Modern Heuristic Search Methods

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Author: V. J. Rayward-Smith | I. H. Osman | C. R. Reeves | G. D. Smith

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Modern Heuristic Search Methods Rayward-Smith, V. J. John Wiley & Sons, Ltd. (UK) 0471962805 9780471962809 9780585356808 English Heuristic programming, Combinatorial optimization. 1996 T57.84.M63 1996eb 658.4 Heuristic programming, Combinatorial optimization.

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1 Modern Heuristic Techniques Colin R Reeves Abstract This book is concerned with applications and developments of heuristic techniques for solving complex optimization problems. The three most popular methods which go beyond simple local search are simulated annealing, tabu search, and genetic algorithms, and each is used in several papers in this volume. Because most of these assume a fairly high level of knowledge of the terminology and practice, this introductory chapter has been written with the aim of explaining the basics of the methods and of setting each of them in context. Experts in the field will find little new here, but this chapter should provide a useful summary of some of the underlying concepts. 1.1 Introduction Many 'hard' OR problems involving areas such as resource allocation, packing and scheduling have traditionally used methods based on linear and/or integer programming. Such approaches inevitably place restrictions on the form of the objective function and/or the constraintsusually linear, although they have been adapted to deal with slightly more general situations. More complex non-linear problems can be solved by calculus-based methods, but these rely on differentiability. Furthermore, such traditional methods all assume deterministic quantities, and have problems dealing with any stochastic effects. There is no denying that these methods have produced excellent results over the past 40 years, but precisely because of their success, there may be a tendency to model problem situations using a framework which permits a solution by such means. Yet the assumptions inherent in such models are often only approximately true at best, and in recent years several very good heuristic methods have been developed which have no need of such assumptions. This opens up the possibility of building more realistic models with a real chance of obtaining 'good' solutions. Three methods which have become particularly popular recently have all arisen at least in part from a study of natural processes which perform an analogy of Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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optimizationsimulated annealing, tabu search, and genetic algorithms. While many other heuristic methods have been proposed in the last 30 years, they have often been tailored to a particular problem. In contrast, these more modern approaches are capable of very wide application. 1.2 Combinatorial Optimization Many problems can be structured as a function of some decision variables, perhaps in the presence of some constraints. Such problems can be formulated generally as follows:

Here, x is a vector of decision variables, and f(·) and gi(·) are general functions. This formulation has assumed that the problem is one of minimization, but the modification necessary for a maximization problem is obvious. There are many specific classes of such problems, obtained by placing restrictions on the type of functions under consideration, and on the values that the decision variables can take. Perhaps the most well known of these classes is that obtained by restricting f(·) and gi(·) to be linear functions of decision variables which are allowed to take fractional (continuous) variables, which leads to problems of linear programming. The term combinatorial is usually reserved for problems in which the decision variables are discretei.e. where the solution is a set, or a sequence, of integers or other discrete objects. The problem of finding optimal solutions to such problems is therefore known as combinatorial optimization. Some examples of this kind of problem are as follows: Example 1 (The Assignment Problem) A set of n resources is available to carry out n tasks. If resource i is assigned to task j, it costs cij units. Find an assignment {π1, . . ., πn} which minimizes

Here the solution is represented by the permutation {π1, . . ., πn} of the numbers {1, . . ., n}. Example 2 (The 01 Knapsack Problem) A set of n items is available to be packed into a knapsack with capacity C units. Item i has value vi and uses up ci units of capacity. Determine the subset I of items which should be packed in order to maximize

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Page 3 such that

Here the solution is represented by the subset I ⊆ {1, . . ., n}. Example 3 (The Set Covering Problem) A family of n subsets collectively contains n items such that subset Si contains ni (≤ n) items. Select k(≤ m) subsets such that

so as to minimize

where ci is the cost of selecting subset Si. Here the solution is represented by the family of subsets

.

Combinatorial problems, such as those described above, have close links with linear programming (LP), and most of the early attempts to solve them used developments of LP methods, generally by introducing integer variables taking the values 0 or 1, in order to produce an integer programming (IP) formulation. For example, in the case of the 01 knapsack problem, we define

The problem then reduces to the following integer program:

Not all combinatorial problems are so easily formulated in IP terms, and even when they can be, solution by this means may not be computationally feasible. 1.3 The Case for Heuristics

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A naive approach to solving an instance of a combinatorial problem is simply to list all the feasible solutions of a given problem, evaluate their objective functions, and pick the best. However, it is immediately obvious that this approach of complete enumeration is likely to be grossly inefficient; further, although it is possible in principle to solve any problem in this way, in practice it is not, because of the vast

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number of possible solutions to any problem of a reasonable size. To illustrate this point, consider the famous travelling salesman problem (TSP). This problem has exercised a particular fascination to researchers in combinatorial optimization, probably because it is so easily stated, yet so hard to solve (the book by Lawler et al. [LLKS85] and the more recent monograph by Reinelt [Rei94] contain a wealth of detail about the various methods used to find solutions to this problem). For the record, the problem is as follows: a salesman has to find a route which visits each of N cities once and only once, and which minimizes the total distance travelled. As the starting point is arbitrary, there are clearly (N 1)! possible solutions (or (N 1)!/2 if the distance between every pair of cities is the same regardless of the direction of travel). Suppose we have a computer that can list all possible solutions of a 20 city problem in 1 hour. Then, using the above formula, it would clearly take 20 hours to solve a 21-city problem, and 17.5 days to solve a 22-city problem; a 25-city problem would take nearly 6 centuries. Because of this exponential growth in computing time with the size of the problem, complete enumeration is clearly a non-starter. In the early days of operational research, the emphasis was mostly on finding the optimal solution to a problemor rather, to a model of a problem which occurred in the real world. To this end, various exact algorithms were devised which would find the optimal solution to a problem much more efficiently than complete enumeration. The most famous example is the Simplex algorithm for linear programming problems1. At first, while such algorithms were capable of solving small instances of a problem, they were not able to find optimal solutions to larger instances in a reasonable amount of computing time. As computing power increased, it became possible to solve larger problems, and researchers became interested in how the solution times varied with the size of a problem. In some cases, such as the 'Hungarian' method [Kuh55] for solving the assignment problem, or Johnson's method [Joh54] for 2-machine sequencing, it can be shown that the computing effort grows as a low-order polynomial in the size of the problem. However, for many others, such as the travelling salesman problem, the computational effort required was an exponential function of the problem size. In a sense, therefore, these exact methods (such as branch-and-bound or dynamic programming) perform no better than complete enumeration. The question that began to exercise the minds of researchers in the late 1960s was the following: is there a 'polynomial' optimizing algorithm for a problem such as the TSP? Nobody has yet been able to answer this question one way or the other, but in 1972, Karp[Kar72] was able to show that if the answer to this question is 'yes' for the TSP, then there is also a polynomial algorithm for a number of other 'difficult' problems. As no such algorithm has yet been found for any of these problems, it strongly suggests that the answer to the original question is 'no'. However, the actual answer is still unknown, some 20 years after Karp's original work. It is for this reason that attention has recently focussed more and more on the use of heuristics. Given the diversity of methods to which the label 'heuristic' has been applied, an exact definition of the term on which everyone can agree is perhaps not possible; the following is an attempt to provide one that at least captures the salient features. 1Actually, Simplex is now known to show exponential behaviour in the worst case, although on average its performance is excellent.

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Definition 1 A heuristic technique (or simply, a heuristic) is a method which seeks good (i.e. near-optimal) solutions at a reasonable computational cost without being able to guarantee optimality, and possibly not feasibility. Unfortunately, it may not even be possible to state how close to optimality a particular heuristic solution is. Although in the light of this definition it might appear that there are severe problems inherent in the use of heuristics, it should be emphasized that many modern heuristic techniques do give high-quality solutions in practice. 1.4 Neighbourhood Search A fundamental idea of heuristic methodology is that of neighbourhood search (NS). This concept has been around for at least 40 years; one of the earliest references to it is by Croes[Cro58] who used the idea to get good solutions to the TSP. Neighbourhood search (NS) is a widely used method in solving combinatorial optimization problems. A recent introductory treatment can be found in Reeves [Ree93c]. Table 1.1 Neighbourhood Search Method 1

(Initialization)

1.1 Select a starting solution xnow∈ X. 1.2 Record the current best-known solution by setting xbest = xnow and define best_cost = c (xbest). 2

(Choice and termination)

2.1 Choose a solution xnext∈ N(xnow). If the choice criteria used cannot be satisfied by any member of N(xnow) (hence no solution qualifies to be xnext), or if other termination criteria apply (such as a limit on the total number of iterations), then the method stops. 3

(Update)

3.1 Re-set xnow = xnext, and if c(xnow) < best_cost, perform Step 1.2. Then return to Step 2. Here we assume a slightly different statement of the problem of combinatorial optimization than that given in Section 1.2. In the context of neighbourhood search, we shall assume that a solution is specified by a vector x, where the set of all (feasible) solutions is denoted by X, and the cost of solution x is denoted by c(x)often called the objective function. Each solution x ∈ X has an associated set of neighbours, N(x) ⊂ X, called the neighbourhood of x. Each solution x′∈ N (x) can be reached directly from x by an operation called a move, and x is said to move to x′ when such an operation is performed. A description of the method is given in Table 1.1 (adapted from Glover and Laguna [GL93]), where it is assumed that we are trying to minimize the cost. Here we assume the choice criteria for selecting moves, and termination criteria for ending the search, are given by some external set of prescriptions. By specifying

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these prescriptions in different ways, the method can easily be altered to yield a variety of procedures. Descent methods, for example, which only permit moves to neighbours that improve the current c(xnow) value, and which end when no improving solutions can be found, can be expressed by the provision in Step 2 shown in Table 1.2. Table 1.2 Descent Method 2

(Choice and termination)

2.1 Choose xnext∈ N(xnow) such that c(xnext) < c(xnow) and terminate if no such xnext can be found. The failing of NS is well known: its propensity to deliver solutions which are only local optima. Randomized procedures, including the important technique of simulated annealing, offer one way of circumventing the localoptimum problem, and they can similarly be represented by adding a simple provision to Step 2 (Table 1.3). The next section will explore this idea in more detail. Table 1.3 Randomization Method 2

(Choice and termination)

2.1 Randomly select xnext from N(xnow). 2.2 If c(xnext) ≤ c(xnow) accept xnext (and proceed to Step 3). 2.3 If c(xnext) > c(xnow) accept xnext with a probability that decreases with increases in the difference ∆c = c(xnext) c(xnow). If xnext is not accepted on the current trial by this criterion, return to Step 2.1. 2.4 Terminate by a chosen cutoff rule.

1.5 Simulated Annealing Simulated annealing (SA) is a technique which first became popular about a decade ago, and has since proved itself as an effective approach to a large number of problems. It works by searching the set of all possible solutions, reducing the chance of getting stuck in a poor local optimum by allowing moves to inferior solutions under the control of a randomized scheme. Specifically, if a move from one solution x to another neighbouring but inferior solution x′ results in a change in value ∆c, the move to x′ is still accepted if

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where T is a control parameter, and R ∈ [0, 1] is a uniform random number. The parameter T is initially high, allowing many inferior moves to be accepted, and is slowly reduced to a value where inferior moves are nearly always rejected. There is a close analogy between this approach and the thermodynamic process of annealing in physics; it was this analogy which originally motivated the development of the method. The ideas that form the basis of simulated annealing were first published by Metropolis et al. [MRTT53] in 1953 in an algorithm to simulate the cooling of material in a heat batha process known as annealing. If solid material is heated past its melting point and then cooled back into a solid state, the structural properties of the cooled solid depend on the rate of cooling. For example, large crystals can be grown by very slow cooling, but if fast cooling or quenching is employed the crystal will contain a number of imperfections. The annealing process can be simulated by regarding the material as a system of particles. Essentially, Metropolis's algorithm simulates the change in energy of the system when subjected to a cooling process, until it converges to a steady 'frozen' state. Thirty years later, Kirkpatrick et al.[KGV83] suggested that this type of simulation could be used to search the feasible solutions of an optimization problem, with the objective of converging to an optimal solution. In Kirkpatrick's proposal, the control parameter T takes the rôle of the temperature, and it becomes important to devise a 'cooling schedule' whereby this parameter is decreased. In order to complete the specification of the procedure, we must also specify initial and final values of T ('temperatures'). These are what Dowsland[Dow93] calls generic parameter choices, while there will also usually be specific decisions depending on the context of the given problem, such as the choice of the space of feasible solutions, the form of the cost function and the neighbourhood structure employed. Both types of decision need some care, as they may affect both the effectiveness and the efficiency of the algorithm. Since Kirkpatrick's original paper considerable strides have been made in research into the theory of simulated annealing. This uses results based on modelling SA as a Markov chain which show that to guarantee convergence to a global optimum will, in general, need more iterations than exhaustive search. This may not sound too helpful, but these theoretical studies also give some guidance in how to make SA function better simply as a heuristic. Dowsland [Dow93] gives a brief review and discussion of the theory, while a deeper treatment of theoretical questions can be found in Van Laarhoven and Aarts[LA88]. 1.5.1 A Simple Example Before considering these decisions in more detail, it might be helpful to consider a short example to clarify the SA methodology. We consider the problem:

where x is coded as a 5-bit binary integer in the range [0,31]. This has a maximum at (0 1 0 1 0), i.e. x = 10 (where f = 4100). In digitized form, it can be shown that there are also 3 local maxima using a 'greedy' 1-bit neighbourhood search. To illustrate SA here (Table 1.4), we start from the string (1 0 0 1 1), binary code for x = 19, f = 2399, which under a greedy 1-bit NS would lead to the local optimum (1 0 0 0 0), i.e. x = 16, f = 3236. We choose neighbours (i.e. the bits to change)

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randomly. Table 1.4 First Attempt: T=100 ∆f

move?

2287

112

N

10111

1227

1172

N

5

10010

2692

<0

Y

72.9

2

11010

516

2176

N

65.6

4

10000

3236

<0

Y

59.0

3

10100

2100

1136

N

T

bit

string

100

1

00011

90

3

81

f

new string 10011 10011 10010 10010 10000 10000

Unfortunately, the temperature is unlikely to be high enough now to move out of this local optimum: at any rate in a further 50 trials, nothing would move it! Table 1.5 Second Attempt: T=500 T

bit

string

f

∆f

move?

500

1

00011

2287

112

Y

450

3

00111

3803

<0

Y

405

5

00110

3556

247

Y

364.5

2

01110

3684

<0

Y

328.0

4

01100

3988

<0

Y

295.2

3

01000

3972

16

Y

265.7

4

01010

4100

<0

Y

239.1

5

01011

4071

29

Y

215.2

1

11011

343

3728

N

new string 00011 00111 00110 01110 01100 01000 0 1 0 1 0** 01011 01011

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In the second attempt (Table 1.5) the initial temperature was high enough to get out of the attraction region of 3236, and the optimum was found. Note however that the search did move away again, but in a further 150 trials most of the states visited were either the optimum or one of its immediate neighbours. 1.5.2 Generic Decisions The choice of cooling schedule and initial temperature in the above example might seem (and in fact were!) rather arbitrary. We now consider in more detail the generic decisions that need to be taken in implementing SA in practice. Initial Temperature The process must start in such a way that most if not all moves can be acceptedi.e. the initial temperature must be 'high'. In practice

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this may require some knowledge of the magnitude of neighbouring solutions; in the absence of such knowledge, one may choose what appears to be a large value, and run the algorithm for a short time and observe the acceptance rate. If the rate is 'suitably high', this value of T may be used to start the process. What is meant by a 'suitably high' acceptance rate will vary from one situation to another, but in many cases an acceptance rate of between 40% and 60% seems to give good results. More sophisticated methods are possible, but not often necessary. Cooling Schedule Perhaps the most important factor in practical application is the cooling schedule. Here we should note that there are basically two types of schedule, having analogies to homogeneous and inhomogeneous Markov chains respectively. In the homogeneous case, annealing is carried out at a fixed temperature until 'equilibrium' is reached. (This involves a further problem in that one has to decide when this has occurred.) Once this state is judged to have been reached, the temperature is reduced, and the procedure repeated. The number of attempted moves at each temperature may be quite large, although the temperature steps can be relatively large also. In the inhomogeneous case, the temperature is reduced (but by a very small amount) after every move. This is less complicated than the homogeneous case, and is the one more commonly used in practice. In either case, one has to decide on the 'shape' of the cooling curve. Two methods are popularthe simplest is a geometric schedule:

where α is a constant close to 1 (typically in the range 0.9 to 0.99). The other method is due to Lundy and Mees[LM86]:

where β is a constant near to zero. Final Temperature In theory the procedure should be continued until the final temperature Tf is zero, but in practice it is sufficient to stop when the chance of accepting an uphill move has become negligible. To some extent this is problem-dependent, and as in the case of selecting an initial temperature, may involve some monitoring of the ratio of acceptances. Lundy and Mees proposed stopping when

where S is the solution space. This is designed to produce a solution which is within ε of the optimum with probability θ. Number Of Iterations It should be noted that the number of iterations Nit is effectively fixed by the above three choices. In the homogeneous case, it also depends on how equilibrium is detected at each stage, but in the inhomogeneous case α, β and Nit are related by

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and

respectively. 1.5.3 Problem Specific Decisions The most important decision in any problem is specifying the neighbourhood structure. Theoretical results require that every solution be reachable from every other, but this is a condition that is usually easy to ensure. However, in the case of constraints on feasibility, one may need to use penalty terms which, while they ensure reachability, may create an undesirably rough solution 'landscape'. Constraints and penalty factors may thus require careful attention and some finetuning. It is often an advantage to define neighbourhoods in such a way that the calculation of the effect of a move can be carried out in an economical manner. For example, in evaluating the effect of a 'λ-exchange' for the TSP (where λ links of the tour are broken and reconnected), it is not necessary to compute the whole tour-length each time, but simply to evaluate the effect of the λ links that have been exchanged. 1.5.4 Other Factors The acceptance function is usually assumed to be exponential:

It has been pointed out that calculating this function (or equivalently, evaluating the logarithm of R) is comparatively expensive on a digital computer. Approximating by 1 ∆c/T often saves computer time at little cost in effectiveness. Other possibilities are discussed by Dowsland[Dow93]. It is worth mentioning here also the idea of 'threshold acceptance' which simply accepts any uphill move below a certain levela threshold which is slowly reduced as the search proceeds in a similar manner to the SA cooling schedule. Reannealing is an idea which has also been used successfully. Connolly[Con90] carries out an initial SA pass, during which he records the temperature at which the best solution was found. He then 're-heats' the process and carries out a more lengthy search at this temperature until some stopping condition is satisfied. Several other authors have suggested carrying out a sequence of heatings and coolingsin fact this was done by Kirkpatrick et al. in the very first trials of the SA method. Dowsland[Dow93] also proposes an adaptive version of the procedure: on accepting a move, the temperature is reduced according to

whereas when a move is rejected, the temperature is increased according to

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Restricted neighbourhoods have been found beneficial in several applications. Problem-specific information may suggest that some moves are so unlikely to be accepted that they can be ignored, and the potential disadvantages of destroying reachability may be outweighed by the advantages of spending less computer time in searching unproductive areas of the solution space. As an example, candidate moves in solving the TSP may usefully be restricted to those which relate to 'close' points. The order of searching a neighbourhood may sometimes be varied to advantage. Most descriptions of SA assume neighbourhoods are sampled randomly, but there are several reported instances where sampling in a systematic or adaptive way has produced better results. 1.5.5 Summary Simulated annealing (SA) is a simple procedure to apply, yet there are several decisions to be made in applying it. This section has described some of these in outline, and given some idea of possible modifications and enhancements. All these are covered in much greater detail by Dowsland[Dow93], who also describes an extensive list of applications. It is usually true that implementing SA with a simple neighbourhood, a starting temperature suggested by some smallscale initial experiments, and a geometric or Lundy and Mees cooling schedule will produce results which are better than a simple neighbourhood search. Further improvements can often be made by more detailed analysis of problem characteristics, and/or by combining SA with other techniques. 1.6 Tabu Search Tabu search, like simulated annealing, is based on neighbourhood search with local-optima avoidance, but in a deterministic way which tries to model human memory processes. Memory is implemented by the implicit recording of previously-seen solutions using simple but effective data structures. These centre on the creation of a 'tabu list' of moves which have been made in the recent past of the search, and which are 'tabu' or forbidden for a certain number of iterations. This helps to avoid cycling, and serves also to promote a diversified search of the solutions. The modern form of tabu search was originally developed by Glover[Glo86], and a comprehensive account of the basic concepts, and of recent developments, is given by Glover and Laguna[GL93]. What follows here should be adequate as a basic introduction, but the reader is referred to Glover and Laguna[GL93] for a more detailed account. 1.6.1 Basic Concepts If we recall the prototype neighbourhood search description in Section 1.4, TS can be simply described in terms of the way it modifies the neighbourhood. In the TS method, a history record H is kept of the states previously encountered during the

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search, so that the neighbourhood N(xnow) is modified to N(H, xnow). The basic NS method is thus modified as follows: Table 1.6 Tabu search method 1

(Initialization)

1.1 Begin with the same initialization used by Neighbourhood Search, and with the history record H empty. 2

(Choice and termination)

2.1 Determine Candidate_N(xnow) as a subset of N(H, xnow). Select xnext from Candidate_N (xnow) to minimize c(H, x) over this set. (xnext is called a highest evaluation element of Candidate_N(xnow).) 2.2 Terminate by a chosen iteration cut-off rule. 3

(Update)

3.1 Perform the update for the Neighbourhood Search Method, and additionally update the history record H. The simplest way in which H can be defined is in terms of a prohibition on re-visiting certain states in N(xnow). Such states are termed tabu, and H thus has the effect of restricting the search. In enhanced versions of TS, H may also include states which are not members of N(xnow) itself, thus allowing both diversification and intensification of the search. Furthermore, H may contain information which serves to modify the evaluation of the objective function, as is indicated in the above description by the notation c(H, x). These various prescriptions for identifying history-based characteristics are often described in terms of four 'dimensions' in tabu searchrecency, frequency, quality and influence. Of these, recency and frequency are the most important, and we now examine them in greater detail. 1.6.2 Recency One objective in TS is to encourage exploration of parts of the solution space that have not been visited previously. This can be achieved in practice by prohibiting the reversal of previous movesthat is, these (reverse) moves become 'tabu'. However, to make this prohibition absolute would confine the search to a straitjacket, so we prohibit the reversal of the most recent moves only. Recency may be construed as a fixed parameter (the tabu tenure of a move), or it may be allowed to vary dynamically during the search. It is also often beneficial to focus on some component or attribute of a move rather than on the complete move itself. Finally, in many implementations, tabu moves are allowed to be overridden under certain conditions specified by an aspiration criterion. In order to clarify the recency-based TS method, we next present a simple worked example.

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A Simple Example Maximize f(x) = x3 60x2 + 900x + 100. Table 1.7 Maximizing f (x) Iteration

bit

string

f

1

1

00011

2287

2

11011

343

3

10111

1227

4

10001

2692

5

10000

3236

1

00001

941

2

11001

725

3

10101

1801

4

tabu

5

tabu

1

00000

100

2

11000

964

3

11100

4

tabu

5

tabu

1

01000

212

3972

2

tabu

3

01100

3988

4

01010

4100

5

tabu

(bit 4 tabu until iteration 9) 6

1

** 01000

(bit 1 tabu until iteration 8) 5

** 11000

(bit 2 tabu until iteration 7) 4

**

10000

(bit 5 tabu until iteration 6) 3

** 10001

(bit 4 tabu until iteration 5) 2

new string

** 01010

tabu

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2

tabu

3

01110

4

tabu

(bit 3 tabu until iteration 10)

3684

** 01110

Again we start from (1 0 0 1 1), using a neighbourhood of size 4, sampled in cyclic order, and with a tabu tenure of 3 iterations (Table 1.7). Here the use of tabu restrictions has effectively moved the search out of the attraction region of 3236, and enabled it to find the optimum. However, as in the case of SA, the heuristic doesn't know the optimum has been found and carries on exploring. Unlike SA, however, the search may not remain close to the optimum.

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Practical Considerations In practice, the 'short-term' memory used in the concept of recency can be implemented in several ways. Perhaps the most obvious is to maintain a list of all solutions previously seen, and make any move tabu which would lead to a solution on this list. This would entail keeping an ever-increasing store of information, and searching it could be very expensive in computer time. This can be a problem even when only the most 'recent' solutions are stored. (More efficient methods based on hashing have been used with some success.) However, it is usually convenient, and often more desirable, to focus on certain attributes of a solution. An attribute of a move from xnow to a tentative solution xtrial may mean any aspect that changes as a result of the move. Some natural types of attributes are shown in Table 1.8. Table 1.8 Illustrative Move Attributes for a Move xnowto xtrial (A1) Change of a selected variable xj from 0 to 1. (A2) Change of a selected variable xk from 1 to 0. (A3) The combined change of (A1) and (A2) taken together. (A4) Change of c(xnow) to c(xtrial). (A5) Change of a function g(xnow) to g(xtrial) (where g may represent a function that occurs naturally in the problem formulation or that is created strategically). (A6) Change represented by the difference value g(xtrial) g(xnow). (A7) The combined changes of (A5) or (A6) for more than one function g considered simultaneously. What is meant by assigning values to a selected variable xj can be understood in a more general sense than the literal setting of its value to 0 or 1, although this may often be the context. However, it may also be used to refer to any operation that is used to define a neighbourhood, such as adding or deleting edges from a graph, assigning or removing a facility from a particular location, or changing the processing position of a job on a machine. We note that a single move can give rise to multiple attributes. For example, a move that changes the values of two variables simultaneously may generate each of the three attributes (A1), (A2), and (A3). Attributes that represent combinations of other attributes may sometimes provide more information. Attributes (A5) to (A7) are based on a function g that may be strategically chosen to be completely independent from c. For example, g may be a measure of distance (or dissimilarity) between any given solution and a reference solution, such as the last local optimum visited or the best solution found so far, in which case attribute (A6) would indicate whether a trial solution leads the search further from or closer to the reference point. Move attributes are often used in tabu search to impose restrictions that prevent moves from being chosen that would reverse the changes represented by these attributes. More precisely, when a move from xnow to xnext is performed that contains an attribute e, a record is maintained for the reverse attribute which we denote by ,

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in order to prevent a move from occurring that contains some subset of such reverse attributes. Examples of some common kinds of tabu restriction are shown in Table 1.9. Table 1.9 Illustrative Tabu Restrictions A move is tabu if: (R1) xj changes from 1 to 0 (where xj previously changed from 0 to 1). (R2) xk changes from 0 to 1 (where xk previously changed from 1 to 0). (R3) at least one of (R1) and (R2) occur. (This condition is more restrictive than either (R1) or (R2) separatelyi.e. it makes more moves tabu.) (R4) both (R1) and (R2) occur. (This condition is less restrictive than either (R1) or (R2) separatelyi.e. it makes fewer moves tabu.) (R5) both (R1) and (R2) occur, and in addition the reverse of these moves occurred simultaneously on the same iteration in the past. (This condition is less restrictive than (R4).) (R6) g(x) receives a value v′ that it received on a previous iteration (i.e. v′ = g(x′) for some previously visited solution x′). (R7) g(x) changes from v'' to v′, where g(x) changed from v′ to v" on a previous iteration (i.e. v′ = g(x′) and v" = g(x") for some pair of solutions x′ and x" previously visited in sequence.) Again, combinations of attributes may be used, and an auxiliary function g may be defined independently from c. A simple way of using such attributes in practice is to define arrays tabu_start(e) and tabu_end(e) which define the iterations on which attribute e begins and ceases to be active. This length of time is often called the tabu tenure, denoted by t. The choice of t may influence the quality of the solution obtained: having it too small increases the risk of 'cycling', while having it too large may restrict the search unduly. A value t = 7 has often been found sufficient to prevent cycling; other values commonly used are t = √n where n is some natural measure of problem size. Dynamic rules may be useful toousually this means choosing lower and upper limits tmin and tmax on the tabu tenure, and allowing t to vary in some way between them. Finally, we should also mention the use of aspiration criteria. It may happen that a trial move that is tabu would lead to a new xbest. In such circumstances it would seem odd to reject this move, so an aspiration criterion may be invoked to deal with such cases. This is the simplest and most obvious case where an aspiration criterion may be useful, but Glover and Laguna[GL93] suggest several others. 1.6.3 Frequency Recency simulates short-term memory; Glover and Laguna[GL93] suggest that a form of long-term memory can be implemented by the use of a variety of frequency measures. Two types of frequency measure are identified: residence measures and transition

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measures. The former relate to the number of times a particular attribute is observed, the latter to the number of times an attribute changes from one value to another. Frequency measures are normally ratios, where the numerators represent counts of the number of occurrences of a particular event (e.g. the number of times a particular attribute belongs to a solution or move), as in Table 1.10. Table 1.10 Example Frequency Measures (Numerators) (F1) |S(xj = p)| (F2) |S(xj = p for some xj)| (F3) |S(to xj = p)| (F4) |S(xj changes)|, i.e. |S(from-or-to xj = p for some p)| (F5) (F6) Replace S(xj = p) in (F5) with S(xj≠ to xj = p) (F7) Replace c(x) in (F6) with a measure of the influence of the solution attribute xj = p Here S is a set of solutions seen during the search so far (not necessarily all the solutions seen), and symbols such as S (xj = p) denote the subset of S for which it is true that xj = p. Measures (F1) to (F4) are fairly obvious; the measures (F5) to (F7) are weighted measures, created by reference to solution quality in (F5) and (F6), and by reference to move influence in (F7). Measure (F5) may be interpreted as the average c(x) value over S when xj = p. This quantity can be directly compared to other such averages or can be translated into a frequency measure using appropriate denominators. Table 1.11 Denominators (D1) The total number of occurrences of all events represented by the numerators (such as the total number of associated iterations). (D2) The sum of the numerators. (D3) The maximum numerator value. (D4) The average numerator value. The denominators represent quantities such as those in Table 1.11. Denominators (D3) and (D4) give rise to what may be called relative frequencies. In cases where the numerators represent weighted counts, some of which may be negative, (D3) and (D4) are expressed as absolute values and (D2) is expressed as a sum of absolute values (possibly shifted by a small constant to avoid a zero denominator).

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In either case, the frequency measures are usually used to generate penalties which modify the objective function c(x) to c(H, x) as discussed above. Diversification is thereby encouraged by the generation of solutions that embody combinations of attributes significantly different from those previously encountered. Conversely, intensification is promoted by incorporating attributes of solutions from selected élite subsets (implicitly focussing the search in subregions defined relative to these subsets). 1.6.4 Summary We have not formally discussed the concepts of quality and influence, although they have already been seen implicitly in some of the discussion above. It may thus be helpful to provide a brief reiteration of the basic notions. Quality in TS usually refers simply to those solutions with good objective function values; a collection of such solutions may be used to stimulate a more intensive search in the general area of these élite solutions, as already mentioned above. Influence is, roughly speaking, a measure of the degree of change induced in solution structurecommonly expressed in terms of the distance of a move from one solution to the next. It is an important aspect of the use of aspiration criteria, and is also relevant to the development of candidate list strategies. A candidate list is simply a list of, for example, the most promising moves to be explored at a given iteration, which are usually identified with respect to moves either used or ignored at previous iterations. Reeves[Ree95a] found such an approach highly beneficial in the context of a machine sequencing problem. These ideas are treated comprehensively in Glover and Laguna [GL93], along with a comprehensive review of applications which suggests that tabu search is a very effective way of navigating through large and complex solution spaces. From a practical point of view, a recency-based approach with a simple neighbourhood structure which is searched using a restricted candidate list strategy will often provide very good results. 1.7 Genetic Algorithms Genetic algorithms (GAs) have become perhaps the most well-known of all these modern heuristics, having had many column-inches of newsprint and several TV programmes devoted to them. They can also be viewed as a form of neighbourhood search (Reeves[Ree94]; Rayward-Smith[RS95b]), although their original inspiration comes from population genetics. Unlike SA and TS, GAs use a collection (or population) of solutions, from which, using selective breeding and recombination strategies, better and better solutions can be produced. Simple genetic 'operators' such as crossover and mutation are used to construct new solutions from pieces of old ones, in such a way that for many problems the population steadily improves.

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1.7.1 Crossover Crossover is a simple idea: suppose we have 2 strings a and b, each consisting of 6 variables, i.e.

representing two solutions to a problem. A crossover point is chosen at random from the numbers 1, . . ., 5, and a new solution produced by combining the pieces of the the original 'parents'. For instance, if the crossover point was 2, then the 'offspring' solutions would be

In many applications, the component vector, or chromosome, is simply a string of 0s and 1s. Goldberg[Go189] suggests that there are significant advantages if the chromosome can be so structured, although other arguments [Ant89] have cast some doubt on this. Nevertheless, much of the theoretical development is easier to understand if it is thought of in this way. Continuing the genetic analogy, variables are often called genes, the possible values of a variable alleles, and the position of a variable in a string is called its locus. In simple cases, the locus of a variable/gene is usually irrelevant, but in more complex problems it becomes important. A further distinction is drawn, in genetics, between the chromosome (or a collection of chromosomes) as genotype, meaning the actual structure (or structures), and the phenotypethe physical expression of the structure (or structures). In terms of a GA, we may interpret the genotype as the coded string which is processed by the algorithm, while the decoded set of parameters represents the phenotype. 1.7.2 Mutation The other most commonly used operator is mutation, which provides the opportunity to reach parts of the search space which perhaps cannot be reached by crossover alone. Each gene of a string is examined in turn, and with a small probability its current allele is changed. For example, a string

if the 3rd and 5th alleles are mutated. 1.7.3 Reproduction Crossover and mutation are the basic tools for creating new solutions (reproduction). However, which chromosomes are chosen as a basis for the reproductive step (or selected for reproduction) is clearly critical in what happens to a population as a whole. Holland[Hol75], who was responsible for the initial development of GA concepts, suggested that at least one 'parent' should always be chosen on the basis of its 'fitness'in terms of combinatorial problems, this implies some monotonic function of the objective function value. With this he was able to prove the important Schema

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Theorem which provides one way of understanding how a GA works. Details of this can be found in Goldberg's book [Gol89], and also in Reeves[Ree93a], but the basic idea is fundamental to traditional GA theory, so we describe it here. If we take two strings such as

we notice something they have in common: they both have a 0 for the first gene, and a 1 for the second. In GA jargon, they are both instances of the schema (01∗∗∗∗), where the ∗ symbol is essentially used as a 'don't care' character. Schemata thus represent similarity subsets of the hypercube over which the search is taking place. In this case the subset consists of all 16 (= 24) strings generated by substituting particular values for the ∗ symbols. The order of a schema is the number of defined (0 or 1) genes, while its length is the distance between the outermost defined genes. (In the above example, the order is 2, the length is 1.) Using this way of grouping strings together, Holland showed that, given a fixed number of instances of a schema S in the population at generation t, the expected number of instances of the schema at generation t + 1 would increase provided that its fitness ratio f(S) (its fitness divided by the average population fitness) exceeded

where l(S) is the length and k(S) the order of S, and Pm the probability of mutation. This is often interpreted as saying that low-order, short and above-average schemata will propagate at an exponentially increasing rate, but this is incorrectall it says is what is expected to happen in the next generation. The actual number of instances of S may be quite different from the expected number. Nevertheless, it provides a basis for an explanation of how GAs work. Holland also put forward other arguments, but they are not really germane to this discussion. 1.7.4 A Simple Example To make the action of a GA more transparent, we again use the problem of maximizing the function

Suppose now we generate five random strings with the results as shown in Table 1.12.

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Page 20 Table 1.12 Five random strings No.

String

x

f(x)

P[select]

1

10011

19

2399

0.206

2

00101

5

3225

0.277

3

11010

26

516

0.044

4

10101

21

1801

0.155

5

01110

14

3684

0.317

Average fitness

2325

Treating the f(x) values simply as fitness values, the probability of selecting each string as Parent 1 is in direct proportion to its value, and is given in the P[select] column. By drawing uniform random numbers between 0 and 1, we select Parent 1 using this distribution, and Parent 2 using a discrete uniform distribution. Again, by drawing uniform random numbers, we proceed to carry out crossover (with Pc = 1) at a randomly determined site, followed by mutation (with Pm = 0.02 at each locus). The results of a typical experiment were as shown in Table 1.13. Table 1.13 A typical experiment. Step

Parent 1

Parent 2

Crossover point

1

1

2

4

2

5

3

3

5

4 5

Mutation?

