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• K is convex. (It is further a linear program if P is polyhedral and 9? is a linear function.) To be well-defined, a further requirement on the problem is that P n T is non-empty. The convexity and non-emptiness conditions are assumed to hold for the present. We shall further assume throughout that P is bounded, and that every vector p 6 P satisfies J2a£A [^o(/*) + Pa] > 0 for every compatible cycle Ac in the network, that is, there is no feasible toll vector such that a traveller can benefit by travelling a full cycle. These natural conditions ensure the existence of an optimal solution to [TOP] (under the above conditions), and that the algorithm proposed for solving it is well-defined. Example 5.1 Several optimization models of the form [TOP] are conceivable. We may wish to levy the least total toll in the network, whence we choose (p(p) = Y^atAPafa-i or Pre" cisely the opposite; the function 9? may also include a measure of the cost of implementing toll stations on the different links, which we naturally would like to minimize. D 5.2
An algorithm for the toll optimization problem
Solving the problem [TAP-VIP-SC-F]. We consider solving [TAP-VIP-SC-F] using a nonlinear pricing scheme which simultaneously generates /* and a vector (3 of multipliers for the side constraints. One such example is the (proximal) method of multipliers of Rockafellar (1978). Reasons for choosing such a scheme are, at least, threefold: (1) in the optimization setting, the corresponding augmented Lagrangean dualization schemes have been found to work very well in numerical experiments (Hearn and Ribera, 1980; Larsson and Patriksson, 1995a); (2) a vector /3 (and hence, through relation (4.1), a toll vector p € T) is obtained through a convergent sequence of dual vectors; and (3) even in the case of inconsistency (that is, P Pi T = 0), a toll vector with interesting properties is produced. (More on the last point below.) We refer the reader to Hearn and Ribera (1980), Patriksson (1994) and Larsson and Patriksson (1994 and 1995a) for further details on the properties of this class of schemes in the context of traffic problems. Within this scheme a sequence of subproblems of the form [TAP-VIP-F] are constructed and solved; in these problems, the mapping t is augmented by primal-dual terms associated with the side constraints. When selecting an efficient algorithm for their solution, we note
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these important points: (1) the algorithm should have efficient primal reoptimization capabilities, to utilize that successive subproblems differ only in the travel cost mapping; this suggests using an efficient primal algorithm; and (2) when solving the toll problem [TOP], we must have an explicit description of the set T of optimal tolls, or a good approximation thereof; this suggests the use of column generation/cutting plane approaches. The algorithm chosen is the disaggregate simplicial decomposition (DSD) method, which has been successfully applied in previous works on optimization versions of [TAP-VIP-F] (Larsson and Patriksson, 1992) and [TAP-VIP-SC-F] (Larsson and Patriksson, 1995a).3 We refer the reader to those references and Patriksson (1994, Sections 4.2.3, 4.3.5 and 5.3.5) for details on this method, but mention here that, at termination, the algorithm stores those routes in each OD pair that have a positive flow at some optimal route flow solution. In the application to [TAP-VIP-SC-F], then, the DSD algorithm will provide, for each OD pair (p, q), the routes in Tlpq which are utilized in one route flow solution h to [TAP-VIP-SC-F]; these routes will also [cf. (4.2a)] all be least-cost routes given the fixed link travel cost vector t ( f * ) + p, where p = Vg(f*)ft and ft is the vector of multipliers for (3.1) which is obtained in the limit of the multiplier scheme. A regularity condition. Using linear programming duality on (4.2a), the set T can be described as the set of solutions in p of the following linear system in p, ft and TT:
(5. la)
fFg(n ft
= 0, > 0;
(5.1d) (5.1e)
(5.1a)-(5.1b) is a dual description of the normal cone inclusion of (4.2a), and (5.1c)-(5.1e) is an explicit description of the corresponding inclusion of (4.2b). The constraints (5. la) render this system intractable in practice, since it contains as many constraints as there are routes in the network (cf. the dimensions of AT and F); the constraints actually needed, however, are defined by the subset corresponding to the union of the routes being at minimum cost for some multiplier vector /?, and should be comparatively very small. Larsson and Patriksson (1994) introduce a regularity condition on the solution to [TAPVIP-SC-F], so that the set of least-cost routes is invariant over the set of multipliers ft for the side constraints, and hence over the set p G T (cf. ibid, Thm. 2.6). The condition says that, for each such vector, the Wardrop conditions are satisfied with strict complementarity, that is, that all routes with the minimal cost are actually used. In this case, using the DSD algorithm for the traffic equilibrium subproblems in the multiplier scheme will yield all the information necessary to describe the (interesting subset of the) toll set T explicitly. If this is not the case, then the least-cost routes are not the same for all values of the multiplier vector. Although, as remarked earlier, the description of the toll set T requires the union of the least-cost routes over all values of the multiplier vector ft in the solution to [TAP-VIP-SC-F], the set of routes provided by the DSD algorithm will most probably 3 0ne complication with the DSD algorithm, as with any pricing scheme which includes shortest-route calculations, is that for some non-optimal multiplier values, negative cycles may appear. In such cases, one may either derive and incorporate a dual constraint in the scheme, or, preferably, continue the algorithm, ignoring the route generation stage.
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be a very good approximation of the total number of least-cost routes needed (it is, for example, quite conceivable that regularity holds at least for some OD pairs), and that is the main motivation for using the DSD algorithm in this context. Our limited numerical experience, which is described in Larsson and Patriksson (1998), also supports this belief. The case of primal inconsistency. If the traffic manager has set unattainable goals, then the result will be F D G — 0. Under this circumstance, the algorithm will still provide the manager with useful information. In fact, the algorithm will provide the smallest possible adjustment of the manager's aspiration levels so that the goals become achievable, and also provide the toll vector p = \/g(f)/3 that will achieve these slightly less ambitious aims. More formally, in the optimization setting, we may apply Theorem 4.1 of Golshtein and Tretyakov (1996, Sec. 3.4) to conclude that (a very small adjustment of) the algorithm stated above will yield a primal-dual pair (/, /5) which solves a perturbation of [TAP-VIPSC-F], in which the original restrictions (3.1) have been replaced by
9k(f) < £*,
k € /C,
the vector of e^ being that of smallest Euclidean norm which makes the problem consistent.4 Example 5.2 A traffic manager can utilize this scheme to evaluate several scenarios, possibly involving goals which are deliberately chosen to be unattainable. In contrast to the consistent case [cf. Remark 4.2.c], the resulting toll vector is sensitive to the introduction of scaling factors on the goals. Should the manager be uncertain if the goals are attainable, he/she can fulfill the most important goals while sacrificing others, through the use of different scaling factors on the goals according to their respective importance (cf. the use of aspiration levels and reference goals in multi-criteria optimization). d Having observed that the method detects primal inconsistency, and also in this case yields information that will be of use to the traffic manager, we move on to consider the problem [TOP] under the assumption that the problem [TAP-VIP-SC-F] is consistent. An algorithm for [TOP]. The above discussion can be concluded with the remark that, at termination of the multiplier algorithm, the link flow solution /* to [TAP-VIP-SC-F] and a toll vector p 6 T are given. The subset of routes generated within the DSD algorithm corresponds to a subset of the constraints in (5.la). With this as the starting solution and information about the toll set T, a conceptual algorithm for [TOP] proceeds as follows. We begin by noting that although the toll vector p is in T, it is not necessarily in the set P of feasible tolls. The fact that we do not know the set T explicitly further complicates the search for a toll vector which is simultaneously feasible (i.e., belongs to P) and optimal (i.e., belongs to T), should such a vector exist. The starting point in the solution of [TOP] is a Phase-I method, which attempts to find such a vector. With the vector p as a starting point, we search for a point in the intersection of P and the set of tolls described by the subset of constraints in (5.1) currently known. (We shall use the notation T for this set.) This is an example of the problem of finding a point in the intersection of a finite number of convex sets, and a variety of methods exist for its solution (e.g., Bauschke and Borwein, 1996). Preferably, we should always stay in the set P, since it is to be considered to be described by hard constraints, while the set T is described by 4
The perturbation e found is actually the projection of the origin onto the effective domain of the perturbation function.
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soft constraints. 5 We note that in the case where P is described by linear constraints, the Phase-I problem is an ordinary linear Phase-I problem, as solved within a simplex method. Two outcomes are possible from this procedure. In the first case, the procedure reports that there is a positive distance between the sets P and T. Since T C T holds, it must therefore be the case that the problem [TOP] is inconsistent, that is, P D T = 0. (We refer to this property as dual inconsistency.) The Phase-I method described above will, at termination, then have at hand a vector p which belongs to the set P (and is hence feasible), but which does not belong to the set T. This implies that the link flow f* solving [TAP-VIP-SC-F] is not possible to obtain through the use of a link toll scheme which is feasible with respect to the set P. (The toll vector obtained through the solution of [TAP-VIP-SC-F], for example, then clearly is infeasible.) We can, however, expect that by using the toll vector obtained, at least a subset of the goals described by (3.1) will be achieved, and hence, the toll vector obtained will achieve, by way of the Phase-I procedure, the highest aspiration levels possible within the limits set by the toll constraints. The second outcome possible is that the distance is reported to be zero, and then the output is a toll vector belonging to P fl T. The reader is, however, warned against interpreting this as a confirmation that the problem [TOP] is consistent. This toll vector, p, together with the vectors j3 and TT, obtained from the solution to the above Phase-I problem, is next used to investigate if p G T actually holds, by checking whether any constraints in (5.la) are violated. With the link costs £(/*) + p at hand, the most violated, if any, constraint in (5.la) for an OD pair (p, q) is obtained by finding the shortest route, r € Tlpq say; the constraints (5.la) for the OD pair are satisfied if and only if Sae^ <Wa[^a(/*) +Pa] ^ ^pq'j otherwise, the most violated constraint has been identified, and the corresponding route is added to the subset of lZ,pq currently known. Performing this test for each OD pair will either result in termination of the Phase-I procedure, with a toll vector p (E T at hand, or to the generation of constraints (routes) in some OD pairs. In the latter case, a better outer approximation T of T has been found, and the Phase-I problem is resolved. This repeated procedure will terminate finitely, since the number of routes is finite. At termination, the procedure either reports that P n T = 0, or provides a vector p in this set. Having already discussed the case of dual inconsistency, we henceforth assume that the Phase-I procedure was terminated with a toll vector in PnT, 6 and hence the problem [TOP] is consistent. We next turn to the Phase-II part of the algorithm for [TOP]. In the second phase, we solve the modification of [TOP], where T is replaced by T.7 Let (jo, /?,7r) be a solution to this problem.8 We next investigate if this solution solves [TOP], by checking whether any constraints in (5.la) are violated. This is done in precisely the same manner as described above in the Phase-I procedure. Performing this test for each OD pair will either result in termination, with (p,/3,vf) being a proven optimal solution to [TOP], or to the generation of constraints (routes) in some OD pairs. In the latter case, a better outer approximation of T has been found, and (p, /3,7f) is used as an infeasible starting solution when solving the improved approximation of [TOP]. Obviously, since the 5 We can envisage the set P to be described by much simpler constraints than those describing T, and hence such a procedure should be realistic. 6 This solution also provides an upper bound on the optimal value of [TOP]. 7 We do not discuss particular methods for the solution of these restrictions in this paper; the adequate method will be a function of the properties of the side constraints and the toll restrictions. 8 Since (p is optimized over the intersection of P and an outer approximation of T,
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number of routes is finite, this procedure is finite, and its output is a feasible and optimal toll vector, p*, to [TOP].9 The algorithm stated here is by no means the only one possible; an alternative constraint generation approach would be the result of using an aggregate simplicial decomposition (SD) algorithm for the [TAP-VIP-F] type subproblems in place of the DSD algorithm. The route generation process is then replaced by the generation of all-or-nothing solutions. Convergence of the disaggregate algorithm given should, however, be much faster, since the latter algorithm corresponds to aggregating constraints generated in the former.
6 6.1
APPLICATIONS Capacity constraints
In the case of capacity constraints (see Example 3.1), the toll pa equals the Lagrange multiplier (3a associated with the corresponding constraint in (3.2); see Example 4.1. Algorithms for the generation of /* solving this special case of [TAP-VIP-SC-F] appear in Hearn and Ribera (1980) and Larsson and Patriksson (1995a). Both of these are multiplier methods which simultaneously provide a multiplier vector /?; the latter utilizes the DSD algorithm for the subproblems, and is therefore an instance of the algorithm for [TAP-VIP-SC-F] described above. Ferrari (1995) considers an elastic demand model with environmental capacity constraints, and establishes the existence of an equilibrium which satisfies the capacity constraints in the presence of proper additional costs on links that would otherwise produce unacceptable environmental damage. 6.2
Obtaining a given flow solution
An interesting special case of a goal described by a set of side constraints is that of achieving an a priori given flow solution /*. We may consider several instances of such solutions; for example, /* may be a calculated system optimal flow or be based on an observed flow. The model [MPEG] applies to such models, with the set of side constraints described by &(/«):=/„-/.' = (),
a€A.
(6.1)
Tracing the above analysis, we find that the normal cone mapping NG is non-empty exactly at /*, where it equals Jftl^L Hence, the requirement (4.2b) poses no further restrictions on p. This implies, in turn, that the solution of [TOP] is simplified substantially. Example 6.1 Let /* be the solution to [TAP-VIP-SC-F]. Using model (6.1) will result in a larger freedom in choosing the tolls compared to that given by the toll set T; it will then be possible to levy a toll on a link which does not contribute to the saturation of any flow restrictions. d If the given flow /* does not belong to F, then it is not possible to achieve with any toll vector. In this case, the use of the multiplier scheme outlined in Section 5.2 will provide the toll which achieves the demand-feasible flow which is closest in Euclidean norm to f * . So, again we observe that the traffic management scheme given in this paper provides valuable information also in the case where the goals are unattainable. 9 Convergence of this algorithm can also be monitored by the bounds on the optimal value of [TOP] generated within the algorithm, and which are remarked upon in previous footnotes.
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95
System optimal flows
Obtaining a system optimal flow by means of link tolls is a classic topic, with contributions tracing back to Pigou (1920). The traditional approach is marginal cost pricing, in which the toll levied equals the marginal cost minus user cost. Dafermos and Sparrow (1971) observe that this toll is not the only one possible, and that it may have certain drawbacks. (For example, a toll will be levied on every link.) They proceed to describe the set of toll vectors which will yield the desired link flow solution, and discuss possible secondary goals to direct the choice of toll vector. Alternative tolls for achieving a system optimal flow have much later been considered also by Bergendorff et al. (1995 and 1997). BergendorfT (1995) describes a particular case of capacitated model, wherein the upper bounds are the system optimal link flows. Based on this model, she develops a toll optimization problem similar to the one in this paper, and presents a method, working in the space of commodity link flows and node prices, for solving linear instances of [TOP]. Bergendorff et al. (1997) present characterizations of the set of toll vectors additional to those of Dafermos, under various assumptions on the travel costs. It is to be noted that the results reached in these references are, in essence, special cases of the results obtained in this paper, obtained by choosing side constraints of the form (6.1) with /* being a system optimal solution. By the generality of the traffic management scheme described in this paper, we can consider more general considerations than the achievement of system optimal flows; for example, we can introduce restrictions of the form (6.1) only on a subnetwork, thus achieving an approximate system optimal flow on a subset of the links, while the rest of the network is governed by the user optimality principle; the resulting model is reminiscent of the combined equilibrium models used in the modelling of vehicle guidance systems. 6.4
Network improvement, subsidies and the redistribution of resources
Consider a situation in which the traffic manager would like to achieve a given link flow solution /*. The absence of any conditions on the toll vector p other than (4.2a) makes it possible that some link tolls may be negative. An interesting network design model emanates from this fact. Indeed, while interpreting a positive value pa as a toll levied on link a, a negative value of pa may be interpreted as the travel time improvement that is necessary in order to make link a more attractive to some travellers; this information may be used, for example, in the calculation of necessary investments in network capacity. Note that the network design problem stated here has the reverse viewpoint to that of the traditional one. Example 6.2 Side constrained traffic equilibrium models can be used to derive actions that favour public transport. Denoting the OD flow for private (public) transport by dpr (d pu ), side constraints describing a favourable situation for public transport could, for example, be constraints of the form
7P,< - <£" < 0,
(p,g)€C,
where jpq > 0 is the least ratio of public transport flows over that of private transport in OD pair (p, ); other possibilities include constraints describing maximal travel times for public transport vehicles. The Lagrange multipliers associated with these constraints in the resulting side constrained two-mode traffic equilibrium model are then used to derive (1) an additional
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queueing delay for private vehicles, or (2) a subsidy to the public transportation sector. If, further, toll constraints of the form P = {p 6 ^\c\ \ E(p,,)ec[p?J + tf£\ = 0 } are introduced in the model, then the subsidy takes the form of a reallocation of resources from the private to the public sector; the subsidy is realized, for example, through tolls levied on private vehicles only, through the design (or, improvement) of designated lanes for the public transport vehicles, or through lower ticket prices. D 6.5
Extensions and further research
In Larsson and Patriksson (1998), the present model is extended in various directions; it is, in particular, shown how elastic demands will affect the toll optimization problem, and how commodity-specific traffic management goals can be fulfilled through the use of consistent, commodity-specific, toll schedules; a particularly interesting model in the latter case is the stohastic user equilibrium model. The paper also discusses some additional properties of the sets of tolls for the fixed and elastic demand problems, and present limited numerical experience with the fixed demand model on some small and medium scale problems. The price-directive traffic management schemes described here may possibly be applied in other areas, where society through prices (taxes or subsidies) may influence markets. Examples of areas of application include protective duties in international trades, and environmentally derived taxes and subsidies. This paper provides the basis for several interesting paths of research, both in directions of theoretical research and applications, several of which are mentioned.
ACKNOWLEDGEMENTS The authors were sponsored by the Swedish Transport and Communications Research Board (Grants KFB 95-118-63 and 96-206-33). The authors thank Pia Bergendorff for inspiring work, and Drs. Jan T. Lundgren and Laura Wynter for constructive remarks; the second author thanks Prof. R. Tyrrell Rockafellar for valuable, continuing discussions.
REFERENCES Bauschke, H. H. and Borwein, J. M. (1996). On projection algorithms for solving convex feasibility problems. SI AM Review, 38 (to appear). Bazaraa, M. S., Sherali, H. D., and Shetty C. M. (1993). Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York, NY, second ed. Bergendorff, P. (1995). The bounded flow approach to congestion pricing. Master's thesis, Department of Industrial & Systems Engineering, University of Florida, Gainesville, FL. Bergendorff, P., Hearn, D. W., and Ramana, M. V. (1997). Congestion toll pricing of traffic networks. In: Network Optimization, Proceedings of the Network Optimization Conference, University of Florida, Gainesville, FL, February 12-14, 1996 (P. Pardalos, D. W. Hearn, and W. W. Hager, eds.), pp. 51-71. Lecture Notes in Economics and Mathematical Systems, vol. 450, Springer-Verlag, Berlin. Dafermos, S. and Sparrow, F. T. (1971). Optimal resource allocation and toll patterns in useroptimised transport networks. Journal of Transportation Economy and Policy, 5, 184-200. Ferrari, P. (1995). Road pricing and network equilibrium. Transportation Research, 29B, 357372. Golshtein, E. G. and Tretyakov, N. V. (1996). Modified Lagrangians & Monotone Maps in Optimization. John Wiley & Sons, New York, NY.
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Hartman, P. and Stampacchia, G. (1966). On some non-linear elliptic differential-functional equations. Ada Mathematica, 115, 271-310. Hearn, D. W. and Ribera, J. (1980). Bounded flow equilibrium problems by penalty methods. In: Proceedings of the 1980 IEEE International Conference on Circuits and Computers, pp. 162-166. Larsson, T. and Patriksson, M. (1992). Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transportation Science, 26, 4-17. Larsson, T. and Patriksson, M. (1994). Equilibrium characterizations of solutions to side constrained asymmetric traffic assignment models. Le Matematiche, 49, 249-280. Larsson, T. and Patriksson, M. (1995a). An augmented Lagrangean dual algorithm for link capacity side constrained traffic assignment problems. Transportation Research, 29B, 433-455. Larsson, T. and Patriksson, M. (1995b). On side constrained models of traffic equilibria. In: Variational Inequalities and Network Equilibrium Problems, Proceedings of the 19th Course of the International School of Mathematics "G. Stampacchia," Erice, Italy, June 19-25, 1994 (F. Giannessi and A. Maugeri, eds.), pp. 169-178. Plenum Press, New York, NY. Larsson, T. and Patriksson, M. (1996). Side constrained traffic equilibrium models—analysis, computation and applications. Revised for Transportation Research, B. Larsson, T. and Patriksson, M. (1998). Side constrained traffic equilibrium models—traffic management through link tolls. In: Equilibrium and Advanced Transportation Modelling (P. Marcotte and S. Nguyen, eds.). Kluwer Academic Publishers, New York, NY (to appear). Migdalas, A. (1995). Bilevel programming in traffic planning: models, methods and challenge. Journal of Global Optimization, 7, 381-405. Nagurney, A. (1993). Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers, Dordrecht, The Netherlands. Patriksson, M. (1994). The Traffic Assignment Problem: Models and Methods. Topics in Transportation Series, VSP, Utrecht, The Netherlands. Pigou, A. C. (1920). The Economics of Welfare. Macmillan & Co, London. Rockafellar, R. T. (1978). Monotone operators and augmented Lagrangian methods in nonlinear programming. In: Nonlinear Programming 3, Proceedings of the Nonlinear Programming Symposium 3, University of Wisconsin-Madison, Madison, WI, July 11-13, 1977 (O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds.), pp. 1-25. Academic Press, New York, NY. Yang, H. and Bell, M. G. H. (1997). Traffic restraint, road pricing and network equilibrium. Transportation Research, 31B, 303-314.
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IMPROVED ALGORITHMS FOR CALIBRATING GRAVITY MODELS
Nanne J. van Zijppl Transportation and Traffic Engineering Section Delft University The Netherlands Benjamin G. Heydecker Centre for Transport Studies University College London LONDON United Kingdom
-Abstract- Knowledge of Origin-Destination (OD) trip matrices is needed in many stages of transport planning. As direct observations of OD-flows are usually not available, a commonly used approach is to estimate OD-matrices from traffic counts. The under-specification that characterizes this estimation problem can be resolved by requiring that OD-matrices conform to a model of travel demand. In this paper we consider a class of methods based on the gravity model, with either an exponential or a discretized deterrence function. We focus on solution methods and establish a unifying analysis of them. A key result presented in the paper is an expression for the matrix of second derivatives of the likelihood objective function. Based on this result, a minimization method is developed with a greatly improved speed and convergence. Numerical experiments based on a set of empirical data obtained from a French tollroad have shown that speed has improved by a factor that typically is in the range 6 - 1 5 and can in some cases be as large as 400 relative to existing methods. The new method has good robustness in that it converges successfully under a wide variety of circumstances including those under which other methods fail. Moreover, this result provides an effective convergence criterion that can also be applied in combination with other solution methods, such as Gauss-Seidel and gradient search methods.
INTRODUCTION The paper deals with the calibration of gravity models as a means to estimate Origin-Destination (OD) trip matrices from aggregated data sources such as traffic counts and trip-end con1. A large part of the work on which the paper is based was done while the first author stayed at the Center for Transport Studies, University College London
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N. J. van Zijpp and B. G. Hey decker
straints. Generally, many OD-matrices can be constructed that reproduce a given set of observations. Therefore the under-specification that characterizes the OD-estimation problem needs to be resolved. This can be done by requiring that OD-values conform to a model of travel demand. Several approaches of this kind have been discussed in the literature and are being used in practice. In the following we consider a class of methods based on the gravity model, with either an exponential or a piecewise constant deterrence function. We focus on numerical methods to calibrate these models from traffic counts and establish a unifying analysis of them. In the past it has been shown that the gravity model can be underpinned in a number of ways, see e.g. Erlander and Stewart (1990). We confine ourselves here to specifying the gravity model and the related distributions of both model residuals and observation errors. A consequent likelihood expression is established which we seek to maximise. We concentrate on methods that can be used for this purpose. Existing methods are discussed, and it is shown that some of the difficulties that occur when applying them can be resolved by a change of variables to logarithms of the model parameters. A key result is the derivation of the matrix of second derivatives of the likelihood function. This result is then applied and leads to a new optimisation method. In a series of experiments this method is shown to be robust and computationally efficient.
MODELLING Model Description The description of the gravity model that is used here is given by:
/=1,2,... 7,7=1,2,... J
(1)
where: Tjj Iff T™. By Oj, Dj y(.) Cjj
OD-flow from zone / to zoney number of origins, number of destinations model estimated OD-flow modelling error production ability, attraction ability deterrence function generalized travel costs
We use a normally distributed error term with variance equal to the associated mean as motivated by a Poisson process for the traffic counts. Various functional forms for the deterrence function^-) can be used; see e.g. Ortuzar and Willumsen (1994) for an overview. Two frequently used classes of deterrence functions are considered:
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Case 1: The exponential deterrence function
= exp[ -a Cjj ]
fay)
(2)
Where a is the dispersion parameter. Case 2: The piecewise constant deterrence function H c
^ (/)
=
ZF*8*
(3)
h =l
where H is the number of cost ranges, F/, is the range specific cost, and &h(cij) is given by: 1, if Cjj is in cost range h. 0, otherwise.
Traffic Counts The model parameters are to be estimated from traffic counts or other aggregated flow observations. In our analysis, traffic counts are summarised in a vector >> and are assumed to be linear combinations of OD-flows: y = AT+t,
(4)
with: y T x(ij) ctyk A £,
vector of traffic counts (with K elements) vector of OD-flows. ordering of the matrix elements 7). in the vector T. proportion of flow from / toy that traverses link k matrix of assignment proportions, ^,xOV) ~ aijk> ^=1,2,.. Jf vector of counting errors
For the counting error E, a zero mean multivariate normal (MVN) distribution with covariance matrix Q is assumed. It should be noted that the above definition of traffic counts allows for a wide variety of data to be used, including trip-ends, screenline counts, and even survey data and prior matrices.
Maximum Likelihood Approach to Model Calibration As a result of the assumptions given in (1) the conditional distribution of OD-vector T, given the model parameters is given by:
\
(5)
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with: O, D, F T"1
vector of model predicted OD-flows vectors of production abilities, attraction abilities and deterrence function parameters) respectively. diagonal matrix with the elements of J1" on its main diagonal.
i.e. the distribution of Tis given by a multivariate normal distribution with expected value Tm, and covariance matrix T"1. Combining equations (5) and (4) results in an expression for the conditional distribution of the traffic counts given the model parameters: y ~ MVN(A TAT" A '+Q)
(6)
where J™ should satisfy the requirements that follow from (1). In theory, the parameters O, D and F can be found by maximising the joint likelihood. In that case these parameters are given by: argmax D = JL „ MVN(4 T"A l^A'+O) |
(7)
Simplifying the Likelihood Objective Function In (7) both the expectation and the covariance matrix of the density that is to be maximised depend on the model parameters. In the present paper we discuss algorithms that can be used to maximise the likelihood if the covariance matrix is approximated with a fixed covariance matrix Z. The approximation S may be obtained by substituting a prior estimate of the OD-flows into equation (7) or by assuming that Z is a diagonal matrix with the observed values >> on its diagonal. In Van Der Zijpp and Heydecker (1996b) an iterative extension of these algorithms is described that can be used solve (7). The problem with the fixed covariance matrix corresponds to:
0>0,D>0,F>0
(8) There are several advantages in formulating the OD-estimation in this way: • The traffic counts that are used may represent arbitrary linear combinations of OD-cells, • The objective function offers the possibility to calibrate a gravity model, even if no explicit data are available on trip ends and/or trip length distribution, • The possibility exists to specify the variance of counting errors, or to weigh the observations according to the confidence in them, • Inconsistencies in traffic counts as a result of error need not be removed prior to calibration, • Spatial dependencies that occur due to the fact that the model prediction error, e^-, is observed at several locations can be accommodated (see Van Der Zijpp and Heydecker, 1996b).
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EXISTING ALGORITHMS FOR CALIBRATING THE GRAVITY MODEL The method implied by equation (8) has a number of desirable properties such as its flexibility with respect to input data and the possibility to weigh various data items according to their assumed accuracies. However, a problem is that in order to solve (8) an algorithm is needed for a highly non-linear minimization problem subject to non-negativity constraints. The absence of an algorithm that performs this task under a sufficiently wide variety of circumstances might very well explain the relative unpopularity of this method compared to, for example, entropy maximising methods. This section discusses a number of numerical methods that can be used to solve (8). For a more elaborate overview, see Van Der Zijpp and Heydecker (1996b).
Balancing Methods The bi-proportional method starts with an initial or prior matrix. This matrix is subsequently scaled row by row and then column by column to yield the observed or given trip end totals. The iterations stop if no more substantial changes occur in the estimated matrix, i.e. when the observations are reproduced closely. An equivalence with the gravity model with exponential deterrence function exists if in the initial tableau the cells are replaced with values derived from an exponential deterrence function. Note that the bi-proportional method cannot be used to find appropriate values for a in (2). Instead Hyman's method (Hyman, 1969) can be used to estimate a. A prerequisite to apply Hyman's method is that the average trip length is known. Over and above the assumptions of the bi-proportional method, in the tri-proportional method the prior matrix is scaled to force the estimated matrix to match an observed trip length distribution (OTLD). An equivalence with the gravity model with piecewise constant deterrence function is reached if the prior matrix is replaced with a matrix of ones. In a balancing method, the iterations continue until the estimated matrix reproduces all observations. A requirement for convergence of the bi- and tri-proportional balancing methods is that such a solution exists, i.e. that the traffic counts that are used are mutually consistent. The analogy between balancing methods and other gravity model calibrating methods disappears if an initial matrix is scaled to meet measurements that do not represent trip ends, as is done in entropy maximizing approaches such as ME2 (Ortuzar and Willumsen, 1994; Willumsen, 1991; Van Zuylen and Willumsen, 1980). Similarly, if an initial tableau is derived from a prior matrix the solution resulting from applying a balancing method no longer satisfies the gravity model. Balancing methods can hence only be applied to find a candidate solution to (8) in cases where: • observations and constraints represent trip ends or observed trip length distributions, and • observations are mutually consistent. In this case, the solution that will be found will reproduce the observations exactly and will hence correspond to a zero value of the objective function (8), which is its global minimum. Balancing methods are popular due to their ease of implementation. However, convergence can be slow (see Maher, 1983) if not uncertain. Proofs of convergence apply only to special cases, e.g. only trip-ends are given and the deterrence function is kept constant, see Evans (1971) and
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Kirby (1974). In the more general case where observations represent traffic counts and may not be mutually consistent, other techniques should be considered, two of which are discussed below.
Gradient Search Methods If we consider a model with a piecewise constant deterrence function, see (1) and (3) it is possible to find an analytical expression for the derivatives of the objective function (8) with respect to the model parameters Of, Dj, Ff, and a. The resulting gradient vector can be used as a basis for an iterative scheme to minimise (8). The simplest way this can be done is to apply a steepest descent method. However, steepest descent method tend to converge very slow. A gradient search methods with a higher speed of convergence is the conjugate gradient method, see Bazaraa et al (1993). An easy way to find conjugate gradient search directions was proposed by Fletcher and Reeves (1964). There are however a number of difficulties that occur when a conjugate gradient method based on Fletcher and Reeves is applied to minimise (8): • The solution to the line searches cannot be expressed in an analytical way. Hence, the line searches must be done numerically, • The parameter space is constrained to non-negative values. This makes the line search more complex, and it also invalidates the Fletcher-Reeves method to find conjugate gradients, • It is difficult to establish an effective convergence criterion, • Convergence can be extremely slow. These factors make it difficult to construct a gradient search method that operates in a reliable way under all circumstances.
Gauss-Seidel Methods Another way to tackle the minimization problem (8) is to divide the model parameters in groups and to solve the parameters in each group separately, while keeping the parameters in other groups temporarily constant. This iterative process is continued until a fixed point is found. For problem (8) it seems advantageous first to solve the vector of production abilities (0), then to solve the vector of attraction abilities (£>), and then to solve the vector of deterrence function values (F). This procedure can only be used to calibrate a gravity model with a piecewise constant deterrence function (case 2). Each step in this sequence corresponds to minimizing a quadratic function. If the nonnegativity constraints can be ignored, this can be solved at the expense of one matrix inversion, so computationally costly numerical line searches can be avoided. The details of the method are described in Van Der Zijpp and de Romph (1996). Although the method works well in a large number of cases and is easy to implement with the help of modern software packages, there are a number of technical difficulties: • No theoretical proof of convergence exists, • The minimization problems can be expensive to solve if the nonnegativity constraints come into play, • The method does not calibrate gravity models with exponential deterrence functions,
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• It is difficult to establish an effective convergence criterion.
A UNIFYING ANALYSIS OF THE ML-OBJECTIVE FUNCTION We now show how, by using a logarithmic transformation, the gravity model formulations presented here can be unified and solved conveniently. It has been shown earlier (see e.g. Sen and Smith, 1995) that both the gravity model with exponential deterrence function (equations (1) and (2)), and the gravity model with piecewise constant deterrence function (equations (1) and (3)) can be summarized in the following form:
(9) where exp(.) q
C
is the exponentiation operator, applied element by element is the vector of logarithms of the model parameters, i.e. D F is a matrix with three nonzero elements per row. The first and second nonzero elements are given by respectively:
Cx(iJ)J+j = 1
(10)
The way the third nonzero element is defined depends on whether the deterrence function is specified to be exponential or piecewise constant: Case 1: The exponential deterrence function
(H) Case 2: The piecewise constant deterrence function C",. , , , +,h = 5,(c..) x(i,j),I+J h ^ ij' ,' h=\,2,...H ' '
v (12) '
In summary, the problem of estimating OD-matrices from traffic counts using a gravity model combined with either an exponential or a piecewise constant deterrence function is equivalent to the minimization of the following objective function:
(13) Compared to the formulation given in (8) this change of variables has the following practical advantages:
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• A single objective function can be used to represent methods based on exponential and piecewise constant deterrence functions, • The minimization is not subject to nonnegativity constraints, • Ease of notation and analysis. This advantage will become apparent in the next section, where expressions for the gradient and the Hessian (the matrix of second derivatives) of the objective function with respect to the vector q will be derived.
DERIVING THE GRADIENT AND HESSIAN OF THE OBJECTIVE FUNCTION Definitions, Lemmas and Proofs For a scalar valued function/of a vector valued variable x we will denote the gradient and the Hessian with respect to x with the symbols Vx[/"(x)] and V2x[/"(x)] respectively. Note that the Hessian of/ i.e. the matrix of second derivatives off, is iheJacobian of the gradient of/, i.e.: V2,[X*)]=Vx[Vx[Xx)]]
(14)
where the outer Vx is applied elementwise. Define the diag(.) operator as the operator that transforms a vector into a diagonal matrix by putting its elements on the main diagonal. Furthermore, define the array multiplication with: C=A®B o Cjj^AjjBpVij
(15)
In connection with these definitions, the following lemma's are proven: Lemma 1: for any two vectors a and b and matrix C of appropriate size, the following holds: diag(a) C diag(A) = (a b')®C (16) Proof (Lemma J) D = diag(a) C diag(2>) o Dtj = afcCy, ViSJ oD = (a 6')®C
( 1 7)
Lemma 2: let/be a vector valued function of x of appropriate size, then: Vx[ diag(x)/x) ]=diag(Xx))+Vv[./(x) ]diag(x)
(18)
Proof (Lemma 2): According to lemma 1: diag(x)Xx)=diag(x)./(x) diag(l)=x®/x)
(19)
Hence:
iag (*)/(*>). dx Tdx.
a/.(x)
+x Jj^J - = dx. ^fj^ dx.
(20)
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Or, in matrix notation, with e being a vector of ones, and lemma 1 applied once more: VJ diag(x)Xx) ]=VJ x ] 0 (X*XH« x') ®vJ>W 1 x[/x) ] diag(x) x) ]diag(x)
(21)
Computing the Gradient of J(q) The gradient of the objective function J(q) defined in (13) satisfies:
(22) Referring to equation (13), we substitute Vqt(q) =C'diag(0 and V, J (i) =2A' ~L'l(At-y) in (22). This yields: V, J(q) = 2C"diag(/K Z '\At-y)
Computing the Hessian
(23)
ofj(q)
Using equation (23) it can be seen that the Hessian ofJ(q) satisfies:
' Z -\At-y) ] =C'diag(OV/[ 2C'diag(/M' S '1(^->') ]
(24)
Letg(/)=diag(f>4' Z"1^^), then: V^tJlKTdiagCOV,! 2C'g(t)] =C'd\ag(t)Vt[g(t)]Vg(C'g] =2C'diag(/)V,[ diag(?M' S '\At-y) ]C
(25)
Now we evaluate V,[ diag(0^' Z "k^^) ]• Applying lemma 2 gives: V,[ diag(/>4' Z -\At-y)] - diag(^' Z '\At-y)) + Vt[ A' Z \At-y) ] diag(0 = diag(,4' Z •1(^/->'))+>4' Z ~1A diag(/)
(26)
Hence: ]=2C'[diag(/)diag(^' Z -1(^^))+diag(?M' Z '1A diag(/)]C
(27)
Equations (23) and (27) give insight into the performance of existing solution methods, and provide a foundation for new ones.
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Implications for Existing Methods The gradient search methods and Gauss-Seidel methods discussed in this paper can be used effectively to find stationary points of an objective function. Whether such a point represents a local minimum or a saddle point can easily be determined by evaluating the Hessian at that point: a local minimum only exists if the Hessian is positive definite (PD). Therefore result (27) can be used as a useful addition to existing methods: it supplies a powerful convergence criterion. However the analysis needed to determine if a particular local minimum to objective function (13) is also a global minimum is more intricate; this problem is still open to investigation.
A NEW ALGORITHM TO CALIBRATE GRAVITY MODELS It is possible to define a new numerical method to minimize (13) using expression (27) for its Hessian. This method is based on a quadratic approximation of the objective function of the form: J*(<7+e)=J(<7) + 6'. V?[ J(q) ] + V2 e' V\[ J(q) ] s
(28)
This expression can be minimised over the vector s at the cost of one matrix inversion, provided that the Hessian of ,/ evaluated at point q is PD. An iterative method based on this strategy is known as the Newton method. The advantage of this method is that each successive iterate can be found analytically, so no numerical line search procedure is needed. Two factors should be taken into account when applying this method to minimize objective function (13): • The Hessian V2g[ J(q) ] need not necessarily be strictly PD for all q. Therefore the quadratic approximation of the objective function, J*, may not be bounded below. In cases where it is not, the next iterate would not be defined. Therefore a modified Newton will be used to ensure that each successive iterate exists and will reduce the objective function. In the present paper, we adopt the Levenberg-Marquardt type of method, see Bazaraa et al (1993), p312. This method will be explained below. • The vector q contains the logarithms of the model parameters. This ensures that the nonnegativity requirements are satisfied. However there does not exists a bounded value of q that corresponds to a solution to (8) that contains one or more zero parameter values. Although the value of zero can be approximated arbitrarily closely, it may not be possible to represent such small numbers internally in a computer, resulting in problems of convergence caused by rounding errors or underflow. Therefore the following penalty function is added to the objective function: ,-c 2 -g.) 3
(29)
This function is continuous in its first and second derivatives, and prevents the problems mentioned above. The values that were chosen for c\ and c^ were 100 and 20 respectively. The idea behind the Levenberg-Marquardt method is that even if the Hessian in a particular point is not PD, a next iterate with a lower objective function value can be found by performing a small step in the direction of the negative gradient. This is effected by adding a multiple ji of
Improved algorithms for calibrating gravity models
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the identity matrix I to the Hessian in (28), before minimizing this expression. The values of u, are modified between iterations using a strategy that is illustrated in figure 1. The next value of u, depends strongly on the ratio R of the true improvement and the predicted improvement. If this ratio is favourable then the value of |i is decreased; otherwise it is increased. The thresholds that were used are given in Bazaraa et al (1993).
NUMERICAL EXPERIMENTS Numerical experiments have been performed on a set of empirical trip data obtained from a French toll road. These data consist of an observed trip table for eight different days between 29 origins and 26 destinations, resulting in 570 OD-flows that contribute to link-flows observed in the study area, and 107 traffic volumes have been synthesized from them. The variance of each of these traffic counts was taken to be equal to its observed value. There is no route choice in this network because it is linear. Two types of experiments were undertaken. The first series of experiments involved estimating an OD-matrix by calibrating a gravity model on the traffic counts. This was done for a model with a piecewise constant deterrence function, see eq. (3). The number of distance classes that were distinguished was increased progressively from 1 to 16. Each additional class was formed by sub-dividing an existing one, so that a hierarchical sequence of models was constructed. From the methods discussed in this paper, the conjugate gradient, Gauss-Seidel and modified Newton method were implemented to minimise the objective function. The conjugate gradient search method turned out to be extremely slow, which prevented the completion of a full set of experiments for this method. In table 1 the results were summarized for the Gauss-Seidel method and the modified Newton method. Each row in this table contains the results of calibrating the model using a deterrence function with the corresponding number (N) of classes, averaged over the eight different days for which observed data were available. The model prediction error could be computed by comparing the model estimated OD-flow with the value contained in the toll-ticket database, and is expressed using the Root Mean Squared Error (RMSE). The log-likelihood value is denoted with L, and decreases with the addition of each extra class. The other columns in the table contain the average CPU-time (on a Pentium 90) and the number of cases for which the algorithm did not converge before a preset maximum number of 2000 iterations was reached (indicated with the symbol E). In all cases, a solution with an equal or higher likelihood value was computed with the modified Newton method. Moreover, with this method a solution that satisfied the convergence criterion was reached without exception, whilst the Gauss-Seidel exceeded the maximum number of iterations in 12 of the 128 cases. The main advantage of the modified Newton method is its speed: none of the cases that were considered took more than 290 seconds, whilst the Gauss-Seidel took 2975 seconds in one case; the average values were 106 and 626 seconds respectively. The experiments also revealed that for this particular dataset the best fit according to the RMSE measure is obtained if a deterrence function with six classes is used. Unfortunately, in practice it is not possible to determine this number as one needs a completely observed trip matrix to compute this measure. This issue is discussed in Van Der Zijpp and Heydecker (1996a).
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initialize:
no
yes
no
yes
no
yes export q^^
Figure 1:
Overview ofLevenberg-Marquardt method (modifiedNewton method)
111
Improved algorithms for calibrating gravity models Table 1: Estimation results based on traffic counts for the number of classes ranging from 1 to 16 N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Gauss-Seidel RMSE L av.CPU 537.33 -17443.93 321 485.57 -11308.71 245 470.82 -8184.56 255 478.93 -7514.64 984 387.47 -5176.33 506 380.23 -4940.86 202 381.40 -4934.20 113 639.94 -4539.26 1181 805.85 -4229.48 1648 783.25 -3533.62 1099 840.49 -3467.37 634 840.47 -3467.37 56 843.22 -3459.97 436 872.92 -3427.43 439 700.29 -2999.34 724 777.60 -2719.70 1176 tot.av.CPU 626 max. CPU 2975
E
3 5 2
2
RMSE 536.91 486.09 470.81 480.46 387.45 380.18 381.46 643.58 806.40 784.37 839.90 839.90 843.65 874.67 699.89 777.74
Newton L -17443.90 -11308.68 -8184.55 -7182.98 -5176.31 -4940.86 -4934.19 -4538.52 -4227.91 -3533.28 -3467.30 -3467.30 -3459.91 -3427.38 -2998.97 -2719.48
av.CPU E 77 223 151 86 166 61 37 94 64 150 61 33 94 51 137 206 106 289
Table 2: RMSE fit of the gravity model for the number of classes ranging from 1 to 16 Gauss-Seidel N RMSE av.CPU 1 329.36 307 2 273.12 408 3 235.05 946 4 224.08 769 5 216.94 807 6 215.03 1191 7 214.74 1392 8 213.83 837 9 212.45 1055 10 212.37 1093 11 207.30 1664 12 205.97 1697 13 203.40 1891 14 202.53 2422 15 196.48 4165 16 196.23 3180 tot.av.CPU 1489 max. CPU 4671
E
1
1 2 1 2 2 7 3
RMSE 329.36 273.12 235.05 224.08 216.94 215.03 214.74 213.83 212.45 211.28 206.71 205.87 203.20 202.49 196.35 196.22
Newton av.CPU 14.82 10.29 11.08 9.75 10.38 8.23 7.83 7.74 10.80 8.86 9.25 8.63 9.20 8.33 13.13 8.82 9.82 16.42
E
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In a second series of experiments each of the 570 observed CD-flows was used as a measurement. This is similar to the case in which a prior matrix is available. When calibrating the gravity model on these data the solution with the highest likelihood value and that with the lowest RMSE coincide: the resulting RMSE is known as the model specification error, see Van Der Zijpp and Heydecker (1996a). The results of the calibration are shown in table 2. In general, the computation time needed for the Gauss-Seidel method was greater than the corresponding one in the first series, whilst the average computation time needed for the modified Newton method was substantially less. Again the Newton method converged in all cases, whilst the Gauss-Seidel did not meet the convergence criterion in 19 of the 128 cases. The maxima of the computation times over these 128 computations are 4671 seconds for the Gauss-Seidel method and 16.42 seconds for the Newton method; the average values were 1489 and 9.82 seconds respectively.
CONCLUSIONS It has been shown that the problem of calibrating a gravity model is equivalent to the minimization of a highly non-linear objective function. This formulation has a number of advantages over more traditional ones, including the possibility to use arbitrary traffic counts and constraints, not requiring the presence of an observed trip length distribution (OTLD) and taking into account counting errors and spatial dependencies. However, the minimization of this objective function is intricate. Specific difficulties are the presence of the nonnegativiry constraints, slow convergence, the absence of an adequate convergence criterion, and the complexity of implementation. The problem of the nonnegativity constraints can be avoided by a change of variables to the logarithms of the model parameters. This formulation unites methods with exponential and piecewise constant deterrence functions. A key result presented in the paper is an expression for the matrix of second derivatives of the objective function. Based on this matrix, a minimization method is presented with a greatly improved speed of convergence. Moreover, this result provides an effective convergence criterion that can also be applied in combination with other solution methods, such as Gauss-Seidel and gradient search methods. -acknowledgements-^^ work was supported by an EC grant in the Human Capital and Mobility programme, contract ERBCHGCT930412
REFERENCES Bazaraa, M.S., Sherali, H.D. and Shetty, C.M. (1993) Nonlinear Programming, Theory and Algorithms, second edition, Wiley Erlander, E. and Stewart, N.F. (1990) The Gravity Model in Transportation Analysis, Theory and Extensions, VSP, Utrecht, the Netherlands Evans, A.W. (1971) The Calibration of Trip Distribution Models with Exponential or Similar Cost Functions, Transportation Research, Vol. 5, pp. 15-38 Fletcher, R. and Reeves, C.M. (1964) Function Minimization by Conjugate Gradients, The Comp.J.,l,pp. 149-154 Hyman (1969) The Calibration of Trip Distribution Models, Environment and Planning, 1(3),
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pp.105-112 Kirby, H.R. (1974) Theoretical Requirements for Calibrating Gravity Models, Transportation Research, Vol. 8. pp. 97-104 Maher, M.J. (1983) Speeding up the Furness Method, Traffic Engineering and Control, pp.485486 Ortuzar, J. D. and Willumsen, L.G. (1994) Modelling Transport, Second Edition, John Wiley and Sons, Chi Chester, England, ISBN 0 471 94193 X Willumsen, L.G. in: Papageorgiou, M. (1991) Concise Encyclopedia of Traffic & Transportation Systems, Pergamon Press, pp.317 Sen, A, and Smith, T.O. (1995) Gravity Models of Spatial Interaction Behavior, Springer, ISBN 3-540-60026-4 Van Der Zijpp, N.J. and Heydecker, B. (1996a) How many Parameters should a Traffic Model have?, UTSG, ann.conference, University of Huddersfield, unpublished Van Der Zijpp, N.J. and Heydecker, B. (1996b) Some Investigations into the Use of the Gravity Model, Proc. 24rd European Transport Forum, PTRC, Seminar D&E - Part 1, Brunei University, 2-6 September Van Der Zijpp, N.J. and de Romph E. (1996) A Dynamic Traffic Forecasting Application on the Amsterdam Beltway, Int. Journal of Forecasting, 1996, forthcoming Van Zuylen, H.J. and Willumsen, L.G. (1980) The most Likely Trip Matrix Estimated from Traffic Counts Transportation Research-B, Vol. 14B, pp. 281-293
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THE STABILITY OF STOCHASTIC USER EQUILIBRIUM WITH A GIVEN SET OF ROUTE INFORMATION Toshihiko Miyagi Faculty of Regional Studies Gifu University
ABSTRACT The main purpose of this paper is to model how travelers utilize the additional routeinformation in their route-choice decisions, as the information through some form of routeguidance system is given. The basic idea of the paper is to combine the information-choice process into a stochastic network equilibrium. We assume that there are two types of travelers with different attitudes toward information in the systenroptimists and pessimists. Their information choice process is different, which in turn affect route choice behavior. The newly obtained route information is accumulated as personal knowledge, being further utilized for updating driver's information to select the best route. The overall choice process is formulated as an equivalent mathematical optimization program. Furthermore, this paper discusses the stability of a stochastic network equilibrium in the case of two type of route information; fixed travel time and variable travel times.
INTRODUCTION The development of intelligent transportation systems (ITS) is now widely studied in Japan, USA and Europe to improve traffic efficiency, road capacity and safety and environmental conditions. In-vehicle equipment, which typically takes the form of navigation or route guidance systems (often called driver information systems), may play the key role in ITS and has a function of providing route information on the best route to desired destination based on current traffic and road conditions. In order to examine efficiency of driver information systems, we need to study on how travel information affects the decision-making on driver's route choice. When information on routes available for each origin-destination pair is given, how does each driver behave for determining his or her route ? Which information does each driver regard as important, information provided by a traffic information center or information based on one's prior experience on driving ? In this paper we try to formulate a simultaneous decision-making 115
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process of information choice and route choice under the condition that each driver can get information on routes available from information provider through the in-vehicle equipment. Route choice problem in this case is partly related with the type of information provided and partly concerned with the type of utility functions of drivers. As for information type, we treat two types of route information: fixed information on route travel times and variable information on route travel times which may fluctuate depending upon traffic conditions. In addition, the paper assumes that two types of drivers distinguished by their attitudes toward information choice exist in the system, namely optimistic drivers and pessimistic drivers. In this case, information updating processes will be necessarily individual specific in the sense that even in an identical traffic condition, individuals make different choices. Therefore, route choice behavior can be naturally described by a stochastic assignment principle. In the usual stochastic network equilibrium model, the perceived travel -time by a driver for a route is assumed to be the sum of the measured travel-time and the random term. In this paper, however, we assume that even the measured part of travel time is replaced by the subjected value, which is built up in accumulation of traveler's past experiences, being updated after the information choice process. Based on which decision-making on route choice is done. Traveler's knowledge about routes is evolved in this manner. Such a route information updating process with route choice process may generate a certain type of dynamics in transportation networks. A study on the stability properties of stochastic equilibrium with a given set of route information is then required. Dynamics presented in this paper lay the theoretical foundation for studying the day-to-day travelers' behavior in the adjustment of their travel route choice. The previously published literature (Horowitz, 1984; Friesz et al, 1993; Nagurney and Zhang, 1996, for example) on the day-to-day dynamics on transportation networks made an oversight both a dynamic behavior incurred by information choice process and its stability analysis. In order to predict the effectiveness of ITS, we can not edge away from the subject. The paper's main objective is to model the simultaneous information/route choice process (IRCP) and to investigate the stability of stochastic user equilibrium (SUE), where route information is exogenously given and drivers are assumed to update their personal information on routes available. We begin with the framework of the information/route choice process and propose the perceived (subjective) travel time function with a given set of information on travel times, then proceed to the properties of the subjective travel time function. In the third section the stability of the process is analyzed; an equivalent mathematical optimization problem for the IRCP is first presented and it is shown that the IRCP process is asymptotically stable when link performance functions are monotone. Numerical examples are demonstrated in the final section to show the convergence properties of equilibrium obtained from the mathematical optimization program.
