to the optimals from the IP model. Table 3 presents results for optimal runs and for 2 and 5% gap from the LP relaxation of the MIP model. In terms of objective value MIP and IP solutions are very similar. It is only in the 5% gap results that it is possible to notice some signiﬁcant diﬀerence which, however, does not exceed 5.12%. In contrast, runtime results are drastically diﬀerent. It is now possible to generate complete stowage plans for nearly all instances within 10 minutes. Only 4 of the 20 instances could not be solved within this time frame, and 14 could be solved to optimality. Experiments have shown no diﬀerence in objective value w.r.t. overstowage. Also we see a very small diﬀerence in crane utilization, clearly due to the increased ﬂexibility of the decision variables. This important observation can be used to

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Table 3. Multi-Port Master Planning with MIP. The second column describes the gap between the IP and MIP optimal solution. For the other columns see Table 2. MIP Results Inst. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Optimal Gap Time (%) (sec.) 0.000 1.75 0.000 3.57 0.000 1.28 0.004 272.90 0.002 384.69 timeout 0.001 8.32 0.009 0.97 0.001 211.48 0.004 7.15 timeout 0.004 1.95 timeout 0.002 6.93 timeout 0.000 0.73 0.005 0.67 0.20 timeout timeout

2% Gap (%) 0.002 0.007 0.033 1.647 0.095 0.001 0.295 0.001 0.133 0.004 0.805 0.000 1.744 -

Gap Time (sec.) 1.75 2.38 0.65 84.27 234.02 timeout 8.32 0.67 211.48 4.52 3,708.52 1.95 311.33 3.25 251.42 0.73 0.52 0.20 timeout 3,636.18

Gap (%) 0.002 0.007 0.033 2.478 0.095 0.001 0.295 0.001 3.545 0.004 0.805 0.000 1.744 -

5% Gap Time Total (sec.) (sec.) 1.75 5.22 2.38 15.59 0.65 4.71 35.04 51.48 234.02 242.96 timeout timeout 8.32 15.14 0.67 2.18 211.48 212.47 1.24 10.43 3,708.52 3,709.47 1.95 8.91 311.33 321.01 3.25 16.82 251.42 256.63 0.73 2.52 0.52 10.25 0.20 1.13 timeout timeout 2,060.71 2,070.46

advocate the use of the MIP model in exchange for IP. The IP model has to face the combinatorial problem of deciding whether one single container should be stowed in one location or another. Using a MIP we can split this container and avoid the combinatorial puzzle. Even a single container could, however, cause a large amount of overstowage, but we do not see this happening in practice. We thus believe the MIP model can be used in practice by the industry. Using a MIP also gives the industry the ability to use standard solvers and eases the process of adding side constraints. 5.2

Slot Planning Experiments

The slot planning experiments discussed in this section are based on master plans obtained from the IP and MIP multi-port master planning experiment with 5% optimality gap. Master plans based on MIP may have fractional numbers of containers to stow in locations which is physically impossible. We attempt to stow a fractioned container in one of the locations where a fraction of it has been assigned by the master plan. If none of these locations have capacity left, the container is rolled out. Based on previous experiments ([8,21]), we set up a time limit of one second for slot planning each location of a vessel. The quality of each slot plan is evaluated by comparing it with the best slot plan generated by CP for the same location within twenty minutes. In the cases where the number of containers of one or more types were reduced by LS, the slot plan is evaluated against the best plan generated by CP within twenty minutes, stowing the same containers as LS. Table 4 summarizes the results of our experiments.

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Table 4. Slot Planning MIP (Cont.) and IP (Int.) master plans with 5% gap. The ﬁrst and second columns are the id of the instance and the number of containers to stow in the ﬁrst port. The next columns show grouped results of slot planning based on MIP and IP master plans. The third and fourth columns show the runtime for the slot plans, ﬁfth and sixth columns is the number of locations. The seventh and eighth columns totalize the number of rolled out containers by fractionality and odd number of 20’ types, the ninth and tenth columns are the containers rolled out by LS, and eleventh and twelfth columns are the percentage of total containers rolled out. The last two columns show the average gap of the slot plans. A dash indicates that no master plan was provided. Slot Planning Current Port ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Conts. 990 2,222 783 1,876 573 862 238 334 1,064 314 757 894 2,190 901 302 1,618 200 1,261

Time(s) Locs. Frac.+Odd LS rolled Rolled out(%) Gap(%) Cont. Int. Cont. Int. Cont. Int. Cont. Int. Cont. Int. Cont. Int. 3.48 13.21 4.07 16.44 8.94 6.83 1.52 0.99 9.20 0.95 6.97 9.69 13.57 5.21 1.79 9.73 0.94 9,82

4.96 13.10 4.94 13.64 6.08 5.26 4.08 0.66 11.06 7.96 9.19 10.73 1.46 13.98 0.91 7,73

34 61 33 69 29 27 12 14 29 12 37 24 61 30 16 50 12 38

34 57 32 70 25 35 14 13 32 27 23 58 14 50 13 37

4 15 3 14 10 5 1 4 8 1 7 0 7 14 2 4 0 6

2 6 7 5 5 6 1 4 5 5 2 5 6 6 0 4

0 42.3 0 19.1 0 3 0 3 41.8 0 0 0 21.5 8.6 0 5.8 0 0

0 39.4 0 15.2 0 0 0 0 58.7 0 0 0 0 41.6 0 1

0.40 2.58 0.38 1.76 1.75 0.93 0.42 2.10 4.68 0.32 0.92 0.00 1.30 2.51 0.66 0.61 0.00 0.48

0.20 2.04 0.89 1.08 0.87 0.70 0.42 1.20 5.99 0.66 0.22 0.23 1.99 2.94 0.00 0.40

0.00 0.00 3.05 4.60 0.36 0.00 12.77 10.47 1.41 0.45 1.25 0.00 0.00 0.00 1.57 0.00 13.86 16.44 0.00 0.63 0.59 0.44 0.67 0.78 0.03 6.73 0.00 0.00 0.38 1.54 0.00 0.00 2,01 9,75

Vessels are slot planned fast by our approach. For the instances with a master plan available, slot plans are generated within 17 seconds in total. There is no time-wise dominance of slot plans generated from MIP and IP master plans, indicating that integrality constraints do not aﬀect the complexity of slot planning. When slot planning IP master plans, a reduction in the number of containers rolled out due to fractionality and odd number of 20’ types in locations is observed in most of the instances. This is, however, a very small (0.4% max) fraction. The number of containers rolled out by LS diﬀers in average only in 1.6 containers (9.7 for the IP and 8.1 for the MIP master plans), and the average percentage of total containers rolled out is the same (1.2%) for both models. These facts indicate no considerable impact of using MIP master plans. Moreover, it was possible to generate slot plans for two extra instances using MIP master plans. The maximum roll-out of an instance is 5.99%, a reasonable number given the amount of containers typically rolled from a loadlist by stowage coordinators.

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Only two instances have an average optimality gap over 10%, due to the presence of outliers, and the median of all instances is always 0%. CP generated optimal slot plans within one second for 91.8% of the locations of the MIP master plans and 94.8% of the locations of the IP ones. Moreover, CP was able to prove optimality in 83% of the slot plans generated for the locations for both, the MIP and IP master plans.

6

Summary

This paper presented a 2-phase stowage planning optimization approach able to solve industry instances to near-optimality within the time limits required for practical usage. In particular, we introduced an IP model of multi-port master planning that includes all major aspects of the problem. Our experiments have shown that multi-port master planning can be solved fast to optimality for a signiﬁcant number of instances, and that runtime can be drastically reduced by relaxing integrality constraints without compromising solution quality. We have shown that in 16 of our 20 test instances, complete stowage plans can be computed in less than 330 seconds. The remaining four instances take longer due to the time needed for master planning. In future work we plan to develop heuristic master planning algorithms to handle such hard cases. We are also interested in optimizing ballast water and compare the performance of our system with human stowage coordinators. Acknowledgments. We would like to thank Stephen Barraclough, Jon Adam Freeman, Robert John Milton, and Mikkel M¨ uhldorﬀ Sigurd at Maersk Line and Nicolas Guilbert, Kim Hansen, Esben Mose Hansen, Benoit Paquin, Marc Cromme, and Hans Schou at Ange Optimization for their extensive support of this work. This research is sponsored in part by the Danish Maritime Fund under the BAYSTOW project.

References 1. Ambrosino, D., Anghinolﬁ, D., Paolucci, M., Sciomachen, A.: An experimental comparison of diﬀerent heuristics for the master bay plan problem. In: Proceedings of the 9th Int. Symposium on Experimental Algorithms, pp. 314–325 (2010) 2. Ambrosino, D., Sciomachen, A.: A constraint satisfaction approach for master bay plans. Maritime Engineering and Ports 36, 175–184 (1998) 3. Ambrosino, D., Sciomachen, A.: Impact of yard organization on the master bay planning problem. Maritime Economics and Logistics (5), 285–300 (2003) 4. Aslidis, A.H.: Optimal Container Loading. Master’s thesis, Massachusetts Institute of Technology (1984) 5. Avriel, M., Penn, M., Shpirer, N., Witteboon, S.: Stowage planning for container ships to reduce the number of shifts. Annals of Oper. Research 76, 55–71 (1998) 6. Botter, R., Brinati, M.A.: Stowage container planning: A model for getting an optimal solution. In: Proceedings of the 7th Int. Conf. on Computer Applications in the Automation of Shipyard Operation and Ship Design, pp. 217–229 (1992)

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7. Davidor, Y., Avihail, M.: A method for determining a vessel stowage plan. Patent Publication WO9735266 (1996) 8. Delgado, A., Jensen, R.M., Schulte, C.: Generating optimal stowage plans for container vessel bays. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 6–20. Springer, Heidelberg (2009) 9. Dillingham, J., Perakis, A.N.: Design of an expert system for container stowage planning. In: Proceedings of the Fleet Management Technology Conference, Baltimore (1987) 10. Dubrovsky, O., Penn, G.L.M.: A genetic algorithm with a compact solution encoding for the container ship stowage problem. J. of Heuristics 8, 585–599 (2002) 11. Flor, M.: Heuristic Algorithms for Solving the Container Ship Stowage Problem. Master’s thesis, Technion, Haifa, Isreal (1998) 12. Gecode Team: Gecode: Generic constraint development environment (2006), http://www.gecode.org (accessed July 12, 2011) 13. Giemesch, P., Jellinghaus, A.: Optimization models for the containership stowage problem. In: Proceedings of the Int. Conference of the German Operations Research Society (2003) 14. Guilbert, N., Paquin, B.: Container vessel stowage planning. Patent Publication US2010/0145501 (2010) 15. Gumus, M., Kaminsky, P., Tiemroth, E., Ayik, M.: A multi-stage decomposition heuristic for the container stowage problem. In: Proceedings of the 2008 MSOM Conference (2008) 16. Imai, A., Sasaki, K., Nishimura, E., Papadimitriou, S.: Multi-objective simultaneous stowage and load planning for a container ship with container rehandle in yard stacks. European Journal of Operational Research (171), 373–389 (2006) 17. Jensen, R.M.: On the complexity of container stowage planning: the capacitated zero-shift problem and the hatch overstow problem. Tech. Rep. ITU-TR-137, IT University of Copenhagen (2010) 18. Kang, J., Kim, Y.: Stowage planning in maritime container transportation. Journal of the Operations Research Society 53(4), 415–426 (2002) 19. Li, F., Tian, C.H., Cao, R., Ding, W.: An integer linear programming for container stowage problem. In: Bubak, M., van Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2008, Part I. LNCS, vol. 5101, pp. 853–862. Springer, Heidelberg (2008) 20. Nugroho, S.: Case-based stowage planning for container ships. In: The Int. Logistics Congress (2004) 21. Pacino, D., Jensen, R.M.: A local search extended placement heuristic for stowing under deck bays of container vessels. In: The 4th Int. Workshop on Freight Transportation and Logistics, ODYSSEUS 2009 (2009) 22. Sciomachen, A., Tanfani, A.: The master bay plan problem: a solution method based on its connection to the three-dimensional bin packing problem. IMA Journal of Management Mathematics 14, 251–269 (2003) 23. Shields, J.J.: Container stowage: A computer aided pre-planning system. Marine Technology 21(4) (1984) 24. Tupper, E.C.: Introdution to Naval Architecture. Elsevier, Amsterdam (2009) 25. Webster, W.C., Van Dyke, P.: Container loading. a container allocation model: I - introduction background, II - strategy, conclusions. In: Proceedings of Computer-Aided Ship Design Engineering Summer Conference, University of Michigan (1970) 26. Wilson, I.D., Roach, P.A.: of combinatorial optimization applied to container-ship stowage planning. Journal of Heuristics 5, 403–418 (1999)

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27. Yoke, M., Low, H., Xiao, X., Liu, F., Huang, S.Y., Hsu, W.J., Li, Z.: An automated stowage planning system for large containerships. In: Proceedings of the 4th Virtual Int. Conference on Intelligent Production Machines and Systems (2009) 28. Zhang, W.Y., Lin, Y., Ji, Z.S.: Model and algorithm for container ship stowage planning based on bin-packing problem. Journal of Marine Science and Application 4(3) (2005)

Game Theoretical Aspects in Modeling and Analyzing the Shipping Industry Xiaoning Shi1,2 and Stefan Voß1 1

2

University of Hamburg, Institute of Information Systems, Von-Melle-Park 5, 20146 Hamburg, Germany [email protected], [email protected] Shanghai Jiao Tong University, Shanghai 200240, P.R. China

Abstract. The shipping industry is known for providing transport service in terms of deploying vessels and accessing ports, making shipping one of the network-based services. From the perspective of traditional as well as neo-economics, shipping is assumed to pursue proﬁt maximization subject to scarce resources, e.g. capital, assets, seafarers, or binding constraints derived from schedules, etc. Players could be any of the following: linkage operators, e.g. liner shipping carriers, port operators, freight forwarders, customs, hinterland haulage carriers, inland navigation carriers, market regulators, etc. Taking into account interdependencies and interrelations, game theory provides a meaningful way to model and analyze behaviors of the involved players. In this paper we provide a survey on game theoretical approaches within the shipping industry.

1

Introduction

Game theory (hereafter GT) is a methodology of decision making involving multiple parties such as persons, companies or agents. For instance, each company must consider what other companies will do. Classical literatures [48,73] together with applications of GT in industrial organizations [21,50,70] usually discuss four classes of games: static as well as dynamic games of complete information and static as well as dynamic games of incomplete information. Corresponding to these four classes of games there are four notions of equilibrium in games: Nash equilibrium (NE), subgame-perfect NE, Bayesian NE, and perfect Bayesian equilibrium. The NE is a solution concept of a game, in which each player is assumed to know the strategies to be taken by the others and no player can be better oﬀ by changing his or her own strategy unilaterally. A subgame-perfect NE is a reﬁnement of a NE used in dynamic games if it represents a NE of every subgame of the original game. Bayesian NE is a solution concept of Bayesian games where at least one player is unsure of the type (and so the payoﬀ function) of another player, which might result in some implausible equilibria in dynamic games. To reﬁne the equilibria generated by the Bayesian Nash solution concept or subgame perfection, one can apply the perfect Bayesian equilibrium solution concept. The characteristics of each game can thus be summarized in Table 1. J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 302–320, 2011. c Springer-Verlag Berlin Heidelberg 2011

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Table 1. Brief summary on solution concepts Solution Concepts

Nash equilibrium

Bayesian Nash Equilibrium

Subgame-perfect Nash equilibrium

Perfect Bayesian equilibrium

Proposed by Applications

John F. Nash Reinhard Selten John C. Harsanyi N/A (see, e.g. [39]) Static games Dynamic games Static games Dynamic games Pure strategy Mixed strategy Sequential games Expressions Normal form Extensive form Extensive form Extensive form Extensive form Approaches Fixed point Backward induction Bayes’s rule Sequential rationality theorem based on updated beliefs Information set Complete Complete Incomplete Imperfect

We start with a game in normal form which is a possible way of describing a game. Unlike extensive form, normal-form representations are not demonstrating a game by a graph or tree, but rather represent the game by way of a matrix or formulations. A game in this form consists of (1) players denoted by i = 1, 2, ..., n, (2) strategies or, more generally, a set of strategies indexed by xi available to player i and (3) payoﬀs πi (x1 , x2 , ..., xn ) achieved by each player. A player in a game is a person or a business community making decisions or choosing a strategy from a set of given options. One player’s decision aﬀects that of the others. In a static game, players make decisions simultaneously without knowing information of other’s decision. In a dynamic game, players make decisions at diﬀerent moments, i.e., a sequential decision making process happens due to the fact that other’s decisions have been disclosed. A strategy in a game is one of the options from which a player may select. Such decision making process may be based on historic experience of himself and/or information disclosed by other players. Traditional applications of GT attempt to ﬁnd equilibria. In an equilibrium, each player of the game has adopted a strategy that none of the players involved is likely to deviate from. Payoﬀ means what a player gets after choosing a strategy. Pursuit of payoﬀ maximization, usually, is the utmost goal of a player. In this paper, we emphasize ways of thinking when decision makers face the shipping business. Therefore, the category of games and their applications are from an industrial viewpoint taking into account that shipping is a network based service. Within this framework, a player can be, e.g., a liner shipping operator, or a tramp shipping operator, or a community of liners – an alliance – behaving as a whole in the market. A set of strategies can include whether to cooperate with other competitors or deviate from the current situation, etc. Payoﬀ of a player is the commercial beneﬁt when a player chooses one of his strategies, e.g., the revenue after choosing to cooperate with his competitor. The shipping industry provides transport services among ports by ships at sea. Its service is based mainly on the networks built by carriers, representing the supply side by providing transport services while following regulations and policies. Shippers represent the demand side by booking transport services. Besides freight rates, a shipper can decide to accept one of the carriers’ oﬀers taking into account his expectation of other shippers’

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decisions to avoid congestion, peak season pricing, risk, etc. Carriers similarly attempt to avoid overcapacity, cut-throat competition, lack of diversiﬁcation, and other negative factors. Therefore, shippers or carriers can be regarded as players in games as they will not take action without considering what their competitors do. In addition, both shippers and carriers must act subject to regulations and policies in the shipping industry. We must also consider the regulator designing regulations and policies with an eye towards how shippers and carriers will react to them. In this sense, a regulator can also be viewed as player, especially in principal-agent relations. The interactions among players in the shipping industry have a considerable impact on each player’s strategy set. The relationship between homogeneous players, e.g. diﬀerent carriers, is horizontal whereas the relationship between heterogeneous players, e.g. shipper and carrier, is vertical. Mechanism design is one of the branches of GT where protocols are designed for players by regulators. Sometimes also called reverse GT, it studies solution concepts for a class of private information games. More speciﬁcally, it is a case of vertical games; we may also categorize it as principal-agent. In addition to the aforementioned players, there is a growing trend for related service providers to integrate. For instance, truck haulage carriers integrate their business with shipping carriers so that door to door service can be achieved. Thus, games such as price auctions and principal-agent incentive games also need to be considered, and these may be classiﬁed as either heterogeneous relations or principal-agent games as mentioned before. Therefore, GT can be a helpful tool in the analysis of the shipping industry given features of the industry that the decisions of multiple players aﬀect each players payoﬀ. When observing the literature, the authors ﬁnd that many meaningful tools are spread over a variety of papers and books, and not so many well groomed surveys exist on systematic application of GT in the shipping industry. Our goal is to provide a survey on how the existing literatures deal with behaviors of either homogeneous or heterogeneous players in the shipping industry. In terms of discussing them step by step, i.e., from horizontal relations to vertical relations, we shed lights to kinds of interactions among the players. The related discussion can be helpful for readers who intend to analyze GT aspects in the shipping industry. Due to the need for short explanations, we focus only on the intuition behind the business patterns discussed in this paper. Our second goal is to provide ample but by no means exhaustive references on speciﬁc applications of various GT techniques. The remainder of this paper is organized as follows. In Section 2, operational business and strategic behaviors are categorized. In Section 3 and Section 4, we introduce and demonstrate in various dimensions how players in the shipping industry interact with each other taking into account both individual and collective rationality. Moreover, in Section 5 we show how market regulators might improve market eﬃciency by means of game theoretical mechanism design.