Offspring String

f(x)

NNNYN

10001

2973

2

NNNNN

01010

4100

2

3

NNNNN

01101

3857

4

2

1

NYNNN

11101

129

2

5

4

NNNNN

00100

2804

Average fitness

2773

The search would continue by recalculating the selection probabilities and carrying out a new cycle of selection, crossover and mutation. This is a very simple GA, but it has been effective in raising the population average quite substantially, and it has actually found the optimum at step 2.

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1.7.5 Modifications and Enhancements The basic GA has been developed and improved to an enormous extent over the past 20 years. Some of the most important of these are described in Reeves[Ree93a]; here we give a brief description, without giving a detailed justification for the various proposals. Selection in proportion to fitness is common to nearly everything that can be called a GA. However, the exact mechanism for accomplishing it can vary significantly. The simplest is roulette wheel selection, where chromosomes are sampled using a cumulative probability distribution which reflects their relative fitnesses (as in the example above). In practice this introduces too much noise into the selection process, and normally it is better to use something like stochastic universal selection, which can be thought of as a roulette wheel which is sampled with M equally spaced pointers. Hancock[Han94] gives a useful summary of the effects of many different selection methods. However, the fitness values usually have to be repeatedly re-scaled as the search progresses, and in many implementations, it is better to carry out selection using the ranks rather than the actual fitnesses. This can be carried out very simply using the probability distribution

where [k] is the kth chromosome when they are ranked in ascending order. This distribution seems to give a sensible selective pressure, in that the best (chromosome [M]) will have a chance of 2/(M + 1) of being selected, roughly twice that of the median, whose chance of selection is 1/M. However, there are many other ways of using ranks. Another approach is tournament selection where groups of T chromosomes are selected from the population and compared, the best being selected. When T = 2, this is very similar in its effect to the ranking approach suggested above. Crossover has been developed from the simple 1-point crossover described above, to 2-point, 3-point and indeed multipoint crossover. There are certainly some situations where 1-point crossover is less effective, but it is hard to give a definitive statement as to which is best in general. Uniform crossover, where each bit is equally likely to come from either parent, is often useful, as it makes no assumptions about the compactness of the coding of the strings. (A biased version where selection is preferentially from one parent is also possible.) Many other special-purpose recombination operators have been devised; perhaps the most important non-binary family comprises those developed for sequence-based coding. Many problems of interest in OR can be most naturally represented as a permutation, the obvious one being the TSP. Unfortunately, the ordinary crossover operator is no use in such situations, as can easily be seen from the following example.

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Goldberg and Lingle[GL85] defined an operator called PMX (partially mapped crossover), which used two crossover points. The section between these points defines an interchange mapping. Thus, in the example above, PMX might proceed as follows:

Here the crossover points X and Y define an interchange mapping {3 ↔ 6,4 ↔ 2,5 ↔ 7}. Goldberg[Gol89] describes two other types of crossover for sequence representations, while Reeves[Ree95] used yet another which performed well in the context of a flowshop sequencing problem. This operator works by choosing a crossover point X randomly, taking the pre-X section of the first parent, and filling up the chromosome by taking in order each 'legitimate' element from the second parent. For the example above it might generate the following offspring:

The rationale for this operator is that it preserves the absolute positions of the jobs taken from P1, and the relative positions of those from P2, which seems to provide enough scope for modification of the chromosome without excessively disrupting it. It should perhaps be noted here that neither of these operators is actually very suitable for the TSP; some reasons for this, and the consequent development of criteria for better operators, are contained in the work of Radcliffe (in, for example, Radcliffe[Rad91] and Radcliffe and Surry[RS95a]). Mutation is usually employed with a very low probability per bit. There is some evidence however that a somewhat higher rate of around 1/n (where n is the string length) should be used. Some experimental studies have also found that it may be beneficial to change the rate of mutation as the search progresses, and also that it may be better to have different rates for different bits. Replacement is often carried out en bloc, and this was the underlying assumption in the description of the basic GA given above: M new strings are generated from the current population and become the next generation. This generational approach is popular, but a significant fraction of the GA community prefer to use incremental replacement. In this case a few strings are generated (often just one), and then used to replace some of the existing population. This approach (sometimes called a steady-state GA) is almost always quicker, but may lead to premature convergence with small populations. For this reason, it is important to consider carefully the deletion strategy: one approach is always to delete the worst, but this may often converge far too quickly, and something less severe may be better overall unless the population is really large. The previously-cited paper by Hancock [Han94] is a useful source of information on the possibilities.

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1.7.6 Summary Like SA and TS, GAs provide the user with several parameters to tune. But also like the other heuristics, GAs are often fairly robust, and simple one-point or uniform crossover, a 1/n mutation rate, and an incremental replacement policy with a mild deletion strategy will often give good results. 1.8 Conclusion These notes have given a brief introduction to three important modern heuristic methodssimulated annealing, tabu search and genetic algorithms. The focus of the first two has traditionally been that of combinatorial optimization, while the last has more commonly dealt with problems where real-valued parameters are discretized. However, this distinction is really artificial by discretizing a set of continuous parameters, the problem does become combinatorial. GAs can be (and have been) used successfully for combinatorial problems, while reports are starting to appear where SA and TS have been applied to discretized versions of continuous problems. This points up a remarkable fact about all of these methods: we do not need to know anything about the problem we are trying to solve, for the method will work so long as we have a 'black box' which will evaluate the solution proposed. Nor do we have to assume all quantities are exact or deterministicthere are several reported applications in the case of GAs (in, for example, Grefenstette and Fitzpatrick[GF85]; Reeves[Ree92]) which use approximate and/or noisy evaluation functions, and still get solutions of high quality. There is also considerable scope for parallelization of the methods, and in the case of GAs in particular such concepts have led to interesting new ideas of 'spatial' as well as 'temporal' population interaction. Having said all that, it is nonetheless true that in many cases the performance of the techniques can be considerably enhanced by building in some problem-specific knowledge. The use of heuristic methods should not be a substitute for thought! For instance, it should be realized that constrained problems may give rise to some difficulties for these methods. The conventional practice is to incorporate constraints into the objective by using penalty factors. However it can be difficult to find appropriate penalty terms, and alternatives such as a modified definition of a neighbourhood (for SA and TS), or modified recombination operators (for GAs) should be explored. Finally, it is becoming clear that there is plenty of scope for traditional OR methods to be used in conjunction with their more modern counterparts, and much research is going into building more effective hybrids. There are already several examples in the specialist literature of cases where combining elements of GAs with SA, or of SA with TS, has been found helpful, although an introductory chapter is not the place to describe these developments. However, it is hoped that sufficient has been said here for interested but previously unaware readers to be able to understand the remaining chapters of this volume, and also to encourage them to explore the wider literature for ideas that will help them to find better solutions to messy real-world optimization problems.

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References [Ant89] Antonisse J. (1989) A new introduction of schema notation that overturns the binary encoding constraint. In Schaffer J. D. (ed) Proceedings of 3rd International Conference on Genetic Algorithms. Morgan Kaufmann, Los Altos, CA. [Con90] Connolly D. T. (1990) An improved annealing scheme for the qap. Euro.J.OR 46: 93100. [Cro58] Croes G. A. (1958) A method for solving travelling salesman problems. Opns.Res. 6: 791. [Dow93] Dowsland K. (1993) Simulated annealing. In Reeves [Ree93c], chapter 2. [GF85] Grefenstette J. J. and Fitzpatrick J. M. (1985) Genetic search with approximate function evaluations. In Grefenstette J. J. (ed) Proceedings of an International Conference on Genetic Algorithms and their applications. Lawrence Eribaum Associates, Hillsdale, NJ. [GL85] Goldberg D. E. and Lingle R. (1985) Loci and the travelling salesman problem. In Grefenstette J. J. (ed) Proceedings of an International Conference on Genetic Algorithms and their applications. Lawrence Erlbaum Associates, Hillsdale, NJ. [GL93] Glover F. and Laguna M. (1993) Tabu search. In Reeves [Ree93c], chapter 3. [Glo86] Glover F. (1986) Future paths for integer programming and links to artificial intelligence. Computers & Ops. Res. 5: 533549. [Gol89] Goldberg D. E. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, Mass. [Han94] Hancock P. J. (1994) An empirical comparison of selection methods in evolutionary algorithms. In Fogarty T. C. (ed) Evolutionary Computing: AISB Workshop, Leeds, UK, April 1994: Selected Papers, number 865 in Lecture Notes in Computer Science. Springer-Verlag, Berlin. [Hol75] Holland J. H. (1975) Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor. [Joh54] Johnson S. M. (1954) Optimal two- and three-stage production schedules with setup times included. NRLQ 1: 6168. [Kar72] Karp R. M. (1972) Reducibility among combinatorial problems. In Miller R. E. and Thatcher J. W. (eds) Complexity of Computer Communications. Plenum Press, New York. [KGV83] Kirkpatrick S., Gellat C. D., and Vecchi M. P. (1983) Optimization by simulated annealing. Science 220: 671680. [Kuh55] Kuhn H. W. (1955) The Hungarian method for the assignment problem. NRLQ 2: 8397. [LA88] Laarhoven P. V. and Aarts E. H. L. (1988) Simulated Annealing: Theory and Applications. Kluwer, Dordrecht. [LLKS85] Lawler E., Lenstra J., Kan A. R., and Shmoys D. (eds) (1985) The Travelling Salesman Problem. John Wiley & Sons, Chichester. [LM86] Lundy M. and Mees A. (1986) Convergence of an annealing algorithm. Math.Prog. 34: 111124. [MRTT53] Metropolis N., Rosenbluth A. W., Teller A. H., and Teller E. (1953) Equation of state calculation by fast computing machines. J. of Chem. Phys. 21: 10871091. file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_24.html（第 1／2 页）2009-7-7 16:56:10

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[Rad91] Radcliffe N. J. (1991) Equivalence class analysis of genetic algorithms. Complex Systems 5: 183205. [Ree92] Reeves C. R. (1992) A genetic algorithm approach to stochastic flowshop sequencing. In Proceedings of IEE Colloquium on Genetic Algorithms for Control and Systems Engineering. IEE, London. [Ree93a] Reeves C. R. (1993) Genetic algorithms. [Ree93c], chapter 4. [Ree93b] Reeves C. R. (1993) Improving the efficiency of tabu search in machine

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sequencing problems. J. Opl. Res. Soc 44: 375382. [Ree93c] Reeves C. R. (ed) (1993) Modern Heuristic Techniques for Combinatorial Problems. Blackwell Scientific Publications, Oxford. [Ree94] Reeves C. R. (1994) Genetic algorithms and neighbourhood search. In Evolutionary Computing: AISB Workshop, Leeds, UK, April 1994: Selected Papers, number 865 in Lecture Notes in Computer Science. SpringerVerlag, Berlin. [Ree95] Reeves C. R. (1995) A genetic algorithm for flowshop sequencing. Computers & Ops. Res. 22: 513. [Rei94] Reinelt G. (1994) The Travelling Salesman. Springer-Verlag, Berlin. [RS95a] Radcliffe N. J. and Surry P. (1995) Formae and the variance of fitness. In Whitley D. and Vose M. (eds) Foundations of Genetic Algorithms 3. Morgan Kaufmann, San Mateo, CA. [RS95b] Rayward-Smith V. J. (1995) A unified approach to tabu search, simulated annealing and genetic algorithms. In Rayward-Smith V. J. (ed) Applications of Modern Heuristic Techniques. Alfred Waller Ltd., Henley-on-Thames, UK.

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with an associated domain D1, . . ., Dn. A complete label for a CSOP is an assignment of a value to each variable from its associated domain. A consistent label is one that satisfies all the constraints. A solution is a complete and consistent label. An optimal label is a solution that optimizes the objective function. This is achieved by using an objective constraint which is associated with the objective function; this constraint is soft in the sense that its admissible values are not fixed but are to be optimized. Many constraint satisfaction and constraint optimization techniques have been developed over the years. The focus in this paper is on a stochastic technique which has proved to be effective for large-scale applications. Simulated annealing (SA) [KGV83] is an iterative stochastic search method derived from the physical annealing process whereby material is gradually cooled to achieve a minimum energy state. It involves a process of moving from one solution to a neighbouring solution under the control of an annealing schedule. The process starts with a high temperature and gradually drops the temperature after a certain number of iterations at each temperature. The success of SA on a given problem typically depends on devising an appropriate annealing schedule. In general, approximating a stationary distribution arbitrarily closely requires an exponential-time execution of the simulated annealing algorithm [AvL85]. Since annealing schedules must be short for practical reasons, they may not produce a good approximation to the stationary distributions, which guide the search procedure to the global minima. By the nature of a practical annealing schedule, the cooling rate will slow down with the decrease in temperatures; consequently the majority of search time will be spent at low temperatures. Spending a lot of time on greedy local search at low temperatures will overlook many good solutions if the approximate distributions are not close to the stationary distributions at these temperatures. It will be worse if the search is trapped by local optima. In summary, given a limited search time there are two issues in devising a proper annealing schedule: 1. Closely approximating stationary distributions at high temperatures could occupy all the search time, and leave no time for finding good or optimal solutions. 2. Greedily focusing on the better solutions in current local spaces will sacrifice the approximation of stationary distributions, and this will degrade the probability of finding globally good solutions. There is a fundamental need for 'compromise': the tasks of approximating stationary distributions and approaching globally good solutions are in conflict due to the practical limitation on search time. An annealing schedule represents a compromise in this conflict; but there are difficulties in finding a good one. 1. Devising a suitable annealing schedule is highly problem-dependent. Good annealing schedules vary from problem to problem. 2. The criteria for judging an annealing schedule are empirical. It is typical that a good annealing schedule is confirmed only after testing a set of parameters. Nevertheless, there is a tendency to use certain general principles in devising an annealing schedule for practical applications [CEG87], in particular, the following.

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2 Localized Simulated Annealing in Constraint Satisfaction and Optimization Y. H. Li, E. B. Richards, Y. J. Jiang and N. A. S. Azarmi Abstract Simulated annealing (SA) is a stochastic technique for solving constraint satisfaction and optimisation problems. Research on SA usually focuses on neighbourhood operators and annealing schedules. While SA can theoretically converge to a global optimum with stationary distributions, this requires an exponential-time execution of the annealing algorithm; in practice only a short annealing schedule can be used. To devise an appropriate annealing schedule one must consider how much time to spend at high temperatures for approximating stationary distributions and how much time to spend at low temperatures for greedily approaching better solutions. Unfortunately these two things are always in conflict. Annealing schedules are problem-dependent, and deciding a good one for a specific problem relies on empirical studies. This makes it difficult to balance the conflict. We argue that the process of balancing the conflict should be a dynamic procedure rather than a static one. We present a refined SA strategy that can adjust annealing schedules according to the solution distribution formed in past search. The basic idea is to divide the search space into disjoint subspaces which are then processed by a localized SA strategy. Different temperatures and annealing speeds can be maintained in the different subspaces depending upon certain evaluation criteria. To assess the performance of the proposed strategy, we apply it to two real-world applications: flight scheduling problems and freight train regrouping problems. Experimental results demonstrate that the proposed SA strategy is consistently better than standard SA. 2.1 Introduction A constraint satisfaction and optimisation problem (CSOP) [Tsa93] consists of an objective function and a number of constraints on a set of variables X1, . . ., Xn each Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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A very high initial acceptance probability or a very high starting temperature Cooling rates usually from 0.8 to 0.99 Iterations for each temperature which typically equal the number of neighbours A terminating criterion which could be transferred to a very low temperature For a given problem, different annealing schedules will produce different qualities of final solutions. This suggests that an annealing schedule interacts with certain problem features during the problem-solving procedure. It is our assumption that the quality of the interaction will be enhanced by applying different annealing schedules to different parts of the search space. We propose a new version of simulated annealing called 'localized simulated annealing (LSA)'. The basic idea is to divide the search space of a problem into disjoint subspaces. These subspaces provide local information to influence the annealing characteristics in the subspaces. During the search process, each subspace is evaluated according to criteria that relate to the features of the subspace, and the annealing procedure in this subspace is adjusted accordingly. This facilitates different annealing speeds in different subspaces. Unlike standard simulated annealing, the proposed method keeps repairing the probability distributions by maintaining high temperatures in no-good subspaces. It also allows the move from a low-temperature subspace to a hightemperature subspace and vice versa. When a global annealing schedule efficiently guides the search process, most subspaces will have similar temperatures. In this case the behaviour of the annealing schedule will be similar to that of standard SA. However, when the global annealing schedule does not efficiently guide the search process, LSA can still escape from local minima that would normally trap standard simulated annealing. In this way it increases the search efficiency by gradually improving the convergence direction. We shall offer empirical justification for the effectiveness of LSA. We shall apply the method to two real-world applications: the flight scheduling problem and the freight train regrouping problem. Empirical results on these problems have shown that the proposed SA consistently reaches a higher-quality solution than standard SA. This paper is organized as follows. Section 2.2 introduces the formulation of subspaces and their evaluations. Section 2.3 describes the localized SA strategy. Section 2.4 provides a theoretical analysis of the proposed SA and justifications. Section 2.5 presents the empirical results of the proposed strategy on some real-world applications. 2.2 Subspaces and Their Evaluation The search space of a combinatorial optimization problem consists of the possible combinations of values from the domains of all variables of the search problem. A subspace of the problem is characterized by certain sub-domains of a subset of the variables of the search problem. Ideally, the division of a search space should reflect the problem structure of an application. This may not always be possible to determine

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in practice. For the purpose of this paper, we choose a division of the domain of a particular variable into a manageable number of disjoint subdomains. In simulated annealing, once the temperature is fixed, the only thing that determines the acceptance probability is the distance between the cost of the current solution and the cost of its neighbouring solution. Any neighbouring solution that has a similar cost to the current best cost will obviously have a high probability of being accepted. So the current best solution can be regarded as the representative of these solutions and can be treated as a quasi-equilibrium point; all solutions around this point have a higher probability of forming the quasi equilibrium. Similarly, the local best can represent the quasi equilibrium point of a subspace and the global best represents the quasi equilibrium point of the entire search space. The best solutions in individual subspaces can be used to assess these subspaces and the gap between a local best and the global best will reflect the goodness of a subspace for the global search. Here we introduce a new term 'approachability' to characterize a subspace. Definition 1 We define the approachability of a subspace as a measure of the goodness of the subspace with respect to the global best. The evaluation procedure for the approachability of a subspace is roughly as follows. In carrying out a search of the whole search space, randomly jumping among all subspaces according to the neighbourhood generator, the procedure records the local best S* for all subspaces and the current global best solution Sb. At the end of each temperature stage, the procedure measures the distances between Sb and each S*. If |cost(S*) cost(Sb)| > εT (where εT is a control parameter for the evaluation of approachability), in other words the local best solution is not reasonably near to the global best solution, we say the subspace represented by S* is no-good. Otherwise we say this subspace is good. The control parameter εT determines what LSA 'looks like'. For example, if the values of εT are large, LSA will behave like standard simulated annealing, because the approachabilities of all subspaces are similar at any temperature. On the other hand, if the values are very small, the behaviour will more likely be random search. Since LSA is essentially SA, small values should initially be avoided. Ideally, εT should be set to maximize the goodness of the approachability of each subspace at each temperature. 2.3 Localized Simulated Annealing (LSA) The motivation behind the approachability measure of subspaces is to reduce the conflict between approximating stationary distributions and approaching globally good solutions. The most important aspect of the LSA algorithm is its flexible annealing schedule. Standard SA uses the same annealing schedule for the whole search space. This kind of annealing procedure may 'overlook' some good parts of the search space if the annealing schedule is not designed carefully. In most real-world applications, it is impractical to admit a large number of iterations at each temperature or a sufficiently slow annealing speed. This means that the annealing procedure inevitably 'cuts of' some good search areas. The different distributions of feasible solutions among different subspaces call for different annealing schedules in the local spaces. The proposed SA

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thus employs a localized annealing strategy that allows different annealing speeds in different subspaces. The LSA annealing strategy contains two parts. 1. A global annealing schedule. This schedule is exactly the same as that of standard SA. It involves an initial temperature, a rate of temperature decrease, a number of iterations at each temperature and a terminating criterion. (a) The initial temperature is the starting temperature for all subspaces. (b) The temperature in the subspace that contains the current best solution is always equal to the globally current temperature. (c) After completing the fixed number of iterations at a given temperature, the algorithm evaluates the subspaces and at least the subspace that contains the current best solution will decrease its temperature to the new globally current temperature. (d) When the terminating criterion is satisfied, the algorithm will stop. 2. A localized annealing procedure among the subspaces. The localized annealing procedure applies different annealing speeds to different subspaces. It involves the following. (a) A procedure for evaluating the approachability of subspaces. Note that the evaluation parameter εT of the approachability of a subspace is dependent on the temperature, and its value decreases with the reduction of the temperature. (b) A procedure for decreasing the temperatures of subspaces. The approachability of a subspace is a criterion for decreasing the temperature of this subspace. The size of the temperature decrease is decided by εT. For example, suppose a subspace S1 has the temperature Ti while the subspace that contains the current global best solution has where Ti > Tk≥ Tj (and there is no temperature Tj and Ti > Tj. Given that |cost(S1best) cost(globalbest)| temperature lower than Tk which satisfies the given conditions), the algorithm decreases the temperature of the subspace from Ti to Tk. This procedure is applied to all subspaces. (c) The use of the temperatures of subspaces. Suppose a new solution j, generated from a solution i by a neighbour generator move, is in a different subspace than the one containing i; the algorithm uses the temperature of the subspace that contains i to calculate the acceptance probability. If the current subspace containing i has a higher temperature than the new subspace containing j, the search procedure has a relatively high probability of going to the new subspace. Conversely, the search procedure has a relatively low probability of going to the new subspace if the temperature is lower. It is important to note that LSA employs the same neighbourhood operator that is employed by a standard SA procedure. The difference is that the neighorhood operator can move from one subspace to another subspace.

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The detailed procedure of LSA is described in Figure 2.1. In this procedure, K denotes the Kth temperature stage, TK the temperature of the Kth temperature stage, LK the iterations at the temperature TK, f the cost function over solutions, lbest( subspace i) the local best solution of the subspace in which solution i occurs, gbest the current global best solution for all subspaces, Tsubspace i the temperature of the subspace of the solution i, and Tglobal(k) the global temperature at the Kth temperature stage of the LSA schedule. The procedure starts from an initial random solution istart. The initial temperature of all subspaces is T0. Suppose j is the neighbour solution of solution i; if j is better than lbest(subspace j), j will become lbest(subspace j). Furthermore, if j is better than the current gbest, j will become the gbest and the temperature will be immediately dropped to the current global temperature that is the temperature of the subspace containing the current best solution. After each LK iteration the global temperature is decreased, which is done by Calculate TEMPERATURE(Tglobal(K)). The Calculate TEMPERATURES(subspaces) calculates what temperature each subspace should be dropped to based on the evaluation of the subspace by Evaluate(subspaces). 2.4 Analysis and Justification Simulated annealing uses the same annealing speed for the whole search space. From a theoretical point of view the global optimal convergence of simulated annealing requires a sufficient amount of search time which is mainly related to two factors: (1) speed of decreasing temperature and (2) number of iterations at each temperature stage [JMS89]. In practice, we often cannot achieve a sufficiently slow rate of decreasing temperatures or sufficiently many iterations at each temperature. This is especially so for large-scale problems in reactive environments where one can allow only a very limited amount of search time at each temperature Tk with a limited number of temperature-decreasing stages. This produces two problems. Let qj(Tk) denote the stationary distribution at temperature Tk and aj(Tk) denote the quasiequilibrium at temperature Tk [AK89]. 1. At each temperature Tk, the quasi equilibrium aj(Tk) is not sufficiently close to the stationary distribution qj(Tk). At the next temperature Tk+1, aj(Tk) cannot be used to generate an aj(Tk+1) that is close to qj(Tk+1). In other words aj(Tk) cannot be relied upon to produce a good aj(Tk+1). 2. Because of the nature of the practical annealing schedule, the cooling speed at low temperatures is much slower than at high temperatures. Thus a majority of search time will be spent at low temperatures. The lower a temperature is, the fewer will be the solutions the SA algorithm will visit at that temperature. Moreover at low temperatures good quasi equilibriums will require longer Markov chains than at high temperatures. So finding better solutions than the local optima supported by good quasi equilibriums at low temperatures is not likely. As a result, the SA algorithm will more likely stay in the local spaces around the local optima. Because of the low quality of the quasi

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begin INITIALIZE (istart,T0,L0); K := 0; i := istart; repeat for l := 1 to LK do begin j := Neighbour(i); if f(j) < f(lbest(subspace j)); then lbest(subspace j) := j if f(j) ≤ f(i) then i := j if f(j) < f(gbest) then gbest := j endif; if f(j) ≤ f(gbest) then Tsubspace j := Tglobal(K) endif else if e ( f (i) f ( j )) / Tsubspace i >random(0,1) then i := j endif endif end; K := K + 1; Calculate LENGTH(LK); Calculate TEMPERATURE(Tglobal (K )); Evaluate(subspaces); Calculate TEMPERATURES(subspaces); until stopcriterion end Figure 2.1 Localized simulated annealing.

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equilibriums, many good solutions may not be found and the local optima found may not be good enough. Unlike SA, LSA keeps repairing the quasi equilibria of previous temperatures while it runs the global annealing schedule. If the solution distribution in a subspace is not good, LSA will leave the temperature at a higher level than the temperatures in those subspaces that have better solution distributions. Once the search is back to this subspace, LSA will visit more solutions than SA in this subspace and these solutions can improve quasi equilibrium for the temperature of the subspace. If the currently global temperature is low, LSA can use the available search time more effectively than would be the case if search were to continue in the local spaces that are around current local optima. In conclusion there are at least two beneficial results arising from LSA. LSA uses search time more effectively by identifying and avoiding no-good local spaces. LSA revises and improves approximate stationary distributions. In effect LSA can dynamically enhance the 'balance' of approximating stationary distributions and approaching good or optimal solutions. To support these claims let us analyse how LSA moves from one subspace to another. Suppose that LSA generates a neighbour j of i and j is accepted. Let T(i) and T(j) denote respectively the temperatures of the subspace containing i and the subspace containing j. Then there are several cases. 1. T(i)

T(j). In this case the search procedure is not different from the standard SA procedure.

2. T(j) > T(i) but f(i) f(j). This means that j is probably a good solution for subspace j and better than the local best of subspace j, because the lbest(subspace i) is better than lbest(subspace j) and solution i has a higher probability of being near lbest(subspace i) than being distant from it. The temperature of subspace j will be reduced in the next global temperature stage. 3. T(j) > T(i) and f(j) > f(i). In this case there are two possibilities. (a) If the search procedure has been trapped in a local optimum, using the higher temperature T(j) will provide a higher probability of escaping from this optimum. (b) If the search procedure is not trapped, because decreasing temperature has guided the search procedure correctly to globally good solutions or optimal solutions, the solution distributions at high temperatures are good and the approximate probability distributions for finding good solutions are close to the stationary distributions. This implies that many subspaces should have the same or similar low temperatures. So it has a very high probability of going to these subspaces. Two key points should be noted. 1. If global annealing schedule leads to optimal solutions, LSA does.

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2. LSA has a better chance than SA of escaping local optima. What is it that allows LSA to work using the information arising from previous search? The answer is the predefined subspaces. In SA the cost of solutions in a trial cannot be used individually to judge a proper annealing schedule that should be carried out thereafter because the solutions do not individually provide significant information for assessing a part of the search space. Moreover a local space is a set of the neighbours of a current solution and in most cases each solution has its own distinguishable neighbours. This means that the number of local spaces is equal to the number of solutions in the search space. Obviously using cost to assess a local space is not different from assessing the solution itself. In LSA the task of capturing information from previous search can be distributed among these subspaces. From the discussion of cost in Section 2.2, we know that the local best can represent the quasi equilibrium point of a subspace and the global best represents the quasi equilibrium point of the entire search space. So the cost of the best solutions in the individual subspaces can be used to assess these subspaces and the approachability of a subspace can be used to measure its goodness. This allows LSA to adjust the annealing procedure in each subspace. In effect LSA utilizes more valuable information than SA. 2.5 Empirical Results To evaluate the effectiveness of LSA against SA, we first used a set of travelling salesman problems (TSP). We chose TSPs because of their importance in real-world applications and because they are a good measure of the effectiveness of a search algorithm. For simplicity, we only tested LSA on TSPs of size 64 cities (the data is randomly generated on the board (0, 400) × (0, 300)). A TSP is represented as a sequence of 64 different variables, with the domain of all variables being the set of 64 cities. A label is an assignment to the variables such that no two variables have the same value. The neighbourhood operator chooses randomly two positions from the sequence and reverses the subsequence delimited by the two positions. For the given problem structure, dividing the domain of one variable cannot form disjoint subspaces. As a result we use two variables to build the subspaces. The value of the first variable is fixed as city 1. The domain of the second variable X is divided into subdomains which characterize the subspaces. For example, if X's value in a label is in the range of [2,8], then the label is in one subspace. If X's value in a label is in the range of [9,16], then the label is in a different subspace. Our initial results are summarized in Table 2.1. It contains the average costs and the best costs of TSP tours produced by SA and LSA respectively over 35 runs for 7 instances. It also lists the percentage improvement of LSA over SA. The initial results of LSA compared to SA are very encouraging; they are even more encouraging for two real-world problems: the flight scheduling problem and the train grouping problem. The specifications and empirical results of these problems are described in turn as follows.

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Page 36 Table 2.1 Computation results for TSP problems SA Av.

LSA Av.

Improvement

SA Best

LSA Best

Improvement

2411

2324

3.61

2275

2197

3.43

2.5.1 The Flight Scheduling Problem The flight scheduling problem consists of a set of flight requests and a set of planes. Each flight request (e.g. from London to Paris) demands a plane to satisfy the request. This plane can either be at the airport (London) in the beginning (in this case, it is a new plane) or it is brought by some other flight request (e.g. another flight from Amsterdam to London). If a flight request is assigned a new plane, it incurs a cost of 30 units. If a flight request is assigned a plane that is not brought by a request in the same tour, it incurs a cost of 10 units. There is a preference that requests in the same tour share the same plane. An example is given below. Request number 3: London to Paris Flight from 10 am to 11:25am Tour number of this request: 16 Possible planes for this request: (plane1, newplane), (plane2, newplane), (plane3, from request 1 or request 2), (plane4, from request 1), (plane5, from request 2), (plane6, from request 1 or request 2). Some planes have defects which matter to some flight requests thereby generating a cost. A hard constraint is that no two flight requests can take the same aircraft unless the plane assigned to one request arrives, prior to the departure time of the other request, at the airport of the other request. The objective of the flight scheduling problem is to allocate each flight request with a plane in such a way that the overall allocation minimizes cost. Curiously, there is no need to specify the hard constraint which states that no one flight request can take more than one plane. This is because we are minimizing the overall costs. So if a flight request is assigned two planes that do not violate any other hard constraint, then choosing one of the two planes will never increase the cost and will probably reduce it. The label of a flight scheduling problem is simply the assignment of planes to flight requests from their domains. Here each flight request is represented by a variable and the domain of the variable is the set of aircraft that can be assigned to the flight request. The neighbourhood operator involves first repairing those flight requests that violate a hard constraint and then any flight request that violates a soft constraint. Repairing a flight request simply assigns an alternative aircraft from the domain of the flight request. Clearly, a label may not be a solution label. By selecting flight requests violating hard constraints to repair first, the neighbourhood operator is geared towards feasible solutions while moving towards more optimal solutions. To allow comparisons

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Page 37 Table 2.2 Computation results for sample BA scheduling problems Class B3 Problem

SA Best

Success Rate

LSA Best

Success Rate

b3a

475

90

475

99

b3b

515

89

515

99

b3c

525

87

525

98

Table 2.3 Computation results for sample BA scheduling problems Class B5 Problem

SA Best

Success Rate

LSA Best

Success Rate

b5a

690

82

690

95

b5b

765

78

765

93

b5c

755

75

755

90

between labels with different sets of hard constraint violations, each hard constraint violation incurs a cost that is equal to the upper bound of the total costs of all soft constraints in the flight scheduling problems. We divide the search space of the problem into 6 disjoint subspaces based on the value of the flight request variable (call it X) that has the largest domain size. Instances of the flight scheduling problem are of two types. The first type B3 consists of 45 requests and 16 aircraft. The second type B5 consists of 100 requests and 26 aircraft. The problem instances of each class only differ on their soft constraints. The same annealing schedules are used for both LSA and SA on the same machine implemented in the same language. In this schedule, the initial acceptance probability X0 is 0.95, the cooling rate is 0.9 and the number of iterations at each temperature is 600 for B3 instances and 2000 for B5 instances. The empirical results are summarized in Table 2.2 and Table 2.3. Each table lists the best costs found by LSA and SA for 100 runs. It also gives the average success rate and the number of runs within each 100 runs that achieves the best optima. Like TSPs, LSA produces better-quality solutions than SA within the time limit of the annealing schedule. The time limit is determined by the number of iterations at each temperature and the rate of temperature decrease. Although the best optima found by LSA and SA are the same in 100 runs each time, the success rate of LSA in finding the optimum solution is about 10% better than SA. This is because SA is more easily trapped at local optima. For example, in b3a both SA and LSA reach cost value 495 at all runs, but SA sometimes is trapped there while LSA is seldom trapped. This difference in success rate is vitally important when one considers the scalability of the LSA algorithm. LSA is likely to produce better optima than SA for larger size problems within a given time limit. We are currently investigating this claim.

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2.5.2 The Freight Train Regrouping Problem The freight train regrouping problem(FTRP) is a scheduling problem for cargo transportation. It involves regrouping cargo carriages from different arrival trains in the marshalling yard of a train station in such a way that carriages from the same direction of destination are put on the same marshalling track. If the capacity of a marshalling track is to be exceeded by the carriages of an arrival train with the same direction as that of the track, then an unused marshalling track (if there is one) will be used. Usually, the capacity of the marshalling yard is much less than the total number of carriages of the arrival trains to be regrouped. Clearly in this case, different regrouping sequences of arrival trains in the marshalling yard will affect the total number of carriages marshalled given the limited capacity of the yard. The objective of FTRP is to find a label that has maximum marshalling of cargo carriages for the marshalling yard. The total number of carriages in the arrival yard is defined as follows:

This expression represents the distribution of trains that will be regrouped. Here i is the ith incoming train in an arrival track and j is the jth direction of a destination of a carriage. The term aij represents the number of carriages on the ith track with jth direction. K is the number of trains in the arrival yard and M is the total number of directions. Let di denote the number of carriages on the ith marshalling track. Let L be the total number of tracks in the marshalling yard and N be the maximum number of carriages that a track can take. Then we have the following constraints. 1. di≤ N (i = 1, . . ., L) 2. L < K 3. Each track in the marshalling yard can only take carriages that go in the same direction. 4. aij is the basic regrouping unit. This means that in the regrouping procedure each element of aij cannot be separated. The objective function is

The resource constraints in FTRP include the limited number of tracks in the arrival yard, the number of carriages that can be carried by a train, the number of available marshalling tracks and the capacity of each marshalling track. In FTRP the key thing that decides the number of carriages accommodated in a marshalling yard is the sequence of regrouping trains. We use a label to represent such a sequence and each label is a solution. In this context the variables represent the positions of the label. The domain values are the regrouping trains. The hard constraint for a feasible solution is that no train appears in a label twice. In FTRP the search space is composed of all possible regrouping sequences of trains in the arrival yard. We choose an arbitrary position of the label to divide the subspaces and select a configuration that distinguishes what set of trains represents a subspace.

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Page 39 Table 2.4 Computation results for sample FTRP problems Data Size

SA Av.

LSA Av.