PERCEIVED TRAVEL TIME AND INFORMATION UPDATING PROCESS Information/Route Choice Process
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As time evolves, information relevant to forecasting the future is received and may be used by drivers in revising their views on routes that ones want to choose. Suppose, with no loss of generality, that the time origin t=0 represents the current time, and the existing information available to, and recognized by, a driver is denoted by the Initial information set: Do. This represents all the available relevant starting information which is used to form initial views about the future. We suppose the initial information set consists of initial perceived travel times on routes available, { CQ }• At any time t, statements made concerning the future are conditional on existing information at that time. Generally, we have the Information set at time t: Dt Obtaining the values of the present information on route travel times at time t, Vt, implies that Dt includes both the previous information set Dt-i and the present information. Now suppose travel times of routes available for a user at a time period t, V t , can be described by Vt=vt+et
(1)
where vt represents the expectation of estimated values of the travel times (or the objective travel times obtained from the observation) given by information provider at time period t as route information, which includes the error terms, et, representing the fluctuation of the travel times caused by various traffic conditions. On the other hand, we assume that each user has estimates of the travel times for the routes, based on his previous experience: C t -i=Ct-i+Q>t-i
(2)
where (ot_i denotes the perception error of a traveler, representing the uncertainty arising from the fact that the perception of the travel time distribution varies from traveler to traveler. Thus, the information set at time t consists of Vt and C t -i; Dt ={Vt, C t -i) Drivers may decide the most relevant information for choosing their routes1 \ The error sequences et and wt are assumed to be independent over time and mutually independent. In addition, we assume that those are distributed according to the Gumbel (or the double exponential) distribution with dispersion parameters, M-t and 0 t , being independent of the initial estimate of link travel time, c0. It may be relevant to assume that £t and w, have the normal distribution, respectively, however, we choose the Gumbel distribution because (1) it's frequency has almost the same general bell shape as the normal frequency and (2) it is stable under maximization (or minimization) operations in the sense that the maximization (or minimization) of two independent random variables distributed according to Gumbel is again a random variable with Gumbel distribution. The second properly of the Gambel is analogous to that the normal family is stable under additive operations. The information choice process may be described as follows. We assume that there are two groups of travelers in the system; optimistic and pessimistic. We call a group of travelers the optimistic traveler if they tend to choose, but with uncertainty to some extent, the smallest value among travel-time information
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available to him. For instance, today's travel time information provided by in-vehicle navigator exceeds yesterday's travel time he experienced, he expects a today's faster journey according his yesterday's experience. For the optimistic travelers, their choices on travel times for routes are formally described by
and the expectation and the dispersion parameter of it are given by ct0p=E[Cmin]= - J-log[exp(-Xct.i)+exP(-Xvt)] JT
(3) (4)
66
where CQP is the mean value of the travel time, as perceived by the optimistic decision-making population, Ilop, and cr2 the variance of the travel time provided as information. In the above expressions, we assume that dispersion parameters, A. and 6, are independent from time periods. The greater value the variance of route information becomes, the smaller value X. takes. Thus, A. may account for a sensitivity (or reliability) of travelers for the route information. Inversely, some drivers are pessimistic, which implies that they tend to be on the safe side and choose the greater value of travel time provided in the system. The selected travel time for the route is given by
and the expected value of it is given as: Cpe=E[Cmax]= J-logtexpCXct.O+expCXvt)]
(5)
Cpe is the mean value of the travel time, as perceived by the pessimistic decision-making population, npe The variance of Cmax is given by the same equation as (4). We assume that the total population consists of two mutually exclusive populations: H =npeunop. The mean value over the selected choices is then defined as: Ct=atE[Cmax]+ (l-at)E[Cmm], at<E(0, 1)
(6)
where a, represents the proportion of the pessimistic population to the total population, which is not known to us. The updated information of the total population is now described as: Ct=ct+o)t (7) Route choice will be made based on the relevant information set: Dt ={Ct}.
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In the stochastic equilibrium model proposed by Daganzo and Sneffi (1977), followed by Sheffi and Powell (1982), travel times denoted by c, in (7) are assumed to be deterministic and the objective values. Mirchandani and Soroush (1987) proposes a probabilistic travel-time model where ct is treated as random variables. On the other hand, as is shown in (6), we assume in this paper that q are random variables before obtaining route information, however, that a traveler takes those as deterministic before the route choice process, through the information choice process. Therefore, while the basic structure of the model presented in this paper is the same as the existing SUE model, q is the subjective value and updated at each time period unlike the existing SUE model. One may suggest that the route choice process should be applied to each population group separately, not to the total population. However, the idea may result in the unrealistic information updating process: Since CQP is monotone decreasing for additional travel-time information, whatever the additional information is, the perceived travel time by the optimistic population is getting smaller and smaller. On the other hand, that of the pessimistic population is getting greater and greater. This may suggest another interpretation for at being possible. Suppose that each driver does not always take the same attitude every day. If so, he or she encounters the unrealistic estimate for travel times some day. Therefore, even if one is a optimistic driver in nature, he sometimes takes the pessimistic attitude and modifies the bias of his estimate. Thus, at is the attitude parameter representing the rate of taking the pessimistic attitude of a driver. Since we assume that the random term in (7) is distributed according to Gumbel distribution, for the given set of routes available, K , the choice probability for route r is, as is well known, given by: Pl=
ex
P(-6c*) 2 expf-BcJ)
(8)
rGK
where pf is the choice probability of route r at time t, 6 the dispersion parameter for the route choice and cr the mean of perceived travel time of route r at time t defined by (6). Perceived Travel-Time Function and Its Properties Equation (6) together with (3) and (5) gives the sequential updating equation that define the evolution of information about the travel time estimated by users. The expected travel-time function representing the estimate of route travel time perceived by a driver is now expressed as: ^log[exp(-Xct.1)+exp(-Xvt)]> ate(0,l)
ct=atct.1+(l-at)vt+
*]og{l+exp[X(v t -Ct.i)]}, a t e(0,l)
(9)
(91)
X
Drivers do not always take the same attitude for route choice. Some drivers tend to take the optimistic attitude and some drivers do not. Thus, the form of perceived travel time function largely depends on the value of the attitude parameter at- Another important parameter included in the model is the sensitivity parameter ^ which represents a driver's evaluation for
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the difference between the perceived (subjective) value based on the driver's past experience and the objective value given by driver information systems. The less value of x, implies a less sensitivity for newly obtained information. Fig. 1 shows a surface of the expected (subjective) travel times under given set of the attitude parameters and the objective travel times. As is expected from figure 1, the subjective travel time function is concave for at
Fig. 1 A surface of the perceived travel time function The above statement is verified by calculating the second derivative of the expected travel time. Define the right hand side of (9) by z(ct-i)=atCpc(ct-i,v)+ (l-at)c0p(ct.i,v), ate(0,l) For the moment we ignore suffix t and suppose that the objective value of travel time is fixed.. The derivative of z with respect to c is given as: exp(-Xc) dc
exp(Xc)+exp(Xv)
exp(-Xc)+exp(-Xv)
or exp(-Xv) ^c
,j ,
u
i v
exp(-Xc)+exp(-Xv)
exp(-Xc) exp(-Xc)+exp(-Xv)
where we use the relation exp(Xc) exp(Xc)+exp(Xv) It follows from (10) that
_
exp(-Xv) exp(-Xc)+exp(-Xv)
(10)
Stability of stochastic user equilibrium with given route information
121
zc =an+( 1 -a)T], and 0
(11)
where r| is defined by exp(-Xc) exp(-Xc)+exp(-Xv) and T]=l-r|.
Therefore, if
The second derivative is obtained as: j2z
dr] dt]
dc2
dc ' dc
aa=l/2, then z is convex :if a
It should be noted that from the symmetry of z with respects to c and v, the same derivative properties as the ones with respect to c can be obtained for the objective travel times v. For the attitude parameter, we have the relation za=J-log[exp(Xc)+exp(Xv)]+J4og[exp(-Xc)+exp(-Xv)]=Cpe-c0p2:0 X X
(12)
Equal sign holds when all of travel times for available routes are equal. It follows from eqs. (11) and (12) that z(a,c) is strictly monotone on S, where S=(a,c) ER++ be the attitude parameter-perceived travel time plane, and the symbol "++ " represents the positive orthant. If we assume that the attitude parameter takes the value of 1/2, then it is straightforward to show that the perceived travel time converges the objective one: {(l--L)v+|}=v
(13)
For other values of the attitude parameter, the recursive equation (8) also converges, however, it can also be shown that in these cases the subjected travel time never converges to the objective values: If at > 1/2, then limt_oo ct>c* if at
*
CONVERGENCE PROPERTIES IN STOCHASTIC EQUILIBRIUM WITH ROUTE INFORMATION An Associated Network Equilibrium Model Consider a transportation network [N, L] of a set of nodes N and a set of directed links L, with the total number of links in the network being q. The nodes i, iGN, represent origins,
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destinations and interactions; the links a, aEL, represent the transportation links. The origin to destination (O-D) demands give rise to the link flow pattern, {fa}, and the resultant cost of traveling on a link is given by a travel-time (or a performance) function, xa (fa)We let w denote an O-D pair and W the set of all the O-D pairs for the network. We assume that there are J O-D pairs. The demand between origin destination pair w, w€EW, may use directed path k. Let K denote the set of all paths, that is, K=UwewKw. We assume that there are mw paths connecting the O-D pair w, with the total number of paths in the network being m= 2,wewmw. Let nk represents the traffic flow on path k. The path flow pattern is a mdimensional nonnegative column vector that can be described as h= {hk,
The path flow satisfies the conservation law of flow : dw= J h k kacw where dw represents the traffic demand between O-D pair w, being nonnegative and given. The corresponding link flows are given by
kGK
where 8^=1 , if link a is contained in path k, and 0, otherwise. The objective travel-time vk of each path k is the sum of the travel times on the links that belong to k. v k (h)=2 8akTa(f(h))
(14)
aEL
Consider the situation where the objective travel times depending on traffic congestion are given by route information provider. Then, the simultaneous information and route choice problem (ERCP) may be formulated as the following mathematical optimization program: [IRCP] For given c° and
n=0, 1,2, ...... determining h*E:Rfand v(h*)ERi , satisfying: 1
dwlog
exp (- 0ck)-
*
(15)
where cjj represents the expected value of perceived travel time of route k at time n, which is expressed by (9). You should note that in the optimization program [IRCP] path flows for the previous time period are assumed to be given. For each time period, [IRCP] induces the necessary and sufficient conditions: (^ _ dwexp(-6cD W kEKw
,
= o, for all kEK and n=0,1,2,...
Stability of stochastic user equilibrium with given route information
123
Since dck / 5vk >0, the necessary and sufficient conditions are expressed by the following fixed point equation: exp[-6c£(h)] kGKw
with c£(h*) = ^log[exp(Xcir VexpftvgCh*))] X
(17)
v^h))], anE(0,l) A.
where v£(h*) is given by (14). The above fixed point equation is sequentially solved because that ck included in ck is given and fixed. Suppose that the initial distribution of perceived travel times, {$}, is given, then we can determine the corresponding equilibrium flow pattern, {h]<J, and the resultant travel times, {c£(h )}, by solving (16) together with (17). The next step is followed by setting to c£ =4(h*) for all k£K to obtain {cj^h*)}. Succeeding steps are carried out in this manner. If we can guarantee that for a small positive constant e the relation Icg-cg'1 <e , for all holds, then it can be ensured that the information/route choice process converges.
STABILITY ANALYSIS We begin with the definitions associated with the stability of network flows. In the following, the first two definitions are due to Nagurney and Zhang (1996). Definition 1: Stability of the System (Nagurney and Zhang, 1996) The route choice adjustment process is stable if for every initial flow pattern ho ^d every equilibrium flow pattern |1*, the Euclidean distance, |h*-ho(t)|, is a monotone nonincreasing function of time t. Definition 2: Asymptotical Stability of the System (Nagurney, 1996) The route choice adjustment process is asymptotically stable if it is stable and for any initial flow pattern ho> there exists some equilibrium flow pattern ^*, such that ho(t)-»h*, ast-»°° where ho(t) solves the initial value problem with h(0)=hoDefinition 3: The information updating process is complete (or the equilibrium is complete) if the route adjustment process is asymptotically stable and the subjective travel times converge to the objective travel times over time.
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Even if the route adjustment process is asymptotically stable, drivers may not be able to get complete information on route travel time in our contexts that the perceived travel time may be different from the information on travel times provided by the operator of the system. Definition 3 claims that users are in the state of complete information if the information updating process is complete. The Stability of SUE: Flow Independent Model Suppose the travel time information on each route provided by the operator of the system is constant and fixed, then route choice by users solely depends on their perceived travel time on that route. In this situation, the convergence to equilibrium state and its stability are examined by the convergence property of the perceived travel function and its stability. If f,g :R—*R are two functions, then we write fAg(x) for f(g(x)), where A represents the composition operator, i.e., for the composition of f with g. Taking a point xo€ER, we want to find xi=f(xo), X2=f(xi), • • -,xn=f(xn_i). Thus, xn=fA- • -Af(xo) where we take the composition off with itself n times, which we call the nested function of n-th order. Now, we recall the perceived travel function mentioned in the previous section, replacing it by q = z(ct. i ), where z(ct_ ] )=atCpe(ct- 1 )+ ( 1 -at)c0p(ct. i ), at£(0, 1 )
(18)
where the route information is assumed to be given and fixed, thus eliminated in z(). We want to find CI=Z(CQ), C2=z(ci), • • •,cn=z(cn.i) or cn=zA- -AZ(CO). If a sequence of information updating process approaches to an equilibrium state, then choice probabilities do so. Proposition 1: The information updating equation (18) converges for every initial state CD if the objective travel time is exogenously given and fixed. In particular, if the attitude parameter takes a value of 1/2, then the equilibrium is complete. Proof. For 0
where cn=zn(cn-i). Suppose that Cn is the equilibrium point c*, then c*is the equilibrium point with |z'(c*)|
Stability of stochastic user equilibrium with given route information
125
The Stability of SUE: Flow Dependent Model It may need a few pages to give a rigorous proof for the stability of the flow dependent model. We just give a rough sketch about the stability of our model here. Although Liapunov functions have been proposed for the traffic network equilibrium problem as early as Smith (1984), existence and uniqueness results for the adjustment process have been provided by Dupuis and Nagurney (1993). Theorem concerning with asymptotical stability in network flow proved by Naguraey and Zhang (1996) is roughly described as follow: Assume that there exists some equilibrium path flow pattern. Suppose that the link travel time function and the negative disutility functions are both continuous and strict monotone. Then, the route choice adjustment process is asymptotically stable. In our case, since we assume that the travel demand is fixed, monotonicity on link travel function is only required. As is shown in the previous section, monotonicity of the perceived link travel time function is ensured if the performance function T is monotonic function with respect to f The asymptotic equilibrium implies that for any initial flow pattern the system converges to a stable equilibrium. Therefore, the theorem seems to ensure that [IRCP] induces stable equilibrium flow patterns for different distributions of perceived travel times.
NUMERICAL EXAMPLES Applications of the information updating process with flow dependent travel times were demonstrated on the single origin-destination network connected by two links depicted in Fig. 2. The following BPR type-performance functions were used to represent delays incurred by traffic congestion. r
where Ha capacity on each link.
r
, a= 1,2
The travel demand for O-D pair was set equal to 100. Route 1
Route2 < * = / ! + / 2 = 10° T j - 1 5 [1+ 0 . 5 3 (A; J 2 =20 [1+ 0.53 (A;
Fig.2
A network for numerical testing
Two cases were examined for comparison purpose. In the case 1, the attitude parameter at was changed from 0.2 to 0.8 with the dispersion parameters 9, A. being equal to 1.0. Three cases for different attitude parameters are depicted in Fig. 3. These three cases describe the situation that drivers' confidence for a given route information is relatively high. Case 1 (b) shows that if
T. Miyagi
126
half of drivers are optimists (or the rate of taking optimistic attitude is 1/2) , then the resultant equilibrium is the Wardropean , being complete in the information updating process. Other two cases also converge to stable equilibrium points. While even in these cases the objective travel times on two routes are equal, those values are substantially different from the subjective travel times. Case 1 (a) [a =0.2 ; 9 = X=1.0] Travel t mc 80 70 60 50 40 30 20 30 . 40 50 60 70
*»•
1
Route 1
L_
—i
2
^Tj
*-
.-I
*>
«
3
t
5
e
<
>
^«
>^^J 7
8
15— jJ
J
—^
Iteration numbei -•-The perceived value -•—The objective value
Route 2 Casel(b) [a =0.5 ; 9 = X=1.0] Travel time
50 45
,
Route 1
\
40 35 30 35
_
'
1
^
1
•
_
*-^~~"~'1 \
2
3
4
5
O^i_^ 40 ' T
45 50
\
^
7
Iteration numbei
!
—«— The perceived value —•— The objective value
\
Route 2 Casel (c) [a =0.8 ; 8 = X=1.0] Trav el
time
Route 1
43
^V
1 . ..
41
^r \>^^^
39
\
37
1
39 41
2
3
4
v .: ^V-dfc^4_-
5
e
7
£
Iteration numbei
i
riI
11
1
-•-The perceived value -•—The objective value
~<
43
Route 2
Fig. 3
Convergence results in the case of high confidence for route information
The second case describes the situation that drivers' confidence for route information is rather dispersed. In those cases, the system never approach to the Wardrop equilibrium, essentially
Stability of stochastic user equilibrium with given route information
127
being in dispersed equilibrium in the usual sense. Even in those cases, if the rate of taking optimistic attitude of drivers is 1/2, drivers' perceived travel times tend to close to the objective travel times by obtaining route information (Fig, 4(b)). In other two cases, the difference between the subjective travel time and the objective one on each route becomes greater compared with the previous case with high confidence for route information. Case2(a) [a=0.2; 0 = X=0.1] Travel time
70 60 50 40 30 1 20 30, 40 50 60 70
Route 1
j 1
M
^
2
3
^se2(b )
4 I
k
^ ^
»
1
ft
1
6
7
8
Iteration number
r — t —i ,F • •
! *——-1
Route 2 9 = X=0.1]
[a =0.5
Travel time 55
Route 1
V
50 45
11
\
^-^ik=J
40 35
^
40
A
3
2
35 ,
5
5
?
8
^3'=* =-.«——,>
">
•
Iteration number
45 50 55 ise 2 (c1
Route 2
[a =0.8
Travel time <<
e = x=o.i]
50 45
C^"""-, r1
40 1
i
Route
|
1
:
|
2
4
^ —<
1
»
«
•
1
^1
f
f
5
6
7
8
40
Iteration number -*— The perceived value
45
r
' ~T—<|
HI1
*—
f
Route 2
Fig. 4
Convergence results in the case of less confidence for route information
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T. Miyagi
CONCLUSION The paper has concluded that if driver's attitude for information choice is in the neutral position, then the system results in an asymptotic, stable equilibrium and the perceived travel times are complete and identical with those provided by drivers information system regardless of drivers specific, personal information and that it depends on the sensitivity parameter whether the system approaches to the Wardrop equilibrium or not: if drivers' reliability for route information is great, then the system realizes the Wardrop equilibrium conditions. However, it has also shown that if the driver population is biased to optimists or pessimists, then the resultant travel times (subjected values) under these conditions can be different from the objective travel times given by information systems. These results indicate that constructing an efficient driver information system solely depends on how reliable route information we can provide. We assumed in this paper that variances of perceived travel times are constant, however, uncertainty on travel times of routes may be improved by learning behavior of drivers who use driver information systems. We did not treat this benefit of the driver information systems in this paper. In order to remedy this problem, Bayesian information-updating process should be taken into account in the model presented in this paper. This is the further research to be addressed.
REFERENCES Daganzo, C.F. and Sheffi, Y. (1977). On stochastic models of traffic assignment. Trans. Sci., 11, 253-274. Dupuis, P., and Nagurney, A. (1993). Dynamical systems and variational inequalities. Annals of Operations Research, 44, 9-42. Fisk, C.S. and D.E. Boyce (1983). Alternative variational inequality formulations of the network equilibrium-travel choice problem. Transpn Sci., 17 , 454-463. Friesz, T.L., D. Bernstein, T. Smith, R. Tobin, B.W. Wie (1993). A variational inequality formulation of the dynamic network user equilibrium problem. Opn. Research, 41, 8091. Horowitz, J.L. (1984). The stability of stochastic equilibrium in a two-link transportation network. Transpn. Res., 18B, 13-28. Mirchandani, P. and Soroush, H. (1987). Generalized traffic equilibrium with probabilistic travel times and perceptions. Trans. Sci., 3, 133-152. Nagurney, A., and D. Zhang (1996). Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic Publishers, Boston, Massachusetts. Robinson, C. (1995). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton, Florida. Sheffi, Y. and Powell, W. (1981). A comparison of stochastic and deterministic traffic assignment over congested networks. Trans. Res., 15B, 53-64. Smith, M.J.(1984). The stability of a dynamic model of traffic assignment-an application of a method of Liapunov. Transpn. Sci. 18, 245-252.
TRAFFIC EQUILIBRIUM IN A DYNAMIC GRAVITY MODEL AND A DYNAMIC TRIP ASSIGNMENT MODEL Xiaoyan Zhang, Napier Univeristy, EDINBURGH, United Kingdom and David Jarrett, Middlesex University, United Kingdom Abstract This paper is concerned with the analysis of traffic network equilibria in a dynamic gravity model and a dynamic logit-based trip assignment model. Both models were suggested by Dendrinos and Sonis (1990) as modifications of, respectively, the conventional static gravity model (Ortuzar and Willumsen, 1990) and the logit-based trip assignment model (Dial, 1971), but without giving any detailed analysis. The two models are considered as discrete-time dynamical systems. They describe the variations of traffic flows in an Origin-Destination (O-D) network or a road network at discrete time periods. In this paper, the existence, uniqueness, and stability of the equilibrium in the two traffic models will be analysed by the methods in nonlinear dynamics. The notions of User Equilibrium (UE) and Stochastic User Equilibrium (SUE) that have been used in trip assignment models will be introduced to the dynamic gravity model to describe the equilibrium in an O-D network. It will be shown that the equilibrium in the dynamic gravity model with the power or exponential deterrence function, and in the dynamic assignment model is unique. However, the equilibrium in the dynamic gravity model with the combined deterrence function is not necessarily unique; multiple equilibria, periodic orbits, and chaos will be shown to exist in the model.
1. INTRODUCTION This paper is concerned with the analysis of traffic network equilibria in a dynamic gravity model and a dynamic logit-based trip assignment model. These two models have a similar mathematical form and so are put together for consideration. Both models were suggested by Dendrinos and Sonis (1990) as modifications of, respectively, the conventional static gravity model (Ortuzar and Willumsen, 1990) and the logit-based trip assignment model (Dial, 1971), but without giving any detailed analysis.
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X. Zhang and D. Jarrett
There has been hardly any previous dynamic consideration of trip distribution. Although Dendrinos and Sonis (1990) suggested the dynamic gravity model as a simple extension of the classical gravity model for trip distribution, the resulting model can be considered as a combined trip distribution and assignment model, with the assignment process being very much simplified. As for dynamic analyses of trip assignment, there have been two main types of them: within-day dynamics and day-to-day dynamics (Cascetta and Cantarella, 1991). The dynamic assignment model to be considered here belongs to the second type. This second type has been considered by several authors (Smith, 1979, 1984; Horowitz, 1984) but with different models. The two dynamic traffic models to be investigated here are both discrete dynamical systems. They describe the variations of traffic flows in an O-D network or a road network at discrete time periods. For traffic systems, an equilibrium is the most important steady state; User Equilibrium (UE) and Stochastic User Equilibrium (SUE) are special types of equilibrium. In general, however, an equilibrium may not be the only possible steady state of a dynamical system; there may also be other types of steady state, such as oscillations or chaos. (We refer to the long-term behaviour of a dynamical system as steady states, though the system at steady states need not be motionless, except at an equilibrium.) In addition, any equilibrium need not be unique, nor always stable. Multiple equilibria will be shown to exist in the dynamic gravity model in this paper. Whether or not a (desired) equilibrium would prevail in the system depends on the road and traffic conditions, and on the socio-economic conditions of the system. By treating the traffic models as dynamical systems, we are able to investigate the time evolution of the flow patterns in the system and to identify possible types of behaviour of the system under various conditions. This approach should be contrasted with the more usual one presuming the system to be in an equilibrium state, with the implicit assumption that the equilibrium is stable and is the only possible type of steady state behaviour in the system. In this paper, the existence, uniqueness, and stability of the equilibrium in the two traffic models will be investigated. The notions of UE and SUE that have been used in trip assignment models will be introduced to the dynamic gravity model to describe the equilibrium in an O-D network. The two dynamic models will also be studied numerically to identify different types of steady state for different values of parameters.
2. THE DYNAMIC GRAVITY MODEL 2.1. The model Dendrinos and Sonis (1990) introduced iterative dynamics to the conventional gravity model. In the dynamic gravity model, the O-D flows at each time period are generated from the travel costs at the previous time period. The travel costs are assumed to be functions of O-D flows. Let t be a trip matrix with ty being the number of trips from origin i to destination j. Denote the set of all possible values of t by S. This will be defined by non-negativity
Traffic equilibrium in a dynamic gravity model
131
constraints and appropriate marginal constraints on the ty. Then the dynamic gravity model is defined by a mapping F: S—>S, FiJ(t) = V i j ( t ) f ( c i j ( t i j ) ) ,
i = 1,2,...,/,
7 = 1,2,...,J.
(1)
Here, y/y(t) is an appropriate normalising factor determined from the marginal constraints, Cy is the travel cost which is normally assumed to be an increasing function of ty-, /(•) is called the deterrence function which relates the number of trips to the travel costs, I is the number of origins, and J is the number of destinations. The map (1) defines a discrete-time dynamical system. If n is the discrete time, and t(n) the O-D flow pattern at time n, then t(n+l) = F(t(w)) is the O-D flow pattern at time n+l. For non-work trips, such as trips for shopping, we may consider the variations of O-D flows over time periods like each day or each week because the number of trips between each O-D pair is the result of daily or weekly decisions of trip makers. For work trips, on the other hand, a longer time slice may be more appropriate since the choices of origins (for example, residence locations) and/or destinations (for example, work place) are normally based on longer-term decisions. In order to model the congestion effect between each O-D pair, Dendrinos and Sonis (1990) suggested the following cost function S-(^) = c,°[l + a(f,/^,r],
i = l, 2,...,/,
7 = 1,2,...,J,
(2)
where c^ is the uncongested travel cost from origin / to destination j, qy is the corresponding capacity of the roads, and a and y are positive constants. This cost function can be recognised as the BPR (Bureau of Public Roads) link performance function used in trip assignment models (Branston, 1976). Clearly, this is not the only form of cost function; other suitable functions may also be used. In the dynamic gravity model, the cost function serves as a coarse-level description of the congestion effect between an O-D pair; it relates the cost and flow between an O-D pair without referring to a road network. One way of looking at the model is that if we assume that at each time period the O-D flows are assigned to a road network by UE assignment, then the cost between each O-D pair would be the cost on all used routes between the O-D pair. (Note that at UE the costs on all used routes are equal.) Therefore, the model can be considered as a simple combined trip distribution and assignment model. It is possible to include an explicit assignment process in the dynamic gravity model by replacing (2) by an assignment process. However, the resulting model will be much more complicated and so will not be considered in this paper. Three types of deterrence function are usually considered (Ortuzar and Willumsen, 1990): (a) exponential function, (b) power function, and (c) combined function. They can be written as:
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where (j, and p are constants. When ^=0 and P>0, / is an exponential deterrence function; when n<0 and P=0 it is a power deterrence function; and when u>0 and P>0 it is a combined deterrence function. The normalising factor ^y(t) in (1) is chosen so that one or more of the marginal constraints of an O-D matrix are satisfied. In terms of relative quantities, the constraints may be written as
where ot is the total relative number of trips originating from zone i, and dj is the total relative number of trips terminating at zone j. Replacing \f/y(i) by appropriate factors in (1) we may then have three types of models with different phase spaces defined by appropriate constraint(s), as follows. (1) Unconstrained model
(2) Singly constrained model. E.g., an origin-constrained model /r (t) = 0, -£S»M--, " Z/M'i,))
tell ((,,,,,, 2,A ,tu\ t,>0, T ti!=o\ . 1M V "" 2 ' "" • '^J' 'I
(3) Doubly constrained model Fij(t) = a i ( t ) b J ( t ) f ( c i j ( t i J ) ) ,
t€J[r,]: tv >0; Zfi =o,,Vi; ^ , =rf y > V 7 ],
(4)
where a,(t) and 6/(t) satisfy the equations > '
i = l , 2 , ...,/,
(4a)
;
,. (t) = =-^- , 7 = 1,2, ..., /.
(4a)
7
Note that in the above singly constrained model Fy depends only on flows from origin i. Thus, the unconstrained and singly constrained models can be written in the more general form, using single subscripts for simplification, ,« (5)
Traffic equilibrium in a dynamic gravity model
133
When K=IJ the equation represents an unconstrained dynamic gravity model; while when K equals 7 or J the equation represents one component of a singly constrained model with oi or dj being set to 1 for further normalisation. The doubly constrained model, however, is different and is more complicated. It contains two sets of normalising factors which are interdependent. An iteration procedure such as the Fumess iteration is needed to solve the model. The map (1) is continuous and the phase spaces of the three types of models are closed convex sets since they are defined by linear equalities and inequalities. Therefore, according to Brouwer's fixed point theorem, the map has at least one fixed point or an equilibrium in the phase space. An equilibrium te in (1) is given by te = F(te). The equilibrium cannot be found analytically unless F is one dimensional and is such that the equation is linear or quadratic. However, it can be obtained by numerical calculations.
2.2. The uniqueness of the equilibrium The uniqueness of the equilibrium in the dynamic gravity model is examined through a related mathematical programming problem, although other methods of proof may be possible. The problem to be considered is Minimize subject to
z(t) = £.. tv Infj, + £.. j[' [/? cy (s) - p Inc.. (5)] dj ta > 0;
J] . t~ = o, , Vz;
]T ty = d} , Vj .
(6a) (6b)
When tjj=0, the expression fylnfy in the objective function is defined to be zero. We have considered the doubly constrained model here. The unconstrained or the singly constrained models can be considered by replacing the marginal constraints in (6b) by the appropriate one in (3). The feasible region in the programming problem is the same as the phase space of the dynamic gravity model. It will be shown that the solution of the program is identical to the equilibrium in the dynamic gravity model. The Lagrangian for the above problem is £(t,u, v) = z(t) + 2(ii, (o, - £/,) + £/,( dj - £/,) , where u and v are vectors of Lagrange multipliers. Setting the partial derivatives of L to zero, a stationary point t* can be found to be y )] exp(- 1 + u, ) exp(vy ) , Vi, j.
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The factors exp(-l + ^-) and exp(v7-) determined from the marginal constraints in (6b) are exactly the same as the normalizing factors in the doubly constrained model (4). Therefore, * P t =t e . If the objective function is strictly convex at t*, then t* is a local minimum. If the objective function is convex elsewhere in the feasible region and if the region is convex as well, then t* is a global or unique minimum in the region, and the equilibrium, too, is unique in the phase space of the dynamic gravity model. As has been mentioned, the phase space or the feasible region is convex. According to the Theorem 1 in Evans (1973), the first term of the objective function is strictly convex in the feasible region. The remaining question is whether the second term of the objective function is strictly convex at t* and is convex elsewhere. Denote the second term by z2(t). Then the Hessian matrix for z 2 (t) is a diagonal matrix with non-zero elements
*,
The condition for Z2(t) to be (strictly) convex is that the derivatives are (strictly) positive so that the Hessian matrix is positive semi-definite (positive definite). Since Cy (fy)>0, the conditions are B
7^7 >0, and /?-—^->0,
Vfc,.
These are automatically satisfied if the exponential or the power deterrence function is used. In both cases, there is equality only when ty=Q, that is, at the boundary of the feasible region. Therefore, when the exponential or the power deterrence function is used, the stationary point is a unique minimum in the feasible region and the equilibrium in the gravity model is unique. With the combined function, if the values of parameters are such that the above conditions are satisfied then the optimum solution and so the equilibrium is unique. Otherwise, the solution and the equilibrium is not necessarily unique. If there are multiple equilibria in the model, the system may stick to any one of them, depending on the initial conditions. An example of multiple equilibria in the model will be shown in the numerical analysis.
2.3. The stochastic user equilibrium The notions of the UE and SUE in trip assignment models can be used to describe the equilibrium in the dynamic gravity model. Consider the objective function of program (6) with u=0, that is In/
,f
Traffic equilibrium in a dynamic gravity model
135
It is clear that the solution of the corresponding program gives the equilibrium of the dynamic gravity model with the exponential deterrence function. This objective function has the same form as that of the mathematical program formulated by Fisk (1980) to solve the logit-based SUE traffic assignment. In that objective function, the variables are the route flows and link flows in a road network, while here they are O-D flows in an O-D network. The unique minimum point of that objective function gives SUE for trip assignment; the unique optimum here is the SUE in an O-D network, which is also the unique equilibrium in the dynamic gravity model with the exponential deterrence function. The first term of the above objective function is the negative entropy of an O-D matrix. If the costs are constants and do not depend on the flows, the entropy of the O-D matrix is maximized with the cost constraint, and the solution is the same as that given by the static classical gravity model (Ortuzar and Willumsen, 1990). The second term, though does not seem to have a clear physical meaning, has the same form as the objective function of the program formulated by Beckmann et al. (1956) to find the UE for trip assignment. Thus, if P is very large, the above objective function is dominated by the second term and the solution tends to the UE in an O-D network.
2.4. The stability of the equilibrium A general analysis of the stability of the dynamic gravity model is difficult, especially for the doubly constrained model. Therefore, here, the stability of the equilibrium is investigated for the unconstrained and singly constrained models based on the general form (5). An ordinary way to investigate the stability of an equilibrium in a nonlinear, multi-dimensional system is to consider the local stability of the equilibrium by examining the eigenvalues of the Jacobian matrix at the equilibrium. An equilibrium is locally asymptotically stable if the magnitude of all the eigenvalues is less than 1 (Parker and Chua, 1989). Unfortunately, it is not normally possible to evaluate the eigenvalues of the Jacobian matrix of F analytically. Here, a sufficient condition for the stability of the equilibrium is given by bounding the eigenvalues of the Jacobian matrix by the norm of the matrix, based on the theorem of Lancaster (1969, page 201). It is desired that the type of norm to be used is small in magnitude and has a simple form. Based on these considerations, the p-norm is used with p=l. Let J denote the Jacobian matrix of F at the equilibrium, with elements
^(t e ). Then the 1-norm of this
matrix is
llJlL V i> iii = max,'/_(,where ym is the index at which the sum is maximum. According to the theorem, no eigenvalue of the Jacobian matrix is bigger than the 1-norm. Therefore, ||j|| < 1 can serve as a sufficient condition for the stability of the equilibrium. The stability depends on the values of parameters (a, P and the cost at the equilibrium. With the power and the exponential deterrence functions, the equilibrium is unique. The smaller (i or P is, the more likely that the equilibrium is stable. With the combined deterrence function, however, the stability
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depends on both \n and p. If the values of parameters are such that an equilibrium is unstable, then for an arbitrarily chosen initial condition, the system will normally approach another steady state in the phase space, such as another stable equilibrium (if there are multiple equilibria), or a stable periodic orbit. These may only be investigated further by numerical analysis.
2.5. Numerical analysis Numerical calculations involve iterations of the equation t(n+l) = F(t(«)) defined by (1) for given initial conditions and values of parameters. The iteration is made until a steady state is approached. The values of a parameter may be varied gradually to find out how the steady state of the system varies with the values of the parameter. It was found that with the power or exponential deterrence function, dominant behaviour in the gravity model includes an equilibrium when the values of parameters are smaller, and a period-2 orbit when parameter values are larger. Figure 1 is a bifurcation diagram (the diagram of the steady state of a dynamical system against the values of a parameter) of the model with the exponential deterrence function. It can be seen that when p is smaller, there is a stable SUE in the system (indicated by the state of a single point). As p increases, the equilibrium becomes unstable and bifurcates into a stable period-two orbit (represented by the state of two points). The bifurcation occurs when the magnitude of the largest eigenvalue of the Jacobian matrix J becomes bigger than 1 as p increases.
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Traffic equilibrium in a dynamic gravity model
We have seen that when the combined deterrence function is used, the equilibrium in the gravity model may not be unique. In order to identify possible multiple equilibria, we need calculations with the same parameter values but different initial conditions. Figures 2a-2b are two bifurcation diagrams for p with different initial conditions. In these diagrams, when Kp <1 5 there are different equilibria from different initial conditions. In fact, when p>2.6, the whole bifurcation sequence of the two diagrams is different. In Figure 2a, as P increases gradually the system starts with an equilibrium. Then there is a period doubling sequence in which the period of oscillations doubles from 2 to 4, then to 8, and so on. This is a typical route to chaos. And indeed, here it is followed by chaotic state (signified by infinite numbers of points). However, in Figure 2b with a different set of initial conditions, the period doubling is followed by undoubling without going through chaos. These two diagrams mean that for some values of p, there are at least two steady states competing in the phase space, each having its own basin of attraction. Which state the system approach depends on which basin of attraction the initial condition is in. Similar experiments with other parameters in the gravity model showed that the kind of behaviour in Figure 2 is not exceptional; it is rather typical. Multiplicities of steady states including chaos is not rare in nonlinear dynamical systems; they have been found in dynamical systems in other applications, such as forced vibrations of a pendulum modelled by the Buffing's equation.
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Figure 2. Bifurcation diagram of the unconstrained or singly constrained gravity model with two origins and two destinations; n=7, ct=1.0, 7=1.0. (a) Initial conditions are the same for all values of p.
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Figure 2. continued, (b) Initial conditions are the final states of the system at the previous step of P, given the initial conditions for the first step of p.
3. THE DYNAMIC LOGIT-BASED TRIP ASSIGNMENT MODEL 3.1. The model The dynamic logit-based trip assignment model describes, for a given O-D matrix, the adjustments of the flow pattern in a road network from one time instant to another (for example, from day to day). The assignment process at each time period is modelled by the logit choice model. It is assumed that a drivers' route choice is based on the travel costs in the previous time period and that the travel cost on each link depends on the flow on that link. The following notations will be used in this section. L M Pij
x
set of links set of origin and destination nodes set of routes connecting O-D pair (i,j) flow on route r joining O-D pair (i,j) vector of route flows on all routes joining O-D pair (ij);
= (..., Xr, ...)
Traffic
equilibrium in a dynamic gravity model
x
1 39
vector of route flows on all routes in the network; x = (..., x^-, ...)
Cijr cost on route r connecting O-D pair (i,j) yi flow on link / df cost on link / dj(yj) link performance function for link / $ijr=l if link / is in the route r connecting O-D pair (i,j), and <5^r=0 otherwise Let U be the set of x that satisfies the non-negativity constraint and the O-D flow constraints, that is:
u
= ru{ 2u** = <* }•
™
Then the dynamic logit-based assignment model can be written as G: £/-»[/,
G,r(x) = t
exp[-0c,. (x)] L ' ,,
rePij, ijeM,
(8)
where
and 9 is a positive parameter. The map G defines an adjustment mechanism in the dynamic assignment process: the flow pattern x at one time period will become G(x) at the next time period. Let n be the discrete time. Then x(n+l) = G(x(w)).
3.2. The equilibrium in the model As has been mentioned, the dynamic assignment model has a similar mathematical form to the dynamic gravity model. Thus the same analysis method used in the last section may be used here. First of all, the existence of an equilibrium xe, given by xe = G(xe), is assured by Brouwer's fixed point theorem. This equilibrium can be recognised as the logit-based SUE. It has been shown by Fisk (1980) that the logit-based SUE is the unique solution of a related convex programming problem. The phase space defined in (7) of the dynamic assignment model (8) is the same as the feasible region of Fisk's programming problem. Therefore, the equilibrium in the dynamic assignment model is unique as well. Both the flow pattern given by Dial's model and UE are included as special cases of the equilibrium. See Fisk (1980) for detailed discussions. A sufficient condition of the stability of the equilibrium can be derived by the same method used in the last section. This will not be repeated here. Instead, stability will be investigated by numerical calculations.
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140 3.3. Numerical analysis
Numerical calculations were made by iterating x(«+l) = G(x(/z)) for artificial road networks. It was found that the dynamic assignment model has similar behaviour to that of the dynamic gravity model with the power or the exponential deterrence function. When the values of parameters are small, the system approaches the SUE. As the values of parameters increase, the equilibrium loses its stability and a stable period-2 orbit emerges. Consider a road network shown in Figure 3. The trip matrix for the network is given in Table 1. The uncongested travel costs and link capacity of the network are listed in Table 2. The steady states in the assignment model with the BPR link performance function (Branston, 1976) is summarized in the parameter space (a-y-0) in Figure 4. The equilibrium is stable for the parameter values underneath the surface and is unstable for the parameters values above the surface. Calculations were also made with some other networks and link performance functions; similar results were found. Therefore, the behaviour outlined so far is the dominant behaviour in the model.
Figure 3. A road network (modified from Potts and Oliver, 1972).
Table 1. Trip matrix for the network Destination Origin 1
1 35
2 20
Total 55
2
15
30
45
Total
50
50
100
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Traffic equilibrium in a dynamic gravity model
Table!. The link data of the network uncongested link cost link capacity link number 3 20 1 2
30
1
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I
5
25
5
6
40
2
7
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1
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1
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40
1
11
43
3
Figure 4. Stability of the equilibrium in the dynamic trip assignment model.
4. CONCLUSIONS In this paper, traffic equilibria in two dynamic models of traffic flows in an O-D network and a road network have been investigated. It has been shown that the equilibrium in the dynamic
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gravity model with the power or exponential deterrence function, and in the dynamic assignment model is unique. The unique equilibrium in the gravity model with the exponential deterrence function gives the SUE in an O-D network while the unique equilibrium in the trip assignment model gives the SUE in a road network. The stability of the equilibrium depends on the interdependence of travel costs and traffic flows. If the dependencies are mild enough, indicated by smaller parameter values, then the equilibrium is stable. Otherwise the equilibrium is unstable and the O-D flows or route flows oscillate. The equilibrium in the dynamic gravity model with the combined deterrence function is not necessarily unique. Multiple equilibria, periodic orbits, and chaos have been found by numerical calculations. Empirical studies are needed to determine the values of parameters in the dynamic models so as to find out if an equilibrium would be stable or if the system would oscillate or would be chaotic.
Acknowledgement—The authors wish to thank Mike Maher and Chris Wright for helpful discussions and suggestions. The first author is grateful to Middlesex University for providing a studentship which supported the major part of the research described in this paper.
REFERENCES Beckmann M, C. B., Mcguire, and C. Winsten (1956). Studies in the Economics of Transportation. Yale University Press, New Haven, Conn. Branston D. (1976). Link capacity functions: a review. Transportation Research, 10(4), 223-236. Cascetta E. and G. E. Cantarella (1991). A day-to-day and within-day dynamic stochastic assignment model. Transportation Research, 25A(5), 277-292. Dendrinos D. S. and M. Sonis (1990). Chaos and Social-Spatial Dynamics. Springer-Verlag. Dial R. B. (1971). A probabilistic multipath traffic assignment model which obviates path enumeration. Transportation Research, 5(2), 83-111. Evans S. P. (1973). A relationship between the gravity model for trip distribution and the transportation problem in linear programming. Transportation Research, 7, 39-61. Fisk C. (1980). Some developments in equilibrium traffic assignment. Transportation Research, 14B, 243-255. Horowitz J. L. (1984). The stability of stochastic equilibrium in a two-link transportation network. Transportation Research, 18B, 13-28. Lancaster?. (1969). Theory of Matrices. Academic press, New York, London. Ortuzar Jde D. and L. G. Willumsen (1990). Modelling Transport. Wiley Parker T. S. and L. O. Chua (1989). Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag. Potts R. B. and R. M. Oliver (1972). Flows in Transportation networks. Academic press. Smith M. J. (1979). The existence, uniqueness, and stability of traffic equilibria. Transportation Research, 13B, 289-294. Smith M. J. (1984). The stability of a dynamic model of traffic — application of a method of Liapunov. Transportation Science, 18(3), 245-252.
10 FOUNDATIONS OF A THEORY OF DISEQUILIBRIUM NETWORK DESIGN Terry L. Friesz Departments of Systems Engineering and Operations Research & Engineering, George Mason University, Fairfax, VA 22030, USA Samir Shah Price Waterhouse LLP, Arlington VA 22209, USA
ABSTRACT This paper is meant as a primer on a new class of models, so-called disequilibrium network design models. These models maintain the usual design objective of maximizing some measure of social welfare, but recognize that traffic on a network is not necessarily in equilibrium and that capacity changes to the network must-induce transient phenomena not captured by the invocation of Wardrop's First Principle (user equilibrium). Such models by their very nature avoid temporal versions of Braess's paradox known from static equilibrium design models.
INTRODUCTION The network design problem is a mathematical abstraction and simplification of a problem often faced by agencies responsible for managing highway networks: namely how to optimally enhance network capacity to meet constantly changing travel demands. This problem becomes increasingly important when regional economic growth is closely intertwined with the capacity of the supporting transportation infrastructure while resources available to enhance capacity remain limited.
143
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T. L. Friesz and S. Shah The traditional equilibrium network design problem seeks an optimal network design
in terms of additional facilities or capacity enhancements when the network flow pattern is constrained to be a static equilibrium. The much reported occurrence of Braess' paradox requires that design models have Wardropian user equilibrium constraints.
The trans-
portation research literature includes a number of such equilibrium design models wherein equilibrium constraints are generally articulated as an equivalent optimization problem (Abdulaal and LeBlanc (1979), LeBlanc (1975)) or as an equivalent variational inequality problem (Friesz et al. (1990), Friesz et al (1992), Friesz et al. (1993), Marcotte (1986)). Unfortunately such equilibrium network design models presuppose a static environment and completely ignore the impacts of transient phenomena or potential disequilibria which can arise due to perturbations in the capacity of the network infrastructure. Such a static perspective may lead to the occurrence of a temporal version of Braess' Paradox. The "temporal Braess' Paradox" does not yet enjoy a standard definition, but in the context of the present discussion it can be viewed as occurring when a capacity altering action lowers (or leaves unchanged) overall delay (or some general measure of network performance) in the present and near future but increases overall delay at some more distant time. Such outcomes are possible when the present value of disequilibrium impacts is substantially negative, as can occur when the immediate disequilibrium response is a sharp congestion increase or when the disequilibrium response is a mild congestion increase of relatively long duration. An example is provided by a highway construction project intended to enhance capacity but which produces traffic congestion for weeks or months prior to its completion; in this case the present value of construction impacts may be sufficiently negative to offset all positive benefits in the post-construction period. In light of the above remarks, this paper reviews and explains the "disequilibrium network design" paradigm first proposed by Friesz et al. (1996b). This new paradigm differs from the historical static one in the sense that it takes into account both the underlying time varying nature of the network and the disequilibrating effects that capacity enhancements to the network may produce. In the sections which follow we mathematically articulate such a disequilibrium network design paradigm as a special type of optimal control problem. To achieve this control theoretic formulation we employ the concept of a network traffic disequilibrium, by which is meant a flow pattern which may fail to satisfy flow conservation (transportation market clearing) constraints and for which network
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users may experience clear advantages from changing their paths, but which evolves from a previously realized (dis)equilibrium in accordance with some plausible behavioral laws. A few words about the organization of this paper are in order. The main sections of the paper, in order of their appearance, are listed below, along with a brief description of their content:
• Introduction. Contains an overview of the paper's aims and content. • Notation and Assumptions. Contains an explanation of the notation that will be used and comments briefly on regularity assumptions which are employed. • Disequilibrium Dynamics. Explains the concept of global projective dynamics and briefly reviews alternative dynamics. • Objective Function. Shows how to mathematically articulate net economic benefits as the total consumers' surplus less transportation costs. • Optimal Control Formulation of Disequilibrium Design. States the disequilibrium network design model originally proposed by Friesz et al. (1996b). • Solution Methodology. Describes how mathematical programming may be used to solve the model. • Conclusion and Future Research. Discusses future enhancements and applications of the disequilibrium network design paradigm.
NOTATION AND ASSUMPTIONS Following the traditional transportation literature let us assume a network represented by a directed graph G(JV, A), where N is the set of nodes and A is the set of arcs connecting them. We assume \A\ = m and \N\ = n. In general, we will use the index a to denote an arc and the index p to denote a path. The set No C N is the set of nodes which are trip origins and NO C N is the set of nodes which trip destinations. We denote the set of all paths connecting node i to node j as P^. The set of all network paths is P = U i
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with cardinality given by p =| P . Throughout our analysis we will consider a fixed planning horizon of length T; that is, all activity occurs at some time t € [0, T]. We employ hp(t) to denote flow on path p at time t. Following the standard vector notation, h(t) = (hp(t) : p G P) is the full vector of path flows at time t. Each path p e P will have a unit cost function cp[h(t}} associated with it at time t. Although transportation costs may be explicitly time dependent, a feature which can in principle be incorporated into the model, we ignore such explicit time dependence for the sake of simplicity. Instead transportation costs are implicitly time varying. Let W = (w^) and F = (jap) denote path-OD incidence matrix and arc-path incidence matrix respectively. The w^ are the binary elements of the path-OD incidence matrix; specifically, w^ = 1 if path p connects origin-destination (OD) pair ( i , j ) and w^ = 0 otherwise. Similarly ^ap is an element of the arc-path incidence matrix; ^ap = 1 if a 6 p and ^ap = 0 otherwise. We let Uij(t) denote minimum average travel costs from origin i to destination j reported to users of the network by the underlying information system at time t. We consider a generalized setting wherein travel demands are elastic and invertible.