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Fig. 1. Structure of the shipping industry and its associated markets

2

Games in Shipping Networks

From an industrial perspective, a transport network is a spatial system of nodes and links over which the movement of cargo and passengers occurs [68], so is a shipping network. A node is a center in a transport network from which cargo and passenger movements emanate. A physical link between two nodes is the transportation way (e.g., waterway, highway, railway, and airway) over some distance between the nodes. From a theoretical perspective (cf. graph theoretical concepts) a network can be represented as a graph, consisting of a number of nodes (vertices) and links (edges). Furthermore, a path is a trail with neither repeated edges nor repeated nodes [23]. However, in shipping practice service providers may also design service routes with repeated linkages as well as repeated portsof-call within one service. In addition, a decision maker representing a link also considers directions and capacities of other links. The same applies to decision makers representing nodes, which inevitably underlies primary principles of GT. Therefore, instead of just applying the path game, the problems investigated in this research are deﬁned by means of link games and node games within networks (Section 3). In order to better understand this mature and complex industry, we ﬁrst outline major business issues. Then the informal business description is translated into formal game theoretical problems. Figure 1 shows the varieties of business relations comprising the shipping industry. The primary shipping industry consists of liner shipping, tramp shipping, the tanker business and the ferry business (Box 1 of Figure 1). Liner shipping is a service following announced and scheduled ports-of-call, regardless of whether it is ocean sea transport or short sea shipping. Tramp shipping is a transport

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service that does not rely upon repeatedly scheduled ports of call, but rather on pick-up and delivery of cargos according to demands and either voyage charter or time charter-based contracts. The tanker business shares similar characteristics with tramp shipping, the main diﬀerence being that the cargo in this case is either crude oil/oil product or bulk. Finally, the ferry business provides service to passengers, which is beyond the scope of this paper taking into account the fact that the behavior of humans is relatively erratic compared to that of cargo. The interested reader could simply replace passengers, to some extent, with cargos and then apply the same GT thinking as discussed below. Within the primary shipping industry, the competition and strategic cooperation among the homogeneous carriers arise as horizontal games. Considering the fact that carriers act as links connecting diﬀerent ports, the carrier related competition and cooperation can be viewed as one of horizontal relations at the macro level (i.e., a link game in service networks). Therefore, a link game is dealing with the construction of links and reconstruction of paths by means of either consolidating or deconsolidating linkage supply, so that demand could be better satisﬁed. As shown on the left side of Figure 1, the service providers component of the shipping service includes port operators, consolidation/distribution centers as well as hinterland service operators such as truck haulage, railway operators and 3rd party logistics providers. The ports and consolidation/distribution centers are nodes contributing to comprehensive service networks [68]. In a ﬁgurative sense, a node makes eﬀorts to attract more links by means of amplifying throughput and storage capacity of the node as well as hinterland connections, where this could relate to various aspects including, e.g., available infrastructures to avoid congestion regarding hinterland traﬃc. Once there are other competitive nodes within the same trading zone or graphical region, the nodes compete with each other in order to sustain as hub [47]. Or the nodes have to cooperate with the existing hub because of their limited capacities [42]. Therefore, the port and consolidation/distribution center related competition and cooperation can also be viewed as a center of horizontal relations among homogeneous players. However, it belongs to the node game. In sum, a node game is aiming to adjust the attractiveness of associated nodes to links in terms of changing the status of the node, so that better accessibility and capacity can be achieved. Furthermore, links and nodes select each other in order to obtain better performance in tandem than in isolation. On one hand, the links select eﬃcient nodes so that the waiting time and total voyage time could be shortened as well as to avoid potential risks. Sometimes, the links observe existing nodes and choose among them, as in the port selection problem of liner carriers [54]. Sometimes, the links even propose and invest in new nodes when it is worthwhile to do so. On the other hand, nodes select weighted links so that the capacity of the nodes can be better utilized and higher proﬁts can be achieved. In this case vertical relations among heterogeneous players occur [60]. In addition, the accessibility and connections with other service providers are also vital to both the links and nodes from the aspect of strategic sub-network integration at the macro level. Therefore, the problem is presented as a network game.

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The right hand side of Figure 1 depicts how associated resource markets support the shipping industry. If containers are built or leased and workers are employed, bargaining and auction games arise, (links 3 and 5 in Figure 1). If ships are to be deployed, schedule optimization and network games are considered (link 4). Finally, regulations and policies (Box 6) direct and control as far as possible behaviors of the associated players who simultaneously account for interdependencies with others (related to mechanism design; see Section 5). Besides demonstrating how players in shipping interact with each other we also mention how market regulators might improve market eﬃciency by means of game theoretical mechanism design. Assume there is a shipping carrier (maybe a tramp shipping carrier operating bulk in the past), who attempts to diversify its business in terms of entering the liner shipping market. First, it might face a market entry barrier game when starting the liner business (Section 3.1). The strategy set consists of two options, i.e., to enter or not. The strategy set of existing liners would include two actions: to defend (and if so to which extent) against the new entrant or not. The pay-oﬀs for both the new liner and the existing liners would be reﬂected in their utility matrices. In case the new liner succeeds in the ﬁrst game, this new liner might further consider being a member of existing liner strategic alliances. Then it faces the problem of fair allocation of proﬁts (Section 3.2). If it does not have enough capital and ﬂeets to cooperate fully with other liners, it might consider cooperation in terms of slot-chartering which can be modeled as a kind of auction game (Section 3.3). However, as the shipping industry keeps changing continually, the liner adjusts its competitive strategy as well as its long-term cooperation partners by means of learning (Section 4.3) undiscovered information (Section 3.5) in iterated and evolutionary sub-games (Section 3.4). If this liner eventually performs well enough, it might invest in related industries, such as container building and leasing business (Section 3.6) as well as the port operation business (Section 4). In addition, it needs to widen its thinking based on the idea of customer relationship management: How would its customer (shipper) behave given various alternative liner carriers in the shipping network (Section 4.1)? The shipper might maximize its own utility by avoiding congestion on sea routes (Section 4.2). Similarly, the liner itself may achieve cost savings by avoiding congestion at ports (Section 4.2), too. Therefore, it could be better for this liner to investigate the whole shipping network by taking into account network games (Section 4.4). In addition, it needs to follow the relevant regulations and policies as well as motivate its freight forwarders to grasp the market share as much as possible (Section 5). Furthermore, even if it becomes more sophisticated compared to when it ﬁrst started the business, it might also obtain fast responses on how other players act by building an algorithmic system (Section 6), which can approach an equilibrium, in case of any.

3

Horizontal Relations among Homogeneous Players

We start with an example as a motivation to depict a simple game in a servicebased network. Assume, some liner companies are providing nearly identical

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service to the market, i.e., these services are pre-arranged as well as announced with ﬁxed vessels, ﬁxed schedule, ﬁxed freight rate, ﬁxed sequence of port-ofcalls. Each service and port-of-calls are eventually constructing a network, while service routes can be regarded as sets of edges and port-of-calls as nodes. Then, we consider a directed network G = (V, E) with ﬁnite sets of nodes v ∈ V and edges e ∈ E. In this case, the liner carriers are the network owners because the shipping service network is built by them, i.e., l1 , l2 , ..., ln . A liner is denoted as li . n Then, the entire capacity of a certain edge is Q = i=1 qi where qi denotes the capacity of liner li along that edge. Given capacity provided by all liner ncarriers, the price of the shipping service of that edge is p = p(Q) = a − b i=1 qi . In this expression, a and b are positive parameters to be deﬁned in price-supply relation. Furthermore, once a liner carrier is regarded as a player, its strategy set consists of diﬀerent options of providing certain amounts of capacities. The bigger the supply (Q) the lower is the market price (p), and vice versa. So, a and b can be estimated by a series of values Q and p in diﬀerent scenarios. The capacity is denoted as qi . The payoﬀ of this player is ui = qi p(Q) − ci qi for i = 1, 2, ..., n(i = j), where ci denotes its cost function. From the perspective of this liner carrier, which is assumed to be selﬁsh, the aim of successfully playing the game is to maximize the output based on above mentioned payoﬀ function. It should be noted that link games together with node games could by no means be separated no matter what niche market of the shipping industry is to be observed. In this section, horizontal competition and cooperation among homogeneous players is considered. We start with the market entry game before deepening the business issues into more sophisticated games. 3.1

Market Entry Barriers Game

Once a liner attempts to enter a certain route, it becomes a competitor of the existing liners operating this route. Therefore, the market entry barriers game may apply. Such games happened when liner shipping conferences functioned before the mid 1980s. Then, the conference deployed a “battle ship,” operating on the same route, following the same schedule as the newly entered liner, but at a much lower price. The aim of such a deployment was to build up entry deterrence [19] and prevent new entries by means of grasping shippers providing lower prices. Losses from these lower prices would be commonly allocated among cartel members of the ﬁghting conference. It is doubtful whether the new entry could survive under a situation of low competitive price. However, the protecting company has to predict the rival’s cost function compared to its own in order to set up entry barriers as well as to better prepare for the competition [43]. 3.2

Fairness and Power Indices

It should be noted that fair cost allocation can be applied to the liner conferences. Fairness does not automatically mean the allocation of costs or proﬁts on an average base. In most cases, there exist leading players and follower players

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based on their performance powers. In other words, there exist partner asymmetries in the strategic alliances [24]. When all the involved players accept a cost allocation method taking into account power diﬀerentiations, it can be viewed as fairness, too. As for the power diﬀerentiation among the players we refer to at least ﬁve indices [1]: the Shapley-Shubik index [58], the Banzhaf index [3], the Johnston index [30], the Deegan-Packel index [10], and the Public Good index [29]. Furthermore, players within alliances or collusion might consider stability of the organization not only since the formation of the organization but also along the iterated procedure in terms of designing an eﬀective mechanism. For instance, instability of shipping cartels is a standard feature of elementary economic theory [62,63,64]. Generally speaking, the empty core can be applied to explain why previous shipping conferences and current alliances exist, so it works for any industrial cartels alike [78]. In addition, the fairness among the partners needs to be considered as one of the main factors of associated mechanism design. 3.3

Auction Game

Together with the decline of traditional liner conferences, innovative means of cooperation arise, which can be classiﬁed along three levels of cooperation: vesselsharing, slot-exchanging and slot-chartering. These various forms of cooperation are of importance for the liner carriers as well as of interest for researchers. The main diﬀerence between vessel-sharing, slot-exchanging and slot-chartering is that vessel-sharing focuses more on network integration [14] and ﬂeet deployment [69], whereas slot-exchanging focuses more on route implementation, and slot-chartering more on price-and-quantity decisions [59]. Concerning the competition and cooperation among the liner carriers, i.e. the link game, much research has been done based on a variety of observations including shippers choice [6], oligopoly market status, overcapacity issues [19] and additional proﬁt allocation approaches as well as fairness based on the Shapley value [56,57] derived from cooperative GT [74,66,41]. For the analysis of liner strategic alliances we also refer to [65,61]. When relations among homogeneous liner carriers are to be discussed in terms of non-cooperative games, there are diﬀerent perspectives which might lead to various models. From the perspective of capacity limitation, the slotchartering problem could be viewed as two persons zero-sum game because the reduced slot allocation of one player after bargaining comprises the added slots that the other player would achieve based on the negotiation. In this sense the decision variables are related to the quantities of slots allocated to each player. From the perspective of bargaining processes, the slot-chartering problem may also diﬀer depending on whether there is an eﬀective long-term replenishment mechanism. The mechanism design on slot-chartering price together with the respective quantity of the slots is valuable to be observed. Regarding the replenishment existing in other areas, e.g. in military management, see [40]. In order to design an appropriate mechanism, the following two situations should be considered, respectively: the slot owner has superior bargaining power over the related charterer, and the reverse situation. Furthermore, during the iterated negotiation processes, the higher the position of any given player the more he tends to

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push down the position of others. For long-term replenishment contracts based on GT analysis see, e.g., [34]. From the perspective of proﬁt sharing resulting from a slot-chartering cooperation, which is one form of link games, it could be viewed as non-zero-sum game since additional proﬁts might occur when liners provide more options to shippers. To the best of our knowledge, so far no such research has been done in the ocean sea shipping industry. For similar research considering the hinterland trucks picking-up and delivery tasks see [37,38]. 3.4

Iterated and Evolutionary Games

From the perspective of negotiation processes, iterated sub-games aﬀect the repeated assignment of either capacities or extra proﬁts [61]. One output of iterated games is to emphasize the evolution of ongoing processes. Players either turn out to be more rational, or capture more information and knowledge on the games. In addition, the mechanism of sub-games might turn out to be more truthful, or just the other way around. Those possibilities could be realized only based on iterations of the game. Fundamental knowledge of evolutionary GT can be found in [76]. Whether the iteration is converging is of great importance to all players involved. The iteration during the negotiation process can be interpreted as a sequence of best responses provided by all players. Therefore, contraction mapping maybe applicable to ﬁnd a solution to the ﬁxed point equation x = f (x), x ∈ R1 [7]. One can think of a contraction mapping in terms of iterative play: player 1 selects some strategy, based on this player 2 selects a strategy, etc. If the best response mapping is a contraction, the NE obtained as a result of such iterative play is stable but the opposite is not necessarily true, i.e., no matter where the game starts, the ﬁnal outcome is the same. See [49] for an extensive treatment of stable equilibria. The feature of contraction iteration of a game can be well applied as an explanation on formation of slot-sharing agreements among liner carriers. However, market circumstances are vital for the stability of the agreements. Once the market changes, e.g. freight rates increase sharply, the previous stable equilibria set by a certain agreement turns to be unstable, or the contraction process deviates from its way. 3.5

Asymmetric Information

From the perspective of incomplete and imperfect information among the involved players of shipping agreements, the true cost and individual prediction on the market share are, to great extent, the business conﬁdentialities of each player. Thus the involved players do not have full information about quantity options and consequently each possible quantity of capacity depends on the probabilities that other players perform diﬀerent actions/behaviors. The unequal status of information derives from the fact that some shipping related companies may pose superior information regarding their own costs and operating procedures. In addition, a shipping related company may know that another company may have better information, and therefore, choose actions that acknowledge this information shortcoming [7]. In some cases, the players simultaneously choose

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actions or one of the players chooses an action without knowing actions of the other players. Then the game could be viewed as static. If one liner already knows actions of the others before making a decision, the game could be viewed as dynamic. Furthermore, a player would use a mixed strategy when he is indifferent between several pure strategies. When the mixed strategy is mentioned, it means that one player is uncertain about what another player will do. So he gives each action of his strategy a probability to better respond to other players’ actions. Related research in supply chain management, however, shows that under certain acceptability assumptions, asymmetric information need not imply decisions which are too far from optimal; see, e.g., [16]. This is valid for the case where asymmetric information and opportunistic behavior is taken into account together with a mediator who supports the negotiation process. This mediator repeatedly generates new candidate contracts, which need to be accepted or rejected by the agents according to particular strategies. Some research based on a GT framework suggests that learning processes might imply a contingence of the equilibrium [51]. Thus, iterated negotiations on price and quantity of the slotchartering agreements deserve further research. Actually because of asymmetric information, the players are not exact homogeneous anymore. Those players can then be categorized into either leader or follower of a game. After player 1 observes the information on another player and/or his decision, the game turns to be a Stackelberg game. In a Stackelberg duopoly model, player 1, the Stackelberg leader, chooses a strategy ﬁrst and then player 2, the Stackelberg follower, observes this decision and makes his own strategic choice. Since in many scenarios the ship operator as an upstream ﬁrm, e.g. the wholesaler, possesses certain power over the typically freight forwarder as a smaller downstream ﬁrm, e.g. the retailer, the Stackelberg equilibrium concept can be applicable in many shipping related business. Besides dynamic games and leader-follower games, asymmetric information may also result in signaling games and Bayesian Games. 3.6

Asset Flows

The assets available for providing transport services, exist in the whole network, but not all of the assets are utilized. In this section, the utilized asset ﬂows are discussed, while the unused ones are discussed in the next paragraph. In some recent research, e.g., [52] assigned involved players right to choose ﬂow rate, energy amount, or bandwidth to reach equilibrium subject to the maximum ﬂow, amounts or width, respectively. Taking into account the characteristics of the special products related to transport capacity, one might realize that the transport capacity does not always equal the customers’ requirements in terms of quantities. Therefore, either overcapacity or lack of service might occur. In both situations, imbalance between the transport capacity and customers’ requirements would result in extra ‘costs.’ In case of overcapacity, the wasted capacity can be interpreted in terms of the wellknown newsboy problem, which belongs to classical Operations Research [33]. Transport capacity not occupied when the ship (or truck) starts its trip, i.e., the empty part of the capacity is just like yesterday’s newspaper – it usually exists

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without being of any value any more. This asks for eﬃcient revenue management tools as they are available in the airline industry; see, e.g., [25]. In case of lack of service, the extra costs can be interpreted as opportunity costs. The carrier might regret as it might be more proﬁtable to prepare more capacity to grasp the market share instead of accounting for lost sales because he does not have the capability to fulﬁll the requirements. Furthermore, the imbalance of trade ﬂows results in an empty container inbound dilemma: empty containers have to be shipped back to export origins so that they can be used for further shipments. Thus, when setting parameters of the network games, empty containers should be labeled as certain load but without proﬁts or even with negative proﬁts, so that more realistic models could be derived.