Improvement

SA Best

LSA Best

Improvement

12×20

387

392

1.51

398

405

1.75

25×40

906

925

2.08

942

961

2.01

45×60

1606

1657

3.17

1681

1724

2.55

The FTRP consists of three types. In the first type, the number of marshalling tracks is 12 and the capacity of each track is 20 trains. In the second type, the number of marshalling tracks is 25 and the capacity of each track is 40 trains. In the third type, the number of marshalling tracks is 45 and the capacity of each track is 60 trains. The empirical results are summarized in Table 2.4. This table lists the average and best results found by LSA and SA for 100 runs. It also lists the improvement rate of LSA over SA in each case. From the table, it can be seen that LSA always finds better solutions than SA although the improvement rate varies between 1.51 and 3.17. 2.6 Conclusion In this paper we have presented a flexible simulated annealing strategy LSA that is based on the division of a search space. We have shown that by exploiting the information of each subspace, the refined strategy provides better control over temperature and annealing speed in different subspaces. We have demonstrated the effectiveness of LSA over SA by considering two real-world applications: flight scheduling and freight train regrouping problems. References [AK89] Aarts E. and Korst J. (1989) Simulated annealing and Boltzman Machines. John Wiley and Sons, Chichester. [AvL85] Aarts E. and van Laarhoven P. (1985) Statistical cooling: a general approach to combinatorial optimization problems. Philips Journal of Research 40: 193226. [CEG87] Collins N., Eglese R., and Golden B. (1987) Simulated annealing an annotated bibliography. Computer 21. [JMS89] Johnson D. C. A., McGeoch L., and Schevon C. (1989) Optimisation by simulated annealing: an experimental evaluation; part i, graph partitioning. Operation Research 37: 865892. [KGV83] Kirkpatrick S., Gelatt Jr. C., and Vecchi M. (1983) Optimization by simulated annealing. Science 220: 671680. [Tsa93] Tsang E. (1993) Foundations of Constraint Satisfaction. Academic Press, New York.

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3 Observing Logical Interdependencies in Tabu Search Methods and Results Stefan Voß Abstract We survey some tabu search approaches for combinatorial optimization problems with special emphasis on dynamic tabu list management. In the latter approach the most important aspect is to observe logical interdependencies between the solutions encountered throughout the search. Various successful applications from the area of location including new results are reviewed. 3.1 Introduction Due to the complexity of a great variety of combinatorial optimization problems, heuristic algorithms are especially relevant for dealing with these problems. As an extension of simple heuristics, a large number of local search approaches, such as deterministic exchange methods, have become increasingly important to improve given feasible solutions. The main drawback of these local search approaches is their inability to continue the search upon becoming trapped in a local optimum. This leads to consideration of modern heuristic search methods for guiding known heuristics to overcome local optimality. Following this theme, we investigate the tabu search metastrategy for solving combinatorial optimization problems. In the following sections we first give a brief introduction to the necessary basic ingredients of tabu search as well as some hints in use from the relevant literature. Most of all, however, we focus on a survey of extensions for observing logical interdependencies and corresponding applications in the area of location problems as they may be derived from a general problem in the field of network design. Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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3.2 Tabu Search Tabu search is a metastrategy for guiding local search approaches to overcome local optimality. This metaheuristic has been reported in the literature during the last few years as providing successful solution approaches for a great variety of problem areas. In the sequel we investigate the basic ingredients of tabu search as well as some issues that have proved successful for various applications. These advanced issues include specialized methods that incorporate different means of intelligence in providing history based search trajectories and allowing the inclusion of learning capabilities. Before going into detail, however, we provide a rough historical development including some of the most important references on tabu search. 3.2.1 Historical Development Following Glover and Laguna [GL93], 'tabu search has its antecedents in methods designed to cross boundaries of feasibility or local optimality standardly treated as barriers, and to systematically impose and release constraints to permit exploration of otherwise forbidden regions'. That is, the real foundation for tabu search may be sought in concepts that systematically violate feasibility conditions as in heuristic procedures based on surrogate constraints (Glover[Glo77]) or even in cutting plane algorithms. Note, however, that tabu search nowadays has become a separate powerful tool where the differences, for instance, to cutting plane algorithms are much more important than the resemblance. Most early references to tabu search in its present form are Glover[Glo86] and Glover and McMillan [GM86] where some foundations and a first application are provided, respectively. In addition, there are a number of contributions that have influenced the exploration of tabu search to the tool it is today: Glover[Glo89, Glo90a], Glover and Greenberg [GG89], de Werra and Hertz[dWH89]. But even older references both of conceptual (Glover[Glo77]) and of application oriented nature (Lin[Lin65]) may be useful to explain the inspirations for tabu search. Most influential is the basic two-part work of Glover[Glo89, Glo90a] and, in particular, the first part. (It seems that up to now, mainly aspects of the first part have been investigated in the literature with only a few exceptions referring to the second part.) Early applications that have greatly influenced the subsequent literature are Skorin-Kapov[SK90] with a tabu navigation approach for the quadratic assignment problem and Widmer and Hertz[WH89] providing an example with respect to flow shop scheduling. For a survey on successful applications of the tabu search metastrategy and a comparison to other search techniques the reader is referred to Glover and Laguna[GL93] as well as Voß[Voß96] and the references given therein. References that clarify concepts by means of extensive but easy examples in the area of network design and location are, especially, Glover[Glo90b] and Domschke et al.[DFV92]. Moreover, tabu search has received considerable interest in the operations research community, e.g. through special issues on applied local search or tabu search of various journals (Annals of Operations Research, Computers & Operations Research, OR Spektrum among others, see Glover et al.[GLTdW93], Laporte and Osman[LO96], Erenguc and Pirkul[EP94] and Pesch and Voß[PV95] or even through promoting a specialized metaheuristics conference (see Osman and Kelly[OK96]).

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3.2.2 Guiding Process versus Application Process The basic principle of any iteration guided by tabu search for solving a problem like

may be stated as follows: Transform the actual solution to a neighbour solution and try to guide the search into a meaningful direction by 'intelligently' utilizing the knowledge or memory about the previous iterations. That is, the search history is used to incorporate adaptive memory and responsive exploration into problem solving. In the easiest way the guidance is accomplished by the idea to avoid visiting solutions yet explored. Following a steepest descent / mildest ascent approach an admissible transformation may, in a simplified way, result in either a best possible improvement or a least possible deterioration of the objective function value. The transition to a neighbour solution is called a move and may be described by one or more attributes or elements. The attributes of the moves (when properly chosen) can become the foundation for creating a so-called attribute based memory. Depending on the application the abstract term move has to be specified. For example, in a zero-one integer programming context the attributes may be the set of all possible value assignments (or changes in such assignments) for the binary variables. With respect to location problems this may refer to opening or closing a facility, respectively. Two attributes e and , which denote that a certain binary variable is set to 1 or 0, are called complementary to each other. That is, the complement of a move cancels the effect of the considered move. If a move and its complement are performed, the same solution is reached as without having performed both moves. This is independent from the number of moves that have been explored between the two complementary moves. Moves or attributes eventually leading to a previously visited solution may be stored in a so-called tabu list and are hence forbidden or tabu for a certain number of transformations or iterations (called tabu tenure). The tabu list is derived from the running list (RL), which is an ordered list of all moves (or their attributes, respectively) performed throughout the search. That is, the RL represents the trajectory of solutions encountered. Whenever the length of RL is limited the attribute based memory of tabu search based on exploring the RL is structured to provide a recency based or short-term memory function. Note that other kinds of memory are important features, too. For instance, recording complete solutions may be referred to as explicit memory. If we not only keep track of attributes but also of the number of transformations they are involved with, we enter considerations of longer-term aspects by means of a frequency based memory. Corresponding long-term memory functions become especially relevant in intensification and diversification strategies. Now, each iteration consists of two parts: The guiding or tabu process and the application process. The tabu process updates the tabu list hereby requiring the actual RL; the application process chooses among the moves that are not tabu and updates the RL. (For faster computation or storage reduction both processes are often combined.) The application process is in principle the same for all tabu search methods except for some additional information provided to the tabu process, whereas the latter process varies. The application process is a specification on, for example, the neighbourhood definition and has to be provided by the user. Furthermore,

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making judicious choices of moves is often influenced by candidate list strategies. Those strategies may be used to restrict the number of moves to be evaluated, e.g., to reduce the number of computations or to identify subsets of promising influential moves seeming to be more attractive than others. Subsequently we will refer to three tabu search methods: static tabu search (STS), the reverse elimination method (REM) and the cancellation sequence method (CSM). For extensive background on these methods see Glover[Glo89, Glo90a]. Specifications used throughout the sequel are partly given in Dammeyer et al. [DFV91] and Voß[Voß96] whereas a comparison of all three methods with a specific simulated annealing implementation for a quadratic optimization problem can be found in Domschke et al.[DFV92]. With the considerations and concepts given in more detail below we can summarize the basic ideas of tabu list management, the very heart of tabu search, by the visualization given in Figure 3.1. This distinguishes the main ideas used throughout the following discussion. A different scheme might discard 'static' and 'dynamic' and replace them with 'ad hoc' and 'exact'. (Then the dashed line to random variation of s could be dropped.) Distinguishing between shortterm and long-term memory may be based on the question of whether the RL observes the whole search history or whether, for example, a frequency based measure is incorporated (cf. Glover and Laguna[GL93]).1

Figure 3.1 Tabu list management scheme (simplified). 1 Basically the different views of the ingredients and most influencing aspects of tabu search may result in the following inequality (which of course may be adapted to other search methods as well): TABU ≠ TABU.

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3.2.3 Tabu Navigation Method Static Tabu Search The STS, i.e. the tabu navigation method, is a rather simple approach requiring one parameter s called tabu list size. STS disallows the complements of the s most recent moves of the running list. That is, the tabu list (or tabu buffer) consists of a (complementary) copy of the last part of the RL. Older moves are disregarded. The tabu status derived from the s most recent moves forces the algorithm to go s moves away from any explored solution before the first step backwards is allowed. Obviously, this approach may disallow more moves than necessary to avoid returning to a yet visited solution. This encourages the intention to keep s as small as possible without disregarding the principal aim of guiding into meaningful directions without endless repetition: If s is too small the algorithm probably will return to a local optimum just left. This process may result in a so-called cycling process: If a solution is re-visited the same sequence of moves may be repeated consecutively until the algorithm eventually stops. Here, we should realize that tabu search (in its pure form) is a deterministic method. However, probabilistic versions of tabu search also exist and have been shown to be a versatile approach within the tabu search concept (see, e.g., Glover [Glo89], Faigle and Kern[FK92], Lokketangen and Glover[LG95]. The danger of cycling favours large values for s. An adequate value for s has to be adopted to problem structure, the characteristics of the considered problem instances (especially problem size) and the values of cost or other coefficients. The parameter s is usually fixed but could also be randomly or systematically varied after a certain number of iterations. Finally, we should note that instead of using a tabu list size one could think of a so-called prohibition period (which indeed is more independent from specific implementations, see Battiti and Tecchiolli [BT95]). The fact that STS disallows moves which are not necessarily tabu led to the development of a so-called aspiration level criterion (ALC) which may override the tabu status of a move. The basic form of the ALC is to choose a move in spite of its tabu status (or the tabu status of the encompassed attributes) if it leads to an objective function value better than the best obtained in all preceding iterations. The reason is obvious: such a move cannot lead to a known solution because otherwise the new value would already have been known. 3.2.4 Reverse Elimination Method and Cancellation Sequence Method The REM and the CSM are based on cancellation sequences which utilize some sort of memory structure imposed on the sequence of moves or attributes yet performed throughout the search. In general, any connected part of the running list is a cancellation sequence (CS): just the sequence of moves between two solutions. It may contain pairs of complementary moves. A residual cancellation sequence (RCS) results from a CS by cancelling (i.e. eliminating) some of its moves. REM and CSM differ in their way to carry out these cancellations but they match in their fundamental principle: Any move that would cancel an RCS is set tabu. REM and CSM compute and maintain RCSs in different ways: REM in each iteration recomputes all RCSs between the current and any previous solution. CSM in one iteration computes and stores the RCSs that are created by the actual move. Older RCSs are updated which may cause tabu settings.

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Page 46 The REM is based on the fact that each iteration starts at the current solution and then step by step traces in reverse direction through the RL. At each step an RCS between the current solution and the actually considered older solution is computed. Each iteration starts with the tabu list and the RCS being empty. For ease of exposition let us temporarily assume that a move consists of exactly one attribute. First, in each step this sequence is expanded by one move. Then, the algorithm checks whether the respective complementary move is part of the RCS, too. If this is true the complementary pair is eliminated and the RCS is reduced. If the complement is not included, the expanded RCS remains unchanged (Figure 3.2 gives the corresponding pseudo code to visualize the concept of REM for iteration iter). Each time the RCS reduces to one move the respective complementary move is set tabu with a tenure of 1. That is, it stays tabu only for that iteration immediately following. If the RL is considered in its full length each move that should be tabu is found. Also each move that is set tabu in fact would lead to a yet explored solution. Therefore, no better (or even optimal) solution can be 'forgotten' because of being tabu without need. Thus, an ALC is not necessary. Finally, we should note that the REM provides a necessary and sufficient condition to prevent from revisiting yet explored solutions, as long as there is no maximal length of the RL limiting the backtracing. for all elements k do status(k)=.free. (* comment: designate all elements 'not tabu' *) (* initialize RCS *) for i = iter, 1, if

step 1 do

(* backtracing of the running list RL *)

then

(* RL(i) cancels its complement on RCS *) else RCS:=RCS ∪ {RL(i)}

(* RL(i) is added to RCS *)

endif if | RCS| =1 then status(k):=.tabu. for k ∈ RCS (* designate tabu status *)

Figure 3.2 The concept of the REM Up to now, with respect to REM we have left the case of multi-attribute moves purposefully unspecified. Consider paired-attribute moves where we have a clear distinction, e.g., between add- and drop-attributes (i.e. defining a binary variable as 1 - add - or 0 - drop). In this case there is a straightforward adaptation of the REM and the consideration of necessity and sufficiency conditions in avoiding revisiting previously encountered solutions is the same as in the simpler case (cf. Glover[Glo90a], Voß[Voß96]). More generally, we may define arbitrary multi-attribute moves by means of classes of attributes (the above add- and drop-attributes provide examples of such classes). Whenever the sequence of the attributes is not critical the same observations are made as in the simpler case. Designating a tabu status (cf. the last line of Figure 3.2) is performed whenever RCS constitutes an admissible move. Only in cases where different sequences of the same set of attributes define different moves adaptations to

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a certain block structure have to be performed where necessity need not be guaranteed in all cases. The CSM embodies the concepts of the static method and the REM. Moves are set tabu in two ways: The simple tabu process is identical with the STS. The main tabu process, similar to the REM, disallows any move that would cancel an RCS. The following description of one iteration of CSM will show the handling of the RCSs. We run the CSM with three parameters c1, c2 and c3: c1 is the maximal length of the running list. c2 is the tabu tenure based on the main CSM tabu process. c3 is the tabu tenure corresponding to the tabu list length s of STS. The value c3=1 prevents the immediate return to the just visited solution. Higher values are also reasonable because for limitations of RL imposed by c1 the main CSM tabu process might not find every move that should be tabu. Usually, c3 < s because of this additional way of setting tabu. Assume that a move mj has just been performed (with j < c1), then one iteration of the CSM is as follows: 1. Store mj in the RL which becomes (m1, . . ., mi, . . ., mj) with the trajectory (S0, S1, . . ., Si, . . ., Sj) of solutions with the starting solution S0 and the current solution being Sj. 2. Set

tabu for the next c3 iterations.

, then: Create a new RCS between Si and Sj as (mi 3. If there exists an mi in the RL that is cancelled by mj, i.e. mj = +1, . . ., mj1) and cancel mi from RL. (Note that RL may be modified in the CSM, i.e., it may be viewed as a so-called active tabu list ATL.) This shortens existing RCSs containing mi. If an RCS now consists of only one move mt, set tabu for the next c2 iterations and cancel the respective RCS. Note that mt as well as mj remain in RL. With respect to multi-attribute moves eventually the attributes of the moves have to be treated within blocks to get reasonable adaptations (cf. Glover[Glo90a], Voß[Voß96]). Let us again compare REM and CSM: the definition of an RCS, the duration of the tabu status, and the treatment of RCSs at the end of each iteration are different. REM in each iteration computes the tabu moves for only the next iteration. According to the implementation imposed on Figure 3.2 only one RCS is known at a time it is changed within the tabu process. (Note that different implementations are possible where corresponding to each iteration an actual RCS is maintained and modified, thus utilizing far more storage space.) When the RL has been traced back the last RCS becomes useless and is disregarded. The next iteration again starts with an empty RCS no information despite the RL is kept from the previous iteration. Contrary to this the CSM keeps each RCS until it reduces to exactly one move. A newly created RCS corresponds to the current difference of two 'boundary' solutions. When cancellations occur later on this is no longer true. As long as an RCS is not completely cancelled it is prevented from returning to any solution attained before the solution from the original start of the RCS. Even more than c3 iterations after the last move of an RCS has been cancelled the search trajectory need not return to a previous solution (depending on the whole sequence of moves performed before and after the RCS). It is reasonable to forbid the complementary move of the last move of an RCS for the

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'next' sufficiently large number of iterations c2 and to disregard it later on. No matter how c2 is chosen: CSM in general may forbid more moves than necessary without guaranteeing prevention of returning to a known solution. 3.2.5 Logical Interdependencies Most successful applications of tabu search reported in the literature so far apply the static tabu list management concept. Nevertheless, with this method cycling may still occur. Note that strategic considerations may enforce the search to revisit previously explored solutions as may be the case when so-called elite solutions are reconsidered for a restart of the search. Various considerations in this respect investigate the interplay of intensification and diversification (see Battiti[Bat95] for the feedback mechanism using hashing in reactive search, Sondergeld and Voß[SV95] and Voß [Voß95] for the REM where a diversification measure is introduced to assure the solutions recorded differ from each other by a desired degree, and Rochat and Taillard[RT95] for probabilistic tabu search). To overcome the deficiency that cycling may still occur, different concepts may be used (assuming for the time being that cycling is not desirable). One idea for so-called cycle detection is to incorporate some hashing function which can be implemented even in cases where moves are not represented in a proper way, i.e., where they do not satisfy a sufficiency property (see, e.g., the reactive tabu search of Battiti and Tecchiolli[BT95] and Woodruff and Zemel[WZ93] for some references). As the STS as described above usually assumes fixed values of s, more flexibility is obtained by varying the tabu list size. If this variation is predetermined in a certain way such as in random modification of s, the attribute 'dynamic', however, may be doubtful. This becomes especially relevant when the treatment of REM and CSM is considered, where logical relationships are exploited as underlying the RL, i.e. the sequence of attributes performed throughout the search. In that sense logical interdependencies may be viewed as relationships between the transformations observed. In REM and CSM these interdependencies are even more fully exploited as the immediate interplay of the attributes enforces tabu restrictions. Both methods have been shown to be purely dynamic approaches in the sense that tabu settings (especially the number of attributes or moves designated tabu) are only developed in the course of the search. Although successful in most applications, one drawback of tabu search sometimes mentioned is the lack of theoretical foundations. Of course, there are a few theoretical results that have to be taken into account like the necessity and sufficiency of the REM. (That is, based on the above-mentioned interdependencies, no such drawback is appearing for the basic idea of REM.) Furthermore, when probabilistic ideas are incorporated even into the static tabu search concept sound theoretical results may be obtained, too (see, e.g., Faigle and Kern[FK92]). 3.2.6 Moving Gap In this survey, dynamic tabu list approaches are emphasized more and other approaches less because most attention in the literature is focused on the latter methods. Seeking to fill a gap by disclosing more fully the relevance of the dynamic methods is motivated by different aspects. Nevertheless, there are variations of the

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navigation approach that might deserve the name dynamic. We explain this by the moving gap idea, leading to a tabu list of changing size and composition (cf. Hübscher and Glover[HG94] as well as an application to the quadratic assignment problem provided by Skorin-Kapov[SK94]).

Figure 3.3 A moving gap tabu list. A moving gap tabu list consists of a static part, i.e. a partial list considering a fixed number of the most recent moves whose complements obtain a tabu status. Furthermore, a second part observes the next recent moves and defines a tabu status for all or for a subset of them. This subset changes in a time-dependent way in that a fixed part (gap) is omitted for a certain number of iterations. Figure 3.3 shows the procedure in more detail. There the static part consists of a list with length s = 4 whereas the moving gap part is partitioned into four equal sublists (of length 1 in our example), where the choice of a subset of these lists varies over time. Like any parameter, these can be modified with respect to specific applications, too. Finally, however, one has to admit that the moving gap tabu list again does not consider any logical interdependencies.

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3.3 Applications in Location Planning In this part we discuss several applications of the above tabu search methods as they have been applied mainly based on our experience in the dynamic tabu list management surrounding. The area under consideration is that of network design and location. Respective problems are formulated by different means and motivation concerning the design and analysis of networks (cf. Magnanti and Wong[MW84]). The outcome of a design process may be a set of locations in connection with a production and distribution problem, a telecommunications or traffic network, just to mention some of the areas where network design either plays or should play an important role. Some of the questions concerned with the design and analysis of networks involve the following issues: How many sites or locations and how many connections should be chosen and where should they be located etc.? In the sequel we investigate Steiner's problem in graphs and focus on some new results for a set of benchmark problems. In addition, we consider a class of warehouse location problems and provide a short review of further location problems. 3.3.1 Network Design and Steiner Trees In this section we first introduce Steiner's problem in undirected and directed graphs and present some move definitions from the literature. Then some new best results for some benchmark problems with respect to the rectilinear Steiner problem are reported. Given a graph with weights on its edges, the Steiner tree problem in graphs (SP) is to determine a minimum cost subgraph spanning a set of specified vertices. Correspondingly, the Steiner tree problem in directed graphs (SPD) is to find a minimum cost directed subgraph of a given graph that contains a directed path between a root node and every vertex of a set of specified vertices. Real-world problems arising in the layout of connection structures in networks as in, for example, VLSI-design may often be decomposed into a number of well-known combinatorial optimization problems. SP and SPD are included within this context. More formally with respect to the SPD, let G = (V, A) be a directed graph with a vertex set V and a set A of arcs with nonnegative weights. V consists of a root w and two disjoint sets Q of q basic vertices and S of possible Steiner vertices (i.e., V = {w} ∪ Q ∪ S; to simplify matters we may refer to w as a basic vertex with w ∈ Q and V = Q ∪ S). SPD is to find a minimum cost subgraph of G including a directed path from w to every basic vertex. In order to achieve this minimum subgraph additional vertices from the set S may be used. Since all arc weights are assumed to be nonnegative there is an optimal solution which is a directed tree (a Steiner tree) or arborescence with root w (provided that a feasible solution exists). Note that any SP instance may easily be transformed into an SPD instance by replacing each undirected edge with two arcs between the same nodes directed opposite each other both having the same weight as the original edge. Recently, it turned out that a specific set of 78 problem instances for SP available through e-mail (see Beasley[Bea90]) has been considered by several authors (data set B with 18 and sets C to E with 20 instances each). These instances comprise data with the number of vertices ranging from 50 to 2500 and the number of edges ranging from 63 up to 65 000. With respect to our idea of using binary variables for

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problem representation we have a good testbed for comparing the different approaches. With respect to a proper neighbourhood definition for SP two main ideas may be distinguished (see, e.g., Duin and Voß[DV94]): Edge-oriented Transformation 1. Given a feasible solution T = (VT, ET) randomly select an edge (i, j) ∈ ET and delete it from E. 2. While there exists a leaf of T being a Steiner vertex do delete that leaf and its incident edge. 3. Let Ti and Tj be the two disconnected components of T. Find nearest vertices i* and j* belonging to Ti and Tj, respectively, and add the vertices and edges of the shortest path between i* and j* to T (eventually disallowing to use (i, j), i.e. defining a tabu status for (i, j)). Node-oriented Transformation 1. Given a feasible solution T = (VT, ET). Let (x1, . . ., X|S|) be a binary vector with xi denoting that i is included (xi = 1) or excluded (xi = 0) from the solution. 2. Choose a vertex i ∈ S and define xi := 1 xi. Recompute a minimum spanning tree of the graph induced by Q ∪ {i ∈ S | xi = 1}. Within the local search literature usually an edge-oriented transformation is described (see e.g. Verhoeven et al. [VSA95]). For numerical results comparing different tabu search concepts see, e.g., Duin and Voß[DV94] as well as Voß[Voß96]. Results are reported on all data sets B to E. For instance, considering the REM in its pure form as described above for at most 200 iterations following the node-oriented transformation idea the following results are obtained for data sets B (number of nodes: from 50 to 100) and C (number of nodes: 500). Starting from initial feasible solutions obtained by a cheapest insertion heuristic an average deviation from optimality of 0.19 % and 1.75 % is gained for data sets B and C, respectively. An additional set of benchmark instances for Steiner's problem that has received considerable attention in the literature throughout the last two decades is provided by Soukup and Chow[SC73]. They put together 46 test instances by giving the coordinates of a number of basic vertices within the unit square. The instances arising are Steiner problems in the plane with the underlying metric being the Euclidean or the rectilinear metric. For a survey on heuristics for the rectilinear Steiner problem the reader is referred to, e.g., Servit[Ser81] and Hwang et al.[HRW92]. Following a result of Hanan[Han66] the rectilinear Steiner problem may be formulated as a special case of SP. For a given set Q of basic vertices there exists an optimal solution which uses only vertices and edges of the grid graph induced by the set of basic vertices by drawing a horizontal and a vertical line through each basic vertex and retaining only the finite segments of these lines. The vertices of the graph will be the set of all interconnections of any two lines. Applying this result we are able to solve even rectilinear problems with the above-mentioned tabu search heuristics by means of node-oriented transformations as well as edge-oriented transformations.

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Table 3.1 Numerical results for the Soukup and Chow benchmark problems problem

|Q|

previously best known solution

(new) solution

15

14

1.48b

1.48

18

62

4.06b

4.04

19

14

1.90b

1.88

31

14

2.60b

2.59

32

19

3.23b

3.13

33

18

2.69b

2.68

34

19

2.54b

2.42

35

18

1.54b

1.52

38

14

1.66b

1.66

39

14

1.66b

1.66

41

20

2.24b

2.24

43

16

2.66b

2.42

44

17

2.56k

2.56

45

19

2.24k

2.20

Table 3.1 shows results for a subset of 14 out of the 46 test instances with the rectilinear metric where optimal solutions are not known or not proved within the literature (to the best of our knowledge). All of these test problems have been converted into equivalent instances of SP and tried to be solve by one of the best-known exact algorithms for SP of Beasley[Bea89]. Nevertheless, following Beasley[Bea92] and Khoury and Pardalos[KP93] the optimal solutions to these problems are still not known. The first two columns of Table 3.1 indicate the problem number and the number of basic vertices (indicating that the problems should be easily solvable the interested reader might check this with respect to the problem instances number 18 and number 32 which are provided in Figures 3.4 and 3.5 together with the best solutions we have found by applying tabu search; basic vertices are given as black circles). Then the previously best known solutions found in the literature are given as they are provided by Beasley[Bea92] and by Khoury and Pardalos [KP93] (indicated by the lettersb and k, respectively). Furthermore, we provide our own results indicating

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Figure 3.4 Problem instance 18 with 62 basic vertices.

Figure 3.5 Problem instance 32 with 19 basic vertices. that our approach (again the REM with at most 500 iterations starting with a cheapest insertion solution after having performed a simple reduction technique) finds the best-known solutions in all cases within a few seconds and is able to improve them for most of the problems (cf. the above neighbourhood definitions). The overall best-

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known values are shown in italics. Similar results are obtained for CSM. For STS with different fixed values of s the results are slightly worse. Further extensive computations have been performed with respect to the rectilinear Steiner arborescence problem, i.e. the directed counterpart of the rectilinear Steiner problem. Although a complexity result for this problem still needs to be discovered for the case that all arcs are directed away from the origin with no cycles arising we investigated tabu search ideas for this problem superimposed on the solutions found by any of the algorithms described by Voß and Duin [VD93]. The case of additional obstacles has been treated, too. For these problems the algorithms even starting with STS tend to behave very efficiently. 3.3.2 Warehouse Location Problems As may be deduced from the above considerations it is obvious that the warehouse location problem (WLP) is a special case of Steiner's problem in directed graphs. In that sense the algorithms for the SPD may be applied to the WLP as well. The only difference worth noting is the fact that the node-oriented transformation outperforms the edge-oriented transformation due to its simplicity while exploiting the underlying problem structure. Subsequently, we provide some information on corresponding implementations in a more general problem setting, i.e. the capacitated warehouse location problem with concave production costs. The problem is to satisfy the demands bj of n customers j = 1, . . ., n. Decisions are performed on the production quantities as well as on the transportation quantities. For each of m possible supplier locations i there is some fixed cost Fi and a maximum quantity of ai units that may be produced (whenever a supplier is located at i). Furthermore, we have to observe concave cost functions fi for the production at location i as well as linear transportation costs cij (for the transport of one unit supplied from i to customer j). Considering binary variables xi indicating whether a supplier is located at i (xi = 1) or not (xi = 0) and continuous variables (yij) for the amount of goods supplied from i to customer j, a mathematical model for such a problem is as follows (cf. Francis et al.[FJW92]):

subject to

Due to (3.4) we have nonnegativity for the amount of production to be transported.

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Then for each supplier located at a location i the amount of goods produced at i may be calculated as Σjyij. With this the objective function (3.1) minimizes the sum of the fixed costs plus the transportation costs plus the production costs. The latter are described by a function fi(Σjyij) a monotonously increasing continuous concave function for the production costs at location i (with fi(0) := 0). Constraints (3.2) ensure that all demands are satisfied and capacity restrictions with respect to the quantities produced are imposed by restrictions (3.3). Whenever the binary variables are fixed the resulting problem becomes a transportation problem. Therefore, considering the binary variables as the only decision variables directly handled by tabu search all our programs available for the SP/SPD based on the node-oriented transformation (or even the multiconstraint zero-one knapsack problem, MCKP, cf. Dammeyer and Voß[DV93] and Thiel and Voß[TV94]) may be applied to this problem, too. Most important, following Schildt[Sch94], who used our implementations in her study with respect to the above problem (3.1) to (3.5), the same ranking between the different procedures may be concluded as, e.g., provided by the above works for the MCKP, even when comparing with genetic algorithms or simulated annealing as well as hybrids. On a basis of 20 representative hard problem instances with n ≤ 30 and m ≤ 30 for the street network of Germany, Austria and Switzerland she finds average relative deviation values for STS (with s = 5), CSM (with c2 = 2 and c3 = 3), and REM of 0.145 %, 0.169 % and 0.141 %, respectively. At most 50 iterations had been performed for all algorithms. For REM a tabu tenure of five had been designated instead of one to have some diversification for the price of eventually losing the necessity property. Due to the latter modification for REM an ALC as described above had been applied for all three tabu search approaches. For simulated annealing and genetic algorithms much worse results were obtained (cf. Schildt[Sch94]). Nevertheless, the dominance of REM and/or CSM over STS need not be mandatory. For instance, Captivo[Cap94] reports that for some benchmark instances of the ordinary capacitated WLP the picture may change (see Beasley [Bea90] on how to obtain the data). The reason for this may be sought in the strictness of the capacity constraints, i.e., in (3.1) to (3.5) the capacity constraints in principle do not restrict the number of neighbours which might happen in different implementations tailored for the ordinary capacitated WLP. 3.3.3 Further Location Problems It should be noted that we have applied the different versions of tabu search to various network design and location problems such as, for example, the capacitated minimum spanning tree problem (CMST), the quadratic (semi-) assignment problem (QAP and QSAP) and the p-median problem. Again the problem with capacity constraints becomes obvious when considering the CMST. Furthermore, certain ideas of diversification become necessary, e.g., when considering the QAP and the p-median problem. For these two problems we have proposed different diversification strategies based on the REM. For the p-median problem a learning approach based on the idea of modifying the condition when to designate a tabu status (cf. Figure 3.2) proved to be successful (Voß[Voß94]). For the QAP a strategy building on elite solutions as well as a parallel development of solution trajectories (cf. Sondergeld and Voß[SV95]) may be

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Table 3.2 Literature survey Problem

Authors

Approach / comments

CMST

Amberg et al.[ADV95]

STS / CSM / REM, comparison to simulated annealing

QAP

Skorin-Kapov[SK90] STS Battiti and Tecchiolli[BT94] reactive tabu search comparison to simulated annealing diversification strategies moving Kelly et al.[KLG94] gap Skorin-Kapov[SK94] REM / elite solution approach Voß[Voß95]

QSAP

Domschke et al.[DFV92]

STS / CSM / REM, comparison to simulated annealing

SP/SPD

Duin and Voß [DV94] Voß [Voß96]

REM STS / CSM / REM

WLP

Captivo[Cap94] Schildt[Sch94]

STS / CSM / REM STS / CSM / REM, comparison to genetic algorithms and simulated annealing

p-median Voß [Voß96] Rolland et al.[RSC95] Voß [Voß94]

STS / CSM / REM STS, incl. random variation of s REM

considered. The minimum weighted p-cardinality tree problem asks for a minimum cost connected acyclic subgraph with p nodes of a given graph. An application appears in location and oil lease management in the Norwegian part of the North Sea (cf. Jörnsten and Lokketangen[JL94]). The authors describe a moving gap approach. Kincaid[Kin95] compares STS with a simulated annealing approach for the damper placement problem for large flexible space truss structures which is to determine those p truss members of a structure to replace with dampers so that the modal damping ratio is as large as possible for all significant modes of vibration. Murray and Church[MC95] compare STS with a simulated annealing approach for an operational forest planning problem. For a survey on relevant references from the literature see Table 3.2. 3.4 Conclusions The basic foundation of incorporating some sort of intelligence into search methods is to emulate human behaviour. Human beings are creatures of memory, using a variety of memory functions to help guide their way through a maze of problem solving considerations. Therefore, it seems to be natural to try to endow solution methods with similar capabilities including a transparent and natural rationale. Its goal may be to emulate intelligent uses of memory, particularly for exploiting structure. This led to the development especially of tabu search, a metastrategy or general purpose approach

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superimposed on local search approaches. One of the goals of tabu search is to permit good moves in each iteration, while including some sort of intelligence into the search, e.g. by means of adaptive memory and responsive exploration. Preventing the search from revisiting previous solutions may be based on some sort of memory function. The basic idea of tabu search is to store previous solutions (the search trajectory) or, more efficiently, the corresponding neighbourhood changes which led to these solutions and then to forbid the reversal of these exchanges for a certain amount of time. In this paper special attention is given to a class of tabu strategies that take account of dynamic interdependencies among the memory elements. These interdependencies rely on logical structure as well as on heuristic elements. With these methods, i.e. REM and CSM, we were able to improve the best-known solutions to some benchmark problems that had previously been investigated by a large variety of researchers. From a theoretical point of view REM discloses the gap between a short-term memory and a long-term memory framework in that it allows for sufficiency and necessity to prevent revisiting previously encountered solutions. Due to this fact it becomes obvious that REM has an intensificational behaviour and different aspects may be taken into account to allow for diversification. Corresponding methods have been developed and investigated with respect to some examples. To judge the merit of this work our main goal may be re-read in the following sentence: We have emphasized the dynamic tabu search approaches more and other approaches less because most attention in the literature is on the other methods, and we were seeking to fill a gap by disclosing more fully the relevance of the dynamic methods. Future directions shall be based on ideas presented, among others, in Glover and Laguna[GL93] and some of the contributions in Glover et al.[GLTdW93] as well as other edited volumes as indicated above. Furthermore, we strongly emphasize hybrids with ideas of Glover[Glo90a] and Voß[Voß96]. References [ADV95] Amberg A., Domschke W., and Voß S. (1995) Capacitated minimum spanning trees: algorithms using intelligent search. TH Darmstadt and TU Braunschweig. [Bat95] Battiti R. (1995) Reactive search: Toward self-tuning heuristics. University of Trento. [Bea89] Beasley J. (1989) An SST-based algorithm for the Steiner problem in graphs. Networks 19: 116. [Bea90] Beasley J. (1990) OR-library: distributing test problems by electronic mail. Journal of the Operational Research Society 41: 10691072. [Bea92] Beasley J. (1992) A heuristic for euclidean and rectilinear Steiner problems. European Journal of Operational Research 58: 284292. [BT94] Battiti R. and Tecchiolli G. (1994) Simulated annealing and tabu search in the long run: a comparison on QAP tasks. Computers & Mathematics with Applications 28: 18. [BT95] Battiti R. and Tecchiolli G. (1995) Local search with memory: Benchmarking RTS. OR Spektrum 17: 6786. [Cap94] Captivo E. (1994) Private communication. [DFV91] Dammeyer F., Forst P., and Voß S. (1991) On the cancellation sequence method of tabu search. ORSA Journal on Computing 3: 262265.