Tij(u(t]
is the travel demand between origin i and destination j at time t and 0j,-(T[w(i)]) is the inverse demand. Furthermore for the present analysis we consider "continuous" network design as opposed to "discrete" network design. That is, we consider continuous capacity enhancements ya(t) for arc a at time i; and TTa(ya(t)) are continuous, nonnegative, increasing, differentiable convex functions describing the cost of making an improvement of magnitude ya(t) at time t. We assume B to be a fixed exogenous improvement budget and r to be a nonnegative constant discount rate. Although, establishing/upgrading network facilities may take several time periods, we assume for sake of simplicity that network facilities can be upgraded instantaneously as of time t. This assumption involves no loss of generality as one can compute the equivalent present value relative to any date of interest of any time stream of cash flows. For convenience Appendix A summarizes the notations introduced above in a tabular format.
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DISEQUILIBRIUM DYNAMICS The aforementioned disequilibrium network design paradigm depends on a valid description of the disequilibrium adjustment process which guides the network of interest from one disequilibrium state to another, eventually settling down to a conventional steady state (static) equilibrium. Friesz et al. (1995, 1996) and Dupuis and Nagurney (1993) introduced a class of dynamic systems based on minimum-norm projections, termed "projective dynamic systems," for studying the time varying evolution of various competitive systems. Following the terminology coined by Smith et al. (1996), the class of dynamic systems proposed by Friesz et al. (1995, 1996) we call "global" projective dynamics, corresponding to the interior, anticipatory nature of their trajectories, while the class of projective dynamic systems proposed by Dupuis and Nagurney (1993) we call "local" projective dynamics in accordance with their boundary following nature. Some debate has arisen in the transportation and spatial economic research community regarding the relative merits of local versus global projective dynamics. In particular, Nagurney (1994) has claimed that our global projective dynamics are "incorrect" and that only the variety of local projective dynamics she has studied are valid for describing behavior. Her main point in this regard has been that global projective dynamics anticipate constraint boundaries and avoid those boundaries, so that only at a steady state do any constraints bind. As Smith et al. (1996) have written, there is an entire body of literature on anticipatory systems in physics and economics with exactly this property of anticipation shown by global projective dynamics; furthermore, there is no reason to believe that all common physical and economic adjustment processes proceed by moving along constraint boundaries as is the case with local projective dynamics. Moreover, the urban network disequilibrium model discussed here and in the Friesz et al. (1995) has only two constraints: nonnegativity of path flows and nonnegativity of perceived costs. That such constraints should not bind - i.e., should not cause path flows and perceived costs to be zero - is not only plausible but also observed. That is:
1. flows on paths actually considered viable by travellers continue to be used at some generally diminishing level until the learning process represented by the disequilibrium adjustment process is complete, and
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2. perceived costs are simply never zero for any realistic travel options.
It was to rigorously establish these and other distinctions between global and local projective dynamics which are not initially apparent that the paper by Smith et al. (1996) was written; the interested reader is strongly encouraged to consult that paper. Both of the aforementioned classes of projective dynamic systems are characterized by polyhedral constraint sets and are constrained in such a way that their steady states (stationary points) are solutions of constrained finite dimensional variational inequalities (VI). Both projective dynamic systems provide intrinsically interesting frameworks for creating models describing disequilibrium trajectories of real economic and physical process prior to reaching steady states. Because of its central importance to the description of disequilibrium design, the following subsection presents a brief introduction to disequilibrium adjustment processes proposed by Friesz et al. (1995) which are realizable network generalizations of the traditional aspatial tatonnement models
1
reported in the economics literature. The versions
of these models discussed below emphasize day-to-day adjustments of flows and costs. See Friesz et al. (1996) for a discussion of how these models may be extended to include within day dynamics.
Global Projective Dynamics for Urban Network This section condenses the presentation in Friesz et al. (1995) of a day-to-day tatonnement model used to describe disequilibrium dynamics. Friesz et al. (1995) develop a model with incomplete information by assuming that the travel demand functions Tij[u(t}} are not invertible. Thus, there is no ability to express the minimum average transportation costs Uij in terms of the T,j and hp variables in closed form. In place of a direct means of imputing the minimum average travel costs, one may imagine that a driver information system facilitates the estimating of the u^ by monitoring excess demands. In particular, :
A tatonnement process is a trading process among all agents (sellers and buyers) in an economy which
is conducted by a super-auctioneer. Specifically, in the trading process the auctioneer calls out a set of prices and receives transaction offers from the agents. If these offers do not match (amount demanded is not equal to amount supplied), he calls another set of prices by following some rules and the process continues without any transaction being allowed to take place until transaction offers match.
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the perceptions of minimum average travel costs have time rates of change which are proportional to excess travel demands. Only when an equilibrium is attained are these perceptions identical to the actual minimum average costs.
Furthermore, Friesz et al.
(1995) posit that path flows have time rates of change which are proportional to excess travel costs. [This latter concept for flow adjustments is similar to an idea put forward by Carey (1980), although he did not treat congestion.] For this discussion it is convenient to assume that the relevant constraints for a disequilibrium adjustment process on an urban network are the non-negativity of flows and perceived costs, and that the flow conservation constraints (since they are market clearing statements) are not in force prior to attaining an equilibrium. The meaning of non-market clearing is discussed in some detail by Friesz et al. (1995), but suffice it to say now that it corresponds to a situation in which the desire to travel expressed through travel demand functions does not match actual travel owing to a discrepancy between perceived and real travel costs (incomplete information). The methodology described below can be used to analyze disequilibrium problem constrained in such a way as well as other more complicated problems. The excess transportation demand between i G NO and j 6 N& is expressed as ETDij(u(t)t h(t)} = T^t}} - £ hp(t) ptPii
(1)
The excess travel cost on path p 6 Pij is expressed as ETCp[Uij(t), h(t}} = cp[h(t)} - ul}(t)
(2)
Further, Friesz et al. (1995) define vl3(t] = Prn{uij(t) + aETDZJ[u(t), h ( t ) } } , a e ^,
(3)
where the operator Pr^{ • } denotes a projection to avoid infeasibilities onto the closed set of constraints Q pertinent to the analysis and is defined by PrnM = argmin||i/-y|| yen
(4)
for an arbitrary vector v. As such Vij(t) may be viewed as the instantaneous revision of the perceived cost in accordance with continuously provided (excess) demand information. Friesz et al. (1995) further postulate that future perceptions of O-D travel costs are formed
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through adjustment of current cost perceptions at a rate proportional to demand induced change in perceived cost [%(£) — "%]• That is,
Kiij € &i, ~r' fc
(5) V /
It is immediate from (3) and (5) that the "price" dynamics are
A similar argument may be used to develop flow dynamics. Specifically, Friesz et al. (1995) observed that
Furthermore, one also must impose the initial conditions u(t = 0) = u° e 5R+
D
'
and h(t = 0) = h° € 3f^.. Using the obvious vector notation £?TD[u(t), h(t)] = (ETDi^t), h(t)} i G No, 3 e A^D)' and ETC7[u(<),/i(<)] = (^TCP[^(^), /i(«)] : « e ^o, j e ND, p e P zj )', the system (6) and (7) is readily stated as it(t) = K[{u(t) + aETD[u(t), h(t)}}+ - u(t)]
V t e [0, T]
h(t) = r)[{h(t) - pETC[u(t),h(t)}}+ - h(t)}
V t € [0, T]
where K = diag(Kjj : i 6 A^o, j € N/j) and 77 = diag(^p : p € P). It is convenient to refer to (8) as a Wardropian user equilibrium tatonnement model (TM). Friesz et al. (1996) have pointed out that when there are additional disequilibrium trajectory constraints, beyond the non negativity restrictions considered above, that (8) becomes:
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where the operator PTQ{-} denotes the projection onto an arbitrary set of constraints and is denned, for an arbitrary vector v, by
Additional constraints requiring the use of formulation (9) could arise when there are technical, economic, or regulatory limitations on the underlying information system not reflected in the idealized story told here. The following key theorem holds: Theorem 1. The steady states of TM (8) are static user equilibrium 2 when arc costs are positive and continuous and travel demands are continuous. Proof: See Friesz et al (1995). It is also significant that Smith et al. (1996) has established existence and uniqueness and containment of the solutions for (9) under quite plausible regularity conditions. 2
Wardropian static user equilibrium problem is to find a nonnegative vector of (h*,u*) such that the
following conditions are satisfied:
where the " * " superscript denotes a steady state or equilibrium value.
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Furthermore Friesz et al. (1995) have provided a proof of asymptotic stability for (8).
THE OBJECTIVE FUNCTION Consumer surplus is generally accepted as a means of assigning a monetary value to the benefits that users derive from transportation investment projects (Jara-Diaz and Friesz (1982)) when travel demand is elastic. The Marshallian concept of consumer surplus as the "area below the demand curve and above the price line" can only be applied to the case of transportation systems if great care is used in extending the notion of economic surplus to demand functions which are nonseparable. This is because of the presence of "shifting demand" caused by the availability of alternative modes or paths to users. In order to avoid the ambiguity that can arise with respect to the value of the consumer surplus integral, Hotelling (1938) provided an extension to Marshallian consumer surplus to general equilibrium by employing a path dependent line integral of demand rather than an ordinary integral. That is, the Hotelling measure for change in consumer surplus i corresponding to a change in perceived costs of transportation (u° to u1) is
(11) where "f" denotes a path dependent line integral and Wij is dummy variable of integration. In the case of the dynamic disequilibrium network design problem, we wish to maximize the net present value of benefits to users of the transportation network over a fixed planning horizon [0, T]. From the preceding discussion, the present value of the gross benefits to the users of the transportation network is the present value of consumer's surplus calculated from these travel demands. Hence, the net benefits accruing to users at time t will be the instantaneous change in consumer surplus less system wide congestion costs:
The net benefits to be reflected in our objective function will be the integration of properly discounted Z[-] over time. If we use r to denote a constant discount rate, the present value of net benefits to the users of the transportation network can be expressed as
Foundations of a theory of disequilibrium network design
153
OPTIMAL CONTROL FORMULATION OF DISEQUILIBRIUM DESIGN For simplicity of exposition, we take the capacity enhancement for arc a at time t to be a continuous variable y Q (t); while fa(i] will denote the flow on arc a at time t. Consequently, the unit cost function for arc a is of the form ca[f(t),y(t)},
where f ( i ) = (fa(t)
'• a £ A)
and y(t) = (ya(t) '• a 6 A). It follows from the identities / — Th and cp = ^a 7apca that the unit cost on path p is of the form cp[h(t),y(t)] with c[h(i),y(t)] = {cp[h(t),y(t)]
: p £ P}.
Moreover, Smith et al (1996) shows that for plausible regularity conditions TM (8) is always contained in the feasible region denned by the constraints of interest. Also observe that since the travel cost vector depends on improvements, namely
the desired disequilibrium network design model has the form
subject to:
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y(t) > 0
V t € [0,T]
(20)
This model is evidently an optimal control problem. In the terminology of optimal control theory, u(t) and h(t) are called state variables and y(t] control variables. We seek the trajectories through time of both the state and the control variables. Equations (15) and (16) are the dynamics of the state variables u and h respectively, while (18) and 19) are their initial conditions. Constraint (17) is a budget constraint. Constraint (20) ensures that variables y ( t ) are non-negative, which prohibits dis-investment.
Alternative Formulation Model (14)-(20) can be further simplified and restated by exploiting the "price" dynamics (15). In particular it is useful to use the notation :[{u(t) + aETD[u(t),h(t)]}+-u(t)]
V te[0,T]
(21)
as a shorthand for the right hand side of dynamics (15), so that du = KG(ETD)dt
(22)
One may unambiguously evaluate consumers' surplus, despite it's original articulation as a line integral, by using the flow dynamics to express it entirely in terms of ordinary integrals as follows: rult)
E
E £
ft
7i>(*)]«K-= E
E JQ Tl][u(t'}}G\ETD(u(t'},h(t'}}}dt'
(23)
As a result the optimal control formulation of (14)-(15) can now be restated as: max J ( u ( t ) , h ( t ) , y ( t ) )
= PVrt| £ ^ f Tlj[u(t')]G[ETD(u(t'), 0
h(t'))]dt'-
\ieN0j€ND''0
(24) peP
)
subject to: u(t) = K[{u(t) + aETD[u(t), h(t)}}+ - u(t)]
h(t) = ri[{h(t) - pETC[u(t),h(t), y(t)}}+ - h(t)] V i6[0,T]
V t € [0, T]
V t e [0, T]
(25)
(26) (27)
Foundations of a theory of disequilibrium network design
155
(30) This last formulation involves only ordinary integral and as discussed below, suggests a natural discrete time mathematical programming formulation for numerical calculations.
Necessary Conditions and Economic Interpretation The necessary conditions for a solution of a classical optimal control problem are given by Pontryagin's maximum principle (Pontryagin et al. (1962)). However, the state equations (15) and (16) are nonstandard. Because of the presence of projection operators, the right hand sides are non-differentiable (non-smooth). Hence, the necessary conditions are not amenable to analysis via classical theory. One has instead system with discontinuities in the state variables and system equations. Shah (1996) analyses in detail the necessary conditions using results from Bryson and Ho (1975). The necessary conditions for disequilibrium network design problem can also be analyzed using generalized subgradients (Clark (1983)) or differential inclusions (Aubin(1989)). Shah (1996) also shows that the necessary conditions derived from the optimal control formulation are economically/behaviorally meaningful in the sense that they imply that net present value of the marginal total costs equals the net present value of the marginal total benefits. This analysis is mathematically quite tedious and is omitted for the sake of brevity.
A SOLUTION METHODOLOGY In this section we discuss briefly a possible solution methodology that can be employed to solve the optimal control formulation of the disequilibrium network design model. Optimal control problems may be viewed as special cases of mathematical programming problems. Since discrete optimal control problems are optimization problems in finitedimensional spaces and, hence, require less mathematical sophistication in their treatment
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than continuous optimal control problems which are in infinite-dimensional spaces, we choose to discretize our disequilibrium network design model. Furthermore, it seems possible to use traditional methods of non-differentiable optimization such as classical search techniques to solve small and medium size mathematical programs created in this fashion. To discretize the present continuous optimal control model, first rearrange the terms of equation (24) to obtain J(u(t),h(t),y(t)}
= f
e~rtdt\Y. E
•^0
/
I „-,- \r_ ^^^/•„ •'0
Tlj[u(t')}G{ETD(u(t'),h(t'))}dt'(31)
pep
For the sake of convenience we define *"(*)= E
E
i£N0 j€ND
J0
lijWmETDfrWMWW I)
(32) (33)
We can then rewrite (31) as J = /oTe-rtdtl-j2
£ F(t)-C(h(t),y(t)]\
(34)
The "inner" integral in F(t) can be expressed in terms of discrete time steps as E'E ET^(t'k}}G(ETD(u(t'k),h(t'k))}(tk+,-tk)
(35)
k=0
In the present model, the initial time t0 is known a priori, and we assume a fixed time step, Ai — tk+i — tk, Vk € [0, K - 1], where K is a integer parameter such that KAt = i and t e [0, T]. That is, K is the number of time steps for the "inner" integral. Similarly one can discretize (34) as follows: J = £ e - rt 'Ffe)fa+i - t,) - C(/i(i / ),«^))(^ + i - <j)>
(36)
where T = I/At and L is another integer parameter. If we define KiAt = tt, then (35) can be expressed as = E' E
E WftWETDWJMmt-,
k=0 ieN0i£ND
(37)
Foundations of a theory of disequilibrium network design
157
Substituting into (36) and rearranging terms, we get
Assuming the time steps At to be unity
The equivalent discrete disequilibrium network design model can then be stated as
subject to:
The above mathematical program (40)-(46) is a nonlinear mathematical program, where h(ti) and u(t{) are vectors of decision variables. It is also obvious that the number of decision variables in the above math program is finite, as the optimal control problem is now in discrete time and has a fixed horizon. Such a mathematical program can in principle be solved by methods of non-differentiable optimization, such as classical search or even artificial intelligence based techniques.
CONCLUSIONS AND FUTURE RESEARCH The paradigm of "disequilibrium network design" based on "disequilibrium dynamics" derived from the tatonnement paradigm of microeconomics was used to present an optimal
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control formulation of network design problem. Indeed it appears that this model, because of it's ability to address disequilibrium issues, holds the potential to significantly impact the transportation planning process. Further theoretical and numerical analysis are required to assess the potential impacts.
REFERENCES Abdulaal, M., and LeBlanc, L. (1979), Continuous Equilibrium Network Design Model. Transportation Research, vol. 13B, pp. 19-32. Aubin, J. and Cellina, A. (1984), Differential Inclusions, Springer-Verlag, Berlin. Bryson, A.E. and Y.C. Ho, (1975), Applied Optimal C'on£ro/(revised), John Wiley and Sons, New York. Carey, M. (1980), Stability of Competitive Regional Trade with Monotone Demand/Supply Functions. Journal of Regional Science, 20, 489-501. Clark, F. (1983), Optimization and Nonsmooth Analysis, Wiley-Interscience, New York. Dupuis P. and A. Nagurney, (1993), Dynamical Systems and Variational Inequalities. Annals of Operations Research, 44, 9-42. Friesz, T.L. (1985), Transportation Network Equilibrium, Design and Aggregation: Key Developments and Research Opportunities. Trans. Res. 19A, 413-427. Friesz, T.L., R. L. Tobin, H. J. Cho and N.J. Mehta (1990), Sensitivity Analysis Based Heuristic Algorithms For Mathematical Programs With Variational Inequality Constraints. Mathematical Programming 48, 265-284. Friesz, T.L., H.J. Cho, N.J. Mehta and R. L. Tobin, (1992), Simulated Annealing Methods for Network Design Problems with Variational Inequality Constraints.
Transporta-
tion Science 26(1), 18-26. Friesz,T.L., R. L. Tobin, S.J. Shah N. J. Mehta, and G. Anandalingam, (1993), The Multiobjective Equilibrium Network Design Problem Revisited: A Simulated Annealing Approach. European Journal of Operations Research 65, 44-57. Friesz, T.L., D. H. Bernstein, N.J. Mehta, R. L. Tobin and S. Ganjalizadeh (1995), Dayto-Day Dynamic Network Disequilibrium and Idealized Driver Information Systems. Operations Research, 43, 1120-1136.
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Friesz, T.L., D. Bernstein, and R. Stough. (1996), Dynamic Systems, Variational Inequalities and Control Theoretic Models for Predicting Urban Network Flows. Trans. Sci. 30(1), 14-31. Friesz, T.L., S. Shah and D. Bernstein (1996b), Dynamic Disequilibrium Network Design: A New Paradigm for Transportation Planning and Control, submitted to Network Infrastructure and the Urban Environment. Hotelling, H., (1968),The General Welfare in Relation to Problems of Taxation and Railway Utility Rates. Econometrica, 6, 242-269. Jara-Diaz, S. R., and T. L. Friesz, (1982),Measuring the Benefits Derived from a Transportation Investment. Transportation Research B, 16B(1), 57-77. LeBlanc, L. (1975) An Algorithm for the Discrete Network Design Problem. Transportation Science 9, 283-287. Marcotte, P. (1986), Network Design Problem with Congestion Effects: A Case of Bilevel Math Program. Mathematical Programming, 34, 142-162. Murchland, J. (1970), Braess' Paradox of Traffic Flow. Transportation Research, 4, 391394.
Nagurney, A., (1994), Comments during The Conference on Network Infrastructure and the Urban Environment - Recent Advances in Land-Use/Transportation Modeling, Sweden. Pontryagin, L.S., V. A. Boltyanskii, R. V. Gramkrelidze and E. F. Mischenko, (1962), The Mathematical Theory of Optimal Processes, John Wiley & Sons, New York. Shah, S.J., (1996), Dynamic Disequilibrium Network Design Model. Ph.D. Dissertation, George Mason University. Smith, T. E. (1993), A Comparative Analysis of Two Minimum-Norm Projective Dynamics and Their Relationship to Variational Inequalities, submitted to SI AM Journal of
Optimization.
Wardrop, J. (1952), Some Theoretical Aspects of Road Traffic Research. Proc. Inst. Civ. Eng., 1, Part II, 325-378.
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Appendix A
Notation A
is the set of arcs; | A \ = m
N
is the set of nodes; \ N \ = n
NO C TV
is the set of nodes which are trip origins
ND C N
is the set of nodes which are trip destinations
Pij
is the set of paths connecting node i to node j
P = U
PIJ
is the complete set of network paths; | P \ = p
3£ND
[0, T]
is the closed time interval of initial interest
hp(t)
is the flow on path p at time (e.g. day) t
h(t) = (hp(t) : p 6 P)
is the full vector of path flows on day t
w?j
is an element of the path-OD incidence matrix; specifically, to?- = 1 if path p connects origin-destination (OD) pair (i, j) and w\, = 0 otherwise
W = (w^ :ieNo,j£ ND,P e Pij)' jap
is the path-OD incidence matrix is an element of the arc-path incidence matrix; specifically 7ap = 1 if a € p and ~fap = 0 otherwise
F = ("fap : a € A,p € PIJ)
is the arc-path incidence matrix
Uij(t)
is the travel costs reported by the ATIS for origin i and destination ion j on day t.
u(t) = (uij(t) : i 6 NO, j 6 NE>)
is the full vector of minimum average travel costs at time t
cp[h(t)}
is the unit cost of flow on path p at time t, as
c[h(t)} = (cp[h(t)] : p e P)'
is the full vector of path costs at
a function of the full vector of path flows h(t)
time t r
constant discount rate
Tij[u(t)}
is the travel demand between i and j at time t
T[u(t)} = (Tij[u(t)] : i 6 No, j e ND)'
is the full vector of travel demands
Foundations of a theory of disequilibrium network design
©ij(T[u(t)])
161
is the inverse travel demand between i and j at time t
Q(T[u(t)]) = (Qij(T[u(t)])
: i € No,j € ND)'
is the full vector of inverse travel demands at time t
ya(t)
is the magnitude of the improvement to link a, i.e., the increase in the effective capacity of link a at time t
n
is the cost of making an improvement of
B
is the fixed exogenous budget available for
(ya(t))
magnitude ya on link a at time t
improvement
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11 THE CONTINUOUS EQUILIBRIUM OPTIMAL NETWORK DESIGN PROBLEM: A GENETIC APPROACH N.D. Creel, SIAS Ltd, Edinburgh, M.J. Maker andB. Paechter, Napier University, Edinburgh.
ABSTRACT A genetic algorithm (GA) program for providing a solution to the Continuous Equilibrium Network Design Problem (NDP) is introduced following a general discussion of the network design problem and genetic algorithms. A description of the current GA operators used in the program are described and early preliminary results shown. While the program is in its early stages of development the results have been encouraging and so further development is planned utilising more of the characteristics of the continuous NDP to reduce the computational burden.
INTRODUCTION The Network Design Problem (NDP) is concerned with improving an existing road network by adding new links or by improving existing ones. With ever decreasing budgets it is becoming more important to obtain the maximum benefit from these improvements by selecting those schemes that provide the highest return (in terms of reduced journey time) on investment. There are primarily two groups of people involved in the NDP: the network planner and the network user. The planner aims for a system optimal solution to any network design, so that the total costs involved in travelling through the network are minimised, whilst remaining within the available budget for road construction or improvement. The system users aim to minimise their own travel costs; the resulting routing pattern is called the user optimal solution, which is in general not the same as the system optimal solution. These two conflicting demands, and the accurate prediction of the effects of a network improvement on a traveller's route choice, are the crux of the NDP. f
This paper was written while the primary author was a Research Assistant at Napier University, Edinburgh. 163
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The NDP may be separated into two broad categories, discrete and continuous. The discrete NDP is concerned with the addition of new links only; i.e. for each proposed link development the scheme is either implemented or not. Hence for a scheme with n proposed link improvements there are 2" feasible solutions. In the continuous case not only is a decision made as to whether a new link is included but also how much extra capacity should be allocated to new and existing links to obtain the maximum return. There are therefore an infinite number of feasible solutions to the continuous NDP. This paper reports on an investigation into the viability of applying genetic algorithms to the continuous network design problem.
NOTATION The following notation is used to define the problem: fa = equilibrium traffic on arc a f = (..../„,..) ca = unit cost of travel on arc a ya = capacity improvement for arc a
y = Uy.,...) Ga = improvement cost for arc a 6 = user defined weighting factor
THE CONTINUOUS EQUILIBRIUM NETWORK DESIGN MODEL AND EXISTING HEURISTIC MODELS The continuous equilibrium network design problem may be written as a bi-level programming problem: Minimise
subject to ya >OVa(a = l,2,...,n)
(1)
f (y)is a user optimised equilibrium flow pattern The deterministic user equilibrium flow pattern is found by solving: fa
min P(f) = ^\ \ca((O,ya)d(O " 0
The continuous equilibrium optimal network design problem
165
The majority of current solution methods to the NDP are based upon iterative heuristic algorithms. Existing algorithms may be classified according to the assignment model used and the way in which investment is treated in terms of whether continuous or discrete investment is chosen and if investment is considered in the objective function or as an additional constraint. In broad terms many heuristic algorithms follow the methodology outlined below.
STEP 1: Initialise/ (this might be set equal to zero to represent the 'do-nothing' case). STEP 2: Assign the trips to the network and obtain a measure of total network cost Z°.
STEP 3: Change y° by some vector of step values, 8. y1=y°+&. STEP 4: Re-assign the trips to the network and obtain a new estimate of the total network costs Z1. IF Z1 < Z°, set Z°=Z',/=/ and goto STEPS. ELSE IF 8 > lower limit on step length, set 8new<8, set 8=8new and goto STEP3. ELSE STOP and keep/.
There are many different methods of choosing 8 and of selecting whether to add capacity to all or some links in any one iteration. Many of the heuristics are based on methods of steepest descent (e.g. Suwansirikul et al. 1987, Abdulaal et al. 1979) and, because the problem is non convex, do not guarantee an exact solution as they 'home in' on local optima which may be globally sub-optimal. To be confident of finding the global optimum a very large number of runs of the heuristic would be required with different starting positions. This is both impractical and expensive. However by limiting ourselves to a practicably small number of runs there is no guarantee of finding the global optimum.
GENETIC ALGORITHMS Genetic algorithms (GA's) attempt to duplicate some of the processes that may be observed in natural evolution. Processes of natural selection cause those chromosomes that encode successful structures to reproduce more often than those that do not. However mutations may occur causing the 'child' chromosome, created by the combination of two 'adult' chromosomes, to be quite different. Through natural selection, the information to produce 'fit1 individuals is carried through to the next generation.
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Genetic algorithms borrow much of their terminology from natural genetics. Within a population of feasible solutions there are individuals (or genotypes, chromosomes) which in turn are made up of units, called genes, arranged in a linear string. Each gene controls the inheritance or otherwise of a certain characteristic and each chromosome represents a potential solution (its phenotype) to a problem (the meaning of which is defined by the user). For genetic algorithms to be used successfully great care has to be taken encoding the solutions to a problem, i.e. converting solutions to chromosome representations, and using an evaluation function that returns a value of the worth of any chromosome. In traditional GA programs the problem is manipulated such that the function variables can be represented by a chromosome of binary digits - 1's and O's (for the purposes of introducing GA's to the reader it will be assumed that all chromosomes consist of binary strings, although this is not the case since GA's have been developed to consider chromosomes of other forms). Broadly, each new generation of chromosomes is created by mating current chromosomes and applying a selection of GA operators such as mutation and crossover. The procedure involved in GA programs has been defined by a number of authors. For example, the model described by Michalewicz (1994) can be seen below.
PROCEDURE Genetic Algorithm BEGIN t = 0; Initialise P(t); (Where P(t) is generation number t of a population of chromosomes) Evaluate structures in P(t); WHILE termination condition not satisfied DO BEGIN t = t+l; Select P(t) from P(t-l); Recombine structures in P(t); Evaluate structures in P(t); END END.
From the above procedural diagram it can be seen that there are four elements to be considered in the use of genetic algorithms: those of initialisation, evaluation, selection and recombination. Firstly a decision has to be made on how to represent the problem. It is possible, given the number of parameters, their domain and precision to encode any feasible solution as a binary chromosome (for details see Michalewicz (1994)). Care must be taken when designing a mapping to ensure that not only are all feasible solutions possible but that all non-feasible solutions are excluded. Generally some form of random binary initialisation is used to generate an initial population. After the initialisation step each chromosome is evaluated to find its fitness value. The evaluation involves decoding the chromosomes and then calculating the function value.
167
The continuous equilibrium optimal network design problem
From the initial evaluation we need somehow to select a parent population of the fittest individuals thus giving a greater chance of reproduction to those solutions which are deemed to be the best so far and which might thus lead to a better solution. Although there are many techniques available roulette wheel parent selection is one which is commonly used. The roulette wheel is constructed as follows: • Calculate the fitness value eval(\\) for each chromosome v, (i= 1,..., pop_size}. • Find the total fitness of the population
F = • Calculate the probability of a selection p, for each chromosome vj (i=l,.. .,pop_size}:
Calculate a cumulative probability q; for each chromosome v; (i= 1,... ,pop_size).
Once the roulette wheel has been created it is just a matter of 'spinning' the roulette wheel pop_size times; at each 'spin' we select a chromosome for the new population by: • Generating a random number r in the range [0.. 1 ] • If r
e.g. In the example below there is a population size of 10 and the total fitness, F=76 Chromosome Fitness
Pi
qi
1 8 0.105 0.105
2 2 0.026 0.131
3 17 0.224 0.355
4 7 0.092 0.447
5 2 0.026 0.473
6 12 0.158 0.631
8 7 0.092 0.868
7 11 0.145 0.776
9 3 0.040 0.908
10 7 0.092 1.000
We spin the roulette wheel ten times and obtain: Random Number
0.295
0.105
0.324
0.542
0.462
0.828
0.961
0.775
0.304
0.224
Chromosome chosen
3
1
3
6
5
8
10
7
3
3
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Each new parent's chance of selection is proportional to its fitness, however due to the random nature of the technique it is still possible for less fit chromosomes to be selected. In the above example the best chromosome (number 3) has been selected 4 times, however due to the random nature of selection the worst chromosome (number 5) has also been selected. Although the random nature of the method allows less fit solutions to be selected (as occurs in nature and is desirable in the GA process), over a number of successive parent selections lessfit chromosomes are excluded from new populations. In nature 'children' inherit characteristics from both parents and also through the process of mutation may also have genes not particular to either parent. Through this evolutionary process it is hoped that some children would inherit the best characteristics of both parents and so in some way 'improve'. Similarly the mutation may introduce genes that are better than any thus far encountered by the species. Genetic Algorithms have both these mating and mutation processes and they are applied after the parent population has been selected in the recombination module. 'Mating' or crossover is a simple process of recombining the genetic material of two parents to make two children. In the technique the breakpoint where crossover takes place is selected randomly.
Parent 1: 1 1 1 1 I 1 1
Child 1: 1 1 1 1 0 0 =>
Parent 2: 0 0 0 0 1 0 0
Child 2 : 0 0 0 0 1 1
The above is an example of one-point crossover due to the fact that there is just one breakpoint in the chromosome (n-point crossover is also possible). While crossover acts at the chromosome level mutation only affects individual genes. Mutation randomly selects a gene and then, in the binary string case, inverts it, i.e. turns a 0 into a 1 and vice versa. Other mutation operators do exist which act on real and integer genes. The user is responsible for selecting the number of occurrences of mutation by defining the probability of mutation occurring, it is therefore possible that in any one chromosome more than one gene is mutated, however a mutation rate of 1/ra (where m denotes the chromosome length) is normally used. Once the population has undergone mutation and crossover the new population is evaluated again and the process is repeated beginning with selection. This loop is then repeated for a number of generations. As stated earlier one of the problems with existing solutions to the NDP is that many of the models rely on methods of steepest descent which can lead to globally sub-optimal solutions. Genetic algorithms (GA's) have the attraction that they combine exploitation of existing solutions with exploration of the solution space to increase the chances of finding the global optimum.
The continuous equilibrium optimal network design problem
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Genetic algorithms are not so much a defined method as a philosophy with a great deal of flexibility of choice left to the user dependent upon the problem being solved. The user has to choose which genetic operators are most appropriate for their problem as well as decide upon population size and number of evaluations before termination. There are currently no defined methodologies for selecting appropriate parameters and the choice of parameter settings is problem specific and very much a 'black art' (see Schaffer (1989)).
THE GENETIC NETWORK DESIGN PROBLEM The authors have developed the Genetic Algorithm Network Design program GANDES* which has the network design problem function (1) as its fitness function. For the purposes of introducing GA's chromosomes were defined binary strings with each digit representing a parameter value to a problem function. This would suggest that GA's are well suited to dealing with the discrete case of the NDP, which indeed they are. However previous work done by the authors led to the decision to examine the continuous case using chromosomes consisting of strings of real numbers with each gene representing the additional capacity associated with a particular link. Real chromosomes may be used in GA programs without altering the procedure as outlined in the previous section, however modifications must be made to the processes of crossover and mutation to reflect the attributes of the real data type. Real chromosomes, and the genetic operators that act upon them are very similar to binary chromosomes and their associated operators. The crossover operators behave much the same except two parents randomly contribute their real number genes to create two new children. Mutation operators are slightly different in that they are not limited to switching a gene between 0 and 1 and visa versa. Real gene mutation operators give the user far greater flexibility in that they change the selected gene by assigning to it any real random number in the solution space. This means that mutation operators may be written which take advantage of what is known about the solution space to bias the success rate of the mutation operator in favour of generating fitter chromosomes. GANDES is initialised by creating a population of chromosomes, each chromosome having m genes, where m is the number of links being considered for improvement. Each gene i is a real number representing the additional extra capacity associated with improvement link i. Currently there are three traditional crossover methods available, one-point, two-point and uniform. Two-point crossover is similar to one-point as described earlier except that there are two cut points in the parent chromosomes, rather than one, selected at random and genes are swapped between the two points. With uniform crossover, for every gene in the chromosome, a random decision is made as to which parent contributes its gene to which child.
* Based on the GA developer by Mihai Petriuc, Gh. Asachi Technical University of lasi, Romania
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GANDES has three mutation operators, random, creep and step. The random mutation operator receives no information about the NDP solution space, it simply replaces the selected gene with a random number in the range of feasible solutions (currently between 0 and the maximum allowable extra capacity which is determined by the user). In this way new random variation is introduced into the population ensuring that other regions of the solution space are investigated, thus reducing the possibility of selecting a sub-optimal solution. Creep is similar to random except that it only changes the gene's value by ±10%; the decision of whether to increase or decrease the value is taken randomly. Again, while no information is utilised about the solution space, in later generations 'creep' is more likely to improve the chromosome than 'random' is. However the possibility of moving towards another minimum in the solution space is permitted by randomly deciding whether to increase or decrease the value of the gene. Step is slightly more intelligent than the other two mutation operators in that it examines the gradient of the Z-function (1) and then moves in the direction which is thought to reduce the Zfunction value further. The step size taken is a random Normally distributed number with a user defined mean step size and a standard deviation of ten times the mean. As mentioned in the introduction to GA's there are no guidelines in GA writing as to what are good parameter values and so those used are experimental with further research required to investigate the effects of changes in them. In the evaluation stage of GANDES for each chromosome, which represents a feasible solution to additional capacity, a deterministic user equilibrium assignment is carried out to determine the new flow pattern given the suggested capacity improvements, and the Z function (1) is then evaluated. In GANDES each link is assumed to have a cost flow function of the form:
Where:
f a is the deterministic user equilibrium flow on link a. Ka is the initial capacity of link a. ya is the additional capacity on link a (gene a of each chromosome of GANDES).
The cost of investment is measured by investment functions of the form:
Experiments have been performed to compare GANDES with the method proposed by HookeJeeves (1961) which involves the initialisation of a feasible solution as a starting point followed by a series of exploratory moves around the solution space at a fixed step size 8. This is followed by a pattern shift with step size a (the Hooke-Jeeves method used is described in Abdulaaletal. (1979)).
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EXPERIMENTAL RESULTS Initial experiments were carried out to investigate whether or not GA's could be used to find consistently superior results as compared with more traditional continuous NDP solution methods. For this GANDES was compared against the Hooke-Jeeves method described earlier on a small sixteen link network (taken from Suwansirikul et al. (1987)). The Hooke-Jeeves method as coded by the authors was validated by comparing results obtained in experiments with those that are published in Suwansirikul et al. (1987). For the 16 link network 30 HookeJeeves experiments were carried out with differing initial solutions, a stopping tolerance of 0.0001 and an initial step length 0.3. The results obtained were then compared with 30 runs of GANDES. The parameters set in GANDES were: a population size of 20, and one gene per chromosome having a 33% chance of undergoing one of the three forms of mutation described earlier. The probability of chromosomes undergoing crossover was 60% overall with a 20% chance it being one-point, 40% chance of it being two-point and 40% chance of uniform crossover. GANDES was set to terminate after 10,000 generations. Figure 1 was used as the initial network with which to compare Hooke-Jeeves and GANDES, Table 1 shows the travel cost functions and the investment functions for each link of the network. The assumed demand for travel on the 16-link network was 10 trips from node 1 to node 6 and 20 trips from node 6 to node 1. Figure 2 shows that GANDES consistently produces better results than the Hooke-Jeeves method. GANDES produced Z-function values between 518.46 and 520.05 with a mean of 519.03 and a standard deviation of 0.403, while Hooke-Jeeves produced values in the region of 519.16 to 674.36 with a mean of 573.12 and a standard deviation of 39.253. One of the problems with GANDES is that it takes approximately four times as long to run as the traditional Hooke-Jeeves method. One run of Hooke-Jeeves takes approximately 45-90 minutes depending upon the initial capacities and GANDES takes approximately 300 minutes for 10,000 generations. However the results show that the traditional Hooke-Jeeves method does not guarantee the best solution and so would need to be repeated a number of times with differing starting solutions, thus extending the computation time required and with no guarantee that any of the final solutions is indeed the optimal. Conversely, GANDES finds a good suboptimal solution very rapidly (see figure 3) and hence a good estimate would be available after far fewer than 10,000 generations and consequently less time.
CONCLUSIONS AND FURTHER WORK At this stage GANDES is a fairly simplistic application of common GA operators, it is only in the 'step' mutation operator that information about the characteristics of the solution space are utilised to assist in the search for an optimal solution. However, even in this primitive form GANDES has shown that a genetic algorithm approach to solving the NDP is a viable consideration worthy of further research. While a GA approach may not be quicker than traditional heuristic methods, it is not affected by the problems associated with steepest descent methods. Whereas the accuracy of traditional
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Link a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Aa 1.0 2.0 3.0 4.0 5.0 2.0 1.0 1.0 2.0 3.0 9.0 4.0 4.0 2.0 5.0 6.0
Ba 10.0 5.0 3.0 20.0 50.0 20.0 10.0 1.0 8.0 3.0 2.0 10.0 25.0 33.0 5.0 1.0
Ka 3.0 10.0 9.0 4.0 3.0 2.0 1.0 10.0 45.0 3.0 2.0 6.0 44.0 20.0 1.0 4.5
da
2.0 3.0 5.0 4.0 9.0 1.0 4.0 3.0 2.0 5.0 6.0 8.0 5.0 3.0 6.0 1.0
Cost function Ca(fa,ya) = Aa + Ba(fa/(Ka+ya))4 Ga(ya) = da*ya Table 1 - Data for 16-Iink network
Link no. 1
Figure 1 - 16-Link test network
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/uu •
»
650 •
600 -
Z-f unction
* ^ ***
***
»
* Hooke-Jeeves • GANDES
* * » »
AZC\ -
Test Number Figure 2 - Comparison of results for H-J and GANDES on the 16-link network
800T 750
§
700
•j3 U
|1
N
• Z-function value
65
° "
600 -550 -
500 0
2000
4000
6000
8000
No. of Generations
Figure 3 - Typical GANDES Best-so-far diagram
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methods is affected greatly by the initial starting solution and therefore often require a number of runs to be sure of finding a best solution, GANDES has shown the potential to consistently find optimal solutions. The trade-off between accuracy and time taken will be investigated further in tests on larger networks. There have already been a few tests carried out on the 76link, Sioux Falls network (LeBlanc(1975)) and results are very encouraging. As stated earlier, there are no guidelines for selecting the optimum parameter settings for genetic algorithm programs and the ones used in the experiments described were 'guessed' at by the authors based on good practice and professional expertise. It is thought that GANDES could be pushed much harder to find the optimum solution far quicker by altering the parameters and future investigations aim to research these parameter settings. New GA operators are being developed which utilise information related to the partial derivatives, with respect to additional extra capacity, of the NDP function (1) as proposed in the work by Davis (1994). Another option being considered is to form a hybrid method utilising the advantages of GA's in the early stages of the model but once the population begins to diverge to a global optimum then switching to a more traditional methodology for the final stages. It is hoped that in doing this the computational burden and consequently the processing time can be significantly reduced.
REFERENCES Abdulaal M. and LeBlanc L.J. (1979). Continuous Equilibrium Design Models. Transpn Res. 13B, 19-32. Davis G.A. (1994). Exact local solution of the continuous network design problem via stochastic user equilibrium assignment. Transpn Res. 28B, 61-75 Hooke R. and Jeeves T.A. (1961). Direct search solution of numerical and statistical problems. Journal, of the association for computing machinery 8, 212-229. LeBlanc L.J. (1975). An Algorithm for the Discrete Network Design Problem. Transpn Sci. 9, 183-199. Michalewicz Z. (1994). Genetic Algorithms + Data Structures = Evolution Programs, 2nd Edition. Springer-Verlag, New York. Schaffer J.D. et al. (1989). A study of control parameters affecting online performance of genetic algorithms. Proceedings of the 3r international conference on genetic algorithms, George Mason university. Suwansirikul C. et al. (1987). Equilibrium decomposed optimisation: A heuristic for the continuous equilibrium network design problem. Transpn Sci. 21, 254-263.
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CONTINUOUS EQUILIBRIUM NETWORK DESIGN PROBLEM WITH ELASTIC DEMAND: DERIVATIVE-FREE SOLUTION METHODS Hai-Jun Huang School of Management, Beijing University of Aeronautics and Astronautics, Beijing, 100083, P.R.China Michael G. H. Bell Transport Operations Research Group, Department of Civil Engineering, University of Newcastle, Newcastle Upon Tyne, NE1 7RU, U.K.
ABSTRACT Three representative algorithms, namely the Hooke-Jeeves method, the equilibrium decomposed optimization heuristic and simulated annealing, which have been applied to the equilibrium network design problem (ENDP) with fixed demand, are compared by numerical examples. The comparison is extended to cover the ENDP with elastic demand. A bi-level formulation of the ENDP with elastic demand is presented and validated. Experiments on simple and complex networks reveal some interesting and important phenomena, for instance, these three algorithms can give very different results when different values of the algorithm's parameters are used. The simulated annealing approach can always obtain the best solution while its computation time requirement is the most too. The equilibrium decomposed optimization heuristic is the most computationally efficient, but the solutions it yields are not quite as good as those given by the Hooke-Jeeves method, in most cases. A warning and suggestions are given in the last section.
1. INTRODUCTION Transportation planners are frequently confronted with the fundamental problem of how best to extend or improve a transportation network, give a limited budget. In order to choose the best option, planners need to assess the impact on social welfare. Economic theory suggests that social welfare is the difference between the price trip-makers are willing to pay (i.e., their benefit) and the full cost to society of constructing and using the network. Thus the network design problem (NDP) is to select link improvements or add new links to an existing network such that social welfare is maximized while accounting for the route choice behavior of network users. Two sets of decision makers with different objectives are involved in the NDP. The network users individually select their routes such that their individual travel costs are minimized, while the planners aim to make the best network improvements for reducing traffic congestion, energy consumption, pollution or other appropriate targets (the proxy index is generally the total travel time), subject to a budget 175
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constraint. Although the network users' behavior may be influenced by changing the network configuration, the planner has no direct control over the users' route choice. This problem is analogous to a leader-follower, or Stackelberg, game in which the planner is the leader and the users are the followers. The NDP could be formulated as a bi-level programming problem. For an overview of bi-level problems in transportation research, see Migdalas(1995). The equilibrium network design problem (ENDP), where flows are constrained to be in deterministic user equilibrium, is considered in this paper. Furthermore, the study focuses on the continuous case, where the link capacity enhancements are continuously divisible. The continuous network design problem via stochastic user equilibrium assignment was studied recently by Davis (1994). The ENDP has received much attention and the literature is still growing. As is well known, it is particularly difficult to solve the ENDP for a global optimum within an acceptable computational time. The reason is that the relationship between the planner and the network users is intrinsically nonconvex, and is impossible to describe by an explicit function. The exact solution of a standard user equilibrium assignment is required to evaluate the effects of the updated network each time* . The development of efficient algorithms which can handle realistic sized problems is still at the frontier of current transportation network analysis (see the remarks in Migdalas (1995)). Yang and Bell (1996) gave a general survey of existing algorithms. The branch and bound approach is generally used in solving the discrete ENDP (see for example, LeBlanc, 1975; Boyce and Janson, 1980; Magnanti and Wong, 1984; Chen and Alfa, 1991). LeBlanc and Boyce (1986) specifically described the discrete ENDP as a bi-level program. For solving the continuous ENDP, Abdulaal and LeBlanc (1979) adopted direct search techniques such as Powell's (1964) method and the Hooke-Jeeves (H-J, 1961) method. A number of heuristic algorithms with reduced computational burden have been proposed for finding an approximate solution to the ENDP, but none of them can be guaranteed to converge to the global or a near global optimum. For example, the iterative optimizationassignment algorithm (IOA), iteratively solves the system optimal network design problem with fixed link flows and the user equilibrium route choice problem with fixed network parameters. This algorithm, originally proposed by Allsop (1974) for traffic control problems and by Steenbrink (1974) for the NDP, is promising in terms of computational efficiency, but may converge to a solution dramatically different from the optimal equilibrium network design, as pointed out by several authors (Steenbrink, 1974; Tan et al., 1979; Marcotte, 1981; Barker and Friesz, 1984; Friesz and Harker, 1985). The IOA is an exact algorithm for a Cournot-Nash game in which the planner is myopic with regard to the reaction of network users to network improvements. Hence, the IOA is not appropriate to solve the ENDP, which is in fact a Stackelberg game. Suwansirikul et al. (1987) suggested a new heuristic, equilibrium decomposed optimization (EDO) in which the direction of the search for the optimal design of each link being considered for improvement is guided by
* The standard user equilibrium assignment problem can be solved either by the link flow-based Frank-Wolfe algorithm (LeBlanc et al., 1975) or by the path flow-based disaggregate simplicial decomposition algorithm (Larsson and Patriksson, 1992). The latter requires greater computer storage capacity but fewer iterations in achieving convergence than the former.
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reevaluating the reaction of the network users to the design changes. Yang (1995) proposed a sensitivity analysis-based heuristic. To avoid implicit functions in ENDP, Tan et al. (1979) presented a single level optimization method by formulating the user equilibrium assignment as a set of constraints involving path variables. The similar single level model was used by Friesz et al. (1993) but solved by simulated annealing. In general, this method is appropriate only to very small networks (Heydecker, 1986). Marcotte (1983) developed an exact algorithm based on constraint accumulation of a sequence of nonconvex-constrained sub-problems with two types of variables (link flows and planner's decisions). This was reported to be very difficult to solve (Marcotte, 1983). Marcotte (1981, 1983, 1986) proposed a family of heuristic algorithms and two of them were efficiently implemented and validated on the benchmark "Sioux Falls" network (Marcotte and Marquis, 1992). Harker and Friesz (1984) developed an efficient method to bound the solution of the ENDP. In spite of the various ingenious attempts made to satisfactorily solve the ENDP, these algorithms are unfortunately either incapable of finding the global optimum and/or impractical for problems of realistic size. Meanwhile, the sensitivity of existing algorithms to the parameters used has not been investigated. Because of the inherent complexity of and computational burden imposed by the ENDP, it seems mandatory to investigate heuristic algorithms having random searching characteristics. Recently, simulated annealing (SA) has been applied to solve the ENDP and some encouraging results have been obtained (Friesz et al., 1992; Friesz et al., 1993). In principle, SA is a universal probabilistic optimization approach which combines the gradient descent method and the random searching method to find the global or near global optimum. Of course, SA is computationally expensive, so it should only be employed to solve important problems. The computational burden will be gradually relaxed with the developments of powerful computers and various improvements to SA itself. Up to now, the best studied ENDP assumes that the transportation demand is given and fixed (Friesz, 1985; Yang and Bell, 1996). However, long-term investment made to the road network will certainly influence the demand for travel. It may be inappropriate to use the ENDP with an assumed fixed travel demand, since the future demand itself depends, partially at least, on the network capacity to be determined. This paper investigates three algorithms, namely H-J, EDO and SA, through comparing the solutions and computational efficiencies on numerical examples. Section 2 presents the bilevel formulation of the ENDP with elastic demand. The user equilibrium assignment problem with elastic demand of the lower-level is transferred into that one with fixed demand by introducing dummy links into the basic network, so that all existing algorithms developed for solving the ENDP with fixed demand, especially the EDO heuristic, are equally effective for solving the ENDP with elastic demand. Section 3 briefly states the three algorithms, basically without the modifications adopted and implemented in the programs produced for this study. In Section 4, two numerical examples, both with fixed and elastic demands, are presented to judge these three algorithms, especially their performance with respect to their parameters. Section 5 concludes the paper with a warning and suggestions in solving the ENDP.
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2. THE ENDP WITH ELASTIC DEMAND
Let A = the set of links in the network R = the set of paths in the network W = the set of O-D pairs in the network a = the index denoting a link, r = the index denoting a path,, RW - the set of paths connecting O-D pair we W fr = the flow on path r va = the flow on link a qw = the travel demand between O-D pair w ya = the capacity increase (enhancement) on link a = §a(ya) the cost of constructing or improving link a ca(va, ya) = the unit cost of travel on link a cr = the unit cost of travel on path r cw = the minimum unit cost of travel between O-D pair w DW(CW) = the demand function between O-D pair w D~l(qv) - the inverse of the demand function &ar = 1 if link a is on path r, 0 otherwise 0 = the conversion factor converting construction cost to travel cost The bold letters used in the following sections represent vectors. In the case of elastic demand, the total network user benefit (UB) from travel can be measured as UB = J] [~ D~\x)dx, weW
where the demand function qw = Z)w(cw) is assumed to be dependent on cw between that OD pair alone. The total social cost (SC) incurred by all network users and link capacity increases is given by ae/f
ue/f
Therefore, the economic benefit (EB) is evaluated by EB = UB-SC = The optimal network design requires that the economic benefit (EB) is maximized, or a system optimum in this sense is achieved. Since maximizing {EB} is equivalent to minimize {-EB} and the user equilibrium flow pattern (q,v) is implemented under link capacities (Ka+ya) where Ka is the existing capacity of link a, a bi-level formulation of the ENDP with elastic demand can then be presented below min Z(y,q(y),v(y)) = Y ca(vfl(y),.y0)v0(y) + eV ga(ya) - Y P* " D~w\x)dx y>0
i—t
"
"
t—t
"
"
f—i
(1)
J)
where (qw(y)) and (va(y)) are obtained by solving min F(q, v) = ^ > I c"a(x,*"' ya)dx - V I Z ) :w 1V(x)dx q,v>0 Z-i k ' VM
k
(2)
^ >
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subject to
(3) (4)
As is shown above, the elasticity of travel demand is incorporated explicitly into ENDP so that the demand is determined endogenously. The lower-level (2)-(4) is a standard user equilibrium assignment with elastic demand (Gartner, 1980; Sheffi, 1985) for a given network capacity (K+y). There does not exist reasonable economic interpretation for the objective function (2), see Sheffi (1985). But, the first-order optimality conditions of the problem (2)-(4) generate the following conditions required for an elastic demand user equilibrium: c, = Zc.(v»>*.)8«r = cw = D-\qJ,
if fr >0,
(5)
aeA
c, = Zc>«'^)5^c-
if/ r =0,
(6)
if fr =0 for all r&Rw (i.e. qw =0).