4

Vertical Relations among Heterogeneous Players

Basically any player of the shipping industry might play either cooperation games or competition games or both, within the designed mechanism and market circumstances including free market, monopoly, duopoly and even oligopoly. In this section, we pay attention to those actors who actually provide diﬀerent services; i.e., they are rather suppliers or customers to each other than competitors in a certain niche market. In the transportation industry, there are games to be investigated among vertically related players. The players who share the same value chain sometimes team up with each other so that better integrated service can be provided to the ﬁnal customer. Once the service provider selection is to be involved, see Figure 1, the vertical relations become apparent. Such vertically related players might be liner carriers and port operators, shippers and freight consolidation/distribution centers as well as hinterland haulage carriers, etc. Leader-follower models can be used to simulate the relationship among players, because some players who have either more experience or higher negotiation power distinguish themselves from their peers becoming leaders in games. In contrast, those players who have relatively less experience or know-how may become followers in games. In case of inter-company management, human resources arrangement and the salary bargaining problem [13] can be interpreted as the process of learning labors’ potentials, hence a principal-agent model might be of use to motivate the labors in terms of design bonus and penalty mechanism. The ships owned by a carrier, without intracompany cooperation, could also be viewed as inter resources, and apply a principal-agent model, too. 4.1

Route Choice Game

The transportation industry is part of the service industry, i.e., the actual revenue must come from the ﬁrst order customers who have the requirements to move cargoes or passengers themselves from origins to destinations taking into account the resource constraints together with customer preferences. The eﬀorts of vertically related games, horizontally related games and network games are to smooth the resource constraints and to fulﬁll customer preferences. In that sense,

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the customers themselves need to “play the game.” They use their common sense on the current service network, collect up-to-date information, and learn how to optimize the utilities in terms of route choice and modal choice, as traditional discrete choice models show [31]. For the freight transport sector, the shipper might choose air transportation as the preferred mode if he needs a faster link to avoid any delay caused by shipping; however, he might also keep in mind the shipping line as an alternative for network reliability reasons. Recently, there has been research considering network users as non-cooperative selﬁsh players who seek to optimize their experienced network latency [28,17]. 4.2

Congestion Game

For the competition among regional ports, zoning techniques can be used to supplement non-cooperative games. The congestion game and the Price of Anarchy are also suitable ways to build pricing policies of associated ports within zones. The congestion game is a game where a player’s payoﬀ only depends on its strategy as well as the number of other players choosing the same strategy [46]. On the other hand, shippers are typically viewed as players in a congestion game, and they are assumed to be non-cooperative and to choose routes selfishly. Shippers of a certain zone ﬁrst choose origins and destinations for their shipments, and then select several carriers as options for fulﬁlling the shippers requirement. Considering the number of shippers, this game could be regarded as an atomic game with a large number of players. Shippers may have the intention to avoid overloaded links, while carriers have the intention to avoid congested nodes. Shippers might consider the reliability and stability of networks by having more links as options in case that overpayment occurs. So far there is no detailed research on this in the shipping industry. However, similar ideas have been observed in telecommunications including wireless networks (see, e.g., [12]). Considering appropriate customer relationship management, as network providers, certain carriers give diﬀerent weights especially to very important shippers so that they have priority to access the required links. This could also be reﬂected in the graph of the shipping networks. 4.3

Learning Processes

We do not claim that there is no learning process in horizontal relations and principal-agent relations; see [9] on how partners in alliances turn to trust each other and/or control the cooperation based on expectations. Rather we include the learning process here mainly because vertical relations are based on heterogeneous players, and it might be harder to learn and control the expected cooperation among heterogeneous players. For instance, service providers forecast the customers’ requirements and never know exactly what customers choose. Therefore, the learning mechanism involved due to business experience, information sharing and exchanging could not reach pure transparency in shipping business reality. Obviously, a liner may learn the performance of a port operator so that he can choose whether to visit. Concerning information sharing among players

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Fig. 2. Two layers of the shipping service network

in shipping, it is not surprising that information is asymmetric. For instance, the owner of the slots gets better access to the information of the costs of slots while the charterer gets less relevant information. The eﬀective transparency of the information sharing among owners and charterers might improve the eﬃciency of the whole agreements. However, with regard to the published vessel-sharing, slot-exchanging and slot-chartering agreements (see, e.g., [15]), the amount and price of the slots are by no means transparent since they are regarded and protected as business conﬁdentiality. Generally speaking, the liner shipping service cooperation among heterogeneous players can be regarded as stochastic resource allocation problems, usually characterized by “curses of dimensionality.” 4.4

Network Dynamics

The node transit capacity and the link weight could change over time, resulting in network dynamics. Shippers own links in a network and sell transport service; at the same time shippers aim to minimize the prices they pay with respect to linkages between origins and destinations taking into account network dynamics. Once the capacity is interpreted as bandwidth of edges, some research done in the telecommunications industry [8] can also be applied to the shipping industry. The shipping service network is quite dynamic. From the shippers’ standpoint, they are atomic, selﬁsh, and non-cooperative, though sometimes shipment consolidations occur. From the standpoint of carriers, they are monopolistic, almost rational, attempt to have more coalitions, which recently resulted in liner shipping strategic alliances and continuously change the capacities of diﬀerent links. From the standpoint of port operators, they are aggressive, struggling for leadership within nearby zones, which push them towards increasing handling speeds and attract more links to connect them. Furthermore, in case hinterland distribution is included, the schedules of either carriers or shippers turn out to be more or less unreliable [72]. All the above mentioned intentions cause dynamics in the shipping network. The shipping network distinguishes two main layers: the links layer and the nodes layer, as shown in Figure 2. Competition and cooperation games among the liners can be described in the links layer as input and a collection of liners’ actions are represented as port-of-call and shipment

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load of each link. This collection of liners’ actions is actually the input of the nodes layer. In addition, given extra investments and encouraging policies, some ports may have extension opportunities. Then, in the base layer of Figure 2, the leader-follower situation of the zones in which promising ports are located, might change. There is valuable research from the perspective of network users. [4] studies reliability of a transport network by constructing a non-cooperative game with two players including a network user and a network provider. The author not only considers link failure of a network, but also the scenario that a connected network fails to provide an adequate level of service. The latter aspect is often faced by ocean sea network service providers, especially when the market is booming and there is not enough capacity available immediately. [5] study the route-choice behavior performed by risk-averse network users by formulating a non-cooperative game. These network users can be transformed to shippers as they can be regarded as ocean sea network users. Such kind of risk-averse users tend to take into account the route costs and their consequent uncertainty on the cost when making decisions on route choice.

5

Mechanism Design

In this section, we start to observe what an industry regulator should do to be better oﬀ compared to anarchy. Mechanism design is one of the branches of GT, where protocols are designed for players by regulators. Generally speaking, mechanism design can be viewed as the ‘inverse problem’ of games, where the input is a game’s outcome and the output is a designed game guaranteeing the desired outcome [44]. Regulators in the shipping industry observe the market turbulence and involved players’ behaviors from a GT perspective, and later ﬁgure out better mechanisms to regulate and motivate shippers, carriers, port operators, etc. Similar ideas have recently been implemented as road pricing policies from the viewpoint of a regulator; see, e.g., [75,77,71]. Ships in our settings usually cause emissions. In this respect a market regulator, e.g. the transport commission of the European Union, regulates the member states in terms of emission quota allocation [20,22]. Once we realize values of cost, proﬁt, emission, slot, and the load share of networks of the same nature, the emission quota allocation problem is actually similar to the cost sharing problem. Regarding proﬁts and the resources sharing problem we refer to [53], where a similar idea could be applied in emission quota allocation, too. The vital problem of a Principal-Agent incentive game is how the principal could motivate agents to perform as eﬀectively as possible [2] by taking advantage of more information about the agents actual eﬃciencies. In the shipping industry, a shipping carrier as principal has more information about its own total costs to control the marketing but less information on how eﬀective its related forwarder agents could be. The carrier authorizes forwarder agents to grasp freights on behalf of the principal. As a dominant principal, the carrier can design mechanisms in such manners, e.g., setting bonus of good performance in monetary terms, setting penalty of laziness or ineﬀectiveness to either motivate or control its agents. Thus, the key points are price setting, contract design

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based on interdependent players, etc., where the Stackelberg model [55] and a worst-case NE together with the Price of Optimum [32] are vital. Another application of mechanism design is to harmonize partnerships between public and private sectors of transport projects. One may refer to [45].

6

Algorithmic Game

Algorithmic GT is a promising subﬁeld and experimental method to ﬁgure out or simulate the players’ behaviors, and later the optimized collective outcome. A NE can be interpreted as the best response of each player so that no other improvement can be better oﬀ. While attractive, numerous criticisms of the NE exist. Two particularly vexing problems are the non-existence of an equilibrium and the multiplicity of equilibria. Therefore, what is the complexity for computing and searching the NE? This is a relatively new subﬁeld which captured research interest over the last couple of years. For the computation of network complexity, we refer to [44] as well as, e.g., contributions in [36]. Multi-agent based simulation for the evaluation of container terminal management operations is considered in studies summarized in [27]. The approach aims at planning and coordinating the processes within the terminal by mapping the terminal’s objects and resources. The agents strive to complete their speciﬁed goal by searching, coordinating, communicating, and negotiating with other agents by means of a market based mechanism such as a series of auctions; see also [67]. In [26], experiments applying multi-agent systems for investigating the impact of diﬀerent policies for sequencing, berthing, and stacking on the performance of container terminals are proposed. Numerical experiments based upon real data are conducted to evaluate eight transshipment policies. Shorter vessel turnaround times can be achieved with good decisions on yard stacking (e.g., using the stacking-by-destination policy) and berth allocation. [18] propose an integrative cost estimation concept and a multi-agent system approach for managing container terminals. The holistic objective is the minimization of the average terminal-eﬀected costs of container handling. The paper presents diﬀerent market mechanisms for resource allocation by coordinating the market with bilateral polypolies. Techniques proposed for the container barge handling in the port of Rotterdam by [35,11] may be seen in the spirit of GT approaches, too.

7

Summary and Conclusion

In this paper, we discussed from various perspectives how to observe the shipping industry by means of game theoretical applications. We categorized the link game, the node game and the network game. In addition, we integrated shipping carriers, port as well as consolidation/distribution operators in terms of interdependent network games. Concerning the nature of the shipping industry, horizontal relations among homogeneous players and vertical relations among heterogeneous players were discussed, too. From the supply perspective, once existing carriers make decisions based on cooperative games, overcapacity would not happen so

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often or severe. In that respect, ideas discussed in this paper propose food for thought for developing liner shipping service in a sustainable manner. We may further observe transport carriers of other modes than shipping, together with more consolidation/distribution options within associated hinterland instead of only ports at sea. The discussion of this paper can be extended to other means of transport as long as they share the same natures of networks. To conclude, game theoretical concepts can help better understand the liner shipping industry and equally support them in decision making.

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Scheduling Yard Cranes Considering Crane Interference Ulf Speer1, Gerlinde John2, and Kathrin Fischer3 1

Hamburger Hafen und Logistik AG Bei St. Annen 1 20457 Hamburg [email protected] 2 HHLA Container-Terminal Altenwerder GmbH Bei St. Annen 1 20457 Hamburg [email protected] 3 Institut für Quantitative Unternehmensforschung und Wirtschaftsinformatik Technische Universität Hamburg-Harburg Schwarzenbergstrasse 95 D 21073 Hamburg [email protected]

Abstract. Automated stacking cranes form the heart of modern container terminals. Hence, their productivity has a major influence on the performance of the terminal. In the first part of this paper, the yard crane scheduling problem and its practical relevance from the point of view of the Container Terminal Altenwerder (CTA) in Hamburg, Germany, is described. In Altenwerder, 26 yard blocks orthogonal to the quay with transfer areas at both ends of each block are operated with double rail mounted gantries (DRMG). In the second part of the paper, an outline of a new scheduling algorithm for yard cranes on this particular layout is given. The procedure minimizes delays for the jobs and the cycle times of the cranes. In addition to in-motion times also other parts of the cycle time, as waiting and blocking times resulting from other cranes, are taken into account in the scheduling approach. A branch and bound algorithm is used to create sequences of jobs for each crane. Using a simulation model, both the influence of the length of these sequences and the impact of technical breakdowns on the results are analysed. Finally, the results are verified with operational data and the applicability for practice at the CTA is evaluated.

1 Introduction Container terminals, as one of the main links in intercontinental supply chains, are highly affected in periods of strongly growing container flows. Such periods have occurred repeatedly in the past, and this trend is also currently apparent in container shipping. It is influenced not only by global economic aspects, but also by the increase of deep-sea vessel sizes. This leads to an increasing volume of containers that has to be handled by the terminal in a limited time period. Guaranteed service times and vessel handling rates are required from the terminal operator. This results in J.W. Böse et al. (Eds.): ICCL 2011, LNCS 6971, pp. 321–340, 2011. © Springer-Verlag Berlin Heidelberg 2011

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an increasing demand for higher berth productivity and leads to higher peak volumes in the storage area of the terminal. Thus, the container yard and the yard handling equipment most likely become the bottleneck of a terminal. Automated yard cranes are the way to go for more productivity at a higher density and lower costs [15], as they do have a strong impact on overall terminal performance. Furthermore, an effective yard crane scheduling with reduced crane interference and less waiting times can lead to a higher service level as well as lower energy consumption, which means that economical and environmental objectives are not in conflict. Hence, this development is also a valuable contribution with respect to sustainability. Therefore, a yard scheduling problem is studied in this work, i.e., the sequencing of jobs and their assignment to the yard cranes with the objective of minimizing service times and crane cycle times. A new scheduling algorithm which considers crane interference is compared to other approaches by means of simulation. Moreover, the results are compared to operational data from the CTA and the impact of the new approach for practice is estimated by the evaluation of selected key figures of performance.

2 Yard Crane Scheduling at Container Terminal Altenwerder Description of the Terminal The yard crane scheduling problem and its relevance for terminal performance will be described by means of the process flow of the CTA in Hamburg. CTA is a state-ofthe-art highly automated container terminal (see Figure 1) with • a quay length of 1,400 meters equipped with 14 semi-automated double trolley quay cranes and 1 smaller feeder crane • 84 automated guided vehicles (AGV) for the horizontal transport between the quay and the yard • 26 yard blocks orthogonal to the quay with transfer positions at both ends, at the waterside for the AGVs and at the landside for internal and external trucks • 52 automated stacking cranes (rail mounted gantry cranes, RMGs), 2 per yard block • semi-automated truck handling facilities and • a rail facility with seven tracks, four semi-automated manned gantry cranes and horizontal transport by manned terminal trucks and chassis between yard and rail head. CTA is a greenfield terminal, which started operations in June 2002. The terminal is designed to handle a total throughput of about 3.0 Mio. TEU per year. The development of CTA, selected logistical optimisations and the experience gained from the extensive automatic cargo-handling processes during eight years of operation are described by John and Witte [7].

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Fig. 1. Terminal layout CTA (HHLA, 2011)

The trade mark of CTA is to reach this throughput on a very tight layout, using a highly sophisticated equipment and control system of the yard. All yard blocks virtually are of identical layout. In each block, 10 rows contain 37 ground slots arranged longitudinally. Within the blocks, containers are stacked up to four high, in the outer lanes up to five high. Each yard block is equipped with one pair of automated RMGs. Each pair is designed in such a way that the large outer crane can pass the smaller, inner crane at any time (see Figure 2).

Fig. 2. DRMG cranes (CTA, 2011)

In this way, the entire block can be serviced by both cranes simultaneously and continuously and all slots are accessible at any time. This arrangement enhances availability of the entire facility, since if one crane fails, service of the affected block is still possible, even if at reduced handling rates.

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The Process Flow Deep-sea vessels are handled at CTA by the double trolley quay cranes. A crane driver operates the semi-automatic main trolley. The main trolley transports the container between the vessel and a so-called lashing platform on the quay crane. The second trolley operates fully automated between the lashing platform and the AGV (see Figure 3). The quay cranes are designed for a technical capacity of 60 moves per hour.

Fig. 3. Process flow at CTA (HHLA, 2011)

In the quayside transfer area, there are four transfer lanes for the handover between quay crane and AGV. The AGVs move the containers to and from the stack. They are suitable for twin-carrying (two 20-foot boxes at the same time). The AGVs always take the shortest route to the stack, which leads to a high AGV performance. At the stack, a yard crane takes the container from the AGV and moves it to the location indicated by the terminal system (and vice versa). Typically, the second crane works on the landside, because two RMGs cannot work in the same transfer area of a block simultaneously. In an ideal process, stacking becomes a continuous process in which both the AGVs on the quayside and the vehicles on the landside do not have to wait. On the landside, trucks directly drive to the transfer lanes. The yard cranes take care of the loading and discharging of containers to and from the trucks. Rail containers are moved between the stack and the rail terminal on CTA owned chassis pulled by internal trucks. For outbound containers, additional reshuffles can occur for the crane, when the outbound container is not accessible for the crane because one or more containers are stacked on top of it and have to be removed to another position in the block prior to the outbound move. Reshuffles mostly occur with respect to the pick-up of import containers. Due to unknown destinations at the time of stacking, inefficient yard locations are assigned and this leads to many reshuffling moves when delivering these containers (see, e.g., [15], p.202). Another kind of moves within the block are housekeeping moves. They are done for optimisation purposes, usually during times of low workload [8].

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Loading and unloading of AGVs, as well as stacking, reshuffling and housekeeping in the yard and the transfer to and from terminal chassis are handled by the RMG fully automated. Manual interference by a remote crane driver is required for the handling of boxes to/from external trucks for safety reasons. Up to ten remote crane drivers are available per shift. CTA is a typical import-export terminal ([15], p.11) with a transhipment percentage of about 25%. Around 30% of the total volume is delivered by rail, 40% by trucks (source CTA). Thus, a high number of boxes will have to pass the whole length of a yard block between waterside and landside transfer point during their dwell time. This leads to long travelling distances for the RMGs and to crane interference which increases the crane cycle times significantly. Description of the Crane Scheduling Problem The crane scheduling problem is defined in this work as follows: A set of available cranes C which work in a yard block and a set of jobs J for this block, i.e., for stacking of inbound containers, delivery of outbound containers and additional reshuffles, are given. The problem is to find an assignment of the jobs to the cranes and a sequence of the jobs for each crane that is feasible according to predecessor relations (e.g., in case of reshuffles), such that the service times for the vehicles at the transfer lanes and the travel times of the stacking cranes are as small as possible. These two objectives are also suggested by Steenken et al. [19]. This leads to a multicriteria decision making problem. The resulting problem is solved by the weighted sum model ([3], pp.28-30), as is explained in more detail below. Each job consists of an empty drive, i.e., the unloaded crane drives to the starting position of the container, for example an inbound container standing on an AGV. The empty drive ends with the pick of the container and is followed by a loaded drive, for example to a stacking position in the yard block. The loaded drive ends with the setdown of the container. For the control of the service times, a so-called ‘due date’ is determined for each job and a job is called ‘late’ when the set down of the container extends this due date. For the terminal in Altenwerder, C typically contains two cranes, but the scheduling also has to work if there is only one crane available due to maintenance or a breakdown of the second crane. Housekeeping moves are not considered for the scheduling because they are usually only performed in times of low workload, especially when no external jobs are available and thus minimization of service times is not an issue.