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[DFV92] Domschke W., Forst P., and Voß S. (1992) Tabu search techniques for the quadratic semi-assignment problem. In Fandel G., Gulledge T., and Jones A. (eds) New Directions for Operations Research in Manufacturing, pages 389405. Springer, Berlin. [DV93] Dammeyer F. and Voß S. (1993) Dynamic tabu list management using the reverse elimination method. Annals of Operations Research 41: 3146. [DV94] Duin C. and Voß S. (1994) Steiner tree heuristics a survey. In Dyckhoff H., Derigs U., Salomon M. and Tijms H. (eds) Operations Research Proceedings 1993, pages 485496. Springer, Berlin. [dWH89] de Werra D. and Hertz A. (1989) Tabu search techniques: a tutorial and an application to neural networks. OR Spektrum 11: 131141. [EP94] Erenguc S. and Pirkul H. (eds) (1994) Heuristics, genetic and tabu search. Computers & Operations Research 21 (8). [FJW92] Francis R., McGinnis Jr. L., and White J. (1992) Facility Layout and Location: an Analytical Approach. Prentice Hall, Englewood Cliffs, NJ, 2nd edition. [FK92] Faigle U. and Kern W. (1992) Some convergence results for probabilistic tabu search. ORSA Journal on Computing 4: 3237. [GG89] Glover F. and Greenberg H. (1989) New approaches for heuristic search: a bilateral linkage with artificial intelligence. European Journal of Operational Research 39: 119130. [GL93] Glover F. and Laguna M. (1993) Tabu search. In Reeves C. (ed) Modern Heuristic Techniques for Combinatorial Problems, pages 70150. Blackwell, Oxford. [Glo77] Glover F. (1977) Heuristics for integer programming using surrogate constraints. Decision Sciences 8: 156166. [Glo86] Glover F. (1986) Future paths for integer programming and links to artificial intelligence. Computers & Operations Research 13: 533549. [Glo89] Glover F. (1989) Tabu search - part i. ORSA Journal on Computing 1: 190206. [Glo90a] Glover F. (1990) Tabu search - part ii. ORSA Journal on Computing 1: 432. [Glo90b] Glover F. (1990) Tabu search: a tutorial. Interfaces 20(4): 7494. [GLTdW93] Glover F., Laguna M., Taillard E., and de Werra D. (eds) (1993) Tabu search. Annals of Operations Research 41. [GM86] Glover F. and McMillan C. (1986) The general employee scheduling problem: an integration of MS and AI. Computers & Operations Research 13: 563573. [Han66] Hanan M. (1966) On Steiner's problem with rectilinear distance. SIAM Journal on Applied Mathematics 14: 255265. [HG94] Hübscher R. and Glover F. (1994) Applying tabu search with influential diversification to multiprocessor scheduling. Computers & Operations Research 21: 877884. [HRW92] Hwang F., Richards D., and Winter P. (1992) The Steiner Tree Problem. North-Holland, Amsterdam. [JL94] Jörnsten K. and Lokketangen A. (1994) Tabu search for weighted k-cardinality trees. Molde College. To appear.

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[Kin95] Kincaid R. (1995) Solving the damper placement problem via local search heuristics. OR Spektrum 17: 149158. [KLG94] Kelly J., Laguna M., and Glover F. (1994) A study of diversification strategies for the quadratic assignment problem. Computers & Operations Research 21: 885893. [KP93] Khoury B. and Pardalos P. (1993) A heuristic for the Steiner problem in graphs. University of Florida, Gainesville. [LG95] Lokketangen A. and Glover F. (1995) Probabilistic move selection in tabu search for zero-one mixed integer programming problems. Molde College. To

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appear. [Lin65] Lin S. (1965) Computer solutions of the traveling salesman problem. Bell Syst. Tech. J. 44: 22452269. [LO96] Laporte G. and Osman I. (eds) (1996) Metaheuristics in combinatorial optimization. Annals of Operations Research, to appear. [MC95] Murray A. and Church R. (1995) Heuristic solution approaches to operational forest planning problems. OR Spektrum 17: 193203. [MW84] Magnanti T. and Wong R. (1984) Network design and transportation planning: models and algorithms. Transportation Science 18: 155. [OK96] Osman I. and Kelly J. (eds) (1996) Proceedings of the Metaheuristics International Conference. Kluwer, Dordrecht, to appear. [PV95] Pesch E. and Voß S. (eds) (1995) Applied local search. OR Spektrum 17. [RSC95] Rolland E., Schilling D., and Current J. (1995) An efficient tabu search procedure for the p-median problem. University of California at Riverside. [RT95] Rochat Y. and Taillard E. (1995) Probabilistic diversification and intensification in local search for vehicle routing. Universite de Montreal. To appear. [SC73] Soukup J. and Chow W. (1973) Set of test problems for the minimum length connection networks. ACM / SIGMAP Newsletter 15: 4851. [Sch94] Schildt B. (1994) Betriebliche Standortoptimierung in der strategischen Produktions- und Distributionsplanung bei degressiv verlaufenden Produktions-kosten. PhD thesis, TH Darmstadt. [Ser81] Servit M. (1981) Heuristic algorithms for rectilinear Steiner trees. Digital Processes 7: 2132. [SK90] Skorin-Kapov J. (1990) Tabu search applied to the quadratic assignment problem. ORSA Journal on Computing 2: 3345. [SK94] Skorin-Kapov J. (1994) Extensions of a tabu search adaptation to the quadratic assignment problem. Computers & Operations Research 21: 855866. [SV95] Sondergeld L. and Voß S. (1995) A star-shaped diversification approach in tabu search. TU Braunschweig. To appear. [TV94] Thiel J. and Voß S. (1994) Some experiences on solving multiconstraint zero-one knapsack problems with genetic algorithms. INFOR 32: 226242. [VD93] Voß S. and Duin C. (1993) Heuristic methods for the rectilinear Steiner arborescence problem. Engineering Optimization 21: 121145. [Voß94] Voß S. (1994) A reverse elimination approach for the p-median problem. TH Darmstadt. [Voß95] Voß S. (1995) Solving quadratic assignment problems using the reverse elimination method. In Nash S. and Sofer A. (eds) The Impact of Emerging Technologies on Computer Science and Operations Research, pages 281296. Kluwer, Dordrecht. [Voß96] Voß S. (1996) Intelligent search. Springer, Berlin, to appear.

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[VSA95] Verhoeven M., Severens M., and Aarts E. (1995) Local search for Steiner trees in graphs. Eindhoven University of Technology. [WH89] Widmer M. and Hertz A. (1989) A new heuristic method for the flow shop sequencing problem. European Journal of Operational Research 41: 186193. [WZ93] Woodruff D. and Zemel E. (1993) Hashing vectors for tabu search. Annals of Operations Research 41: 123137.

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4 Reactive Search: Toward Self-tuning Heuristics Roberto Battiti Abstract Local (neighborhood) search can be guided to go beyond local minima through meta-heuristic methods that use the information obtained in the previous part of the run. The reactive search (RS) method proposes the integration of simple history-based feedback schemes in local search for the on-line determination of free parameters, therefore endowing the algorithm with the flexibility needed to cover a wide spectrum of problems but avoiding a human trial-and-error adjustment. In particular, in the reactive tabu search (RTS) algorithm a simple feedback scheme determines the value of the prohibition parameter in tabu search, so that a balance of exploration versus exploitation is obtained that is appropriate for the local characteristics of the task. This paper summarizes the reactive search framework, reviews RTS and its applications, and presents novel results about the behavior of different tabu search schemes when escaping from a local attractor. 4.1 Reactive Search: Feedback Applied to Heuristics The reactive search (RS) framework proposes the introduction of feedback schemes in heuristics for discrete optimization problems. RS belongs to the meta-heuristic class of algorithms, where a basic scheme (local neighborhood search) is complemented to avoid its known defects. A simple form of reinforcement learning [Lin92] is applied: the machine-learning agent is the discrete dynamical system that generates the search trajectory, whose internal parameters are changed through simple sub-symbolic mechanisms. Reactive search is intensively history-based, in fact learning can be defined as the influence on the behavior of a system of the past experienced events. Let us now summarize the context and the main motivations of the approach. The local search heuristic for minimizing a cost function f starts from an admissible solution and tries to obtain a better one in an iterative way by 'looking in the neighborhood'. A search trajectory is generated in a greedy way if the algorithm Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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chooses at each step the best neighbor of the current tentative solution. Local search can be effective if the neighborhood structure matches the characteristics of the problem, but it stops when the current point is a local minimizer, i.e., when all neighbors have higher f values. Some possible actions to go beyond local minima while aiming at better suboptimal points are: i) the repetition of local search after restarting from different (possibly random) points, ii) the acceptance of upward-moving steps to exit local minima (e.g., in simulated annealing [KJV83], tabu search [Glo89a]), iii) the consideration of more complex schemes based on the combination and selection of a set of suboptimal solutions (genetic algorithms [Hol75] are based on this principle). There are at least a couple of potential drawbacks of simple implementations of the above schemes: parameter tuning: some schemes require a careful selection of parameters by the user before competitive results are obtained for specific problems. This requires either a deep knowledge of the problem structure or simply a lengthy 'trial and error' tinkering process, that is not always fully reproducible (see for example the discussion in [BGK+ 95]). search confinement: after a local minimizer is encountered, all points in its attraction basin lose any interest for optimization. The search should avoid wasting excessive computing time in a single basin and diversification should be activated. On the other hand, in the assumptions that neighbors have correlated cost function values, some effort should be spent in searching for better points located close to the most recently found local minimum point (intensification). The two requirements are conflicting and finding a proper balance of diversification and intensification is a crucial issue in heuristics. Can one obtain wide spectrum algorithms covering many applications without parameter tuning and, at the same time, avoiding excessive wastes of computing resources? We claim that this can be done in some cases by using reactive schemes, where the past history of the search is used for: feedback-based parameter tuning: the algorithm maintains the internal flexibility needed to cover many problems, but the tuning is automated, and executed while the algorithm runs and 'monitors' its past behavior. automated balance of diversification and intensification: the 'exploration versus exploitation' dilemma is present in many heuristics: is it better to intensify the search in the promising regions, or to diversify it to uncharted territories? An automated heuristic balance can be obtained through feedback mechanisms, for example by starting with intensification, and by progressively increasing the amount of diversification only when there is evidence that diversification is needed. The focus of the RS method is on wide spectrum heuristic algorithms for discrete optimization, in which local search is complemented by feedback (reactive) schemes that use the past history of the search to increase its efficiency and efficacy. Related approaches in a different context are, for example, the self-adaptation of the generating

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distributions of Evolutionsstrategie algorithms in [Sch], or the use of adaptive annealing schedules proposed by [HRSV86] and by other researchers. The paper is organized as follows. Some underlying principles of Tabu Search (TS) are summarized in Section 4.2 and a classification of some TS-based algorithms is illustrated in Section 4.3. The application of reactive schemes in TS is summarized in Section 4.4. The results obtained by different versions of TS on a paradigmatic problem defined on binary strings are discussed in Section 4.5. The paper is concluded by a review of RS applications and future projects (Sections 4.6 to 4.7). 4.2 Tabu Search: Beyond Local Search The tabu search meta-heuristic [Glo89a] is based on the use of prohibition-based techniques and 'intelligent' schemes as a complement to basic heuristic algorithms like local search, with the purpose of guiding the basic heuristic beyond local optimality. It is difficult to assign a precise date of birth to these principles. For example, ideas similar to those proposed in TS can be found in the denial strategy of [SW68] (once common features are detected in many suboptimal solutions, they are forbidden) or in the opposite reduction strategy of [Lin65] (in an application to the travelling salesman problem, all edges that are common to a set of local optima are fixed). In very different contexts, prohibitionlike strategies can be found in cutting planes algorithms for solving integer problems through their linear programming relaxation (inequalities that cut off previously obtained fractional solutions are generated) and in branch and bound algorithms (subtrees are not considered if the leaves cannot correspond to better solutions), see the textbook [PS82]. The renaissance and full blossoming of 'intelligent prohibition-based heuristics' starting from the late eighties is greatly due to the role of F. Glover in the proposal and diffusion of a rich variety of meta-heuristic tools [Glo89a,Glo89b], but see also [HJ90] for an independent seminal paper. A growing number of TS-based algorithms has been developed in the last years and applied with success to a wide selection of problems [Glo94, Voß93]. It is therefore difficult, if not impossible, to characterize a 'canonical form' of TS, and classifications tend to be short-lived. Nonetheless, at least two aspects characterize many versions of TS: the fact that TS is used to complement local (neighborhood) search, and the fact that the main modifications to local search are obtained through the prohibition of selected moves available at the current point. Local search is effective if the neighborhood is appropriate to the problem structure but it clearly stops as soon as the first local minimizer is encountered, when no improving moves are available. TS acts to continue the search beyond the first local minimizer without wasting the work already executed, as is the case if a new run of local search is started from a new random initial point, and to enforce appropriate amounts of diversification to avoid the search trajectory remaining confined near a given local minimizer. In our opinion, the main competitive advantage of TS with respect to alternative heuristics based on local search like simulated annealing (SA) [KJV83] lies in the intelligent use of the past history of the search to influence its future steps. On the contrary, SA generates a Markov chain: the successor of the current point is chosen stochastically, with a probability that depends only on the current point and not on the

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previous history. The non-Markovian property is a mixed blessing: it permits heuristic results that are much better in many cases, but makes the theoretical analysis of the algorithm difficult. In fact, the practical success of TS techniques should motivate new search streams in mathematics and computer science for its theoretical foundation. Incidentally, the often cited asymptotic convergence results of SA are unfortunately irrelevant for the application of SA to optimization. In fact, repeated local search [FZ93], and even random search [CC88] has better asymptotic results. According to [AKZ95] 'approximating the asymptotic behavior of SA arbitrarily closely requires a number of transitions that for most problems is typically larger than the size of the solution space . . . Thus, the SA algorithm is clearly unsuited for solving combinatorial optimization problems to optimality.' Of course, SA can be used in practice with fast cooling schedules, but then the asymptotic results are not directly applicable. The optimal finite-length annealing schedules obtained on specific simple problems do not always correspond to those intuitively expected from the limiting theorems [SK91]. Let us define the notation.

is the search space (as an example, the set of binary strings with a given length

), X(t) is the current solution along the trajectory at iteration ('time') t. N(X(t)) is the neighborhood of point X(t), obtained by applying a set of basic moves µ0, µ1, . . ., µM to the current configuration:

In the case of binary strings, the moves can be those changing (complementing) the individual bits, and therefore M is equal to the string length L. Some of the neighbors are prohibited, a subset NA(X(t)) ⊂ N(X(t)) contains the allowed ones. The general way of generating the search trajectory that we consider is given by:

The set-valued function Allow selects a subset of N(X(t+1)) in a manner that depends on the entire search trajectory X (0), . . ., X(t+1). 4.3 Some Forms of Tabu Search There are several vantage points from which to view heuristic algorithms. By analogy with the concept of abstract data type in computer science [AHU83], and with the related object-oriented software engineering techniques [Cox90], it is useful to separate the abstract concepts and operations of TS from the detailed implementation, i.e., realization with specific data structures. In other words, policies (that determine which trajectory is generated in the search space, what the balance of intensification and diversification is, etc.) should be separated from mechanisms that determine how a specific policy is realized. An essential abstract concept in TS is given by the discrete dynamical system of equations (4.1) and (4.2) obtained by modifying local search. A taxonomy of tabu search is now presented where the abstract point of view of the dynamical systems is separated from the different implementation possibilities. Only a limited subset of the existing TS algorithms is cited in the presented classification.

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4.3.1 Dynamical Systems A classification of some TS-related algorithms that is based on the underlying dynamical system is illustrated in Figure 4.1.

Figure 4.1 A classification based on discrete dynamical systems. A first subdivision is given by the deterministic versus stochastic nature of the system. Let us first consider the deterministic versions. Possibly the simplest form of TS is what is called strict-TS: a neighbor is prohibited if and only if it has already been visited during the previous part of the search [Glo89a] (the term 'strict' is chosen to underline the rigid enforcement of its simple prohibition rule). Therefore equation (4.2) becomes:

Let us note that strict-TS is parameter-free. Two additional algorithms can be obtained by introducing a prohibition parameter1 T that determines how long a move will remain prohibited after the execution of its inverse. The fixed-TS algorithm is obtained by fixing T throughout the search [Glo89a]. A neighbor is allowed if and only if it is obtained from the current point by applying a move such that its inverse has not been used during the last T iterations. In detail, if LastUsed(µ) is the last usage time of move µ (LastUsed(µ) = ∞ at the beginning):

If T changes with the iteration counter depending on the search status (in this case the notation will be T(t)), the general dynamical system that generates the 1 The term prohibition parameter is chosen instead of the more traditional list size because list size refers to a specific implementation: prohibitions can be obtained without using any list.

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search trajectory comprises an additional evolution equation for T(t), so that the three defining equations are now:

Let us note that µ = µ1 for the basic moves acting on binary strings. Now, possible rules to determine the prohibition parameter by reacting to the repetition of previously-visited configurations have been proposed in [BT94a] (reactive-TS, RTS for short). In addition, there are situations where the single reactive mechanism on T is not sufficient to avoid long cycles in the search trajectory and therefore a second reaction is needed [BT94a]. The main principles of RTS are briefly reviewed in Section 4.4. Stochastic algorithms related to the previously described deterministic versions can be obtained in many ways. For example, prohibition rules can be substituted with probabilistic generation-acceptance rules with large probability for allowed moves, small for prohibited ones, see for example the probabilistic-TS [Glo89a]. Stochasticity can increase the robustness of the different algorithms, in addition [Glo89a] 'randomization is a means for achieving diversity without reliance on memory,' although it could 'entail a loss in efficiency by allowing duplications and potentially unproductive wandering that a more systematic approach would seek to eliminate.' Incidentally, asymptotic results for TS can be obtained in probabilistic TS [FK92]. In a different proposal (robust-TS) the prohibition parameter is randomly changed between an upper and a lower bound during the search [Tai91]. Stochasticity in fixed-TS and in reactive-TS can be added through a random breaking of ties, in the event that the same cost function decrease is obtained by more than one winner in the Best-Neighbor computation. At least this simple form of stochasticity should always be used to avoid external search biases, possibly caused by the ordering of the loop indices. If the neighborhood evaluation is expensive, the exhaustive evaluation can be substituted with a partial stochastic sampling: only a partial list of candidates is examined before choosing the best allowed neighbor. 4.3.2 Implementations While the above classification deals with dynamical systems, a different classification is based on the detailed data structures used in the algorithms and on the consequent realization of the needed operations. Different data structures can possess widely different computational complexities so that attention should be spent on this subject before choosing a version of TS that is efficient on a particular problem.

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Figure 4.2 The same search trajectory can be obtained with different data structures. Some examples of different implementations of the same TS dynamics are illustrated in Figure 4.2. Strict-TS can be implemented through the reverse elimination method (REM) [Glo89b,DV93], a term that refers to a technique for the storage and analysis of the ordered list of all moves performed throughout the search (called 'running list'). The same dynamics can be obtained in all cases through standard hashing methods and storage of the configurations [WZ93, BT94a], or, for the case of a search space consisting of binary strings, through the radix tree (or 'digital tree') technique [BT94a]. Hashing is an old tool in computer science: different hashing functions possibly with incremental calculation are available for different domains [CLR90]. REM is not applicable to all problems (the 'sufficiency property' must be satisfied [Glo89b]), in addition its computational complexity per iteration is proportional to the number of iterations executed, while the average complexity obtained through incremental hashing is O(1), a small and constant number of operations. The worst-case complexity per iteration obtained with the radix tree technique is proportional to the number of bits in the binary strings, and constant with respect to the iteration. If the memory usage is considered, both REM and approximated hashing use O(1) memory per iteration, while the actual number of bytes stored can be less for REM, because only changes (moves) and not configurations are stored. Trivially, fixed-TS (alternative terms [Glo89a,Glo89b] are: simple TS, static tabu search STS or tabu navigation method) can be realized with a first-in first-out list where the prohibited moves are located (the 'tabu list'), or by storing in an array the last usage time of each move and by using equation (4.4). Reactive-TS can be implemented through a simple list of visited configurations, or with more efficient hashing or radix tree techniques. At a finer level of detail, hashing

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can be realized in different ways. If the entire configuration is stored (see also Figure 4.4) an exact answer is obtained from the memory lookup operation (a repetition is reported if and only if the configuration has been visited before). On the other hand, if a 'compressed' item is stored, like a hashed value of a limited length derived from the configuration, the answer will have a limited probability of error (a repetition can be reported even if the configuration is new, because the compressed items are equal by chance an event called 'collision'). Experimentally, small collision probabilities do not have statistically significant effects on the use of reactive-TS as a heuristic tool, and hashing versions that need only a few bytes per iteration can be used. The effect of collision probabilities when hashing is used in other schemes is a subject worth investigating. 4.4 Reactive Tabu Search (RTS) The RTS algorithm [BT94a] represents the first realization of the principles introduced in Section 4.1 in the framework of F. Glover's tabu search (TS). Let us first note that the prohibition parameter T used in equation (4.4) is related to the amount of diversification: the larger T, the longer the distance that the search trajectory must go before it is allowed to come back to a previously visited point. But T cannot be too large, otherwise no move will be allowed after an initial phase. In detail, if the search space is given by binary strings of length L, an upper bound T ≤ (L 2) guarantees that at least two moves are allowed at each iteration, so that the search does not get stuck and the move choice is influenced by the cost function value (it is not if only one move is allowed!). Now, if only allowed moves are executed, and T satisfies T ≤ (L 2), one obtains: The Hamming distance H between a starting point and successive points along the trajectory is strictly increasing for T + 1 steps.

The minimum repetition interval R along the trajectory is 2(T + 1).

Some problems arising in TS that are worth investigating are: 1. the determination of an appropriate prohibition T for the different tasks, 2. the robustness of the technique for a wide range of different problems, 3. the adoption of minimal computational complexity algorithms for using the search history. The three issues are briefly discussed in the following subsections, together with the RTS methods proposed to deal with them. 4.4.1 Self-Adjusted Prohibition Period In RTS the prohibition T is determined through feedback (reactive) mechanisms during the search. T is equal to one at the beginning (the inverse of a given move

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is prohibited only at the next step), it increases only when there is evidence that diversification is needed, it decreases when this evidence disappears. In detail: the evidence that diversification is needed is signaled by the repetition of previously visited configurations. All configurations found during the search are stored in memory. After a move is executed the algorithm checks whether the current configuration has already been found and it reacts accordingly (T increases if a configuration is repeated, T decreases if no repetitions occurred during a sufficiently long period).

Figure 4.3 Dynamics of the the prohibition period T on a QAP task. Let us note that T is not fixed during the search, but is determined in a dynamic way depending on the local structure of the search space. This is particularly relevant for 'inhomogeneous' tasks, where the statistical properties of the search space vary widely in the different regions (in these cases a fixed T would be inappropriate). An example of the behavior of T during the search is illustrated in Figure 4.3, for a quadratic assignment problem task [BT94a]. T increases in an exponential way when repetitions are encountered, it decreases in a gradual manner when repetitions disappear. 4.4.2 The Escape Mechanism The basic tabu mechanism based on prohibitions is not sufficient to avoid long cycles (e.g., for binary strings of length L, T must be less than the length of the string, otherwise all moves are eventually prohibited, and therefore cycles longer than 2 × L are still possible). In addition, even if 'limit cycles' (endless cyclic repetitions of a given set of configurations) are avoided, the first reactive mechanism is not sufficient to guarantee that the search trajectory is not confined in a limited region of the search

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space. A 'chaotic trapping' of the trajectory in a limited portion of the search space is still possible (the analogy is with chaotic attractors of dynamical systems, where the trajectory is confined in a limited portion of the space, although a limit cycle is not present). For both reasons, to increase the robustness of the algorithm a second more radical diversification step (escape) is needed. The escape phase is triggered when too many configurations are repeated too often [BT94a]. A simple escape consists of a number of random steps executed starting from the current configuration (possibly with a bias toward steps that bring the trajectory away from the current search region). With a stochastic escape, one can easily obtain the asymptotic convergence of RTS (in a finite-cardinality search space, escape is activated infinitely often: if the probability for a point to be reached after escaping is different from zero for all points, eventually all points will be visited clearly including the globally optimal points). The detailed investigation of the asymptotic properties and finite-time effects of different escape routines to enforce long-term diversification is an open research area. 4.4.3 Fast Algorithms for Using the Search History The storage and access of the past events is executed through the well-known hashing or radix-tree techniques in a CPU time that is approximately constant with respect to the number of iterations. Therefore the overhead caused by the use of the history is negligible for tasks requiring a non-trivial number of operations to evaluate the cost function in the neighborhood.

Figure 4.4 Open hashing scheme: items (configuration, or compressed hashed value, etc.) are stored in 'buckets'. The index of the bucket array is calculated from the configuration. An example of a memory configuration for the hashing scheme is shown in Figure 4.4. From the current configuration phi one obtains an index into a 'bucket array.' The items (configuration or hashed value or derived quantity, last time of visit, total

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number of repetitions) are then stored in linked lists starting from the indexed array entry. Both storage and retrieval require an approximately constant amount of time if: i) the number of stored items is not much larger than the size of the bucket array, and ii) the hashing function scatters the items with a uniform probability over the different array indices. More precisely, given a hash table with m slots that stores n elements, a load factor α = n/m is defined. If collisions are resolved by chaining, searches take O(1 + α) time on average. 4.5 Different Ways of Escaping from a Local Attractor Local minima points are attractors of the search trajectory generated by deterministic local search. If the cost function is integer-valued and lower bounded it can be easily shown that a trajectory starting from an arbitrary point will terminate at a local minimizer. All points such that a deterministic local search trajectory starting from them terminates at a specific local minimizer make up its attraction basin. Now, as soon as a local minimizer is encountered, its entire attraction basin is not of interest for the optimization procedure, in fact its points do not have smaller cost values. It is nonetheless true that better points could be close to the current basin, whose boundaries are not known. One of the problems that must be solved in heuristic techniques based on local search is how to continue the search beyond the local minimizer and how to avoid the confinement of the search trajectory. Confinements can happen because the trajectory tends to be biased toward points with low cost function values, and therefore also toward the just abandoned local minimizer. The fact that the search trajectory remains close to the minimizer for some iterations is clearly a desired effect in the hypothesis that better points are preferentially located in the neighborhood of good sub-optimal points rather than among randomly extracted points. Simple confinements can be cycles (endless repetition of a sequence of configurations during the search) or more complex trajectories with no clear periodicity but nonetheless such that only a limited portion of the search space is visited (they are analogous to chaotic attractors in dynamical systems). An heuristic prescription is that the search point is kept close to a discovered local minimizer at the beginning, snooping about better attraction basins. If these are not discovered, the search should gradually progress to larger distances (therefore progressively enforcing longer-term diversification strategies). Some very different ways of realizing this general prescription are illustrated here for a 'laboratory' test problem defined as follows. The search space is the set of all binary strings of length L. Let us assume that the search has just reached a (strict) local minimizer and that the cost f in the neighborhood is strictly increasing as a function of the number of different bits with respect to the given local minimizer (i.e., as a function of the Hamming distance). Without loss of generality, let us assume that the local minimizer is the zero string ( [00 . . . 0] ) and that the cost is precisely the Hamming distance. Although artificial, the assumption is not unrealistic in many cases. An analogy in continuous space is the usual positive-definite quadratic approximation of the cost in the neighborhood of a strict local minimizer of a differentiable function. In the following parts the discussion is mostly limited to deterministic versions.

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4.5.1 Strict-TS In the deterministic version of strict-TS, if more than one basic move produces the same cost decrease at a given iteration, the move that acts on the right-most (least significant) bit of the string is selected.

Figure 4.5 Search trajectory for deterministic strict-TS: iteration t, Hamming distance H and binary string. The set of obtained configurations for L = 4 is illustrated in Figure 4.5. Let us now consider how the Hamming distance evolves in time, in the optimistic assumption that the search always finds an allowed move until all points of the search space are visited. If H(t) is the Hamming distance at iteration t, the following holds true:

This can be demonstrated after observing that a complete trajectory for an (L 1)-bit search space becomes a legal initial part of the trajectory for L-bit strings after appending zero as the most significant bit (see the trajectory for L = 3 in Figure 4.5). Now, a certain Hamming distance H can be reached only as soon as or after the H-th bit is set (e.g., H=4 can be reached only at or after t = 8 because the fourth bit is set at this iteration), equation (4.8) trivially follows. In practice the above optimistic assumption is not true: strict-TS can be stuck (trapped) at a configuration such that all neighbors have already been visited. In fact, the smallest L such that this event happens is L = 4 and the search is stuck at t = 14, so that the string [1101] is not visited. The problem worsens for higher-dimensional

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strings. For L = 10 the search is stuck after visiting 47 % of the entire search space, for L = 20 it is stuck after visiting only 11 % of the search space.

Figure 4.6 Evolution of the Hamming distance for deterministic strict-TS ( L = 32). If the trajectory must reach Hamming distance H with respect to the local minimum point before escaping (i.e., before encountering a better attraction basin) the necessary number of iterations is at least exponential in H. Figure 4.6 shows the actual evolution of the Hamming distance for the case of L = 32. The detailed dynamics is complex, as 'iron curtains' of visited points (that cannot be visited again) are created in the configuration space and the trajectory must obey the corresponding constraints. The slow growth of the Hamming distance is related to the 'basin filling' effect [BT94a] of strict-TS: all points at smaller distances tend to be visited before points at larger distances (unless the iron curtains prohibit the immediate visit of some points). On the other hand, let us note that the exploration is not as 'intensifying' as an intuitive picture of strict-TS could lead one to believe: some configurations at small Hamming distance are visited only in the last part of the search. As an example let us note that the point at H = 4 is visited at t = 8 while one point at H = 1 is visited at t = 11 (t is the iteration). This is caused by the fact that, as soon as a new bit is set for the first time, all bits to the right are progressively cleared (because new configurations with lower cost are obtained). In particular, the second configuration at H = 1 is encountered at t > 2, the third at t > 4, . . . the n-th at t > 2(n1). Therefore, at least a configuration at H = 1 will be encountered only after 2(L1) have been visited. Let us note that the relation of equation (4.8) is valid only in the assumption that strict-TS is deterministic and that it is not stuck for any configuration. Let us now assume that one manages to obtain a more 'intensifying' version of strict-TS, i.e.,

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that all configurations at Hamming distance less than or equal to H are visited before configurations at distance greater than H. The initial growth of the Hamming distance is in this case much slower. In fact, the number of configurations CH to be visited is:

It can be easily derived that CH > > 2H, if H < < L. As an example2, for L = 32 one obtains from equation (4.9) a value C5 = 242 825, and therefore this number of configurations have to be visited before finding a configuration at Hamming distance greater than 5, while 25 = 32. An explosion in the number of iterations spent near a local optimum occurs unless the nearest attraction basin is very close. The situation worsens in higher-dimensional search spaces: for L = 64, C5 = 8 303 633, C4 = 679 121. This effect can be seen as a manifestation of the 'curse of dimensionality': a technique that works in very low-dimensional search space can encounter dramatic performance reductions as the dimension increases. In particular, there is the danger that the entire search span will be spent while visiting points at small Hamming distances, unless additional diversifying tools are introduced. 4.5.2 Fixed-TS

Figure 4.7 Evolution of the Hamming distance for both deterministic and stochastic fixed-TS (L = 32, T = 10). 2 Computations have been executed by Mathematica

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The analysis of fixed-TS is simple: as soon as a bit is changed it will remain prohibited ('frozen') for additional T steps. Therefore (see Figure 4.7), the Hamming distance with respect to the starting configuration will cycle in a regular manner between zero and a maximal value H = T + 1 (only at this iteration the ice around the first frozen bit melts down and allows changes that are immediately executed because H decreases). All configurations in a cycle are different (apart from the initial configuration). The cycling T behavior is the same for both the deterministic and the stochastic version, the property of the stochastic version is that different configurations have the possibility of being visited in different cycles. In fact all configurations at a given Hamming distance H have the same probability of being visited if H ≤ T + 1, zero probability otherwise. The effectiveness of fixed-TS in escaping from the local attractor depends on the size of the T value with respect to the minimal distance such that a new attraction basin is encountered. In particular, if T is too small the trajectory will never escape, but if T is too big an 'over-constrained' trajectory will be generated. 4.5.3 Reactive-TS The behavior of reactive-TS depends on the specific reactive scheme used. Given the previously illustrated relation between the prohibition T and the diversification, a possible prescription is that of gradually increasing T if there is evidence that the system is confined near a local attractor, until a new attractor is encountered. In particular, the evidence for a confinement can be obtained from the repetition of a previously visited configuration, while the fact that a new attraction basin has been found can be postulated if repetitions disappear for a suitably long period. In this last case, T is gradually decreased. This general procedure was used in the design of the RTS algorithm in [BT94a], where specific rules are given for the entire feedback process. In the present discussion we consider only the initial phase of the escape from the attractor, when increases of T are dominant over decreases. In fact, to simplify the discussion, let us assume that T = 1 at the beginning and that the reaction acts to increase T when a local minimum point is repeated, in the following manner:

The initial value (and lower bound) of one implies that the system does not come back immediately to a just left configuration. The upper bound is used to guarantee that at least two moves are allowed at each iteration. Non-integral values of T are cast to integers before using them (the largest integer less than or equal to T). The evolution of T for the deterministic version is shown in Figure 4.8, repetitions of the local minimum point cause a rapid increase up to its maximal value. As soon as the value is reached the system enters a cycle. This limit cycle is caused by the fact that no additional attraction basins exist in the test case considered, while in real-world fitness surfaces the prohibition T tends to be small with respect to its upper bound, both because of the limited size of the attraction basins and because of the complementary reaction that decreases T if repetitions disappear.

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Figure 4.8 Evolution of the prohibition parameter T for deterministic reactive-TS with reaction at local minimizers (L = 32).

Figure 4.9 Evolution of the Hamming distance for simplified reactive-TS (L = 32).

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The behavior of the Hamming distance is illustrated in Figure 4.9. The maximal Hamming distance reached increases in a much faster way compared to the strict-TS case. Now, for a given T(t) maximum Hamming distance that is reached during a cycle is Hmax = T(t) + 1 and the cycle length is 2(T(t) + 1). After the cycle is completed the local minimizer is repeated and the reaction occurs. The result is that T(t) increases monotonically, and therefore the cycle length does also, as illustrated in Figure 4.10 which expands the initial part of the graph.

Figure 4.10 Evolution of the Hamming distance for reactive-TS, first 100 iterations (L = 32) Let us now consider a generic iteration t at which a reaction occurs (like t = 4, 10, 18, . . . in Figure 4.10). At the beginning equation (4.10) will increase T by one unit at each step. If the prohibition is T just before the reaction, the total number t of iterations executed is:

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where the relation T = Hmax 1 has been used. Therefore the increase of the maximum reachable Hamming distance is approximately O(√t) during the initial steps. The

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increase is clearly faster in later steps, when the reaction is multiplicative instead of additive (when T × 1.1 T + 1 in equation (4.10)), and therefore the above estimate of Hmax becomes a lower bound in the following phase. Let us note that the difference with respect to strict-TS is a crucial one: one obtains an (optimistic) logarithmic increase in the strict algorithm, and a (pessimistic) increase that behaves like the square root of the number of iterations in the reactive case. In this last case bold tours at increasing distances are executed until the prohibition T is sufficient to escape from the attractor. In addition, if the properties of the fitness surface change slowly in the different regions, and RTS reaches a given local minimizer with a T value obtained during its previous history, the chances are large that it will escape even faster. 4.6 Applications of Reactive Search: A Review The reactive search framework and, in particular, the RTS algorithm have been and are being applied to a growing number of discrete optimization problems (Section 4.6.1), continuous optimization tasks (Section 4.6.2), and different problems arising in the context of sub-symbolic machine learning (Section 4.6.3). Special purpose VLSI systems have been designed and realized for real-time pattern recognition applications with machine learning techniques (Section 4.6.4). 4.6.1 Combinatorial Tasks Reactive search has been applied to the following list of problems. quadratic assignment problem (QAP) [BT92, BT94a, BT94b] Nk model, a model of biological inspiration [BT92, BT95b] 01 knapsack [BT94a] multi-knapsack (with multiple constraints) [BT95b] max-clique [BP95] biquadratic assignment problem (Bi-QAP) [BC95] In many cases the results obtained with alternative competitive heuristics have been duplicated with low computational complexity, and without intensive parameter and algorithm tuning. In some cases (e.g., in the max-clique and bi-QAP problems) significantly better results have been obtained. A comparison of RTS with alternative heuristics (repeated local minima search, simulated annealing, genetic algorithms and mean field neural networks) is presented in [BT95b]. A comparison with simulated annealing on QAP tasks is contained in [BT94b]. A simple parallel implementation is presented in [BT92]. It has to be noted that RS is not a rigid algorithm but a general framework: specific algorithms have been introduced for unconstrained and constrained tasks, with different stochastic mechanisms and prohibition rules. As usually happens in heuristics, the more specific knowledge about a specific problem is used, the better the results. Nonetheless, it was often the case that simple RS versions realized with very limited effort could duplicate the performance of more complex schemes because of their simple embedded feedback (reinforcement learning) loop. A

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long-term goal of RS could be stated as the progressive shift of the learning capabilities from the algorithm user to the algorithm itself, through machine learning techniques. 4.6.2 Continuous Optimization A simple benchmark on a function with many suboptimal local minima is considered in [BT94a], where a straightforward discretization of the domain is used. A novel algorithm for the global optimization of functions (C-RTS) is presented in [BT95a], in which a combinatorial optimization method cooperates with a stochastic local minimizer. The combinatorial optimization component, based on RTS, locates the most promising boxes, where starting points for the local minimizer are generated. In order to cover a wide spectrum of possible applications with no user intervention, the method is designed with adaptive mechanisms: in addition to the reactive adaptation of the prohibition period, the box size is adapted to the local structure of the function to be optimized (boxes are larger in 'flat' regions, smaller in regions with a 'rough' structure). 4.6.3 Sub-Symbolic Machine Learning (Neural Networks) While derivative-based methods for training from examples have been used with success in many contexts (error backpropagation [RHW86] is an example in the field of neural networks), they are applicable only to differentiable performance functions and are not always appropriate in the presence of local minima. In addition, the calculation of derivatives is expensive and error-prone, especially if special-purpose VLSI hardware is used. We use a radically different approach: the task is transformed into a combinatorial optimization problem (the points of the search space are binary strings), and solved with the RTS algorithm [BT95c]. To speed up the neighborhood evaluation phase a stochastic sampling of the neighborhood is adopted and a 'smart' iterative scheme is used to compute the changes in the performance function caused by changing a single weight. RTS escapes rapidly from local minima, it is applicable to non-differentiable and even discontinuous functions and it is very robust with respect to the choice of the initial configuration. In addition, by fine-tuning the number of bits for each parameter one can decrease the size of the search space, increase the expected generalization and realize cost-effective VLSI, as it is briefly described in the next section. Representative papers are [Bat93] [BT95c] [BTT95] [BST+95]. 4.6.4 VLSI Systems with Learning Capabilities In contrast to the exhaustive design of systems for pattern recognition, control, and vector quantization, an appealing possibility consists of specifying a general architecture, whose parameters are then tuned through machine learning (ML). ML becomes a combinatorial task if the parameters assume a discrete set of values: the reactive tabu search (RTS) algorithm permits the training of these systems with low number of bits per weight, low computational accuracy, no local minima 'trapping', and limited sensitivity to the initial conditions [BLST94a, BLST94b].