(7)
aeA
cw > £T'(0),
Most existing ENDP models attempt to minimize the total social cost (SC) only in the upper-level, which is obviously inappropriate in the case of elastic demand because this minimization might be achieved through minimizing the demand (Yang, 1995). Kocur and Hendrickson (1982) had studied the local bus service design problem with variable demand by maximizing the economic benefit (EB) too, but where the lower-level problem did not exist. In this paper, the following BPR type link cost function (the same as that used by Abdulaal and LeBlanc (1979), Suwansirikul et al. (1987), Friesz et al. (1992, 1993), etc.) and exponential O-D demand function are adopted: C
a(Va,ya) =
A
a+ 5>« / (K-a + X,))' >
°
GA
(8)
AXO = qw exp(-7twcw), w eW (9) where Aa and Ba are constants associated with link a; qw represents the potential level of travel demand between O-D pair w and nw is a positive parameter of the demand function. Set qw = qw in eqn (3) for a fixed demand O-D pair w. Now, we transfer the user equilibrium problem with elastic demand into an equivalent one with fixed demand (Gartner, 1980; Sheffi, 1985). There are two motivations for this transformation: first, the well known Frank- Wolfe algorithm for fixed demand equilibrium assignment problems can then be used in our study directly; second, and most importantly, the equilibrium decomposed optimization heuristic (see Section 3.2) is effective for fixed demand network design problems only. The integration of the inverse demand function in eqns (1) and (2) can be written as ~\x)dx ,
(10)
where the first term of the right-hand side is a constant and denoted by C. Let x = qw-u and substitute it into the second term of the right-hand side in eqn (10), we have
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D~w\qw-u)du,
(11)
where ew = qw- qw and is called surplus demand. Define a function for the surplus demand Ew(ew) = E(?w~(lw) = £*'(#*) ' which is clearly an increasing function with respect to ew because of the exponential form of the demand function given in eqn (9). Since D~\qw - u) = Ew(qw - (qw - it)) - Ew(ii) , eqn (11) becomes D-\x)dx = C -
" Ew(u)du .
(12)
Substitute eqn (12) into eqns (1) and (2), delete the constant C and replace qw in eqn (3) by qw-ew, and obtain an equivalent formulation of the ENDP with elastic demand, as below min Z(y,e(y),v(y)) = y
w
ca(va,ya)va + aeA
Ew(u)du +
weW
ga(ya}
(13)
aeA
where (ew(y)) and (vfl(y)) are obtained by solving mm F(e,v) = £ f ca(x,ya)dx+ £ [" Ew(u)du '
aeA
subject to ew = qw, = vfl,
(14)
wefT
weW
(15) (16)
at A.
It can be seen that, if the second term of eqn (13) is regarded as the sum of all costs on dummy links, the problem described by eqns (13)-(16) is the usual ENDP with fixed demand (qw) . Here for each elastic demand O-D pair we W, one dummy link w with flow ew and cost function Ew(ew) is introduced. This treatment method is often used in solving standard elastic demand user equilibrium problems* (Sheffi, 1995). Note that each dummy link just accepts the flow loaded by its corresponding O-D pair only. At the solution point given by eqns (5)-(7), cw = minr{cr} = Ew(ew) = D~l(qw) should hold for all we Wiff^O or 0<ew
2^.
r!
|
1y llT/l Zj
(17)
,..-«/
where \f¥\ and \W\ are the cardinalities of the set W (aW) of all elastic O-D pairs and the set ^respectively, / is the iteration indication and r\ is a small positive number. It is obvious that much more CPU time will be required to achieve an equilibrium between the shortest path costs and the values of inverse demand functions.
There are two types of treatment methods in solving elastic demand user equilibrium problems. One optimizes the link flows and O-D flows directly on basic network and the other optimizes the link flows only in an augmented network including dummy links. These two methods require the same computer storage capacity since the former's O-D flow variables correspond with the latter's dummy link flow variables. This paper selects the latter for reasons explained as before.
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3. THREE ALGORITHMS FOR SOLVING THE ENDP The Hooke and Jeeves method (H-J) was first applied to the ENDP with fixed demand by Abdulaal and LeBlanc (1979). An alternative equilibrium decomposed optimization heuristic (EDO) was proposed by Suwansirikul et al. (1987). Simulated annealing (SA) was applied by Friesz (1992, 1993). The steps of these algorithms are given below. 3.1. Hooke and Jeeves algorithm Define A (o4) as the set of links considered for improvement in the basic network. The cardinality of the set A is denoted by A . Step 1. (Initialization Step) 1.1. Choose an initial solution y°. Put y=y° and solve the elastic demand user equilibrium problem (EUE) with link capacity improvement y, then compute Z(y). 1.2. Choose an initial step size 8 and the acceleration factor a. Set (3=1. 1.3. Set the indices7=l and k-0. Step 2. (Exploratory Search) 2.1.If(/-I|)>0,gotoStep3. 2.2. y =y+(38ey (ey is a vector with 1 in the^'th position and 0 elsewhere), solve the EUE with y and compute Z(y). 2.3. If Z(y)
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Step 1. Choose the initial lower and upper bounds, y^ and yu. Choose a stopping criteria s and let r=0.618034. Step 2. Set yl=y^+r2(yu-y^), y2=y^+r(yu-yl), solve the EUE with y1 and y2, respectively, then compute Za(y!) and Za(y2) for each link ere A . Step 3. For each link ae A, if Z^y'^Z^y2), set y'a = y'a and y"a = y2a ; otherwise, set y'a = yla and y" = y". For each link ae A, if (yua-y'a)
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candidate solution is generated by randomly perturbing the current solution and accepted with a probability given by the Metropolis criteria (Steps 2.3 and 2.4). Since the probabilities of accepting worse solutions are always positive, SA can escape a local optimum. Consequently, in an asymptotic sense, SA potentially provides a global optimum. In order to efficiently implement the random searches, Vanderbilt and Louie (1984) suggested, and Friesz et al. (1992) adopted, an equation y=y+QAy to replace the equation (19), where Q is a matrix to control the step size distribution. By using a so-called selfregulating mechanism, the Q matrix was modified at each "temperature" stage so as to insure an efficient choice of step size. This technique was tested in this study but the advantage in terms of computational time saving was not observed; the reasons for this are currently being investigated.
4. NUMERICAL EXAMPLES In this section, we present the numerical results of the three algorithms on two examples. These two examples have been used by many authors to test various algorithms proposed for solving the ENDP with fixed demand (see, for example, Marker and Friesz, 1984; Suwansirikul et al., 1987; Friesz et al., 1992, 1993; Marcotte and Marquis, 1992). The first is a 16 link, 6 node and 2 O-D pair network shown in Figure l(a). The second is a 76 link, 24 node and 552 O-D pair network shown in Figure l(b), which is the aggregated network of the city of Sioux Falls.
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(a)
(b)
Fig. 1. Test networks: (a) the first example; (b) the second example. The three algorithms were coded in double precision FORTRAN 77 and run on a PC486. Moore's (1957) shortest path algorithm and Fibonacci search using the 16 numbers sequence were employed in Frank-Wolfe algorithm to solve the fixed or elastic demand user equilibrium subproblems. 4.1. The first example All links are considered for improvement with the linear construction cost function ga(ya)=daya> a&A. Set the link construction cost conversion factor 9=1.0. Table 1 gives the data relating to these links, where (and hereinafter) a link from node i to node j is denoted by (ij). The potential levels of travel demands for O-D pairs (1,2) and (2,1) are 10 and 20 and the parameters nw of the demand functions are 0.03 and 0.01 respectively. In the fixed demand case, the demand is set to the potential level.
Link a (1,3)
1.0
Table 1. Data for the first example Link a 10.0 3.0 2.0 (4,3) 1.0 10.0
1.0
4.0
Continuous equilibrium network design problem with elastic demand
Link a (1,4) (2,5) (2,6) (3,1) (3,4) (3,5) (4,1)
A a 2.0 5.0 6.0 3.0 4.0 5.0 2.0
B
K
5.0 5.0 1.0 3.0 20.0 50.0 20.0
10.0 1.0 4.5 9.0 4.0 3.0 2.0
a
a
da Link a 3.0 6.0 1.0 5.0 4.0 9.0 1.0
(4,6) (5,2) (5,3) (5,6) (6,2) (6,4) (6,5)
Aa 1.0 9.0 2.0 3.0 2.0 4.0 4.0
Ba 1.0 2.0 8.0 3.0 33.0 10.0 25.0
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Ka 10.0 2.0 45.0 3.0 20.0 6.0 44.0
da 3.0 6.0 2.0 5.0 3.0 8.0 5.0
The solutions by the three algorithms with respect to the different parameters are shown in Tables 2, 3, 4 and 5. The user equilibrium (UE) subproblems encountered during the solution process were solved using 10 Frank-Wolfe iterations (it is a very accurate assignment for this example since the tests showed that 2000 Frank-Wolfe iterations generated almost the same flow pattern!). Table 2 shows that the solutions by H-J are very sensitive to the initial step size 8 and acceleration factor a. When 8=0.3 and a=5.0, the solution with lowest Z-value is obtained. Normally the solution computed by Suwansirikul et al. (1987) should be better than this, since accurate equilibrium flows were not generated during the solution process. Table 3 shows the solutions by H-J for the case of elastic demand, where the number of FW iterations for solving each elastic demand user equilibrium (EUE) subproblem is not pregiven but controlled by the stopping criteria stated by eqn (17). This principle is also adopted in the elastic demand cases in Tables 4 and 5. It can be seen in Table 3 that, the solutions, the y-values, are very different when different r|-values and a-values are used, although the Z-values are relatively stable. The EDO heuristic is the most computationally efficient, as shown in Table 4. It is very interesting to note that the value of>?(6,5) depends heavily on the value of d(6,5) but the link flow v(6,5) does not. This is because the Golden-section search discards the good interval of X6,5) when the small value of d(6,5) leads to an ambiguity of this link's evaluation function (i.e. eqn (18)). The value of d(6,5) does not affect the final link flow pattern and the solutions on others-variables, so it should be set to be a large value. Table 4 also shows that the upper bounds of y affect the solution too. Comparatively speaking, the SA approach gives more stable solutions than either H-J or EDO, as shown in Table 5. It is interesting to note that 10 F-W iterations are not enough to solve each user equilibrium subproblem encountered during the solution process sufficiently accurately, a fact that does not emerge in using the H-J and EDO algorithms. The reason is that positive link flows appear on links (1,3) and (3,4) which make it very difficult to achieve an equilibrium between the costs of routes (1,3,4,6,2) and (1,4,6,2). After running 2000 F-W iterations in the network with the y-value obtained by SA using 10 F-W iterations for each UE inside, a very accurate equilibrium flow pattern is generated and the upper-level objective function is evaluated again, see Z^ in Table 5. Clearly, Z^>Z, but this does not contradict the fact that the SA approach has found the lowest Z-value. Table 2. Solutions by H-J for the first example with Fixed demand 0.4 0.2 0.3 0.3 0.3 3.0 5.0 5.0 5.0 6.0 2.175 1.500 0.075 0.009 0.028
0.3
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H.-J. Huang and M. G. H. Bell 2.850 4.613 2.025 1.884 1.363 XU 4) 5.40 4.112 0.591 0 1.250 1.847 X2,5) 8.10 17.288 18.225 12.588 14.334 13.938 X2,6) 8.40 7.800 9.872 13.275 10.263 11.972 8.18 X3,l) 0 0 0.725 0 0 X3,4) 0 0 0 0 0 0 0 X3,5) 5.859 6.900 10.337 10.341 8.188 8.10 X4,l) 0 0 0 0 0 0 X4,3) 1.088 0.019 0 1.725 0.613 X4,6) 0.90 0 0 0 0 0 0 X5,2) 0 0 0 0 0 0 X5,3) 0 0 0 0 0 0 X5,6) 0.816 2.025 2.225 1.038 0.019 3.90 X6,2) 0 0 0 0.063 0 0 X6,4) 0 0 0 0 0 0 X6,5) 538.764 549.647 Z 544.812 542.844 526.495 557.22 364 466 379 No. UE 134 435 440 1.517 1.233 1.200 CPU(m) 1.432 1.433 Note 1. Set y^=0 and 5/=0.001 for each. All y-values in this and following tables denote the link capacity increases. Note 2. The last column is taken from Suwansirikul et al. (1987) who used 0.5 as the stopping tolerance for Z and carried out 319 F-W iterations in all.
4.2. The second example
The complete data relating to this example can be found in Suwansirikul et al. (1987). The link construction cost function adopted is ga(ya) = day2a. Set the link construction cost conversion factor 0=0.001. There are 10 links (5 sets of two-way links marked by bold lines in Figure l(b)) being considered for improvement. The total of trips in the O-D matrix given by Suwansirikul et al. (1987) is 3J96.76. In this study, all O-D pairs connecting the central node 10 are regarded as elastic O-D pairs and the data offered in that matrix for these 46 OD pairs as the potential levels of demands with a total of 99.33. The values of the parameters 7rw of the demand furctions are all set to be 2.05. All results obtained for this example are given in Tables 6, 7 and 8. In the elastic demand case of this example, achieving an accurate equilibrium flow pattern is so difficult that the number of F-W iterations for solving each EUE inside has to be fixed at 100 which can basically make the value of the left-hand side of eqn (17) less than 0.6. In principle, the parameter TT^ of each O-D pair's demand function should be specified alone, /ould be useful to allow El JE subproblems inside to be more efficiently solve Table 3 Solutions b; f H-J for the first example with elastic demand T! a XU) XM) X2,5) X2,6) X3,l) X3,4)
0.01 3.0 0.009 1.172 0.009 13.978 5.503 0
0.01 5.0 0.122 1.959 2.259 9.947 5.409 0
0.02 3.0 0.056 2.972 0.075 15.544 5.288 0
0.02 5.0 0 0.525 0.375 11.925 4.875 0
0.05 3.0 1.500 3.975 1.538 14.700 4.425 0
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0 0 0 0 0 X3,5) 8.972 4.378 8.475 4.125 8.419 X4,D 0 0 0 0 0 X4,3) 0 0 0 0 1.875 X4,6) 0 0 0 0 0 X5,2) 0 0 0 0 0 X5.3) 0 0 0 0 0 X5,6) 0 0 0 0 0 X6,2) 0 0 0.375 0.591 0 X6,4) 0 0 0 0 0 X6,5) -1884.26 -1883.72 -1897.51 Z -1894.53 -1882.93 8.043 7.968 7.932 8.078 8.017 9(1,2) 16.756 16.829 16.667 16.851 16.748 9(2,1) 484 304 622 466 456 No. EUE 2.183 5.417 CPU(m) 2.483 1.517 5.950 Note 1. Set yu=0, 8=0.3 and 8/=0.001 for each. Note 2. q(\,2) and q(2,l) are the demands implemented between O-D pairs (1,2) and (2,1), respectively.
Table 6 presents the solutions by H-J algorithm to the different initial values of y. The results obtained by Suwansirikul et al. (1987) are really better than ours, which further shows that the local optimal solutions by H-J would strongly depend on the selection of parameter values. Tables 7 and 8 demonstrate that the number of F-W iterations used to solve each UE inside also affects the final solutions. This implies that the tentative idea of trying to substantially alleviate the computational overhead by having a loose stopping criterion for the F-W iterative procedure may fail. Table 9 shows the changes of the values of upper-level objective function and its component parts and the total demand implemented with respect to the parameters 7rw of the demand functions. As we can seen, when nw (refer to the 46 O-D pairs mentioned above) becomes smaller, the total demand implemented (QT) gets close to the potential level, the total network travel cost (Z]) increases too, the total weighted link construction cost (Z2) fluctuates (it should be attributed to the multi-local optima of y-solutions), and the user benefit (Z3) tends to a huge number. These results are just what we expect. Table 4. Solutions by EDO for the first example with elastic and fixed demands 20.0 20.0 22.0 20.0 20.0 20.0 J(6,5) 5.0 30.0 5.0 9.0 30.0 30.0 0.005 0.005 0.005 0.005 0.005 0.005 XU) 0.827 0.560 4.616 4.616 4.616 4.616 Xl,4) 0.011 0.005 0.005 0.005 0.005 0.005 X2,5) 19.990 19.995 19.995 19.995 19.995 21.995 X2.6) 13.586 7.333 12.341 7.605 12.356 12.356 X3,l) 0.005 0.005 0.005 0.005 0.005 0.005 X3,4) 0.005 0.005 0.005 0.005 0.005 0.005 X3,5) 7.664 7.644 7.644 7.673 7.659 8.408 X4,l) 0.005 0.005 0.005 0.005 0.005 0.005 X4,3) 0.594 0.014 0.005 0.593 0.593 0.593 X4,6) 0.005 0.005 0.005 0.005 0.005 0.005 X5,2)
y"
20.0 5.0 0 4.88 0.26 12.52 8.59 0 0 7.48 0.26 0.85 0
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X5,3) 0.105 0.0 35 0.014 0.034 0.005 0.015 0 X5,6) 0.005 0.0 35 0.005 0.005 0.005 0.005 0 X6,2) 0.014 0.0 35 1.314 1.314 1.314 1.315 1.54 X6,4) 0.025 0.0 54 0.005 0.005 0.019 0.005 0 X6,5) 12.318 0.0 35 12.356 0.072 0.005 0.005 0 Z -1835.04 -1896. 11 584.214 526.453 525.975 529.611 540.74 No. EUE 33 3 No. UE 33 33 33 33 12 ^(1,2) 7.934 7.9 1 4(2,1) 16.995 16.9 50 CPU(m) 1.433 1.4 57 0.10 0.10 0.10 0.10 Note 1. The second and third columns are for the elastic demand cases, where r|=0.01. Note 2. Set y'=0 and e=0.01 for e ch. Note 3. The last column is taken from Suwansirikul et al. (1987) who used 0.5 as the st( pping tolerance for Z and carried out 68 F-W iterations in all.
5. SUMMARY AND CONCLU SIGNS In this paper, a bi-level formulati on for the equilibrium network design problem (ENDP) with elastic demand is proposed. ' 'he upper-level is aimed at maximizing the net economic benefit and the lower-level solve ; a standard user equilibrium assignment problem with elastic demand. Three representati 'Q algorithms developed for solving the ENDP with fixed demand are investigated and empl oyed to solve the ENDP with elastic demand. Numerical results on both small and large tes networks are presented and some interesting phenomena are revealed. In summary, our find ngs are: 1 . The model in the form of bi-1 5vel formulation proposed in this paper can capture the basic and main characters of the E 4DP with elastic demand (see Table 9). 2. The existing algorithms develo] ed for solving the ENDP with fixed demand can be used in the case of elastic demand. 3. All three algorithms tested are sensitive to the values of the parameters, with the H-J algorithm being the most sensiti re and the SA method the least. If suitable values of parameters are selected, each al§ orithm may give satisfactory solutions. Thus, it seems necessary to solve the model man; r times with different values of the parameters, no matter which algorithm is used. Table 5. Solutions by SA f )r the first example with elastic and T| 0.01 0.02 0.02 kB 0.001 ( .001 0.003 0.003 Tf 250 250 100 100 83.87 83.87 T 204.80 2( 4.80 0 0.123 0 XU) 0.012 .288 0.234 1.324 XI, 4) 0.521 X2,5) 0 0 0.048 0.057 X2,6) 18.958 1! .879 14.502 19.829 6.714 10.948 X3,l) 5.964 ( .271
fixed demands 0.001 250 256.00 0 1.230 0.022 19.472 10.432
100 0 0 0 17.2786 10.1740
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0.040 0 0 0 0 0.011 X3,4) 0 0 0 0.008 0 0 X3,5) 6.832 7.332 6.626 5.7769 6.878 6.903 X4,l) 0.027 0 0 0 0.020 0.031 X4,3) 0.784 0.075 0.207 0 0 0 X4,6) 0 0 0 0 0 0 X5,2) 0 0 0 0 0 0 X5,3) 0.019 0 0 0 0 0.003 X5,6) 1.381 2.238 0 0 0 0 X6,2) 0 0 0 0.036 0 0 X6,4) 0 0 0 0 0 0 X6,5) 520.496 520.317 528.497 -1898.28 -1899.26 -1897.86 Z 8,769 8,422 7,817 No. EUE 10,861 9,499 24,300 No. UE 7.780 7.977 7.907 9(1,2) 16.843 16.860 16.885 9(2,1) 0.5861 0.5114 0.6514 0.7453 0.8486 CPU(h) 531.934 531.931 Z# Note 1. The second, third and fourth columns are for the elastic demand cases. Note 2. Set y^=0, y"=20, 7^=500 and M=300 for each. T is the "temperature" at which the algorithm stopped. Note 3. Z# is the Z-value computed after 2000 W-F iterations, using the y-value given in the same column. Note 4. The last column is taken from Friesz et al. (1992).
4. The SA approach always gives the best solutions, but its computation times are always the most too. The EDO heuristic is the most computationally efficient, but the solutions by it are not quite up to that by H-J algorithm in most cases (see Table 10). 5. The accuracy of solving the lower-level problem also affects the final solutions, thus the idea of trying to substantially alleviate the computational overhead by having a loose stopping criterion for the solution process of the lower-level problem may fail, especially in complex networks as these may have many more local optima (see the figures in brackets shown in Tables 6,7, and 8). Finally, the best solutions obtained by the three algorithms investigated in this paper, for the two examples with fixed and elastic demands, are summarized in Table 10. It is worth noting that the solutions of the first example with fixed demand are the best thus far, to the knowledge of the authors.
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Table 6. Solutions by H-J for th 2 second example with elastic and fixed demands a 5.0 3.0 3.0 3.0 5.0 yO 2.0 1.0 2.0 3.0 3.0 2.0 1.0 X6,8) 3.088 5.545 5.588 4.507 4.247 4.8 3.8 X7,8) 1.758 2.194 2.113 4.509 3.863 1.2 3.6 X8,6) 2.989 4.725 3.888 4.520 4.149 4.8 3.8 X8,7) 0.961 1.825 3.588 4.052 3.976 0.8 2.4 X9,10) 1.805 3.850 0 4.299 3.979 2.0 2.8 XI 0,9) 1.939 1.600 2.113 2.949 3.054 2.6 1.4 X10.16) 2.145 1.150 3.800 3.000 3.018 4.8 3.2 XI 3, 24) 2.202 2.200 2.913 3.601 3.154 4.4 4.0 4.050 3.006 2.784 4.8 4.0 XI 6, 10) 2.003 2.603 X24,13) 2.302 1.002 2.000 3.200 3.012 4.4 4.0 84.170 83.316 83.572 80.78 81.21 Z 19.882 85.433 (82.977)
(83.274)
506 349 322 58 108 No. UE 437 CPU(m) 95.650 20.95 24.217 16.717 15.433 Note 1. The second column is fo the elastic case. The total demand implemented is 357.03, number of EUE solved is 34'. , each EUE is solved using 100 F-W iterations. Note 2. Set 5=0.2 and 8/=0.001 for each. Ii fixed demand case, each UE is solved using 20 F-W iterations. Note 3. The last two colu nns are taken from Suwansirikul et al. (1987) who used 0.5 as the stopping tolerance foi Z and carried out 147 and 247 F-W iterations in all, respectively. The figures in bracke s are the Z-values computed by running 20 F-W iterations using the y-values given in he columns.
Table 7. Solutions by EDO i >r the second example with elastic and fixed demands KT 5 10 20 3.336 4.276 4.59 X6,8) 2.399 5 .377 X7,8) 1.189 1 .027 2.314 2.288 1.52 4.133 4.080 5.45 X8,6) 2.225 5 207 X8,7) 1.044 3 .171 3.146 1.618 2.33 X9,10) 1.577 2 .661 3.337 1.654 1.27 XI 0,9) 2.300 2 .989 2.66 1.130 2.33 X10,16) 2.121 1 667 1.769 3.219 0.41 XI 3,24) 2.605 i 287 1.628 3.326 4.59 X16,10) 1.438 1 183 3.241 1.981 2.71 X24,13) 2.764 3 688 3.354 3.190 2.71 Z 20.492 8^ .585 84.866 83.703 83.08 (84.775) No. UE 45 45 45 12 CPU(m) 12.050 ( .550 1.067 2.083 Note 1. The second colum i is for the elastic case. The total demand implemented is 357.128, number of EUE solved is 45, each EUE is solved using 100 F-W iterations. Note 2. Set y/=0, y"=25.0 and e=0.01 for each. KT is the number of F-W iterations to soh e each UE. Note 3. The last column is taken from Suwansirikul et al. (1987) \ 'ho used 0.5 as the stopping tolerance for Z and carried out 89 F-W iterations n all. The number in bracket is the Z-value computed by running 20 F-W itei ations using the y-values given in the column.
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Table 8. Solutions by SA for the second example with elastic and fixed demands
^
ooi
T KT X6,8) X7,8) X8,6) X8,7) X9,10) X10,9) X10,16) X13,24) X16,10) X24,13) Z
2.577 4.184 1.173 4.643 1.302 1.703 2.016 1.952 3.051 1.926 2.686 19.330
6 ~ o l 6 ~ o l 6 ~ b l 12.288 10 3.972 4.336 5.664 1.306 4.189 3.605 4.272 4.898 3.592 4.237 84.520
3.221 10 4.731 3.464 4.445 1.173 4.812 2.506 3.929 5.000 4.263 4.670 84.723
4.027 20 5.322 2.596 5.664 1.309 2.498 2.732 4.123 4.508 3.736 3.903 81.983
0.013 5.38 2.26 5.50 2.01 2.64 2.47 4.54 4.45 4.21 4.67 80.87 (82.077)
3,900 12,541 15,627 9,994 No.UE 10.079 6.357 4.058 66.418 CPU(h) Note 1. The second column is for the elastic case. The total demand implemented is 357.346, number of HUE solved is 14,882, each EUE is solved using 100 F-W iterations. Note 2. Set y°=6.25, y'=0, y«=25, 7^=3, 7/=3 and A/=200 for each. KT is the number of F-W iterations to solve each UE. T is the "temperature" at which the algorithm stopped. Note 3. The last column is taken from Friesz et al. (1992). The number in bracket is the Z-value computed by running 20 F-W iterations using the y-values given in the column. Table 9. Results vs different values of the parameters nw of the second example's demand functions (solved by H-J)
QT Z
z\ Z2 Z3
0.01 396.446 -9850.320 78.461 4.163 9932.944
nw (we W , W - the set of 46 O-D pairs specified) 1.0 2.05 0.1 0.5 393.549 373.482 357.030 383.928 19.882 -909.582 -118.520 -25.310 62.048 79.942 75.391 66.495 4.411 3.212 2.915 1.575 43.741 992.736 196.826 96.216
3.0 349.658 32.537 58.534 2.557 28.554
Note: QT= ]>>„, Z, = ^cjqw, Z2=Q^tga(ya), Z3 = £ f"D?(x)dx . Table 10. Summary of the best numerical results for two examples First example Fixed demand Elastic demand H-J EDO SA H-J EDO SA 526.495 525.973 520.317 Z -1897.51 -1896.11 -1899.26 CPU 5.417m 1.467m 0.65 14h 1.517m 0.100m 0.5114h Second example Fixed demand Elastic demand (nw =2.05) H-J EDO SA H-J EDO SA Z 19.882 20.492 19.330 83.316 83.703 81.983 CPU 95.650m 12.050m 66.418h 16.717m 2.083m 10.079h
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Acknowledgements: This stud; was carried out while the first author was a visiting scholar at the Transport Operat: ans Research Group, Department of Civil Engineering, University of Newcastle. He wou d like to thank the State Education Commission of China for supporting this visit of one y ar. The helpful comments of the anonymous referees are gratefully acknowledged.
REFERENCES Abdulaal, M., & LeBlanc, L. J. (1979). Continuous equilibrium network design models. Transpn. Res., 13B, 19-32. Allsop, R. E. (1974). Some I ossibilities for using traffic control to influence trip distribution and route choic( . In: Proceedings of the 6th International Symposium on Transportation and Traffic 7 reory, pp. 345-375. Elsevier, New York. Boyce, D. E., & Janson, B. N. (1980). A discrete transportation network design problem with combined trip distribute m and assignment. Transpn. Res., 14B, 147-154. Chen, M., & Alfa, A. S. (1991). i L network design algorithm using a stochastic incremental traffic assignment approach. Transpn. Sci., 25, 215-224. Davis, G. A. (1994). Exact loca solution of the continuous network design problem via stochastic user equilibrium a: signment. Transpn. Res., 28B, 61-75. Friesz, T. L. (1985). Transport^tion network equilibrium, design and aggregation: key developments and research o jportunities. Transpn. Res., 19A, 413-427. Friesz, T. L., & Harker, P. T. (1985). Properties of the iterative optimization-equilibrium algorithm. Civil Engin•eering Systems, 2, 142-154. Friesz, T. L., Cho, H-J., Mehfe , N. J., Tobin, R. L., & Anandalingam, G. (1992). A simulated annealing appro .ch to the network design problem with variational inequality constraints. Transp •«. Sci., 26, 18-25. Friesz, T. L., Anandalingam, G., Vlehta, N. J., Nam, K., Shah, S. J., & Tobin, R. L. (1993). The multiobjective equilib ium network design problem revisited: a simulated annealing approach. Europec n J. of Operational Research, 65, 44-57. Gartner, N. H. (198). Optimal tra fie assignment with elastic demands: a review; Part 1 and Part 2. Transpn. Sci., 14, 174208. Harker, P. T., & Friesz, T. L. (1< 84). Bounding the solution of the continuous equilibrium network design problem. I Proceedings of the 9th International Symposium on Transportation and Traffic Theory, pp. 233-252. VNU Science Press, The Netherlands, Heydecker, B. G. (1986). The eq lilibrium network design problem: a critical review. Paper presented at NATO Advance 1 Workshop, Capri, October, 1986. Hooke, R. & Jeeves, T. A. (1< 61). Direct search solution of numerical and statistical problems. Journal of the Ass Delation for Computing Machinery, 8, 212-229. Kirkpatrick, S., Gelatt, C. D., & ^ ecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671-680 Kocur, G., & Hendrickson, C (1982). Design of local bus service with demand equilibration. Transpn. Sci., 6, 149-170. Larsson, T., & Patriksson, M. (1992). Simplicial decomposition with disaggregated representation for the traffic issignment problem. Transpn. Sci., 26, 4-17. LeBlanc, L. J. (1975). An algorithm for the discrete network design problem. Transpn. Sci., 9, 183-199.
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LeBlanc, L. J., Morlok, E. K., & Pierskalla, W. P. (1975). An efficient approach to solving the road network equilibrium traffic assignment problem. Transpn. Res., 9, 308-318. LeBlanc, L. J., & Boyce, D. E. (1986). A bi-level programming algorithm for exact solution of the network design problem with user-optimal flows. Transpn. Res., 20B, 259-265. Magnanti, T. L., & Wong, R. T. (1984). Network design and transportation planning: models and algorithms. Transpn. Sci., 18,1-55. Marcotte, P. (1981). An analysis of heuristics for the network design problem. In: Proceedings of the 8th International Symposium on Transportation and Traffic Theory, pp. 452-468. University of Toronto Press. Marcotte, P. (1983). Network optimization with continuous control parameters. Transn. Sci.,11, 181-197. Marcotte, P. (1986). Network design problem with congestion effects: a case of bilevel programming. Mathematical Programming, 34, 142-162. Marcotte, P. & Marquis, G. (1992). Efficient implementation of heuristics for the continuous network design problem. Annals of Operations Research, 34, 163-176. Migdalas, A. (1995). Bilevel programming in traffic planning: models, methods and challenge. J. of Global Optimization, 7, 381-405. Moore, E. F. (1957). The shortest path through a maze. In: Proceedings of the International Symposium on the Theory of Switching, pp. 285-292. Harvard Univ. Press, Cambridge. Powell, M. J. D. (1964). An efficient method for finding the minimum of a function of several variables without calculating derivatives. Computer Journal, 1, 155-162. Steenbrink, P. A. (1974). Optimization of Transportation Network. John Wiley & Sons, London. Suwansirikul, C., Friesz, T. L., & Tobin, R. L. (1987). Equilibrium decomposed optimization: a heuristic for the continuous equilibrium network design problem. Transpn. Sci., 21, 254-263. Sheffi, Y. (1985). Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Cliffs, New Jersey. Tan, H., Gershwin, S., & Athans, M. (1979). Hybird optimization in urban traffic networks. Technical Report No. Dot-TSC-RSPA-79-7, MIT, Cambridge. Vanderbilt, D., & Louie, S. G. (1984). A Monte Carlo simulated annealing approach to optimization over continuous variables. J. of Computational Physics. 56, 259-271. Yang, H. (1995). Model and algorithm for the network design problem with variable demand. Unpublished working paper, Department of Civil and Structural Engineering, The Hong Kong University of Science and Technology. Yang, H. & Bell, M. G. H. (1996). Models and algorithms for road network design: a review and some new developments. Unpublished working paper, Department of Civil and Structural Engineering, The Hong Kong University of Science and Technology.
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A COLUMN GENERATION APPROACH To Bus DRIVER SCHEDULING Sarah Fores, Les Proll and Anthony Wren Scheduling and Constraint Management Group School of Computer Studies University of Leeds Leeds, LS2 9JT, UK
ABSTRACT Mathematical programming approaches to solving the driver scheduling problem have become successful with improvements in computer technology but heuristics are also necessary to reduce many problems to a manageable size. A column generation method is described which allows much larger problems to be solved than is currently possible. The approach allows problems to be solved more quickly than with the current approach and encourages better solutions to be found due to the availability of a larger set of potential duties.
INTRODUCTION The problem of scheduling public transport vehicles and their drivers has been the subject of six international workshops (see, e.g., Desrochers and Rousseau, 1992; Daduna et al., 1995). Research has progressed to a point where there are several computer scheduling systems in widespread commercial use, e.g. HASTUS (Rousseau and Blais, 1985) and IMP ACS (Smith and Wren, 1988), which were originally developed for use in the bus industry. This paper outlines an alternative solution method which has been incorporated into a system which originated from IMP ACS. Improved results on a selection of real bus driver problems are presented.
THE DRIVER SCHEDULING PROBLEM Although some systems attempt to schedule vehicles and drivers simultaneously, or even to schedule drivers first, it is generally the case that a vehicle schedule is built initially to cover a set of predetermined journeys. At various stages during the work of each vehicle there are convenient driver changeover locations with associated times and these are known as relief opportunities. An indivisible period between any two relief opportunities is known as apiece of 195
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work. A shift usually consists two or three spells of work which each cover several consecutive pieces of work on the same vehicle. The formation of shifts is governed by a set of labour agreement rules to ensi ire that there is adequate provision for mealbreaks and acceptable working hours etc. The driver scheduling problem is then to define a set of shifts in such a way that all the vehicle wo c is covered and some measure of the number of drivers and the costs of shifts is minimised. I sually this requires minimisation of the number of shifts as a priority, and with that minimum finding a good compromise between cost and some subjective measure of quality. Mathematical programming solut: in methods applied to driver scheduling have become more dominant and successful with im rovements in computer technology but the problem is still too large to be able to guarant an optimal schedule in most cases. Thus mathematical programming is frequently comb ned with heuristic approaches to provide a viable solution method.
THE TRACS II MODEL TRACS II is the driver scheduling system which was developed at the University of Leeds and retained for research purposes, This system uses the same general approach as the commercially available IMP ACS ystem (Smith and Wren, 1988). TRACS II first generates a set of valid shifts, then reduces th set to a manageable size and finally selects from it a subset of shifts which cover the bus wor . This final stage in the solution process involves solving a set covering model which ensures hat each piece of work is covered by at least one driver and that the overall cost is minimal, As the introduction of each driver incurs a large cost the primary objective is to minimise he number of shifts. The remaining shift cost involves the actual wage cost of the shift plus some subjective cost reflecting any penalty for shifts order properly to combine the two objectives a Sherali containing undesirable features, strategy (Sherali, 1982) was adopt d by Willers et al. (1995), viz : N
N
7=1
7=1
Mil imise
where X is 1 if they'th generatec ihift is used, and 0 otherwise, is the combined wage i nd penalty cost of shifty'i, and Cj is W is the weight that priori ses the reduction of shifts in the solution. The Sherali weight W is calculatec N
= \ + UB
Z 7=1
where UB[] is an upper bound on 1 ic sum of the shift costs
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The upper bound can be taken to be the sum of the 5" largest C ,• values, where S is itself an upper bound to the number of shifts which might be used in a sensible solution and can be set to be the number of shifts in the initial solution. We can now define a new objective : N
Minimise
X ^ jxj 7=1
where Dj represents the combined wage and penalty cost of shifty plus the Sherali weighting. It should be noted that in a practical problem there are generally many million potential shifts, too many to be treated by the model. For this reason restrictions are placed on the form of the shifts to be generated and retained. These restrictions are governed by parameters designed to ensure that the most likely shifts are available to the above model, and that a good choice of shifts is available for each piece of work. Nevertheless, it is quite possible that some apparently inefficient shifts may be required in a good solution, and that these will have been excluded by the parameters. For this reason TRACS II uses a set covering approach which allows pieces of work to be overcovered, i.e. covered by more than one shift. As overcover involves a wage cost for more than one shift the optimisation will deter overcover from being formed. Where overcover remains in the final schedule, the relevant shifts can be edited to form shorter shifts excluded from the generation process. There are instances where the existence of overcover would not be sensible, e.g. on the first and last few pieces of work on any bus, and so these corresponding constraints can be defined as equality constraints to be covered by only one driver. The equality constraints cut down the problem size when forming an initial solution and are necessary in one of the branching strategies of the branch and bound algorithm. The TRACS II constraints can thus be defined as follows :
7=1 where E is the number of equality constraints and AJI = 1 if shifty covers piece of work i. It is then possible for the user to add side constraints which typically are used to limit the number of shifts of a particular type or the total number of shifts. Such constraints are of the form:
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N jXj<
=1
or
j=\
where 8j = 1 if the correspondinj x j is of the type that is to be constrained, = 0 otherwise.
Method of Solution of the Model 1) Create an initial solution by se ecting shifts to cover pieces of work in increasing order of the number of shifts available to cover them; each successive shift is selected so as to minimise the cost per currently uncovered workpiece covered by this shift. 2) Solve the linear programming relaxation of the TRACS II model using a primal steepest edge algorithm. 3) If the minimum number of duties is fractional, add a constraint which increases the number of duties to the next highest intt gral value and re-solve if necessary using a dual approach. 4) Find an integer solution using s serialised Branch and Bound techniques.
COLUMN GENERATION Column generation is an approa< h which can be used to solve mathematical programming problems involving many variable 5, which would otherwise have to be decomposed or reduced before being solved. Scheduling p oblems are an important class of combinatorial optimisation problems which frequently contain many variables. TRACS II uses heuristics to reduce the number of shifts in order to enable conventional mathematical programming solvers to produce a solution. Further investigatior has taken place into incorporating column generation techniques within TRACS II. It is shown that column generation methods overcome many of the difficulties which arise in so ving large problems, and also improve upon the solution and/or speed of the process. Column generation systems norm illy use the Revised Simplex Method to solve the problem over a subset of the columns. Th process then repeatedly adds further columns which will potentially improve the solution 0 the subset and re-solves over the new set. This process continues until no further columns can be found which will improve the objective. Assuming that all the columns arc available in the matrix of constraints, the Revised Simplex Method begins with an initial olution and iteratively improves the objective value by swapping a basic variable for a no i-basic variable with favorable reduced cost, until no further improvement can be made. In the case of column generation only a subset of the columns is available at the outset. The Revi led Simplex Method is used to find the solution which is optimal over the subset. One the 1 generates new columns or searches through columns not previously considered. If there are any new columns which have favorable reduced costs then
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some, or all, of them are added to the subset as non-basic variables so that the current solution is no longer optimal. Using the current basic solution, the Revised Simplex Method continues to swap columns where necessary to reoptimise over the current larger subset. For any LP relaxation which is optimal over its available subset the overall optimal solution is attained when no more columns which would improve the objective can be added to the set. Iterations of the Revised Simplex Method within a particular shift subset are known as minor iterations; augmentations of the shift subset are known as major iterations.
COLUMN GENERATION AND DRIVER SCHEDULING The major disadvantage of using most computerised scheduling systems, including TRACS II, is that the search for optimality is limited to an heuristically reduced shift set, and so for some problems the ILP fails to find a feasible solution. In TRACS II the constraint introduced in step 3 of the solution process must then be relaxed, thereby allowing more shifts in the solution. Large problems can be decomposed (Wren and Smith, 1988) but this is usually deemed undesirable by users. Heuristics often limit the formation of shifts to include only efficient shifts, but because it may be the case that some inefficient shifts are required to link with them to produce the optimal solution, ideally we would wish to consider allowing all possible valid shifts which could be formed to enter the set covering model. In the context of driver scheduling a column generation method implicitly considers many more valid shifts than standard linear programming approaches whilst retaining a much smaller working subset of shifts which are available to the mathematical program solver. The HASTUS scheduling system (Rousseau and Blais, 1985) contains a module, Crew-Opt (Desrochers et al., 1992), which uses column generation techniques to form bus driver schedules and has produced encouraging results (Rousseau, 1995). A subset of valid shifts is identified and further shifts are added to the subset based upon utilising a shortest path algorithm to identify shifts which will improve the current solution. Once no new shift can be found which will reduce the schedule cost the current solution is optimal over the relaxed model. A branch and bound technique which also incorporates column generation is then applied to find a good driver schedule. The model works well on smaller problems but takes substantial time to perform the shift additions and the optimality can be compromised in the branch and bound phase as the search often stops at a good solution.
A COLUMN GENERATION MODEL WITHIN TRACS II A column generation approach to solving the driver scheduling problem requires strategies for resolving the following three issues : • How to choose the initial shift subset. • How to generate or search for further shifts. • How to produce an integer solution from the LP solution.
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These issues will be discussed h an attempt to find the most appropriate technique to be incorporated into the TRACS II sy tern, and an outline of the model will follow.
Shift Generation Incorporating column generation tc hniques into the solution strategy of the set covering model potentially allows all valid shifl s to be considered. However, to guarantee the overall continuous optimum one of the fol owing two methods would have to be used: I. AH valid shifts would have to b ? formed initially and their reduced costs calculated during the column generation process lo ensure optimality. II. A subproblem must be availab to generate any remaining shifts with favorable reduced costs once optimality has been/bund over the current shift subset. Crew-Opt uses the second metho 1, with a subproblem defining a constrained shortest path technique to generate further shifts as paths through a network. In the current TRACS II system the shift costing not only depends upon the duration of the shift, but may also include a subjective weighting to deter some shifts from being included in a schedule. This is because many organisations seek to minimise unpopular or administratively difficult shifts. The penalty costs are currently defined in a subroutine specific to each organisation and typical requirements include: • penalties added to three-part shifts • penalties added to shifts whose meal breaks are not close to the middle of the shift • penalties added to early shifts which start after the earliest time for split shifts. Although these penalties can be added cumulatively as a shift is being constructed, they are not added as resource constraints and cannot be built into a shortest path network. If the shortest path through the network corresponds to a shift with undesirable features its reduced cost may not be the most favorable once the penalty cost has been added to the shift cost. Also, although Crew-Opt hopes to be able to handle larger problems, published work (Rousseau, 1995) states that it is restricted. Indeed, larger problems will require large networks and for each set of labour agreement rules there will be a certain amount of time required to convert them into network constraints. In order to speed up the process of column generation it is usual for methods to add more than one column at each iteration, and for larger problems it will become complex and time consuming to traverse a network to generate such shifts. Decomposing a larger problem will compromise optimality. For the reasons given above, the technique which has been explored is that of removing the heuristics which reduce the size of the generated shift set in TRACS II, hence allowing more shifts to be considered by the set covering model. A network formulation is not used, and so shifts from this larger set are selected to enter a working subset by means of a less sophisticated
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enumeration method. An improvement in continuous LP optimum may be achieved by using this larger set, and so for each bus company both a larger shift set and an heuristically reduced shift set are used in the experiments reported here. As the current system produces an LP solution, the column generation method on the same data set will not improve the cost, as both will be optimal. However, the column generation method may prove to be faster, and the larger data set may produce a lower LP cost than the smaller data set, which would normally have been used in TRACS II.
Integer Solution If the overall continuous optimum solution has been found, then in order to guarantee an optimal integer solution it would be inadequate to terminate the branch and bound phase as soon as a good solution had been found. Crew-Opt uses a column generation technique within a branch and bound structure in order to ensure that the integer solution is not limited to the shifts formed in finding the optimal LP solution. Continuing through the search tree in this way would guarantee an overall optimal integer schedule, although in practice Crew-Opt pauses at the first integer solution and uses the continuous optimum to analyse whether the expected improvement in integer solution cost justifies the probable extra computational cost of further searching. At each node the column generation method used in the branch and bound phase limits the shifts which are formed to those relevant. Results given for Crew-Opt showed that a large number of new columns are added overall in the branch and bound phase even though only one path through the tree will provide the integer schedule. In TRACS II, because the shift costs include subjective weightings, many integer solutions may be equally as good as or possibly preferable to the apparently optimal schedule with respect to shift content, and so spending time searching for the optimal schedule may be wasted. Whilst the solution time of computerised systems is acceptable compared to previous manual methods, bus companies would tend to give execution time reduction a higher priority than improving a good integer solution. Faster solutions would allow a user more opportunity to analyse the effect of relaxing certain conditions or altering parameters. Since a method of producing schedules from a larger shift set cannot guarantee the overall optimal LP solution, the optimal schedule cannot be guaranteed either. For this reason the branch and bound phase used in TRACS II is adequate, and although it considers only some of the shifts retained in the model, this has not detracted from its ability to find a good solution.
Initial Shift Set Crew-Opt uses an initial set consisting of short shifts in order to provide a feasible solution quickly. In the work reported here a large set of potential shifts known as a superset is generated initially and a subset of these is used in such a way as to provide an initial subset of good shifts, and hence a good feasible initial solution. Details of the method of selecting a shift subset are given in the next section.
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A Column Generation Model There are many possible ways of implementing a column generation strategy, but the method of solution which will be adopted is outlined as follows: • • • • •
Step 0 Generate a shift superset. Step 1 Create an initial solution and form an initial shift subset. Step 2 Solve the LP over the current shift subset. Step 3 Add a set of shifts to the current subset which will improve the solution. Step 4 If no favorable shifts can be found then the LP solution is optimal, otherwise go to Step 2. • Step 5 Find an integer solution using branch and bound.
IMPLEMENTATION WITHIN TRACS II Having outlined the column generation model and decided that a larger set is to be generated at the outset, it is now necessary to define which shifts are to be included in the initial subset and the factors which determine how many further shifts are added at any intermediate solution stage. Full details of the implementation strategies and results are given by Fores (1996).
Initial Solution and Initial Shift Subset Given that all reasonable shifts are available at the outset, the method chosen to form an initial solution is the same as that currently used by TRACS II. As pieces of work are being considered in increasing order of the number of available shifts covering them, shifts are added to an initial subset if there are fewer than a specified number of other shifts, currently in the subset, which cover that piece of work. Research has shown that a value of 10 provides a sufficiently large and varied initial subset to reduce the number of major iterations required. A larger set would take more time to form and contain many more inefficient shifts, which would slow down the mathematical program solver.
Addition of Further Shifts Once an initial shift subset has been formed, the Revised Simplex Method is used to optimise the model. This produces simplex multipliers which are used to calculate the reduced cost of any potential shift which exists in the superset. The reduced cost of a shift k is defined to be : M i=\
where : Mis the number of constraints Dj, is the cost of shift k
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KJ is the simplex multiplier for constraint i a^ is the coefficient of shift k in constraint / A shift which will improve the current objective will have a negative reduced cost. Experiments show that the simplex multipliers can vary significantly more than the Dk due to the presence of the prioritising weight. It is likely therefore that shifts which cover the pieces of work associated with the highest simplex multipliers may have the lowest reduced costs. The simplex multipliers are therefore considered in decreasing order and shifts covering their corresponding pieces of work are added to the subset if they have a negative reduced cost and if they are allowed to be added based upon the following rules, which limit the number of shifts which can be added during any major iteration.
Total Shifts Added Per Major Iteration As it is time consuming to consider every simplex multiplier at each iteration, and the simplex multipliers have been ordered so that the later values are less likely to produce further good shifts, a limit has been imposed on the percentage of simplex multipliers used to produce further shifts, viz: P% of the total number of pieces of work for which additions can be made For many simplex multipliers no further shifts will be added but there is a limit on the number of remaining simplex multipliers which can be considered in any major iteration. Experimentation has shown that a value of 20 for P adds sufficient further efficient shifts. Total Number of Shifts Added There is also a parameter limiting the total number of shifts which can be added to the subset defined by : M- MAX{50,X% of the total number of shifts remaining} The constant value 50 is used to ensure that if the number of remaining shifts decreases to a very low value, a reasonably large number of shifts can still be added in order to avoid extra major iterations. The parameter X reflects the need for fewer shifts to be added towards the end of the process but it is observed that P is normally a tighter limit than M and so X is set to 10 to allow it to be useful in very large data sets. Total Shifts Added Per Piece of Work Each piece of work is potentially covered by a very large number of shifts and so a limit is placed upon the number of shifts which can be added which cover a particular piece of work : / = MAX{5,Y% of the total shifts remaining/total no. pieces of work} The parameter Y again reflects the need to consider fewer shifts as the process continues and the constant value 5 deters more major iterations from having to be performed as the process nears its conclusion. Setting Y to 50 provides a sufficient number of shifts per piece of work without compromising the time taken to evaluate the reduced costs of the shifts involved.
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COMPUTATIONAL RESULTS Seven problems were tested, each with two sizes of shift superset. The smaller is a subset of the larger and is of a size which would normally be used within TRACS II. At the time of this research, TRACS II was limited to deal with problems containing fewer than 22000 shifts but this limit was increased in order to compare results and times with those produced by the column generation approach on all problems, the largest of which contains over 90,000 shifts and over 400 pieces of work. Details of these problems are given in Table 1.