3 Yard Crane Scheduling Algorithms Literature Review Optimisation at container terminals is an important area of research and numerous articles on different aspects have been published. Steenken et al. [19] give an extensive overview and classification based on the ship planning process, storage and stacking logistics, transport optimisation, to which also the crane scheduling belongs, and simulation systems. An update is provided by Stahlbock and Voß [18]. In the

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crane scheduling part of their literature overview, articles for rubber tired gantries (RTG), single RMG, DRMG and twin RMG (two cranes on the same rail) are provided. Apart from the RTG, these variants are also considered by Saanen et al. [16], who compare the three crane systems applying different scheduling heuristics. Crane productivities are determined, which are measured by simulation on an isolated stack module. Hartmann [5] describes an algorithm for generating scenarios for simulation and optimization of container terminals, considering different parameters for the distribution of delivery dates and container attributes, which allows generating scenarios of practical relevance. Saanen [15] provides comprehensive recommendations for the design of automated container terminals, also with respect to the modelling of a crane system. Also a visual representation of a typical movement pattern of stacking cranes is shown (p. 209), which comes close to the crane movement used at CTA. Dorndorf and Schneider [2] extend the list of examined crane systems by a scheduling approach for triple stacking cranes, where job assignment is combined with a crane routing algorithm. They compare crane productivities measured during the simulation of an isolated yard block. Stahlbock and Voss [17] provide a simulation study based on operational data from CTA. They compare priority rulebased methods and simulated-annealing. They develop one of the few approaches where crane interference is considered. Vis and Carlo [20] present a scheduling approach for two passing cranes (DRMG) using simulated-annealing. They also develop a lower bound for the total duration of the scheduled jobs, assuming that the sequence of jobs has no influence on their duration and crane interference is neglected. Park et al. [12] compare a heuristic-based and a local-search-based scheduling approach for twin RMGs on several yard blocks. They evaluate the potential of the cooperation of both cranes with regard to reshuffles. Dell and Royset [1] develop a procedure in which the block is sub-divided into two areas. One crane is responsible for the waterside and the other crane for the landside of the block. All contributions on crane scheduling mentioned so far focus on a layout with stacking blocks orthogonal to the quay and transfer positions at both ends of the block, as considered in this paper. In contrast, for transhipment terminals, a layout with yard blocks parallel to the quay is commonly used. These blocks can be operated by RMGs or RTGs which can also work in adjacent blocks. Commonly, groups of containers with the same destination are stacked in certain sections of the block. Various research was published on the scheduling of these crane systems. As an example, the work of Petering et al. [14] should be mentioned, where different control strategies for yard cranes are compared by means of simulation and the impact on the quay crane performance is analysed. Narasimhan and Palekar [11] also focus on the routing of cranes inside a block optimising the loading process of the ship. Mak and Sun [10] evaluate genetic algorithms and tabu search for the crane scheduling and explicitly include crane interference in their model. Crane interference for several multiple yard cranes on the same rail is also considered by Javanshir and Seyedalizadeh [6], who present a mixed integer programming model for the yard crane scheduling problem.

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In addition to layout aspects, Petering and Murty [13] also investigate the problem of yard crane deployment, which results from the ability of RTGs to work on different yard blocks. This problem is also evaluated in earlier articles, e.g., by Linn and Zhang [9], but it is not in the scope of this work. A more general approach has been provided by Hartmann [4], who presents a framework for the scheduling of different equipment types on a container terminal as well as the scheduling of human resources. Priority rule-based heuristics and a genetic algorithm are used within this framework for the scheduling of straddle carriers, AGVs, yard cranes and reefer workers. Outline of Algorithms In this section, three scheduling algorithms for the yard crane scheduling problem are described. The scheduling routine is activated by the following events: • Publishing of a new job or update of an existing job • End of the empty drive of a job • End of a job The scheduling at the end of an empty drive is necessary because, in case of a reshuffle, the other crane can start the subsequent job (next reshuffle or outbound move) from the same stack. The simultaneous execution of two jobs at the same stack is not permitted with the implemented scheduling algorithms. FIFO: First In First Out In this approach, jobs are assigned to the cranes in the same order they are published to the block. In our model, for inbound moves this is done at the moment when the horizontal transport arrives at the block. Outbound moves are generated in advance, because possible reshuffles have to be done prior to the outbound move. The required reshuffle moves are generated simultaneously with the announcement of the outbound move. MU2J: Most Urgent Two Jobs In this approach, the most urgent two jobs are taken into account for the scheduling. Each of these jobs is assigned to each of the cranes and durations and end times for the jobs are determined without considering crane interference or waiting times. Based on the end times, costs for the solutions are calculated as the weighted sum of the duration of the empty drive and the lateness relative to the due date. The assignment with the lowest cost is selected. BBCI: Branch and Bound with Crane Interference For this approach, a set of n jobs is considered for the scheduling, initially the n most urgent jobs in relation to their due dates. Based on this set of jobs, a Branch and Bound algorithm is used, which creates job sequences for each of the cranes by iteratively assigning a job to the next available crane.

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Objective function: As the objective function for the Branch and Bound procedure, the weighted sum of three components is used. These components measure the following properties of a solution: 1. Lateness: the lateness relative to the due date, summed up for all jobs 2. Total duration: the difference between the time of the scheduling and the estimated end of the last job 3. Cycle times: the sum of the durations for each job This agrees with the two objectives suggested by Steenken et al. (see above and [19]), as components 1 and 2 ensure low service times. Instead of the travel times of the cranes, in this approach the cycle times (part 3) are considered which means that the waiting times are included. Adequate weights for the three objectives have been determined with respect to different key figures by extensive sensitivity analysis; however, the results have not yet been published and this aspect is not in the focus of this study. The solution of the resulting single-objective problem leads to a non-dominated solution of the multiobjective problem [3]. Branching: The Branch and Bound algorithm operates on a set of partial solutions. Such a partial solution (node of the search tree) consists of a job sequence for each crane and a set F of jobs which have not yet been assigned to a crane. F is named ‘free jobs’. Initially, F contains the n most urgent jobs (root of the search tree). For each branching step the earliest available crane of a partial solution is considered. Each of the free jobs is assigned to this crane, resulting in |F| new partial solutions. Another partial solution is developed by the assumption that no further job will be assigned to the earliest available crane. To avoid crane waiting times at the transfer areas, for the assignment at the first position of the scheduling sequence only jobs can be chosen for which the horizontal transport (truck, rail chassis or AGV) has already arrived at the transfer position. A complete solution (leaf of the tree) is found, when the set of free jobs is empty, i.e., each job is assigned to a crane at a specific position in the sequence. Hence, the leafs of a partial solution are all solutions whose job sequences start with the same job assignments as in the partial solution. Figure 4 illustrates the development of a new partial solution. Each rectangle represents a job and its length d displays its duration. Cranes with assigned jobs

Set of free jobs

j1 j2

Crane 2 j3

Crane 1 d

t

Fig. 4. Development of a new partial solution

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Unlike in most approaches found in the literature, the duration d of a job is not assumed to be constant. Instead, in this approach the duration of an assigned job is calculated subject to the following aspects: • Technical parameters of crane, trolley, lift and spreader (e.g., speed, acceleration and retardation) • Additional driving times resulting from interference with other cranes (particularly in the transfer areas of the block, but also when jobs with intersecting routes are executed simultaneously) • Blocked time: time when a crane is standing away from its target and waiting for the other crane to complete its job or to give way Of course these kinds of interferences can occur both during the empty and during the loaded drive of the crane. As analysis has shown, the duration of a single job can grow up to a multiple of the duration which would result without crane interference. This can occur, for example, when one crane is working at a certain position of a yard block for a while (e.g., while executing several reshuffles) and the other crane has to wait because it has been assigned a job in the same area of the block. Bounding: Taking into account these occasionally long job durations, the calculation of upper bounds for the duration of the free jobs and hence for the objective function value of a partial solution is difficult because no sufficient upper bound for the duration of the free jobs could be found. Therefore, a starting solution is created by a heuristic approach which always assigns the most urgent job to the next available crane. The cost of this solution is used as an upper bound. Of course, the upper bound is adjusted any time a better solution is found during the Branch and Bound process. For each partial solution created during the Branch and Bound procedure, a lower bound for the objective function is calculated. Therefore, for each assignment of a job to a crane, times for empty and loaded drive and for any kind of interference and the resulting costs are exactly determined. For the free jobs, only a lower bound for the job duration can be calculated because, for example, the duration of the empty drive depends on the predecessor in the sequence and can not be calculated until the job is assigned to a crane. The same applies to interference times during empty or loaded drives. Partial solutions with lower interferences, resulting in lower durations, are preferred with respect to the objective function, while partial solutions with higher crane interference are cut off during the Branch and Bound process. The solution with minimum cost is used to select the next job to start. But in special cases, the next job can not be started immediately after the previous one. This can occur due to predecessor relations of jobs, arising from reshuffles. For example, let job r1 be the reshuffle for an outbound move j1, because the container for r1 stands on top of the container for j1. Assuming that crane 1 has already started the empty drive for move r1, the other crane can not start j1 until the empty drive for r1 is finished, because two cranes can not operate at the same stack simultaneously. This leads to unproductive crane times, illustrated in Figure 5.

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Non-productive time Crane 2 Crane 1

j1 r1

t End of empty drive r1 Fig. 5. Non-productive crane times caused by reshuffles

To avoid such solutions, these unproductive crane times are considered in the calculation of job durations within the Branch and Bound process. It has turned out during the evaluation, that the second part of the objective function, i.e., the total duration, is essential. Especially solutions where all jobs are assigned to one crane while the other crane is idle, are avoided by this objective. These solutions particularly occur when there are no tight due dates. Cycle times are often low, when one crane stands at a parking position causing no interference. Of course, such solutions are not preferable, because service times should be as low as possible and additional jobs can arise at any time.

4 Simulation Model Description and Assumptions In this section, the simulation model developed with Plant Simulation and its assumptions are described. To simplify the model and to allow for a sufficient number of simulation runs, taking into account computing times, only one yard block is simulated. This approach is also found in Saanen et al. [16]. Due to the fact that cranes are scheduled for each block separately, results can be easily transferred to multiple yard blocks. The scheduling algorithms are implemented within the simulation model. Hence no – possibly asynchronous – communication via a network connection is necessary, and simulation results become deterministic. However, a lower efficiency of the programming language is accepted because some data structures (e.g., heaps) are not supported in SimTalk. Layout of the Modelled Yard Block As in Altenwerder, the simulated block is equipped with two simulated gantry cranes, one small gantry and one large gantry, which can cross the small crane. Each crane is equipped with a spreader with adjustable size. Technical parameters like dimension, maximum speed depending of loading condition, acceleration, deceleration, resizing time for the spreader, times for twist lock movement and positioning of containers are adjusted close to reality. The horizontal transport surrounding the DRMG is represented in the simulation model by vehicles moving on a detailed layout, but caused by the limitation to only one yard block and the resulting low traffic density, particularly at the quayside, the simulated AGVs have lower driving times. However, the transfer area on both sides of the block, which has main influence on the crane behaviour and their time consumption, is modelled in detail. Figure 6 shows the yard block layout from the top view.

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Landside

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Quayside

3 lanes for rail chassis 4 lanes for external trucks

4 AGV lanes

Fig. 6. Layout of a yard block (CTA, 2011)

Announcement and Due Dates for Different Move Types The definition of due dates depends on the underlying process, e.g., loading of a ship or rail delivery. Usually jobs for the quayside have the highest priority. For loading containers, the due date is determined by the estimated time of loading at the quay crane. The estimated driving time of the AGV and an additional buffer time is subtracted. If reshuffles are necessary, their due date is set about 30 seconds before the successor, e.g., if three containers stand above the outbound container, the due date for the reshuffle for the topmost container is set 90 seconds before the outbound due date. Of course, the reshuffle moves are handled as independent moves and can be executed by both cranes, as also suggested by Park et al. [12] and practiced at CTA since its start of operation. AGVs with discharged containers are to be unloaded within 10 minutes after arrival at the block. The storage of the container is not urgent, but long unproductive AGV times and occupation of the transfer lanes must be avoided. Hinterland moves have lower priorities. For the service of trucks, a time frame of 15 minutes is used. For rail containers handled on chassis, a time frame of 30 minutes is defined. Required reshuffles have to be done in advance as on the quayside. In general, a crane will not start a job until the corresponding vehicle has arrived. This is done to avoid waiting times at the transfer area and is designed in contrast to the situation at CTA due to the focus on better crane performance in peak situations. It should be noted that sufficient time frames for the jobs are an important requirement for acceptable scheduling results. The algorithm needs degrees of freedom for a flexible scheduling, which is also mentioned by Hartmann [3]. This becomes impossible, if the sequence is mostly determined by very strict due dates. Scenarios Two scenarios are used for the following investigation: One scenario with an average workload and one scenario simulating a long lasting peak situation. In the standard scenario, the simulated time covers a period of four weeks of terminal operation, which corresponds with approximately 7,000 inbound and outbound moves and 2,500 additional reshuffles. As this study focuses on mean cycle times calculated based on a large number of jobs, it was refrained from doing several repetitions of the simulation runs to reach statistical confidence. Moreover, the results are compared with operational data to be able to judge their quality. The container movements are extracted from a characteristic period of real operations and are used in the simulation model with the same chronology. The simulation starts

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with an empty stack and the stack allocation is updated with every container movement. The first two weeks are used as a stabilisation phase. Thereafter an average utilisation and state of the block is reached and data for crane operations is collected which can be used for the statistical evaluation presented in the next section. In the standard scenario, extensive idle times for the cranes occur during the nights or in times of low workload for a block. Hence, in many situations crane scheduling is not very difficult, particularly when only one job is executable and therefore only one crane is working, while the other is idle and sent to a parking position. In contrast to the standard scenario, in the peak scenario the simulation moves are generated consecutively and without idle times in between, resulting in a continuously high workload for the cranes. Of course, the jobs for the crane system are limited by the number of transfer positions of each block plus additional reshuffles. The peak scenario is an important instrument for simulation purposes, because it allows detailed analysis of a situation that also occurs in real operations very often for some hours. Especially in these situations, an efficient crane operation is important to ensure sufficiently low service times for quay operation and landside traffic. Of course, in these peak situations the differences between scheduling algorithms become most apparent.

5 Simulation Experiments In this section, the simulation results of the described algorithms and scenarios are presented. First of all, the number of jobs to be considered for the BBCI approach is evaluated, which is necessary to reach an appropriate quality of solutions. Afterwards, the influence of technical breakdowns on the algorithms is analysed. The resulting cycle times are compared with operational data from the CTA and the frequency of breakdowns is adjusted until the cycle times in simulation match with the operational cycle times. Based on this comparison, the impact of an improved scheduling on operational cycle times is estimated and the possibility to use the BBCI algorithm in practice is evaluated. To avoid the publishing of internal CTA data, most results are shown in relative values only. As a reference for the initial value (100%), if nothing else is mentioned, the MU2J algorithm is used which comes close to the approach actually used at CTA. Evaluation of the Necessary Sequence Length for BBCI In the first step, the necessary length of the generated job sequences for the BBCI approach is evaluated. Taking into account the number of transfer lanes at both ends of the block, the occurrence of reshuffles and additional quayside outbound moves, which are known before the AGV has reserved a transfer lane, the number of jobs to be handled by the scheduling algorithm can grow up to more than 50 jobs in the worst case. However, experience with simulation has shown, that in fact the number of jobs rarely exceeds 10 to 12 jobs in the investigated scenarios, because of comparatively long travel times of horizontal transport and low reshuffle rates at the quayside. Nevertheless, in particular cases, such as long-lasting crane breakdowns or highly allocated stacks, the number of jobs can grow and the scheduling has to determine a

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solution within a few seconds in these cases as well. Therefore, a subset of the scheduled jobs is generated and used for the Branch and Bound process to ensure reasonable computing times. Of course, the resulting solution is not optimal for the problem with the entire number of jobs. An obvious approach is to consider the most urgent jobs for the scheduling with Branch and Bound. Therefore the jobs are ordered by ascending due dates and the first n jobs are selected for scheduling. If less than two of these n jobs can be started immediately, e.g., according to reshuffles, which have to be done before, additional executable jobs are added if available, until there are at least two executable jobs, or more generally, until the number of executable jobs equals the number of cranes. This selection ensures that each crane can start with a job immediately. To measure the influence of the number of jobs n on the quality of the results, several simulation runs were carried out. Every dot in Figure 7 represents the result of a simulation run. On the x-axis the number of jobs considered for simulation is shown and on the y-axis the mean lateness relative to the due dates is measured. Two charts are plotted, one for the standard and one for the peak scenario. The presented values are normalised relative to the mean lateness in the standard scenario for the BBCI approach with 7 jobs. Mean Lateness 300% 250%

BBCI Peak

200% 150%

BBCI Standard

100% 50% 0% 0

2

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Number of Jobs Considered for Scheduling 8

Fig. 7. Influence of the number of scheduled jobs on the mean lateness

Obviously, for the standard scenario the number of scheduled jobs has only little influence on the lateness of jobs. This may be caused by the fact that in many situations only few jobs are executable or the algorithm selects one of the most urgent jobs in any case. This is in contrast to the peak scenario, where the quality of the solution (measured again by the mean lateness) is drastically reduced when less than four or five jobs are considered. Scheduling of more than the most urgent six jobs has no further positive impact on the results. To ensure reasonable computing times, for the following evaluations, the most urgent four jobs are selected for the scheduling with the BBCI approach.

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Influence of Technical Breakdowns In the next step, the influence of technical breakdowns on the simulation results is evaluated. Technical breakdowns occur because of many reasons, such as damage on crane or containers, drawdown or unevenness of the ground surface in the block, resulting in inclined stacks, failures with sensor systems or problems caused by software or staff. In operational practice, usually some minutes elapse until the failure is recognized by the control system and analysed by the operator. Depending on the severity, it can take between some seconds and several hours to solve the problem and to continue working with the affected crane. For the simulation, an average of 5 minutes for the total duration of a breakdown is assumed. At first sight, this seems to be an optimistic value, but in most cases, after this period an operational workaround is found, which may lead to manual intervention of a remote crane driver or the manual assignment of the respective job to another crane. In the maintenance of machines, this duration of a breakdown is constituted as ‘mean time to repair’ (MTTR). A second parameter which must be determined for the simulation of breakdowns is the ‘availability’, by which the downtime percentage in the long run is measured. The availability is difficult to measure in practice due to the various reasons and characteristics of breakdowns. Hence, simulation runs for different crane availabilities, resulting in different frequencies of breakdowns, were carried out to analyse their effects and to compare them with operational data by means of crane cycle times. In Figure 8, the influence of different crane availabilities on the crane cycle times in the standard scenario is shown. The cycle time is defined as the average time between the assignment of a job to a crane and its completion, including empty and loaded drive. For the calculation of the cycle times, inbound, outbound and reshuffle moves of both cranes in the DRMG block are considered. The cycle times are shown on the y-axis, measured relatively to a normalized value based on the MU2J approach with the standard scenario and 100% crane availability. The availability is scaled on the x-axis.