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Figure 4.11 Layout of TOTEM chip: top level view of 32 processors, each containing the RAM cells (left), decoders (center), the MAC unit (bottom-right) and the output register (top-right). The control block is at the left. Our project aims at developing special-purpose VLSI modules to be used as components of fully autonomous massively parallel systems for real-time 'adaptive' applications. Because of the intense use of parallelism at the chip and system level and the limited precision used, the obtained performance is competitive with that of state-of-the-art supercomputers (at a much lower cost), while a high degree of flexibility is maintained through the use of combinatorial algorithms. In particular, neural nets can be realized. In contrast to many 'emulation' approaches, the developed VLSI completely reflects the combinatorial structure used in the learning algorithms. The first chip of the project (TOTEM, funded by INFN and designed at IRST [BLST94b, BLST95]) achieves a performance of more than one billion multiplyaccumulate operations. The chip layout is shown in Figure 4.11. Applications considered are in the area of pattern recognition (optical character recognition), events 'triggering' in high energy physics [ABL+95], control of non-linear systems [BT95c], compression of EEG signals [BST+95]. 4.7 Software A two-year project 'Algorithms and software for optimization, and models of complex systems' started in 1995. The aim of the project is to apply optimization and machine learning algorithms in different domains and therefore particular attention is devoted to modern software engineering methods [Car95]. A code fragment is illustrated in Figure 4.12, that shows the basic RTS loop in the C programming language. On-line

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information about the project as well as additional information about reactive search are available through an Internet WWW archive (http://rtm.science.unitn.it/). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

void

RTSLoop(long long

max_time, report_every)

{ while (Task.current.time < max_time) { if (check_for_repetitions(&Task.current) == DO_NOT_ESCAPE) { evaluate_neighborhood(); choose_best_move(); make_tabu(Task.chosen_bit); update_current_and_best(); } else { escape(); } if ((Task.current.time % report_every) == 0) PrintStateCompressed(); } }

Figure 4.12 RTS basic loop, in the C programming language. 4.8 Acknowledgements This research was funded in part by the Special Project 199596 of the University of Trento (Math. Dept.), the initiative RTS of INFN (Istituto Nazionale di Fisica Nucleare), and MURST 'Progretto 40%-Efficienzo di algoritmi e progretto di structure informative'. Dr. G. Di Caro provided assistance in the computational tests. References [ABL+95] Anzellotti G., Battiti R., Lazzizzera I., Lee P., Sartori A., Soncini G., Tecchiolli G., and Zorat A. (1995) Totem: a highly parallel chip for triggering applications with inductive learning based on the reactive tabu search. International Journal of Modern Physics C 6(4): 555560. [AHU83] Aho A. V., Hopcroft J. E., and Ullman J. D. (1983) Data Structures and Algorithms. Addison-Wesley, Reading, MA. [AKZ95] Aarts E., Korst J., and Zwietering P. (1995) Deterministic and randomized local search. In P. Smolenskij M. M. and Rumelhart D. (eds) Mathematical Perspectives on Neural Networks. Lawrence Erlbaum Publishers, Hillsdale, NJ. to appear. [Bat93] Battiti R. (1993) The reactive tabu search for machine learning. In Mauri G. (ed) Atti del Quarto Workshop del Gruppo AI*IA di Interesse Speciale su Apprendimento Automatico, pages 4155. Milano. [BC95] Battiti R. and Çela E. (1995) Reactive tabu search for the biquadratic

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assignment problem. University of Trento. Working Paper. [BGK+95] Barr R. S., Golden B. L., Kelly J. P., Resende M. G. C., and Stewart W. (1995) Designing and reporting on computational experiments with heuristic methods. In Proceedings of MIC-95. Breckenridge, CO. in press. [BLST94a] Battiti R., Lee P., Sartori A., and Tecchiolli G. (June 1994) Combinatorial optimization for neural nets: Rts algorithm and silicon. Technical Report UTM 435, University of Trento. [BLST94b] Battiti R., Lee P., Sartori A., and Tecchiolli G. (1994) Totem: A digital processor for neural networks and reactive tabu search. In Proceedings of the Fourth International Conference on Microelectronics for Neural Networks and Fuzzy Systems MICRONEURO'94, pages 1725. IEEE Computer Society Press, Torino, Italy. [BLST95] Battiti R., Lee P., Sartori A., and Tecchiolli G. (1995) Special-purpose parallel architectures for highperformance machine learning. In Proc. High Performance Computing and Networking, Milano Italy, page 944. Springer-Verlag, Berlin. [BP95] Battiti R. and Protasi M. (1995) Reactive local search for the maximum clique problem. Technical Report TR95052, ICSI, Berkeley, CA. [BST+95] Battiti R., Sartori A., Tecchiolli G., Tonella P., and Zorat A.(1995) Neural compression: an integrated approach to eeg signals. In Proc. of the International Workshop on Applications of Neural Networks to Telecommunications (IWANNT*95), pages 210217. Stockholm, Sweden. [BT92] Battiti R. and Tecchiolli G. (1992) Parallel biased search for combinatorial optimization: Genetic algorithms and tabu. Microprocessor and Microsystems 16: 351367. [BT94a] Battiti R. and Tecchiolli G. (1994) The reactive tabu search. ORSA Journal on Computing 6(2): 12640. [BT94b] Battiti R. and Tecchiolli G. (1994) Simulated annealing and tabu search in the long run: a comparison on qap tasks. Computer and Mathematics with Applications 28(6): 18. [BT95a] Battiti R. and Tecchiolli G. (1995) The continuous reactive tabu search: blending combinatorial optimization and stochastic search for global optimization. Annals of Operations Research in press. [BT95b] Battiti R. and Tecchiolli G. (1995) Local search with memory: Benchmarking rts. Operations Research Spektrum 17(2/3): 6786. [BT95c] Battiti R. and Tecchiolli G. (1995) Training nets with the reactive tabu search. IEEE Transactions on Neural Networks 6(5): 11851200. [BTT95] Battiti R., Tecchiolli G., and Tonella P. (1995) Vector quantization with the reactive tabu search. In Proceedings of MIC-95. Breckenridge, CO. in press. [Car95] Caro G. D. (1995) Regole di stile e di organizzazione per lo sviluppo di programmi in c. Technical Report UTM 462, University of Trento. [CC88] Chiang T.-S. and Chow Y. (1988) On the convergence rate of annealing processes. SIAM Journal on Control and Optimization 26(6): 14551470. [CLR90] Cormen T. H., Leiserson C. E., and Rivest R. L. (1990) Introduction to Algorithms. McGraw-Hill, New York. [Cox90] Cox B. J. (1990) Object Oriented Programming, an Evolutionary

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Approach. Addison-Wesley, Milano. [DV93] Dammeyer F. and Vofi S. (1993) Dynamic tabu list management using the reverse elimination method. Annals of Operations Research 41: 3146. [FK92] Faigle U. and Kern W. (1992) Some convergence results for probabilistic tabu search. ORSA Journal on Computing 4(1): 3237. [FZ93] Ferreira A. G. and Zerovnik J. (1993) Bounding the probability of success of stochastic methods for global optimization. Computer Math. Applic. 25: 18. [Glo89a] Glover F. (1989) Tabu search part i. ORSA Journal on Computing 1(3): 190260. [Glo89b] Glover F. (1989) Tabu search part ii. ORSA Journal on Computing

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2(1): 432. [Glo94] Glover F. (1994) Tabu search: Improved solution alternatives. In Birge J. R. and Murty K. G. (eds) Mathematical Programming, State of the Art 1994, pages 6492. The Univ. of Michigan. [HJ90] Hansen and Jaumard B. (1990) Algorithms for the maximum satisfiability problem. Computing 44: 279303. [Hol75] Holland J. (1975) Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI. [HRSV86] Huang M., Romeo F., and Sangiovanni-Vincentelli A. (1986) An efficient general cooling schedule for simulated annealing. In Proceedings of IICAD 86, pages 381384. [KJV83] Kirkpatrick S., Jr. C. D. G., and Vecchi M. P. (1983) Optimization by simulated annealing. Science 220: 671680. [Lin65] Lin S. (1965) Computer solutions of the travelling salesman problems. BSTJ 44(10); 224569. [Lin92] Lin L.-J. (1992) Self-improving reactive agents based on reinforcement learning, planning and teaching. Machine Learning Journal, Special Issue on Reinforcement Learning 8: 293322. [PS82] Papadimitriou C. H. and Steiglitz K. (1982) Combinatorial Optimization, Algorithms and Complexity. PrenticeHall, Englewood Cliffs, NJ. [RHW86] Rumelhart D., Hinton G., and Williams R. (1986) Learning internal representations by error propagation. In Parallel Distributed Processing Vol 1, pages 318362. MIT Press, Cambridge, MA. [Sch] Schwefel H.-P. Numerische optimierung von computer-modellen mittels der evolutionsstrategie. Interdisciplinary systems research 26. [SK91] Strenski P. N. and Kirkpatrick S. (1991) Analysis of finite length annealing schedules. Algorithmica 6: 346366. [SW68] Steiglitz K. and Weiner P. (1968) Some improved algorithms for computer solution of the traveling salesman problem. In Proceedings of the Sixth Allerton Conf. on Circuit and System Theory, pages 81421. Urbana, Illinois. [Tai91] Taillard E. (1991) Robust taboo search for the quadratic assignment problem. Parallel Computing 17: 443455. [Voß93] Voß S. (1993) Intelligent search. Manuscript, TH Darmstadt. [WZ93] Woodruff D. L. and Zemel E. (1993) Hashing vectors for tabu search. Annals of Operations Research 41: 123138.

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5 Combinatorial Optimization by Genetic Algorithms: The Value of the Genotype/Phenotype Distinction J. David Schaffer and Larry J. Eshelman Abstract To date, genetic algorithms (GAs) have been generally more successful at solving parameter optimization problems than combinatorial problems. The reason seems to be that a string of numeric parameters, represented as bit strings or numbers, do naturally induce schemata that can be readily discovered and exploited for many problems. Combinatorial problems, on the other hand, call for discrete patterns or arrangements of objects which often exhibit highly discontinuous performance changes with small changes in the pattern. These problems seems to lack the schemata needed for GA success. We advocate an approach in which combinatorial problems are cast into parameter optimization problems by developing domain-specific algorithms that produce solutions (phenotypes) from strings of parameters (genotypes). After considerable experience with this approach, we recommend it over alternative approaches like developing specialized representations of operators for specific combinatorial problems. We illustrate the advantages with a successful industrial application: balancing an assembly line of robots placing chips on circuit boards. 5.1 Background It is generally recognized that combinatorial problems are difficult to solve with genetic algorithms (GAs). Why? Because representations that induce good schemata are hard to find [RS95]. The usual approaches are to either devise a special representation, hoping it will induce good schemata (most do not), or devise special operators designed to preserve and recombine schemata that the user 'knows' are important Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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for the problem at hand (e.g. subtours for the travelling salesman problem) (or both). Unfortunately, success has been limited to date, largely because there is yet no clear theory of schemata (or formae [Rad90] ) for combinatorial problems to guide these efforts. There is an additional difficulty. Even if those working on finding ways to represent classical combinatorial problems for GAs were to succeed, the results would still be difficult to apply. The reason is that real world combinatorial problems are usually plagued by engineering details that are abstracted away in the well-studied versions of these problems (e.g. traveling salesman or bin packing). Fortunately, there is an alternative approach that depends only on the existing body of theory and practical experience with GAs on parameter optimization problems and allows one to deal with all the necessary engineering messiness. That is, cast the problem as a parameter optimization one. The key to this approach is to devise an algorithm that produces solutions to the combinatorial problem (e.g. an arrangement of objects) that vary considerably when some parameters are varied. This step may be considered as a development process that produces a phenotype from a genotype. It seems to be easier to devise such an algorithm that induces good schemata in the genotype space than it is to find a more direct representation of the phenotypes that do so. One element of success in this approach seems to be to avoid the temptation to include in this algorithm very strong (optimum-seeking) heuristics. After all, if we possessed optimum-seeking heuristics that were strong enough, then the development algorithm would be a sufficient optimizer by itself. We are not the only ones to employ this approach [KN90, Sys91], but we have come to believe it is an approach with wide practical applicability, given the large number of combinatorial problems of economic significance. We will illustrate the approach with a class of industrial problems: balancing an assembly line of robots that place surface mount devices (SMDs) on printed circuit boards (PCBs). 5.2 The Line Balancing Problem There is a well-studied problem called the 'line balancing' problem [FD92]. For SMD assembly, the problem can be stated as follows: Given: A set P {pi, i = 1, 2, . . . n} of parts to be placed each with a time ti needed to pick and place it on a PCB, and a set M {mj, j = 1,2, . . ., m} of machines for picking and placing the parts,

The X matrix contains the assignments of parts to machines and the first constraint specifies that each part must be assigned to one and only one machine. The maximum machine time is often called the cycletime of the production line.

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The classical 'line balancing' problem usually also includes precedence constraints which specify that some parts must be placed before others, but this is usually not the case for SMD assembly problems. It can be shown that the line balancing problem is NP-complete since it contains the bin packing problem [GJ79] as a special case.

Figure 5.1 A sketch of the FCM SMD assembly machine. This problem has been tackled with GAs [AF90, FD92], but unfortunately, this formulation of the problem is not sufficiently close to the real problems one finds in the circuit assembly factories. For instance, a Philips SMD assembly machine, the fast component mounter (FCM), has 16 independent robot heads as illustrated in Figure 5.1. It looks like a line balancing problem until you realize that the PCBs are advanced on a transport system called a walking beam which must advance all PCBs in lock step. This forces synchronization of all robots at each index step of the transport system. Furthermore, the working area of each robot is smaller than most PCBs which means that only a portion of the PCB real estate is accessible to each robot at any one time. Obviously, only parts in the reachable area can be placed. There is also a limited amount of space for the component feeders inducing a feeder packing problem on each robot. Add to this the need to allow the assembly line managers to impose constraints on the solutions, because of limited supplied of the needed components, and the problem takes on additional complexity. A further requirement is that optimizers solve what is called the family of boards problem. In this problem one is given the

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specifications for a group of printed circuit boards (PCBs) and the goal is to find a solution that minimizes the group's assembly time while requiring no alterations in the robots' setups (no changeovers). So the real optimization task is as follows: Given: a PCB (or family of PCBs) and a production line specification Subject to: user-specified constraints Find: a complete setup and action specification To Minimize: the assembly line cycle time. A complete setup means an assignment of part feeders to the robots' feeder slots, and part gripping and alignment tooling to each robot head. A feasible setup must provide a combination of tooling that covers the required parts (a set covering problem, also NP-hard). This problem arises because of the wide variety of tooling available, each covering a different subset of part types. But the covering constraint is only a feasibility constraint. The numbers of robots assigned to each tooling type in order to give a well-balanced solution is not known. The assignment of part feeders to feeder slots involves another constraint satisfaction problem because feeders vary in size as do the parts they feed and feeder slots are often at prespecified locations. Again, the minimum number of feeders required for a feasible solution is known (at least one feeder must be provided for each part type), but the optimum number for a well-balanced solution is not known. An action specification is the detailed specification for every robot action needed to pick and place all the parts assigned to it. This may be as simple as specifying the sequence of pick and place locations, but may also involve specifying a route (a TSP problem). 5.3 An Inadequate Approach One approach that has been applied to this problem is the well-established divide-and-conquer approach. Since the problem consists of a small number of identifiable subtasks, solve them one by one. To do this, an algorithm is developed to solve the tooling assignment task to optimality, then the feeder assignments are solved and finally the action specification is produced. The difficulty that must be overcome (besides the NP-hard nature of these tasks) to make this approach work is to define performance measures for the subtasks that can be computed in isolation. This is where this approach fails; it is usually not possible to evaluate, say a feeder assignment, without knowing which parts are picked and placed from each one. The true cost of any complete setup can easily be computed, given an accurate timing model of each robot which is easy to build, but the worth of any part of a setup cannot be accurately computed in isolation. Nevertheless, algorithms have been developed using

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this approach and proxy measures, but in our experience, they cannot compete with the approach we describe next. 5.4 A GA Approach

Figure 5.2 A schematic of the solution strategy. As described above, this problem can be cast as a parameter optimization problem by developing a domain-specific solution-generator to be run with chromosome-supplied parameters. For this domain we call this algorithm a heuristic layout file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_89.html（第 1／2 页）2009-7-7 16:56:46

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Page 90 generator (HLG). The concept is shown schematically in Figure 5.2. The HLG consists of a collection of decision-making routines, one for each of the main decisions needed and each using simple greedy heuristics with a proxy measure of desirability. The key point is that these greedy heuristics contain embedded parameters, the values for which are supplied by a GA. The concept is that by altering these parameters, the GA can induce the greedy heuristics to yield a wide variety of layouts even with the same proxy measures of desirability. As stated above, this proxy desirability cannot be depended upon to give optimum decisions, but this is not needed. At the end of each call to the HLG, a timing model is run giving an accurate cycle time that reflects the ensemble of all layout decisions. By allowing the GA to call the HLG with thousands of different parameter sets, it can find sets that give overall good performance. A schematic of what occurs inside the HLG is shown in Figure 5.3. To illustrate, we will give some details for one version of an HLG for the FCM machine. The rest of this section necessarily contains much detail which pertains only to the specifics of optimizing FCM layouts and therefore may be of little interest to most readers. It is presented here by way of illustration of the general concept. The FCM HLG contains the following steps: Step Step Step Step

1: 2: 3: 4:

assign assign assign timing

grippers feeder types parts model

Each of these steps makes irrevocable decisions about part of a production line layout for a given PCB. The goal of this step is to make an assignment of machine tooling (a combination of a suction nozzle for picking up the parts and an alignment mechanism (PPU) for accurately centering the part on the nozzle) to each of the robot heads available. Among the task descriptors computed when the PCB is defined is an accommodation matrix as illustrated in Table 5.1. Table 5.1 An example accommodation matrix PPU: PIP

pkt

p_types

LPIP

MPIP

PIP

MPIP

LPIP

noz: F3

F3

F3

F2

F2

F2

GT_max: 16

16

8

16

8

16

n_parts

1210R

3

3 1

1

1

0

0

0

2010R

4

6 0

0

0

1

1

0

SOT23

2

3 1

1

1

0

0

0

1206R

39

68 1

1

1

0

0

0

1206C

9

53 1

1

1

0

0

0

0805R

8

16 0

0

0

1

1

1

0805C

4

6 0

0

0

1

1

1

SMD parts come in different package types which define the external dimensions of the plastic envelope that contains the circuit elements. The part handling equipment

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Figure 5.3 A schematic of the Heuristic Layout Generator.

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is only concerned about the package type since this is what must be picked up by the nozzle and aligned by the PPU and then accurately placed. In the example, there are six different package types (1210R, 2010R, . . .) each with the specified number of different part types (specific circuit elements, say a resistor of a given magnitude) and the specified number of individual parts of that type. Table 5.1 lists the 6 gripper types (combination of PPU and nozzle) that might be used for this task. The body of the accommodation matrix shows which package type may be handled by each gripper type by the presence of a ''1". There is an algorithm that examines the accommodation matrix and identifies partitions which allow the breakdown of the entire task into subtasks. A partition, called a phantom gripper, is a set of grippers which handles a separable group of package types. Table 5.1 shows two phantoms, one comprising package types 2010R, 0805R and 0805C covered by grippers PIP-F2, MPIP-f2 and LPIP-F2, and the other comprising the remaining package types and grippers. Some problems cannot be partitioned and so are 1-phantom problems. Notice that within a phantom, some grippers may be completely interchangeable (they cover the same package types) while others are not. For the problem illustrated in Table 5.1, there are 16 robot heads available and most of the grippers are available in unlimited numbers. However, only 8 of the PPU type 'MPIP' are available so the GT_max for grippers using this type are similarly limited. The desirability calculus used in this step depends on two numbers computed from the accommodation matrix for each phantom. An estimate of the minimum number of robot heads (hmin) is needed to accommodate at least one feeder for each of the part types it covers. This figure is necessarily an estimate since to know the exact number would require the solution of the problem of packing the feeders on the feeder bars of each robot head. The method used is to compute a lower bound for min-heads and then to increment it if experience shows it is too low. The other number precomputed is hideal. This too is an estimate and represents the number of heads that would result in a perfect balance of the pick-andplace workload between the phantoms, if it could be achieved. Note that the min-heads is not necessarily less than the target-heads. It sometimes happens that a particular phantom has only a few parts to place (a small workload), but these parts are of many different types (each requiring a separate feeder) and they are large package types (requiring wide feeders that take up a lot of the available feeder bar space). Armed with the minimum (hmin) and ideal (hideal) number of heads, the assign-grippers algorithm is as follows: assign_grippers while (there are heads left to assign) compute desirability of each gripper for most desirable gripper, choose most desirable of the remaining heads assign best gripper to best head if a feasible assignment was found return zero else return a measure of how close feasibility was approached

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The desirability of each gripper is a straightforward (if tedious) calculation based on head-shortfall. A shortfall is the difference between hmin (or hideal) and the number currently assigned. But since many grippers can cover the same package types, shortfalls must be computed for phantoms and package types and the worst value used for each gripper. The algorithm begins using shortfalls from the hminS, and only when these are all zero (indicating a feasible assignment has been achieved) does it begin using shortfalls from hidealS. The key aspect of this algorithm is that the desirability actually used is the product of the computed shortfalls and a chromosome-supplied gripper weight (a binary coded value between zero and one). Thus, some chromosomes may have such a poor set of weights that they cannot induce this algorithm to find a feasible gripper assignment. Others may find feasible assignments, but do so so inefficiently that only poorly balanced solutions result. However, a main point of our recommendation for this approach is that a GA is needed for this problem because a good gripper assignment cannot be assessed based only on the numbers of heads assigned. All the remaining steps of the HLG must be executed before the cycle time of the production line can be accurately computed. Once heads have grippers assigned, feeders must be assigned to the feeder slots provided. There is an independent subproblem of this type for each phantom. The packing of feeders of different geometries into a given amount of space can be a difficult problem in general, but again we resort to a simple greedy algorithm, outlined below: assign_feeders for each phantom while there are feeders with non-zero desirability compute desirability of each feeder for most desirable feeder, choose most desirable slot assign best feeder to best remaining slot if a feasible assignment was found return zero else return a measure of how close feasibility was approached Again, desirability is computed based on the precomputed minimum and maximum number of slots for each feeder, and again, these numbers are multiplied by chromosome-supplied feeder weights which allows the behavior of this simple algorithm to be varied producing a potentially wide variety of different feeder assignments for any given problem. Perhaps the core of the HLG is this step. It works with the assignments made in the previous steps and assigns the individual parts to specific robot heads, either assigning a new feeder (to the part type needed) or reusing an existing feeder (if one is already available supplying the needed part types). There are chromosome-supplied parameters which dictate the order in which parts are sorted for placement. The desirability in this step is based on each head's slack time, i.e. the difference between its current pp_workload (the sum of the pp-time for the parts currently assigned to it) and the line's heartbeat time (i.e. the pp-time of the slowest head). There are also other chromosome-supplied parameters called bucket_choice_bits, one for each part. These dictate whether slack time alone should determine desirability or whether the choice should be restricted to those heads which would not require giving away a free

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feeder. Thus, the algorithm can be made to behave as a complex mixture of assign-to-minimize-slack and assign-toconserve-feeders under chromosome control. This is a straightforward calculation of the time it takes and FCM to execute the task as described in the layout resulting from the previous steps. The output from this routine is treated as the fitness of the chromosome. 5.5 How Well Does it Work? This domain is one where small improvements in the cycle time of a production line can be very valuable and also where fairly long computation times (say overnight) are tolerable. Fortunately, the algorithm described has the property of being quite strongly guided towards feasible solutions, so it usually reports a feasible solution quickly and, when given more time, reports steadily improving solutions. The user need only wait until a solution of acceptable quality appears. By way of estimating the value of the solutions provided by this algorithm we point out that a 40 hour week contains 144 000 seconds. If a manufacturer plans a production run of 144 000 PCBs (not an overly large number for major electronics or automotive producers), then a one second smaller cycle time layout can save the cost of a 40 hour shift. With these figures in mind we illustrate the performance of this algorithm in Figures 5.4 and 5.5 for two different PCBs. Each figure shows several replications of the GA/HLG each initiated with different random seeds. The performance measure is called the optimality gap and is the difference between the current best cycle time and the estimated lower bound cycle time (i.e. perfect balance). The GA used is Eshelman's CHC [Esh91], Note that there is some variation in the best solution achievable from PCB to PCB. For instance, there is quite a small gap for PCB 1, while PCB 2 has a large one. PCB 2 is a particularly challenging test case because there is very limited scope available for balancing. There are so many different part types that only one feeder slot can be allocated to each one and the challenge is mainly to find any feasible packing of the feeders. Once this is achieved, the line is limited to the slowest robot which in turn is the one with the most instances of its most populous part type. However, there are still many permutations among the one-slot-per-part-type solutions.

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Figure 5.4 Six replicated learning curves for PCB 1.

Figure 5.5 Six replicated learning curves for PCB 2.

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A version of this software has been supplied with Philips FCM machines for about two years and reports from the field have been positive. Efforts to improve the solutions provided by this algorithm by both human experts and other computer programs have consistently failed. Note that no tuning of the GA is needed to handle different PCBs, or even when going from single PCBs to families. We should point out that the approach advocated is not limited to a genetic algorithm. In principle, any powerful search method could be coupled to the HLG with good results. However, our own efforts in this direction have convinced us that simulated annealing is unacceptably slow and a variety of hillclimbers are either hopelessly slow or they become trapped by local optima quite far from the best solutions. 5.6 Conclusions The balancing of production lines of real robots is known to be a difficult problem. This is so for many reasons including the significant differences between the real-world situations and the idealized versions studied academically. Their search spaces are very large (10100 is not uncommon) and complex (including several NP-hard subproblems). They usually do not readily admit to a divide-and-conquer strategy. The success we have described of a GA solving problems of this class might be taken as an illustration of the general power of GAs for such problems, but there is a more important lesson we wish to stress. The approach that allowed a combinatorial problem to be solved by the GA was the development of the HLG; a procedure analogous to the development of a phenotype from the genotype. This procedure, comprising fairly simple greedy solution-generating algorithms with embedded behavior-modifying genes, is not a optimization algorithm by itself; it needs the GA to tune it for each problem it tries to solve. What it does do is allow the established power of GAs for solving fixed parameter optimization problems to be applied to combinatorial problems. We are optimistic that this approach can be replicated on other combinatorial problems which have heretofore resisted the GA. References [AF90] Anderson E. and Ferris M. (1990) A genetic algorithm for the assembly line balancing problem. In Proceedings of the Integer Programming/Combinatorial Optimization Conference. University of Waterloo Press, Waterloo, Ontario, Canada. [Esh91] Eshelman L. (1991) The chc adaptive search algorithm: How to have safe search when engaging in nontraditional genetic recombination. In Rawlins G. (ed) Foundations of Genetic Algorithms, pages 265283. Morgan Kaufmann, San Mateo, CA. [FD92] Falkenauer E. and Delchambre A. (1992) A genetic algorithm for bin packing and line balancing. In Proc. IEEE 1992 Int'l Conference on Robotics and Automation (RA92). Nice, France. [GJ79] Garey M. and Johnson D. (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York. [KN90] Kadaba N. and Nygard K. (1990) Improving the performance of genetic

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algorithms in automated discovery of parameters. Technical report, Dept. of SC and OR, North Dakota State University. [Rad90] Radcliffe N. (1990) Genetic Neural Networks on MIMD Computers. PhD thesis, Dept. of Theoretical Physics, University of Edinburgh, Edinburgh, Scotland. [RS95] Radcliffe N. and Surry P. (1995) Fitness variance of formae and performance prediction. In Whitley D. and Vose M. (eds) Foundations of Genetic Algorithms 3, pages 5172. San Mateo, CA. [Sys91] Syswerda G. (1991) Schedule optimization using genetic algorithms. In Davis L. (ed) Handbook of Genetic Algorithms, pages 332349. Van Nostrand Reinhold, New York.

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6 Integrating Local Search into Genetic Algorithms Colin R Reeves and Christian Höhn Abstract Genetic algorithms (GAs) have proved to be a versatile and effective approach for solving combinatorial optimization problems. Nevertheless, there are many situations in which the simple GA does not perform particularly well, and various methods of hybridization have been proposed. These often involve incorporating other methods such as simulated annealing or local optimization as a post-processor to the basic GA strategy of selection and reproduction. In this paper, rather than keeping the local optimization separate from recombination, we intend to explore the possibilities of integrating it more directly into the standard genetic algorithm. The graph partitioning problem and the no-wait flowshop sequencing problem will be used to illustrate the approach. 6.1 Introduction Genetic algorithms (GAs) as developed by Holland[Ho175] have frequently been applied to combinatorial optimization problems, such as bin-packing (Falkenauer and Delchambre[FD92], Reeves[Ree93], machine-sequencing (Reeves [Ree95b]), vehicle routing (Blanton and Wainwright[BW93]), and of course the travelling salesman problem (Jog et al. [JSG91], Whitley et al.[WSS91], Homaifar et al.[HGL93]). All of these problems have serious practical applications, but they typically require some fairly major adjustments to the 'classical' GA paradigm in order to find high-quality solutions. In what follows, we assume the reader has a basic understanding of the way in which GAs operateif not, we refer them to Goldberg[Go189] or Reeves[Ree93a] for an introduction. A discussion of GAs in the particular context of combinatorial optimization can be found in Reeves[Ree95b]. The successful application of GAs requires a good adaptation to the problem to be solved. A promising approach is when existing methods such as constructive heuristics or local search methods can be incorporated into the algorithm. This idea has been Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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used by Jog et al.[JSG91] (amongst others) in the context of the TSP. Other examples of this type of approach are in Mühlenbein[Müh91], Ulder et al.[UAB+91] and Prinetto et al.[PRR93]. An alternative strategy is to run the GA for a fixed number of cycles and then apply neighbourhood search to all the final population. However, in all these cases, the application of local optimization is kept distinct from the recombination operation. However, recent work (Reeves[Ree94]; Culberson[Cul93]) which has started to explore GAs from a more traditional neighbourhood search perspective has suggested that local search methods can be integrated in a more fundamental way. It is assumed that readers will have at least a basic understanding of both GAs and neighbourhood search (NS). Further details can be found in Reeves[Ree94], but in the next section we briefly discuss the relation between the two approaches. 6.2 GAs and NS Traditional genetic algorithms have three main characteristics by which they are commonly distinguished from classical and modern local search techniques. Population-based selection: In point-based NS methods selection is restricted to a choice between the parent and one or more offspring, but with GAs the presence of a population gives a spatial dimension to selection. Recombination: By recombining two parent solutions, offspring usually inherit those variables for which both parents share the same value. Thus recombination essentially results in a reduction of the size of the search space. Crossover: Traditionally crossover involves two (often binary) strings. Having defined a set of crossover points, the intermediate bits are swapped. While selection and recombination represent general search techniques, crossover essentially stands for a special class of operators and thus could be replaced by an abstract type of operator. In the context of local search an operator w defines a neighbourhood N(x,w) ⊆ S for all points x in the search space S. This general definition holds for a 1-bit Hamming hill climber as well as for many more sophisticated heuristics. However, crossover as introduced by genetic algorithms seems to be an exception in so far as two points, say x and y, are involved in the operation such that N = N(x, y,w), x, y ∈ S. Some implications of this are explored in Reeves[Ree94], but it is obvious that defining a neighbourhood by more than one point complicates the analysis. Furthermore, in this way the crossover operation is inevitably linked to recombination leading to a superposition of both effects. As a consequence the terms are often used synonymously. However, having two points effectively reduces the neighbourhood size by the total of all elements the two points have in common. This is an effect of recombination which occurs before any operators are applied. Following the ideas of Culberson[Cul93] the crossover effect can then be implemented by a unary operator which always pairs a point x ∈ S with its complementary point in the reduced neighbourhood.