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AUC CTJ CTR GMB RI2 STK SYD
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Smaller Subset(S) Larger Subset(L) #Pieces Superset Final #Pieces of Superset Final of Work Size Size Subset Work Subset Size Size 413 3740 480 43798 6112 10529 429 439 18307 3494 2586 10775 4112 443 3603 10690 451 13792 127 138 1126 920 4199 6956 167 180 1160 5176 7997 1461 250 90234 2421 10678 418 8297 302 1978 8866 396 43150 4853
Table 1 The aims of the research were twofold: • •
to investigate whether better solutions could be found by making more potential shifts available to the optimisation process; to determine whether for a given number of presented shifts, the optimisation process could be made faster.
Results of the experiments on the test problems are given in Tables 2 and 3. The experiments were performed on a Silicon Graphics Iris Indigo workstation with a 33MHz R300 MIPS processor. In Table 2, * means an attempt was made to find a solution with the given number of shifts and \§v means that the maximum node limit was reached in the branch and bound phase before an integer solution was found.
Data Set
Continuous # Shifts
AUC(S) AUC(L) CTJ(S) CTJ(L) CTR(S) CTR(L) GMB(S) GMB(L) RI2(S) RI2(L) STK(S) STK(L) SYD(S) SYD(L)
86.41 85.33 87.50 86.42 86.62 86.54 33.50 32.85 44.19 43.79 60.55 57.83 56.00 56.00
Final Major # Shifts Iteration s 87 7 4 87 4 88 5 87 88* 8 88* 7 34 4 33 4 45 5 44 4 61 6 6 59 56 6 56 5
Time to solve LP (mins) Column TRACS Generation II 12.8 144.6 10.0 18.3 17.1 17.1
13.9 20.7
0.6 1.3 0.9 1.7 5.2 2139.8 5.4
0.5 0.7 0.8 1.0 3.7
83.7
6.0 9.0 18.3 17.8
20.5
4.0 15.0
Total Solution Time(mins) Column TRACS Generation II 35.5 166.4 16.4 46.1 *?•> *?•>
50.3 91.2 19.8 18.6
0.9 2.1 2.2 2.8
0.6 0.9 1.2 1.2 7.8
15.7 2143.9 5.8 94.6
^ 53.8
24.2
4.3 17.5
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Table 2 shows that for AUC, CTR, and STK, the continuous solution produced by the larger shift set indicated too few shifts to enable a feasible schedule to be produced. However, the addition of a side constraint which specified a shift total at the next highest integer enabled a schedule to be found which was better than that produced by the smaller shift set. STK found a solution with two fewer shifts, AUC produced a solution with the same number of shifts but at lower cost, and CTR produced a solution where none had been found either with the smaller data set or with TRACS II on the larger set. Schedules with fewer shifts were also found by using the larger shift set on three out of the remaining four problem instances. Table 2 also presents the computer time necessary to solve the relaxed LP and the total time to the ILP solution for each of the runs. Overall, results show that the larger shift sets used in the column generation approach can yield better schedules in terms of the minimum number of shifts necessary to cover the work. In addition, the results show a reduction in execution time using column generation for 10 of the 12 data sets for which a comparison is meaningful. As a major aim of the research has been to determine whether better solutions could be obtained by using column generation on a larger subset than used by TRACS II, we compare in Table 3 the column generation solutions to the larger problems with the TRACS II solutions to the corresponding smaller ones. The small increases in computing time are justified by the potential savings. Where there has been a large increase (AUC), all but eight minutes of this is attributable to a longer route having been taken, by chance, through the branch and bound tree.
Data set AUC CTJ CTR GMB RI2 STK SYD
#shifts
87 88 ~ 34 45 61 56
TRACS II (S) run time (mins) relaxed LP total to ILP 36 13 10 16 0.6 0.9 0.9 2.2 5.2 16 5.4 5.8
Column generation (L) #shifts run time (mins) relaxed LP total to ILP 87 21 91 87 9 19 88 18 54 33 0.7 0.9 44 1.0 1.2 59 21 24 56 15 18
Table 3 It was surprising that the new approach should, on three occasions, achieve a significantly better result than TRACS II (saving a single shift is very significant to a bus company). TRACS II consistently achieves better results than those obtained by conventional methods, and when tested against other computer systems has proved more efficient. Users of TRACS II are very happy with the quality of solution obtained. We have to conclude from the above that there are opportunities for significantly better savings than are currently being realised.
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CONCLUSION Although optimal solutions still cannot be guaranteed using the column generation approach described, results show that the technique can produce better solutions more quickly than a successful and widely used driver scheduling system. Increased solution speed opens up the possibility of solving larger problems without decomposition which, in turn, may lead to better solutions. TRACS II is now being used successfully to solve large train driver scheduling problems (Kwan et al., 1996a, 1996b), using a form of decomposition where necessary. The new column generation approach is currently being adapted for the production version of the system, and is expected to enable TRACS II to yield better and faster solutions to such problems.
REFERENCES Daduna, J. R., I. Branco and J. M. P. Paixao (1995). Computer-Aided Transit Scheduling Proceedings of the Sixth International Workshop on Computer-Aided Scheduling of Public Transport. Springer-Verlag, Berlin. Desrochers, M. and J.-M. Rousseau (eds.) (1992). Computer-Aided Transit Scheduling Proceedings of the Fifth International Workshop on Computer-Aided Scheduling of Public Transport. Springer-Verlag, Berlin. Desrochers, M., J. Gilbert, M. Sauve and F. Soumis (1992). CREW-OPT: Subproblem modelling in a column generation approach to urban crew scheduling. In: Computer Aided Transit Scheduling (M. Desrochers and J.-M. Rousseau, eds.), pp. 395-406. Springer-Verlag, Berlin. Fores, S (1996). Column Generation Approaches to Bus Driver Scheduling. Ph.D. Thesis, University of Leeds. Kwan, A. S. K., R. S. K. Kwan, M. E. Parker and A, Wren (1996a). Producing train driver shifts by computer . In: Computers in Railways V - Volume 1: Railway Systems and Management (J. Allan, C. A. Brebbia, R. J. Hill, G. Sciutto and S. Sone, eds.), pp.421-435. Computational Mechanics Publications, Southampton. Kwan, A. S. K., R. S. K. Kwan, M. E. Parker and A, Wren (1996b) Scheduling train drivers and investigating alternative scenarios by computer. Presented at 4th Meeting of the EURO Working Group on Transportation, 9-11 September, University of Newcastle. Rousseau, J.-M. (1995). Results obtained with Crew-Opt, a column generation method for transit crew scheduling. In: Computer-Aided Transit Scheduling (J. R. Daduna, I. Branco and J. M. P. Paixao eds.), pp. 349-358. Springer-Verlag, Berlin. Rousseau, J.-M. and J.-Y. Blais (1985). HASTUS: An interactive system for buses and crew scheduling. In: Computer Scheduling of Public Transport 2 (J.-M. Rousseau ed.), pp. 45-60. North-Holland, Amsterdam. Sherali, H. D. (1982). Equivalent weights for lexicographic multi-objective programs. European Journal of Operations Research, 18, pp. 57-61.
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Smith, B. M. and A. Wren (1988). A bus crew scheduling system using a set covering formulation. Transportation Research, 22A, pp. 97-108. Willers, W. P., L. G. Proll and A. Wren (1995). A dual strategy for solving the linear programming relaxation of a driver scheduling system. Annals of Operations Research, 58, pp. 519-531. Wren, A. and B. M. Smith (1988). Experiences with a crew scheduling system based on set covering. In: Computer-Aided Transit Scheduling (J. R. Daduna and A. Wren eds.), pp. 104-118. Springer-Verlag, Berlin.
14 STOCHASTIC NETWORK MODELS AND SOLUTION METHODS FOR DYNAMIC FLEET MANAGEMENT PROBLEMS Raymond K. Cheung Department of Industrial Engineering and Engineering Management Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong
Dynamic networks have been used in a variety of transportation and logistics problems that involve both the spatial and time dimensions (Aronson, 1989; White 1972; Magnanti and Simpson, 1978). In real-world applications, most of these problems concern decision making in an uncertain envirnoment. With the recent advances of information technology, making decisions by using real-time information is now possible. This possibility has created the need to develop, solve, and analyze new stochastic, dynamic models. Our goal is to review recent developments using stochastic dynamic networks to tackle the problem of managing a fleet of vehicles or containers over time, which is referred to as the dynamic fleet management problem. In this paper, we first describe these problems in the contexts of vehicle allocation for truckload carriers and empty container repositioning for ocean shipping. Second, we show the stochastic network formulations of these problems. Third, we compare solution methods that take advantage of the network structure. Finally, we propose directions of developing new solution methods.
MOTIVATING APPLICATIONS In stochastic, dynamic models, decisions have to be made over time and once new information is available, we can make new decisions or adjust the previously made decisions. The commonly used objective for these models is to minimize the expected total cost over a finite horizon. In general, these problems are very difficult to solve. 209
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Fortunately, problems in transportation planning often exhibit special structures. By taking advantage of the structure, we can develop some computationally feasible solution techniques. Below, we consider two motivating applications and illustrate their special network structure.
Dynamic vehicle allocation problems Dynamic vehicle allocation problems (DVA) arise when motor carriers need to allocate a fleet of vehicles to the right place at the right time in anticipation of future market demands. Detailed description of these problems can be found in Powell (1987) and Cheung and Powell (1996). In short, the main decisions of these problems include a) repositioning empty vehicles between cities, b) moving loaded vehicles, and c) refusing loads. The output of these problems can also guide sales and telemarketing groups to solicit additional freight. The challenge of these problems is that we need to make decisions about the movement of vehicles now while anticipating the downstream impact of the current moves. The problem can be modelled as a dynamic network where a node represents a city at a specific time and an arc represents the movement between cities. There are three types of arcs, namely the loaded movement arc, the empty movement arc, and the inventory arc. Load movements arcs are used to represent the loaded movement of vehicles. These movements are revenue generating. Realising that our objective is to minimise cost, the cost coefficient for a loaded movement arc is the negative of the net profit for the move. On the other hand, the number of loaded movements between a particular pair of cities is limited by the market demands. Thus, the market demands can be represented by arc capacities (which limit the number of revenue-generating moves between the particular pair of cities within a time period). Since future market demands are known with uncertainty, the corresponding arc capacities are random variables. Secondly, empty movements are used to represent the pure repositioning of vehicles. Such moves are necessary due to the imbalance of in-bound and out-bound demands of a city. Empty movements involve the moving cost but have no market restriction. Typically, there are no arc capacities for these arcs although including arc capacities does not alter the model. Finally, inventory arcs represent vehicles that are held at their current locations from one time period to the next one. The cost for these arcs are the holding costs per time period. Such a network is shown in figure 1.
Stochastic network models for dynamic fleet management problems 2
3
4
5
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6
City 3
Arcs
Cost
Capacity
-revenue
market demand
moving cost
00
inventory cost
00
Figure 1: A network model for dynamic vehicle allocation problem
Dynamic empty container relocation problems In maritime industry, the imbalance of in-bound and out-bound containers at ports has forced liner operators to relocate their owned empty containers or to lease containers from vendors in order to satisfy customer demands. We refer to this problem as the dynamic empty container relocation (DCR) problem (Crainic et al. 1993). Similar to the DCA problems, DCR problems can also be represented as dynamic networks where more types of nodes and arcs are needed. Figure 2 illustrates how such a problem can be modeled as a dynamic network and shows a voyage going from port 3, to port 2 and then to port 1. The ship leaves port 3 on day 1, arrives port 2 on day 2. After spending a day at port 2, it leaves port 2 on day 3 and arrives port 1 on day 4. Major activities in the problem include: loading containers to ships (arc a), unloading containers from ships (arc d), taking containers from ports to customers (demand), returning containers from customers to ports (supply), transporting containers between ports (arc b), keeping empty containers on ship when the ship stops at a port (arc c), picking up leased containers from and returning leased containers to vendors, and storing empty containers as inventory at ports (arc e). In the model, we assume that customers' demands must be met: any unsatisfied
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demands will be met by leased containers. We use a demand arc (arc i) to model this situation where the arc cost is the negative of the leasing cost and the capacity is the demand. When a container arrives at a demand arc, it will be considered leaving the system and will re-enter the system as a stochastic supply in a later time period.
2
3
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Cost holding cost loading cost unloading cost storage cost on ship transportation cost - leasing cost salvage cost
Capacity oo 00 00 00
available space demand at port oo
Figure 2: A network model for dynamic container relocation problem
In practice, the availability of empty containers is subject to several uncertain parameters. First, once a loaded container arrives at a port and is picked up by the consignee, the exact returning time of the container (empty) is uncertain. Therefore, the supplies of empty containers at ports are random variables. Second, the exact number of empty containers required to meet the customer's demand at each port is also known with uncertainty - the farther away from now, the higher the degree of uncertainty. Thus, the demands of empty containers are also random variables. Finally, when loaded containers are ready at a port, their weight and their contents may affect how many of them and how they are being put in the vessel. For example, containers with certain contents cannot be put together to avoid hazardous situation. In other words, the available space for empty containers is also known with certain degree of uncertainty. In the network context, the random supply of
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empty containers is modelled as the random supply of flow to a node; the random demand is modelled as the random arc capacity for a demand arc (such as arc i). Finally, the random ship available space for empty containers can be represented by the arc capacities for the transportation arcs (such as arc b).
STOCHASTIC NETWORK MODELS Despite of some differences, common to both the DVA and the DCR problems are the network structure and that decisions have to be made over time subject to uncertainty. The common objective is to minimise the cost for the current stage, known as stage 1, plus the expected future cost. Let (Q,
is: min < c1, x > +EQ(vl, co) subject to: A].x] B].x] 0
= b] Vie# ! W, = v] Vie N, < x1 < w1
In this formulation, the expected future cost, denoted by.E^v1, co), is a function of v1 and is known as the expected recourse function. Depending on how we treat future
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decision making, stochastic network models can be classified as either two-stage models or multistage models.
Two-stage model If we assume that future random parameters are realised at once, then EQ(vl, co) is the expectation of the minimisation problem: Q(vl, co) = min < c2, x2 >
subject to: A?.x? B^.x] 0
= v,1 + S,(co) V/eTV'W, = bf + S,(o>) V/e N, < x 2 < £'(to)
For simplicity, we let X(v', co) be the set of stage 2 constraints. For problems with large number of independent variables, finding optimal solution is computationally infeasible. However, it can be shown that the function EQ(vl, co) is convex. Hence, this property can help develop some approximation methods as we describe later.
Multistage model If we assume that future random parameters are realised over time, then EQ(v}, co) is the expectation of a recursively defined minimisation problem embedded in expectation. Let X' (v' ~ ' , co') be the stage t constraint set, Q' (v' ~ ' , co') is given by: e'(v'- ] ,co') =
min
< c',x'
> +EQ' +l(v', co / +1 )
x' e X ' ( v ' ~ ' , c o ' )
where X' (v' ~ ' , co') is the set of stage t flow conservation and capacity constraints. The main difference between the two-stage model and the multistage model is that in the twostage model, we assume that once we are in stage 2, all the random parameters are observed at once, and then we make the decision. On the other hand, in the multistage model, in stage 2, we only realise the random parameters for stage 2, the parameters in the future stages are still uncertain. In other words, in stage 2 of the two-stage model, we
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have the perfect information for making the decision; whereas, in stage 2 of the multistage model, we only have partial information.
SOLUTION METHODS In stochastic programming literature, the above models are known as stochastic programs with recourse which are very difficult to solve. Approximation methods have appeared and fall into three categories: (a) scenario methods that use scenarios to approximate the underlying population, turning a stochastic program to a large-scale deterministic program which is be solved via decomposition methods (see, for examples, Van Slyke and Wets, 1969; Rockafellar and Wets, 1991); (b) stochastic quasi-gradient methods which use sample subgradients to approximate^ and solve a sequence of linear programs (see Ermoliev, 1988); and (c) nonlinear approximation methods which use analytical nonlinear functions to approximate EQ and solve one or more simple nonlinear programs (see Nazareth and Wets, 1988; Powell and Cheung, 1994a, 1994b). Below we review and compare the methods that utilise the network structure of our problems.
Stochastic linearization method Note that if the random quantities are discrete random variables, then EQ(v\ co) is not differentiable at all values of v 1 . Thus, the partial derivative of EQ(v}, co) with respect to v 1 is not unique and is called a subgradient. Let g(v')be an unbiased estimate of the subgradient of EQ(v\ co). If
g(v') is based on sampling, we call it a stochastic
subgradient. For the two-stage model, the stochastic linearization (SL) method uses a sequence of linear programs to approximate the original nonlinear programs. For each linear program, the objective function involves a weighted average of the stochastic subgradients which linearly approximate EQ(v}, co). If the stage 2 problem is solved for an outcome co , then the vector of the optimal dual variables that are associated with the interface nodes is a valid choice for g(v]). The method generates a sequence of solutions xjas:
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x\
= arg min < c 1 , x] > + < gk, v 1
where g\ is the stochastic subgradient at iteration k . With some conditions on ak and p\ , jc^ will converge to the optimal solution almost surely (see Gupal and Bazhenov, 1972). This method only needs to solve linear programs and has the convergence property. For general stochastic programs, however, the convergence method is believed to be very slow. A recent study in Cheung and Chen (1996) suggests that few iterations of this method can produce a reasonably good solution for DCR problems (although getting a very good solution requires substantially more iterations). One drawback is that the method destroys the integrality of the solution. Furthermore, how this method can be applied for solving multistage models is unclear.
Nonlinear approximation method A nonlinear approximation (NLA) method for DVA problems (which are networks with random arc capacities where supplies are deterministic) is proposed by Powell and Cheung (1994b). The idea is to structurally decompose a network into a set of trees with random arc capacities. Each tree is rooted at an interface node and the trees are overlapping. For each tree, we can obtain the expected recourse function exactly (see Powell and Cheung, 1994a). For example, in stage t+l, the expected total cost is approximated by Q' ( v ' ~ l ) which is the sum of the expected cost of the individual trees. In other words, Q'(v1 ~ ' ) i s a separable, convex, piecewise linear function and has the form Q' (v' ~ ' ) = ^] Q' (v,'"') where Q' (v' ~') is the exact total cost for each tree as a i parametric function of the supply v,'"'. Notice that the result of the decomposition of a network into a set of overlapping trees is that some of the capacity constraints (that is, x, < £, ' where £ ' is the vector of random arc capacities in stage i) are violated. Thus, we can use a set of multipliers to penalise the constraint violation and then update these multipliers iteratively. Thus, in general, the stage / problem can be written in the form of
Stochastic network models for dynamic fleet management problems
min
x' sX" ( v ' - 1 , e > ' )
217
< (c + A.)', x1 > +0' + V)
where X is the vector of multipliers and X" represents the set of trees. Since each component function Q' (v' ') is convex, piecewise linear with integer break points, it is easy to see that this function can be represented by a set of parallel arcs. As a result, the method can produce integer solutions. Furthermore, the Q' (v,'"') are obtained independently of decisions made in earlier stages. Thus, we can successively obtain Q' (v,' ~ ' ) starting from the last stage back to the second stage. Nevertheless, this method is limited to the structure only of networks with random arc capacities. Furthermore, no convergence result has been obtained.
Hybrid method For two-stage networks, a hybrid method, known as the stochastic hybrid approximation procedure (SHAPE), is proposed recently by Cheung and Powell (1995). In SL, the sequence of solutions produced are believed to be unstable because of the extreme point solution property of the linear programs. Therefore, the hybrid method tries to impose a nonlinear function for stabilising the optimisation process. The idea is to first obtain an initial nonlinear function Q0 to approximate EQ and then update Q0 linearly using stochastic subgradients iteratively. Suppose that we are at iteration k and Qk is the nonlinear approximation at iteration k . The update mechanism involves first finding a stochastic subgradient at the current solution point, say x\ , then comparing the slope of this stochastic subgradient (denoted by g\ ) and the slope of the nonlinear approximation function at x\ (denoted by qk ). The difference of the slopes, g\ - qk , is used to linearly adjust the slope of the whole nonlinear approximation function Qk by
Qk + \x) = Qk(x) + ak < gl - qk,x > where ak satisfies the condition:
ak > 0, £ ak = co, £ a]
< co
k
Given the new nonlinear function, we find a new solution x\ + , and continue the process. After some simple algebra, the problem at iteration k can be written as
218
R. K. Cheung a
mn < c, x > +0(v)+ <
Jt eX
'
'
/ =!
where a is a sequence of positive numbers and depends on the history of the process. This method can produce an optimal solution almost surely and it can reduce the number of iterations to reach a good solution. However, this method produces fractional solutions, is sensitive to the choice of Q0 , and needs to solve a nonlinear network in each iteration. Modifications of the basic method are investigated in Cheung and Chen (1996) where the numerical experiments suggest that solving the nonlinear subproblems only approximately instead of accurately can save substantial amount of computational time and produce solutions with good quality.
NEW DIRECTIONS The above models and methods raise two interesting research issues. First, although SL and SHAPE can be used for solving two-stage models, how to extend them for solving multistage models remains a challenging question. Second, the above models have some restrictive assumptions such as no back-logging is allowed for unsatisfied demands. Two promising alternatives are discussed as follows.
Stochastic nonlinear approximation method Consider the network problem in stage 2 when an outcome co is given. Suppose that the problem is solved optimally and the dual prices n2 associated with the interface nodes are obtained. The dual price nf for node / can estimate the marginal cost of having flow arriving at node / . However, dynamic network problems are highly degenerate, this estimate can be poor. Alternatively, we can use parametric programming techniques to estimate a convex, piecewise linear function Q2 (v,1) that tells us the marginal cost for each incremental unit of flow arriving node / . To generate this function, we can use multiple samples for the stage 2 problem to get a set of it 2 for different samples of the vector v 1 . Then, using these n2, we estimate the nonlinear function. The detail of the construction of the nonlinear function has yet to be developed.
Stochastic network models for dynamic fleet management problems
219
This method uses sampling information in the future to construct a nonlinear function. The construction itself is independent of the decisions made in stage 1, meaning that it can be used for solving multi-stage models. Furthermore, the method is more robust than the method of Cheung and Powell (1996) which is limited to networks with random arc capacities. The nonlinear approximation can be applied to problems with various sources of randomness.
Logistics queuing network approach We can view the dynamic fleet management problem as a queuing network system where containers or vehicles are waited to be dispatched. In transportation applications, such a system is typical non-stationary and the planning horizon is finite. Therefore, classical queuing theory methodology may not be applied directly. However, they can be used as a guideline to develop new solution methods. One promising alternative is the dynamic control techniques for discrete event systems. First, we define a set of control variables and then optimise the control variables. The optimisation is based on the gradient information estimated by simulations. For example, for DVA problems, we can use the limits of empty movements on arcs as control variables and use simulation to estimate the effects of changing these limits. Initial testing for the case of deterministic DVA problems shows that the approach is promising as shown in Powell et al. (1994). This approach is quite flexible that many assumptions required by the existing mathematical models (such as no back-logging for unsatisfied demands) can be relaxed. The challenge for this approach is (a) how to identify and to update the set of control variables for general dynamic fleet management problems and (b) how to apply this approach in a stochastic setting.
ACKNOWLEDGEMENT The research was partially supported by the National Science Foundation of the United States under the Grant DMII-9501446 and by the Research Grant Council of Hong Kong.
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REFERENCES Aronson, J. E. (1989). A survey of dynamic network flows. In Bala Shetty, editor, Annals of Operations Research, volume 20, pages 1-66. J.C. Baltzer AG. Cheung, R. K. and C. Chen (1996). A two-stage stochastic network model and solution methods for dynamic empty container allocation problem.
Working paper,
Department of Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, Iowa. Cheung, R. K. and W.B. Powell (1996). An algorithm for multistage dynamic networks with random arc capacities, with an application to dynamic fleet management. Operations Research, Vol. 44, No.6, 951 -963, 1996. Cheung, R. K. and W. B. Powell (1995). A stochastic hybrid approximation procedure, with an application to dynamic networks. Working paper, Department of Industrial and Manufacturing Systems Engineering, Iowa State University. Crainic, T. G., M. Gendreau, and P. Dejax (1993). Dynamic stochastic models for the allocation of empty containers. Operations Research, 41(1):102~126. Ermoliev, Y. (1988). Stochastic quasigradient methods. In Y. Ermoliev and R. Wets, editors, Numerical Methods in Stochastic Programming. Springer-Verlag. Gupal, A. M. and L. G. Bazhenov (1972). A stochastic method of linearization. Cybernetics, 482-484. Magnanti, T. L. and R. W. Simpson (1978). Transportation network analysis and decomposition methods. Report no. dot-tsc-rspd-78-6, U.S. Department of Transportation. Nazareth, J. L. and R. J-B Wets (1988).
Nonlinear programming techniques. In
Numerical Techniques for Stochastic Optimization. Springer-Verlag.
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Powell, W., T. Carvalho, G. Godfrey, and H. Simao (1994). Dynamic fleet management as a logistics queueing network. Working paper, SOR-94-18, Department of Civil Engineering and Operations Research. Powell, W. B. An operational planning model for the dynamic vehicle allocation problem with uncertain demands. Transportation Research, 218:217-232. Powell, W. B. and R. K. Cheung (1994a). Stochastic programs over trees with random arc capacities. Networks, 24:161 -175. Powell, W. B. and R. K. Cheung (1994b). A network recourse decomposition method for dynamic networks with random arc capacities. Networks, 24:369-384. Rockafellar, R. T. and R. J-B. Wets (1991). Scenarios and policy aggregation in optimization under uncertainty. Mathematics of Operation Research, 16(1). Van Slyke, R. M. and R. J-B. Wets (1969). L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math., 17(4):638663. White, W. W. (1972). Dynamic transshipment networks: An algorithm and its application to the distribution of empty containers. Networks, 2(3):211-236.
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15
PERIODIC SHIPPING STRATEGIES FOR THE MINIMIZATION OF THE LOGISTIC COSTS
Luca Bertazzi and Maria Grazia Speranza Dept. of Quantitative Methods University of Brescia, Italy
Abstract In this paper we study the often neglected relationship between inventory and transportation costs in several logistic networks. In particular, we first consider the problem of shipping several products in the single link case. In this problem a set of products has to be shipped from a common origin to a common destination; the aim is to determine a periodic shipping strategy which minimizes the sum of the transportation and inventory costs. We present a general framework of analysis from which we derive the known approaches with a continuous frequency and with given frequencies as particular cases. Moreover, we consider two more complex logistic networks, the one origin-multiple destinations case and the sequences of links case, and we show a property of the inventory.
INTRODUCTION In this paper we analyze the trade-off between transportation and inventory costs in different types of logistic networks. In particular, we consider some structured networks: the single link case, the one origin-multiple destinations case and the sequence of links case. In the single link case, a set of products has to be shipped from a common origin to a common destination; in the one origin-multiple destinations case, a set of products has to be shipped from a common origin to several destinations and, finally, in the sequence of links case a set of products has to be shipped from a common origin to a common destination through one or several intermediate nodes. 223
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L. Bertazzi and M. G. Speranza
In each of these networks the problem is to determine a periodic shipping strategy which minimizes the sum of the inventory and the transportation costs. This problem has been mainly studied with two different approaches: the approach with a continuous frequency (the continuous case) and the approach with given frequencies (the case with given frequencies). The continuous case has been studied in Anily and Federgruen (1990), in Blumenfeld et al. (1985) and in Burns et al. (1985). In this approach the time between shipments can assume any positive real number and all products are shipped at a unique frequency; therefore, the aim is to determine the unique shipping frequency which minimizes the total cost. The main drawback of this approach is that the obtained solution can be infeasible from a practical point of view, as discussed in Hall (1985), Maxwell and Muckstadt (1985), Jackson et al. (1988), Muckstadt and Roundy (1993). The approach with given frequencies has been introduced by Speranza and Ukovich (1994b) for the single link case on the basis of the motivations given in Speranza and Ukovich (1992), where a Decision Support System for logistic managers has been presented. In this approach, a minimum time between shipments is given, for instance one day: This means that at most one shipment, possibly with several vehicles, can be made every day. Moreover, a set of frequencies is given; each frequency is such that the corresponding time between shipments is a multiple of the minimum time between shipments. In this approach, the aim is to determine the percentage of each product to ship at each frequency in order to minimize the total cost. As proposed in Hall (1985), a simple way to obtain a feasible solution of the problem with given frequencies is to round off the continuous shipping time provided by the model for the continuous case; unfortunately, as pointed out in Speranza and Ukovich (1994a) on the basis of large computational experiments, this way to impose feasibility often fails to produce the optimal solution. In Speranza and Ukovich (1994b) exact optimization models for the single link case with given frequencies have been studied; their properties and computational methods have been presented in Speranza and Ukovich (1996) and in Bertazzi, Speranza and Ukovich (1995, 1996). A first application of the approach with given frequencies to more complex logistic networks can be found in Speranza and Ukovich (1994a) and a detailed study of the one origin-multiple destinations case and of the sequence of links case can be found in Bertazzi, Speranza and Ukovich (1997) and in Bertazzi and Speranza (1996, 1997a, 1997b), respectively. This paper has two objectives. On one side we survey the results obtained in the analysis of periodic shipping strategies. On the other side we give some theoretical contributions. First of all, we present a unifying framework of analysis of the periodic shipping problem for the single link case, from which the two known approaches are derived. Through such unifying framework the underlying assumptions of the approaches are made precise and justified. Besides, for the one origin-multiple destinations and for the sequence of links cases a property of the inventory is shown. The focus of the paper is on the single link problem. The unifying framework is described in Section 1, while in Sections 2 and 3 the continuous case and the case with given frequencies are derived, respectively. In Section 4 we survey the results obtained for the one origin-multiple destinations case and the sequence of links case with given frequencies, showing for both cases a property of the inventory.
Periodic shipping strategies for minimization of logistic costs
1
225
THE SINGLE LINK PROBLEM
In the single link problem a set I = {1,2,..., |/|) of products has to be shipped from a common origin A to a common destination B over an infinite horizon. Each product i e / is made available at the origin A and absorbed at the destination B in a continuous way {continuity assumption) with a constant rate <& (steady-state assumption) equal in A and in B (equilibrium assumption). Each product i £ / has a volume u>i and an inventory cost hi per time unit. The total inventory cost is proportional to the sum of the transportation time and of the idle time both at the origin A and at the destination B. The transportation time is supposed to be constant and, therefore, equal to zero without loss of generality. The inventory of each product i e / in A and in B must be non negative in each time instant (stock-out exclusion assumption). In order to satisfy this assumption, a starting inventory is made available at time 0 in A and in B. Shipments from A to B are performed by a fleet of vehicles V. The set V is the union of a finite number of subsets Vi, I E L = {1,2,..., |L|}; each subset Vi contains an infinite number of vehicles of given transportation capacity r\ and transportation cost c\. The transportation cost is charged for each journey independently of the quantity loaded on the vehicle. Our aim is to find a periodic shipping strategy which minimizes the sum of the transportation and inventory costs in the network. A shipping strategy is a rule which defines when shipments have to be performed, how products have to be shipped from A to B, how much starting inventory in A and in B must be made available at time 0. A shipping strategy is periodic if we can find a time horizon H such that every operation made in each time instant t, 0 < t < H, is repeated in the time instants t + nH, n = 1,2,...; in this case, the optimal solution for the infinite horizon problem is simply obtained by repeating at infinity the optimal solution for the problem with finite horizon H. Each periodic shipping strategy is defined on the basis of a minimum time between shipments p, that can be unknown or given. In the first case, p can be interpreted as the unique continuous time between shipments t. This means that all products i € / are shipped at the unique frequency / = 1/t. Therefore, the time horizon H is simply equal to t and only one shipment is performed during the time horizon H at time 0; in other words the set of shipping times K is {0}. The aim is to determine the shipping frequency / that minimizes the total cost. We refer to this case as the continuous case. In the second case p is given and therefore can be normalized, without loss of generality, to 1. If a set of shipping frequencies is given and each frequency is such that the corresponding time between shipments is integer, then we have the so called approach with given frequencies. In this case, the time horizon H is simply the minimum common multiplier of all times between shipments and the set of shipping times K is a subset of T = {0,1,2,...,#-!}. In this section we present some general considerations which hold for both these two cases; in particular, we formally define the concept of periodic shipping strategy and we derive the total inventory cost. In the following two sections we specialize these concepts to the continuous case and to the case with given frequencies, viewed as particular cases of the situation with p unknown and the situation with p given, respectively.
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Periodic shipping strategy Given a time horizon if, a Periodic shipping strategy is a rule which defines: 1. The minimum time between shipments p, if it is not given; 2. the quantity suk of each product i G / to ship from A to B by using vehicles of type I e L at each shipping time k G K; 3. the number of vehicles y\k of each subset Vj, I G L, to use at each shipping time k£K4. the starting inventory df at time 0 in A, for each product i 6 J; 5. the starting inventory d? at time 0 in B, for each product i e I; on the basis of the following constraints: 1. Demand constraints: for each product i e /, the total quantity shipped from A to B during the time horizon H must be equal to the total quantity made available in A and absorbed in B; more precisely, the demand constraint of each product i e I can be formulated as: E E SM = QiH
i 6 I;
(1)
2. Capacity constraints: for each vehicle type Z € L , the total number y/^ of vehicles used in each shipping time k e K , must be sufficient to load the total quantity shipped; more precisely, the capacity constraints can be formulated as: E
w s
i M ^ riyu
I E L
k e K;
(2)
.67
3. Stock-out constraints: these constraints guarantee that no stock-out can occur during the time horizon H, i.e. guarantee that the inventory of each product i e / in A and in B is always non negative. The inventory QfT of product i e / in A at time T e [0, H) is equal to the sum of the starting inventory df at time 0, plus the cumulative quantity made available in A up to time r, minus the cumulative quantity shipped from A to B up to time r; more precisely, Qir = d f +
q i
T-
EES^
(3)
where KT = {k e K : k < r}. At the origin A, given that each product is offered at a constant rate, no stock-out occurs during the time horizon if a stock-out constraint is satisfied in each of the shipping times of K. Therefore, the set of the stock-out constraints in A can be formulated as: Q£>0
ieI,t£K.
(4)
Periodic shipping strategies for minimization of logistic costs
221
The inventory QfT in B of product i at time r 6 [0, H) is equal to the sum of the starting inventory df at time 0, plus the cumulative quantity shipped from A to B up to time r, minus the cumulative quantity absorbed in B up to time r; more precisely,
At the destination J3, given that each product is absorbed at a constant rate, no stock-out occurs if a stock-out constraint is satisfied in each of the the shipping times of K. Therefore, the set of the stock-out constraints in B can be formulated as: (6)
The objective function The objective function expresses the minimization of the sum of the inventory cost and the transportation cost in the time unit. The total inventory QiT of each product i G / in each time instant r £ [0, H) is the sum of the inventory QfT at the origin A and the inventory QfT at the destination B, where Qfr and QfT are defined by the (3) and (5), respectively. In the following remark we state that the inventory QiT of each product i 6 I is constant in each time instant r £ [0, H ) and equal to the sum of the starting inventory at time zero in A and in B. The proof is simply obtained by summing up the inventory in A and in B in each time instant. Remark 1 The total inventory Q±T of each product i £ / in each time instant r £ [0, H) is equal to df + df . As a consequence, the total inventory cost of product i £ I in the tune unit is h,(df + df). The formulation of the transportation cost in the time unit is straightforward. In fact, it can be expresses as follows:
In conclusion, the single link shipping problem is the problem of finding a periodic shipping strategy that minimizes the objective function
subject to the constraints (1), (2), (4) and (6).
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2
L. Bertazzi and M. G. Speranza
THE CONTINUOUS CASE
A classical approach in the formulation of periodic shipping strategies is referred to as the continuous case. This approach has been applied by Anily and Federgruen (1990), Blumenfeld et al. (1985) and Burns et al. (1985) to the single link network and to other types of logistic networks. The continuous case can be seen as a particular case of the general problem presented in the previous section. This allows us to formulate an explicit opimization model. The continuous case is based on the following additional hypotheses: 1. Unique and unknown shipping frequency: all products i e / are shipped at the same unknown unique frequency / = 1/t, where t is the unknown time between shipments; 2. Continuous and unknown time between shipments: the unknown time between shipments t can take any positive real value; 3. Single vehicle: the fleet of vehicles V is composed of only one vehicle with transportation capacity r and transportation cost c. Given these assumptions, the minimum time between shipments p and the time horizon H are equal to t\ therefore, during H only one shipment is performed from A to B at time 0. The quantity of each product i e / shipped in 0 is the whole quantity made available in A during a time interval of length t, that is <&£; therefore, given that the time horizon H is equal to t, the demand constraints (1) are obviously satisfied. Given that there is only one shipment during the time horizon H, the stock-out constraints in A and in B must be evaluated only in 0; in particular, given that qrf is the quantity of product i shipped at time 0, the minimum starting inventory in A at time 0 which satisfies the stock-out constraints is df* = qrf and the minimum starting inventory in B at time 0 is df* = 0 . Therefore, given the result stated in Remark 1, the inventory cost in the time unit is £)» h^t and the stock-out constraints can be omitted in the formulation of the problem. The problem is to determine the shipping frequency /* at which all the products i 6 / have to be shipped in order to minimize the sum of the transportation and inventory costs. The optimization problem can be formulated as follows. Problem H it + c/t
(7) (8)
ie/
t >0
(9)
The objective function (7) expresses the minimization of the sum of the inventory cost and of the transportation cost in the time unit; the constraint (8) is the capacity constraint and the constraint (9) defines the unique decision variable of the problem.
Periodic shipping strategies for minimization of logistic costs
229
Problem "R, is a nonlinear constrained optimization model; it has the following closed solution: /
i
:
_
\
(10) The first term is the classical Wilson's formula and the second part guarantees that the capacity constraint (8) is satisfied.
3
THE CASE WITH GIVEN FREQUENCIES
In this section we describe the model with given frequencies proposed by Speranza and Ukovich (1994b) as a particular case of the problem described in Section 1. In this way the assumptions upon which the model is based are clarified and justified. This model is based upon the following additional hypotheses: 1. Given minimum time between shipments: without loss of generality;
p is given and normalized to one,
2. Given discrete frequencies: shipments from A to B can be performed only with frequencies which belong to a given set F = {f}; : j € J}, where J={1,2,...,|J|}. Each frequency fj, j £ J, is such that the corresponding time between shipments tj = I/fj is an integer. The time horizon H is equal to the minimum common multiplier of all times between shipments, that is H — mcm{tj,j e J}, and the set of shipping times K is a subset of {0,1,..., H — 1}; 3. Phasing of all frequencies in zero: for each frequency fj, a set Kj is defined as the subset of K having for elements the times t e K in which a shipment can be performed. The set Kj has cardinality H/tj\ therefore, the first shipment must be done in one of the time instants which belong to the set P}• = {t E K : 0 < t < tj}. We define phase tpj of the frequency fj the time instant in Pj in which the first shipment occurs, if frequency fj is actually used. Given the phase rfj, the set Kj is defined as follows: Kj = {t £ K : t = t/)j + ntj, n integer, 0 < n < H/tj — 1}. In principle, all possible combinations of phases should be considered in order to obtain the minimum cost. In this paper, we analyze only the situation in which all frequencies are phased in 0; in other words, ipj = 0, Vj €E J. This situation has several applications from a practical point of view. It corresponds to the case in which the decision-maker has no degrees of freedom in the phasing of the frequencies; a typical example is the case in which shipments are performed by public transportation systems or carriers; 4. Frequency consolidation: only the products shipped at the same frequency can share the same vehicle; 5. Fixed shipping quantity: for each frequency /;, the quantity of each product i €E / shipped at the corresponding times t £ Kj is constant; in particular, if o;^ is the percentage of product i G J shipped with frequency fj, then the quantity shipped in each time t 6 Kj is
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L. Bertazzi and M. G. Speranza
6. Fleet composition: the fleet of vehicles is composed of at most | J| different sets Vj] each set contains an infinite number of vehicles which can be used only with frequency fj ; each vehicle v € Vj has a given transportation capacity TJ and a given transportation cost GJ. Given these assumptions, the aim is to determine the percentage & ^ of each product i 6 / to ship with frequency fj, j £ J, the number of vehicles of each type Vj, j £ J, to use and the minimum starting inventory in A and in B to make available at time 0 in order to minimize the sum of the transportation and inventory costs.
The stock— out constraints In this section we present a particular formulation of the stock-out constraints (4) and (6) for the Problem with given frequencies which allows us to derive, in a closed form, the minimum starting inventory at time 0 in A and in B. At the origin A, being p = 1, the stock-out constraints (4) must be evaluated in the time instants which belong to the set K = {0, 1, . . . , H — 1} in which a shipment can be performed. We recall the definition of Kt — {k £ K : k < t}. In each time instant t £ K, the stock-out constraint in A for the product i 6 / can be expressed as: df + Qit- £ £ ^ i ^ > 0
(11)
fcetftje./*
where Jk = {j G J : k/tj is integer} is the set of frequency indices for which a shipment can be performed at tune k. At the destination J5, the stock-out constraints (6) must be evaluated in the time instants which belong to the set K — {1,2, .. . ,-ff}. For each time instant t E K, the stock-out constraint in B for the product i £ J can be expressed as:
d
f + E E tjwn - &* > o
(12)
The following lemma allows us to write the stock-out constraints (11) and (12) in a nicer form. Lemma 1 For each time instant t, t G T = {0, 1, . . .}, the following equality holds:
£ E *W*ij = After rewriting the stock-out constraints (11) and (12) on the basis of the Lemma 1 we can state the following propositions.
Proposition 1 The minimum starting inventory in A at time 0 which satisfies the stock-out constraints (11) is
Proposition 2 The minimum starting inventory df* in B at time 0 which satisfies the stock-out constraints (12) is zero.
Periodic shipping strategies for minimization of logistic costs
231
For the proof of the above propositions we refer to the Appendix. Given these results, the stock-out constraints (11) and (12) can be omitted in the formulation of the problem and the total inventory cost in the time unit is Xlie/ ^j&J hiqitjXij. This cost is in product form: The coefficients can be expressed as the product of a factor (fo^i) which depends only on the product i e / and a factor (tj) which depends only on the frequency /;. Moreover, given this formulation, the inventory cost accounted only for the origin or for the destination is simply IX^e/X^e./^i&t^i;'-
The optimization problem As stated in Speranza and Ukovich (1994b), the optimization problem for the case with given frequencies can be formulated as a mixed integer linear programming problem as follows. Problem P
ie/ j€.J
j'6.7
n=1
iG/ j < rjyj
(15) j GJ
ieI,j&J yj > 0
integer
(16) (17)
j £ J.
(18)
The objective function (14) represents the sum of the inventory cost and the transportation cost in the time unit; the inventory cost has been obtained by summing df* and df*, as stated in Remark 1; the transportation cost in the time unit has been obtained considering that, during the time horizon H, there are H/tj shipments with each frequency fj, j 6 J. The constraints (15) are the Demand constraints; they have been obtained from (1) by considering that, given the Fixed quantity assumption, a quantity tjqiXij of product i e I is shipped H/tj times during the time horizon H. Constraints (16) are the capacity constraints; they have been obtained from (2) by considering that, given the Fixed quantity assumption, the quantity shipped with each frequency fj is constant. We shortly recall that Problem P has the following properties, shown in Speranza and Ukovich (1996): 1. Problem P is NP-hard; 2. Finite divisibility property: the optimal value of the variables x^ belongs to the finite set {0, 1/tfj, 2/Wi, ...,!}; 3. Iso-ranking of frequencies and unit values of the items: the items with higher hi/Wi ratio are shipped more frequently;
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4. Saturation property: all used vehicles have a full load, with the possible exception of the one which travels at the lowest used frequency; 5. Optimality of frequency consolidation: the total cost is not improved by consolidating products shipped with different frequencies on the same vehicle. The above properties made it possible to devise some efficient algorithms for the solution of Problem P. In fact, the saturation property implies that, given the number of vehicles to use with each frequency ft, j 6 J, the optimal assignment of the products i e / to the frequencies is obtained by the following simple procedure: Rank the products i e / in the non-increasing order of the ratio hi/Wi and the frequencies ft, j € J, in the non-decreasing order of the times between shipments t j , j € J; then, on the basis of these orders, assign the products to the frequencies. Therefore, the optimal solution of Problem P can be obtained by a branch-and-bound algorithm that works on a search tree in which each level corresponds to a frequency ft and each node at each level represents the number of vehicles yj used with frequency ft. In Speranza and Ukovich (1996) a first branch-and-bound algorithm with this structure has been implemented; computational results show a good performance in problems of small/medium size. In Bertazzi, Speranza and Ukovich (1995, 1996) an improved version of the algorithm has been presented for the case in which the transportation capacities are constant; in other words, TJ = r, Vj 6 J, and, therefore, GJ — c, Vjf e J. For this case, dominance rules, which allow to tighten the bounds on the problem variables and to avoid dominated solutions, have been derived and applied. Moreover, in order to obtain an initial good upper bound on the total cost, a set of heuristics for Problem P has been solved at the beginning of the algorithm in each instance. Large computational experiments, in which the two branch-and-bound algorithms have been compared, showed that the efficiency of the former algorithm has been substantially improved in situations with up to 10,000 products and 15 frequencies. Moreover, the heuristics have been compared with the optimal solution in order to assess their performance: The results show that one of the proposed heuristics generates the smallest error which is almost always below 0.4% and much lower in most cases.
4
MORE COMPLEX NETWORKS WITH GIVEN FREQUENCIES
In this section we summarize the results obtained from the approach with given frequencies on two more complex networks than the single link network: The one origin-multiple destinations case and the sequence of links case. Moreover, for both networks, we show a property of the inventory.
The one origin—multiple destinations networks In these networks a set / of products is made available at a common origin A; in particular, each product i € / is offered in A at a given constant rate q±. A subset Im C I of products is absorbed at each destination Bm, m € M = {1,2,...,|M|}, at a given constant rate qmi such that £meM qmi = <7i- A set F of shipping frequencies is given;
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233
shipments with frequency fj from node A to node Bm, m G M, are made by vehicles of given transportation capacity TJ and transportation cost cmj. In the following remark, we state that the inventory during the time horizon H is constant and equal to the sum of the starting inventory at time 0 at the origin (df ) and at the destinations (df m , m G M). The proof is simply obtained by summing the inventory in each node and the inventory in transit in each time instant. Remark 2 In the One origin-multiple destinations networks the inventory of each product i G / in each time instant r G [0, H) is equal to df + Y^,m&M ^i m • As in the single link case, this remark stresses the importance of the starting inventory at time 0. In order to minimize the total cost, the minimum starting inventory must be found which satisfies the stock-out constraints; for its derivation, slight modifications of Propositions 1 and 2 hold. In the One origin-multiple destinations networks the problem is to determine for each link (A, Bm), m G M, the percentage xmij of product i G Im to ship with frequency /y, j G J, the total number of vehicles to use in the network with each frequency /y, j G J, the subset of destinations that each vehicle has to visit and the route that each vehicle has to follow in order to minimize the sum of the transportation and the inventory costs. Given that the simpler single link network has been proved to be NP-hard, a unique optimization model for this problem would be impractical. For this reason, in Bertazzi, Speranza and Ukovich (1997) a basic heuristic procedure and several variants have been proposed. In the basic heuristic each link (A, Bm), m G M, is first optimized independently by solving Problem "P\ then the destinations JBTO, m G M, are clustered into subsets and local search techniques, which involve modifications in the selected shipping frequencies and the solution of routing problems, improve the obtained solution. In the variants of this heuristic the preliminary zoning of the destinations and the possibility of phasing the frequencies have been considered. All these heuristics have been compared on a large set of randomly generated instances with a discretized version of the solution method for the continuous case presented in Burns et al. (1985).
The sequence of links networks Let S = (1,2, . . . , 1^1}, |5| > 3, be the set of nodes in the sequence and (s, s + 1), s = 1,2,...,|5| — 1, bea link. A set of products J has to be shipped from 1 to |5| through the \S\ — 2 intermediate nodes. Each product i G J is made available in 1 at a given constant rate <£ and absorbed at the destination |5| at the same given rate. Each product can be partially shipped on the link (s,s + 1) at any of the given frequencies f j ' s + , j e j(s,s+i) = {1,2,..., | J(s's+1>|}. The problem is to determine, for each link (s, s + 1), s = 1,2,. ..,151 — 1, the percentage x^'s+ of product i G J to ship with frequency f j + , j G j(s's+1), and the number yf' of vehicles of transportation capacity TJS+I' to use s s+ s s+1 with frequency /j ' , J G j( ' )) in order to minimize the sum of the transportation and the inventory costs on the network. In this problem the inventory cost in the intermediate nodes s, s = 2, 3, . . . , \S\ — 1, requires a different formulation with respect to the origin 1 and the destination \S\; in fact, in the intermediate nodes the products are not offered and absorbed with a given constant rate. In Bertazzi and Speranza (1996, 1997a, 1997b) a formulation of
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L. Bertazzi and M. G. Speranza
the inventory cost in the intermediate nodes is given for the general case and several particular situations have been considered. Two types of sequences can be identified: The Multiple decision-makers sequences and the One decision-maker sequences. In the first case, each node s, s = 1,..., \S\ — 1, has a decision-maker and the decisions are hierarchically ranked. The sth decision-maker, given the frequencies selected on the previous link (s — l,s), has to determine for each product i e / the shipping frequencies which minimize the sum of the inventory cost in the node s and the transportation cost in the link (s, s + 1). In this problem, the optimal solution can be found by optimizing separately each link; for the optimal solution of each link, a mixed integer linear programming problem has been proposed in Bertazzi and Speranza (1996). In the One decision-maker sequences, a unique actor has to determine, for each link (s, s + 1) and for each product i e /, the shipping frequencies that minimize the sum of inventory and transportation costs on the network. In the following remark we state that the inventory during the time horizon H is constant and equal to the sum of the starting inventory at time 0 on all nodes in the networks (d?, s G S).
Remark 3 In the One decision-maker sequences the inventory of each product i £ / in each time instant r 6 [0, H) is equal to £]ses
CONCLUSIONS In this paper, we summarized the results obtained for the analysis of periodic shipping strategies. Moreover, we gave some theoretical contributions, by presenting in particular
Periodic shipping strategies for minimization of logistic costs
235
a unifying framework for the two known models for the single link case. For the approach with given frequencies, further research can be carried out with the following aims: Find good heuristics for the case in which the transportation capacity is different at different frequencies; reduce the total cost by removing some hypotheses, as the Phasing of all frequencies in zero assumption or the Fixed quantity assumption.
APPENDIX In this appendix we present a formal proof of the propositions stated in the previous sections.