Mean Cycle Time 160% 150% 140% 130% 120% 110% 100% 90% 80%

FIFO MU2J BBCI

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Crane Availability [%]

Fig. 8. Influence of crane availability on the crane cycle times in the standard scenario

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As expected, crane cycle times seem to develop inversely proportional to the crane availability. The FIFO approach provides the longest cycle times. The MU2J procedure leads to lower cycle times, which was expected, because minimization of the empty drive times is part of the objective function. The lowest cycle times among the compared variants are provided by the BBCI. Surprisingly, the approach seems to work very robustly also with disturbances, which was not obvious, because the estimated crane positions and their interference used for the BBCI lose validity, when breakdowns occur and cranes do not reach their target in time. But apparently, preventing crane interference in the scheduling approach leads to lower interference also in the case with lower availability, possibly caused by preferring situations where cranes work far away from each other; therefore the non-disturbed crane can continue its work for a while without being affected by the breakdown of the other crane. Comparison with Operational Data To evaluate to what extent the results are transferable to the real world, the cycle times measured in the simulation are compared with the cycle times collected at CTA. Again it is distinguished between a standard scenario, for which the data was collected as an average of all CTA blocks over several consecutive days, and a peak scenario, consisting of a statistical evaluation for one DRMG block for several hours. Longer peaks are difficult to find because of operational workload variations. Hence, the statistical confidence is lower for the operational cycle times at peak situations. The operational scenario data from CTA is not identical to the simulation scenario, but the data have a similar structure and are comparable due to the large number of jobs, especially for the standard scenario. The crane availability in the simulation was adjusted until the cycle times for the MU2J procedure in the simulation best match with the operational data, both for the standard and the peak scenario. Figure 9 shows the different cycle times in detail. In addition to times for empty and loaded drive, also interference and blocking times with the other crane, waiting times for horizontal traffic and for remote crane drivers are pictured in a stacked bar chart. These times are defined as follows: • Move time: time when the crane is moving between the assignment of the job and the twist lock operation, including times for gantry, trolley and lift movement as well as times for adjustment and operation of the spreader • Interference time (determined only for simulation, as these times are not known for the real data): Part of the move time that extends the driving time which would be necessary for driving without any crane interference or breakdown • Blocked time: time when the crane is standing away from its target and waiting for the other crane to complete its job or to give way • Waiting time for HZT: time when the crane stands near the transfer area waiting for horizontal transport • Waiting time for remote crane driver: time when the crane stands near the transfer area waiting for a remote crane driver

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In Figure 9, the different parts of the cycle time are distinguished for empty and loaded drive. Waiting times for remote crane drivers are not mentioned, because they are too small to appear in the chart. Mean Crane Cycle Times 43%

Operation Standard

53%

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18%

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Empty Blocked Time

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Fig. 9. Comparison of crane cycle times from operation and simulation

It can be stated that not only the cycle times, but also the composition of the different parts of the cycle times in operation and simulation with MU2J are similar, for the standard as well as for the peak scenario. Blocking times are only a small part of the cycle time. The same applies for the waiting times for horizontal transport. In contrast to the real world, they do not occur in the simulation, because in the simulation inbound and outbound moves are executed only if the horizontal traffic has arrived. This design is chosen with regard to the focus of better crane performance in peak situations. Especially in these situations, the crane should not waste its time waiting for an AGV. If the operational waiting times for horizontal transport (HZT) are subtracted from the cycle times, the average length of a cycle nearly exactly matches with the simulation approach MU2J for the standard scenario. Obviously the crane processes in the simulation match the real processes very well. For the peak scenario minor differences can be observed for the duration of the empty drive between operation and the MU2J approach. This may be due to the small database of the peak scenario. Interference times represent an extensive part of the driving times. They can only be measured in the simulation, since operational reports for this item do not exist. But the sum of moving and interference time matches well with the operational moving times. Regarding the BBCI approach, it can be stated that most parts of the cycle time can be reduced compared to MU2J, leading to about 10% to 15% shorter cycle times. As estimated above, interference times are considerably reduced for empty and loaded drive. But also empty move times are significantly lower with BBCI. While with MU2J, cycle times grow in peak situations compared with the standard scenario, they can be reduced by BBCI. This is probably due to the fact that times of moving and

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interference can be reduced more successfully under high workload, when there are several jobs that can be started. This leads to more flexibility in sequence creation. On the other hand, especially in these peak situations an efficient crane scheduling is important and the BBCI approach shows promising results. This can be confirmed by comparing some additional key figures for the MU2J and the BBCI approach in the peak scenario (see Table 1). Table 1. Comparing BBCI and MU2J with the peak scenario according to different key figures Key figure

Mean lateness

Relative improvement of BBCI compared to MU2J

Number of combinations of Number of crane outbound with inbound interferences move -23% +63% -34%

As these key figures show, considering crane interference has some advantages in addition to the reduced cycle times: The BBCI approach also leads to 23% better compliance of due dates and 63% more combinations of outbound and inbound moves are reached, i.e., after an outbound move, the crane stays in the transfer area and continues with an inbound move. This effect is ascribed to the minimisation of the empty drives contained in the objective function. Furthermore, 34% less occurrences of crane interference were measured in the peak scenario simulation. Applicability of the Results for Practice The evaluation of the BBCI approach shows promising results, but for practical application of the approach at the CTA some additional aspects have to be considered. First of all, it must be assured that computation times do not exceed a few seconds. Therefore, the number of jobs which build the subset of jobs considered for scheduling (cf. Figure 7), is varied and the computation times were measured. Table 2 shows the mean and the maximum computation times and the 99-percentile for computing the best job sequence and hence for deciding on the next move of a crane on a common workstation with 3 GHz clock frequency. Table 2. BBCI Computation times for the peak scenario Number of scheduled jobs 2 3 4 5 6 7

Computation times [s] Mean value Percentile P99 Maximum value 0.27 1.25 4.66 0.45 1.96 6.78 0.77 3.59 14.77 1.44 6.87 33.33 2.78 13.68 72.20 4.67 30.04 113.52

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The mean calculation times for the scheduling of up to four or five jobs are less than 2 seconds, which is a very low value. Also the 99th percentile is not critical, but the maximum values considerably exceed the reasonable waiting times for a scheduling result. A thorough analysis revealed two main reasons: • The BBCI computation time for a partial solution is rather high because acceleration and retardation of different crane components are taken into account, and because of the detailed estimation of interferences. For each job up to fifty steps and more have to be calculated. • In special situations with non-critical due dates, the lower bound does not work properly and therefore a large part of the search tree has to be investigated by the Branch and Bound procedure, as illustrated in the following example. • Example: There are several inbound jobs from the landside with non-critical due dates. All these inbound containers have already reached the transfer position and all containers have a destination at the quayside of the block. Both cranes are available. As the due dates are not critical, most of the possible jobs sequences can be executed without any lateness. Due to the long travel distances of the cranes through the entire block, also long empty drives are necessary and heavy crane interference will occur. In such cases, the lower bound for the free jobs provides very poor results, because the major portion of a job duration can not be estimated before the job is assigned to a crane. Hence, the pruning takes effect only near the leaves and in particular cases more than 50% of the search tree is investigated. This explains the long calculation times because the number of possible solutions is (n+1)! for two cranes, which is, e.g., 5,040 for only 6 jobs. If this issue cannot be resolved, e.g., by improving the lower bound for the free jobs or the calculation times in general, one of the following alternatives can be taken into consideration: 1. The scheduling can be executed as a continuous process, which holds a scheduling solution ready at any time. Difficulties may occur with this approach, because job times must be continuously updated. 2. The scheduling process can be started prior to the end of the current job. When lowering of the spreader has begun before a pick-up or set-down, the remaining job time can be estimated with a certain accuracy. Typically the remaining period is between 10 and 20 seconds, but nevertheless disturbances can occur during that period, resulting in a bad solution. If the scheduling process is activated by the publishing of a new job while at least one crane is idle, no forward calculation is possible, because the new job is not known in advance. But in this case the number of jobs is usually very low. Alternatively or in addition to the lowering time of the spreader, the lift-up time can be used for computing a new scheduling solution. 3. The scheduling process may be terminated after some seconds or after a certain quality of the solution is found with regard to the current bound. If no solution exists, a heuristic solution can be considered. Obviously, the second approach seems to be the most promising and can be examined in future research. In addition, the functionality of the approach should be investigated for situations where only one crane is active in a block (of course, crane

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interferences will not occur in this situation, however, a high crane productivity is important) or where long waiting times for remote crane drivers occur, e.g., due to employees’ breaks or load peaks at the truck dispatching. Furthermore, a scenario for the entire terminal should be used to validate the results, which were presented here for only one yard block.

6 Summary and Future Prospects A new Branch and Bound procedure for scheduling yard cranes was developed, which focuses on the detailed estimation of crane driving times. It is one of the first approaches which considers crane interference and waiting times explicitly instead of assuming constant times for the processing of each job. For two different scenarios, the approach was compared with other heuristics. By varying crane availability, the crane cycle times were adjusted to operational data of the CTA. Based on the comparison of cycle times in simulation and operations, the impact of an improved scheduling on operational cycle times was estimated. It turned out that only a small number of jobs has to be considered for scheduling, leading to significant improvement of crane cycle times as well as better compliance of due dates for the jobs, especially in peak situations. The approach has shown to be quite robust against crane breakdowns and average computing times vary in the range of several seconds, which is both very important for the applicability in practice. As the approach is applicable also to other crane systems as twin or triple cranes, further work should focus on these evaluations. Regarding the increased probability of crane interference for triple cranes in particular, the potential is estimated to be higher than for the DRMG system. Furthermore, considering waiting times for remote crane drivers and the inclusion of housekeeping moves in the scheduling might be aspects of future research. Prior to an introduction at CTA, the long computing times occurring in particular situations must be resolved and results should be verified for further scenarios and for the entire terminal.

References 1. Dell, R.F., Royset, R.O., Zyngirdis, I.: Optimizing Container Movement using One and Two Automated Stacking Cranes. Journal of Industrial and Management Optimization 5, 285–302 (2009) 2. Dorndorf, U., Schneider, F.: Scheduling automated triple cross-over stacking cranes in a container yard. OR Spectrum 32(3), 593–615 (2010) 3. Eiselt, H.A., Sandblom, C.-L.: Decision Analysis, Location Models, and Scheduling Problems. Springer, Berlin (2004) 4. Hartmann, S.: A General Framework for Scheduling Equipment and Manpower at Container Terminals. OR Spectrum 26, 51–74 (2004) 5. Hartmann, S.: Generating Scenarios for Simulation and Optimization of Container Terminal Logistics. OR Spectrum 26, 171–192 (2004) 6. Javanshir, H., Seyedalizadeh Ganji, S.R.: Yard crane scheduling in port container terminals using genetic algorithm. Journal of Industrial Engineering International 6(11), 39–50 (2010)

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7. John, G., Witte, I.: Acht Jahre CTA – Erfahrungen mit einem automatisierten Containerterminal. In: Jahrbuch der Hafentechnischen Gesellschaft, Projektierung, Bau und Betrieb von Containerterminals, vol. 56, pp. 103–113 (2010) 8. Kemme, N.: Effects of storage block layout and automated yard crane systems on the performance of seaport container terminals. OR Spectrum (2011); doi: 10.1007/s00291011-0242-7 9. Linn, R.J., Zhang, C.Q.: A heuristic for dynamic yard crane deployment in a container terminal. IIE Transactions 35, 161–174 (2003) 10. Mak, K.L., Sun, D.: A Scheduling Method for Cranes in a Container Yard with Inter-Crane Interference. In: Electronic Engineering and Computing Technology. LNEE, vol. 60, pp. 715–725 (2010) 11. Narasimhan, A., Palekar, U.S.: Analysis and algorithms for the transtainer routing problem in container port operations. Transportation Science 36, 63–78 (2002) 12. Park, T., Choe, R., Ok, S.M., Ryu, K.R.: Real-time scheduling for twin RMGs in an automated container yard. OR Spectrum 32(3), 593–615 (2010) 13. Petering, M.E.H., Murty, K.G.: Effect of block length and yard crane deployment systems on overall performance at a seaport container transshipment terminal. Computers & Operations Research 36(5), 1711–1725 (2009) 14. Petering, M.E.H., Wu, Y., Li, W., Goh, M., de Souza, R.: Development and simulation analysis of real-time yard crane control systems for seaport container transshipment terminals. OR Spectrum 31, 801–835 (2009) 15. Saanen, Y.A.: An approach for designing robotized marine container terminals. Phd thesis, Technical University of Delft (2004) 16. Saanen, Y.A., Valkengoed, M.V., Kuhl, M.E., Steiger, N., Armstrong, F.B., Joines, J.A.: Comparison of three automated stacking alternatives by means of simulation. In: Proc. 2005 Winter Simulation Conf., pp. 1567–1576 (2005) 17. Stahlbock, R., Voß, S.: Efficiency considerations for sequencing and scheduling of doublerail-mounted gantry cranes at maritime container terminals. International Journal of Shipping and Transport Logistics 2(1), 95–123 (2010) 18. Stahlbock, R., Voß, S.: Operations research at container terminals – a literature update. OR Spectrum 30, 1–52 (2008) 19. Steenken, D., Voß, S., Stahlbock, R.: Container terminal operations and operations research – a classification and literature review. OR Spectrum 26, 3–49 (2004) 20. Vis, I.F.A., Carlo, H.J.: Sequencing Two Cooperating Automated Stacking Cranes in a Container Terminal. Transportation Science 44(2), 169–182 (2010)

Solving the Resource Allocation Problem in a Multimodal Container Terminal as a Network Flow Problem Elisabeth Zehendner, Nabil Absi, St´ephane Dauz`ere-P´er`es, and Dominique Feillet Ecole des Mines de Saint-Etienne, CMP, site Georges Charpak, 13541 Gardanne, France {zehendner,absi,dauzere-peres,feillet}@emse.fr

Abstract. Continuously increasing global container trade and pressure from a limited number of large shipping companies are enforcing the need for eﬃcient container terminals. By using internal material handling resources eﬃciently, transfer times and operating costs are reduced. We focus our study on container terminals using straddle carriers (SC) for transportation and storage operations. We assume that SCs are shared among maritime and inland transport modes (truck, train, barge). The problem is thus to decide how many resources to allocate to each transport mode in order to minimize vehicle (vessel, truck, train, barge) delays. We present a mixed integer linear programming model, based on a network ﬂow representation, to solve this allocation problem. The modular structure of the model enables us to represent diﬀerent container terminals, transport modes and service strategies. We present parts of our model and exemplary applications for a terminal at the “Grand Port Maritime de Marseille” in France. Keywords: Container terminal, resource allocation, intermodal transportation, mixed integer linear programming.

1

Introduction

In intermodal transportation, container terminals play the role of exchange hubs. They oﬀer transfer facilities to move containers from vessels to trucks, trains and barges and vice versa. Due to the pressure from a limited number of large shipping companies, terminals, especially geographical close ones, face strong competition. Their competitiveness is particularly marked by vessel turnaround times. But recently, the competitiveness of a container terminal is becoming more and more a function of its delays and costs for transport to and from the hinterland. One reason is that inland transportation costs account for a large part of total costs for container shipping (40% to 80% according to [6]). Other reasons are an increased interest in door-to-door services and environmental aspects (e. g., CO2 emissions). An eﬃcient allocation of internal material handling resources to vessels, trucks, trains and barges reduces the time these vehicles J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 341–353, 2011. c Springer-Verlag Berlin Heidelberg 2011

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have to spend at the terminal and renders the terminal more competitive. In this study, we present a mixed integer linear programming model to determine an optimal resource allocation minimizing vehicle (vessel, truck, train, barge) delays. Numerous studies deal with diﬀerent optimization problems arising at container terminals. Only a few deal with the resource allocation problem and they mostly neglect inland transport modes. [3] formulates the allocation problem of quay cranes, transport vehicles and yard cranes to vessels as a network design problem. The objective is to determine the cheapest allocation to serve all vessels. [1] presents a predictive control approach based on a queuing model to allocate available resources to vessels, truck and trains. The aim is to minimize the turnaround time of vehicles. [5] presents, among other, a Markovian decision model to determine optimal policies for ﬂeet management in real time, minimizing ﬂeet operating costs and vessel waiting costs. [7] determines the number of vehicles required to transport containers between the quay and the yard by a disjoint paths problem. Other studies aim at minimizing truck delays but consider scheduling rather than allocation problems. [4] develops a general model to assign jobs to container terminal resources and to temporally arrange jobs with precedence constraints and sequence-dependent setup times. [2] develops an assignment algorithm that dynamically matches SCs to waiting trucks. The aim is to minimize truck serving times and empty travels of SCs. In this paper, we propose a mixed integer linear program to allocate internal material handling resources (SCs) to diﬀerent vehicles arriving at a terminal. Our objective is to minimize vehicle delays with focus on landside transport modes (trucks, trains, barges). Section 2 presents the resource allocation problem at a terminal. In Section 3, a basic resource allocation problem is formulated as a network ﬂow model. Possible service strategies applied to serve diﬀerent vehicles are indicated in Section 4. The basic network ﬂow model may be adapted to these service strategies. We present an exemplary extension of the core model to represent a terminal at the “Grand Port Maritime de Marseille” in Section 5. In Section 6, we show and discuss experiments of the model to determine a resource allocation minimizing vehicle delays and to determine the number of resources needed to cope with the expected demand. Section 7 concludes the paper.

2

The Resource Allocation Problem

A container terminal has to provide transfer facilities for two types of container ﬂows: Import and export containers. Import containers arrive on vessels, are unloaded, stored at the yard, and ﬁnally transferred to landside gates where they are loaded on trucks, trains or barges. Export containers pass the terminal in the opposite direction. We concentrate our study on terminals using manned SCs for transportation (between the quay, the yard and landside transfer points) and storage operations. Figure 1 illustrates the layout of such a container terminal. It presents the diﬀerent areas of the terminal. Vessels and barges are loaded/unloaded at the quay by quay cranes. Trucks and trains are

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Fig. 1. Schematic view of a container terminal

loaded/unloaded in speciﬁc areas. Containers are temporarily stored at the yard. In most cases, import and export containers are stored separately in dedicated areas. SCs move containers within the terminal. Normally, dockers driving SCs are hired on short-term (e.g., the day before). This allows terminal operators to adapt their capacity from day to day via the number of hired dockers. For organizational reasons, manned SCs are assigned to one type of tasks (e.g., serve trucks or serve a vessel) for a certain period. The arising allocation problems are to determine the needed capacity for a given day and to allocate available SCs to diﬀerent vehicles arriving over the day. We focus our study on the landside part of the terminal. Our aim is to minimize the time the vehicles of the diﬀerent landside transport modes stay at the terminal. We assume that arrival times and due dates (i.e. departure times) for vessels are given (e.g., imposed by the carrier or determined by the terminal operator) and that each vessel has to be served within its time window. Hence, no delays are allowed for vessels. However, our model can be extended to also minimize potential delays of vessels.