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This operator, which we term complementary crossover, thus fits into the general definition above. As a consequence, the traditional crossover may be decomposed into neighbourhood reduction and complementary crossover. In traditional GAs the search space defined by both parents is usually explored by evaluating only one or two randomly selected offspring. However, this is not a necessary part of a GAsome have suggested, for example, evaluating several randomly selected offspring and picking the best. In fact, one could take this to its logical conclusion and carry out a local optimization within the reduced neighbourhood. The types of problem where this approach is likely to be most effective are those where the objective function of the neighbouring points can be evaluated quickly and cheaply: if every point has to be evaluated from scratch, this is not likely to be a computationally efficient scheme. 6.3 No-Wait Flowshop Problem We have tried this out on some combinatorial problems. Initially, the problem studied was the n/m/P, nowait/Cmax problem (i.e. there are n jobs, m machines and the objective is to minimize the makespan), which can be modelled as an asymmetric TSP. This has the advantage that the cost of a small change (here an exchange of two job positions) is easily calculated, so it meets the criterion described above. Nine sets of 10 problem instances were investigated, with either 20, 50 or 100 jobs, and 5, 10 or 20 machines. The neighbourhood search was carried out by a simple exchange operator as indicated above, and this was compared with the C1 and PMX crossovers'traditional' GA crossover operators for sequence-based problems. The total amount of computation was fixed at that used in Reeves[Ree95b]. The results were as shown in Table 6.1 (the results of G1 and PMX were so similar that only C1 is shown). The figures shown are the mean and standard deviation of the difference in makespan values; a positive difference means C1 gave better results, a negative one that the embedded NS operator was superior. Table 6.1 Mean and standard deviation of differences in makespan (C1-NS), averaged over 10 instances in each group Problem group

mean diff

stdev diff

20/5

8.50

19.40

20/10

5.60

32.07

20/20

79.90

85.00

50/5

9.0

44.0

50/10

63.7

59.3

50/20

94.7

132.8

100/5

45.9

69.1

100/10

107.5

145.5

100/20

284.6

154.5

n/m

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It can be seen that for the smallest problem instances (20/5, 20/10 and 50/5) there was little difference between the methods. However, for the larger cases, the average makespan obtained from the embedded NS was (statistically) significantly less than those obtained by C1 or PMX. What is not shown here is the relative amounts of computer time consumed; each technique was allotted what was nominally the same fixed amount of timea function of n and m (for further details see Reeves[Ree95b]. However, because the test for exceeding the stopping time was only carried out after each new string was generated, which is more frequent for the conventional GA with C1 or PMX than for the embedded NS approach, the results recorded for embedded NS were actually those on the iteration before the stopping criterion was reached. Thus, on average the time taken to achieve the results of Table 6.1 were 5 to 10% less for the embedded NS approach, so that by integrating NS into the GA we have not only found higher-quality solutions, but also achieved them in less computer time. 6.4 Graph Partitioning What the above results do not show is how much of the performance is due to the operators used, and how much to the effect of recombination. This is not easy to disentangle in the context of the flowshop problem, and for a more fundamental study, we looked at the rather simpler problem of bipartitioning a graph which, as it has a natural binary encoding, is more amenable to analysis. A fuller discussion of this work is contained in Höhn and Reeves[HR94]. The uniform bipartitioning of a graph is a special case (i.e. k = 2) of the k-way graph partitioning problem which is as follows: given a graph G = (V, E) with a cost function and a size S(u) attached to each node, partition the nodes of G into k subsets of equal size such that the total cost of the edges cut is minimized. The uniform kway partitioning problem is a constrained optimization problem, i.e. the search space usually contains a non-empty set of illegal solutions. This can pose severe problems for GAs (see Reeves[Ree95b], for a dicussion of these), but happily in this case, the problem can be quite simply transformed into a free optimization problem as in Lee et al.[LPK89] and Tao and Zhao[TZ93]. For the purposes of this investigation we define the following abstract GA: 1. Initialize a population of size N. 2. Apply the local hill climber to each individual until a local optimum is reached or another stopping criterion is met. 3. Choose 2N individuals from the population according to fitness and pair them (randomly) so that each pair defines a reduced neighbourhood. This is passed to the next generation together with the complete bitstring and objective value from one parent. 4. All the offspring define the population for the next generation. 5. If not TERMINATED go to 2, otherwise stop. The scheme is not restricted to a particular hill-climbing technique and thus provides a high degree of flexibility. Furthermore, for a given application several local hill climbing techniques may be evaluated individually before being extended by recombination

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and selection. In the rest of the paper such an approach is illustrated for the graph bipartitioning problem. First we describe the operators. 6.4.1 The LPK Heuristic One of the traditional operators for solving the k-way graph partitioning problem is the heuristic by Lee et al. [LPK89] which is based on the heuristic of Kernighan and Lin[KL70]. Given an initial partition, π = (A, B), consisting of the blocks A and B, a pass starts with moving the vertex having the highest gain to its complementary block. Once moved, the vertex becomes locked for the remainder of the pass and the gains of the remaining vertices are updated. The pass proceeds with the next best move and terminates after all vertices have been moved, i.e. when the initial partition is reached again. Then the best partition encountered during the pass is taken as initial partition for the next pass. The algorithm stops when no further improvement can be achieved, i.e at a local optimum. The LPK-heuristic uses a steepest ascent strategy where the neighbourhood searched forms a closed path. For problems involving l vertices the path is of length l starting and ending at the initial point and always follows the maximum gain. Initializing and maintaining the gains require O(l2) time per pass. 6.4.5 Complementary Crossover Using group-number encoding a solution for the bipartitioning problem of l vertices requires a binary string of length l. Usually when crossover is applied to partitioning problems two common pitfalls arise: The operator is not closed with respect to the search space. Thus illegal solutions may result. The coding is redundant, multiplying the number of local and global optima. Because of the graph transformation the solution space is extended to the whole search space, so that no illegal solutions occur. Furthermore, since in this study the investigations are restricted to the bipartitioning problem any influence of redundancy can be eliminated fairly easily. 1-Point Complementary Crossover (1CX) The implementation of the 1CX is based on the following definition:

Unlike Culberson[Cul93], we allow a crosspoint to occur in front of the first bit resulting in an inversion of the complete bit string. 1CX produces a neighbourhood structure similar to the LPK-heuristic. It forms a closed path of length l starting and ending with the initial partition. Unlike the LPK-heuristic the path is determined by the

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order of the vertices. A steepest ascent strategy will follow all the path where each step requires the update of the associated gains. Thus the whole pass requires O(l2) time. Obviously, although the crossover does not process additional information the time required is the same magnitude as that of LPK heuristic, clearly an advantage for the more sophisticated heuristic. 2-Point Complementary Crossover (2CX) The 2CX is defined as follows:

Defined in this fashion 2CX resembles the 1-bit Hamming hill climber (if j = i + 1) as well as 1CX (if j = l + 1). Since points. Thus the size of the there are l + 1 feasible sides for a cross point the neighbourhood contains neighbourhood is l(l+1)/2, i.e bounded by O(l2). Consequently, for larger problems an exhaustive search is out of the question. Instead a next ascent search is performed with an upper limit of tmax = l trials per pass. Since the complete evaluation of the objective value involves l(l1)/2 operation the time bound is given by O(l2) for a single operation and O (l3) for a complete pass. 6.4.3 1-Bit Hamming Hill Climber (HC) The 1-bit Hamming hill climber can be defined as follows:

Obviously, applying a 1-bit Hamming hill climber combined with a steepest ascent search strategy to the graph bipartitioning problem corresponds to a strategy that always moves the vertex of the highest gain. Since only a single vertex is moved it is convenient to maintain the gains and update the solution after each move which requires O(l) time. Since spotting the best vertex can be done in linear time too the computational complexity of a steepest ascent Hamming hill climber is bound by O(l). The search terminates after a local optimum has been reached. 6.4.4 Mutation (MUT) Mutation traditionally flips a single bit with a given probability p. Thus

The effect of mutation corresponds to moving several vertices at a time. Thus the computational complexity for a single operation will range between O(l) for a low

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Page 105 mutation rate and O(l2) for a high mutation rate. If only a single vertex is moved the move can be suspended if the associated gain is nonpositive. In the following experiments the mutation rate has been set to p = 1/l. The search terminates after the maximum number of evaluations per pass, tmax = l, has been exceeded. Thus the time required for one pass is bounded by O(l3). Table 6.2 Certain characteristic of the operators considered. SA and NA stand for steepest and next ascent. Op

N'hood

Time

Strategy

Termination

LPK

l

O(l2)

SA

local optimum

1CX

l

O(l2)

SA

local optimum

2CX

l2

O(13)

NA

< l trials/pass

HC

l

O(l)

SA

local optimum

MUT

2l

O(13)

NA

< l trials/pass

6.5 Evaluation of the Operators In the first step 1-point and 2-point complementary crossover, the 1-bit Hamming hill climber and mutation are compared against the LPK-heuristic. Therefore, they are implemented in a simple local hill climbing template that uses both next ascent and steepest ascent respectively and terminates when a local optimum has been reached or the maximum number of trials per pass has been exceeded. For evaluation a set of random graphs has been generated. All graphs have uniform weights at the edges and at the vertices. The parameters are summarized in Table 6.3. Table 6.3 Parameter of the graphs Acronym

R.50.5

R.50.10

R.100.5

R.100.10

Vertices

50

50

100

100

Mean degree

5

10

5

10

Edges

124

252

244

472

Max degree

10

18

11

16

Min degree

2

5

0

3

The results of fifty runs are displayed in Tables 6.4 to 6.7. Besides the standard measures (best result, worst result, average and standard deviation) the average number of moves per pass has been measured as well as the time required. While the first gives a rough idea of the distribution of local optima the latter highlights the computational complexity. The results can be summarized as follows: The LPK-heuristic performs best in terms of accuracy and requires a

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Page 106 Table 6.4 R.50.5 Operator

max

min

average

deviation

min cut

average moves

time in s

LPK

572

562

569

3

26

2.280

2.0

1CX

532

488

511

8

46

1.720

1.0

2CX

530

498

515

8

47

6.340

8.0

MUT

552

498

531

13

36

13.060

7.0

HC

544

482

515

13

40

2.920

0.0

Table 6.5 R.100.5 Operator

max

min

average

deviation

min cut

average moves

time in s

LPK

1142

1124

1137

5

51

2.500

7.0

1CX

1058

972

1019

14

93

1.820

2.0

2CX

1060

1002

1029

12

92

7.540

55.0

MUT

1116

1040

1081

16

64

32.460

46.0

HC

1082

972

1030

26

81

4.500

1.0

Table 6.6 R.50.10 Operator

max

min

average

deviation

min cut

average moves

time in s

LPK

600

572

597

5

76

2.220

1.0

1CX

540

478

512

13

106

1.980

0.0

1CX

542

498

518

11

105

6.020

7.0

MUT

580

490

539

18

86

12.360

6.0

HC

558

478

519

17

97

2.920

0.0

Table 6.7 R.100.10 Operator

max

min

average

deviation

min cut

average moves

time in s

LPK

1208

1176

1200

8

132

2.820

6.0

1CX

1074

972

1025

20

199

2.060

4.0

2CX

1080

1012

1048

17

196

7.220

57.0

MUT

1164

1064

1114

23

154

32.260

58.0

HC

1128

966

1038

39

172

4.500

1.0

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reasonable amount of time. 1-bit Hamming hill climbing is the fastest operator and achieves a similar performance to 1CX and 2CX. Mutation shows the second best performance after the LPK. However, it suffers from high computational costs mainly caused by the evaluation procedure. 2CX requires the most computational time which, however, is not justified by the performance achieved. To sum up, operators have to be judged according to the search space they induced (number of local optima, speed of convergence) and the degree up to which they support an efficient evaluation of the objective function. Concerning graph bipartitioning the LPK-heuristic exhibits an efficient tradeoff between both measures. 6.6 Recombination Recombining two binary strings of length l results in a search space Sd = {0,1}d, 0 ≤ d ≤ l where d is the Hamming distance between both strings. Since the time required for performing an operation on a bit string is usually a (polynomial) function of l a reduced search space will speed up the operator and thus save computational time. This effect has been described elsewhere (Reeves[Ree94]; Höhn[Höh94]). However, for some operators recombination exhibits an even more interesting phenomenon which is now considered in more detail. Consider two bit strings, say x, y, both of length l. Applying a 1-bit Hamming hill climber the neighbourhood of both points contains l neighbours. Assuming that the Hamming distance between x and y is 0 < d < l, recombination of x and y generates a neighbourhood N(x, y, HC) containing d points. Obviously, N(x,y,HC) ⊆ N(x,HC) and N(x,y,HC) ⊆ N(y,HC). In contrast, as an easy example reveals (Figure 6.1), the last relation does not necessarily apply for 1-point complementary crossover. Here recombination is able to introduce new points into the reduced neighbourhood and thus it may help to escape out of a local optimum. The same observation may be made for the 2point complementary crossover or the LPK-heuristic. To investigate both effects of recombination a population of 50 hill climbers was implemented in the scheme of the abstract GA. Tables 6.8 to 6.11 display the results for a typical run of five generations. Convergence and diversity of the population can be measured by the size of the search space averaged over the population and the deviation in the cost values. Furthermore, the average number of moves is given which is cumulative over the generation. Thus it can be considered as a measure for how efficient the local optima encountered at the end of one generation are circumvented in the next generation.

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Figure 6.1 Full and reduced neighbourhood of some bit string when 1CX is applied. In the lower picture the second bit from the right is fixed due to recombination. Obviously, the reduced neighbourhood contains two points which are not included in the full neighbourhood.

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Page 109 Table 6.8 Typical run of the GA when applied to R.50.5 Operator

max

min

average

deviation

min cut

average moves

average size

time in s

LPK

572

562

569

3

26

2.280

50.00

1.0

572

566

571

1

26

2.660

12.96

2.0

572

572

572

0

26

2.860

5.72

2.0

572

572

572

0

26

2.860

1.92

3.0

572

572

572

0

26

2.860

2.08

3.0

532

488

511

8

46

1.720

50.00

0.0

536

510

520

6

44

2.460

21.60

0.0

538

520

529

5

43

3.160

21.44

1.0

544

526

535

4

40

3.620

15.36

1.0

550

536

539

3

37

4.100

11.12

1.0

530

498

515

8

47

6.340

50.00

9.0

538

512

524

6

43

7.540

22.12

12.0

544

518

530

6

40

8.680

19.84

14.0

548

526

535

5

38

9.500

20.44

16.0

548

530

538

5

38

10.040

18.12

18.0

552

498

531

13

36

13.060

50.00

6.0

552

514

539

10

36

14.380

20.04

7.0

556

532

547

6

34

15.340

20.40

8.0

558

534

551

5

33

15.860

18.72

8.0

558

544

553

3

33

16.540

17.64

9.0

544

482

515

13

40

2.920

50.00

1.0

544

494

522

12

40

2.920

21.32

1.0

544

512

527

9

40

2.920

21.28

1.0

544

518

533

8

40

2.920

19.92

1.0

544

522

537

7

40

2.920

18.28

2.0

1CX

2CX

MUT

HC

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Page 110 Table 6.9 Typical run of the GA when applied to R.100.5 Operator

max

min

average

deviation

min cut

average moves

average size

time in s

LPK

1142

1124

1137

5

51

2.500

100.00

7.0

1142

1132

1140

2

51

2.920

34.96

8.0

1142

1140

1141

1

51

3.120

29.56

8.0

1142

1142

1142

0

51

3.120

31.16

9.0

1142

1142

1142

0

51

3.120

26.48

10.0

1058

972

1019

14

93

1.820

100.00

3.0

1064

1014

1035

14

90

2.700

46.00

3.0

1064

1030

1052

9

90

3.300

45.64

4.0

1070

1048

1059

5

87

3.900

30.92

7.0

1070

1056

1064

5

87

4.240

26.44

8.0

1060

1002

1029

12

92

7.540

100.00

59.0

1064

1020

1042

11

90

9.120

46.32

72.0

1074

1030

1054

10

85

10.680

43.16

86.0

1078

1040

1061

9

83

11.320

43.12

98.0

1092

1050

1067

7

76

12.320

35.60

112.0

1116

1040

1081

16

64

32.460

100.00

38.0

1116

1068

1092

11

64

34.700

44.52

41.0

1118

1082

1100

8

63

35.980

41.84

43.0

1120

1090

1106

8

62

37.200

38.40

45.0

1122

1094

1110

7

61

38.380

31.96

47.0

1082

972

1030

26

81

4.500

100.00

1.0

1082

994

1041

24

81

4.500

45.40

1.0

1082

998

1053

20

81

4.500

44.60

2.0

1082

1024

1058

18

81

4.500

39.36

3.0

1082

1036

1066

15

81

4.500

41.08

3.0

1CX

2CX

MUT

HC

For the given set of test problems the following observations may be made. Owing to neighbourhood size reduction all the operators are speeded up. Thus five generations require less than double the time of the first generation. As predicted, recombination does not have any effect on the 1-bit Hamming hill climber. The increasing average performance is merely due to selection. Clearly, for 1CX and LPK, recombination leads the search away from the local optima found in the first generation. The increasing average number of moves indicates the progress. file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_110.html（第 1／2 页）2009-7-7 16:56:58

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For mutation and 2CX the local search stops after l trials have been failed. However, in the next generation the selected points will be evaluated again which essentially exceeds the stopping criterion and accounts for the progress of one move per generation. Obviously, this effect is an artefact of the selection scheme and cannot be attributed to

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Page 111 recombination. However, both operators benefit from the speed up. Table 6.10 Typical run of the GA when applied to R.50.10 Operator

max

min

average

deviation

min cut

average moves

average size

time in s

LPK

600

572

597

5

76

2.220

50.00

1.0

600

598

600

0

76

2.460

8.64

1.0

600

600

600

0

76

2.460

3.20

1.0

600

600

600

0

76

2.460

2.72

2.0

600

600

600

0

76

2.460

3.32

2.0

540

478

512

13

106

1.980

50.00

0.0

546

508

526

9

103

2.980

21.28

0.0

556

522

538

7

98

3.640

18.44

1.0

556

534

546

6

98

4.300

15.40

1.0

566

542

554

5

93

5.040

14.32

1.0

542

498

518

11

105

6.020

50.00

5.0

548

516

532

9

102

7.340

22.08

7.0

572

520

540

10

90

8.120

20.28

9.0

588

528

553

12

82

9.220

18.24

10.0

586

540

562

9

83

10.020

16.76

11.0

580

490

539

18

86

12.360

50.00

5.0

580

518

550

15

86

13.300

20.68

6.0

592

540

561

13

80

14.180

20.04

6.0

592

548

569

10

80

14.900

17.12

7.0

594

552

576

10

79

15.300

16.12

7.0

558

478

519

17

97

2.920

50.00

0.0

558

494

528

15

97

2.920

21.80

0.0

558

510

536

13

97

2.920

20.40

1.0

558

520

544

10

97

2.920

17.12

1.0

558

522

548

9

97

2.920

18.64

1.0

ICX

2CX

MUT

HC

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Page 112 Table 6.11 Typical run of the GA when applied to R.100.10 Operator

max

min

average

deviation

min cut

average moves

average size

time in s

LPK

1208

1176

1200

8

132

2.820

100.00

5.0

1208

1196

1206

3

132

3.340

28.52

5.0

1208

1204

1208

1

132

3.700

12.04

6.0

1208

1208

1208

0

132

3.780

7.96

6.0

1208

1208

1208

0

132

3.780

5.12

7.0

1074

972

1025

20

199

2.060

100.00

2.0

1074

1014

1048

17

199

2.920

46.28

3.0

1086

1044

1066

9

193

3.640

43.96

4.0

1096

1064

1075

6

188

4.320

41.52

4.0

1096

1074

1083

7

188

4.680

30.68

5.0

1080

1012

1048

17

196

7.220

100.00

43.0

1098

1026

1066

15

187

8.440

45.16

55.0

1114

1052

1080

14

179

9.520

41.96

66.0

1120

1072

1092

12

176

10.460

36.24

75.0

1120

1078

1101

12

176

11.200

38.28

84.0

1164

1064

1114

23

154

32.260

100.00

35.0

1164

1082

1128

17

154

34.200

41.56

38.0

1172

1106

1136

16

150

35.640

41.04

40.0

1174

1116

1146

14

149

36.920

39.60

41.0

1184

1126

1156

13

144

37.980

34.80

43.0

1128

966

1038

39

172

4.500

100.00

1.0

1128

976

1062

34

172

4.500

45.28

2.0

1128

1010

1079

30

172

4.500

44.60

2.0

1128

1024

1092

23

172

4.500

41.80

3.0

1128

1044

1095

21

172

4.500

38.52

3.0

1CX

2CX

MUT

HC

However, since the LPK-heuristic found the best values always within the first generation, it is not clear whether recombination is more effective than simply trying different initial points. To obtain a preliminary answer on this question a hill climber was compared to the GA when applied to a more ambitious random graph with 500 vertices and 1224 edges corresponding to a mean degree of 5. In 30 runs the CPU time has been fixed for both algorithms and the best values found per run are compared. The results are shown in Table 6.12. It appears that LPK coupled with recombination performs better than LPK alone. In particular, the worst value found in 30 runs with recombination exceeds the average performance of LPK on its own. The last column displays the effective population size, i.e. 80 restarts require the same amount of time as 50 individuals evolved for 6 generations.

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Page 113 Table 6.12 The LPK heuristic applied alone and coupled with recombination Operator

max

min average deviation popsize

LPK

5778 5754

5762

6

80

GA + LPK

5788 5764

5774

5

50

6.7 Conclusion We have shown elsewhere (Reeves[Ree94]) that GAs are closer to NS than is sometimes believed, in that a GA works by first finding a reduced neighbourhood and then searching it in a random fashion. Adding local optimization as an extra 'operator' (as suggested by previous work) has been found to improve the GA's performance, albeit at a cost in terms of the computational requirements. However, integrating local optimization more firmly into the GA can help to maintain the benefits of recombination, and if implemented efficiently can speed up the time to reach a good solution. In this paper an abstract genetic algorithn has been presented and tested on a special case of graph partitioning. The proposed GA couples a local hill climbing technique with recombination and selection in a general way, so that any local search technique may be accommodated within the proposed scheme. In this paper the investigation has focused on several operators and the effects of recombination. Initially the operators have been studied individually when implemented in the framework of a simple local hill climber. Such preliminary investigations may form a first step in the design of more sophisticated algorithms. Secondly, we have separated the effect of the operator used from that of the recombinative effect of a GA, and shown that, at least for larger problem instances, recombination has been proved to save computational time as well as offering a mechanism to circumvent local optima for certain operators. Both effects provide a reasonable explanation why GAs can improve traditional hill climbing techniques. Furthermore, both characteristics may be used to guide the design of application-specific genetic algorithms. However, the efficiency of recombination certainly depends on the population size as well as the selection scheme applied. Both problems deserve further investigation. In future work the performance of the GA will be investigated when more sophisticated hill climbers are incorporated. References [ARS93] Albrecht R. F., Reeves C. R. and Steele N. C. (eds) (1993) Proceedings of the International Conference on Artificial Neural Networks and Genetic Algorithms. Springer-Verlag, Vienna. [BW93] Blanton Jr. J. and R.L. Wainwright (1993) Multiple vehicle routing with

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time and capacity constraints using genetic algorithms. In S. Forrest [For93]. [Cul93] Culberson J. C. (1993) Mutation-crossover isomorphisms and the construction of discriminating functions. Evolutionary Computation to appear. [Dav91] Davis L. (ed) (1991) Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York. [For93] Forrest S. (ed) (1993) Proceedings of 5th International Conference on Genetic Algorithms. Morgan Kaufmann, San Mateo, CA. [FD92] Falkenauer E. and Delchambre A. (1992) A genetic algorithm for bin packing and line balancing. In Proc. IEEE 1992 Int'l Conference on Robotics and Automation (RA92). Nice, France. [Go189] Goldberg D. E. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, Mass. [Hol75] Holland J. (1975) Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor, MI. [Höh94] Höhn C. (1994) Heuristic neighbourhood search operators for graph partitioning tasks. In Proc. 10th Int. Conf. on Systems Engineering, pages 469476. Coventry, UK. [HGL93] Homaifar A., Guan S., and Liepins G.E. (1993) A new approach on the travelling salesman problem by genetic algorithms. In Forrest [For93]. [HR94] Höhn C. and Reeves C. R. (1994) Embedding local search operators in a genetic algorithm for graph partitioning. Technical report, School of Mathematics and Information Sciences, Coventry University, UK. [JSG91] Jog P., Suh J. Y., and Gucht D. V. (1991) The effects of population size, heuristic crossover and local improvement on a genetic algorithm for the travelling salesman problem. In J. D. Schaffer [Sch89] [KL70] Kernighan B. W. and Lin S. (1970) An efficient heuristic procedure for partitioning graphs. Bell System Technical Journal pages 291307. [LPK89] Lee C-H., Park C-I., and Kim M. (1989) Efficient algorithm for graph partitioning problem using a problem transformation method. Computer Aided Design 21: 611618. [Müh91] Mühlenbein H. (1991) Evolution in time and spacethe parallel genetic algorithm. In G.J.E. Rawlins [Raw91]. [MM92] Männer R. and Manderick B. (eds) (1992) Parallel Problem-Solving from Nature, 2. Elsevier Science Publishers, Amsterdam. [PRR93] Prinetto P., Rebaudengo M., and Reorda M. (1993) Hybrid genetic algorithms for the travelling salesman problem. In R. F. Albrecht and N. C. Steele [ARS93]. [Raw91] G.J.E. Rawlins (ed) (1991) Foundations of Genetic Algorithms. Morgan Kaufmann, San Mateo, CA. [Ree93] Reeves C. R. (1993) Hybrid genetic algorithms for bin-packing and related problems. Annals of OR to appear. [Ree93a] Reeves C. R. (ed) (1993) Modern Heuristic Techniques for Combinatorial Problems. Blackwell Scientific Publications, Oxford. [Ree94] Reeves C. R. (1994) Genetic algorithms and neighbourhood search. In Fogarty T. C. (ed) Evolutionary Computing: AISB Workshop, Leeds, UK, April 1994: Selec ted Papers, number 865 in Lecture Notes in Computer Science. Springer-Verlag, Berlin.

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[Ree95a] Reeves C. R. (1995) A genetic algorithm for flowshop sequencing. Computers & Ops. Res. 22: 513. [Ree95b] Reeves C. R. (1995) Genetic algorithms and combinatorial optimisation. In Rayward-Smith V. J. (ed) Applications of Modern Heuristic Methods, pages 111126. Alfred Waller, Henley-on-Thames, UK. [Sch89] Schaffer J. D. (ed) (1989) Proceedings of 3rd International Conference on Genetic Algorithms. Morgan Kaufmann, Los Altos, CA. [TZ93] Tao L. and Zhao Y. (1993) Multi-way graph partitioning by stochastic probe. Computers & Ops. Res. 20: 321347.

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[UAB+91] Ulder N. L. J., Aarts E. H. L., Bandelt H.-J., Laarhoven P. J. M., and Pesch E. (1991) Genetic local search algorithms for the travelling salesman problem. In H.-P.Schwefel and R.Männer (eds) Parallel Problem-Solving from Nature. Springer-Verlag, Berlin. [WSS91] Whitley D., Starkweather T., and Shaner D. (1991) The traveling salesman and sequence scheduling: quality solutions using genetic edge recombination. In L.Davis [Dav91].

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7 Local Search for Steiner Trees in Graphs M.G.A. Verhoeven, M. E. M. Severens and E. H. L. Aarts Abstract We present a local search algorithm for the Steiner tree problem in graphs, which uses a neighbourhood in which paths in a Steiner tree are exchanged. The exchange function of this neighbourhood is based on a multiple-source shortest path algorithm. We present computational results for a known benchmark set of generated Steiner tree problem instances and for a set of instances derived from real-world travelling salesman problem instances, containing up to 18512 vertices. The results show that good quality solutions can be obtained in moderate running times. 7.1 Introduction In the Steiner tree problem in graphs a minimum weight tree has to be found that includes a prespecified subset of vertices of a graph. The Steiner tree problem occurs in several practical applications, such as the design of telephone, pipeline, and transportation networks, and the design of integrated circuits. Although the Steiner tree problem is NPhard [HRW92], several sophisticated optimization algorithms for it exist, which are able to solve instances with up to 2 500 vertices and 62 500 edges at the cost of substantial amounts of computation time, viz., several hours on a powerful workstation or supercomputer. In addition many heuristics have been proposed, most of which have running times that are polynomially bounded at the risk of finding sub-optimal solutions. Hwang, Richards and Winter[HRW92] present an overview of both optimization algorithms and heuristics. In this paper we study local search approaches to the Steiner tree problem. Local search is a generally applicable approximation technique for hard combinatorial optimization problems, and the literature reveals that this approach can find good quality results for a wide variety of problems. For an overview of local search we refer to [Ree93, AL96]. For the Steiner tree problem local search has so far only been applied to relatively small instances containing up to 100 vertices, requiring several minutes Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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Page 118 of running time. Such instances are well within the range of current optimization algorithms, using a smaller or equal amount of running time. So, up to now local search is not competitive with efficient optimization algorithms. In this paper we present a local search algorithm that is able to find solutions with a relative error of a few percent requiring moderate running times. Moreover, it is able to handle very large problem instances in an acceptable amount of time. 7.2 Local Search The use of local search presupposes the specification of an instance of a combinatorial optimization problem and a neighbourhood structure. Definition 1 An instance of a combinatorial optimization problem is denoted as a pair (S, f), where the solution space S is the set of feasible solutions, and the cost function

gives the cost of a solution. The problem, in the case of

minimization, is to find a solution with minimal cost. A neighbourhood structure solution a set of solutions. A solution s∈S is a local minimum of

if f(s′)≥f(s) for all

assigns to each .

A neighbourhood is weakly connected if it is possible to reach an optimal solution from any solution, and it is strongly connected if it is possible to reach each solution from any other solution. The neighbours of a solution are not given explicitly, but can be constructed by a polynomially computable function. Most neighbourhoods are based on the replacement of a few elements that constitute a solution. For this we need the following definition in which we assume that solutions are represented by sets. Definition 2 Let the set of elements constituting solutions in S be given by ε = , and let the exchange be a partial function with . Then, an l-exchange neighbourhood structure function

is

for each s ∈ S.

defined by

Local search algorithms constitute a class of approximation algorithms that are based on continually improving a solution by searching its neighbourhood for a solution with lower cost. The simplest form of local search is the iterative improvement algorithm presented in Figure 7.1. There are many variants on this basic local search algorithm known in the literature; examples are simulated annealing, tabu search, and genetic local search. Simulated annealing and some forms of probabilistic tabu search converge to a global minimum provided that the neighbourhood is at least weakly connected. 7.2.1 The Steiner Tree Problem in Graphs (SP) Formally the Steiner tree problem in graphs (SP) is defined as follows. Definition 3 Let G = (VG, EG) be an undirected graph, a function that assigns a weight to each edge, and let X⊆ VG be a set of terminals. The problem is to find a subtree T = (VT,ET) of G with X⊆ VT for which the sum of the edge weights

is minimal.

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proc Iterative_Improvement (s : S) var w : P(S); begin

while

do

if f(s′) ≥ f(s) then w := w ∪ {s′} else s := s′;

fi

od {s is a local minimum of

}

end

Figure 7.1 An iterative improvement algorithm. Vertices in VG \ X are called non-terminals. Since no non-terminals with degree one are included in the optimal solution, the solution space S consists of all subtrees T of G with X ⊆ VT that contain no non-terminals with degree one. Such a tree is called a Steiner tree. Non-terminals in a Steiner tree T are called the Steiner vertices of T. Steiner vertices with a degree at least three are called key vertices. A key path is a path in a Steiner tree T of which all intermediate vertices are Steiner vertices with degree two in T and whose end vertices are terminals or key vertices. It follows directly that a minimal Steiner tree consists of key paths that are shortest paths between key vertices or terminals. A basic property of the SP is that Steiner trees contain at most |X| 2 key vertices, and consequently they consist of at most 2|X| 3 key paths. 7.3 Neighborhoods for the SP The question of finding appropriate neighbourhoods for the Steiner tree problem in graphs has been addressed by several authors. Duin and Voss[DV93] distinguish between vertex-oriented neighbourhoods based on exchanging Steiner vertices and edge-oriented neighbourhoods in which edges are exchanged.

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Osborne and Gillett[OG91] propose a neighbourhood based on the observation that a Steiner tree T can be represented by its vertices VT, since its edges ET can be determined by computing a minimum spanning tree for the subgraph of G induced by VT. Neighbors are then constructed by adding and removing elements to and from VT. A similar , neighbourhood is used in the genetic algorithm of [KRSS93]. Although the neighbourhood size is verification of local optimality for these neighbourhoods is computationally expensive, as construction of neighbours time complexity. All the above papers present requires a minimum spanning tree computation with only computational results for small instances with up to 100 vertices. Esbensen[Esb95] presents a genetic algorithm based on a vertex-oriented neighbourhood and a heuristic to construct Steiner trees from sets of vertices. This

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Page 120 genetic algorithm gives good-quality solutions, albeit at the expense of substantial running times, viz., several hours for , which makes it difficult to handle larger

instances with 2500 vertices. Moreover, memory requirements are instances.

Voss[Vos92] and Dowsland[Dow91] propose edge-oriented neighbourhoods based on exchanging key paths. Neighbors are constructed by removing one key path from a Steiner tree T and connecting the remaining two components by a shortest path between two arbitrarily chosen vertices, one in each component, such that a Steiner tree is obtained. A disadvantage of this neighbourhood is that the complexity of verification of local optimality, and thus the complexity of a single local search step, is

, since a Steiner tree T contains at most 2|X| 3 key paths. Moreover, a time

pre-processing step is needed to compute shortest paths between all pairs of vertices, and storing all paths space complexity. [Dow91] furthermore presents a second neighbourhood that consists of the requires a above neighbourhood extended with Steiner trees obtained by connecting the two components by shortest paths from any vertex not in the tree to any two vertices, one in each component. This extension is needed to prove strong connectivity of this neighbourhood, i.e., the possibility to transform each feasible solution to any other feasible solution. , which makes this The time complexity to evaluate all neighbours in this neighbourhood is neighbourhood not very suited for local search algorithms in which neighbourhoods have to be enumerated, e.g. iterative improvement or tabu search. In this paper, we present neighbourhood structures and exchange functions that have better time and space complexities and are therefore more adequate for larger problem instances. Next, we present a neighbourhood which is comparable to the 1exchange neighbourhood of [Dow91], but it uses a different exchange function that has a smaller computational complexity, requires no pre-processing, and has smaller memory storage demands. First, we note that removal of a key path splits a Steiner tree into two components. Reconnection of these components can be considered as a new STPG instance in which components are treated as terminals. A SP instance with two components can be solved optimally by computing a shortest path between these components with a straightforward variant of the single-source shortest path algorithm of [Dij59]. This observation gives rise to the following neighbourhood. Definition 4 Let T be a Steiner tree which consists of K key paths, l1, . . . ,lK. Let Si, be the two components that remain after removal of key path li from T. Let the function sp : P(VG) × P(VG)→ P(EG) give a shortest path from a subset of vertices to another subset of vertices. Then, the neighbourhood

is defined by

Neighborhood sizes can be different for different Steiner trees. Removal of a key path which ends in a key vertex with degree three turns that key vertex into a non key vertexSteiner vertex with degree twoand the two remaining key paths are joined to a single key path. Addition of a key path that ends in non key vertex converts that vertex into a key vertex, and the key path that passes through it is split into two key paths. So, |T| 2 ≤ |T′| ≤ |T| + 2 for denotes the number of key paths in T. The number of key paths whose removal has to be considered in a given neighbourhood most 2|X| 3, since a Steiner tree

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, where |T|

(T) is at least |X| 1 and at

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proc sp (W, W′ : P(VG)) begin Y1 :=W; for v ∈VG \ W do m(v) := ∞ rof; for v ∈ W do m(v) := 0 rof; do

while

if v ∉ W′ then for v′ ∈ {w ∈ VG | (v, w) ∈ EG} do if v′ ∉ Y0 then m(v′) := min{m(v′), m(v) + d(v, v′)}; Y1 := Y1 ∪ {v′}; fi rof fi; Y0 := Y0 ∪ {v}; Y1 := Y1 \ {v}; od {m(v) gives the length of the shortest path from W to W′} end

Figure 7.2 A multiple source shortest path algorithm. T contains at least |X| 1 and at most 2|X| 3 key paths. Hence, a neighbourhood contains at most 2|X| 3 neighbours. An interesting observation is that solutions only have neighbours with lower or equal cost, because only paths with at most the length of the removed path are inserted since this length is an upper bound on the length of the shortest path between components. So, an attempted replacement of a key path in a Steiner tree can lead to the same Steiner tree if no shorter path exists. In particular this can imply that local minima have no neighbours. The function sp is computed with the algorithm presented in Figure 7.2. The complexity of this algorithm is if the set Y1 is represented by the classical heap data structure. Identification of the two components that arise when a key path is removed from a Steiner tree can be done in

time.