Lemma 1 For each time instant t, t G T = {0, 1, . . .}, the following equality holds: (19)
j. k&Kt j&Jk
je.7
Proof We prove the Lemma by induction. In t = 0, given the hypothesis of Phasing all frequencies in 0, JQ = J; therefore, equality (19) holds. Suppose the equality holds in t. In t + 1 we have i+l
t
v = Z) £ *jftzv + Z tjWii = ZX1 + [tltjNfl&ij
+ J
The set J can be written as J = Jt+i U Jtc+1 and the last expression as a;v + E *>
The first term can be written as Ej6.71+1(l+ [(*+!) A; J)*jftaJy- In fact, given that t belongs to the interval [t/tjfa < * < ( ! + \t/t-\ }tj, t + 1 is multiple of tjift=(l+ [t/tj\ )tj - 1; therefore, as j e Jt+i, 1+ \t/tj\ = (t + l ) / t j . Given that t + l is multiple of tj, (t + l } / t j = In the second term, given that j 6 Jtc+1, then [*/*;J be written as Z);e./tc t (l + L(*
=
L(* + 1)A;J- Therefore, it can
Proposition 1 T/ie minimum starting inventory in A at time 0 which satisfies the stock-out constraints (11) is df = ^t^Xij. je.7
Proof The stock-out constraints (11), given the result stated in Lemma 1 and taking ixij can be written as follows:
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These constraints are satisfied as f - Y L*/*jJ *ixij > * - Y, t/tjt&ij = 0
teK.
Moreover, these constraints are satisfied at equality in 0; therefore, df* is the minimum starting inventory. D
Proposition 2 The minimum starting inventory in B at time 0 which satisfies the stock-out constraints (12) is zero. Proof The stock-out constraints (12), given the result stated in Lemma 1 and taking df = 0 can be written as follows: 5^(1+ \t/tj\ }tjXij -t>Q
teK.
These constraints are satisfied as .76.7
je.7
REFERENCES Anily, S., and Federgruen, A. (1990), One Warehouse Multiple Retailer Systems with Vehicle Routing Costs, Management Science 36, 1990, 92-114. Bertazzi, L. and Speranza, M.G. (1996), Minimization of Logistics Costs on Sequences of Links, in the Proceedings of the 4th IFIP WG7.6 Working Conference, Noisy-leGrand, May 28-30, 1996. Bertazzi, L. and Speranza, M.G. (1997a), Inventory Control on Sequences of Links with Given Transportation Frequencies, to appear in International Journal of Production Economics. Bertazzi, L. and Speranza, M.G. (1997b) Heuristic Algorithms for a Transportation Problem with Given Frequencies on a Sequence of Links, Tecnical Report 127, University of Brescia, Italy (submitted). Bertazzi, L., Speranza, M.G., Ukovich, W. (1995), Exact and Heuristic Solutions for a Shipment Problem with Given Frequencies, Techical Report 101, University of Brescia, Italy (submitted). Bertazzi, L., Speranza, M.G., Ukovich, W. (1996), An Algorithm for the Transportation Problem with Given Frequencies, In: System Modelling and Optimization, (J. Dolezal and J. Fidler, eds.), pp. 535-542, Chapman & Hall. Bertazzi, L., Speranza, M.G., Ukovich, W. (1997), Minimization of Logistic Costs with Given Frequencies, Transportation Research B 31, 327-340. Blumenfeld, D.E., Burns, L.D. Diltz, J.D., and Daganzo, C.F. (1985), Analyzing Tradeoffs between Transportation, Inventory and Production Costs on Freight Networks, Transportation Research, 19B, 361-380.
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Burns, L.D., Hall, R.W., Blumenfeld, D.E., and Daganzo, C.F. (1985), Distribution Strategies that Minimize Transportation and Inventory Cost, Operations Research 33, 469-490. Hall, R.W. (1985), Determining Vehicle Dispatch Frequency when Shipping Frequency Differs among Suppliers, Transportation Research 19B, 421-431. Jackson, P.L., Maxwell, W.L. and Muckstadt, J.A. (1988), Determining Optimal Reorder Intervals in Capacitated Production-Distribution Systems, Management Science 34, 938-958. Maxwell, W.L. and Muckstadt, J.A. (1985), Establishing Consistent and Realistic Reorder Intervals in Production-Distribution Systems, Operations Research 33, 13161341. Muckstadt, J.A. and Roundy, R.O. (1993), Analysis of Multistage Production Systems, In: Handbooks in Operations Research and Management Science (Graves, S.C., Rinnooy Kan, A.H.G. and Zipkin, P.H., eds.), Vol. 4, pp. 59-131, NorthHolland. Speranza, M.G. and Ukovich, W. (1992), A Decision Support System for Materials Management, International Journal of Production Economics 26, 229-236. Speranza, M.G. and Ukovich, W. (1994a), Analysis and Integration of Optimization Models for Logistic Systems, International Journal of Production Economics 35, 183-190. Speranza, M.G. and Ukovich, W. (1994b), Minimizing Transportation and Inventory Costs for Several Products on a Single Link, Operations Research 42, 879-894. Speranza, M.G. and Ukovich, W. (1996), An Algorithm for Optimal Shipments with Given Frequencies, Naval Research Logistics 43, 655-671.
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16
AN ALGORITHM FOR THE COMBINED DISTRIBUTION AND ASSIGNMENT MODEL Jan T. Lundgren Department of Mathematics, Linkoping University S-581 83 Linkoping, Sweden Michael Patriksson Department of Mathematics, Chalmers University of Technology 96 Goteborg, Sweden
ABSTRACT We consider the problem of simultaneously determining the distribution of trips between origins and destinations in a transportation network and the assignment of trips to routes in each origin-destination pair. We consider a model for such a problem where the distribution follows a gravity model and the assignment a user equilibrium model. The most well-known algorithm for this model is that of Evans (1976). Although this algorithm has been shown to be more efficient than the Frank-Wolfe method, which it generalizes, it builds the sequence of link flow solutions based on the same underlying algorithmic principle, and therefore also is subject to slow convergence. We propose to combine Evans' approach with the disaggregate simplicial decomposition (DSD) algorithm for updating the link flows; this leads to faster convergence, as well as other improvements. For example, the algorithm is less sensitive to inexact solutions of the entropy maximization subproblems; it further provides explicit route flows in each of the origin-destination pairs. Numerical results are presented for four small and medium scale network models.
1 INTRODUCTION We consider the problem of simultaneously determining the distribution of trips between origins and destinations in a transportation network and the assignment of trips to routes in 239
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J. T. Lundgren and M. Patriksson
each origin-destination (OD) pair. We consider a model for such a problem, first formulated by Evans (1973 and 1976), where the distribution follows a gravity model with negative exponential deterrence function, and the assignment follows a classical user equilibrium model, as described by Wardrop (1952) and formalized in a mathematical programming form by Beckmann et al. (1956). Evans (1976) presents an iterative algorithm for this combined distribution and assignment (CDA) problem. One iteration proceeds as follows. Given a feasible demand and link flow, together with the associated link travel costs, a shortest route and its cost is computed for each OD pair. Given these costs, a doubly constrained entropy maximization problem is then solved to obtain an auxiliary OD flow, and an auxiliary link flow is obtained by assigning the auxiliary demands onto the shortest routes. A one-dimensional minimization is then performed on the line segment between the current and the auxiliary demand and link flow solutions, resulting in a new feasible solution, and the procedure is repeated until the solution is near-optimal. Evans' algorithm generalizes the method of Frank and Wolfe (1956) applied to the CDA problem, in that the auxiliary problem is the result of an approximation of the objective function by means of a partial linearization only, and has been observed to be more efficient (e.g., Frank, 1978; and LeBlanc and Farhangian, 1981). The difference in the behaviour of the two methods is, however, often rather small, for reasons that will be explained in more detail in the next section, and Evans' algorithm therefore also suffers from slow convergence. As a means to improve the convergence rate, we propose to combine Evans' algorithm with the disaggregate simplicial decomposition (DSD) method of Larsson and Patriksson (1992), which was originally proposed for the solution of user equilibrium models. In this algorithm, all the routes generated in the shortest route computations are retained and, instead of as in the Frank-Wolfe and Evans algorithms rerouting flow towards one route flow pattern only, it selects the optimal flow among all possible combinations of route flows, given the set of routes generated. Its established efficiency in applications to user equilibrium models suggests the possibility of a successful application also to the CDA model. The rest of the paper is organized as follows. In the next section, we present the CDA model, describe Evans' algorithm, and discuss some its characteristics. In Section 3, the new algorithm is presented. Numerical experiments conducted with the algorithm are reported in Section 4. The paper is concluded with a summary of our experience with the method, and a discussion on possible extensions of it to more general models. 2
THE COMBINED DISTRIBUTION AND ASSIGNMENT MODEL
The urban area under study is represented by a network Q = (J\f,A). The set M of nodes contains subsets corresponding to the origin zones (denoted by P) and destination zones (Q), in addition to the nodes representing the road intersections. Each (directed) link a € A is associated with a generalized travel cost, c a (/ a ), representing the disutility of using link a as a function of its flow /a. This cost may include several additive components, perhaps most importantly the travel time on the link, and it is assumed that, as a result of congestion, ca is a positive, continuous and strictly increasing function of the flow on link a. Between a subset of nodes in T and Q there is a potential demand for transportation; for notational simplicity we will however assume that all pairs (p, q) e P x Q define potential
An algorithm for the combined distribution and assignment model
241
OD pairs. The (marginal) total flow emanating from an origin node p 6 T is known, and equals Op\ correspondingly, the (marginal) total flow terminating at a node q G Q equals Dq. For each OD pair (p, q), we denote the set of simple (loop-free) routes from p to q by Tlpq—this is a set which in general is not known explicitly—and the flow on route r from p to q by hpqr. By defining a link-route incidence matrix (6pqra) for Q, that is, 5pqra = 1 if route r e Jlpq contains link a, and Spqra = 0 otherwise, link flows may be calculated from route flows according to the relation /a
=
/ ., / .,
/ .,
Opqranpqr.
We assume that the assignment of travellers to routes in each OD pair (p, q) satisfies the user equilibrium conditions of Wardrop (1952), hpqr > 0
=3-
Cj,qr = 7Tpq,
W € 7?,pg,
(2.la)
where cpqr — ^2ae^Spqraca(fa) is the cost on route r, and Trpg is the least route cost; the Wardrop conditions (2.1) state that only the routes of least cost are used. We also assume that the distribution of travellers between OD pairs follows a gravity model with negative deterrence function, according to dpq — apOpbqDqexp(-^Trpq),
p <E P,
q e Q,
(2.2)
where 7 > 0 is the dispersion parameter, and ap and bq are positive balancing factors, determined so that (2.2) satisfies the marginal total constraints. The combined distribution and assignment model can be formulated as the following convex optimization problem:
[CDA] /•/a
v* := min T(d, /) = E / nfA
c
1
i(s) ds + ~ E E dpq\ogdpq
®
(2.3a)
T
subject to 5J hpqr
hpqr
=
dpq,
> 0,
E E E <W./»,*r = A.
Vp G 7^,
V*? G Q,
Vr e ftpg, Vp € P, V
«eA
(2.3b)
Vg e Q,
(2.3e)
(2.3f)
This formulation was first proposed by Evans (1973 and 1976), combining works of Beckmann et al. (1956), Murchland (1966) and Wilson (1967). Its optimal solution satisfies the conditions (2.2) for a gravity model, as well as the user equilibrium conditions (2.1), for all OD pairs. In an application, the value of the parameter 7 must be calibrated through some estimation procedure (e.g., through a maximum likelihood estimation); this
242
/. T. Lundgren and M. Patriksson
estimation issue has been considered by Erlander et al. (1979), Fisk and Boyce (1983) and Boyce et al. (1983). Evans (1973 and 1976) proposes and establishes the convergence of a solution algorithm for the problem; the algorithm is described in Table 2.1.
Table 2.1: Evans' algorithm
0. (Initialization): Choose an initial feasible solution (d°,/°), and let k = 0. 1. (Search direction generation): a. Given are the fixed link costs c a ( f k ) . For each OD pair (p, #), compute a shortest route and its corresponding route cost, Trkq. b. Solve the doubly constrained entropy maximization problem of minimizing T(d) := EperEqeQ(rfgdpg + ±dpg\ogdpq) subject to (2.3c)-(2.3d) and d > 0. The resulting auxiliary demand is d . Assign this demand onto the shortest _ f, routes computed in a; the resulting auxiliary link flow is / . The resulting search direction is (d , / ) -
(dk,fk).
2. (Line search): Solve min/e[0>i]{T[(d*, /*) + i f f - dkjk - /*)]}. The resulting step length is 1^. 3. (Update): Let (d*+1, / fc+1 ) := (dkjk) + £k(dk - dkjk - fk). Let Jfc := fc + 1. 4. (Termination criterion): If some termination criterion is satisfied, then stop. Otherwise, go to Step 1.
Remark 2.1 (Notes on Evans' algorithm) (i) An initial OD matrix satisfying (2.3c)(2.3d) and d > 0 can be determined through a simple feasibility heuristic for linear transportation problems, or be based on a predicted or historical OD matrix. An initial feasible link flow can be given by an all-or-nothing assignment of this initial OD matrix based on the link costs at zero flow, c a (0), a G A. (ii) The subproblem in Step 1 is the result of a partial linearization of the original objective. The problem in l.b is traditionally solved using a dual algorithm, which can be interpreted as a balancing procedure with respect to the marginal constraints (2.3c)-(2.3d); see, for example, Lamond and Stewart (1981). Being a dual algorithm, the procedure does not provide a feasible demand matrix if the problem is not solved exactly. The accuracy by which this subproblem is to be solved is a crucial point in the implementation of the algorithm, since errors made in the calculation of the demand vector result in a primal infeasibility, which will accumulate both in the demand and link flow vectors.1 l
A.n infeasible OD matrix calculated in the balancing procedure can be adjusted to a feasible one by
An algorithm for the combined distribution and assignment model
243
(iii) In an implementation of the Frank-Wolfe algorithm, the subproblem in l.b is a linear transportation problem; see Florian et al. (1975) for an application to the CDA model. (iv) The solution to the subproblem provides a lower bound, T_(d ), on the optimal value v* of [CDA]; this bound is strictly better than the corresponding one for the Frank-Wolfe algorithm. A lower bound is also obtained when the subproblem is solved inexactly, if it is defined by the dual functional value at termination. (v) Feasible solutions to [CDA] provide upper bounds on v*, and hence an interval of values, the (relative) length of which, [T(fk+\dk+l)-T(dk)}/\T(dk)\,
(2.4)
can be utilized in a termination criterion in Step 4.
D
As noted above, the approximation utilized in the construction of the auxiliary solution in Evans' algorithm is better than that of the Frank-Wolfe algorithm, since only the term associated with the link flows is linearized. Notwithstanding this principal improvement, the two algorithms suffer from a similar, slow asymptotic convergence behaviour, which manifests itself in a rapidly diminishing step length l^. The reason for this is the following. Most of the formulas in use for the travel costs are such that the first term in the objective function T of [CDA] is highly nonlinear, as compared to the second term of T; to the relatively flat shape of the second term contributes the fact that d is at least an order of magnitude smaller than / in general. Since both algorithms use a linear approximation of the highly nonlinear term and update the link flow in the same manner in a simple line search, both algorithms suffer from slow convergence; moreover, Evans' algorithm behaves similarly to the Frank-Wolfe algorithm, since the second term of T is comparatively unimportant in the minimization of Step 2. Based on this observation, improvements have been made, especially in the line search step. Horowitz (1989) proposes to perform the line search in the link flows only, and Huang and Lam (1992) analyze the convergence characteristics of this heuristic. This algorithm is however based on the same updating formula, and the improvement over Evans' algorithm therefore minor. The unfavourable properties of Evans' algorithm discussed above have lead us to consider utilizing developments made in recent years in the computation of user equilibrium flows in the construction of more efficient and robust methods for the CDA model. The next section is devoted to the description of one such scheme.2
3
A COLUMN GENERATION ALGORITHM FOR CDA MODELS
3.1
Disaggregate simplicial decomposition
The disaggregate simplicial decomposition (DSD) algorithm was developed by Larsson and Patriksson (1992) for the solution of the traffic assignment problem
[TAP] ffa
minimize T(f)= V /
ca(s) ds,
(3.la)
°
means of a primal feasibility heuristic for network flows (e.g., Marklund, 1993), which however, has not been considered so far. 2 An alternative approach, provoked by the above discussion, is to utilize a nonlinear approximation of the first term of [CDA]; this is, however, left as a possible topic for future research.
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J. T. Lundgren and M. Patriksson
subject to 53 hpqr = dpq, V ( p , 9 ) e ^ x Q , re-ftp, hpqr > 0, Vr e npq, V(p, q)ePxQ, 53
H Spgrakpgr
=
/«,
Vtt £ A
(3.1b) (3.1c) (3.1d)
(p,9)ef xQ re-ftp, Suppose that, at some iteration &, subsets 7?.p? of 7^pg are known, and fk solves the restriction of [TAP] obtained when replacing TLm with 7£* . Given this link flow, the shortest routes are calculated for all OD pairs. If all the routes thus generated are included in the corresponding subsets T^kq already, then the current link flow is optimal, since (2.1) then is fulfilled. Otherwise, each new route obtained is added to its corresponding subset Hpq, and a new, larger, restriction is defined; solving this new restriction (approximately) yields a solution fk+l, and the process is repeated with k := k + 1. Convergence of this scheme is finite in the number of restricted problems solved, and the output is a set of routes with flows satisfying the user equilibrium conditions (2.1). See Larsson and Patriksson (1992) and Patriksson (1994) for further details on this and other column generation and simplicial decomposition algorithms for [TAP]. Numerical experiments performed with this method on many known test networks have established that it is robust and much more efficient than the Frank-Wolfe algorithm, while also being at least competitive with state-of-the-art codes for [TAP]. It also has very favourable reoptimization capabilities; thanks to the route information stored, it is easy to reoptimize [TAP] if some model components should change, such as the travel cost functions or the demand matrix. The DSD algorithm was, for these reasons, considered for use in solving [CDA]. 3.2
The new algorithm
The combination of the DSD algorithm with Evans' algorithm works as follows. Assume that, at iteration k, subsets T^kpq are known, and that (dk, fk) is the current iterate. Solving the same subproblem as in Evans' algorithm yields a set of shortest routes and an auxiliary demand, d . Instead of assigning this demand matrix onto the shortest routes and performing a line search, we continue by storing those routes generated that were not stored already. Each new route, say r e 7£pg, receives a designated variable, Xpqr, which denotes the portion of the total demand of OD pair (p, q) that is allocated to route r. The relation hpqr = Xpqrdpq can be used as the definition of the route flow given the weights A and current demand. (Choosing this variable definition in place of the route flow variable hpqr has the advantage that the value of Xpqr needs not be scaled when the demand changes.) The algorithm operates as a column generation algorithm only in the link flow space. The restricted master problem, generalizing that of the DSD algorithm for [TAP], will hence solve the original problem [CDA] over the same line segment in the demand space as Evans' algorithm, while simultaneously optimizing the objective over all possible combinations of weights among the routes currently stored. The restricted master problem has the following form:
[RMP]
,fa := min T(d, /) = E /
i c
"(s) ds + ~ E £ dPi l°SdPi
(3.2a)
An algorithm for the combined distribution and assignment model
245
subject to £ \pqr = 1, Vp 6 P,
Vg e Q,
(3.2b)
re**,
Apgr
£ Z H 5pqra\pqrdpq
> 0,
Vr €
= /Q,
ftj,,
Vp e 7>,
Vg € Q,
Va e .A,
(3.2c)
(3.2d)
perq£Qr£K$q
dpq
= dkq + e(dkpq-dkq),
VpeP,
V
I e [0,1].
(3.2e)
(3.2f)
—fe
Provided that d and d are feasible, the value of vk is an upper bound on v*. The solution to this problem defines the new iterate (dfc+1,/*+1), and if the algorithm is not terminated due to some termination criterion being satisfied, then the process is repeated, with k := k + 1. We summarize the algorithm in Table 3.1. Table 3.1: The Evans/DSD algorithm
0. (Initialization): Choose initial sets 7lpq C 7^pg, p € P, q £ Q, a feasible solution (d°, /°) consistent with the sets 7£°g, and let k = 0. 1. (Subproblem): a. Given are the fixed link costs ca(fk). For each OD pair (p, g), compute a shortest route and its corresponding route cost, Trkq; if the shortest route, r 6 7£pg, is not contained in Kkq, then let ftjg := Hkq U {r}. b. Solve the doubly constrained entropy maximization problem of minimizing T(d) = £p67>£,6e«dM + ^dpqlogdpg) subject to (2.3c)-(2.3d) and d > 0. The resulting auxiliary demand is d . 2. (Restricted master problem): Solve [RMP]. Let the solution be (dk+l, / fc+1 ). 3. (Update): Let k:=k + l. 4. (Termination criterion): If some termination criterion is satisfied, then stop. Otherwise, go to Step 1.
Remark 3.1 (Notes on the Evans/DSD algorithm) (i) In Step 0, an initial set of routes can be defined by the routes generated when performing an all-or-nothing assignment given zero flows. (ii) Step 1 is essentially identical to that of Evans' algorithm, with the exception that routes generated are stored if not known already, and that a one-dimensional search direction is only defined in the demand space. (iii) Because the number of routes in the network is finite, the column generation phase in Step l.a will finitely cease to generate new routes.
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(iv) In contrast to Evans' algorithm, the availability of explicit route information enables the use of termination criteria in Step 4 based on detailed measures of the failure of a solution to satisfy Wardrop's conditions (2.1). D Remark 3.2 (Primal feasibility at termination) The algorithm generates route flows explicitly, in contrast to Evans' algorithm which only provides link flows. The consequences of this fact are several. In contrast to Evans' algorithm, errors in the computation of the demand subproblem solution are not transfered to the link flows, since they are defined implicitly by the weights Xpqr whose values are calculated independently of the demands. As an example of the consequences of this fact, consider a scenario in which we have a near-optimal, but primal infeasible, demand matrix d fc , and wish to terminate the algorithm with a pair of nearoptimal but primal feasible vectors (d,f). We then first either solve the subproblem of Step l.b exactly (if possible) or apply a conservative, primal feasibility heuristic on dk to convert it into a (still near-optimal) feasible matrix, d. Fixing this feasible demand matrix in the restricted master problem (which then is equivalent to a master problem for a traffic assignment problem) then yields a near-optimal and primal feasible set of route and link flows. Evans' algorithm can not perform this task, for the simple reason that the link flow solution / fc , which is the only flow information stored, is contaminated by the errors made in all the previous computations of demand matrices. An immediate effect of this is that the subproblem Step l.b can be solved with almost arbitrary accuracy throughout the algorithm, since primal feasibility can be restored at any iteration. D Remark 3.3 (Schittenhelm's algorithm) Schittenhelm (1990) presents an algorithm for the CDA problem which is related to ours. He improves upon Evans' algorithm by —k replacing the auxiliary link flow solution / , which in her algorithm is an all-or-nothing flow, by the link flow solution obtained when solving [TAP] with the demand matrix fixed at d . This subproblem is solved using a method similar to the DSD algorithm. However, he only utilizes the flow solution obtained in a one-dimensional search as in Evans' algorithm, whereas we make use of all information generated. As such, our algorithm can be expected to converge faster. Also, errors in the solution to the demand subproblems propagate in the link flow solution in Schittenhelm's algorithm as in Evans' algorithm. d Although the restricted master problem [RMP] is more difficult to solve than the onedimensional problem in Evans' algorithm, the Evans/DSD method is expected to converge in much fewer iterations; further, each new restricted master problem is efficiently reoptimized starting from the solution to the former master problem. It still needs to be shown that the much fewer iterations needed in the Evans/DSD method amortizes the additional work performed in each iteration due to the more complex updating steps. This is the purpose of Section 4. 3.3
Solution of the restricted master problem
The problem [RMP] is solved with a coordinate-wise (Gauss-Seidel) search, in which alternately the vector A and the step length t is fixed at its current value while optimizing with respect to the other variable(s). This scheme is globally convergent, since the problem is strictly convex in / and £, and the respective feasible sets are nonempty, convex and bounded.
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When fixing I, that is, when fixing the OD matrix, the problem [RMP] reduces to a traffic assignment problem. For solving this problem, we utilize the same method as for the restricted master problem in the implementation of the DSD method by Larsson and Patriksson (1992), that is, a method based on reduced gradients combined with a Newton method, where line searches are performed using the Armijo step length rule. When fixing A, that is, when fixing the portion of the demands to allocate to each route, [RMP] reduces to a simple one-dimensional problem in the demand space. Preliminary tests indicated that golden section search was better than the Armijo step length rule for this problem, and was subsequently used in our implementation. The iterate dk and the auxiliary demand matrix d define an interval in which an improved matrix is sought. In the first iteration of the solution of [RMP], this matrix is found in a sufficiently smaller interval than the original one; in latter iterations, the interval of search is from the current matrix to one of the endpoints of the original interval, according to the sign of the derivative.
4 4.1
NUMERICAL EXPERIMENTS Test networks and implementational considerations
The starting point for creating test networks for our experiments was four networks used previously in applications of [TAP]. Their sizes are found in Table 4.1. Table 4.1: Traffic network models. Network Sioux Falls Hull Winnipeg Barcelona
# Nodes # Links 24 76 501 798 1052 2836 1020 2522
# OD pairs 7 528 1.0 142 1.0 4344 0.1 79,.'; 0.2
To create instances of [CDA], we used the marginal totals corresponding to the respective network's fixed demand matrix. (Hence, the fixed demand matrix is a feasible, but most probably non-optimal, demand matrix.) The initial solution used in our experiments is the matrix resulting from using the balancing procedure on a problem where the link costs are uniform (equal to 1). (From our experience, different choices of initial matrices did not result in a much different behaviour of the algorithms.) The value of the dispersion parameter 7 was not available, but was instead calculated, for each network, such that the respective optimal demand matrix has a structural similarity to the original, fixed demand matrix. Their respective values are found in Table 4.1. The Evans/DSD and Evans algorithms were coded in Fortran-77, compiled using the -fast switch, and executed on a SUN Ultra SPARC 170E. The main termination criterion used was that of a low relative objective error (2.4); it was checked twice during each main iteration, namely after the solution of the balancing problem, when the lower bound is updated, and after each restricted master problem (Evans/DSD) or line search (Evans), when the upper bound is updated. Common to the two methods are the solution of shortest route and doubly constrained entropy maximization subproblems. The shortest route calculations were performed using the Iqueue implementation of Dijkstra's algorithm (the same as the one used in the original DSD code). The procedure for solving the demand subproblems is a standard dual
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coordinate search (balancing) procedure (e.g., Lamond and Stewart, 1981). The procedure was terminated when the maximal relative difference between any left- and right-hand side in the marginal total constraints fell below 10~5. For any fixed value of t in the restricted master problem in the Evans/DSD algorithm, we have implemented three methods; a reduced gradient algorithm, a diagonalized Newton method, and a combined method which switches from the former to the latter when the relative error (2.4) in the objective falls below 1%. The line searches in these methods were performed using the Armijo step length rule, with acceptance parameter 0.2. More details on these methods are found in Larsson and Patriksson (1992). Initial tests performed with the Winnipeg network on the Evans/DSD algorithm showed that each restricted master problem should be solved rather roughly only; a generally good strategy is to perform only a few iterations of the Gauss-Seidel procedure, and, for a fixed value off, utilize only a few steps of the procedure for the fixed demand problem, regardless of which of the three methods were used. Within reasonable limits of these choices, we found that the total computational effort was roughly constant, so the proper choice is not very crucial. (More work performed on [RMP] results in less main iterations, and hence less work performed on the shortest route and balancing calculations.) In all the numerical experiments reported we have applied 2 iterations of the Gauss-Seidel procedure per main iteration, and 5 iterations of the fixed demand algorithm in each of these. (Hence, in each restricted master problem, a total of 2 (10) line searches are performed in the demand (link flow) space.) The golden section method was used such that the initial interval is reduced to 10% of the previous one. 4.2
Numerical experience
Detailed experiments with the Winnipeg network. To illustrate the performance of the Evans/DSD algorithm compared to Evans' algorithm, we first present results for the algorithms applied to the Winnipeg network. In Evans/DSD, we used the combined version for the restricted master problems. At each iteration of the two algorithms, we recorded the respective lower and upper bounds on the optimal value, and CPU times. In Figure 1, upper and lower bounds are plotted against iteration, while Figure 2 plots the bounds against CPU time (in seconds). In both figures, dashed (dotted) lines correspond to the Evans/DSD (Evans) algorithm. The appearance of Figure 1 is the expected one, since more effort is spent in each iteration of the Evans/DSD algorithm. Figure 2 is therefore more revealing of the performance of the algorithm, which shows that it is indeed more efficient than Evans' algorithm. The reader should note that in the Evans/DSD method, upper bounds are generated throughout the procedure for [RMP]; only the last iterate in each such problem is reported in the figures, however. (This explains why Evans' algorithm, from Figure 2, appears to generate better upper bounds initially.) An experiment was made to investigate the computational effort of the algorithms when demanding different levels of accuracy. In Table 4.2, we report the number of iterations and CPU time needed to solve the Winnipeg network model with the three versions of the Evans/DSD method, and with Evans' method. In this case, the three versions of the Evans/DSD algorithm have similar behaviour. It is also clear that Evans' algorithm is efficient in the first few iterations, but much less so when approaching higher accuracies. Note that in the Evans/DSD algorithm, the accuracy
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I" 2.65
Figure 1: Upper and lower bounds versus iteration.
Figure 2: Upper and lower bounds versus CPU time.
Table 4.2: Computational effort for obtaining various accuracies. Rel ace
10% C 07
1% 0.5% 0.1 %
Red Iter 1 2 3 4 7
Evans/DSD Diag Newton Combined Iter CPU Iter CPU 1 1 1.5 s 1.5s 1.8 s 2.8s 2 2 2.7s 3.3s 4.1 s 2 3.3s 3 4.1 s 4 5.8s 5.5s 3 5.0s 9.9s 5 9.0s 5 8.0s
grad CPU
Evans Iter CPU 3 1.9s 4 2.4s 10 5.3 s 16 8.1 s 42 20.6s
level required is reached earlier than the times reported, since the termination criterion is only checked at termination of the restricted master problems. When solving to the highest accuracy reported in the table, in the Evans/DSD algorithm, the portions of the total time spent on the shortest route, gravity and restricted master problems were (on average over the three method versions), roughly, 25%, 10%, and 75%, while the corresponding figures for Evans' algorithm were 65%, 20%, and 15%. The Evans/DSD algorithm generated, on average, 14,700 routes, of which 9,800 were actually used. (By comparison, Evans' algorithm generated 24, 700 routes during the execution.) An experiment was made wherein the balancing was made to a lesser accuracy. As indicated earlier, Evans' algorithm should be more sensitive to the errors generated through the inexact computations. This was indeed observed: when the accuracy was decreased from 10~5 to 10~3, Evans' algorithm was terminated prematurely due to numerical problems associated with the generation of search directions at infeasible points. As was noted in Section 2, the objective function is relatively flat with respect to the demand variables. A possible consequence of this fact is that a small difference between the upper and lower bounds on the optimal value may wrongly indicate near-optimality of the link flows and demands. Therefore, we also investigated the behaviour of the two algorithms with respect to their convergence in terms of the link flows and demands. Using the combined algorithm version in the Evans/DSD algorithm, and the same algorithm parameters in both methods as in the experiments above, we calculated the Euclidean
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distance from each link flow and demand iterate to the corresponding optimal solution vector. (The "optimal solution" is, in fact, a very accurate solution obtained from the use of the Evans/DSD algorithm, executed with a tolerance in relative objective error (2.4) set to 10~7.) For illustration purposes, we terminated the two algorithms after a longer period of time (25 CPU seconds) than in the previous experiment (which resulted in a relative objective error (2.4) of 2.0 • 10~5 for the Evans/DSD algorithm and 5.0 • 10~4 for Evans' algorithm). In Figure 3, we plot the Euclidean distance from a link flow iterate, normalized by the Euclidean length of the optimal link flow vector, against the CPU time (in seconds). Figure 4 contains the corresponding information for the demand iterates. (As in Figures 1 and 2, dashed (dotted) lines correspond to the Evans/DSD (Evans) algorithm.)
Figure 3: Relative deviation to optimal link flows versus CPU time.
Figure 4: Relative deviation to optimal demands versus CPU time.
The figures show that the Evans/DSD algorithm converges faster than the Evans algorithm with respect to link flows and OD demands; they also indicate that, not surprisingly, convergence is faster in the demand space for both methods. We remark that for any given level of accuracy [defined in terms of the relative objective error (2.4)], the deviations from optimality in the link flow space were similar in the two methods, and the same is true for the demands. To complement Figures 3 and 4, for both methods we also calculated the maximal deviation from the optimal flow among the links, and from the optimal demand among the OD pairs; these values are shown in Figures 5 and 6. The result is in many ways similar to that of Figures 3 and 4, with the exception that the deviation fluctuates more in Evans' algorithm (although this can partly be explained by the fact that the value of the deviation is registered more often in Evans' algorithm than in the Evans/DSD algorithm), and, more importantly, that the maximal deviation from the optimal solution is, relatively speaking, even higher in Evans' algorithm than in the proposed method, compared to the measure used in Figures 3 and 4. To assist in the interpretation of Figures 5 and 6, we remark that the average equilibrium flow on a link is about 450, while an average demand is around 15; thus, from Figure 5, we can note that at termination, the maximal deviation in the link flow is in the order of 10 % in the Evans/DSD algorithm while it is in the order of 30 % in Evans' algorithm; in the case of OD demands, the corresponding figures, taken from Figure 6, are 4 % and 50 %, respectively. Clearly, convergence is not only faster in the Evans/DSD algorithm, but it is also more stable.
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Figure 5: Maximal deviation to optimal link flows.
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Figure 6: Maximal deviation to optimal demands.
An important conclusion that can be drawn from the above tests is that the relative objective error (2.4) can be used as a convergence measure in practice, although the tolerance chosen must be orders of magnitude smaller than the average or maximal errors accepted in the flows and demands; we base this conclusion on the observation that the difference in magnitude between the relative objective error presented in Figures 1 and 2 and the deviations from the optimal solution presented in Figures 3-6 is roughly the same, regardless of the accuracy level studied, that is, the relative error and the deviation from the optimal solution decreases to zero at about the same rate. Further experiments. With the same parameter settings as in the initial experiment, we ran the other three network models, with the results shown in Table 4.3. Table 4.3: Computational effort for different network models. Network Sioux Falls Hull Barcelona
Evans/DSD Evans Red grad Diag Newton Combined Iter CPU Iter CPU Iter CPU Iter CPU 1.3s 13 0.7s 1.2s 9 0.6s 104 0.1 % 16 4 0.2s 0.7s 5 0.3s 4 0.2s 48 0.1 % 24 10.9s 7 7.0s 0.1 %
Rel ace
With regards to the results for Evans' algorithm when applied to the Sioux Falls network model, an important comment must be made. In order to reach the accuracy demanded, we were forced to increase both the accuracy of the line search (so that the length of each consecutive interval was 1% of the former) and of the balancing procedure (to a relative accuracy of 10~7). This is due to the combined effect of generating search directions with small directional derivatives and the propagation of errors in the computations. The Barcelona network model is extreme, in that the network really is capacitated; the polynomial travel time formulas reflect this by being flat for moderate flows and highly nonlinear (with powers in the 10-20 range) for large flows. It was the experience in an earlier work (Larsson and Patriksson, 1992) that second-order methods did not work very well on this model, and our conclusion is the same. The (first-order) reduced gradient
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method worked well, as did Evans' algorithm, but the two second-order ones did not converge; the combined algorithm will in the near future be modified so that the secondorder option is never used for such problems.
5
CONCLUSIONS
The experiments conducted with the proposed algorithm show that it is an interesting alternative to Evans' algorithm, and that it indeed has some advantages over it. For example, we encountered problems with numerical instabilities with Evans' algorithm, while the most robust version of the Evans/DSD approach so far—the one using the reduced gradient approach only—always worked satisfactorily. This method was, on average, twice as fast as Evans' algorithm. We then note that the parameter choices in the algorithm was not calibrated for the different networks, whence one most probably can construct a specialized version with a better performance when using it in a practical application. Further, demanding a higher accuracy poses no problems with the Evans/DSD algorithm, while Evans' algorithm tends to suffer from numerical difficulties. We also experimented with different convergence criteria on a sample network, and found that the use of the relative objective error (2.4) seems to be useful in practice, but that it must be used with caution, in particular since there is a large difference (orders of magnitude, in fact) between this measure and the deviation from an iterate to the optimal solution. The CDA model can be extended to include modal split (see e.g. Florian and Nguyen, 1978). Also, there exists a class of models for location, destination and modal split choices in combination with user equilibrium (see, e.g., Boyce et al., 1983) which are founded on the combined model framework of Evans. The proposed algorithm can be applied also to such extended models.
Acknowledgements The authors were sponsored by the Swedish Transport and Communications Research Board (Grants TFB 92-128-63 and KFB 95-118-63).
REFERENCES Beckmann, M., McGuire, C. B. and Winsten, C. B. (1956). Studies in the Economics of Transportation. Yale University Press, New Haven, CT. Boyce, D. E., Chon, K. S., Lee, Y. J., Lin, K. T. and LeBlanc, L. J. (1983). Implementation and computational issues for combined models of location, destination, mode and route choice. Environment and Planning, A15, 1219-1230. Erlander, S., Nguyen, S. and Stewart, N. F. (1979). On the calibration of the combined distributionassignment model. Transportation Research, 13B, 259-267. Evans, S. P. (1973). Some applications of optimisation theory in transport planning. PhD thesis, Research Group in Traffic Studies, University College London, London. Evans, S. P. (1976). Derivation and analysis of some models for combining trip distribution and assignment. Transportation Research, 10, 37-57. Fisk, C. S. and Boyce, D. E. (1983). Alternative variational inequality formulations of the network equilibrium-travel choice problem. Transportation Science, 17, 454-463. Florian, M., Nguyen, S. and Ferland, J. (1975). On the combined distribution-assignment of traffic. Transportation Science, 9, 43-53.
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Florian, M. and Nguyen, S. (1978). A combined trip distribution and modal split and trip assignment model. Transportation Research, 12, 241-246. Frank, C. (1978). A study of alternative approaches to combined trip distribution-assignment modeling. PhD thesis, Department of Regional Science, University of Pennsylvania, Philadelphia, PA. Frank, M. and Wolfe, P. (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3, 95-110. Hearn, D. W., Lawphongpanich, S. and Ventura, J. A. (1987). Restricted simplicial decomposition: computation and extensions. Mathematical Programming Study, 31, 99-118. Horowitz, A. J. (1989). Tests of an ad hoc algorithm of elastic-demand equilibrium traffic assignment. Transportation Research, 23B, 309-313. Huang, H.-J. and Lam, W. H. K. (1992). Modified Evans' algorithms for solving the combined trip distribution and assignment problem. Transportation Research, 26B, 325-337. Marklund, J. (1993). A study of Lagrangian heuristics for convex network flow problems. Master's thesis, Department of Mathematics, Linkoping Institute of Technology, Linkoping, Sweden. Lamond, B. and Stewart, N. F. (1981). Bregman's balancing method. Transportation Research, 15B, 239-248. Larsson, T. and Patriksson, M. (1992). Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transportation Science, 26, 4-17. LeBlanc, L. J. and Farhangian, K. (1981). Efficient algorithms for solving elastic demand traffic assignment problems and mode-split assignment problems. Transportation Science, 15, 306-317. Murchland, J. D. (1966). Some remarks on the gravity model of traffic distribution, and an equivalent maximization formulation. Report LSE-TNT-38, Transport Network Theory Unit, London School of Economics, London. Patriksson, M. (1994). The Traffic Assignment Problem—Models and Methods. VSP BV, Utrecht, The Netherlands. Schittenhelm, H. (1990). On the integration of an effective assignment algorithm with path and path flow management in a combined trip distribution and traffic assignment algorithm. In: Proceedings of the 18th PTRC Summer Annual Meeting on European Transport and Planning. Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. Proceedings of the Institute of Civil Engineers, Part II, 1, 325-378. Wilson, A. G. (1967). A statistical theory of spatial distribution models. Transportation Research, 1, 253-269.
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17 ASSESSING THE PERFORMANCE OF ARTIFICIAL NEURAL NETWORK INCIDENT DETECTION MODELS Hussein Dia and Geoff Rose Institute of Transport Studies, Department of Civil Engineering, Monash University, Clayton Victoria 3168, Australia.
ABSTRACT This paper explores the performance of a relatively new-generation of algorithms for automated freeway incident detection using Artificial Neural Networks (ANNs). These new models have the potential to provide faster and more reliable incident detection times and faulttolerant operation while being easy to implement on existing and new hardware platforms. The ANN incident detection models were trained on data obtained from two freeways in Melbourne, Australia. Two sources of data were used to assemble the training data sets. The first comprised speed, flow and occupancy data from dual-loop detector stations and the second was an incident log showing the approximate time of incidents. This resulted in a database comprising a set of 100 incidents of varying severity and under a variety of traffic conditions at different locations on the freeway. Sixty of these incidents were used for training the ANN model while the remaining forty incidents were used for model testing. The model developed in this study uses speed, flow and occupancy data measured at dual stations, averaged across all lanes and only from one preceding time interval. The off-line performance of the model is reported under both incident and non-incident conditions. The incident detection performance of the model is reported based on a validation-test data set of forty incidents that were independent of the sixty incidents used for training. The false alarm rates of the model are evaluated based on non-incident data that were collected from a freeway section which was video-taped for a period of 33 days. The results presented in this paper provide a comprehensive evaluation of the performance of the ANN model and confirm that neural network models can provide fast and reliable incident detection on freeways.
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INTRODUCTION The high cost of congestion caused by incidents, mainly in terms of traffic delays, air pollution and deteriorated safety conditions, has prompted a growing world-wide interest in developing efficient and effective automated incident detection methods. Incidents are defined as nonrecurring events such as accidents, disabled vehicles, spilled loads, maintenance work and other events that disrupt the normal traffic flow and result in a capacity reduction of a facility. They are believed to constitute about 50-60 percent of the total delays on US freeways (Lindley, 1987). The adverse effects of incidents are also expected to increase as freeway facilities in major cities around the world become more congested. The benefits to be derived from early incident detection and quick response can drastically reduce traffic delays and improve road safety and real-time traffic control. Automatic incident detection systems involve two main components: a traffic detection system and an incident detection algorithm. The traffic detection system provides the traffic information necessary for detecting an incident while the incident detection algorithm interprets that information and ascertains the presence or absence of incidents or non-recurring congestion.
INCIDENT DETECTION ALGORITHMS A number of AID algorithms have been developed from a variety of theoretical foundations. Their structure varies in the degree of sophistication, complexity, data requirements and the type of surveillance technology used for data collection. Some of the most widely used algorithms include the comparative or pattern comparison algorithms, eg. the California-type algorithms (Levin and Krause, 1979) and the ARRB/VicRoads algorithm (Luk and Sin, 1992), the McMaster algorithm (Persaud and Hall, 1989) and the time series algorithms, eg. the AutoRegressive Integrated Moving Average (ARIMA) algorithm (Ahmed and Cook, 1982).
Artificial Neural Networks Few of the previously developed algorithms have been implemented in practice due to various limitations and varying operational levels in terms of performance criteria such as detection rate, false alarm rate and time-to-detect. Therefore, the need is pressing for more effective realtime incident detection algorithms that maximise detection rate while only generating an acceptable level of false alarms. One promising approach involves the application of Artificial Neural Networks (ANNs). Neural Networks, as the name implies, are loosely modelled after the biological structure of the brain. A neural network is constructed from a set of inter-connected simple processing elements (PEs). Each PE performs only a few simple computations such as receiving inputs from other PEs and computing an output value which it sends to other PEs. A neural network is inherently parallel in that many PEs can carry out their computations at the same time. The
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processing ability of the network, stored in the connection strengths or weights, is obtained by a process of adaptation to, or learning from, a set of training patterns. Ritchie and Cheu (1993), demonstrated the feasibility of using ANNs for incident detection. They tested a multi-layer feed-forward (MLF) ANN on a freeway section using simulated traffic detector data. The results confirmed their hypothesis that spatial and temporal traffic patterns could be recognised and classified by ANNs. However, their results were limited in the sense that they trained the ANN models on simulated traffic detector data, used only volume and occupancy data and did not address operational issues such as the impact of detector malfunction and quality of input data on model performance. The work reported here is part of a research program that aims to address many of these unresolved issues.
Performance Measures for Incident Detection Algorithms The performance of an incident detection algorithm is measured by three criteria: detection rate (DR), false alarm rate (FAR) and time-to-detect (TTD). The DR is defined as the number of incidents detected by the algorithm divided by the total number of incidents known to have occurred during the recorded time. The FAR can be defined in different ways depending on whether it is an on-line or off-line FAR. The on-line FAR (FARon) is defined as the number of time intervals (typically provided in 20 or 30-second cycles) which gave false alarms divided by the total number of time intervals. The off-line FAR (FAR^) is defined as the number of incident-free intervals which gave false alarms divided by the total number of incident free intervals. Finally, the TTD is the difference between the time of occurrence and the time at which the incident was declared or an alarm was raised by the algorithm. When an algorithm is being evaluated, however, it is customary to seek the mean time-to-detect (MTTD) a set of (n) incidents. The occurrence time of an incident is usually not known precisely and an estimate has to be deduced from loop detector data or records kept by police, traffic control centres or towing companies. The above definitions clearly show that both the DR and FAR measure the effectiveness of the algorithm while the MTTD reflects its efficiency. The detection rate and false alarm rates are, unfortunately, positively correlated. In order to detect more incidents, the algorithm thresholds are relaxed which causes some incident-free intervals to be interpreted as alarms. The relative importance of the measures, however, is typically DR, FAR and MTTD.
DATA FOR THE DEVELOPMENT OF ANN INCIDENT DETECTION MODELS In order to train a neural network to perform incident detection, the network must be presented with examples of input detector data (speed, flow and occupancy) and output states for both incident and incident-free conditions. Therefore, the data required should at least have a description of the state of traffic along the freeway in addition to detector data comprising traffic flow measurements at regular time intervals for each detector station. In contrast to previous research which has relied on simulated data for model development (Ritchie and Cheu, 1993), this study relies on real data.
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Data Collection The data required for model development in this study were assembled from two data sources held at the VicRoads Traffic Control and Communications Centre (TCCC) in Melbourne, Australia. The first data source comprised information logged by the operators at the TCCC regarding the incidents that occurred on the freeways. This information is received by the operators from a variety of sources including motorists using either private mobile phones or the emergency phones located near the freeway or from the currently operational ARRB/VicRoads incident detection model. The CCTV surveillance system is then used by the operators in the TCCC to confirm the incident and determine its nature. However, when operators are busy managing incidents, that is notifying response teams and dispatching emergency services, it is not uncommon for important details to be left out from the records. This presented some difficulties when examining the incidents since in many cases the location of the incident, its direction or time of occurrence/clearance remained unknown from the TCCC record. A total of 385 incidents that were logged during the period from January 1992 to March 1994 were extracted from this database. The second data source comprised the loopdetector data which consisted of speed, flow and occupancy measurements in 20-second cycles. Each of the data files obtained consisted of the detector data for the study site for that specific day. The detector data collected also included data files comprising incident-free days. Each of the 385 incidents was then examined individually. The log entry for the incident was compared against the detector data to find out if the incident could be detected from the loop data. This was accomplished by monitoring the immediate upstream and downstream stations from the incident using a graphical computer program that was developed to plot the detector data. Out of the 385 incidents recorded by the operators in the log, only 120 incidents could be confirmed from the loop data. The rest either occurred outside the 8.5 km test segment of the freeway or during light flow conditions and therefore had no effect on traffic conditions. Others could not be confirmed due to missing information about the location or time of incident. However, for sixty of these confirmed incidents, the detector data at the upstream and/or downstream stations were faulty or corrupted. These incidents, although confirmed, cannot be used for model development since data of good quality from all lanes in both stations upstream and downstream of the incident need to be used. Therefore, out of the 385 incidents logged by the operators in the TCCC, only sixty were clearly detectable from the detector data and could be confirmed in the operator's log. These incidents will be used for model development in this study.
Creation of Training and Training-Test Data Sets The next activity involved compiling the training and training-test data sets that will be used for training the ANN incident detection models. The training data set will be used for determining the network parameters while the training-test data set will be used to prevent the network from learning the idiosyncrasies in the training data set and thereby enables the model to generalise better. Therefore, the two data sets are essentially used for training the ANN model and are thus referred to as the 'training data sets'. The ANN models should be trained on a set of incidents that are representative of the population to which the network will ultimately
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be applied. Training an ANN model with a wide range of incidents that include different patterns under a variety of flow conditions and traffic periods helps improve the robustness of the model in detecting incidents under varying conditions. Therefore, the data was stratified according to incident severity (in terms of the number of lanes blocked due to the incident), prevailing flow conditions prior to the occurrence of incidents (heavy, moderate and light), traffic period of the day (peak or off-peak) and incident duration. Two data sets, each comprising 30 incidents, were then selected randomly from the sixty incidents to form the master training and training-test data sets. Further details regarding the determination of start and end times of incidents, the creation of the training data sets and the distribution of incidents in these sets can be found in Dia and Rose (1995).
DATA FOR THE VALIDATION OF ANN INCIDENT DETECTION MODELS In addition to the training data sets required for model development, a third data set is also needed for validating the performance of the trained incident detection models. This data set should be independent of the data sets used for model training. In addition, it is desired that the incidents in the validation-test data set be representative of the population to which the network will ultimately be applied. The training data sets described previously were collected in April 1994 for the purpose of training the ANN models. In March 1995, another set of incident data, the validation-test data set, was collected from two freeways in Melbourne for validating the performance of the developed ANN models. A total of 90 incidents that occurred on the Tullamarine and South Eastern freeways between January 1992 and March 1995 were extracted from the operator's log for examination. The 90 incidents collected from both freeways were then pre-processed using the procedures described previously. A total of 50 incidents were discarded because the loop detectors upstream and/or downstream of the location of incidents were faulty or the data from these detectors were corrupted. This resulted in the compilation of a validation-test data set comprising 40 incidents that were detectable from the detector data. Of these, 25 occurred on the Tullamarine Freeway and 15 occurred on the South Eastern Freeway.
A FRAMEWORK FOR AUTOMATED INCIDENT DETECTION USING ANNS As implied by their name, ANN models can be visualised as a network. Consider the section of freeway shown in Figure l(a) which is defined by upstream and downstream detector locations. A corresponding ANN model structure is shown in Figure l(b). The detector station data form the input to the ANN. The output is a {0,1} variable indicating the absence or presence of an incident in the freeway section, respectively. The parameters of the ANN model are established through a process known as training. In order to train a neural network to perform incident detection, the network must be presented with input detector data and output states for both incident and incident-free conditions. Therefore, the input to the ANN model comprises real-time speed, flow and occupancy measurements in 20-second intervals from each of the upstream and downstream stations. The output of the ANN model is the traffic state within the section. Output State 1 {0} represents incident-free conditions and output State 2
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{1} represents incident-conditions. One of the well-known and widely used neural network models is the back-propagation or multi-layer feed-forward (MLF) network.