3

Core Network Flow Model

We model the resource allocation problem as a network ﬂow problem, where containers to be moved are ﬂows in the network and arc capacities are limited by the number of allocated resources. Our model is inspired by the approach presented in [3]. Before stating the actual implementation as a mixed integer linear program, we explain the underlying idea with the help of Figure 2, which represents a terminal with two vehicles arriving over the day. One network ﬂow model is implemented for each vehicle. The round nodes stand for the discrete time periods of the working day (1-10). The rectangular nodes, which are sources of ﬂow, represent the arrival of vehicles with their associated demand for container

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Fig. 2. Resource allocation problem as a network ﬂow model

movement requests. Each source is connected to the period corresponding to the arrival of the vehicle. The square nodes represent sinks of ﬂow. The ﬂows from a period node to a sink represent the number of container movements executed per period and per vehicle. Capacities on these arcs represent the maximum number of container movements that may be executed per period and per vehicle. These capacities are proportional to the number of resources allocated to the vehicle in the corresponding period. The total number of allocated resources cannot exceed the total number of available resources at the terminal. The ﬂows between two periods represent unexecuted tasks which are transferred to the next period. These arcs exist only if it is permitted to delay tasks. Note that an independent submodel is implemented for each vehicle and that those submodels are only related by one constraint limiting the total number of allocated resources. This modular structure makes it possible to include vehicle speciﬁc constraints to each submodel. We now state the mixed integer linear program formulating these network ﬂows. For our resource allocation problem, the situation at a container terminal may be represented by the expected workload and the terminal’s capacity. The expected workload is determined by the number of vehicles with their arrival times, their due dates and their number of required container movement requests. The terminal’s capacity is given by the number of available SCs and the average number of containers a SC can handle per period. We assume that resources may be reallocated only at discrete points in time and that all tasks have to be executed at the end of the time horizon (e.g., end of the working day). Our objective is to provide the terminal operator with a tool to estimate the number of SCs needed to handle with the expected workload and to propose a possible allocation of these resources to diﬀerent vehicles. The detailed scheduling and routing of SCs is not addressed in this study. We assume that the Terminal Operating System uses the rough allocation of our model to determine

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a detailed schedule. Therefore, we use average values on the number of tasks a resource can handle per shift rather than real travel times. The model is used for the resource allocation problem for the following day. Therefore, information about vehicles arrival and departure times and the number of containers to be moved are quite reliable. This allows considering information related to vehicles as deterministic. Major disruptions, however, can be handled by recomputing the model since this takes only seconds. In addition, we took into account that a container terminal is a highly dynamic and uncertain work environment. We analyzed the quality of the resource allocation obtained by the optimization model in a stochastic environment via a discrete event simulation. Results conﬁrmed that the proposed resource allocation performs well. Details on the simulation model and the validation process are presented in [8]. We use the following sets, parameters and variables to describe the expected workload, the terminal’s capacity and the container ﬂows in the network. Sets and parameters: T Number of time periods describing the time horizon M Number of transport modes being served at the terminal Number of vehicles of transport mode m arriving during the time Im horizon T Set of all time periods, T = {1, . . . , T } M Set of all transport modes, M = {1, . . . , M } Set of all vehicles of transport mode m, I m = {1, . . . , I m } Im st Number of available resources in period t d˜m Period t in which vehicle i of transport mode m has to be ready for i departure Period t in which vehicle i of transport mode m arrives at the terminal rim pm Total number of tasks to be carried out for vehicle i of transport i mode m Average number of tasks a SC serving transport mode m can handle hm per period (hm ≥ 1) Variables: m Number of resources allocated to vehicle i of transport mode m in Xi,t period t m Number of tasks executed in period t for vehicle i of transport mode Wi,t m, depending on the number of allocated resources m Number of non-executed tasks in period t for vehicle i of transport Zi,t mode m which are transferred to period t + 1 The constraints below formulate the basic network ﬂow model for all vehicles arriving at the terminal. This core model determines a resource allocation to serve each vehicle within its time window (if such a solution exists). m m Wi,t ≤ hm · Xi,t

∀m ∈ M, i ∈ I m , t ∈ T

m m ≥ hm · (Xi,t − 1) + 1 Wi,t

∀m ∈ M, i ∈ I m , t ∈ T

(1) (2)

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m Zi,t

=

m pm i − Wi,t ,

t = rim

m m Zi,t−1 − Wi,t ,

∀t = rim + 1, . . . , d˜m i

Zi,md˜m = 0, i

∀m ∈ M, i ∈ I m

∀m ∈ M, i ∈ I m

(3)

(4)

m

M I

m Xi,t ≤ st

∀t ∈ T

(5)

m=1 i=1 m m , Zi,t ∈ N+ Wi,t m Xi,t ∈ N+

∀m ∈ M, i ∈ I m , t ∈ T ∀m ∈ M, i ∈ I m , t ∈ T

(6) (7)

Constraint (1) deﬁnes the arc capacity limiting the ﬂow of executed container tasks. It makes sure that the number of served tasks per vehicle and per period does not exceed the capacity of resources allocated to this vehicle. Constraint (2) imposes that each allocated resource executes at least one task and prevents allocating excess resources. This makes the solution more comprehensible. Constraint (3) formulates the mass balance constraint for arriving, executed and delayed tasks for each vehicle. It also ensures that no container movement requests are executed prior to a vehicle’s arrival. Constraint (4) imposes that each vehicle is completely served prior to its departure. Constraint (5) guarantees that the total number of allocated resources does not exceed the number of resources available at the terminal. This constraint links the otherwise independent models for each vehicle. Constraint (6) ensures that tasks are always completely executed within one period (transported from their origin to their destination) or not at all. Constraint (7) imposes that resources are allocated to exactly one vehicle per period by preventing partial allocations of resources to vehicles. If resources may be shared among diﬀerent vehicles per period constraint (7) has to be replaced by constraint (7a) which allows a partial resource allocation to vehicles. m ∈ R+ ∀m ∈ M, i ∈ I m , t ∈ T (7a) Xi,t

4

Service Strategies at Terminals

Container terminals serve diﬀerent transport modes like vessels, trucks, trains and barges. These vehicles diﬀer with regard to cargo volume, operating costs, knowledge and reliability of arrival dates, due dates and required handling equipment. Therefore, each transport mode is served with a speciﬁc strategy taking into account its characteristics. Service strategies for the same transport mode may also diﬀer from terminal to terminal. We introduce a notation α|β|γ to describe the service strategy for one transport mode. Table 1 presents an overview of diﬀerent values that may be taken by α, β and γ. α speciﬁes if resources may

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Table 1. α|β|γ Notation to describe service strategies

ded shar d p=1 maxv maxm nonincr C T U S

Resource allocation (α) Resources are allocated to exactly one vehicle per period Resources may be shared among vehicles per period Additional constraints (β) Service should be ﬁnished prior to vehicle due dates, but ﬁnishing ˜ later is possible at a cost (d < d) Every vehicle requires exactly one container movement (e.g., trucks) Maximal throughput per vehicle per period is limited (e.g., interaction with other equipment like quay cranes) Maximal throughput per transport mode per period is limited (e.g., space restrictions) Once the service of a vehicle has started, the number of allocated resources cannot increase Objective (γ) Each period a vehicle stays at the terminal is penalized Each period a vehicle spends at the terminal after its due date is penalized Unexecuted container movements are penalized Penalize the number of shifts working on a vehicle

be shared among diﬀerent vehicles or not. β indicates additional constraints like limits on the maximum throughput per period. γ represents the objective we want to achieve. This may be to minimize the time a vehicle spends at the terminal, the number of unexecuted tasks when the vehicle leaves the terminal or the number of shifts used to work on the vehicle.

5

Adaptation to the Case of a Container Terminal in Marseilles

Our core model determines a resource allocation to serve every vehicle within its time window. Thanks to the modularity of our model we can formulate transport mode speciﬁc submodels reﬂecting the speciﬁc service strategies by adding parameters, variables and constraints. Presenting all cases is out of the scope of this paper. We rather present an exemplary adaptation to a container terminal at the “Grand Port Maritime de Marseille” in France. This terminal serves vessels, barges, trains and trucks. For the sake of concision, we present only the adapted submodels for barges (m=1) and trains (m=2). For trucks and vessels, we indicate only their service strategies without detailing their implementation. 5.1

Barge (m=1)

The time a barge spends at the terminal (waiting and service times) should be minimized. Resources are allocated to exactly one barge and are not shared among barges. The maximum number of resources handled per period is limited

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by the throughput of quay cranes. This service strategy may be represented by ded| maxv | C with our notation. We introduce parameter qim to represent the maximum throughput per period of the quay cranes assigned to the barge. Constraint (8) ensures that this limit is satisﬁed. We also introduce a binary m , to measure the delay of a vehicle. It indicates at each period if variable, Yi,t m the vehicle is completely served and may leave the terminal (Yi,t = 0) or not m (Yi,t = 1). Each period the vehicle spends at the terminal is penalized in the objective function. Constraints (9) and (10) together with the objective function m is equal to zero if and only if the service of a vehicle is ﬁnished. assert that Yi,t m

min

I T i=1 t=rim

m Yi,t for m = 1

s.t. Constraints (1)-(4), (6)-(7) for m = 1 m Wi,t ≤ qim

m Yi,t ≥

m Zi,t pm i

m = 1, ∀i ∈ I m , t ∈ T

(8)

m = 1, ∀i ∈ I m , t = rim , . . . , T

(9)

m Yi,t ∈ {0; 1}

5.2

(10)

Train (m=2)

Trains are served at the rail station. Railcars stay at the terminal over the day and are picked up by an engine according to a ﬁxed schedule every day. There is a cost for each container remaining after the departure of its associated train. SCs are shared among trains. They transport containers from the common rail buﬀer to the yard and vice versa. The loading/unloading of trains is done by reach stackers which in the model. This service strategy is equivalent are not included ˜ We allow resource sharing among trains, but not to shar|d| U with d = d. ˆ tm and constraints among diﬀerent transport modes. We introduce variable X (4a) and (11) to limit resource sharing on vehicles of the same transport mode. ˆ tm is the total number of resources allocated to transport mode m in period t. X m We also introduce variable Ui,t to indicate the number of container movement requests remaining unexecuted at the train’s departure. Each unexecuted task is penalized in the objective function. Constraint (4) is modiﬁed to consider unexecuted tasks. m

min

I i=1

Uim for m = 2

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s.t. Constraints (1)-(3), (6)-(7a) for m = 2 m =0 Zi,md˜ − Ui,t

m = 2, ∀i ∈ I m

(4a)

m = 2, ∀t ∈ T

(11)

m

I

m ˆ tm Xi,t ≤X

i=1

ˆ tm ∈ N+ X 5.3

m = 2, ∀t ∈ T

(12)

Vessels and Trucks

Vessels have to be served during their imposed time windows. It is not allowed to plan a delayed departure of a vessel, but there is no incentive to ﬁnish earlier. The objective is to serve the vessel with the smallest number of shifts. SCs are allocated to exactly one vessel and are not shared among vessels. The service of a vessel requires some preparation and coordination. Therefore, no additional SCs are allocated to a vessel once its service has started, but retrieving superﬂuous SCs is possible. The maximum throughput per period is limited by the capacity of quay cranes. This service strategy is equivalent to ded|non-incr,maxv | S. Arriving trucks are assigned to parking slots where they are loaded and unloaded directly by SCs. Trucks should be served as fast as possible. We assume that each truck loads or unloads exactly one container. SCs are shared among trucks. This service strategy is equivalent to shar|p = 1| C.1 5.4

The Entire Terminal

To represent the entire container terminal, we only have to combine the independent transport mode speciﬁc submodels. To do this, we sum the respective objective functions, include all constraints into the combined model and add one constraint limiting the number of allocated SCs. Weights cm may be added in the objective function to represent priorities among diﬀerent transport modes. To illustrate this procedure, we present the combination of the barge and the train submodels. We observe that the objective function is the sum of the delay of barges and the number of containers left at the terminal at train departures. All constraints of the two submodels are included. Constraint (5a) is added to make sure that only available resources are allocated. It takes into account the fact that resources may be shared among trains. The submodels for vessels and trucks may be added in the same manner by including their objective functions, their speciﬁc constraints and by updating constraint (5a). 1

To reduce the problem size we represent all trucks arriving over the working day by one aggregated network ﬂow model. To do so, vehicle speciﬁc parameters and variables have to be aggregated and constraint (3) has to be modiﬁed to include arrivals in several periods.

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1

min c · 1

I T

2

1 Yi,t

+c ·

i=1 t=ri1

2

I

Ui2

i=1

s.t. Constraints (1)-(3), (6) for m = 1, 2 Constraints (4),(7), (8)-(10) for m = 1 Constraints (4a),(7a), (11), (12) for m = 2 1

I

1 Xi,t + Xt2 ≤ st

∀t ∈ T

(5a)

i=1

6

Numerical Experiments and Results

Experiments were run on actual data of one of the terminals in Marseilles. The time horizon is set to one working day and is divided into 21 one-hour periods. The data provides information on the number of vessels (0-2), of trains (1) and of barges (0-1) arriving per day. It also indicates arrival and departure times and the number of container movement requests per vehicle and the aggregated demand for trucks (280-770 requests). Information on the available SCs, the average number of resources a SC can handle per period and the chosen service strategy per transport mode were obtained via discussion with the terminal operator. The analysis was also carried out for a division of the working day into 42 half-hour periods. The analysis below holds for both cases, but we will only present results for the case with one-hour periods. Table 2. Delays for diﬀerent numbers of available SCs Available Infeasible Instances Average delay Average delay costs SCs instances with delays costs (feasible with 8 SCs) 22 16 14 12 10 8

0 0 1 1 3 8

0 1 2 7 9 8

0 261 32 102 143 57

0 0 0 0 4 57

Experiments are run for 20 instances (days) with diﬀerent numbers of available SCs (22, 16, 14, 12, 10 and 8) for each instance. The instances are solved using the commercial solver IBM ILOG CPLEX 12. Each feasible instance is solved in a few seconds. Infeasibility is discovered immediately. Infeasibility may occur

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if the number of available SCs cannot serve all trucks, barges and vessels within their time windows. The results of these experiments are shown in Table 2. The ﬁrst column indicates the number of available SCs used as input. The second column indicates how many instances are infeasible for the given number of available resources. As expected, the number of infeasible instances increases as the number of available SCs decreases. Columns 3 to 5 present the delays for the feasible instances. Remember that delays are deﬁned diﬀerently for each transport mode. For trucks and barges, each period the vehicle spends at the terminal (waiting and service) is penalized. For trains, unexecuted tasks are penalized. For vessels, each shift working on the vessel is penalized. To interpret the results, it must be noted that instances with large delays for a given number of available SCs may become infeasible for a smaller number of available SCs. This has impacts on the measured delays. The third column shows for how many instances delays occur. The fourth column presents the average delay costs for instances with delays. These delays highly depend on the number of feasible instances. Therefore, delays for diﬀerent number of SCs can hardly be compared. The last column shows the average delay costs only for those instances that are feasible with eight SCs. As expected, decreasing the number of available SCs leads to larger delays. Figures 3 and 4 illustrate two resource allocations proposed by the optimization model for 14 and 10 available SCs, respectively, for an instance with 1 vessel, 0 barge, 1 train and 580 trucks. The x-axis represents the discrete time periods of the working day. The y-axis represents the number of allocated resources. Both ﬁgures show how many resources should be allocated to the diﬀerent transport modes (truck, train, barge, vessel) at each period of the working day. They also indicate the total number of allocated resources per period. For 14 SCs, the train is served at periods 3, 5 and 7 and the vessel from periods 11 to 14. All trucks are served during the period in which they arrive. Since no barges arrive, no resources are allocated to barges. From periods 10 to 13, more than 10 SCs are on duty. If the number of available SCs is reduced to 10, some tasks executed in these periods have to be executed in other periods. The vessel is then served

Fig. 3. Resource allocation for 14 available SCs

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Fig. 4. Resource allocation for 10 available SCs

in periods 15 to 17 and the train in period 3. Nevertheless, it is not possible to serve all trucks in their arrival periods and delays occur for 15 trucks. This discussion shows how the model’s output indicates a possible resource allocation minimizing delays. It also illustrates how the model may be used to determine the number of SCs needed for the next working day for the expected workload. For this purpose, the model should be executed for the expected workload several times with diﬀerent numbers of available SCs. Results indicate possible resource allocations but also the number of expected delays. The terminal operator may then determine the number of needed SCs for a desired service quality. Remember that SCs are driven by dockers and that the capacity may thus vary from day to day.

7

Conclusion

In this paper, we deal with the allocation problem of internal container handling equipment (SCs) at container terminals. Our objective was to determine a resource allocation minimizing vehicle delays. Terminals serve vessels, trucks, trains and barges with adapted service strategies. We presented a notation to describe the service strategy used at terminals for diﬀerent transport modes. We formulated a modular mixed integer linear program based on a network ﬂow model for the resource allocation problem at container terminals. For each transport mode, a submodel is implemented respecting its characteristics. To represent the entire terminal, these independent submodels may be easily combined. We presented parts of the model implementation for a container terminal at the “Grand Port Maritime de Marseille”. We discussed some experiments carried out for this terminal to determine an allocation of available resources to diﬀerent transport modes and to determine the number of resources required to serve the expected demand with a desired service level. In continuation of this work, the model may be used at tactical and strategic levels. At a tactical level, the model allows the beneﬁts of a shared resource allocation to be analyzed. Diﬀerent priority rules between trucks, trains, barges

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and vessels may also be compared. Beneﬁts of an increasing number of tasks that can be handled per resource can also be evaluated. Such an increase may for example result from a more eﬃcient yard organization, shorter travel distances or better trained workers. At a strategic level, the model may be used to analyze the impact of an increasing volume of containers being transported via trains and barges. This transfer from road to rail and waterways is a general aim of today’s logistics. Another possible application is to determine the number of SCs that will be necessary in future to cope with the forecasted demands. Another interesting perspective is to analyze the impacts of the possibility to forward or delay arrivals of vehicles on the number of necessary resources and delays. The model may also be coupled with a scheduling and routing problem to replace the average number of movements per SC per shift by a travel and handling times. Acknowledgments. Part of this work has been conducted within the ESPRIT project, ﬁnanced by the Mission Transports Intelligents of the DGITM (Direction G´en´erale des Infrastructures, des Transports et de la Mer) of the MEEDDM (Minist`ere de l’Ecologie, de l’Energie, du D´eveloppement Durable et de la Mer). The authors would like to thank Christophe Reynaud, from Marseille Gyptis International.