An important observation is that the algorithm in Figure 7.3 computes the shortest paths to vertices in ascending order. In this algorithm m(v) gives the length of a shortest path to a vertex v ∈ Y0. The algorithm can be terminated as soon as m(v) ≥ d(li) for is at least d(li) long. Consequently, a vertex v ∈ Y0 and a removed key path li, because then the shortest path from Si to replacement of li cannot lead to a lower-cost tree. This leads to the following upper bound for the time complexity of the function sp. Let T be a Steiner tree from which a path l with length d(l) is removed, which results in two components S, S′. Let W be the largest set of vertices that are within distance d(l) from S for some l in T. Then, the complexity of the function sp is . As a consequence of the insertion of shortest paths between components, the neighbourhood always possible to reach a globally

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Page 122 minimal Steiner tree by a sequence of exchanges, as can be seen by a simple example. Therefore, we present a neighbourhood

that is a small augmentation of

such that it is weakly connected, i.e., it is possible to reach an

(T) also contains neighbours with higher optimal solution from any solution by a sequence of exchanges. Moreover, cost than a Steiner tree T, so here neighbourhoods of local minima are definitely not empty. Such neighbourhoods are needed in tabu search algorithms to escape from local minima. also a single key path is replaced with another key path, as is the case in . The supplement of to In consists of neighbours constructed by adding shortest paths from one component to the other component via vertices that lie at a distance of one edge from the other component. Definition 5 Let T be a Steiner tree which consists of K key paths, l1, . . ., lK. Then, the neighbourhood

The complexity of evaluating the cost of all neighbours in a neighbourhood

(T) is equal to

(T) of a Steiner tree T is

, as all neighbours originating from removal of the same key path, can be evaluated in a single execution of the multiple-source shortest path algorithm. Note that the neighbourhood

(T) is a strict subset of the

second neighbourhood of [Dow91] and that all excluded neighbours have higher cost than those that are included in (T) The average cost of neighbours in

(T) is therefore lower than in Dowsland's second neighbourhood, which also

, viz. . is not strongly connected since it is not has a much larger computational complexity than possible to transform a Steiner tree into another Steiner tree that differs only in a single key path that is not a shortest path between two components. However, we have the following result. Theorem 1 The neighbourhood structure is weakly connected, i.e., it is possible to reach an optimal solution from any solution by a valid sequence of exchanges. Proof. Let T* be an optimal Steiner tree, let T be a Steiner tree, and let sp(A, B) denote a shortest path from component A to B. Partition T* into subtrees such that all leaves are terminals and all other vertices are Steiner vertices. Let S ⊆ T* be such a subtree that is not duplicated in T. We consecutively add adjacent key paths in S to T, starting with the key paths rooted from the leaves in S. We distinguish between two cases based on whether vertex i with which a vertex a ∈ T has to be connected, is included in T. Let i ∈ T, and let sp(a, i) be a key path in T* not in T that is to be added. Remove a key path l from T not in T* such that addition of sp(a, i) would result in a Steiner tree. This is always possible since addition of sp(a, i) to T gives a cycle in T of which at least one key path is not included in T*. Removal of l splits T in components A and B with a ∈ A and i ∈ B. . If sp(a, j) ≠ sp(A, j) = sp(a′, j) Let (j, i) ∈ sp(a, i) and construct then addition of sp(a, j) to T′ results in a cycle. So, there exists a key path in T′ not in T* whose replacement with sp(a, j) would result in a Steiner tree. We can repeat the

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above procedure, adding one edge of sp(a, i) at a time, until the entire path sp(a, i) has been added. Let i ∉ T, then i is a key vertex connected with at least three vertices a, b, c ∈ S that also exist in T. Let sp(a,i) be a key path in T* not in T that is to be added. Remove a key path l from T not in T* such that addition of sp(b, a) would result in a Steiner tree. This is always possible since addition of sp(b, a) to T gives a cycle in T of which at least one key path is not included in T*. Removal of l splits T in components A and B with a ∈ A and b ∈ B. Let (j,a) ∈ sp(a,i) and . Next, remove a key path l′ from T′ not in T*, which results construct in components A′ and C, such that addition of sp(c,j) would give a Steiner tree. Let (j′, j) ∈ sp(a, i) and construct . Repeating these steps of adding edges of sp(a, i) one after the other, starting alternately from b and c, gives a Steiner tree that includes i, at which point the first case applies. Once a key path of T* has been added to a subtree S, other key paths from T* are consecutively added to S in order of ascending number of key paths to the leaves of S. This process gives a valid sequence of exchanges that transforms a Steiner tree T to an optimal Steiner tree T*. and are based on the insertion of a single shortest path between two components. They The neighbourhoods are inspired by the polynomially solvable Steiner tree problem with two terminals. Chen[Che83] presents a polynomial algorithm for the SP with three terminals. This motivates the following neighbourhood in which two paths are removed from a Steiner tree. Reconnection of the remaining three components can be done optimally in polynomial time with the following algorithm which improves upon the time complexity of the algorithm of [Che83]. The essential observation in this algorithm is the following one. Lemma 1 Consider an SP instance with three terminals s,s′,s′′, and let T be a minimal Steiner tree with key vertex w. Then, the length of the path in T from w to s and the length of the path from w to s′ is at most d(sp(s,s′)). Note that it is possible that w coincides with a terminal. If we replace terminals s,s′,s′′ with three components S,S′,S′′ we see that the key vertex w in a minimal Steiner tree that contains the components S, S′, S′′, is included in the set W of vertices for which the shortest path to S and the shortest path to S′ is at most d(sp(S,S′)). This optimal key vertex w can be found as follows. 1. Construct the set of vertices for which the shortest path to S is at most d(sp(S,S′)). This set is given by the final value of the set Y1 in the algorithm in Figure 7.3 for the computation of sp(S,S′). Similarly, the set of vertices for which the shortest path to S′ is at most d(sp(S, S′)) is given by the final value of Y1 in the computation of sp(S′, S). The set W of candidate key vertices is the intersection of these sets. 2. The algorithm in Figure 7.2 is used to determine the shortest path from W to S′′. In this algorithm m(v) for v ∈ W has to be initialized as the distance from S to v plus the distance from S′ to v. The starting vertex w of this path is the key vertex in a minimal Steiner tree T for the components S, S′, S′′.

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Page 124 . The above algorithm adds at most three additional key paths to The time complexity for finding w S,S′,S′′ to construct a minimal Steiner tree that contains S,S′,S′′. Let stp3 (S,S′,S′′) denote this set of key paths. Using the function stp3, we can define the following 2-exchange neighbourhood

.

Definition 6 Let T be a Steiner tree which consists of K key paths, l1, . . ., lk. The function stp3 gives for three components the additional key paths in a minimal Steiner tree that contains these components. Then, the neighbourhood by

The size of this neighbourhood is

is defined

as the number of key paths K in a Steiner tree is at most 2|X| 3. The and the complexity of the algorithm to implement the function stp3 is

complexity of identifying S,S′,S′′ is

. Hence, the complexity of verifying local optimality w.r.t. . The time complexity of the function stp3 can be reduced by terminating its execution as soon as m(v) > li + lj, where m(v) is the summed length of the shortest paths from S, S′ and W to v ∈ VG, because then replacing key paths li and lj cannot lead to a Steiner tree with lower cost. holds that f(T′) ≤ f(T) since neighbours are constructed by computing a minimal For all neighbours Steiner tree to connect the remaining components after removal of two key paths. Note that removal of two key paths can lead to a Steiner tree in which Steiner vertices with degree one exist. This occurs if the two from T removed key paths lj and lj have a key vertex with degree three in T in common. The remaining key path from this vertex can also be removed from T leading to an additional cost decrease. So, in some cases three key paths are removed from a Steiner tree T for the construction of its neighbours in

(T).

(T) is not a subset of (T) for some Steiner trees T, and in some cases the intersection can The neighbourhood even be empty. However, the following result holds. Theorem 2 Let a Steiner tree T ∈ S be given. If T is a local minimum of and

, then T is also a local minimum of

. Moreover, let

with l,l′∈ T and l,l′∈ T and l,l′∉ T′′, then f(T′) ≤ f(T′′).

Proof. Consider a local minimum T of . Let , where key paths l and +l′ are removed from T, which results in three components S,S′,S′′. T′ is a minimal Steiner tree that connects the components S,S′,S′′. Let connects the components S,S′,S′′, so f(T′′) ≥ f(T′) = f(T).

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T′′ is also a Steiner tree that

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A disadvantage of the neighbourhood number of neighbours is

is its considerable time complexity for a single local search step since the

and the time complexity to evaluate the cost of a neighbour is

Therefore, we present the following neighbourhood

which is a restriction of

.

.

Definition 7 Let T be a Steiner tree which consists of K key paths, l1, . . ., lK. Let the function stp3 return, on input of three components, the additional key paths in a

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Page 125 minimal Steiner tree that contains these components. Then, the neighbourhood

(T) is equal to

In only those two key paths are removed that have a vertex in common, which can be a terminal or a key vertex. Typically, if this key vertex has degree three in the Steiner tree, the remaining key path is also removed. The three remaining components are connected using the algorithm stp3 to compute a minimal Steiner tree for three components. The size of the neighbourhood

(T) for a Steiner tree T is

vertex in T. The complexity of verifying local optimality of T for substantially less than the complexity of verifying local optimality for minimum of T ∈ S.

is a local minimum of

, where K, is the maximum degree of a key is then

, which is

. Moreover, it still holds that a local

since

for a Steiner tree

Implementation Issues The time complexity of a single step of an iterative improvement algorithm depends also on the data structures used to represent solutions. A Steiner tree is a set of key paths. This set is given by a list of records that represent key paths. Each record also contains cross references to key paths that share end vertices with this key path. Furthermore, a function is maintained that gives, for each vertex included in the Steiner tree, a key path that contains this vertex. Given the end vertices of a key path it is possible to find the corresponding key path by following the cross references. In this way removal and addition of key paths can be done in a time linearly bounded by the number of vertices in a key path and the degrees of the end vertices. It should be noted that removal or addition of key paths can require that other key paths are joined to one key path or split into two key paths. Another important aspect in the implementation of an iterative improvement algorithm is how neighbourhoods are enumerated. Although this aspect is rarely discussed in the literature it can make a substantial difference in running time how the neighbourhood is enumerated. Essential for enumerating neighbourhoods is that exchanges which do not lead to lower cost solutions do not have to be examined. Of course it is generally not possible to determine beforehand which exchanges do not lead to a lower cost solution, but often exchanges which do not yield any gain when applied to a Steiner tree T also do not lead to a lower cost Steiner tree when applied to a neighbour . This implies that it is profitable to enumerate neighbourhoods in such a way that exchanges which have not yet been examined are explored first. To this end, a Boolean function b is introduced that assigns a Boolean to arguments of the exchange function that indicates whether the corresponding exchange has been evaluated and has led to cost increase. While enumerating the neighbourhood of the current solution in a local search algorithm, first those exchanges are examined for which this Boolean does not hold. If no lower cost neighbour has been found, the remaining exchanges are evaluated. In the neighbourhood

only a single path is removed in an exchange, so Booleans can be associated with key paths,

. In the neighbourhood and the space-complexity for storing the Boolean function b is removed in an exchange, so Booleans have to be associated with pairs of key paths, and here the

two key paths are

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space complexity of b is . To reduce the space complexity of b, it is possible to associate a Boolean only with a single key path and to evaluate those exchanges first for which at least one Boolean associated with one of the two removed key paths is false. A key path's Boolean is set to true if all exchanges which remove this key path have been most exchanges can be identified by marking examined and do not result in cost decrease. In the neighbourhood key vertices, because when a key vertex in a Steiner tree has degree three, neighbours are obtained by removing the three key paths which end in this key vertex. Considering that most key vertices in a Steiner tree have degree three, we can associate Booleans with key vertices to indicate whether they have been exchanged. The space complexity of b can then be reduced to 7.4 Computational Results . Computational results are We have implemented an iterative improvement algorithm that uses the neighbourhood presented for 40, randomly generated, instances from [Bea90]. Furthermore, a number of real-world Euclidean travelling salesman problem instances of the TSPLIB from [Rei91] are transformed to Steiner tree problems, since for many combinatorial optimization problems it is observed that real-world instances typically are much harder to solve than randomly generated instances. This transformation is done as follows. Edges in an SP instance are connections between adjacent Voronoi regions in a Euclidean travelling salesman instance. The resulting graph G is a planar graph and contains at most 3n edges where n is the number of vertices. To construct SP instances with more edges, we add edges between vertices that are reachable from a vertex in 1,2,3 or 4 steps along the edges in G. From each graph constructed in this way three SP instances are obtained by randomly denoting 15%, 25%, or 35% of the vertices as terminals. No reduction techniques to reduce Steiner tree problem instances have been used. Initial solutions are constructed with a shortest path heuristic [TM80]. Tables 7.1 and 7.2 present the results obtained with our iterative improvement algorithm. In these tables |V|, |E|, and |X| give the number of vertices, edges, and terminals, respectively. For all problems but one (e18), in Beasley's series D and E, the optimal solution values are known [Bea89, CGR92] and given by opt. Note that for the Euclidean SP instances optimal solutions are only known for the instances f1 f24 with up to 1000 vertices. These solutions are computed by the exact algorithm of [Dui94], which is among the fastest optimization algorithms available for the SP. It can handle instances with at most 1000 vertices. The best solution found in ten runs of the iterative improvement algorithm is given by best, the average cost of these ten local minima by avg, and their average deviation in percentages from the optimal solution is given by ∆avg. The average running time in seconds on a Sun Classic workstation is given by t(s). We observe that our iterative improvement algorithm finds solutions whose relative excess is within a few percent over the optimal solution for the instances of [Bea90] in a small amount of running time (at most a few minutes, whereas exact algorithms require several hours on supercomputers or powerful workstations for instances with 2 500 vertices). For the Euclidean instances with up to 1000 vertices we are able to find solutions with a relative deviation of 0.5% to 2.3% from optimality in a few seconds.

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Page 127 Table 7.1 Results for Beasley's series D and E. |V|

|E|

|X|

opt

best

∆aug

t(s)

1000

1250

5

106

106

2.64

0.7

d2

10

220

220

0.73

0.3

d3

167

1565

1567

0.22

9.4

d4

250

1935

1939

0.34

11.9

d5

500

3250

3254

0.18

24.7

5

67

70

5.67

0.8

d7

10

103

103

0.00

0.9

d8

167

1072

1082

1.64

8.2

d9

250

1448

1454

0.70

14.5

d10

500

2110

2119

0.70

29.5

5

29

29

4.14

0.8

d12

10

42

42

3.81

0.1

d13

167

500

509

2.46

9.2

d14

250

667

674

1.27

15.0

d15

500

1116

1123

0.79

32.6

5

13

13

3.08

0.1

d17

10

23

23

4.35

0.3

d18

167

223

236

7.40

12.3

d19

250

310

336

9.32

22.8

d20

500

537

558

4.49

53.4

5

111

111

0.00

0.2

e2

10

214

214

2.62

0.5

e3

417

4013

4035

0.64

57.1

e4

625

5101

5118

0.35

85.2

e5

1250

8128

8130

0.11

171.4

5

73

73

2.04

0.2

e7

10

145

145

3.70

0.7

e8

417

2640

2661

0.98

76.2

e9

625

3604

3633

1.03

120.1

e10

1250

5600

5621

0.47

242.9

5

34

34

4.41

0.2

name d1

d6

2000

d11

5000

d16

e1

e6

e11

25000

2500

3125

5000

12500

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e12

10

67

67

1.49

0.6

e13

417

1280

1312

2.82

77.5

e14

625

1732

1756

1.59

139.4

e15

1250

2784

2794

0.69

265.2

5

15

15

6.67

0.3

e17

10

25

25

4.80

0.8

e18

417

568

613

8.87

94.1

e19

625

758

809

7.68

156.3

e20

1250

1342

1398

4.71

384.5

e16

62500

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Page 128 Table 7.2 Results for Euclidean instances. |V|

|E|

|X|

opt

∆avg

t(s)

783

2322

117

2992

1.5

4.4

f2

196

3826

0.7

7.5

f3

274

4401

1.2

8.9

117

2899

1.9

5.1

f5

196

3744

1.8

8.8

f6

274

4368

1.5

10.1

117

2892

1.0

28.9

f8

196

3742

1.4

18.2

f9

274

4368

1.0

19.5

117

2892

1.1

24.9

f11

196

3742

1.5

51.5

f12

274

4368

1.0

53.9

150

6509412

2.3

11.2

f14

250

8123458

1.3

16.8

f15

350

9804964

1.2

24.0

150

6316653

2.3

11.3

f17

250

7946357

0.7

19.5

f18

350

9675663

1.1

26.1

150

6297956

2.0

20.2

f20

250

7931250

0.6

41.8

f21

350

9673224

0.9

98.7

150

6297956

1.6

66.4

f23

250

7931250

0.7

99.1

f24

350

9673224

0.5

151.7

name f1

f4

7532

f7

28365

f10

f13

109895

1000

f16

2981

9699

f19

38080

f22

184597

|V|

|E|

|X|

opt

∆avg

t(s)

3795

11326

569

13266

13288.0

153.2

g11

949

14745

14772.5

276.4

g12

1328

15901

15960.3

367.2

569

13105

13117.8

237.6

g14

949

14628

14634.3

394.4

g15

1328

15716

15775.3

547.9

name g10

g13

52684

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g16

569

13063

13076.2

850.3

g17

949

14612

14629.7

1171.2

g18

1328

15685

15716.9

1770.7

1777

383245

383694.4

2475.5

g23

2962

480145

480879.8

4296.3

g24

4147

561050

561668.0

5740.1

2108

154479

154575.1

2698.5

g26

3513

198609

198667.3

4565.9

g27

4918

235609

235697.7

6177.6

2777

205697

205775.5

4761.3

g29

4628

273631

273712.7

7369.0

g30

6479

324805

324875.1

10338.7

g22

g25

g28

325093

11849

14051

18512

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42128

55510

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We can also handle even larger instances in an acceptable amount of time. Furthermore, we are able to deal with much larger instances than those studied in other local search approaches as presented in [Dow91], [KRSS93], and [Esb95], who all consider instances with at most 100 vertices. Moreover, our approach has limited memory requirements since only a compact representation of an instance is stored. Most other heuristic and exact approaches require space for storing all-pairs shortest paths, which makes it very hard to adapt them for handling large instances efficiently. References [AL96] Aarts E. and Lenstra J. (eds) (1996) Local Search in Combinatorial Optimization. Wiley, New York. [Bea89] Beasley J. (1989) An SST-based algorithm for the Steiner problem in graphs. Networks 19: 116. [Bea90] Beasley J. (1990) OR-library: distributing test problems by electronic mail. Journal of the Operational Research Society 41: 10691072. [CGR92] Chopra S., Gorres E., and Rao M. (1992) Solving the Steiner tree problem using branch and cut. ORSA Journal on Computing 12: 320335. [Che83] Chen N. (1983) New algorithms for Steiner tree on graphs. In Proc. of the IEEE Int. Symposium on Circuits and Systems, pages 12171219. [Dij59] Dijkstra E. (1959) A note on two problems in connection with graphs. Numer. Math. 1: 269271. [Dow91] Dowsland K. (1991) Hill-climbing, simulated annealing and the Steiner problem in graphs. Engineering Optimization 17: 91107. [Dui94] Duin C. (1994) Steiner's Problem in Graphs. PhD thesis, Amsterdam University. [DV93] Duin C. and Voß S. (1993) Steiner tree heuristics a survey. In Dyckhoff H., Derigs U., Salomon M., and Tijms H. (eds) Proc. 22nd annual meeting DGOR / NSOR. Springer, Berlin. [Esb95] Esbensen H. (1995) Computing near-optimal solutions to the Steiner problem in a graph using a genetic algorithm. Networks 26: 173185. [HRW92] Hwang F., Richards D., and Winter P. (1992) The Steiner Tree Problem, volume 53 of Annals of Discrete Mathematics. [KRSS93] Kapsalis A., Rayward-Smith V., and Smith G. (1993) Solving the graphical Steiner tree problem using genetic algorithms. Journal of the Operational Research Society 44(4): 397406. [OG91] Osborne L. and Gillett B. (1991) A comparison of two simulated annealing algorithms applied to the directed Steiner problem on networks. ORSA Journal on Computing 3: 213225. [Ree93] Reeves C. (ed) (1993) Modern Heuristic Techniques for Combinatorial Problems. Blackwell, London. [Rei91] Reinelt G. (1991) TSPLIB: A traveling salesman problem library. ORSA Journal on Computing 3(4): 376384. [TM80] Takahashi H. and Matsuyama A. (1980) An approximate solution for the Steiner tree problem in graphs. Math. Japonica 24(6): 573577. [Vos92] Voß S. (1992) Steiner's problem in graphs: heuristic methods. Discrete Applied Mathematics 40: 4572. file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_129.html（第 1／2 页）2009-7-7 16:57:13

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8 Local Search Strategies for the Vehicle Fleet Mix Problem Ibrahim H. Osman and Said Salhi Abstract The vehicle fleet mix (VFM) problem is the vehicle routing problem with additional practical extensions. The VFM problem involves a heterogeneous fleet of vehicles with their associated variable running and fixed purchasing/renting costs. The objective is to find the optimal fleet composition of vehicles and a set of feasible routes that minimize the total costs. In this paper, two techniques are proposed: a constructive heuristic and a tabu search metaheuristic. The constructive heuristic is an enhanced modification of the Salhi and Rand route perturbation procedure. The tabu metaheuristic is new and it uses a compound-moves neighbourhood with a special data structure. Computational results are reported on a set of 20 test problems from the literature. The proposed methods obtain new results that improve upon the best published results. 8.1 Introduction The vehicle fleet mix (VFM) problem is the vehicle routing problem with a heterogeneous fleet of vehicles. Each vehicle is characterized by the carrying capacity, maximum travel time, variable running cost and fixed purchasing/ renting cost. Each vehicle route originates and terminates at a central depot in order to service a set of customers with known demands. The total cost involved in the VFM problem consists of the fixed vehicle utilization costs as well as the variable routing and scheduling costs. The later costs include the distance and travel time costs in addition to the service (loading/unloading) time costs. The constraints of the VFM problem are: (i) each customer must be supplied by exactly one vehicle route; (ii) the total load carried by any vehicle must not exceed its maximum capacity; (iii) the total length of any route, which includes the inter-customer travel times and service times, must not exceed a pre-specified limit. The objective is to find the optimal mix of heterogeneous vehicles and the associated set of feasible routes that minimizes the total variable and fixed Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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costs. The VFM problem is practically more important than its counterpart, the vehicle routing problem (VRP). The VRP, however, has attracted much more research attention. The lack of VFM research is mainly due to the less obvious way of how to determine the best mix of vehicles which is a medium-term decision problem. The decision problem involves an important cost investment factor. If an unwise decision was made to purchase a fleet of vehicles, it would be difficult to change such a decision. In general, researchers have considered a homogeneous fleet of vehicles. Yet in real life, the appropriate fleet is, by no means, homogeneous and a heterogeneous vehicle fleet is likely to yield better results. Furthermore, the VFM problem still has the difficulty of the operational routing decision which is encountered in the daily operations of vehicles in the classical VRP. Hence, the VFM problem is computationally more complex than the VRP which is known to be NP-hard (Lenstra and Rinnooy Kan[LR81]). Consequently, an exhaustive search of the solution space is impossible for VFM problems of large size and there is no known algorithm that can solve the VFM problem to optimality in a polynomial time. The complexity of the VFM problem therefore necessitates the development of effective heuristics that are capable of providing high-quality approximate solutions within a reasonable amount of computational effort. Local search methods form a class of heuristics that proceed by examining some neighbourhoods of the current solution. The simplest type of local search method is the descent algorithm. It starts with any solution, neighbours are searched until an improved solution is found. A further search is then initiated from the improved solution and the algorithm continues until no further improvement is possible. The solution obtained may deviate significantly from the optimum. Tabu search (TS) is a metaheuristic which is superimposed on the descent algorithm to guide the search process to avoid getting trapped in bad local optima. TS allows the search to proceed to a neighbour even if it causes a deterioration in the objective function value. TS has been proven to be a very effective and successful metaheuristic for solving many combinatorial optimization problems. Particularly impressive results have been obtained for many vehicle routing problems: Eglese and Li[EL96], Gendreau, Hertz and Laporte[GHL94], Potvin, Kervahut, Garcia and Rousseau [PKGR96], Osman [Osm93], Rego and Roucairo [RR96], Salhi and Sari[SS95], Taillard[Tai93], Thangiah, Osman and Sun[TOS94] and Thangiah, Osman, Vinayagamoorthy and Sun[TOVS93]. For recent bibliographies, we refer to Osman and Laporte[OL96] on metaheuristics, to Laporte and Osman[LO95] on routing problems. For an overview and an introduction on metaheuristics, we refer to Osman[Osm95b] and Osman and Kelly[OK96b]. For metaheuristic books, we refer to: Aarts and Lenstra[AL96], Laporte and Osman[LO96], Osman and Kelly[OK96a], Rayward-Smith[RS95] and Reeves[Ree93]. Motivated by tabu search successes, in this paper, we present a tabu search algorithm for solving the mixed fleet vehicle routing problem. We design a special data structure to evaluate efficiently trial solutions generated by the 1-interchange (compound-moves) neighbourhood mechanism[Osm95a, Osm93, OC94]. Finally, we introduce some modifications and refinements to enhance the performance of the RPERT procedure proposed by Salhi and Rand [SR93] for the VFM problem. The paper is organized as follows. The vehicle fleet mix and its relevant literature are reviewed in Section 8.2. The modified RPERT construction heuristic is described in Section 8.3. The important features of the tabu search algorithm with its embedded

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data structures are discussed in Section 8.4. Computational comparisons on a set of 20 test problems with the best published methods in the literature are provided in Section 8.5. Finally, our conclusion and perspectives on future research are presented in Section 8.6. 8.2 The Vehicle Fleet Mix (VFM) Problem 8.2.1 Representation of the VFM Let '0' denote the depot with no service time and zero demand; the remaining notations are defined as follows: K = The set of different vehicle types, K = {1, . . ., kmax}. Fk = The vehicle fixed cost of type k ∈ K. Qk = The vehicle capacity of type k ∈ K. Tk = The maximum travel time for the vehicle capacity of type k ∈ K. v = The decision variable indicating the total number of mixed vehicles used in the final solution of the VFM problem. V = The set of desired vehicles of different types, V = {1, . . .,v}, V ⊆ K. N = The set of customers, N = {1, . . .,n} where n is the total number of customers. qi = The demand of customer for i ∈ N,

δi = The service time of customer for i ∈ N. dij = The distance between customers i and j, dij = dji∋∋i, j ∈ N ∪ {0}.

αk = The variable cost per unit of distance for a vehicle of type k. βk = The time factor per unit of distance for a vehicle of type k. Rp = The set of customers serviced by a vehicle p ∈ V.

σ = The function σ : V → K, where σ(p) indicates the smallest type of vehicles that can serve the customers in the set Rp. πp = The travelling salesman route which serves the set Rp∪ {0}, where πp(i) indicates the position of customer i in the route, πp. D(πp) = The total distance of the route, πp.

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T(πp) = The total travel time of the route, πp. C(πp) = The total variable and fixed cost of the route, πp. S = The feasible solution which is defined as S = {R1 . . .,Rv} Π = The set of all travelling salesman routes in S, Π = {π1, . . ., πv}. Our goal is to solve the following optimization problem:

such that

Constraint (8.2) ensures that each customer is supplied in one route. The set of constraints in (8.3) guarantees that the vehicle capacity is not exceeded. Equations in (8.4) represent the total sum of distance of each route, πp. Since the TSP is a hard problem by itself, we have used approximate procedures to estimate each route, πp. Equations (8.5) guarantee that the maximum travel time is not exceeded. Equations (8.6) represent the total cost per route including variable and fixed cost while Equation (8.1) is the total sum of costs in the solution to be minimized over all routes. 8.2.2 Literature Review of the VFM There are few published works on the VFM problems. In the early 1980s, Golden, Assad, Levy and Gheysens [GALG84] were among the first to address this problem. They presented a mixed integer formulation, generated lower bounds on the optimal solution and described various heuristics based on the savings method of Clarke and Wright [CW64] and the route-first cluster-second algorithm. The later heuristic starts with a giant route that visits all customers but not the depot. This giant route is further improved by the Or-OPT exchange procedure before it is then partitioned into feasible routes. This heuristic is then repeated using a sample of five initial routes, hence leading to its name (MGT +OrOPT)5. Excellent results were reported using the (MGT+OrOPT)5 heuristic on problems ranging from 12 to 100 customers. Gheysens, Golden and Assad[GGA84] developed a cluster-first route-second heuristic. In the

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first stage, a heterogeneous fleet mix was obtained using a lower bound procedure developed in Golden, Assad, Levy and Gheysens[GALG84]. In the second stage, they adopted a generalized assignment based heuristic to solve the VRP problem using the vehicle fleet mix obtained in the first stage. This heuristic was also run for five different times using the fleet compositions associated with each of the five best lower bounds and it was denoted by LB5+VRP. Gheysens, Golden and Assad [GGA84] presented another mixed integer formulation for the VFM problem with time windows and performed a computational comparison of heuristics developed in Gheysens, Golden and Assad[GGA86]. Ferland and Michelon[FM88] showed that an exact method for the VRP with time windows and a homogeneous fleet size can be extended to the VRP with the heterogeneous fleet size. The proposed exact method was based on a column generation approach with relaxed time windows but no computational results were reported. In the 1990s, Desrochers and Verhoog[DV91] presented a new savings heuristic called MBSA which was based on successive route fusions. At each iteration, the best fusion was selected by solving a weighted matching problem. The MBSA heuristic was implemented considering several weights in the matching problem such as the total savings in routing costs, savings in fixed costs, or opportunity savings associated with each feasible combination. Computational results were provided for a number of benchmark problems in order to compare the algorithm's performance to that of other methods. Ronen[Ron92] presented a mixed integer formulation to assign trips to mix fleet vehicles with the minimal cost. A two-step heuristic was proposed based on the assignment of trips to vehicles first, then slide-and-switch of trips between vehicles second. The heuristic provided results within 1% of the linear relaxation bound. Salhi, Sari, Saidi and Touati[SSST92] presented a mixed integer formulation for the VFM problem with fixed and variable running costs. They assessed the effect of neglecting the variable running cost on the solution of different procedures. They modified the route-first cluster-second and the savings algorithms in order to take into account both variable running and fixed costs. This mixed integer formulation is similar to that of Gheysens, Golden and Assad [GGA84] which assumes the same value of the unit running cost across the different vehicle types. Salhi and Rand [SR93] presented a review on the VFM and also developed an interactive route perturbation procedure (RPERT) consisting of a series of refinement modules. New best-known solutions were reported by the RPERT procedure for some standard test problems. Finally, the importance of the strategic decisions on the fleet make-up and the vehicle number of each types was discussed in the context of distribution system design by Bookbinder and Reece[BR88] and Beaujon and Turnquist[BT91]. 8.3 Modified RPERT Procedure (MRPERT) Salhi and Rand[SR93] proposed a successful interactive procedure denoted by RPERT. The RPERT procedure constructs a VFM solution using a series of seven phases. Each phase uses one perturbation module in order to improve upon the solution of its predecessor. If the new solution is better and feasible in terms of vehicle capacities and maximum travel times, it is then accepted as the current solution and passed into the next phase for further improvements. Otherwise the previous solution is retained and

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the search continues until the RPERT procedure is terminated at the end of the last phase. It was noticed that in some modules, the routing cost was increased but with a greater decrease in the fixed cost. The aim was to explore the compromise between having a higher total operating cost and a lower total fixed cost, while achieving a maximum utilization of the whole fleet. We shall briefly describe these modules in the same order as they were implemented in the RPERT procedure. For more details, the reader should refer to Salhi and Rand[SR93]. RPERT begins with the savings module developed for the vehicle routing problem in Salhi and Rand[SR87]. The savings module constructs an initial set of routes using a given vehicle capacity. This savings module uses a 'shape' parameter to modify the savings function used in [Daw86] to compute the savings in distance if pairs of customers were served in a single route. The shape parameter value was varied between 0 and 2 to generate a set of different initial routes. A matching module was then used to determine for each route the smallest type of vehicles that can serve the customers on it. This module was followed by a reduction module which was attempted, whenever possible, to eliminate a given route by relocating its customers and inserting them in other routes or merging the route with another to build a larger single one. A sharing module was also implemented. It attempted to split a given route into few smaller routes to achieve a cost reduction, if possible. Finally, the RPERT procedure improved the cost of routes by a swapping module which exchanged customers between routes. Let us define a cycle of search to be a single execution of the above sequence of the RPERT modules. The RPERT procedure was only performed using one cycle and was terminated at the end of its last module. The best stored solution at the end of this cycle was called the RPERT final solution. In RPERT, a rigid restriction was imposed on the allowed type of moves in order to speed up the procedure. A move is a transition from one solution to another neighbouring solution. More precise, a move was not allowed to be performed if it resulted in a utilization of larger-sized vehicles than the currently used ones when implementing the reallocation and swapping modules, i.e, only feasible moves in terms of capacity and maximum travel were allowed and infeasible moves were prohibited. To alleviate these restrictions in this paper, we introduce the following two modifications. First, to enlarge the neighbourhood size (set of available moves), the feasibility restriction is relaxed, i.e., a move which leads to a utilization of a larger-sized vehicle was allowed. Second, we also allow the RPERT procedure to restart for another cycle starting from the best solution obtained at the end of previous cycle. The search continues for a number of cycles and it is terminated when a cycle is performed without finding any improvements. The reason is that RPERT performed only one cycle before terminating its search. However, if an improvement happened in any (say sharing) modules it may be possible to improve this solution by using an earlier (say reallocate or reduction) module. These further improvements would not be detected unless the RPERT procedure is restarted. The restart process and the allowance of infeasible moves were not implemented in RPERT. These modifications form the basis of the new procedure which is denoted by MRPERT.

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8.4 Tabu Search Tabu search (TS) is a relatively novel technique for solving hard combinatorial optimization problems. Tabu search ideas were proposed by Glover[Glo86]. It is based on selected concepts from artificial intelligence. Its goal is to emulate intelligent uses of memory for exploiting historical information. Tabu search uses memory structures to record in a tabu list, TABL, attributes of the recently accepted solutions, i.e. the changes occurred when performing a move from S to the best solution S′ in its neighbourhood N(S). Moreover, attributes can consist of customers removed, added or repositioned by the moves executed. Sometime, attributes can be strategically combined to create other attributes. Selected attributes, that are stored in TABL, are designated tabu-active. Solutions, that contain tabu-active elements, are designated tabu. The duration that an attribute remains on the tabu list and hence remains tabu-active, is called the tabulist size which is denoted by t. The approach of using attributes to identify the tabu status of future moves is very easy to implement and requires less storage than the approach of using the actual solutions which is very expensive to store and difficult to check and maintain. The attributes-based approach, however, may wrongly forbid moves leading to unvisited solutions that may be attractive. It is therefore necessary to override the tabu status of such moves, if an aspiration level criterion is satisfied, in order to correct this wrong diagnosis. A move is considered admissible if it is not a tabu move or its tabu status is overridden by the aspiration criterion. In general, TS uses an aggressive guiding strategy to direct any local search procedure to carry out exploration of the solution space to avoid getting trapped in local optima. When a local optimum is encountered, the aggressive strategy moves to the best solution S′ in the whole neighbourhood N(S) even if it may cause a deterioration in the objective function value. For situations, where the neighbourhood is large or its elements are expensive to evaluate, candidate list strategies are used to help restrict the number of solutions examined on a given iteration. A candidate list of solutions N′ (S) is generated either randomly or strategically using memory structures[Osm93] in order to identify the best move exactly, or heuristically. Finally, the rule for execution is generally expressed by a stopping criterion. Several stopping criteria can be used to terminate the search: either a pre-specified limit on the number of iterations or on the number of iterations since the last improvement was found. Since TS embeds heuristic rules and different strategies to guide the search, it becomes a metastrategy algorithm or simply a metaheuristic. The basic TS procedure is sketched in Figure 8.1.