(a) Physical System Downstream
Upstream
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The back-propagation algorithm's popularity is due mainly to the solid theoretical foundations on which it rests. The algorithm has been successfully implemented in many pattern recognition applications across many disciplines (Maren et a/., 1990). Cheu (1994) tested three ANN architectures suitable for incident detection and real-time classification problems. These included the multi-layer feed-forward (MLF) neural network, the self-organising feature map (SOFM) and the adaptive resonance theory (ART). The MLF, implemented with the backpropagation (BP) training algorithm, proved to be superior to the other architectures tested. The MLF was chosen for implementation in this study based on its earlier success, especially in real-time pattern recognition problems, and based on its demonstrated superior incident detection performance over the other ANN architectures (Cheu, 1994). In particular, the standard three-layer feed-forward neural network has been chosen for this study, as shown in Figure l(b). Further details regarding the structure of the MLF, the implementation of the back-propagation algorithm, the choice of incident detection and ANN parameters and the model types that were investigated in this study can be found in Dia and Rose (1995).
MODEL EVALUATION USING PERFORMANCE ENVELOPES The basic principle behind the ANN incident detection model being developed in this study is the correct classification of the traffic flow input parameters, provided in 20-second cycles, into either an incident-free or incident category. In other words, the input parameters are to be classified into one of two classes or states: State 1 {0} representing incident-free conditions and State 2 ( 1 } representing incident conditions. Assuming no persistence tests are applied, an incident is then detected when the traffic state changes from State 1 to State 2 or from {0} to
The ANN incident detection models are ultimately going to be used for detecting the presence or absence of an incident within a certain section of a freeway. During the development of the ANN models, the output PEs were trained to produce a low activation level for non-incident conditions and a high activation level for incident conditions within the specific test section. As was discussed previously, the network output ranges were restricted to a range between 0.2-0.8, corresponding to no activation and full activation of the output PE, respectively. Typically, a decision threshold of half-activation (a value of 0.5) is chosen for making the decision regarding the presence or absence of incident conditions. If a vector of input data is presented to the ANN model which results in the output PE being activated to at least a value of 0.5 (halfactivation), it is concluded that incident conditions are present for that time period. Otherwise, the data for that time period is classified as non-incident conditions. Therefore, the value selected for the decision threshold (DT) plays an important role in the classification of the input data and consequently in determining the incident detection performance of the model. A broader performance measure can be obtained if the evaluation criteria is not limited to a single decision threshold (Masters, 1993). A graph that helps to show the relationship the DR and FAR) can be obtained by evaluating the detection rates and false alarm rates for many
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possible decision thresholds. The plot of detection rate against false alarm rate (shown in Figure 2) is called the Performance Envelope Curve (PEC) of the network.
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Fig. 2- A performance envelope curve The lower-left corner (0,0) of the PEC will always be one endpoint of this curve. It corresponds to a threshold slightly larger than 0.8. At this threshold, an observation is classified as an incident condition if it activates the output PE of the network to a value slightly greater than 0.8. Since this is not possible (because the PE's output range is defined between 0.2 and 0.8), all observations will be classified as non-incident conditions. Therefore, both the DR rate and FAR will be zero. The upper-right corner (100%,100%) will be the other endpoint of the PEC. This point corresponds to a threshold of 0.2. At this threshold, an observation will be classified as a non-incident condition if it activates the output PE to a value less than 0.2. Again, since this is not possible, all observations will be classified as incident conditions. Therefore, both the DR and FAR will be 100%. A network that is able to correctly classify all observations would have a PEC that is at a right angle (as shown by the dashed lines in Figure 2). At the ideal performance threshold, the DR would be 100% and the FAR would be zero. For all lower thresholds, the DR would remain at 100%, while the FAR would increase to 100%. For greater thresholds, the FAR would remain at zero, while the DR would drop to zero. The PEC would be two straight lines that intersect at (0,100%), as shown by the dashed lines in Figure 2 (Masters, 1993). The quality of performance of the network is demonstrated by the degree to which the PEC pushes upward and to the left.
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RESULTS OF TRAINING A total of 500 models consisting of different input parameters and hidden units were designed and trained in this study. In order to compare the performance of the different groups of models and determine which model types had better incident detection performance, statistical analysis techniques were used to investigate the trade off in performance between the different groups of models. The results from the statistical analysis and refinement of a selected number of models revealed that the architecture of the best performance model was similar to that shown previously in Figure 1 (model MLF). This model uses dual stations, traffic data averaged across all lanes and only from the current time interval t. The optimal number of hidden PEs for this model was found to be 14. The incident detection performance measures for model MLF based on the application of a two-interval persistence test and a range of decision thresholds (DT) are presented in Figure 3 for the validation-test data set of 40 incidents. It should be pointed out here that a FAR of 0.1% corresponds to about 4.3 false alarms per day per section. Investigation of several techniques with the potential for reducing the FAR (Dia, 1996) revealed that model MLF is best applied using a decision threshold of 0.64 and a twointerval persistence test. This will be referred to as model MLFl and will be used for the evaluation of the incident detection performance. The results for model MLFl, with a decision threshold of 0.64, (shown in Figure 3) will also be used as an indicator of the expected incident detection performance of the model when it is actually implemented in the field. They comprise a DR of 82.5%, a FAR of 0.065% and a MTTD of 203 seconds (3.4 minutes). The PEC in Figure 3 clearly shows that implementing a higher decision threshold results in lower DR and FAR as illustrated for model MLF2 which implements a decision threshold of 0.695 and has a DR of 50% and FAR of 0.013%.
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Fig. 3- PEC for the ANN model based on the validation-test set of 40 incidents
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PERFORMANCE ASSESSMENT OF THE ANN INCIDENT DETECTION MODEL When assessing the performance of an incident detection model, it is important to conduct the assessment in the context of both incident and non-incident conditions. These separate performance assessments are reported in the sections which follow.
Performance Evaluation Under Incident Conditions The evaluation of the ANN model will be based on the application of model MLF1 to the validation-test data set that was not used in the model development. This evaluation will give an indication of the prediction ability of the model in detecting incidents that the model had not previously seen. The validation-test data set, described previously in Section 4, comprised 40 incidents. A total of 25 incidents were collected from same freeway (Tullamarine) that was used as a test-bed for the development of the ANN model. The remaining 15 incidents were collected from the SEA freeway which is a different facility from the one used for the development of the ANN models. These two data sets will be referred to as 'validation test data set- Tulla' and 'validation-test data set - SEA', respectively. The performance evaluation under incident conditions is based on the performance measures discussed previously. The results reported for the false alarm rates in this study are based on the definition of the off-line FAR (FARoff). In addition, and for the purpose of this study, an incident will only be declared as detected if the algorithm takes no longer than 5 minutes for its detection. Therefore, the failure to detect an incident could be either due to the inability of the algorithm to detect the incident at all or the inability of the algorithm to detect the incident within 5 minutes. The incident detection results for model MLF1, based on this criteria, are shown in Table 1. While the DR is high for both subsets of data, these results clearly indicate that the reported FARs vary according to the incident data set under consideration. One possible explanation for this is related to the fact that not all incidents in the data sets had similar characteristics in relation to the 'noise' or general 'cleanliness' of the incident. The presence of a larger number of 'noisy' incidents in a certain data set would therefore affect the performance of the model (especially FAR) on that data set.
Data Set
Number of Incidents
Validation Test Set-Tulla Validation Test Set-SEA Total Set
25 15
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Incident Detection Performance Detection False Alarm Mean Rate Rate Time-toDetect No. % (Second) (%) 19/25 76.0 0.09 188 14/15 93.3 0.00 224 33/40 82.5 0.065 203
Table 1: Incident detection performance of model MLF1 based on the total set of 40 incidents and an upper bound of 5 minutes for the TTD
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The performance of the model was also examined by segmenting the results on the basis of: - incident severity (one, two or three lanes blocked), - prevailing flow conditions (light, moderate or heavy), and - period of day (peak or off-peak). The detailed results from these analyses are provided in Dia (1996), however, the main findings were as follows: - the poorest performance was achieved for incidents which only involved a single lane blockage, in this case the model detected 66.7% of the incidents. - even under light flow conditions the model was able to detect 75% of the incidents however, the false alarms generated by the algorithm were primarily associated with heavy flow conditions, - the model's performance is consistent across the peak and non-peak periods of the day with respect to detection rates (82.4% in the off-peak versus 83.3% in the peak) and MTTD (201 & 216 seconds in the off-peak and peak respectively). The FAR results, however, showed more variability and were ten times as high during the peak period than the off-peak period (0.25 % versus 0.029 %).
Distribution of Incidents Detected According to the Time to Detect Incidents. As was mentioned previously, the ability of an incident detection algorithm to detect incidents quickly such that an appropriate and timely response is initiated to clear incidents and restore traffic conditions has become the most critical requirement of an efficient freeway management systems. In this context, the performance of the ANN incident detection model according to the time taken by the algorithm to detect incidents is of interest. The model detected 72.7% of incidents in less than 4 minutes, 39.4% were detected in less than 3 minutes and 18.2% were detected in less than 2 minutes. It should be mentioned here that model MLF1 implements a two-interval persistence test which results in a minimum delay of 40-seconds before declaring an incident condition. As was mentioned previously, the operators' log files obtained from the VicRoads Traffic Control and Communications Centre included the times when the operators detected or confirmed the incidents. Inspection of these log times revealed that 2 incidents were detected by the operators before their impact on traffic was confirmed from the detector data. Only 18.4% (7 incidents) of the remaining 38 incidents were detected by the operators within 3.0 minutes of their occurrence. The average time taken by the operators to detect the 38 incidents, however, was 6.9 minutes after their estimated occurrence times. The results reported in this study for the MTTD of the ANN model (based on the 40 incidents) was 3.4 minutes (203 seconds as shown in Table 7). This suggests that the ANN model has the potential to provide a 50% improvement in efficiency compared to the average time taken by the operators to detect incidents.
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PERFORMANCE EVALUATION DURING NON-INCIDENT CONDITIONS The FAR results reported in Table 1 for the ANN incident detection model were based on the non-incident conditions immediately before and after the occurrence and clearance of incidents. The unstable traffic conditions during these periods may cause the algorithm to generate a higher number of false alarms than would be expected during normal non-incident traffic conditions. In addition, since the ANN incident detection model was developed to be implemented on Melbourne's freeways, the FAR performance under normal daily traffic conditions (especially during peak-hour periods) needs to be investigated. For the purpose of this study, this evaluation was conducted off-line. A section of the Tullamarine Freeway in Melbourne which produced the highest number of alarms from the ANN model was identified. This section was continuously video-taped (on a 24-hour basis) for a period of 33 days from early November to early December 1995. A time lapse video recorder was used to monitor traffic conditions within the section. The corresponding speed, flow and occupancy data from the detectors were stored for the test section during that period. The real-time video monitoring of the test section (in conjunction with the detector data) and the subsequent results presented here are believed to be a first for this type of analysis with respect to evaluating the performance of freeway incident detection models. Description of the Data for the Off-Line FAR Tests. Figure 4 shows the video camera location and the field of view captured through the camera. Section S4-S5 was continuously video taped (on 24-hour basis) for a period of 33 days from November 6th until December 8th, 1995. In addition to the video data, traffic data from the detector stations at S4 and S5 were also collected for the duration of the evaluation period. In this way, the model could be run on the detector data and when an alarm (either true or false) was raised, the video could be advanced to that same time period to see what traffic conditions had caused the alarm.
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Fig. 4- Schematic of section S4-S5 on the Tullamarine Freeway that was video monitored for the evaluation of off-line FAR
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Analysis of Detector and Video Data. The results from the initial appraisal of the off-line FAR indicated that a large number of alarms were generated during peak-hour periods. Therefore, it was decided to divide each day into five periods, as shown in Table 2, such that the performance of the ANN models is evaluated separately for each period. The analysis of false alarm rates reported in this paper correspond to model MLF1 only.
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00:00-06:00 06:00-10:00 10:00-16:00 16:00-20:00 20:00-24:00 00:00-24:00
Total Number of 20-Second Observations for the 3 3 -Day Test Period 35,446 23,642 35,394 23,714 23,459 141,655
Table 2: Classification of periods of the day for the off-line FAR evaluation
FAR analysis using model MLF1. For the purpose of evaluating the off-line FAR, model MLF1 was run in a 'prediction' mode using the 33-day detector data. In this mode, the ANN model was presented with only the input values (ie. the speed, flow and occupancy data at both the upstream and downstream stations). The output of the model in response to these inputs represented either an incident or incident-free condition for the specific time period under consideration. This procedure was adopted because the model will be run in this mode when it is eventually implemented in the field. The model's output or decisions (every 20-seconds) were monitored and all the alarms generated by the algorithm, along with the time at which they occurred, were written to a file for later inspection. This procedure was implemented for the analysis of all 33-days of data according to the periods described in Table 2. The next step in the evaluation procedure involved the examination of the individual alarms generated by the ANN model and investigating the causes of these alarms by checking the video tapes using the time-lapse video recorder. This was made possible because the video tapes included a time stamp that was synchronised with the time stamp in the detector data files. It should also be mentioned here that the generated alarms were based on the application of a two-interval persistence test of the model. This meant that two alarms were also generated during the previous two 20-second intervals. Therefore, this suggests that the recorded alarms must have been generated due to the presence of some abnormal condition that had lasted for at least one minute and are therefore significant enough to be investigated. A total of five incidents occurred within section S4-S5 during the 33 days of off-line evaluation. These incidents were detected by algorithm MLF1 and were confirmed using the video tapes which also allowed for the determination of the true start and end times of the incidents. The incident detection performance of model MLF1 on these incidents consisted of a DR of 100.0% (5/5) and a MTTD of 108 seconds.
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Table 3 lists the distribution of alarms generated by model MLF1 for the 3 3-days of operation on both the inbound and outbound directions of section S4-S5 according to the cause of the alarm. Causes of Alarms Period of the Day Total % B A D C E Incident Related:
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38',543 1563
402,264,45 162,844,25
278 2
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5 103
1 1 - Other Conditions 867 Total Alarms (33 Days) 281 745 39.9 34.2 % of Total Alarms 12.9 : Period of the day as defined in Table 2 1,2,3,4,5. Denotmg incident ID numbers
3 283 13.0
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Table 3: Distribution of alarms generated by MLF1 on section S4-S5 (both directions) according to the causes of alarms Based on the results obtained from Table 3, the major causes of alarms were found to fall into three main categories: incident related, traffic flow related and non-incident/false alarm related. The nature of the alarms generated within each of these categories and their causes are described in detail below. I. Incident Related Category The alarms that were generated within this category fall into two sub-categories: incident conditions and rubber necking conditions. As was mentioned earlier, a total of five incidents occurred within section S4-S5 during the 33 days of off-line evaluation of the ANN model. The high number of incident conditions shown in Table 3 is due to the fact that the ANN incident detection model (MLF1) generated multiple alarms for each incident. The ANN models were trained to distinguish between incident and non-incident conditions and would therefore continue to generate alarms as long as incident conditions are detected. Therefore, the repeated alarm declarations during an incident are not considered as false alarms.
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During the analysis of the detector data, it was noticed that model MLF1 generated many alarms within the test section when an incident occurred on the other side of the carriageway. When the video tapes were examined, 'rubber necking' conditions became evident when it was observed that drivers were slowing down due to the presence of an incident on the other side of the freeway. The ability of the model to generate 'rubber necking' alarms may be considered as a positive feature of an incident management system. These alarms could be used to automatically switch on variable message signs (VMS) to advise motorists of the traffic disruptions that lie ahead in the 'rubber necking' direction. This strategy has the potential to reduce the risk of rear-end accidents in the incident-free direction where queues would start to build up due to the 'rubber necking' effect. II. Traffic Related Category The alarms that were generated within this category also fall into two sub-categories: absence of traffic and congested conditions. The alarms generated during absence of traffic conditions were easy to discern since the detector stations were not providing any data during these periods due to the absence of traffic. Data for these periods consisted of zero values for all input parameters for both the upstream and downstream stations of the section. This caused the ANN model to generate a false alarm. For these periods, the video tapes did not show any abnormal traffic conditions. In fact, about 50% (278/553) of these alarms were generated during the period from 00:00 to 06:00 in the morning when the traffic volumes inbound were light (Table 3). The 'absence of traffic' alarm conditions cannot be considered as legitimate false alarms because they were basically due to the lack of 'valid' detector data. From a practical perspective, it would be possible to implement procedures that would validate the integrity of the data before they are presented to the model. If the data for a specific 20-second interval are found to consist of only zeros (whether due to absence of traffic or faulty detectors), it is discarded and the model does not make a decision on that data. Alternatively, data from the immediate upstream station could be substituted until the faulty detectors are fixed (Dia, 1996). The congested conditions were identified by checking the prevailing traffic flow conditions on the video tape for the designated alarm periods. A major part of these alarms was detected on the inbound direction of the section for the periods between 06:00 and 10:00 and in some cases for the periods between 10:00 and 12:00. The detector data showed a significant drop in the speeds and an increase in the flow and occupancy values during these periods for which the ANN model generated a series of false alarms. Investigation of the video tapes clearly showed congested conditions and slow moving vehicles for these periods which comprised the major part of the morning peak-hour for the inbound direction of section S4-S5. The ability to distinguish between recurrent and non-recurrent congestion is an important issue that has a great influence on the performance of an incident detection system. Due to the fact that recurrent congestion can be expected in advance (ie. during peak-hour
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III. Non-Incident Related/False Alarm Category The five alarms within this category were mainly those that could not be accounted for or justified from the detector or video data and were therefore considered false alarms. Table 4 below presents FAR calculations that are based on three scenarios. In the first scenario, the FAR is calculated by excluding the legitimate incident-related alarms which included incident and rubber-necking alarms. Basis for FAR Calculations (1) All alarms except: - incident & alarms - rubber-necking alarms (2) All alarms except: - incident alarms - rubber-necking alarms - absence of traffic alarms (3) All alarms except: - incident alarms - rubber-necking alarms - absence of traffic alarms - congestion alarms
FAR Inbound 1453/141335 1.028%
FAR Outbound 303/141555 0.214%
Average FAR 1756/282890 0.621%
163/141045 0.825%
40/141292 0.028%
1203/282337 0.426%
3/139885 0.0021%
2/141254 0.0014%
5/281139 0.0018%
Table 4: Calculations of FAR for the 3 3-days of off-line testing using MLF1 The computed FAR for the inbound direction (1.028%) is about five times the FAR for the outbound direction (0.214%). For both directions of the test section, this resulted in a FAR of 0.621% which is equivalent to about 53 false alarms per day per section. From a practical perspective, this FAR is very high and would undermine the performance of the algorithm. In the second scenario, the FAR is calculated by excluding the incident-related alarms along with the alarms that were caused by the absence of traffic. As was mentioned previously, the alarms that were generated due to the absence of traffic cannot be considered as legitimate false alarms because they were basically due to the lack of 'valid' detector data. The reported FAR for this scenario consisted of a FAR of 0.426% (averaged on both directions of the test section) which is equivalent to about 36 false alarms per day per section. Although the definition of the FAR for this scenario resulted in a reduction of about 32% over the first scenario, the reported FAR is still too high from a practical perspective.
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In the third and final scenario, the FAR calculations are based only on the alarms that could not be accounted for from the video and detector data and were therefore deemed as false alarms. The reported FAR for this scenario is 0.0018% which is equivalent to only 0.15 false alarms per day per section (ie. only 2.1 false alarms per day on both directions of the 14 sections of the Tullamarine Freeway under study). The third scenario clearly shows that excluding the recurrent congestion alarms from the FAR calculations resulted in a significant reduction of FAR (by about 99.6%) over the second scenario.
CONCLUSIONS AND RESEARCH DIRECTIONS The results presented in this paper have demonstrated the feasibility of using 'real-world' data for the development of ANN incident detection models. These results provide a comprehensive evaluation of the ANN incident detection model and confirm that the developed ANN models can provide fast and reliable incident detection on freeways. While the model considered here (MLF1) had excellent performance under incident conditions, the high level of false alarms produced by the model during congested incident-free conditions could be a cause for operational concern. Additional research (Dia, 1996) indicates that improved performance can be achieved by running a model with a high decision threshold during peak period conditions and a lower threshold during off-peak conditions. This suggests there would be merit in developing a dynamic decision threshold model and this is clearly a recommended area for future research work in freeway incident detection modelling. There's also scope in future research projects to perform a comparative evaluation between the ANN model and other operational algorithms. The difficulties associated with choosing the optimal ANN configuration may also be overcome by applying neuro-genetic techniques. This is also a recommended area for future research work.
ACKNOWLEDGMENTS The authors acknowledge the support of VicRoads for providing the data. In particular, we acknowledge the generous assistance of Mr. Francis Sin and Mr. Anthony Snell.
REFERENCES Ahmed S.R. and Cook A.R. (1982). Application of time-series analysis techniques to freeway incident detection. Transportation Research Record, 841, 19-21. Cheu, R.L. (1994). Neural Network Models for Automated Detection of Lane-Blocking Incidents on Freeways. Ph.D. Dissertation. (University of California: Irvine.) Dia, H. (1996). Artificial Neural Network Models for Automated Freeway Incident Detection. Ph.D. Dissertation. (Monash University: Clayton).
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Dia, H. and Rose, G. (1995). Development of artificial neural network models for automated detection of freeway incidents. Paper presented at The 7th World Conference on Transport Research, Sydney, Australia, July 1995 (forthcoming in the selected conference proceedings). Levin M. and Krause G.M. (1979). Incident detection algorithms. Part 1: off-line evaluation; Part 2: on-line evaluation. Transportation Research Record, 722, 49-64. Lindley J.A. (1987). Urban freeway congestion: quantification of the problem and effectiveness of potential solutions. Institute of Transportation Engineers Journal, 57(1), 27-32. Luk, J.Y.K. and Sin, F.Y.C. (1992). The calibration of freeway incident detection algorithms. Australian Road Research Board Working Document No WD TE 92/00l.(ARRB: Vermont South, Victoria) Maren, A.J., Harston, C.T. and Pap, R.M. (1990). Handbook of Neural Computing Applications. (Academic Press, Inc.) Masters, T. (1993). Practical Neural Network Recipes in C++. (Academic Press Inc.: San Diego). Persaud, B.N. and Hall, F.L. (1989). Catastrophe theory and patterns in 30-second freeway traffic data - implications for incident detection. Transportation Research, 23A(2), 103113. Ritchie, S.G. and Cheu, R.L. (1993). Simulation of freeway incident detection using artificial neural networks. Transportation Research, 1C(3), 203-217. Rose, G. and Dia, H. (1995). Freeway automatic incident detection using artificial neural networks. Proceedings of the International Conference on Application of New Technology to Transport Systems, Vol. 1, Melbourne, Australia, 123-140. Rumelhart, D.E., Hinton, G.E. and Williams, R.J. (1986). Learning Internal Representations by Error Propagation. In D.E. Rumelhart, J.L. McClelland and the POP Research Group (Eds) Parallel and Distributed Processing, Vol. 1 318-362. (MIT Press: Cambridge, MA.)
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RELIABILITY MEASURES OF AN ORIGIN AND DESTINATION PAIR IN A DETERIORATED ROAD NETWORK WITH VARIABLE FLOWS YasuoAsakura Department of Civil and Environmental Engineering Ehime University Matsuyama, 790, JAPAN FAX.+81 (0) 89-927-9843 TEL.+81 (0) 89-927-9829 E-mail, [email protected]
ABSTRACT A flow network should be studied for evaluating the reliability of a degraded road network, in which the inconvenience of travel may bring the reduction of travel demand and network flow pattern may change. This paper aims to show reliability measures of an origin and destination (OD) pair in a road network when some links are possibly damaged by natural disasters and may be closed to traffic. The UE model with variable demand and strict link capacity constraints is applied to describe flows in a network with some disconnected links. An approximation algorithm is proposed for estimating the cumulative travel time distribution between an OD pair. Small scale examples are calculated to test the performance of the proposed algorithm.
INTRODUCTION Natural disasters such as earthquakes and heavy rainfall are liable to cause serious damage to road networks. Improving durability against natural disasters and maintaining reliability are recognized as the objectives of road network infrastructure planning and management. In particular, redundancy became a popular keyword of highway system planning after the devastation of road networks by the Kobe Earthquake. A reliable network means a network which can guarantee an acceptable level of service for road traffic if the functions of some links 273
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of the network are degraded by disasters. There are several studies on reliability of transport networks. Turnquist and Bowman (1980) present a set of simulation experiences to study the effect of the structure of the urban network on service reliability. lida and Wakabayashi (1989) develop an approximation method to calculate the connectivity between a node pair in a road network. Assuming the connectivity of each link, they propose a method using partial minimum path and cut sets. These studies concentrated on the calculation of the connectivity measure for a pure network, and thus network flows were not explicitly considered. Asakura and Kashiwadani (1991) discuss road network reliability considering day-to-day fluctuations of traffic flow. They show a traffic assignment simulation model and a statistic estimation method for describing day-to-day dynamics of network flow. They propose a time reliability measure that is the probability of whether travel between an origin and destination pair is possible within an acceptable travel time. The time reliability is suitable for evaluating network performance under normal conditions in which a driver may meet traffic congestion caused by heavy travel demand compared with the capacity of a network. Sanso and Milot (1994) present a reliability model for urban transportation planning considering traffic accidents. They propose three-T's model to describe dynamic behaviour of drivers in an accidental state. Applying static equilibrium package EMME2, they calculate travel time during accident periods and evaluate network plans using the general approach proposed by the same authors. Du and Nicholson (1993) show a theoretical frame to systematically outline the main issues in the analysis and design of Degradable Transportation Systems(DOTS). User Equilibrium (UE) model with variable demand is applied to describe network flows for a degraded transport network. They develop an analytical framework and algorithms for calculating network reliability. Asakura and Kashiwadani (1995) apply the frame proposed by Du and Nicholson and evaluate the reliability of an actual scale road network. Instead of the UE model, they use a Stochastic Assignment model describing the diverting behaviour of the driver and trip canceling behaviour in a deterioration network due to natural disasters. A flow network should be studied for evaluating the reliability of a degraded road network, in which the inconvenience of travel may bring the reduction of travel demand and network flow pattern may change. This paper aims to show reliability measures of an origin and destination (OD) pair in a road network when some links are possibly damaged by natural disasters and may be closed to traffic. The UE model with variable demand and strict link capacity constraints is applied to describe flows in a network with some disconnected links. An approximation
Reliability measures of an origin and destination pair in a road network
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algorithm is proposed for estimating the cumulative travel time distribution between an OD pair. Small scale examples are calculated to test the performance of the proposed algorithm.
RELIABILITY MODEL Reliability Measure between an OD Pair State Vector and State Probability A road network is represented by a directed graph that includes a set of numbered nodes and a set of numbered links. We assume that links may be degraded by disasters, while nodes will not be. If it is necessary to consider the failure of a node like an intersection, a node is represented in detail using some links. We also assume that a deteriorated link is completely closed to traffic and that the failed condition of a link continues for a long time. This assumption is necessary to identify a deteriorated network as a discrete state. That is also applied to describe the behaviour of network users. Adegraded road network is identified by the state vector x={xi,..,x a ,..,XL} whose element x a denotes whether link a is degraded or not; namely x a is equal to 0 if it is closed to traffic, or xa is equal to 1 if the link has its normal functions. The possible maximum number of the state vector is 2L, and X={ x } denotes the state vector space. If any link of the network is not degraded, the state vector is written as\o={l,...,!}. This state is referred as the normal or ordinary state in the following part of this paper. The worst state vector is written as Xw={0,...,0} in which all links are not operating. Here we define the probability of the occurrence of a state x, p(x). Disasters such as earthquakes are not predictable and the degree of the external force due to disaster is quite uncertain. Then, a road link may be durable against natural disasters or may not be. This means the physical conditions of a link are not deterministic. We define pa as the probability of whether the link a is not degraded and not closed to traffic. Although it is very important to develop an estimation method of pa, we assume the value of p a is exogenously given and fixed. When the probability pa is independent each other, the probability of the occurrence of a state x is calculated as:
P(x)=nPaX'(l-Pa)1"X% a£A
We call p(x) as state probability.
(1)
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Y. Asakura
Definition of Travel Time Reliability The travel time between an origin and destination (OD) pair is often used as a performance measure of a network. Even if a road network is kept at a normal state, the OD travel time is fluctuating due to traffic congestion. You may arrive at your destination earlier than you expected or vise versa. In the degraded network, the uncertainty of traffic flow conditions seems higher than in the normal network. We focus the uncertainty of OD travel time and define the reliability measure of traveling between an OD pair. There are several routes between an OD pair and travel times may be different from route to route. We take the minimum travel time of some possible routes in an OD pair. The minimum travel times between origin r and destination s in a deteriorated network x and in the normal network xo are denoted by trs(x) and trs(xo), respectively. The value of trs(x) is infinite if any route between the OD pair r-s is not connected. The OD travel times are the results of the interaction between travel demand and network performance. It is necessary to describe this interaction and resulting network flows fora degraded network x. This is explained in the later section. When the OD travel of a state x is larger than that of the normal state xo, the network performance is decreased. The travel time of an OD pair is generally different from those of the others. In order to compare the performance measure between different OD pairs, we use the ratio of the OD travel of a state x to that of the normal state xo. The travel time performance is acceptable if the ratio is not too large. The reliability measure between an OD pair r-s is defined as the probability of whether the ratio of the OD travel time in a degraded network to that of the normal network is sustained within an acceptable level. The probability is written as: R,.(c)=Piob.[t ll (x)/t 11 (x 0 )ic]
(2)
where c denotes an acceptable level. The criterion c is exogenously determined considering the level of service that should be maintained even in deteriorated situations. Rrs(c) is consistent with the connectivity measure discussed in the previous studies on system reliability when the value of c is given as infinite. The connectivity measure in the previous reliability studies implicitly assumed that network users would divert to any alternative routes. Even if the OD travel time along the diverted route is remarkably large, users would use the route. For the same OD pair, the network covering a wider area becomes more reliable than that covering a smaller area. The nearly infinite diversion will be applied to a communication network, however it may not always be suitable for evaluating transport networks.
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Approaches Estimating Reliability Measure There are two approaches for calculating the value of Rrs(c). One is to evaluate the expected value of the operated/failure function, which was shown by Du and Nicholson(1993). The operated/failure function discriminates whether the ratio of the OD travel time in a degraded network to that of the normal network is within the given level. That is, l if t r s (x)/t r s (x 0 )^c 0 if
t r s (x)/t r s (x 0 )>c
&
The probability Rrs(c) is the mathematical expectation of Zrs(c,x) weighted by the state probability p(x), which is written as: Rre(c) = E[Zre(c,x)] = I p(x) Zre(c,x)
(4)
xex
When the number of all state vectors is not so large, Eq.(4) could be directly calculated. However, the number of state vectors is generally very large and the direct calculation of Eq.(4) requires huge computational cost. An approximation algorithm was proposed and is shown in the next chapter. When we evaluate the reliability measure for different criteria c's using the approach above, it is necessary to calculate the operated/failure function for each value of c. Even if we use an approximation algorithm, this is time consuming work. Here, we will show an alternative approach for evaluating Rrs(c). The occurrence of a state x is stochastic and the OD travel time trs(x) is randomly distributed as well. If we could estimate the cumulative distribution function of travel times between OD pair r-s Frs(t), the probability Rrs(c) is easily calculated for any values of c, such as: Rrs(c) = F ts (ct rs (x 0 ))
(5)
The cumulative distribution function is also approximated using similar methods for approximating the expectation of the operated/failure function. This is explained in the next chapter as well. Network Flows in a Degraded Network In order to obtain the values of OD travel times for a state x, it is necessary to describe traffic
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flow in a network with some degraded links. Compared with the network flow analysis for the normal network conditions, the following problems must be considered. First, the number of available routes between an OD pair will be limited in a degraded network. There may not be a route in the network. Then, travellers may cancel their trip when they expect longer diversion in a deteriorated network. This will result in the reduction of travel demand. Secondly, some links in a network are closed to traffic and the remaining links become congested, because travellers will be concentrated into surviving routes in the deteriorated network. However, each link has its own capacity and the link traffic volume cannot exceed that capacity. The second point means that explicit link capacity constraint must be considered and a link cost function such as the Davidson function should be applied. This constraint excludes considering any traffic assignment methods with fixed OD demand, because feasible network flows may not exist if link capacity constraint is very strict compared with larger amount of travel demand. The most extreme case is that there is no route available between OD pairs. It is obvious that the fixed OD demand cannot be handled in this case. Thus, we employ traffic assignment methods with variable OD demand, in which OD travel demand is determined as a function of the level of service between the OD pair. The User Equilibrium assignment with variable OD demand and explicit link capacity constraint seems one of the simplest assignment methods which satisfies the above conditions. As we have already assumed, degraded links would be completely closed to traffic for a long time and the same network state would be prolonged sufficiently. Thus, the network users will experience the same network state for a long time and the User Equilibrium condition seems appropriate to describe users' behaviour in the deteriorated network as well. The UE assignment model with variable demand and link capacity constraint is formulated as follows. f
qn
1 Z= I Jsa(x)dx - I I jD;1(y)dy -minimize a£A 0
r6R seS 0
subject to I
h* = qre
VreR,seS
keK r>
\°)
h^O
VkeKre,reR,seS
q r s ;>0
VreR,seS
fa
VaeA
where s a (x) is the link cost function with explicit capacity constraint, D'1 (y), is the inverse of
Reliability measures of an origin and destination pair in a road network
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the demand function associated with OD pair r-s. qre is the number of trips between OD pair r-s, h* is the flow on the k-th path between OD pair r-s, f a is the traffic volume of link a and satisfies f a = Z Z Z h" d" a .
The difference between the UE model shown above and that of
r6RS6Sk€Krj
Sheffi(1985) is that the above formulation includes link capacity constraint. This model can be reformulated using the excess demand formulation and solved using Daganzo's algorithm(1977) for calculating fixed demand traffic assignment with link capacity constraint. Approximation Algorithms For a network with L links, the number of possible state amounts to 2L. If the UE traffic assignment is calculated for each network state, the direct calculation of the expected value of Rrs(c) using Eq.(4) requires huge computation cost. This is also true for estimating cumulative distribution function Frs(t). In this section, we show two algorithms for approximating Rrs(c) and Frs(t), respectively. The original idea was presented by Li and Silvester (1984). The algorithm defines the lower and upper bounds using the J most probable state vectors. Approximation of the Expected Value Sorting state vectors in the order of the state probability as Eq.(7): p(Xj) > ... > p(\.)> p(x j + 1 )> ... > p(x N )
(?)
where Xj denotes the j-th most probable state vector, p(xj) represents the state probability for the state Xj and N is the number of all state vectors. Using the state vectors by the J-th most probable state vector {xi,...,xj} and corresponding the values of the operated/failure function {Zrs(c,xi), ...,Zrs(c,xj)}, the upper and lower bound of the expected value can be defined as follows. The upper bound is obtained through the optimistic expectation of the operated/failure function. If we assume that the states Xj for j=J+l to N are equivalent to the normal state XQ, the OD travel times for those states are also equivalent to that of the normal state; namely trs(xj)=trs(xo) for j=J+l to N. This means the values of the operated/failure function ZTS(C,XJ) are equal to 1 for j=J+l to N. Thus, the upper bound of R^(c) is obtained as the expected value of Zrs(c,x) of these conditions. That is,
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Y. Asakura
j-i
j-J+i
Ep(x j )Z r a (c,x j ) + I p(Xj)
(8)
On the other hand, the lower bound is obtained through the pessimistic expectation of the operated/failure function. Assuming that the states Xj for j=J+l to N are equivalent to the worst state xw, the OD travel times for those states are also equivalent to that of the worst state; namely trs(xj)=trs(xw) for j=J+l to N. The travel time of the worst state trs(xw) will probably be infinite or extremely larger than that of the normal state trs(xo). Since the ratio of the two trs(xw)/trs(xo) is also very large, the value of the operated/failure function for the worst state ZrS(c,xw) is equal to 0 for any criterion c. Accordingly, the values of the operated/failure function ZTS(C,XJ) are equal to 0 for j=J+l to N. Thus, the lower bound of R^s(c) is obtained as the expected value of Zrs(c,x) of these conditions. That is, J
N
R-(c)-Ip(x j )Z I i (c,x j )+ I p^Z^xJ j-l
i-J+l
(9)
-IpOgz.fex,)
j-i The expected value of Rrs(c) stays between R^ (c) and R^(c) . Comparing the values of the upper
and the lower bounds of the J-th iteration with those of the (J+l)-th iteration, we obtain the followings:
i i Rfox^Rifex^)
(10)
This means the upper bound and the lower bound converge to the exact expected value of Rrs(c) from the upper side and the lower side, respectively. We take the next most probable state vectors one after another and update the approximated expected value of the reliability measure Rrs(c) until the difference between the upper and lower bounds becomes small enough. This procedure can be represented as the following algorithm.
Reliability measures of an origin and destination pair in a road network
28 1
Step.2 Calculate the OD travel time trs(xj) by solving the UE problem for the state x j. Step.3 Compare the OD travel time trs(x j) with trs(xo) and evaluate the operated/failure function ZTS(C,XJ). l if t n (x J )/t I I (x 0 )sc 0 if t r e (x j )/t r s (x 0 )>c Step.4 Calculate the upper and lower bounds using Eq.(8) and Eq.(9), respectively.
Step.5 Check the convergence. If the difference of the upper and the lower bounds is small enough, go to Step.6. Otherwise, set J=J+1 and return to Step.2. Step. 6 Approximate the expected value of Rrs(c) as:
(13) This algorithm seems efficient for evaluating the expected values of the time reliability measure between OD pairs r-s (r 6 R, s e S) for given criteria c. However, one must calculate the number of iterations again if it is necessary to obtain the expected value of Rrs(c') for different criteria c' . Then, we propose another algorithm to approximate the cumulative distribution function Frs(t) of OD travel times. The same as the approximation of the expected value of Rrs(c), we can approximate the function Frs(t). Using the J most probable state vectors XQ,...,XJ and corresponding values of the occurrence probability p(xo),...,p(*j), the upper and the lower bounds of the cumulative distribution function are represented as: F°(t)- I pfrjWl-IpOi,)) J e H(t)
j-1
Q4\
Fl(t)- I p( Xj ) j6H(t)
where H(t) denotes the subset of [1,...,J], for which travel time trs(xj) is less than or equal to t. The difference between the upper and the lower bounds is: F«(t)-Ft(t) = l - I p ( X j ) j-i
(15)
which is kept constant for any range of t. This is convenient for examining the convergence of the iteration. When we take the next most probable state vector xj+i, the upper bound is lowered and the
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Y. Asakura
lower bound is raised, respectively. That is, Fi(9*Fi*i
( 16 )
Fre j(t) and Fre j(t) denote the upper and the lower bounds for the J-th iteration. Frs,j+i( t ) an dF r s,jti( t ) represent the upper and the lower bounds for the J-th iteration, respectively. This means that the difference of the upper and the lower bounds becomes small enough when a sufficient number of the probable state vectors are included. The cumulative distribution function of travel time Frs(t) is then approximated as: forallUO
(17)
Once the function Frs(t) is estimated, we can evaluate the value of Rrs(c) using Eq.(5) for any criteria of c. This is quite useful for analyzing network reliability measures since it is not necessary to consume additional computational cost for different criteria. The algorithm is similar to the one for approximating the expected value od Rrs(c). That is summarized as follows:
NUMERICAL EXAMPLES Input Conditions Numerical examples are calculated using the test network depicted in Figure 1. The network has 3 nodes and 5 directed links. Table 1 shows the values of pa; the probability of whether link a is not degraded by natural disasters and not closed to traffic. Davidson's function(1966) is used as the link capacity function. That is, t,(Qa) = ta0{l + KQ a /(C a -Q a )} for 0*Q a *C a
(18)
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Reliability measures of an origin and destination pair in a road network
where Qa and Ca denote the traffic volume and the capacity of link a, respectively. K is a parameter of the function and set to 0.5. The travel demand of an OD pair is assumed decreasing for the travel time of corresponding OD pair. An exponential function is employed for the demand function of an OD pair. That is,
where D° and t° denote the upper limit of demand and the free flow travel time of OD pair r-s. Y is a parameter of the function and set to 0.02. We assume the demand exists between origin 1 to destination 2 in the network. The values of D°2 and t°2 are set 100 and 19, respectively.
Figure 1 Test Network (Link and Node Number) Table 1 Link Properties of the Test Network Link Number
Node-Node
Link Probability Free Flow Travel Link Capacity Ca Time tao
Pa
1
1-3
0.95
10
25
2
1-3
0.90
12
30
3
3-2
0.85
9
20
4
3-2
0.93
11
35
5
1-2
0.98
35
40
Computational Results The value of the probability of the normal state xo is 0.662. Solving the variable demand UE problem with link capacity constraints for the normal network xo, we obtain equilibrium travel
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Y. Asakura
time ti2(xo) and demand Di2(ti2(xo)) between the OD pair 1 to 2. Those are 56.06 and 47.94, respectively. Convergence of Algorithm 1 Figure 2 shows the convergence of the upper and the lower bounds of the expected value Ri2(c). The value of c is given 1.5, which means that the OD travel time ti2(x) is acceptable if it is less than or equal to 1.5 times of the OD travel time for the normal state ti2(xo). The number of iterations is 12 when the difference of the upper and the lower bound is set to 0.01. The expected value of Ri2(c) is approximated as 0.983 which is sufficiently close to the exact value 0.985. Probability 1.0
17
25 Iteration Center, J
Figure 2 Convergence of Algorithm 1
Convergence of Algorithm 2 Figure 3 shows the upper and the lower bounds of the cumulative distribution function (c.d.f) Fi2(t) of the OD travel times at the 5th iteration of the algorithm 2. The difference between the upper and the lower bounds is nearly equal to 0.1 and may not be converged enough. Figure 4 shows the approximated function in which the difference of the upper and lower bounds reaches 0.01. As mentioned above, the probability of the normal state XQ is 0.662 and OD travel time ti2(xo) is 47.9. We can easily obtain the value of the reliability Ri2(c) for any value of c, that is the ratio of the OD travel time ti2(x) to the normal travel time ti2(xo).
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Reliability measures of an origin and destination pair in a road network
Using the inverse function F-^t) of the c.d.f function, it will take 1.18 times (56.54) the normal travel time if we travel the OD pair 1-2 with the probability of 0.8. If we accept 1.25 times (59.34) the normal travel time, we can travel the OD pair with the probability of 0.9.
Cumulative Distribution Function, F(t) l.U Upper Bound of CD.F.,F"(t) 0.8
N
i
Jr P
0.6
0.4
Ap prox. C.D.F., F(t) 1.0 0.9
1
J .../ J
'
0.6 Lower Bound of C.D.F., FL(t)
0.4 0.2
0.2
47.94
nn
20
40
60 80 OD Travel Time
Figure 3 Convergence of Algorithm 2
0
20
40
^59.34 \~ /60 56.54
80 100 OD Travel Time
Figure 4 Approximated C.D.F. of OD Travel Time
Comparison with the Non-Congested Case The differences between non-congested and congested cases are compared. For the noncongested case, link capacity constraints are not considered. It is not necessary to solve the UE problem for calculating the travel time between OD pair r-s. The shortest path using the free flow travel time tao of link a is calculated for a state vector in Step.2 of the algorithm above. Figure 5 shows the approximated distribution function of the travel time of the OD pair 1-2. For any criterion c, the value of the reliability measure Ri2(c) for the congested case is smaller than that for the non-congested case. For example, Ri2(c=1.5) is equal to 0.84 for the congested case and 0.98 for the non-congested case, respectively. This implies that the interaction between demand and performance should be considered in evaluating network reliability. The previous reliability analysis using a pure network may overestimate the reliability of the network, and this will result in underestimation of the potential risk.
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Y. Asakura
Reliability 1.0 0.98
0.84 0.8 \
• Without Congestion
. With Congestion
0.6 1.0
Ratio c
Figure 5 Comparison with and without Congestion
CONCLUSION In this paper, a model is proposed for evaluating travel time reliability between an origin and a destination pair in a network deteriorated by natural disasters. The state of a network is not deterministic since some links may be physically damaged by disasters and closed to traffic. The travel time of an OD pair is then stochastic. Reliability measure is defined as a probability of whether one can travel between an OD pair within an acceptable level of travel time. In order to reduce computational cost for estimating the reliability measure, an approximation algorithm is presented through modifying the previously proposed approximation algorithm for estimating the expected value. The cumulative distribution function of OD travel time is approximated using the upper and the lower bounds of the function. Numerical examples shows the convergence of the algorithms. It is found that previous reliability analysis using a pure network may overestimate the reliability of a network, and this will result in the underestimation of the potential risk. Further computational studies for different input conditions are now on going. In particular, the number of required iterations for larger networks are examined. When the value of link probability is not so small, it is expected that the computational cost for a large network is approximately proportional to the scale of the network. The results will be presented in the near future.
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REFERENCES Asakura, Y. and M. Kashiwadani (1991). Road Network Reliability Caused by Daily Fluctuation of Traffic Flow. Proceedings of the 19th PTRC Summer Annual Meeting in Brighton, Seminar G, pp.73-84. Asakura, Y. and M. Kashiwadani (1995).Traffic Assignment in a Road Network with Degraded Links by Natural Disasters. Journal of the Eastern Asia Society for Transport Studies, Vol.1, No.3, pp.1135-1152. Davidson, K.B. (1966). A Flow-Travel Time Relationship for Use in Transportation Planning. Proceedings, Australian Road Research Board, Melbourne, Vol.3,pp. 183-194. Daganzo, C.F. (1977). On the Traffic Assignment Problem with Flow Dependent Costs-I. Transportation Research, Vol.11, pp.433-437. Du, Z. P. and A. J. Nicholson (1993). Degradable Transportation Systems Performance, Sensitivity and Reliability Analysis. Research Report, No. 93-8, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand, lida, Y. and H. Wakabayashi (1989). An Approximation Method of Terminal Reliability of Road Network Using Partial Minimal Path and Cut Set. Proceedings of the 5th World Conference on Transport Research, Vol.IV, Yokohama, Japan, pp.367-380. Lam, Y. and Li, V.O.K. (1986). An Improved Algorithm for Performance Analysis of Networks with Unreliable Components. IEEE Transactions on Communications, Vol.COM-34, pp.496-497. Li, V.O.K. and J.A. Silvester (1984). Performance Analysis of Networks with Unreliable Components. IEEE Transactions on Communications, Vol.COM-32, No.10, pp.1105-1110. Sanso, B. and L. Milot (1994). A Reliability Model for Urban Transportation Planning. Preprints in TRISTAN-II Conference, Capri, Italy, pp.617-622. Sheffi, Y. (1985). Urban Transportation Networks. Prentice-Hall, Inc., Englewood Cliffs, N.J. Turnquist, M.A. and L.A. Bowman (1980). The Effect of Network Structure on Reliability of Transit Service. Transportation Research-B, Vol.l4-B, pp.79-86.
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19
VARIATIONAL INEQUALITY MODEL OF IDEAL DYNAMIC USER-OPTIMAL ROUTE CHOICE David E. Boyce Department of Civil and Materials Engineering, University of Illinois, Chicago, IL 60607 USA Der-Horng Lee Institute of Transportation Studies, University of California, Irvine, CA 92697 USA Bruce N. Janson Department of Civil Engineering, University of Colorado, Denver, CO 80217 USA
ABSTRACT An ideal dynamic user-optimal (DUO) route choice model is described for predicting dynamic traffic conditions, as required for off-line evaluation of Advanced Traffic Management Systems and Advanced Traveler Information Systems. The model is formulated as a variational inequality (VI), a general way of describing a dynamic network equilibrium. Although routebased VI models have an intuitive interpretation, their computational complexity makes them intractable for real applications. Consequently, the proposed model is formulated as a linkbased variational inequality for use in large-scale implementations. Using the diagonalization technique with discrete tune intervals, the model is solved to a specified level of convergence. Computational results for a real, large-scale traffic network are presented.
INTRODUCTION The ideal dynamic user-optimal (DUO) route choice problem is to determine vehicle flows at each small interval of time so that the travel times experienced by vehicles departing at the same time, and with the same origin-destination (O-D) attributes, are minimal and equal. This paper presents a discrete-time variational inequality (VI) formulation and solution algorithm for this ideal DUO route choice problem. Variational inequality models may be formulated either in terms of route flows or link flows to describe a dynamic network equilibrium. For reasons of computational tractability, a link-timebased VI model is proposed. Dynamic network constraints including flow conservation, flow propagation, first-in-first-out (FIFO), nonnegativity and definitional constraints are presented and discussed. Although the proposed model has a discrete-time formulation, temporally289
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D. E. Boyce et al.
correct routes and time-continuous flow propagations are maintained in a quasi-continuous manner. Based on the link-time-based ideal DUO route choice conditions, the equivalent variational inequality is derived A diagonalization algorithm is proposed to solve this VI model to convergence using fiveminute time intervals. First, the inner problem, a multi-interval, time-varying demand route choice problem with fixed node time intervals, defined in the subsection on Flow Propagation Constraints, is formulated. Then, the shortest route travel times and node time intervals constraints are updated in the outer problem. The inner problem and the outer problem are solved iteratively until a specified level of convergence is reached. This model is implemented for the ADVANCE Network which is located in the northwestern suburbs of the Chicago area; the study area includes about 800 square kilometers (300 square miles), has 447 zones, 406,560 trips during the 4-6 PM period, and nearly 10,000 nodes and 23,000 links in an expanded intersection representation. To generate time-dependent traffic characteristics for a real network, realistic traffic engineering-based link delay functions such as proposed by Akcelik (1988) are applied for improved estimation of link delays at various types of intersections. To the best knowledge of the authors, this is the largest dynamic route choice solution obtained thus far. The model was solved using the CONVEX-C3880 at the National Center for Supercomputing Applications (NCSA), University of Illinois at Urbana-Champaign. Convergence and computational results are presented and analyzed.
IDEAL DYNAMIC USER-OPTIMAL STATE The travel-time-based ideal DUO state is defined as (Ran and Boyce, 1996): Travel-Time-Based Ideal DUO State: For each O-D pair at each interval of time, if the actual travel times experienced by travelers departing at the same time are equal and minimal, the dynamic traffic flow over the network is in a travel-time-based ideal dynamic user-optimal state. Under the ideal DUO state, travelers have no reason to change their routes. Therefore, the obtained DUO state can be viewed as an equilibrium.
DYNAMIC NETWORK CONSTRAINTS We consider a multiple O-D network that is represented by a directed graph G=(N, A) where N is the set of nodes and A is the set of directed links. In the following constraint sets, the index r denotes an origin and s denotes a destination. In addition, both d and t (t>d) denote a time interval. However, d denotes the departure time interval and t denotes a specific time interval during the journey.
Variational inequality model of ideal dynamic user-optimal route choice
291
Definitional Constraints Consider a fixed time period T which is long enough to allow all vehicle flows departing during the peak period to complete their journeys. Let xa[t] be the total flow on link a in time interval t and x"[d] be the flow of vehicles from origin r to destination j on link a that departed in time interval d. Therefore, for total flow on link a in time interval t, Equation (1) must hold.