References 1. Alessandri, A., Cervellera, C., Cuneo, M., Gaggero, M.: Nonlinear Predictive Control for the Management of Container Flows in Maritime Intermodal Terminals. In: 47th IEEE Conference on Decision and Control, pp. 2800–2805. IEEE Press, New York (2008) 2. Das, S.K., Spasovic, L.: Scheduling Material Handling Vehicles in a Container Terminal. Production Planning & Control 14, 623–633 (2003) 3. Gambardella, L.M., Mastrolilli, M., Rizzoli, A.E., Zaﬀalon, M.: An Optimization Methodology for Intermodal Terminal Management. Journal of Intelligent Manufacturing 12, 521–534 (2004) 4. Hartmann, S.: A General Framework for Scheduling Equipment and Manpower at Container Terminals. OR Spectrum 26, 51–74 (2004) 5. Kang, S., Medina, J.C., Ouyang, Y.: Optimal Operations of Transportation Fleet for Unloading Activities at Container Ports. Transportation Research Part B 42, 970–984 (2008) 6. Notteboom, T., Winkelmans, W.: Factual Report on the European Port Sector. Report commissioned by European Sea Ports Organisation, ESPO (2004) 7. Vis, I.F., de Koster, R., Savelsbergh, M.W.P.: Minimum Vehicle Fleet Size Under Time-Window Constraints at a Container Terminal. Transportation Science 39, 249–260 (2005) 8. Zehendner, E., Rodriguez Verjan, G.L., Absi, N., Dauz`ere-P´er`es, S., Feillet, D.: Optimizing and Simulating Transport Vehicle Allocation in a Multimodal Container Terminal. Working Paper EMSE CMP-SFL (2011)

A Simulation Study for Evaluating a Slot Allocation Model for a Liner Shipping Network Sebastian Zurheide and Kathrin Fischer Hamburg University of Technology, Schwarzenbergstr. 95 D, 21073 Hamburg, Germany {zurheide,kathrin.fischer}@tu-harburg.de http://www.oris.tu-harburg.de

Abstract. Revenue management (RM) methods are still only rarely used in the liner shipping industry. Usually, skilled employees make the decisions whether to accept or reject a booking and decision support systems are not commonly used. But for maximizing a company’s proﬁt it can be crucial to make the right decisions for each booking request and therefore such a system can make an important contribution to success. In this work, a discrete-event simulation model for container bookings is developed, including a quantitative slot allocation model which takes into account diﬀerent segmentations, the network structure of liner shipping with the possibility of transshipments, and the existence of diﬀerent round trips of ships on the services. With a simulation study, diﬀerent scenarios, networks and input settings can be evaluated regarding their performance, to determine the best strategy for the company. In this study, the incoming bookings are simulated and the decision whether a booking is to be accepted or rejected is based on the capacity that is available in the respective booking class according to the solution of the slot allocation model. This booking limit (BL) strategy is compared to a ”ﬁrst come ﬁrst serve” (FCFS) strategy and to an optimal strategy (OS) which is based on the assumption that all bookings are known in advance. The results show that the BL strategy is recommendable for liner shipping companies, as it leads to signiﬁcant proﬁt increases compared to FCFS.

1

Introduction

In the year 2008, the conference system of the liner shipping industry in Europe ended. Hence, it was no longer allowed to make price agreements. This gives the carriers the opportunity to operate on a market where they can implement revenue management and pricing strategies [2,3]. In the liner shipping industry, usually skilled employees handle the requests for container bookings. Mostly, these employees have long experience in this business and make their decisions without or with little decision support. However, revenue management systems, which are used in other industries with great success, can support these employees in decision making and might lead to a higher proﬁt for the company. Through the use of booking limits, the sales agents get J.W. B¨ ose et al. (Eds.): ICCL 2011, LNCS 6971, pp. 354–369, 2011. c Springer-Verlag Berlin Heidelberg 2011

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support in deciding whether to accept or to reject a booking in a speciﬁc booking class. While in times with low demand, a FCFS strategy and a high utilization rate can be the best strategy, with scarce capacities it can be more beneﬁcial to reject certain bookings even if capacity is available to allow for the acceptance of more proﬁtable bookings later on. Hence, the employees have to make the right decisions to achieve the proﬁt maximizing slot allocation for the company. The basic concept of revenue management is transferable to the liner shipping industry from other revenue management industries (e.g., air cargo), but the speciﬁc models and methods from other industries cannot be used or need to be modiﬁed [1]. However, the main characteristics necessary for a successful application of revenue management deﬁned for example by Hellermann [8] can be identiﬁed in the liner shipping industry. These characteristics are a perishable product, ﬁxed capacity, high ﬁxed costs and low marginal costs, possibility of deﬁning booking segments, advance bookings, stochastic and ﬂuctuating demand and the availability of historical data. A more detailed discussion of these characteristics in liner shipping can be found in Zurheide and Fischer [20]. With all the relevant characteristics present, it is advisable to implement revenue management methods in the liner shipping industry. Of course, as in all the other industries where revenue management is used, demand needs to exceed capacity to apply a useful revenue management system. In the B2B liner shipping industry, these methods are most useful in the spot market when no long-term relation between the carrier and the customer exists. Before applying a new revenue management system in practice it is advisable to evaluate the possible strategies within a simulation study. As performance of diﬀerent revenue management methods and strategies can diﬀer and depends on many factors, it is necessary to simulate diﬀerent scenarios and diﬀerent environments to ﬁnd the best strategy for diﬀerent situations. Hence, it is the aim of this work to develop a concept for a discrete-event simulation of a liner shipping network and the respective booking process and to evaluate the performance of a booking limit strategy in comparison to other approaches. In the following, a literature review on revenue management in liner shipping is given and the basic concept of simulating a revenue management system is presented. Afterwards, liner shipping speciﬁc conditions are explained and a discrete-event simulation speciﬁcally designed for liner shipping revenue management is developed which incorporates a slot allocation model and the speciﬁc network structure. Using the simulation, the performance of diﬀerent strategies is evaluated for diﬀerent scenarios. Finally, some conclusions and an outlook are given.

2

Literature Review

Only a few publications on the topic of revenue management in liner shipping can be found in the literature. In most of these papers, slot allocation models are developed to create optimal booking limits for a liner service. In the following, the literature on revenue management in liner shipping and on simulation concepts for revenue management systems is brieﬂy reviewed.

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Literature on Revenue Management in Liner Shipping

The possibility of using revenue management in liner shipping is mentioned for the ﬁrst time in 1994 by Brooks and Button [2] and Maragos [15]. To the authors’ knowledge, the PhD thesis by Maragos [15] is the ﬁrst work with the core topic of revenue management in liner shipping. In 2004, Ting and Tzeng [18] present an arc ﬂow container slot allocation model. The optimal number of containers to be accepted for each container type and port pair is determined within restrictions for the demand of loaded and empty containers and capacity restrictions of the ship. By Demirag and Swann [4], the capacity limits for liner shipping sales agents in a decentralized logistic network are determined. In the works of Lee et al. [10,11], a stochastic dynamic programming model for a single leg revenue management problem is presented, which includes is the possibility to postpone containers to the next ship. A multi-leg multi-line slot allocation path ﬂow model is formulated and solved with a robust optimization approach by Xiangzhi et al. [19]. Bingzhou [1] develops a stochastic model for dynamic capacity allocation with multiple container types on one leg. A revenue management slot allocation model for a short-haul, multiple-port network is presented by Feng and Chang in [5,6]. The capacity is restricted at the loading port and a port-to-port network structure is used. In Lu et al [13], the potential proﬁt per service is maximized with a quantitative slot allocation model. Moreover, the demand and empty container repositioning restrictions, the slot, deadweight and reefer capacities are taken into account. In Løfstedt et al. [12], an arc ﬂow and a path ﬂow model for container allocation in a liner shipping network are presented and the diﬀerent types of formulation are compared. A time-space network is used to model the liner shipping network over time. Finally, a slot allocation model with a service related segmentation and the possibility of postponement is presented in Zurheide and Fischer [20]. The container market is segmented not only by the well-established segmentation criteria (e.g., type of container), but also into priority and non-priority containers. This leads to more ﬂexibility and for urgent cargo the customer has the possibility to book a priority service. 2.2

Literature on the Basics of Simulation in Revenue Management

The simulation of revenue management systems is mentioned in Talluri and van Ryzin [17] as a powerful methodology to evaluate their performance. By modeling such a system as close to the real world as possible, the simulation can give a good picture of its beneﬁts. The basic principles for simulations in revenue management are described by Frank et al. [7]. It is stated that no standards or standard tools for the simulation of revenue management systems exist. Therefore, they propose an event-driven simulation as the best way to model them, because each event (e.g., a booking) changes the state of the system. The different components which are suggested by Frank et al. [7] for the simulation of revenue management in airline seat booking are shown in Figure 1. It is proposed to generate the demand in advance as the demand is an input factor for the ”inventory control”. The ”inventory control” is the capacity management of the

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revenue management process and provides the bookings to the booking system, the inventory status to the optimization module and the inventory history to the forecasting. The pricing model sets the prices for the optimization and the inventory control. In the optimization model, the forecast, the inventory status and the prices are used to optimize the allocations and to provide them to the inventory control. Other revenue management simulations for the airline industry are mentioned in the literature (e.g., PODS mentioned in [17, p.485]), but to the authors’ knowledge, no other contributions focus on the concept and development of a revenue management simulation.

Demand generation

History

Demand

History building

Inventory control

Allocated seats

Booking system

Fares

Inventory

Fares

Forecasts

Forecasting

Bookings

Optimization

Pricing

Fig. 1. General design of an RM simulation [7]

3

Problem Description

In the liner shipping industry, the carriers oﬀer container transportation services from one port to another. The aim of each company is to maximize its proﬁt and to get the most beneﬁcial bookings. Hence, it may be advantageous to reject an early booking in favor of a later and more beneﬁcial one. To support these decisions, revenue management methods can be used with which, for example booking limits for diﬀerent booking classes, e.g., for diﬀerent types of containers, can be determined. As the beneﬁt of these methods is not always obvious, they need to be evaluated, for example by a simulation study, before implementing them in a company. This is discussed in more detail below. 3.1

Basics on Liner Shipping

The network of a liner shipping carrier can be divided into a short-sea and a deep-sea shipping network. On the short-sea shipping network, containers are brought from smaller regional ports to the bigger deep-sea shipping ports. In short-sea shipping, the services connect the ports of one region, while deep-sea shipping services connect diﬀerent regions or continents [16]. The ships on deep-sea services operate on round trips where a number of ports are called in a predeﬁned order. The trips follow a ﬁxed schedule, and several

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ships are deployed on a service in order to secure a weekly departure at each port. In Figure 2, an example for two services (A-B-C-D-A and B-C-E-F-B) is presented. The services consist of diﬀerent sequences of the ports A to F and consecutive ports are connected by legs (e.g., A to B). Two ports of the same service are connected by a service path which consists of one or more legs, e.g., the service path A to C uses the legs A to B and B to C. Based on a network structure with several services, it is possible to transship containers from one service to another service. In the example in Figure 2, the ports B and C can be used for transshipment. Therefore, it is possible to transport a container between the ports A and E with transshipment in B or C. Such a connection is named global path and consist of one or a sequence of service paths.

A

B

F

D

C

E

Fig. 2. Example for a liner shipping network

3.2

Simulation of a Revenue Management System for Liner Shipping

The basic concepts for revenue management simulation in the airline industry stated by Frank et al. [7] also apply to the liner shipping industry. The system structure is similar, but especially the network structure, as it was described above, is diﬀerent from the airline industry. Moreover, in the simulation of these networks, time is an important factor. The network has to be build for a predeﬁned simulation range. Hence, each leg and each path needs a speciﬁc time reference. In particular, a speciﬁc leg is characterized not only by the service and the vessel, but also by the ship cycle. A ship cycle is deﬁned as one round trip of a ship on one service and a new ship cycle starts when the ship passes the predeﬁned starting port of the service. Each leg can be assigned a speciﬁc ship cycle count which is related to the start of the simulation. The service paths have a diﬀerent time index, named booking cycle. A booking cycle also starts with the predeﬁned starting port and consists of all the possible bookings on the diﬀerent paths of the respective ship cycle. But the booking cycle overlaps with the next ship cycle. This can be illustrated by the following example: In Figure 2, port A is assumed to be the starting port of service 1 and a transport from port D to port B is considered. In this case, the leg from D to A and the leg from A to B are on diﬀerent ship cycles, but the service path belongs to only one booking cycle.

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In addition to the ship cycle and the booking cycle, the ports need a time index to deal with empty container repositioning. The necessity of empty container repositioning results from the imbalance in global trade. Therefore, each port has to be modeled for diﬀerent periods of time in which the empty containers need to be balanced by repositioning, leasing or storing. Furthermore, the simulation needs several additional input parameters. If no real data is available, these input parameters need to be generated. For example, prices, costs and historical data (which are available in a real world scenario) need to be created based on assumptions and using realistic distributions. Moreover, the bookings need to be created to simulate the booking process. In contrast to only calculating slot allocations without testing them as it is done in, e.g., [5], through this approach it is possible to imitate the booking process in the simulation and to get a better understanding of the performance of the model. 3.3

Slot Allocation Models and Segmentation in Liner Shipping

A slot allocation model is solved to determine booking limits for the diﬀerent booking classes. The booking classes result from the diﬀerent possible combinations of the segmentation criteria (e.g., container type and global path). The slot allocation model includes a set of diﬀerent restrictions regarding the capacity, the empty container repositioning and the forecasted demand. It is described in more detail in Section 4.5; its exact formulation can be found in [21]. The booking limits determined by the model can be used as decision support for the sales agents when they have to decide whether a booking is to be accepted or rejected. A booking should be rejected, when not enough capacity is available in the respective booking class, and it should be accepted otherwise. The performance of such a slot allocation model depends on various factors, e.g., the demand structure and forecast or the available capacity in relation to demand. In diﬀerent scenarios, the performance of a revenue management system can vary a lot. Diﬀerent variations of the input factors can be used to create these scenarios. The slot allocation model uses various input data to determine the booking limits. Even when real data is available, e.g., prices and costs for container transports between two ports and the available capacity on the diﬀerent ships, some data are not known, e.g., the unconstrained demand. The unconstrained demand is the sum of all accepted and denied booking requests during a certain period of time, i.e., all bookings in this period. But usually only data for accepted demand from previous round trips is available, and the rejected booking requests are not logged. Therefore, the unconstrained demand needs to be forecasted based on the accepted bookings. An important aspect for slot allocation models is the segmentation of the market. In liner shipping, some segments are already present. The diﬀerent container types (e.g., 20’/40’dry or reefer), diﬀerent customer types (contractual and spot customers) and the diﬀerent global paths build segments which can be used in diﬀerent combinations as booking classes. The content of the contracts that carriers and customers negotiate can vary a lot and as mentioned above,

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the liner shipping industry is a B2B market with sometimes long term customer relationships. Therefore, in this work only the spot market is taken into account and the capacity allocated to contracts and long term customers is assumed as a ﬁxed reduction of each ship’s capacity. In addition to the above-mentioned segments (container type and global path), a time-based service segmentation is used here. This segmentation is similar to the ”Priority Product Upgrade” which was recently introduced by Maersk Line [14]. This segmentation approach by Maersk Line can be seen as the ﬁrst obvious attempt of a carrier to implement a revenue management system. With this segmentation, the market is divided into urgent and non-urgent cargo. The customer with urgent cargo can book a priority loading on the next ship and the carrier assures for an extra charge that the container is loaded on this ship. Hence, in this paper the segmentation is done by container type, container path in the network and by service segment.

4

Simulation Design

A discrete-event simulation with next-event time advance is selected to simulate the revenue management system for the liner shipping industry. The simulation is written in C# including the Microsoft Solver Foundation (MSF) (Version 3.0.1) with the Gurobi solver (Version 4.0.2). The input data ﬁles are provided in the XML format and a Microsoft SQL Server 2008 is used for data storage. Other solver plug in’s (e.g., Cplex) are available in the MSF framework. The basic concept of the simulation is shown in Figure 3. The simulation consists of two main parts. The ﬁrst part is the simulation environment and it includes the network initialization, the preparation of the input data and the analysis of the results. The second part is the core simulation of the booking process and is highlighted in Figure 3 by the dotted line. The main concept of the revenue management system is based on the principles for revenue management simulation proposed by Frank et al. [7]. In the following, the concept of a discreteevent simulation is introduced and the diﬀerent modules of the simulation are described in more detail. 4.1

Discrete-Event Simulation

In a discrete-event simulation, the events change the status of the system at a discrete point in time and no continuous changes over time take place [9, p.6]. In a revenue management system, a booking is such an event that causes a change of the state of the system. Therefore, a discrete-event simulation is the appropriate simulation type for a revenue management system. The next-event time advance is used, because this approach is the most common one. It is a time advance mechanism in which the time advances to the time of an event when the event occurs. Hence, long periods without any event are skipped [9, p.7f]. Two diﬀerent types of events can change the state of the system. First, each booking changes the state of the system when it is either accepted or rejected.

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Segment data

Create historical bookings Create network

Network data

361

Create demand, cost, price data

Data read in

Create forecast Create historical network

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Optimisation to create empty container forecast

Core simulation Reoptimisation to create new booking limits Create new forecast

Yes

Handle next booking or ship event

Start simulation

Optimisation to create booking limits

No

Reopt. time reached?

No

End of simulation?

Yes

End Simulation

Write simulation results

Fig. 3. Simulation process

The second event type are ship events. With these events the ships proceed on the services. A ship event can be either a port or a leg event. The port event represents the arrival of a ship in a port, and the leg event the departure to the next port. 4.2

Network Initialization

Before the initialization starts, the available input data ﬁles are read. The input is split up into three diﬀerent input ﬁles. The ﬁrst ﬁle includes the diﬀerent service segments and the container types. Each service segment has a speciﬁc market share and a speciﬁc extra charge. The container types are deﬁned with their detailed information, e.g., the tare weight or the dimension. The second ﬁle includes the structure of the network with the diﬀerent regions, ports and services. A region (e.g., Europe, Asia, ...) includes the speciﬁc relations to the other regions. The diﬀerent ports of the services are assigned to the regions and the transshipment costs and a maximum number of containers in a speciﬁc segment (e.g., in the priority segment) are deﬁned. For each service, a name, a start time and the duration of stay at each port and on each leg is given in the input ﬁle. The durations of the stays in ports and the durations of trips on legs are ﬁxed. Delays and shorter port or travel times are not taken into account. Furthermore, each leg has a known capacity for 20 foot containers, reefer plugs and the deadweight of the ship. In the third ﬁle, the diﬀerent simulation settings are deﬁned, e.g., the length of the simulation, the reoptimization range and the length of the optimization as the period of time that is included in the optimization. With this information, the network is build for a certain time range. Based on the durations in ports, the durations on legs and the start time of the network,

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a schedule for each service is derived. With this schedule, each port, leg, service path and global path gets a starting and an end time. The time lag between two consecutive ships and the schedule for a service is used to determine a schedule for each ship and the position of the ship at the beginning of the planning period. 4.3

Input Data Generation

When no real data is available, missing input data needs to be generated. If real data was available, all the processes in Figure 3 which are marked with dashed lines could be replaced with a real data read-in process. However, without real data the simulation generates historical booking data, demands, prices and costs. The historical network is modeled for a certain time range previous to the simulation range. This historical network is used for the creation of historical booking data. But before these booking data can be created, the other parameters need to be set. Information regarding demand, price, cost, number of containers, weight of the containers and the booking time is deﬁned for each pair of regions, because it is assumed that the characteristics of the parameters are based on this relation and not on individual port pairs. The demand, the prices and the costs for each global path are deﬁned according to a set of distributions. The other parameters are set for each booking, also based on distributions. With these parameters and the historical network, a set of historical bookings is created. These bookings can be seen as the unconstrained historical demand. Another parameter that can normally be determined from historical data of a carrier is the ”empty container forecast”. In the simulation, this empty container forecast is based on a ﬁrst optimization of the booking limits. According to these booking limits, the number of containers that will be transported to and from a port can be predicted. Hence, the number of containers that are needed or are available at each port in each time period can be determined. 4.4

Bookings and Forecast

Following the recommendation of Frank et al. [7], the generation of bookings is separated from the simulation and the demand stream is saved in advance. With this approach it is possible to test diﬀerent strategies for the same booking set. According to the known market share for the booking classes, a number of bookings is created for each global path. Each booking has individual characteristics (e.g., contribution to proﬁt, number of containers or weight of the booking). It is assumed that each booking includes only one type of container in one service segment. When real data is available, the bookings should be created with a demand function derived from this data. In the simulation, various demand functions can be used which can include seasonal ﬂuctuation of the demand. They can be determined from historical data. The forecast for the simulation is based on the historical bookings. Diﬀerent forecasting methods can be implemented in the simulation (e.g., moving average or exponential smoothing). The bookings, which are taken into account for the

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363

forecast, are selected based on the path, the service segment, the container type and the time period. After the demand is forecasted and all the parameters are set, the ﬁrst set of booking limits is determined with the slot allocation model. 4.5

Slot Allocation Model

The slot allocation model is the core part of the simulation. Only a short description of the model is given here, because the focus of this work is on the simulation environment. A detailed presentation of the model can be found in Zurheide and Fischer [21]. A path ﬂow formulation is used for this model which is based on a four level modeling structure and includes diﬀerent time factors. On the ﬁrst level, the different ports are modeled including a time index for diﬀerent periods. The second level consists of the diﬀerent legs connecting two ports of a service operated by a ship at a speciﬁc time. On the third level, the diﬀerent service paths, as deﬁned in Section 3.1, are located. The fourth level of the model is the global path level, with the global paths also deﬁned in Section 3.1. According to the deﬁnition, it is possible that in a network there is more than one global path between two ports. The service paths and the global paths are build before the model is solved. The diﬀerent levels of the model are linked with indicator parameters. Five diﬀerent indicator parameters are used in this model; the indicator parameters are equal to one if for example a leg is included in a service path, and equal to zero otherwise. The purpose of the model is to maximize the expected proﬁt of the slots which are allocated to the diﬀerent booking classes. Decision variables for loaded and empty containers on global paths and on service paths are used to determine the optimal number of slots within each booking class. In addition, container leasing and storing are included as decision variables for empty container repositioning. The objective function maximizes the sum of the expected proﬁt minus the costs for empty container repositioning. The expected proﬁt is calculated using the average price minus the costs for transporting a container on a global path. For empty container repositioning, only costs for transporting, leasing and storing are taken into account. The model has several restrictions. First, the number and weight of the allocated containers cannot exceed the diﬀerent capacity restrictions on a leg. The dimension of the diﬀerent container types, the average weight per service path and the weight of empty containers are taken into account. The number of containers in the priority segment is limited in each port, because the carrier has to assure that these containers can be loaded even when delays occur. Another restriction is the maximum forecasted demand in each booking class, which should not be exceeded by the number of allocated slots. Moreover, the demand for empty containers needs to be satisﬁed in each port and period. This demand can be satisﬁed by container repositioning from other ports, container leasing and container storage from the previous period. If empty containers are available for repositioning in a port, they can be stored or repositioned to a another port. With a further set of restrictions the global paths and the service

364

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paths are connected. The sum of the slots allocated to the global paths using a speciﬁc service path is equal to the slots allocated to this service path. For empty container repositioning, it is assured that the sum of the number of empty containers allocated to the global paths using a service path is equal to the number of empty containers allocated to this service path. 4.6

Core Simulation

The ﬁrst set of booking limits for each global path and the bookings are the main input for the core simulation. The bookings are handled based on the booking time and the ship events according to the time of arrival at a port or the time of departure on a leg. Each booking event triggers a check for available capacity in the respective booking class. If space is available, the capacity of each leg of the global path is reduced by the values of the booking and the booking information is stored. If no space or not enough space is available, the booking is denied. The booking information is also stored for an analysis. The ship events move the ships on their routes and update the ship time. After each event a check is done whether there are more events in the future and if the reoptimization time is reached. The time for reoptimization can depend on a ﬁxed time range or a number of bookings. Until this time or this number is reached, the simulation proceeds with the next event. At this point in the simulation, a new forecast is created and new booking limits are determined. The slot allocation model is build for a deﬁned period of time. Hence, not all booking limits are reoptimized each time, because only the booking limits in the optimization range need to be considered. With this approach, smaller optimization models can be built. At the end of the simulation, the results of the simulation are analyzed and a result ﬁle is written. This ﬁle consists of the important input parameters, the proﬁt from the accepted bookings and the utilization rates. More information can be determined if necessary. 4.7

Limitations of the Simulation

With this simulation, the performance of the slot allocation model can be analyzed. But with no real data available, some of the input data have to be created based on distributions and based on assumptions. Hence, the data input for the creation of the historical data and the booking data is based on the same set of distributions. But the demand function should be determined from historical data and a forecast should be made based on real historical data. Furthermore, the booking limits can only be used for short term customer relations, because a customer value for long term customers is not yet included. Nevertheless, the simulation leads to a good understanding of the eﬀect a slot allocation model can have in a liner shipping network.

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5

365

Simulation Results

To evaluate the slot allocation model, a basic input scenario is built and the booking limit strategy is compared with the FCFS and the optimal strategy. The optimal strategy is deﬁned as the selection of the most beneﬁcial bookings for the available capacity when all bookings are known in advance. An optimization model which includes the capacity restrictions is used to identify these bookings. Based on the basic scenario, diﬀerent demand functions are used to create three speciﬁc demand scenarios. The scenarios are then tested on two diﬀerent network instances. 5.1

Case Study

The scenario takes the ﬁve main container types (20’ dry and reefer, 40’ dry and 40’ high cube dry and reefer) into account. The speciﬁc input parameters for the container types are shown in Table 1. Moreover, the two service segments ”priority booking” and ”non-priority booking” are used. For each container, in the priority segment an extra charge of 250$ is applied. The market share of the priority segment is assumed to be 20%. The capacity restrictions on each leg are 2500 TEU (twenty-foot equivalent unit), 250 reefer plugs and 35,000 dwt (dead weight tonnage). The transshipment costs in a port are 200$ and the maximum number of priority containers in a port are 400 containers. The capacity parameters can vary from one leg to another and from one port to another, but to simplify the scenario, the values are assumed to be identical here. In Table 2, the set of distributions for each region combination (for two regions (R1 and R2)) is shown. The mean and standard deviation of the normal distributions for demand, price, cost, number of containers per booking and weight of containers are based on assumptions. Only positive values are used in the simulation; i.e., if a negative value is generated from a distribution, it is ignored and a new value is generated. The number of bookings should increase towards departure time. For this reason, the booking time is modeled with an exponential distribution. The demand can be modeled with and without seasonal eﬀects. For the demand on a global path, the normal distribution (see Table 2) provides a value. Afterwards, this value can be adjusted by a seasonal demand eﬀect. Table 1. Container types Container Type

Dimension Tare Maximum Leasing Price Share Weight

20’ 20’ 40’ 40’ 40’

1 TEU 1 TEU 2 TEU 2.25 TEU 2.25 TEU

Dry Reefer Dry High Cube High Cube Reefer

3 3 5 5 5

t t t t t

32.5 30.5 32.5 34.0 32.5

t t t t t

0.70 4.80 1.10 1.20 5.80

$/per $/per $/per $/per $/per

day day day day day

30% 5% 30% 30% 5%

366

S. Zurheide and K. Fischer Table 2. Set of distributions per region combination for a 20’ container

Parameter

R1-R1

R1-R2

R2-R1

R2-R2

Demand (Number of containers) Price (in $) Cost (in $) Number of containers per booking Weight of container (in ton) Booking time before departure

N(50,10) N(600,50) N(300,10) N(5,10) N(17,5) Exp(0.17)

N(300,20) N(1800,50) N(800,30) N(5,10) N(17,5) Exp(0.17)

N(700,40) N(3000,50) N(800,30) N(5,10) N(17,5) Exp(0.17)

N(200,20) N(1000,50) N(400,20) N(5,10) N(17,5) Exp(0.17)

1

ent factor Adjustment

0,9 0,8 0,7 0,6 0,5

Short Seasons

0,4

Long Seasons

0,3

Non-adjusted demand

0,2 0,1 0

0

2

4

6

8

10

12

14

16

18

20

Ti Time Period P i d

Fig. 4. Example demand adjustment factors

In addition to the non-adjusted demand, two diﬀerent adjustment factors (a1 and a2 ) are used. The ﬁrst factor for a short seasonal eﬀect is calculated with a1 = (sin(t) + 2)/3 with t as the time period. For a long seasonal eﬀect, the factor a2 is determined with a2 = (sin(t/2) + 2)/3. In Figure 4, the three diﬀerent adjustment factors used for the three scenarios are illustrated. On the x-axis, the diﬀerent time periods and on the y-axis, the value of the adjustment factor are shown. 5.2

Results

Two diﬀerent instances which are based on real liner shipping services are tested for the three demand scenarios. The ﬁrst instance includes three services in two regions connecting 19 ports. The second instance connects 21 ports with four services in two regions. As the forecasting method, a (moving) average is used and only data of previous ship cycles on the same path with a similar demand situation - from a comparable season - are used in the forecast. In Table 3, results are shown in terms of proﬁt rate (PR), which is represented as a percentage of the OS solution, and utilization rate (UR), for which the number of used slots is divided by the total number of slots. Overall, the results show similar tendencies, especially for the BL strategy. The BL strategy outperforms the FCFS strategy with respect to proﬁt in all cases. However, the performance of the booking limit strategy depends largely on the ratio between available capacity

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367

and demand, as only if demand exceeds capacity it makes sense to decide whether a container should be accepted or rejected, instead of accepting all bookings. Therefore, performance for the unadjusted demand scenario is better, because total demand is higher. On the contrary, for situations with lower demand as in the scenarios with seasonal eﬀects, the advantage of the BL strategy decreases. The diﬀerence between the results for the two scenarios with seasonal eﬀects is rather small, as it is assumed that the seasonal eﬀects are known and integrated in the forecast. This leads to very similar situations and to comparable results. However, it was found that if a normal moving average is used as a forecast, the bad forecast quality results in a major decrease in the performance of the booking limit strategy as well. Hence, a known seasonality of bookings should always be taken into account, and it can be concluded that only with a good forecast and especially when demand exceeds capacity, the BL strategy is a valuable approach. As can be observed in Table 3, the utilization rate of the BL strategy is much lower than the rate for the FCFS and the OS strategy where it is close to 100%. This is due to the use of booking limits, because space is kept free for possible future beneﬁcial bookings which eventually do not materialize. But even with a much lower utilization rate, the BL strategy always outperforms FCFS in terms of the proﬁt rates. Table 3. Results Not adjusted

Short seasons

Long seasons

PR

PR

Instance

Strategy

PR

3 Services 19 Ports

OS BL FCFS

100,00% 98,55% 100,00% 98,39% 100,00% 98,41% 71,43% 62,14% 67,02% 58,74% 66,47% 58,96% 52,35% 98,90% 56,77% 98,43% 56,27% 98,49%

4 Services 21 Ports

OS BL FCFS

100,00% 98,99% 100,00% 98,90% 100,00% 98,98% 73,66% 65,51% 67,50% 61,49% 67,24% 61,61% 52,48% 99,08% 56,52% 98,55% 56,79% 98,72%

6

UR

UR

UR

Conclusion and Outlook

A discrete-event simulation design for evaluating the performance of a slot allocation model is presented in this work. The approach is used to simulate the booking requests and to handle them according to diﬀerent strategies. The results of the newly developed booking limit strategy are compared with a FCFS strategy and an optimal strategy. It is shown that booking limits create good results in diﬀerent demand situations with and without seasonal eﬀects. The simulation also gives valuable insights into the dependencies in a revenue management system. It turns out that the size of the demand and the quality of the forecast are key aspects for a good performance that have to be taken

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into account. Hence, more experiments using diﬀerent data sets and diﬀerent forecasting methods should be carried out, preferably using real data. However, the results of the simulation show that a booking limit strategy can be very useful for a liner shipping company. Further research is needed to include a customer segmentation strategy and other acceptance strategies, like nested booking limits [17, p.28f]. With customer value included in the simulation, a more detailed representation of the B2B market environment in the liner shipping industry can be realized. Nested booking limits would most likely further improve the results of the booking limit strategy, because more beneﬁcial bookings will be accepted with this strategy and the available capacity can be utilized in a better way. Furthermore, cancellations and overbookings should be taken into account, with the possibility of creating new service segments based on diﬀerent cancellation fees.

References 1. Bingzhou, L.: A Stochastic Model for Dynamic Capacity Allocation of Container Shipping Two-Dimensional Revenue Management. In: Proceedings of 2008 International Conference on Service System and Service Management, pp. 1–7. IEEE, Los Alamitos (2008) 2. Brooks, M., Button, K.J.: Yield Management: A Phenomenon of the 1980s and 1990s. International Journal of Transport Economics 21(2), 177–196 (1994) 3. Brooks, M.R.: Sea Change in Liner Shipping: Regulation and Managerial DecisionMaking in a Global Industry. Pergamon Press, Oxford (2000) 4. Demirag, O.C., Swann, J.L.: Capacity allocation to sales agents in a decentralized logistics network. Naval Research Logistics (NRL) 54(7), 796–810 (2007) 5. Feng, C.-M., Chang, C.-H.: Optimal Slot Allocation in Intra-Asia Service for Liner Shipping Companies. Maritime Economics & Logistics 10, 295–309 (2008) 6. Feng, C.-M., Chang, C.-H.: Optimal slot allocation with empty container reposition problem for Asia ocean carriers. International Journal of Shipping and Transport Logistics 2(2), 22–43 (2010) 7. Frank, M., Friedemann, M., Schr¨ oder, A.: Principles for simulations in revenue management. Journal of Revenue and Pricing Management 7(1), 7–16 (2008) 8. Hellermann, R.: Capacity Options for Revenue Management. Springer, Berlin (2006) 9. Law, A.M.: Simulation Modeling & Analysis. McGraw-Hill, New York (2007) 10. Lee, L.H., Chew, E.P., Sim, M.S.: A heuristic to solve a sea cargo revenue management problem. OR Spectrum 29(1), 123–136 (2007) 11. Lee, L.H., Chew, E.P., Sim, M.S.: A revenue management model for sea cargo. International Journal of Operational Research 6(2), 195–222 (2009) 12. Løfstedt, B., Pisinger, D., Spoorendonk, S.: Liner Shipping Cargo Allocation with Repositioning of Empty Containers. INFOR special issue on Maritime Transportation (in press) (forthcoming) 13. Lu, H.-A., Chu, C.-W., Che, P.-Y.: Seasonal slot allocation planning for a container liner shipping service. Journal of Marine Science and Technology 18(1), 84–92 (2010) 14. Maersk Line: The answer is YES. Maersk Line News (June 1, 2010), http://www.maerskline.com/link/?page=news&path=/news/story_page/10/yes, accessed (June 17, 2011)

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15. Maragos, S.A.: Yield Management for the Maritime Industry. Ph.D. Dissertation, Massachusetts Institute of Technology (1994) 16. Stopford, M.: Maritime Economics. Routledge, New York (2009) 17. Talluri, K.T., Ryzin, G.J.V.: The Theory and Practice of Revenue Management. Kluwer Academic Publishers, Boston (2004) 18. Ting, S.-C., Tzeng, G.-H.: An optimal containership slot allocation for liner shipping revenue management. Maritime Policy & Management 31(3), 199–211 (2004) 19. Xiangzhi, B., Rongqiu, C., Li, L.: Container Slot Allocation Model with Liner Shipping Revenue Management. In: Jinlong, Z., Wei, Z., Xinping, X., Jianqiao, L. (eds.) Proceedings of ICM 2007, The 6th International Conference on Management Globalization Challenge and Management Transformation., August 3-5, pp. 130–137. Science Press, Wuhan (2007) 20. Zurheide, S., Fischer, F.: A revenue management slot allocation model with prioritization for the liner shipping industry. In: Hu, B., Morasch, K., Pickl, S., Siegle, M. (eds.) Operations Research Proceedings 2010. International Conference of the German Operations Research Society 2010, September 01-03, pp. 143–148. Springer, Heidelberg (2011) 21. Zurheide, S., Fischer, F.: A revenue management slot allocation model for liner shipping networks. In: Proceedings of the International Association of Maritime Economists (IAME) 2011 Conference, Santiago de Chile, Chile (accepted) (forthcoming)

Author Index

Absi, Nabil 86, 341 Ackermann, Heiner 1

Klaws, Jan 243 Kopfer, Herbert 1, 18 K¨ ufer, Karl-Heinz 1

Bebbington, Tom 286 Bennell, Julia A. 273 Bierwirth, Christian 74 Buer, Tobias 18 Cao, Jin Xin 233 Chen, Jiang Hang 233 Correa, Gabriel 183 Daduna, Joachim R. 29 Dauz`ere-P´er`es, St´ephane 341 Delgado, Alberto 286

Ma, Tai-Yu 59 Malliappi, Fotini 273 Mankowska, Dorota Slawa Mattfeld, Dirk C. 127 Meisel, Frank 74 Nolz, Pamela C.

256

74

86

Pacino, Dario 286 Petersen, Hanne L. 101 Potts, Chris N. 273

Eleyat, Mujahed 170 Ewe, Hendrik 1 Feillet, Dominique 86, 341 Fischer, Kathrin 321, 354 Großkurth, Peter 142 G¨ und¨ uz, Halil Ibrahim 44 Guo, Xi 221 Haijema, Ren´e 160 Haugland, Dag 170 Hetland, Magnus Lie 170 Hsu, Wen Jing 256 Hu, Hao 114 Hu, Lei 209 Huang, Shell Ying 221, 256 Jensen, Rune Møller Jin, Jian Gang 233 John, Gerlinde 321

Lau, Mei Mei 221 Lee, Der-Horng 233 Liu, Fan 256 Low, Malcolm Yoke Hean

286

Ropke, Stefan 101 Ruan, Tingting 114 Ruvalcaba, Loecelia 183 Shi, Xiaoning 209, 302 Speer, Ulf 321 Stahlbock, Robert 142, 243 Vogel, Patrick 127 Voß, Stefan 142, 209, 243, 302 Win, Cho Aye 256 Winkelkotte, Tobias

194

Zanella, Vittorio 183 Zehendner, Elisabeth 341 Zeng, Min 256 Zhang, Weigang 209 Zurheide, Sebastian 354

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