V] #M
Vfl, t;a = (ij)
(1)
whereof/] is the fraction of all flows departing zone r in tune interval d that crosses node / in time interval /. Equation (1) defines total flow on link a in time interval / to be the sum of flows departing from zone r in any time interval d from interval 1 up to and including t (t > d) using link a in time interval t.
Flow Conservation Constraints Flow conservation needs to be considered for different types of nodes including intermediate nodes, origins and destinations for a dynamic route choice model. Define n^ as the minimal travel time actually experienced by flows departing from origin r to node i in time interval d, where n^ denotes the number of time intervals traversed in^ and A t is defined as the duration of each time interval.
n* = CD + 1
if
CD < n* I A/ < a> + 1
(2)
where a> is an integer (0 < CD < T) . Equation (2) makes the actual travel time n% equal to a multiple of the tune increment A / . Define q"d as flows from zone r to node 7 departing in tune interval c?via any route. Let x™p[t] be the flow on link a in time / from zone r to s on route/?. Equation (3) constrains the inflow minus outflow at any intermediate nodey (j #r,s) in each time interval (/ > d) to the proper departure flows in each time interval between all OD pans.
(3) sp
where
d=l_aeB(j)
2>'[d] = *,[]
VrJ.rf.
A(j) is the set of links whose tail node isy, and B(j) is the set of links whose head node is j. Conservation of flow at origin r requires the flow originating at node r in time interval d to
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D. E. Boyce et al.
equal the flow entering the links leaving origin r in time interval d. Equation (4) states the flow conservation at origins. ^™*
^~*
aeA(r)
p
rsvr
Similarly, conservation of flow at destination s requires the flow exiting at node s in time interval / to equal the flow entering destination s in time interval t. The flow conservation at destinations is expressed by Equation (5).
Note that eK[t] is a variable; Equation (5) describes the solution of the model, but does not constrain x"p[t] in this version.
Flow Propagation Constraints The proposed DUO route choice model requires nonlinear mixed-integer constraints with node time intervals (a^|V]) and flow fractions (^[t]) indicating the time intervals where flows originating from each origin cross each node in order to maintain temporally-correct routes and time-continuous flow propagations over time intervals. Define afa] as a [0,1] variable indicating whether the flow departing zone r in time interval d has crossed node / in tune interval /. Each node time interval acts as an if-then operator to activate or deactivate certain constraints. A node time interval only applies to the last trip (strictly, the end of the platoon or pulse of flow) departing hi each departure time interval. The difference between the crossing tunes at node / of the flows departing in successive time intervals, defined as A n^, is applied to equations (6)-(8) to determine the temporal spread of trips crossing node / from the same origin. arl[t]
V& = 0
(6)
for which ^ -(t-l-k)At< 0
(7)
^^ai[t]
(8) 1
V * = l for which n^' -(f-l-A)Af >0 where k.ndri -ndrt -nd^1 and TT° =n\ - A / Vr,/,c? where k is used to count the number of boundaries of tune intervals spanned by the difference in node crossing times of the last vehicle in the platoon between successive tune intervals. We now use Figure 1 to explain Equations (6)-(8). In Figure 1, the y-axis denotes the departure time intervals of platoons and the x-axis denotes the sequence of nodes along an example route. As suggested hi Figure 1, platoon 1 departed from a given origin in departure tune interval 1; the last vehicle of platoon 1 crossed node A in tune interval 3. Similarly, Platoon 2 departed
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from the same origin as platoon 1 in time interval 2 and the last vehicle of platoon 2 crossed node A in time interval 4. Now consider the node crossing times of those two platoons at node D. Equation (6) determines the fraction of platoon 2 crossing node D in time interval 7. This fraction equals the elapsed tune between the starting time of interval 7 and the node D crossing time of the end of platoon 2 (represented by dash line in Figure 1), divided by the elapsed time between the node D crossing times of platoons 1 and 2. Equation (6) is designed for k = 0. The need to take min[l, etc.] is that this calculation can exceed 1 when computing this fraction for platoons departing in time interval 1, since there is no node crossing time for a previous platoon.
Last vehicle of Platoor
E
La t vehicle of Plati on 1
4
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B C D Sequence of nodes along an example route
E
Figure 1: Effect of flow propagation constraints
Equation (7) determines the fraction of platoon 2 crossing node D in the whole time interval (if any) between the node D crossing times of platoons 1 and 2. The whole tune interval is interpreted as follows. Suppose platoon 1 crosses node D in time interval 5 and platoon 2 crosses node D in interval 7 because delays have caused vehicles in platoon 2 to fall farther behind platoon 1. Under this situation, a fraction of platoon 2 crosses node D in interval 5, another fraction of platoon 2 crosses node D with whole time interval 6 and another fraction of platoon 2 crosses node D in interval 7. In this case, however, there are no whole time intervals. Equation (8) determines the fraction of platoon 2 crossing node D in time interval 6. This fraction equals the elapsed time between the crossing time of node D of the last vehicle in platoon 1 (represented by solid line) and the starting time of interval 7, divided by the elapsed time between the crossing times of node D of platoons 1 and 2. Equation (8) is designed for k = 1. The need to take max[Q, etc.] in Equation (8) is that this fraction can be negative when
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computing this fraction for platoons departing in time interval 1, since there is no node crossing tune for a previous platoon. First-In-First-Out Constraints Equations (9)-(12) state FIFO constraints between all 0-D pairs according to their travel times in successive time intervals. Define /?* as the time at which the last flow departing zone r in time interval d crosses node / via its shortest route less FIFO delay time at node /'; 9^\t~\ is defined as the fraction of a tune interval / that the last flow departing zone r in time interval d crosses node i; and ju^[t] is defined as the average travel time on link a of the last flow departing zone r in time interval d. The value / z ( 0 < / z < l ) i s the fraction of a time interval that the end of the platoon (the last vehicle) departing from zone r in tune interval d must follow the end of the platoon departing from zone r in tune interval d-\. Vehicles are assumed to make one-for-one (or zero-sum) exchanges of traffic positions along any link, which is an acceptable and expected feature for any aggregate traffic model (Janson and Robles, 1995). , ndrt~l +AA /) V r,i,d and n°rt = *i - A / ^[/]
Vr,i,d,t
(10) d
M)r a (x fl [5])]a n M Vr,a,d,t,s = t-l V r,a,d,t, s = t-\;
(9)
4«'M^^M«"M where A r'a[s]= ra(xa[s])-tfa[t]
(11) (12)
Equation (9) is a vehicle following constraint that regulates flows departing from the same zone in successive time intervals from passing each other. When solving for n^ on the left hand side of Equation (9), ndri~l on the right hand side is held fixed. If h = 0, a trailing platoon can completely overlay (but not overtake) a leading platoon so that the two platoons become coincident, which is not realistic. If h = 1, a trailing platoon can never partly gain ground on a leading platoon. Since Equation (9) does not insure FIFO ordering between all O-D pairs, Equations (10)-(12) are required. Equations (10) and (11) determine the average travel time on link a of the end of the platoon departing zone r in tune interval d adjusted for the time into interval t versus t-1 that the platoon enters the link. Equations (10) and (11) dampen speed transitions between time intervals in a quasi-continuous manner so that vehicle speeds do not abruptly change if flows enter links just seconds before or after a time interval change. Equation (12) is needed to prevent FIFO violations in cases where link travel times exceed A / . Equation (12) does not entirely replace the need for Equation (9). Equation (12) allows trips between different O-D pairs to become concurrent while sharing the same route. Equation (9) insures a minimum separation of the last platoon departing from the same zone in successive time intervals. Trips from the same zone bunch together and cause excessively dense flows if Equation (9) is removed.
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Nonnegativity Constraints Finally, all variables must be nonnegative at all time intervals. We have x"[t]>0
Vr,s,a,t; Vr,s,t;
(13) (14)
Vr,/,rf,/.
(15)
LINK-TIME-BASED CONDITIONS We now derive the equivalent mathematical inequalities for the travel-time-based ideal DUO state using link variables. For any route from origin r to destination s, link a is defined as being used in time interval / if x"[t] > 0 . Define n** as the minimal travel time actually experienced by flows departing from origin r to node / in tune interval d, the asterisk denoting that the travel time is calculated using DUO traffic flows. For link a = (i,j), the minimal travel tune n** from origin r to nodey should equal to or less than the minimal travel time nd* from origin r to node / plus the actual link travel time Ta[d + ?rd] in time interval [d + n*], where the nd denotes the number of time intervals traversed in ndrt ; see Equation (2). The first time interval of [d + nd} must be the earliest time interval that flow departing zone r in tune interval d can enter link a. It follows that ndn' + ra[d + n*'} > n*
V a = (i,j),r,d .
(16)
For each O-D pair (r, s), if any departure flow from origin r in time interval d enters link a at the earliest tune interval [d + n**], or x™*[d + ?fdi*]> 0, then the ideal DUO route choice conditions require that link a is on the route with minimal travel time. In other words, the minimal travel tune ^*from origin r to node j should equal the minimal travel time Jtdn* from origin r to node / plus the actual link travel time Ta[d + n*"] in time interval [d + n**]. It follows that n£ =x? +Ta[d + xf]
if
x?[d + xdi']>0
V a = (i,j\r,s,d.
(17)
The above equation is also equivalent to the following: ']-x*] = Q V a = (i,j),r,s,d.
(18)
Thus, the link-time-based ideal DUO route choice conditions can be summarized as follows: x?+Ta[d + x?]-rf>0
Va = (iJ),r,d;
(19)
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*T\d + *?] [<* + ra[d + *?*] - <] = 0 V a = (i,j),r,s,d; a = (i,j),r,s,d.
(20) (21)
THE LINK-TIME-BASED MODEL Define O?a* as the difference between the minimal travel tune from zone r to node / plus the travel time on link a and the minimal travel tune from zone r to node j for flows departing from zone r in tune interval d: ^=nd:+Ta[d + ndrt*}-nd;.
(22)
The link-time-based ideal DUO route choice conditions are rewritten as: CC>0
\/a = (i,j),r,d; &* = 0 V a = (i,j),r,s,d; 0 Va = (i,j),r,s,d.
(23) (24) (25)
The equivalent VI formulation of the link-time-based ideal DUO route choice conditions defined in Equations (23)-(25) can now be stated as follows. l*{<^ + ^]-
(26)
where * denotes the DUO state, and where the dynamic traffic flow pattern must satisfy the constraints described in Equations (1)-(15).
SOLUTION ALGORITHM This algorithm consists a sequence of diagonalizatioo. iterations. First, in the initialization step, the node time intervals and shortest route travel times are initialized based on the initial link flows. Then, the algorithm solves a sequence of route choice problems (called the inner problem) using the Frank- Wolfe (F-W) algorithm with fixed node time intervals until the convergence criterion of the route choice problem is satisfied. Next, the node time intervals and shortest route travel times are updated (called the outer problem) based on the most recently assigned link flows from the inner problem. Adjustments of link capacities are made between the inner and outer problems to account for capacity changes caused exogenously (e.g., signal timing changes, incidents and other unexpected capacity reduction events) or generated endogenously (e.g., queue spillbacks). The algorithm terminates when the changes of the node time intervals obtained from two consecutive outer problems are within a prespecified tolerance. Figure 2 shows the steps of the solution algorithm. The steps of the solution algorithm are described as follows:
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Step 1: Initialization. Input the O-D matrix, time-dependent trip departure rates, network data and initial link flows (optional). If initial link flows are used, the shortest route travel times and node time intervals are initialized as well. Step 2: Solve Route Choice Problem (Inner Problem). Using the F-W algorithm, the route choice problem is solved to convergence with the optimal values of the node time intervals from the outer problem. Step 3: Link Capacity Adjustments. Perform exogenous and endogenous link capacity adjustments. Step 4: Update the Node Time Intervals (Outer Problem). Update the node time intervals with time-dependent link flows and FIFO constraints by solving the shortest route problem. Step 5: Convergence Test for Outer Iterations. Sum the total number of differences of the node time intervals (NDIFFS) between consecutive outer iterations. If NDIFFS < allowable percentage of all node time intervals, then the algorithm stops; otherwise, return to Step 2.
2. Inner Problem: Solve the route choice problem with fixed node time intervals using Frank-Wolfe algorithm 4. Outer Problem: Update the node time intervals with time-dependent link flows and FIFO constraints by solving the shortest route problem
Figure 2: Flowchart of the solution algorithm
To update the node time intervals (adrt[t]), the following procedure is applied in Step 4:
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1. find the shortest route travel times (ndri) and time intervals (n^ ); 2. reset values of (ocd[t]) as follows: if (nd
TEST NETWORK AND LINK TRAVEL TIME FUNCTIONS The ADVANCE Network was selected for testing the proposed DUO route choice model. As shown in Figure 3, the ADVANCE Network is located in the northwestern suburbs of Chicago and covers about 800 square kilometers (300 square miles).
Figure 3: The ADVANCE Test Area hi the northwest suburbs of Chicago
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The specific delay function for links at signalized intersections applied in the research has the following form (Akcelik, 1988):
d=
0.5C(l-w)2 1 "2 8(x0.5)1 l — + 9007V x-l + J(x-l) + ' 1-MX
\
C1
\
(27)
where d is the average delay per vehicle (second/vehicle), C is the signal cycle length (second), u = g/C is the green split, g is the green time (second), x = v/c is the flow-to-capacity ratio, T is the duration of the flow (hour) and y - 1 for x > 0.5, and 0, otherwise. The first term, called the uniform delay, reflects the average delay experienced by drivers in undersaturation conditions, that is when the arrival flow does not exceed capacity. In oversaturation conditions, x = 1 is used in the uniform delay term. The second term of Equation (27), called the overflew delay, reflects the delay experienced by the vehicles when the flow rate is close to or exceeds the capacity. The BPR function is temporarily used for estimating delays at major/minor priority intersections with the following form (Bureau of Public Roads, 1964): 4
]
(28)
where / is the link travel time, v is the link flow and c is the capacity of the link at a specified level of service. Although using the BPR function as an intersection delay function is somewhat outside its designated scope, the lack of alternative functions with the desired analytical properties for use in a network equilibrium model mandates the use of the BPR function. As for the delay function for all-way-stop intersections, the following exponential delay model is used (Meneguzzer et al., 1990). d = exp[3.802(v / c)]
(29)
where d is the average approach delay (second/vehicle), v is the total approach flow and c is the approach capacity. Equation (29) is suitable for use in a network equilibrium model, since it is defined for any flow-to-capacity ratio.
COMPUTATIONAL RESULTS AND ANALYSIS The model was implemented on the CONVEX-C3880 at the National Center for Supercomputing Applications (NCSA), University of Illinois at Urbana-Champaign. For the ADVANCE Network with afternoon peak of two hours (4 to 6 PM) divided into 24 five-minute tune intervals, a total flow of 406,560 trips was solved from a zero-flow initial solution. As shown in Figure 4, the rate of changes of the node time intervals is calculated as the changes in the node time intervals divided by the total possible changes of the node time intervals (i.e., nodes x zones x intervals; 104,061,600 total for the afternoon peak period) between
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consecutive outer iterations. Figure 4 reveals that this model was solved quite smoothly for the applied travel demand over the ADVANCE Network. Note that we omit the value of the first outer iteration in Figure 4 to provide a better display of the rest of the iterations.
c
0.08
0.02
5
6
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10
Outer Iteration
Figure 4: Rate of change of node tune intervals for the afternoon peak period The flow profile of a selected approach link and the time-dependent flow departure rates are displayed in Figures 5. In this research, time-of-day (e.g., afternoon peak) O-D tables were factored from CATS (Chicago Area Transportation Study) estimates for 1990. To solve this model, the afternoon peak period is further divided into 24 five-minute tune intervals. The fiveminute flow departure rates for each origin zone (447 zones total) are based on half-hour departure rates provided by CATS. Therefore, the time-dependent travel demand is loaded onto the network by multiplying the time-dependent flow departure rates (which sum to one) and the afternoon peak O-D table. The highest flow departure rates occur between intervals 11 and 12 (4:50 to 5:00 PM). Based on the derived flow departure rates, flows on the selected link are obtained and plotted. In the first few time intervals, the link flows tend to be low because the solution algorithm started from an empty network. The osculation of flows hi the last few time intervals implies the occurrence and dissipation of the queue spillback of this link. Unfortunately, empirical link flow and flow departure rate data are not available for the ADVANCE Network, either hi general, or more specifically for the O-D matrix used in this solution. These data, as well as route flow data, are urgently needed to advance the state of the art of network modeling for ITS.
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0.046 Non-intersection Link 8658 ^•H Row Departure Rate
- Link Flow 0.044
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!n
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-o 0)
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5-minute Time Interval (4 PM to 6 PM) Figure 5: Predicted link flows vs. flow departure rates per time interval
CONCLUSIONS This paper presents a discrete-time formulation of the link-time-based variational inequality model of dynamic user-optimal route choice for evaluating time-dependent traffic characteristics for Advanced Traffic Management Systems and Advanced Traveler Information Systems, such as the recently concluded ADVANCE Project. To date, very few traffic flow prediction models are suitable for ATMS and ATIS applications. Although not yet fully validated, this model is able to predict time-dependent traffic characteristics for a large-scale traffic network which are reasonable and internally consistent. Eventually, dynamic route choice models should be integrated into a traffic management center to support the decisions on the adjustments of arterial signal timing, ramp metering, incident management and future route guidance strategies, etc. This paper describes the largest dynamic route choice model that has been solved to date, to the best knowledge of the authors. Consequently, this research establishes a new benchmark for dynamic route choice modeling. Solving dynamic route choice models on a realistic and largescale traffic network is no longer an intractable task.
ACKNOWLEDGMENTS The authors are pleased to acknowledge the following sources of support in the preparation of this chapter: the Illinois Department of Transportation and the Federal Highway Administration
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through the ADVANCE Project; the National Institute of Statistical Sciences; the National Center for Super computing Applications, University of Dlinois at Urbana-Champaign, for the use of computational facilities. The last two organizations are supported by the National Science Foundation, Arlington, VA.
REFERENCES Akcelik, R. (1988). Capacity of a Shared Lane. Australian Road Research Board Proceedings, 14(2), 228-241. Bureau of Public Roads. (1964). Traffic Assignment Manual. U.S. Department of Commerce, Washington, DC. Janson, B.N. and J. Robles. (1995). A Quasi-Continuous Dynamic Traffic Assignment Model, Transportation Research Record, 1493, 199-206. Meneguzzer, C., D.E. Boyce, N. Rouphail and A. Sen. (1990). Implementation and Evaluation of an Asymmetric Equilibrium Route Choice Model Incorporating Intersection-Related Travel Times. Report to the Illinois Department of Transportation, University of Illinois, Chicago. Ran, B. and D. Boyce. (1996). Modeling Dynamic Transportation Networks. Springer-Verlag, Heidelberg.
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TRAVEL TIMES COMPUTATION FOR DYNAMIC ASSIGNMENT MODELLING C. Buisson2, J.P. Lebacque l, J.B. Lesort2 1 CERMICS, Centre d'Enseignement et de Recherche en Mathematique, Informatique et Calcul Scientifique. ENPC (Ecole Nationale des Ponts-et-Chaussees), La Courtine, 93167 NOISY-LE-GRAND Cedex, FRANCE. Tel: +33 1 64 15 35 81. email: [email protected]. 2 LICIT, Laboratoire d'Ingenierie Circulation Transport, ENTPE (Ecole Nationale des Travaux Publics de l'Etat)-INRETS (Institut National de la Recherche sur les Transports et leur Securite), rue M. Audin 69518 VAUX-EN-VELIN cedex, FRANCE. Tel: +33 4 72 04 77 13. emails: [email protected] & [email protected].
ABSTRACT Travel time is the commonly used cost function for dynamic road traffic assignment. The paper begins with a review of various travel time and aggregation methods used in simulation models. It further attempts to clarify the different- definitions of travel times given in the literature: experienced travel time, predictive travel time, instantaneous travel time and to give mathematical definitions of these quantities, notably for macroscopic models. Finally recursive computational procedures are given for the discretized case.
INTRODUCTION Travel time is the basic criterion used in most assignment modelling processes. However, few examples of systematic in-depth investigation exist in the literature concerning the definition and properties of travel time models that are in use. The aim of the present paper is twofold: to make a cursory review of various existing travel time formulas and concepts, and to introduce some new formulas for reactive dynamic assignment. The contents of the paper are the following: first, a brief description of travel time functions and models used in dynamic assignment modelling (considering both the more theoretical formulations and the more application-oriented computational algorithms used in traffic flow simulation models); second, the analysis in the context of macroscopic traffic modelling of three travel time concepts: experienced travel time, predictive travel time and instantaneous travel time. Emphasis will bear on experienced and instantaneous travel times, which are most useful in the context of reactive dynamic assignment, and for which various computational formulas, many of them recursive, will be developed, suitable for both interrupted and uninterrupted traffic and most macroscopic models. 303
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AN OVERVIEW OF THE LITERATURE Definitions The trajectory of the individual vehicle constitutes the basis of the computation of travel times. For a link or a trip, the experienced travel time (ETT) of the vehicle can be deduced, estimated at the exit of the link or at the end of the trip. Conversely, at the beginning of the trip, or on the entry of the vehicle on the link, one might try to estimate a predictive travel time (PTT), deduced from prevailing traffic conditions. When stationary traffic conditions are assumed, both definitions concur and can be extended in an obvious way to links, since all vehicles crossing a link have the same travel time. Link travel times can then be defined and route travel times computed as the sum of travel times on the links forming the route. When dynamic traffic conditions are considered, the problem becomes more complex and the definition of link or route travel times is no longer unique or straightforward. If the above definitions remain valid for an individual vehicle, the definition and manipulation of link and route travel times depend very much on the type of assignment considered. Definitions of various types of dynamic assignment have been widely discussed in the literature, (see for instance (Papageorgiou 1990) for a very clear presentation). The typology of travel times is closely related to that of assignment problems. These can roughly be classified according to the nature of the optimum, whether user or system optimum, and whether predictive or reactive. Furthermore, every assignment problem is related to a specific traffic flow model and a specific time-scale, which have great impact on the corresponding travel-time model. For a system optimal assignment, the criterion used is the total time spent in the network (i.e. the sum of all individual travel times). This can be considered as an extension of the notion of mean travel time, and is strictly equivalent if only one origin/destination is considered. For a user optimal dynamic assignment, it is necessary to distinguish two cases. If a reactive optimum is considered (or the similar Boston equilibrium as defined by (Friesz et al. 1989)), a notion of instantaneous travel time (ITT) has to be defined, which has no physical meaning but characterizes traffic flow conditions at a given instant. This ITT might be constituted of link PTTs, link ETTs, or be a completely synthetic index of the link or network flow state. If a predictive optimum is considered, predictive travel times must be used, only known at the end of trip time, or predicted, or estimated through an iterative assignment procedure. The simple cases
In traffic conditions that are stationary, or simplified as such, the most usual way to compute travel times is to use some function deriving link travel times from link flows. One can refer to (Branston, 1976) which presents an extensive review of travel time function in use at that time. Some more recent papers on that subject are indicated in the work of Weynmann et al. (1994), in which travel time functions having finite values for over-saturated traffic conditions are briefly presented. Koutsopoulos and Habbal (1994) present a more detailed review, insisting on travel time functions used in practice for traffic equilibrium models. On the contrary, when a system optimal assignment is computed, individual link travel times must be replaced by a global cost function: the total time spent on the network by all users. This quantity can be easily calculated by integrating the number of vehicles present in the whole network over the optimization horizon, as was initially proposed by Merchant and Nemhauser (1978a and b). This work has been re-used in quite a different context: in the case of route guidance, the discretized macroscopic METACOR model enables the operator to maintain a system optimum (Eloumi 1996).
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A large part of the more theoretical literature concerned with predictive optima is represented by models in which the basic component of the traffic flow model itself is constituted of travel time models. These are supposed to be functions of the state of the network, and vehicles are propagated along links according to those travel times which must therefore be considered as PTTs (Friesz et al. 1993, Fernandez and De Cea 1994, Ran and Boyce 1994, Astarita 1996, Ran et al., 1996). The only real difficulty with such models is that of internal coherence. We shall say again a few words later about these models, but for the time being, let us note that they are characterized by: a large time-scale, uninterrupted traffic, the lack of explicit supply constraints (reputedly such constraints are implicit in the travel time function itself), travel times which are a function of the network state.
Interrupted traffic flow: temporal aggregation When a traffic flow model is used in relation to reactive assignment (simulation of information or guidance on the basis of instantaneous travel times) the time scale of this model is typically shorter than a minute. Therefore, some kind of temporal aggregation or averaging must be applied, in order to smoothen the short-term travel time fluctuations resulting from traffic light cycles etc.... Further, if the outflow of the link is nil, most natural travel time estimates yield infinite or unrealistic values. Predictive travel time, estimated on the basis of prevailing traffic conditions, will be underestimated. Experienced travel times will not take any effective value during the period in which the exiting flow is nil (but may be given conventional values, as will be seen later). Instantaneous travel times, when calculated on the basis of the link length divided by the mean speed on that link, will tend to be infinite. These difficulties led most authors to compute directly average travel times, with the choice of the duration of the averaging period depending closely on the specific nature of the problem addressed to by the traffic model. We shall make in the sequel a brief review of the averaging methods mentioned in the literature, in various cases of optimum calculation. But first let us stress a point we deem important: that the computation of travel times and their averaging relate to two distinct processes and should be strictly separated. Indeed they belong to two distinct levels of the traffic model: travel time computation is related to the network state estimation, i.e. the basic level of the model, whereas travel time averaging concerns user information and/or traffic management and control, and is therefore related to the control level of the model, in some broad sense. Hence there are two very distinct problems that must be addressed: the choice of travel time estimates with reasonable properties, both physical and computational, and the temporal aggregation of these travel times. It is the former problem that will be examined in some more detail in the second part of the paper. Let us turn towards the literature on temporal aggregation and examine a few cases. Mesoscopic traffic models divide travel time into two parts: the journey time and the queue waiting time. In the reactive assignment model DYNASMART (Jayakrishnan et al., 1994), the total travel time is calculated at each time step and, to the knowledge of the authors of the present paper, never averaged. In case of fixed time signals, the queue length is calculated on the basis of the effective status of the traffic signal. In the case of traffic actuated signals, at the end of each cycle, an averaged exit capacity of the link is used to compute the exiting flow. CONTRAM is one of the earlier models developed to simulate the user optimum in the case of day-to-day variation (Leonard et al. 1978, Leonard et al. 1989). The peak hour is divided into a dozen periods. For each period and for each link controlled by a traffic signal an average queueing waiting time is calculated. The calculation is made according to time dependent queuing theory, thus reflecting the stochastic nature of vehicle arrivals. A queuing time is calculated for the whole period, based on the total demand during that period and on the ratio of green over the cycle duration. In the more recent mesoscopic model of Weynmann et al. (1994), the delay is calculated for the fluid, desaturating and
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saturated cases of the traffic signal. The averaging period is taken to be the minimum common multiple of all cycle durations of the network. During this period, the inflows are supposed to be constant. To cope with the traffic light signal problem, the recent macroscopic model STRADA (Buisson et al., 1995a, b) uses also an averaging period that is taken to be the minimum common multiple of all cycle durations. During this period, demand can vary. INTEGRATION uses a experienced travel time (ETT) to simulate the on-route guidance of drivers. This ETT is aggregated spatially in order to yield link travel times or route travel times. The link travel time is computed typically every five minutes. No mention is made in the most recent paper (Van Aerde, 1995) of the impact of traffic signal cycles on the average values of travel times or of really severe incidents (remaining flow nil) affecting more than one such averaging period. Travel time estimation in the case of incidents is a relatively new and expanding subject, motivated notably by route guidance problems. If the incident totally blocks the flow on some network link, the travel time on that link tends to be infinite. But this is not a useful piece of information because the assignment model can compute the shortest path without considering that link (which obviously does not belong to the shortest path). On the other hand, for simulation models whose travel time estimate relate to the ETT category, such as INTEGRATION, using the travel time of the last vehicle to exit the link leads to a systematic underestimation of the mean travel time values when the exiting flow is nil. If there is a residual flow, the mean travel time resulting of ITT-like estimates might take on absurd values like one day or one week. Various heuristic methods have been proposed, as for instance in the work of Cremer et al. (1993). These authors propose to estimate the time necessary for drivers to exit the incident-impeded link by summing the total number of vehicles between the entrance and the incident bottleneck and dividing this sum by the remaining bottleneck outflow.
Interrupted traffic flow: spatial aggregation All models must also address the problem of spatial aggregation, i.e. the computation of travel times along paths. Spatial aggregation is trivial for both static models and dynamic system optima, as mentioned above. For dynamic models using PTTs as their basic propagation model, the situation is straightforward enough. CONTRAM for instance (Leonard et al. (1978)), or the model described by Drissi-Kai'touni et al. (1992), intend to reproduce the assignment of commuters on the basis of the travel time experienced the day before. Therefore, these models use a spatial aggregation based on the following rule. To compute the total travel time in a network from one origin to one destination, the mean travel time of the first link at the date of entrance in the network is used. The travel time of the second link is the one computed for the moment following the exit of the first link, and so on from one link to the next. This path travel time is effectively a predictive travel time and reproduces the conditions encountered by a vehicle. Other models cannot easily compute such a predictive travel time and must rely on some link ITTs to be combined in order to yield path ITTs. No car actually experiences such a travel time, but it can be a good estimation of the conditions encountered on the network at that moment. This is the solution retained by the macroscopic assignment models (METACOR and STRADA) which aim to reproduce the effect of a guidance and/or information system. It is also, for the time being, used by DYNASMART. In the next sections we shall try to define such travel times in a rigorous manner.
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Travel times computation for dynamic assignment modelling LINK TRAVEL TIMES: A GENERAL FRAMEWORK Introduction
Our aim in the following sections is to give rigorous definitions and computational procedures for different travel times within the macroscopic approximation of interrupted traffic flow. The starting point of our analysis of possible travel time expressions is the single link, say [a, b], with its associated traffic flow modelled with the help of the usual macroscopic variables Q (flow), K (density) and V (speed).
b Nearly everything we shall say in the sequel does not depend on the specific nature of the underlying traffic flow model, whether first- or second-order etc A crucial role will be played by the speed field V(x,t) = (V(x,t),l), assumed to be integrable, with existence and unicity of the corresponding field-lines. Let us denote by X the vehicle trajectory associated to such a field-line: X(xo,to;t) is the position at time t of the user whose position at time t0 is XQ. X(xo,tQ-,t) is the solution at time t of: x(t) = V ( x ( t ) , t ) x(tQ) = x0 . Since the field-lines associated to trajectories X do not intersect in the (x, t)-plane, this kind of description is intrinsically in agreement with the FIFO hypothesis: in accordance with this representation, vehicles exit the link in the precise order they entered it. Considering a vehicle entering the link at time t, exiting it at time E(t), it follows X(a,t; E ( t ) ) = b. The function E is increasing and admits an inverse /: /(£) is the time at which a vehicle about to leave the link at time t has entered it. If V(b,.) = 0 during some time interval, the definition of E and the relationships between / and E are not completely straightforward since no vehicle may leave the link during such an interval. Indeed it is necessary to define E(t) as: E(t)d= mfs{s/X(a,t-s)>b} . It follows that
I(E(t)) = t \/t , but the converse is not true, indeed:
[ E(I(t)) = t if V(b,t) >0 [ E(I(t)) > t if V(b,t) = 0 . This last inequality reflects the fact that the vehicle about to leave the link (after having entered it at time I ( t ) ) must wait till the speed at exit point b becomes > 0 again. The graphs of E and / are illustrated hereafter; they are of course symmetric. Kt)
308
C. Buisson et al.
Another way yet to express the complications resulting from V(b,.) = 0 is to note that V(b,t) = Q implies f (t) = 0. Experienced and predictive travel times
Then we define: ETT(a, b-1) d=f t - I ( t ) the experienced travel time (of the user about to exit the link at time t). Let us note an important consequence of the preceding considerations: (1)
V(b,t) = 0 =» ^-ETT(t) = 1 . at
Conversely, PTT(a, b; t) d=f E(t) - t defines the predictive travel time of users entering the link at time t. In the case of a first-order model of the LWR (Lighthill-Whitham-Richards) type given by
at
.
ox
.,
with Qe the equilibrium flow-density relationship, the computation of PTT(a, b; t) at time t would require the initial condition K(.,t) on the link and the downstream traffic supply E(6, r) for time r ranging from t to E(t), since it would require the computation in the (x,t)-plane of the field-lines with origin ( x , t ) and x 6 [a, b}. We refer to (Lebacque 1995) for a definition of the twin notions of local traffic supply and demand, and the related definition of boundary conditions for the LWR model. The upstream demand at point a is without influence on the field-line originating at ( a , t ) , because in the LWR model, the propagation speed of information, ^-(.,x) is always less than the vehicle speed V e ( . , x ) (with Ve the equilibrium speed-density relationship). On the other hand, the computation of ETT(a, b; t) can be carried out by solving the following partial differential equation: dx """ dt
(2)
T(a, t) = 0
(V«)
Indeed, dT is equal to dt along trajectories, hence T(x,t) = ETT(a,x;t). This definition of the ETT is important in practice, because it implies that it is not necessary to store past data in order to compute the ETT function, i.e. T, as the formula (??) will illustrate later on.
Considering three points a, ft, c in that order on a line, and considering the trajectory of a vehicle passing through these points, the following functional equations are satisfied: PTT(a, c-1) = PTT(a, 6; t) + PTT(b, c; t + PTT(a, b; t)) ETT(a, c; t) = ETT(b, c; t) + ETT(a, b-1 - ETT(b, c; t}}
,
which show how these travel times are to be combined. These are definitely not additive quantities! Hence the difficulties related to spatial aggregation.
Travel times computation for dynamic assignment modelling
309
Instantaneous travel times
Although experienced travel times may be used for reactive assignment (this is the case in the INTEGRATION model), they do not necessarily constitute the prime choice, since an experienced travel time reflects more what has just happened than what is about to happen. Another possibility is to define directly an instantaneous travel time ITT(a, b; t) for the link. A standard and natural definition (such as the one given by Ran and Boyce (1994)) specifies the instantaneous travel time as the travel time that would result if prevailing traffic conditions remained unchanged. In the present context of macroscopic models, this means that to compute ITT(a,b;t) we have to define a time-constant speed field say Vt(x,r) = (V(x,i), 1) for all instants r > t, and compute its field-lines, which are of course invariant through translations parallel to the time-axis. Therefore the corresponding formula is given by: (3)
This formula is of course additive (a desirable feature if the estimation of trip travel times is required) but regrettably it is only applicable if there exists some strictly positive lower bound for the speed, as in some models. In the general case, the speed may become nil if the traffic is interrupted, and the above integral might diverge, or take on unrealistic values. Therefore, we construct the instantaneous travel time in order to satisfy some set of properties. The properties we retained are that, on a small scale, the ITT should be of the order —dx/V(x, t) at low density and high speed (yielding ITT(x, .; t) w fx dx/V(Xi *)) and of the order dt for strongly congested traffic. The former property reflects simply the idea that the instantaneous travel time should be close to its "natural definition" (??) at low density. The latter property is similar to the analogous property of ETTs (??). It reflects the fact that, for interrupted traffic, dITT(x, .; t) = dt yields the simplest estimate of the interruption duration, especially in the case of an incident in which this duration may not by definition be known beforehand. The simplest model satisfying to these properties is described by the following partial differential equation (Lebacque 1996):
(4)
R(b,t) = 0
(V*)
with Vmax the maximum speed and R(x, t) = ITT(x, b; t) (instantaneous travel time from x to b estimated at time t, labeled backward ITT in the above reference). To explain formula (??), let us note first that at high speed and low density, the model should yield (5)
-Vf*l OX
,
and at low speed and high density, it should yield
i-i . Of course, in all cases, the boundary condition R(b,t) = 0 should be satisfied. Now, (??) is nothing more than a linear interpolation between (??) and (??). Of course, (??) constitutes by no way the only possible model satisfying our requirements. Even the boundary condition might be changed to say S(a,t) = Q ,
310
C. Buisson et al. def
yielding an estimate S(x,t) = ITT(a,x;t) (instantaneous travel time from a to £ estimated at time t, labeled forward ITT in. the above reference). A partial differential equation for S can be constructed according to the same ideas as those used for R, yielding:
& = 1^ 9t (7)
S(a,t) = 0 The choice between R and S is essentially a matter of application. In the sequel, for brevity's sake, we shall concentrate our attention on R. The ITT estimate R can be computed analytically. The method is the following. Denoting u the variable def , U = t—
(8) and W the field
Vm whose field-lines (£(«), r(u)) are given by: (10)
*L — 1 _ v (f T\ du ~ -1 Vmol V^> ' '
it follows that (11)
dR = du along such a field-line. Hence, to compute R, it suffices to compute the field-lines (??). The trivial graphical interpretation of u in the (x, t) plane extends to R along field-lines of W.
A difficulty arises at shock-waves; it is indeed possible that two distinct field-lines of W originate from the same point of a shock-wave, as is illustrated hereafter. t
S hock-Wave
Field-Lines In that case, the shock-wave itself must be considered a field-line of W, along which (??) applies. By considering three consecutive points of a field-line, the following three-point functional equation results: ITT(a, c; t) = ITT (a, b; t) + ITT(b, c; t -
a —b
- ITT(a,b;t)}
V m.a.T
It may be noted that, if the speed is near Vmax on [a, b], i.e. ITT(a, b;t) is nearly equal to ^^-, then the above functional equation becomes nearly additive, i.e.:
ITT(a, c; t) w ITT (a, 6; t) + ITT(b, c; t) , which is precisely what is to be expected. Similar properties are satisfied by the forward ITTs.
311
Travel times computation for dynamic assignment modelling
SEMIDISCRETIZED MODELS By semidiscretized models we mean models continuous in time and discretized in space, with the link as the space discretization unit. As indicated above, the link PTT(t) is essentially a function of the link state K(.,t) at time t and the downstream traffic flow supply E(6, s) for s € [t,E(t)]. In fluid traffic conditions (i.e. downstream traffic flow supply sufficient to accomodate the traffic demand of the link at all times), and at the zero-th order approximation, one might consider PTT(t) as a function of N ( t ) , defined as the total number of vehicles in the link at time t. This is the basis of some flow models for assignment problems (Fernandez and de Cea 1994, Friesz et al. 1993, Astarita 1996, Ran et al. 1996). In such models the link traffic flow dynamics are described by a model of the following kind: dN . . — (t) = u(t)-v(t) (u(t) the link inflow and v(t) the link outflow), supplemented by a model for the PTT(t), called here r(t): r(t) = f(N(t}) and the FIFO condition, which in the present case does not result naturally from the model. This last condition implies (Astarita 1996) that:
u(t)
u(t)
(12)
by expressing that users entering the link at time t exit it at time t + r(t). With the FIFO hypothesis, the following integral relationships result: t
/
u(s)ds = /
fE(t)
v(s)ds
.
These are the same as the relationships that would be obtained within the framework of the preceding section. Indeed, by integrating the conservation equation dK dQ n =0 UT at + 1? ox
over areas (1) and (2) depicted hereafter (and bounded diagonally by a field-line in the ( x , t ) plane),
the following relationships result: . ,
rb
rt
N(t) =f \ K(X, t)dx = / Ja
Jl(t}
Q(a,s)ds=
rE(t) Jt
Q(b,s)ds
,
312
C. Buisson et al.
relating the link inflow and outflow to the number N(t) of vehicles contained in the link at a given time. Let us note at this point some differences in the terminology, since in (Ran and Boyce 1994) for instance, the PTT is called the actual travel time. It can be shown that the only consistent FIFO model of the above kind is the one associated to a linear travel time function: T(t]
= a + 0N(t)
.
This result was suggested in (Daganzo 1995), the sufficiency of this linear form was demonstrated in (Friesz et al. 1993), and its necessity in (Lebacque and Lesort 1996). The linear part represents the average time lost in the queue at the exit of the link, which is somewhat at odds with the hypothesis that the downstream traffic supply can be neglected. It is not known to the authors of the present paper whether non-FIFO models of the above kind can be built. The analysis of such models might prove difficult since they would not admit any closed expressions such as (??) for the link outflow.
FULLY DISCRETIZED MACROSCOPIC MODELS We shall now consider discretized macroscopic models and develop recursive formulas for ETTs and ITTs. The emphasis on recursive formulas is motivated by the need for formulas requiring as little computational effort as possible in order to be suitable for real-time applications. Unless stated otherwise, the report (Lebacque 1996) is the reference for all results described in the present section. For discretized macroscopic models, either there exist no intrinsic estimates of speed (1st order models), or, if such estimates exist, they may lead to unrealistic values of the travel times. Hence we propose the following methodology. In most discretized macroscopic models, links are divided into cells say (s) = [xs-\,xs], of length / s , containing JV* vehicles at time tAt, with the average cell exit flow Q* during time-step [tAt, (t + l)At], estimated at the cell exit point xa. A notable exception to this kind of approach to discretized macroscopic modelling is the particle discretization approach, as illustrated in INTEGRATION.
d~— l*~f\
0
-A-1
1
•">
2
"
i
S 1
"
S
Cell(s)
!
"
U—— •**-£,
S
No hypothesis is made on the macroscopic model itself, nor on the manner in which the above quantities are computed. We denote by Vs* the cell exit speed defined as
N\ with K\ = N*/ls the mean cell density at time tAt. The significance of this choice is that it permits emulation of FIFO behaviour within each cell. We shall consider only what we call proper discretizations i.e. discretizations such that the cumulated cell outflow Q^At during a time-step be less than the number of vehicles N* contained in the cell at time tA.t:
Travel times computation for dynamic assignment modelling
313
Let us define now the coefficients
(13) ft =
The fact that the discretization is proper can be translated as
and in order for the discretization to work at all, it is necessary that
Those inequalities are assumed to hold throughout the present section. For a link containing cells s = 1 to 5", equation (??) for the experienced travel-time T may be discretized according to:
with Ts* denoting the approximate experienced travel time from x0 to xs at time £At. With 5 = 1, this formula is equivalent to the ITT formula introduced in (Buisson et al. 1995a, b) on a completely heuristic basis. The boundary condition is TQ = 0. Introducing the cell travel times ETT* = Ts* — T^, the above formula can be rewriten as: ETTt+i = _
This last formula is important in two ways. First it gives an indication of how to aggregate spatially travel-times. Second, it implies
K* 6
o
showing that the cell-travel time relaxes towards ls/Vg if the traffic flow is not too unstationary. The recursive formula (??) is obtained in the following way. t+1
t+1
(t+1) At
T1t
t At s-1
T(x,t) def = ETT(a,x;t) is approximated by a continuous piecewise linear function whose values at points xs are Ts* at times tAt. The field-lines of V are approximated by lines, and we use the fact that dT = dt along such a line. Let (y*,tAi) denote the origin of the field-line ending at ( x s , (t + l)At), then: rf =
314
C. Buisson et al.
at the first order approximation. It follows:
and the linear interpolation of T(yl,tA.t) between T^ and T*^ yields (??). The discretization of (??) yields:
(15) i f i /s* — >
+(!-!/*)#. + (0
l+a,
The boundary condition is T| = 0. Introducing the cell travel time ITT% = R\ - Rts_l, it follows:
It follows:
showing that the cell-travel time relaxes towards ls/V* if traffic flow conditions approach stationary, and that this relaxation process is much faster for ITT estimates than for ETT estimates, and this is precisely the result that was hoped for: that the ITT should be as close to the ideal formula (??) as possible even in interrupted traffic conditions. It is in this restrictive sence that our definition of an ITT and the above-mentioned definition of Ran and Boyce (1994) may be said to concur. The recursive formula (??) is obtained in the following way.
(t+1) At
t At
R(x,t) = ITT(x,b;t) is approximated by a continuous piecewise linear function whose values at points xs are Rls at times iAt. The field-lines of W are approximated by lines, and we use the fact that dR = du along such a line. Depending on whether v\ > l/(l + as), or v\ < 1/(1 + as), the field-line ending at (x s _i, (t + 1)A£) originates at point (xs, rj) or at point (£*,£A£), with:
ft
-
T
-I- /
ft'
Ss — •*'« — 1 ~r '« 1— i/'
Travel times computation for dynamic assignment modelling
315
Applying dR = du and the linearity of R, it follows: #*+j
=
/?(x s ,T f ) + (t 4- l)At — T* + (ls/Vs max)
and
At/(l - i/i)
if ^ <
Both formulas (??) and (??) are recursive, hence easy to implement and of reduced computational cost. Further, they can be extended to the case where the global flow is split into partial flows, according to destination, path, type of driver (informed or uninformed), depending on the assignment problem. They can also, whithin the framework developed for the STRADA model (Buisson et al. 1995a, b), be extended to movements of intersections. Finally, when the traffic flow is nearly stationary, both formulas imply the convergence of the ETT*s, ITT*s, towards fa on every cell s, as already not^ed. with a better convergence for the ITT.
CONCLUSION Much work remains to be done. Other ITT estimates are conceivable, depending on the properties one deems important for such quantities. The problem of the time-aggregation of travel times must be addressed, especially in the case of adaptative regulation schemes lacking periodicity. The impact of the proposed estimators on traffic assignment and management schemes must be studied as well. Especially since one of the motivations behind such an "axiomatic" definition of travel times as we have given here is to provide some solid ground for assignment computations. Finally, some experimental assessment of the proposed travel time estimators should be attempted, although it would seem difficult to separate the properties of the estimators from those of the associated macroscopic traffic flow model.
REFERENCES Astarita, V. (1996). A continuous-time link model for dynamic network loading based on travel time function. In: Proceedings of the 13th ISTTT (International Symposium on Transportation and Traffic Theory), Lyon 1996 (J.B. Lesort, ed.), pp 79-102. Pergamon. Branston, D. (1976). Link travel time functions: a review. Transportation Research, 10, 223-236. Buisson, C., J.P. Lebacque and J.B. Lesort (1995a). Macroscopic modelling of traffic flow and assignment in mixed networks. In: Proceedings of the 6th International Conference on Computing in Civil and Building Engineering, Berlin 1995 (P.J. Pahl and H. Werner, ed.), pp 1367-1374. Buisson, C., J.P. Lebacque and J.B. Lesort (1995b). STRADA, a discretized macroscopic model of vehicular traffic flow in complex networks based on the Godunov scheme. In: Proceedings of the IEEE-SMCIMACS'96 Multiconference, Lille 1996, Symposium on Modelling, Analysis and Simulation Vol 2, pp 976-981.
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Cremer, M., F. Meissner and S. Schrieber (1993) On predictive control schemes in dynamic rerouting strategies. In: Proceedings of the 12th ISTTT (International Symposium on Transportation and Traffic Theory), Berkeley, 1993 (C.F. Daganzo, ed.), pp 407-426. Daganzo, C.F. (1995). Properties of link travel time functions under dynamic loads. Transportation Research B, 29, 95-98. Drissi-Kai'touni, O. and M. Gendreau (1992). A new dynamic traffic assignment model. Montreal, Centre de Recherche sur les Transports. Elloumi, N. (1996). Modelisation et commande du trafic sur un corridor - Application de la methode LP. Orsay, Universite Paris XI: 179 p. Fernandez, J.E. and J. de Cea (1994) Flow propagation description in dynamic network assignment models. In: Proceedings of the Triennal Symposium in Transportation Analysis (TRISTAN), Capri 1994, PP 119-133. Friesz, T.L., J. Luque, R.L. Tobin and B.W. Wie (1989) Dynamic network traffic assignment considered as a continuous time optimal control problem. Operations Research, 37, 893-901. Friesz, T.L., D.H. Bernstein, T.E. Smith, R.L. Tobin and B.W. Wie (1993). A variational inequality formulation of the dynamic network user equilibrium problem. Operations Research, 41, 179-191. Jayakrishnan, R., H.S. Mahmassani, T.Y. and T.Y. Hu (1994). An evaluation tool for advanced traffic information and management systems in urban networks. Transportation Research C, 3, 129-147. Koutsopoulos, H.N. and M. Habbal (1994). Effect of intersection delay modeling on the performance of traffic equilibrium models. Transportation Research A, 28, 133-149. Lebacque, J.P. (1995). The Godunov scheme and what it means for first order traffic flow models. Complete version: CERMICS Report 48-95. Short version: published in: Proceedings of the 13th ISTTT (International Symposium on Transportation and Traffic Theory), Lyon 1996 (J.B. Lesort, ed.), pp 647-677. Pergamon. Lebacque, J.P. (1996). Instantaneous travel times for macroscopic traffic flow models. CERMICS Report 59-96. Lebacque, J.P. and J.B. Lesort (1996). Simplified traffic flow models for dynamic assignment. Groupe de Travail Modelisation du trafic CERMICS-INRETS. Leonard, D. R., P. Gower and N.B. Taylor (1989). CONTRAM: structure of the model. TRRL Report RR 178, Crowthorne. Leonard, D. R., J. B. Tough and P. C. Baguley (1978). CONTRAM: A traffic assignment model for predicting flows and queue during peak periods. Crowthorne, Transport and Road Research Laboratory. Merchant, D. K. and G. L. Nemhauser (1978a). A model and an algorithm for the dynamic traffic assignment problems. Transportation Science, 12, 183-199. Merchant, D. K. and G. L. Nemhauser (1978b). Optimality conditions for a dynamic traffic assignment model. Transportation Science, 12. 200-207. Papageorgiou, M. (1990). Dynamic modeling, assignment, and route guidance in traffic networks. Transportation Research B, 24, 471-495.
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Ran, B. and D.E. Boyce (1994) Dynamic urban transportation network models. Springer Verlag. Ran, B., R. W. Hall and D.E. Boyce (1996). A link-based variational inequality model for dynamic departure time/route choice. Transportation Research B, 30, 31-46. Van Aerde, M. (1995). INTEGRATION release 2: User's guide. Kingston, Ontario, Canada, Transportation Systems Research Group. Weymann, J., J.L. Farges, Henry, J.J. (1994). Dynamic route guidance with queue and flow dependent travel time. Transportation Research C, 2, 165-183.
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Index Addison, D. 35 Allsop, R. E. 1 Asakura, Y. 273
Lee, D.-H. 289 Lesort, J. B. 303 Li, I. Y. 67 Lundgren, J. T. 239
Bell, M. G. H. 1, 175 Bertazzi, L. 223 Boyce, D. E. 67, 289 Buisson, C. 303
Maher, M. J. 163 Miyagi, T. 115 Paechter, B. 163 Patriksson, M. 83, 239 Proll, L. 195
Cheung, R. K. 209 Cree, N. D. 163
Dia, H. 255
Ran, B. 67 Rose, G. 255
Florian, M. 51 Fores, S. 195 Friesz, T. L. 143
Shah, S. 143 Smith, M. 9
Grazia Speranza, M. 223
van Zijpp, N. J. 99 Wren, A. 195 Wu, J. H. 51
Heydecker, B. G. 35, 99 Huang, H.-J. 175
Xiang, Y. 9 Xu, Y. W. 51
Janson, B. N. 289 Jarrett, D. 129
Yarrow, R. 9
Larsson, T. 83 Lebacque, J. P. 303
Zhang, X. 129
319