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Tabu Search Procedure (TSVFM). Step 1: initialization: Generate an initial solution, S, for the VFM problem. Set the best current solution Sbest = S. Evaluate all the moves in the neighbourhood N(S). Set values for: t, the tabu list size; TABL, the tabu list; DS, the special Data Structure for attributes; MAXBEST, the maximum number of iterations after Sbest. Set iteration counters: nbiter=0 (current iteration number) and bestiter=0 (iteration number of the best solution). Step 2: Candidate list of solutions: Determine strategically using DS, the exact set of the candidate list of best moves in the neighbourhood, i.e., N′(S) ⊆ N(S). Update DS after each iteration. Step 3: Selection Strategy: Choose the best admissible solution S′∈ N′(S). Set S = S′ and nbiter= nbiter +1. If C(S′) < C(Sbest), then set Sbest = S′ and bestiter = nbiter. Record in TABL the changed attributes. Step 4: Stopping criterion: If { (nbiter bestiter) > MAXBEST }, then Stop, Else go to Step 1. Figure 8.1 A basic TS procedure. In the following, we shall give a description of the basic components of the above TS procedure and practical implementation details for solving the vehicle fleet mix problem. For recent details on tabu search, we refer to by Glover [Glo95a]. Initial Solution The initial solution, S, can be generated using any VFM or VRP heuristic. In this study, we have used the VRP heuristic of Salhi and Rand[SR87] to generate the initial set of routes, II. The simple matching module of Salhi and Rand[SR93] is then used to determine for each route, πp, the type of the vehicle, σ(p), that can serve the set of customers Rp. The total cost C(S) of this VFM solution is computed using equations (8.1) and (8.6). An empty route with no customer assigned to it is added to the set of routes with zero costs. The reason for this empty route is to allow infeasible moves to be considered. file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_138.html（第 1／2 页）2009-7-7 16:57:18

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Neighbourhood Generation Mechanism The neighbourhood generation mechanism describes how a given VFM solution S = {R1, . . ., Rp, . . ., Rv} can be altered to generate another solution S′ in N(S), the neighbourhood of S. Here, we adapt the λ-interchange mechanism which was introduced in Osman[Osm93] and successfully used for many problems, e.g., Chiang and Russell[CR96], Hasan and Osman[HO95], Osman[Osm95a], Thangiah, Osman and Sun[TOVS93, TOS94]. It has the property that λoptimal solutions are µ-optimal solutions for any integer µ < λ, Osman and Christofides[OC94]. Given a pair of route sets Rp and Rq in S, a 1-interchange generation mechanism invokes two processes to generate neighbouring solutions: a shift process which is represented by the (0,1), (1,0) operators and an interchange process which is represented by the (1,1) operator. Note that the reallocation and swapping procedures in Salhi and Rand[SR93] have some similarities with the shift and the interchange processes. The (1,0) shift process denotes the reallocation of one customer (say i) from the set of customers, Rp, to another set of customers, Rq. This shift process, if performed, and . The (0,1) shift process would result in a new pairs of route sets: denotes a shift in the opposite direction, i.e., a shift of j from Rq to Rp and the new route sets become and . The (1,1) interchange process combines the two shift processes simultaneously to generate compound moves to reach solutions that can not be generated by consecutively applying any of the shift process alone due to capacity/time restrictions. Each customer i ∈ Rp is systematically attempted for interchange with every other customer j ∈ Rq to get two new route sets

and

. The advantage of the (1,0) and (0,1) shift processes is that an empty vehicle may be produced or a single customer route may be formed. In both cases, savings in fixed costs may be obtained. To allow a single route to be formed using the shift operators, at least one empty set of customers, , at any iteration needs to be available for usage among the other non-empty route sets. If a decrease in the vehicle number has occurred, we may have two empty route sets, one of which is redundant and can be deleted. Another advantage of the empty set is that it allows infeasible moves to be considered when searching the neighbours. Here, we have a non-monotonic increase/decrease in the number of vehicles as well as a non-monotonic search of feasible/infeasible regions. For more details on non-monotonic search strategies, we refer to Glover [Glo95b]. Finally, the neighbourhood, N(S), is defined to be the set of all solutions that can be generated by considering a total number of v(v + 1)/2 pairs which is equal to v(v 1)/2 + v different pairs of route sets. The first term, v(v + 1)/2, determines the number of pairs of route sets (Rp, Rq) where 1 ≤ p < q ≤ v, involving v non-empty route sets, while the , considered by the (1,0) process operating on second term, v, is the number of pairs of route sets, the pair of non-empty and empty route sets. The customers in a given pairs of route sets are searched sequentially and systematically for improved solutions by the shift and interchange processes of the 1-interchange mechanism. Evaluation of the Cost of a Move Let 1IM denote a 1-interchange move from a current solution S to one of its neighbours S′∈ N(S) by either the shift or the interchange processes, i.e., 1IM (S) = S′. Given a pair of route sets Rp and Rq, the

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Page 140 cost of the move, 1IM depends on the type of operators involved. There are three type of operators (1,0), (0,1) and (1,1). (I) The cost of (1,0)-based move. The (1,0)-based move involves shifting a customer i from the route πp of Rp and inserting it in the best possible position on arcs (least cost insertion) of the approximate TSP route πq of Rq. The (1,0) move would and . Let a and b be the predecessor and the result in two route sets successor of i in the old TSP route πp of Rp. Let us assume c and e to be the predecessor and the successor in of the new TSP route computed as follows:

where i is to be best inserted. The costs of the new TSP routes are then

where p′ and q′ are the vehicles that serve the new route sets

and

, respectively.

(II) The cost of (0,1)-based move. The (0,1)-based move involves shifting a customer j ∈ Rq to another route set Rp. This move is similar to the one in (I) except that Rp and Rq are in reversed order. Hence the cost of the (0,1) move is simply obtained by applying the (1,0) process on the route sets Rq, Rp in a similar way. Let (f,g) and (l,m) be the (predecessor, successor) of j in the routes πp and as follows:

respectively. Equations (8.7) to (8.11) can then be similarly rewritten

where p′ and q′ are the vehicles that serve the new route sets

and

, respectively.

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(III) The cost of (1,1)-based move. The (1,1)-based move involves an exchange of i ∈ Rp with j ∈

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Rq. This move can be seen as a combination of two simultaneous shifts: a shift of i into the route set by the (1,0) operator and another shift of j into the route set by the (0,1) operator. Note that, the change in the objective function value ∆(1,1) due to the (1,1) operator can be easily evaluated from the stored data used for computing equations (8.7) to (8.11) and (8.12) to (8.16) bearing in mind some special cases that need careful examinations. For example, let us assume that the best insertion of customer i inside the route

is recorded to be between two customers (c and e) in the old route

is only the πq, where j was included. If c and e are different from j, then the new insertion cost of i into comparison between the old insertion cost of i in πq with that along the newly created arc joining f to g (predecessor and successor of j) due to the removal of j. However, if either c or e is identical to j, then the insertion cost of i into

needs to be evaluated in the same way as indicated in (I). Similar analysis needs to

be conducted to find the insertion cost of j into of the (1,1) move, ∆(1,1).

. Then the computed costs will be used to evaluate the cost

The best 1-interchange move involving either i or j or both is determined from the values of the 1IM costs obtained by using the (1,0), (0,1) or (1,1) operators. Its value, δij, is the most negative or the least positive change in costs and it is computed as:

The best 1-interchange move, 1IMpq, involving the pair of routes (Rp, Rq) is the 1IM move which gives the best change in costs over all δij obtained by searching systematically all possible shifts or exchanges of customers between Rp and Rq. Its value, ∆pq, is computed as:

The change in the objective function values, ∆ = C(S′) C(S), is then set equal to ∆pq, if the best move, 1IMpq, is performed using the set of routes (Rp,Rq). The 1IMpq move can be identified by its attributes: the move value, ∆pq; the customer indices, i and j; the operator type; the route indices and the new/old predecessors and new/old successors. These attributes can be stored in a special data structure, DS, to be retrieved fast and save computational effort. Data Structures for the Candidate List of Moves Data structures (DS) for strategically defining the set of best moves were introduced for the VRP in Osman[Osm93] and resulted in big savings of more than a half of the required computation time. Osman's DS stored only the set of best moves between all pairs of routes. However, it did not make use of any computed information generated in early evaluations in relation to the best insertion/removal positions and their associated

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costs. These attributes could be shared among many moves. Hence, storing them would generate further savings in computational effort. In this paper, we extend the previous data structure to include the above mentioned information. TS selects the best move, 1IMbest, from S to its best neighbours S′ requiring the whole neighbourhood N(S) to be reevaluated after each iteration. The neighbourhood N(S) consists of v(v + 1)/2 pairs of routes which is very expensive to compute for large-sized problems. The set of best candidate moves N′(S) in N(S), which consists of the best 1IMpq moves for all 1 ≤ p < q ≤ v, can strategically be determined and updated using two levels of memory structures. The aim is to identify the changes and intelligently update the candidate list of 1IMpq moves with the minimal effort without scarifying the quality of the 1IMbest move. At each iteration, we are able to find exactly the best move rather than an inferior one which can be obtained by random sampling of the neighbourhood. Hence, the importance of the DS structure can not be over-emphasized to achieve efficient TS implementations. There are two levels of memory structures. The first level stores the costs of all 1-interchange moves, 1IM, in the of serving a route, πp, without a customer, i, as evaluated in (8.7), neighbourhood of S. For instance, the cost can be stored in a cell COST-WITHOUT (p, i) of a v × n matrix. This value will remain unchanged once computed, during all the (1,0) and (0,1) processes of evaluating the cost of shifting customer i into all other different routes, for at least the current iteration. The COST-WITHOUT matrix can also help to evaluate quickly the cost of (1,1) move with a little extra effort as explained earlier. The memory concept is not only used for the move costs but can be extended to store other useful information on restrictions.

, such as its current load, its type, and its status in terms of capacity and time

The second level of memory structure is mainly used to identify the set of all the best v(v + 1)/2 moves, 1IMpq, obtained when all v(v+1)/2 pairs of route sets, (Rp,Rq) for all p < q ∈ V, are considered by the 1-interchange mechanism to generate the neighbourhood N(S) after each iteration. This set of best 1IMpq moves defines the list of elite candidate solutions, N′(S), in the neighbourhood. In Step 2 of the TS procedure, we select the best move, 1IMbest, from the set of best moves and apply it to the current solution S to obtain the next solution S′∈ N′(S). At this stage, S′ is exactly the best solution in N′(S) and in N(S). Once, the best move, 1IMbest, is performed S′ becomes the current solution for the next iteration. It can be seen after performing the best move that the change between S and S′ is only in one pair of routes, say (Rp, Rq), and the other routes remain intact. Consequently, 2 × (v 1) pairs of route sets, (Rp, Rm) and (Rm, Rq) (for all m ∈ V, m ≠ p, m ≠ q), are necessary to be re-evaluated in order to update the candidate list of best neighbouring solutions. Having defined the set of elite moves and solutions, we shall describe our data structure (DS) to update the candidate list of moves and N′(S) with a small number of re-evaluations. The data structure DS consists of two matrices. DSTABLE takes the form of a v × v matrix, whereas DSINF is a {v(v + 1)/2} × A matrix where 'A' is the number of attributes required to identify a move in DS. Each entry of the top triangular part of DSTABLE is used to store the best change in the objective value associated with one of the v(v+1)/2 pairs of routes (Rp, Rq). For example, DSTABLE(p, q) stores ∆pq associated with the best 1IMpq between the pair (Rp, Rq), or a large positive value if no such move exists. The lower triangular part DSTABLE(q, p) is used to

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store a positional index l associated with the pair (Rp, Rq) in the set of possible pair combinations {1, . . ., v(v + 1)/2}. Note that the determination of the index needs special care as the number of vehicles may increase during the computation. Increasing the vehicle number option was not considered in Osman's data structure whereas it is of a primary importance in the VFM. In this study, we have used a value of six for 'A' indicating that six attributes are to be saved in DSINF. For instance, the best move, 1IMpq, between Rp and Rq is identified by its ∆pq value in DSTABLE, the index l associated with the (Rp, Rq) pair that is used to identify other attributes in DSINF as follows: DSINF(l, 1) = l, DSINF(l, 2) = p, DSINF(l, 3) = q, DSINF(l, 4) = i, DSINF(l, 5) = j, DSINF(l, 6) = ∆pq. Note that both DSINF(l, 1) and DSINF(l, 6) could be ignored as they are already recorded in DSTABLE, but are introduced for convenience. If ∆pq, is associated with (1,0) or (0,1) operators, then we replace the values of i or j by in DSINF so that we can identify easily the type of the operators applied to this move. The columns in DSINF can be increased easily to store more information, if necessary. The management of DS occurs in Step 1 of the TS procedure. At the first iteration, all the moves in the neighbourhood N (S) are evaluated and the best moves are stored in the appropriate matrices. When the TS procedure returns to this step, the DSTABLE matrix is updated considering only the 2 × (v 1) pairs of routes that may contain improved moves due to performing the previous move. After this quick update, DSTABLE is scanned to identify the best move, 1IMbest, in N′ (S) for the next iteration. 1IMbest is identified by comparing the ∆pq stored values of the 1IMpq moves. The attributes and , are further checked of the identified move are then retrieved from DSINF. The corresponding new routes, for possible improvements applying the 2-OPT or 3-OPT post optimization procedures of Lin[Lin65]. After the post optimization procedure is applied, this best move, 1IMbest, is performed to update the current solution. The tabu list is also updated before the search continues for another iteration. The selection of the best moves based on the above data structure has an advantage over the random sampling approach as it is performed with the minimal computational effort. Random sampling is normally used to reduce the computational effort as the systematic search is expensive to use without a data structure. Consequently, good moves may be missed and can not be detected unless a systematic search is used. Tabu List TABL takes the form of a (v + 1) × n matrix (v rows: one for each route set Rp, one for the empty route; n columns: one per customer). In the tabu list, we store some attributes of the best 1-interchange move, 1IMbest which was accepted to transform S to S′∈ N (S). To prevent reversing the accepted move from returning back to S, its attributes are stored in TABL for t iterations, in the hope that we would be in a different region. The reverse move can be regenerated by returning immediately i to Rp and j to Rq. Let us assume that the current iteration number is nbiter and the best 1interchange move involves the pair of customers (i, j) in (Rp, Rq). The return to previous solutions during the next t iterations can be prevented by storing in TABL the future value (nbiter + t) in both TABL(i, p) and TABL(j, q) entries. After nbiter+t iterations, customers i and j are allowed to return to Rp and Rq, respectively.

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Tabu Condition One way to prevent cycling is to store exactly all previously accepted solutions, but this approach can be computationally very expensive for checking and maintaining solutions. Another approach is to use hashing functions (Hasan and Osman[HO95]) to represent these solutions. We found in many implementations that tabu conditions based on move attributes provided good results. Therefore, we did not try other types of tabu conditions and the reader may refer to Osman[Osm95a] for a comprehensive computational study on the effect of different types of tabu conditions on the quality of solution. In this paper, we implemented one type of tabu condition. A 1-interchange move, 1IM, is considered tabu if i is returned to Rp and j is returned to Rq. At iteration nbiter, we can check the status of a move 1IM involving the pairs (i, j) and (Rp, Rq) using the simple test in equation (8.19) and the information in TABL as follows. A move is considered tabu if

TABL is initialized with zeros so that all moves are not considered tabu by the test. Tabu List Size Tabu list size seems to be related to tabu conditions, selection strategy of neighbours and problem characteristics, such as the number of customers, the number of vehicles and the ratio of the total required demands to the available capacities. Osman[Osm93] established such linkages and used a statistically derived formula to find an estimate of the tabu list size, t. Then, in a systematic way, the tabu list sizes were varied every 2 × t iterations over three different values (±10% centred at the estimate). In this paper, a simpler way is used with a static value of t such as for p varying between 2 and 7 to watch the best range between these extremes. The aim is to identify the value of t for other problems. A small value may lead to cycling with poor results while a large value may lead to deterioration in solution quality caused by forbidding too many moves. Aspiration Criterion Tabu conditions are based on stored attributes of moves; it may be possible to forbid wrongly unvisited solutions which share these attributes. Aspiration criteria are tests to correct such prevention. We use the following aspiration criterion. If 1IM is a tabu move but gives a new solution which is better than the best found so far, then we drop its tabu status. Therefore, a move from S to S′ is considered admissible, if it is a non-tabu move, or a tabu move which passed an aspiration level criterion, C(S′) < C(Sbest). Stopping Criterion Our TS procedure is terminated after a pre-specified number of iterations MAXBEST= 5 × n is performed after the best iteration number, bestiter, at which the best solution was found without finding any improvement. 8.5 Computational Experience It is a common practice to compare the performance of a heuristic with that of existing techniques on a set of benchmarks (test problems) in terms of solution quality and

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computer requirements. Other measures such as simplicity, flexibility, ease of control, interaction and friendliness are also of interest but are not considered here. Test Problems A set of 20 test problems are used. These problems were proposed by Golden, Assad, Levy and Gheysens[HN93]. They vary in size from 12 to 100 customers, there is no restrictions on maximum travel time (Tk = ∞, for all k ∈ K) nor on the number of vehicles. There are at least three types of vehicles and can be up to six for some problem instances, i.e., k varies between 3 and 6. The unit running cost, αk, and βk are set to a constant equal to one (αk = 1, βk = 1, for all k ∈ K) to be able to compare with published results which do not consider these two factors. However, the fixed cost, Fk, increases with the vehicle capacities. Quality of Solutions A common measure to decide on the quality of a heuristic solution is the relative percentage deviation from the optimal solution. However, if an optimal solution can always be easily obtained, we do not need to use a heuristic. Due to the difficulty of finding an optimal value, either a good lower bound from a linear relaxation of a mixed integer formulation or the best known solution can be used to replace the optimal value in the analysis. For each test problem, the relative percentage deviation over the best known solution is computed as follows:

In this study, we also report the actual solution, the average relative percentage and standard deviations, and the number of times a particular heuristic found the best solution. In Table 8.1, the actual solutions for our heuristics are given. The first and second columns give the problem number and the number of customers, respectively. The results of RPERT are in Salhi and Rand[SR93], our results of the MRPERT (modified RPERT) and the TSVFM (tabu search) with their respective CPU computation time in seconds on VAX 4500 are next in the table. The 'Old Best' are the cost of the best known solutions in the literature. These results can be found in either Golden, Assad, Levy and Gheysens[GALG84] or Gheysens, Golden and Assad[GGA84], Desrochers and Verhoog[DV91], and Salhi and Rand[SR93]. The final column contains the 'New Best' known solutions adjusted to reflect our findings. The new best solutions are indicated in bold while the references of the old best solutions are put between brackets [.]. It can be seen from Table 8.1 that we have found seventeen new or equal best known solutions for the twenty test problems using either the TSVFM or MRPERT heuristics. In fact, twelve out of sixteen are new best known solutions and the other five problems are equal to the published ones. Looking at the performance of each individual heuristic, we notice that the tabu search heuristic, TSVFM, has obtained fifteen best known values out of which ten are new best results. The modified heuristic, MRPERT, has obtained six best known solutions out of which two are new best results. MRPERT provides an improvement over RPERT in thirteen problems and is equal in the remaining instances.

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Page 146 Table 8.1 Comparison of the best known solutions No.

Size

RPERT

MRPERT

CPU TSVFM

CPU Old Best

New Best

1

12

614

606

0.3

602

3

602[TL92]

602 [*]

2

12

722

722

0.2

722

2

722[GALG84]

722 [*]

3

20

1003

971.95

0.7

971.24

5

965[HO95]

965

4

20

6447

6447.80

0.2

6445.10

6

6446[GALG84]

6445 [*]

5

20

1015

1015.13

0.2

1009.15

5

1013[GALG84]

1009 [*]

6

20

6516

6516.56

0.2

6516.56

4

6516[SR93]

6516 [*]

7

30

7402

7377

4.6

7310

15

7298[GALG84]

7298

8

30

2367

2352

3.4

2348

17

2349[GALG84]

2348 [*]

9

30

2209

2209

0.8

2209

14

2209[SR93]

2209 [*]

10

30

2377

2377

0.6

2363

14

2368[GALG84]

2363 [*]

11

30

4819

4787

0.4

4755

19

4763[GALG84]

4755 [*]

12

30

4092

4092

0.3

4092

10

4092[SR93]

4092 [*]

13

50

2493

2462.01

7.8

2471.07

62

2437[GALG84]

2437

14

50

9153

9141.69

9.3

9125.65

71

9132[GALG84]

9125 [*]

15

50

2623

2600.31

2.8

2606.72

46

2621[TL92]

2600 [*]

16

50

2765

2745.04

1.2

2745.01

35

2765[SR93]

2745 [*]

17

75

1767

1766.81

6.3

1762.05

85

1767[SR93]

1762 [*]

18

75

2439

2439.40

4.5

2412.56

116

2432[GALG84]

2412 [*]

19

100

8751

8704.20

8.1

8685.71

289

8700[TL92]

8685 [*]

20

100

4187

4166.03

61.1

4188.73

306

4187[SR93]

4166 [*]

[X]:

x indicates the reference in which the best known solution is reported and, * refers to this paper.

RPERT:

Interactive procedure of Salhi and Rand[SR93].

MRPERT:

The modified RPERT in this paper.

TSVFM:

The Tabu search procedure.

CPU:

The CPU time in seconds on VAX 4500.

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Page 147 Table 8.2 The relative percentage deviation above the best known solution. No

Size

(MGT- OrOPT)5

LB5+ VRP

MBSA RPERT MRPERT TSVFM OLBBEST

1

12

3.32

2.65

0.00

1.99

0.66

0.00

0.00

2

12

0.00

5

0.00

0.00

0.00

0.00

0.00

3

20

0.10

0.31

0.10

3.93

0.72

0.62

0.00

4

20

7.52

0.09

1.70

0.03

0.03

0.00

0.01

5

20

0.39

2.08

2.27

0.59

0.59

0.00

0.39

6

20

7.02

0.03

0.01

0.00

0.00

0.00

0.00

7

30

1.24

0.76

1.69

1.42

1.08

0.16

0.00

8

30

0.80

0.59

0.21

0.80

0.17

0.00

0.04

9

30

0.49

2.39

1.26

0.00

0.00

0.00

0.00

10

30

0.29

1.05

2.28

0.59

0.59

0.00

0.21

11

30

0.16

0.69

2.56

1.34

0.67

0.00

0.16

12

30

1.07

0.97

3.39

0.00

0.00

0.00

0.00

13

50

0.04

-

0.28

2.29

1.02

1.39

0.00

14

50

0.06

0.32

0.06

0.29

0.17

0.00

0.06

15

50

1.53

0.80

0.80

0.88

0.00

0.23

0.08

16

50

2.80

-

2.33

0.72

0.14

0.00

0.72

17

75

1.19

-

6.52

0.28

0.28

0.00

0.28

18

75

0.82

-

3.10

1.11

1.11

0.00

0.82

19

100

0.41

-

0.17

0.75

0.21

0.00

0.17

20

100

0.69

-

1.99

0.50

0.00

0.55

0.50

Average deviation 1.50

0.91

1.54

0.88

0.36

0.14

0.21

Standard deviation 2.10

0.83

1.59

0.94

0.39

0.35

0.28

# best solutions

1

1

2

4

7

14

8

(MGT-OrOPT)5:

Route first-cluster second algorithm of Golden et al. [GALG84].

LB5+VRP:

Cluster First-route Second algorithm of Gheysens et al. [GGA84].

MBSA:

Matching based savings algorithms of Desrochers and Verhoog[DV91].

Bold:

shows the worst instance performance of each algorithm and the best average over all problems.

Others:

As explained in Table 8.1.

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In Table 8.2, we have reported the relative percentage deviation over the new best known solutions for most of the stateof-the-art algorithms in the literature. We have included those heuristics which have produced at least one of the best known solutions. The first row contains the headers of the considered heuristics: (MGT-OrOPT)5 is the route firstcluster second heuristic based on route partitioning approach by Golden, Assad, Levy and Gheysens[GALG84]; LB5 +VRP is the cluster first-route second heuristic based on the generalized assignment approach by Gheysens, Golden and Assad[GGA84]; MBSA is the matching based savings heuristic by Desrochers and Verhoog[DV91] and those remaining are our own algorithms. It should be noted that the relative percentage deviations for the MBSA column are the best relative percentage deviations from several variants (CM, OOM, ROM, ROM-γ, ROM-ρ) of the matching based savings heuristic. Desrochers and Verhoog[TL92] did not report the objective solution values, we had to estimate these results from their best percentage deviations in order to compute the data in the MBSA column. It can be seen from Table 8.2 that TSVFM has the best performance as it produces the largest number of best known solutions and the smallest average relative percentage deviation of 0.14%. The MRPERT, however, produces the second largest number of best solutions, has the second average percentage deviation and the least worst percentage deviation value. These best average and worst relative percentage deviations for each heuristic are shown in bold in Table 8.2. The statistical analysis demonstrates the robustness of our heuristics when compared to the best available algorithms in the literature. Computational Effort The computer programs are coded in Fortran 77 and run on a VAX 4500 computer. In the previous section, we have shown that TSVFM is the best heuristic; however, there is a cost for a such good performance. In Table 8.1, the CPU time for both TSVFM and MRPERT heuristics are reported and the average CPU time over all the twenty instances for TSVFM and MRPERT are 56.2 and 5.7 CPU seconds respectively. Comments In this section, we discuss observations and issues related to our experimental results. In Table 8.1, we observe that MRPERT has produced better solutions than TSVFM in three problems, namely: 13, 15 and 20. We attempted to see if any link may exist between this failure and the existence of many vehicle types since problem 13 has six different vehicle types. We are not yet able to justify the link since TSVFM has produced results better than or equal to MRPERT for other problems of similar types. Further, both methods have failed to find the best known solutions for three problems. The failure may be attributed to the combinatorial structure of these problems. In our view this observation may need a closer look. A further study is needed to find a concrete answer for this failure. The second observation is that there is a good range for t values which vary with the size of problems. Figure 8.2 shows the average relative and standard percentage deviations of the best solutions for different values for the tabu list sizes, for p = 2 to 7, from the best known solutions for all the twenty test problems. From the figure, it can be seen that the performance of tabu search depends on the values of t. A good range for t values seem to lie between

and

. Outside this range t is

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either too small or too big. In the former case, cycling would occur and relatively bad solutions would be generated. In the later case, many moves are prevented leading to unvisited solutions that may be attractive and longer computation time is needed to find good solutions. It was found that when using the right interval of the tabu list size value, the total CPU time required by the TS algorithm was reduced by a half of that needed for other values of p. To derive a better values with p = 3, 4 and 5 and let t takes each range for tabu list size, we could generate a random permutation of value in this permutation. Then t can be kept at each value, say T, for 2 × T iterations before it can be changed to take the other values. Varying tabu list size values has been shown to be effective in Taillard[Tai93] and Osman[Osm93]. Other recent approaches based on a reverse elimination method or the reactive tabu method can be used and merit further investigations. These approaches are reviewed in Battiti[Bat96] and Voß[Vos96].

Figure 8.2 Effect of tabu list size on the quality of solution. The initial starting solution was generated by solving a VRP problem using one vehicle type at a time by Salhi and Rand VRP's heuristic. The best VRP solution is then used to start both MRPERT and TSVFM for further improvements. No other starting solutions were considered for convenience, simplicity and consistency. It may be possible to use other VFM heuristics and lower bounds on the fleet composition

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Page 150 to generate good initial VFM solution, and then feed them into TSVFM or RPERT heuristics to obtain a more refined solution with probably less computational effort. The fleet compositions, which are used by the Salhi and Rand's VRP heuristic together with its saving multipliers to generate initial solutions, are given in Table 8.3. The best fleet compositions for each problem which are obtained either by MRPERT or TSVFM are also shown in Table 8.3. It should be mentioned that since MRPERT is an iterative procedure and depends on the starting solution, changing the savings multipliers and the vehicle types may lead to different initial solutions. Table 8.3 Details of initial and final results for both TSVM and MRPERT heuristics. No Size

VRP (s, V C)

MRPERTf

VRP (s, V C)

TSVFMf Bestiter

1

12

1:C

B2C2

1:C

B2C2

44

2

12

1:C

A3C

1:C

A3C

10

3

20

1.1:E

ABDE2

1.0:E

ABDE2

67

4

20

0.8:A

A6

0.8:A

A6

67

5

20

0.6:E

E3

0.6:E

E3

73

6

20

0.7:A

A6

0.7:A

A6

0

7

30

0.1:B

AB7CD

0.1:B

AB4C3D

155

8

30

1:D

C2D2

1:D

C2D2

122

9

30

1.2:E

DE3

1.2:E

DE3

1

10

30

1.8:D

C2D4

1.8:D

C2D4

10

11

30

0.1:C

BC5

0.1:C

BC5

144

12

30

0.9:E

E6

0.9:E

E6

0

13

50

0.2:F

A4B4F4

0.2:F

A2BEF4

165

14

50

1:A

A7B

1.4:A

A7B

276

15

50

1:B

A8B4

0.5:B

A4B6

154

16

50

1.2:B

B10

1.3:B

B10

55

17

75

0.6:C

C7

0.6:C

C7

2

18

75

0.5:C

B2C13

0.5:C

B4C12

102

19 100

1:A

A15

1:A

A15

419

20 100

1:A

A4B9

1.9:B

A25

491

VRP:

(s, VC) denotes that the initial solution is obtained by using a vehicle capacity of VC, a savings multiplier s at the routing stage of Salhi and Rand VRP heuristic.

f:

A,. . . , F.denote the proposed fleet composition. For instance in problem 1, B2C2 means that two vehicles of type B and two vehicles of type C are proposed to be the final solution suggested by the modified, MRPERT, and the tabu search, TSVFM, heuristics

Bestiter:

the iteration number at which the best solution is found using TSVFM.

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Finally, for each test problem, the iteration number at which the best TSVFM solution is found, is listed in Table 8.3. The solution quality based on the MAXBEST stopping criterion would require more computation time, since it performs 5 × n to confirm that the solution can not be improved further. However, this criterion has the advantage of allowing time requirements to be varied according to problems structure and avoiding underestimated total number of iterations. Moreover, allowing a larger value for MAXBEST would give even better results. Other stopping criteria based on a total number of iterations, or a cut-off time can also be used to terminate the search. 8.6 Conclusion and Future Directions The vehicle routing problem with different vehicle capacities is investigated. A method based on tabu search is developed using an appropriate data structure. Tabu search has obtained interesting results and new best known solutions were found. The modified route perturbation procedure (MRPERT) enlarged the neighbourhood of RPERT by performing moves that were considered infeasible by the old RPERT, in addition to allowing MRPERT to run for a few entire cycles until no further improvement is found at the last cycle. With these modifications, MRPERT has produced significant improvements over RPERT in all the cases, but required extra computing time. It also found two new best known solutions. This makes the total number of new best known solutions obtained by both procedures reach twelve out of twenty problems. Tabu search and the modified route perturbation algorithm are also compared to the best methods in the literature. It was found that the tabu search procedure produces the smallest average relative percentage deviations and the largest number of best known solutions. Our aim of this study was not to show how much we can stretch the limit of tabu search, but to present it as an effective tool which still has a great potential to be discovered. There is a number of research avenues that deserve further investigations. It seems that TS has failed in problems with a large number of vehicle types. A further study is needed to establish the reasons for such a failure. One way would be to extend the neighbourhood by increasing the value of λ from one to two in the λ-interchange mechanism, or considering more than two routes simultaneously. Other tabu search features such as intensification, diversification, and a dynamic tabu list size can be explored. Combining in a hybrid way more than one metaheuristic as in [Osm95b, Osm93, OC94, TOVS93, TOS94], investigating the effect of good initial solutions on the quality of solution, introducing limitations on the maximum available number of vehicles in each type, implementing the data structure within MRPERT, testing other metaheuristics are parts of an on-going research on this problem. Further investigations on the above issues will be reported. 8.7 Acknowledgement The authors would like express their gratitudes to the referees who provided a number of constructive comments that helped to improve the presentation of the paper.

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References [AL96] Aarts E. H. L. and Lenstra J. K. (1996) Local Search in Combinatorial Optimization. John Wiley, Chichester. [Bat96] Battiti R. (1996) Towards self-tuning heuristics. In Rayward-Smith V. J., Osman I. H., Reeves C. R., and Smith G. D. (eds) Modern Heuristic Search Methods. John Wiley, Chichester. [BR88] Bookbinder J. H. and Reece K. E. (1988) Vehicle routing considerations in distribution and system design. European Journal of Operational Research 37: 204213. [BT91] Beaujon G. L. and Turnquist M. A. (1991) A model for the fleet sizing and vehicle allocation. Transportation Science 25: 1945. [CR96] Chiang W.-C. and Russell R. (1996) Simulated annealing metaheuristics for the vehicle routing problem with time windows. Annals of Operations Research 63. Baltzer, Basel. [CW64] Clarke G. and Wright J. W. (1964) Scheduling of vehicles from a central depot to a number of delivery points. Operations Research 12: 568581. [DV91] Deroschers M. and Verhoog T. W. (1991) A new heuristic for the fleet size and mix vehicle routing problem. Computers and Operations Research 18: 263274. [EL96] Eglese R. W. and Li L. (1996) Heuristics for arc routing with a capacity constraint and time deadline. In Osman I. H. and Kelly J. P. (eds) Metaheuristics, Theory and Applications. Kluwer, Boston. [FM88] Ferland J. A. and Michelon P. (1988) The vehicle routing with multiple types. Journal of Operational Research Society 39: 577583. [GALG84] Golden B. L., Assad A., Levy L., and Gheysens F. G. (1984) The fleet size and mix vehicle routing problem. Computers and Operational Research 11: 4966. [GGA84] Gheysens F., Golden B. L., and Assad A. (1984) A comparison of techniques for solving the fleet size and mix vehicle routing problem. OR Spektrum 6: 207216. [GGA86] Gheysens F. G., Golden B. L., and Assad A. (1986) A new heuristic for determining fleet size and composition. Mathematical Programming Studies 26: 233236. [GHL94] Gendreau M., Hertz A. and Laporte G. (1994) A tabu search heuristic for the vehicle routing problem. Management Science 40: 12761290. [Glo86] Glover F. (1986) Future paths for integer programming and links to artificial intelligence. Computers and Operations Research 13: 533549. [Glo95a] Glover F. (1995) Tabu search fundamentals and uses. Technical report, University of Colorado Boulder. Working Paper. [Glo95b] Glover F. (1995) Tabu thresholding: Improved search by non-monotonic trajectories. ORSA Journal on Computing 7: 426442. [HO95] Hasan M. and Osman I. H. (1995) Local search strategies for the maximal planar graph. International Transactions in Operational Research 2: 89106. [Lin65] Lin S. (1965) Computer solutions to the travelling salesman problem. Bell System Technology Journal 44: 22452269. file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_152.html（第 1／2 页）2009-7-7 16:57:28

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[LO95] Laporte G. and Osman I. H. (1995) Routing problems: A bibliography. Annals of Operations Research 61: 227262. [LO96] Laporte G. and Osman I. H. (1996) Metaheuristics in combinatorial optimization. Annals of Operations Research 63. Baltzer, Basel. [LR81] Lenstra J. K. and Rinnooy Kan A. H. G. (1981) Complexity of vehicle routing and scheduling problem. Networks 11: 221227. [OC94] Osman I. H. and Christofides N. (1994) Capacitated clustering problems by hybrid simulated annealing and tabu search. International Transactions in Operational Research 1: 317336. [OK96a] Osman I. H. and Kelly J. P. (1996) Metaheuristics. Theory and

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Applications. Kluwer, Boston. [OK96b] Osman I. and Kelly J. (1996) Metaheuristics. an overview. In Osman I. H. and Kelly J. P. (eds) Metaheuristics, Theory and Applications. Kluwer, Boston. [OL96] Osman I. H. and Laporte G. (1996) Metaheuristics: A bibliography. Annals of Operations Research on Metaheuristics 63. Baltzer, Basel. [Osm93] Osman I. H. (1993) Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem. Annals of Operational Research 41: 421451. [Osm95a] Osman I. H. (1995) Heuristics for the generalised assignment problem. simulated annealing and tabu search approaches. OR Spektrum 17: 211225. [Osm95b] Osman I. H. (1995) An introduction to metaheuristics. In Lawrence M. and Wilsdon C. (eds) Operational Research Tutorial Papers. Operational Research Society, Birmingham. [PKGR96] Potvin J.-Y., Kervahut T., Garcia B., and Rosseau J. (1996) A tabu search heuristic for the vehicle routing problem with time windows. ORSA Journal on Computing, Forthcoming. [Ree93] Reeves C. R. (1993) Modern Heuristic Techniques for Combinatorial Problems. Blackwell, Oxford. [Ron92] Ronen D. (1992) Allocation of trips to trucks operating from a single terminal. Computers and Operations Research 19: 451451. [RR96] Rego C. and Roucairol C. (1996) A parallel tabu search algorithm using ejection chains for the vehicle routing problem. In Osman I. H. and Kelly J. P. (eds) Metaheuristics, Theory and Applications. Kluwer, Boston. [RS95] Rayward-Smith V. J. (1995) Applications of Modern Heuristic Methods. Alfred Waller, Oxford. [SR87] Salhi S. and Rand G. K. (1987) Improvements to vehicle routing heuristics. Journal of Operational Research Society 38: 293295. [SR93] Salhi S. and Rand G. K. (1993) Incorporating vehicle routing into the vehicle fleet composition problem. European Journal of Operational Research 66: 313330. [SS95] Salhi S. and Sari M. (1995) A heuristic approach for the multi-depot vehicle fleet mix problem. Technical report, School of Mathematics and Statistics, University of Birmingham, U.K. [SSST92] Salhi S., Sari M., Saidi D., and Touati N. A. C. (1992) Adaptation of some vehicle fleet mix heuristics. OMEGA 20: 653660. [Tai93] Taillard E. D. (1993) Parallel iterative search methods for vehicle routing problems. Networks 23: 661673. [TOS94] Thangiah S., Osman I., and Sun T. (1994) Metaheuristics for the vehicle routing problems with time windows. Technical Report UKC/IMS/OR94/4, Institute of Mathematics and Statistics, University of Kent, Canterbury. [TOVS93] Thangiah S., Osman I., Vinayagamoorthy R., and Sun T. (1993) Algorithms for vehicle routing problems with time deadlines. American Journal of Mathematical and Management Sciences 13: 323357. [Vos96] Voss S. (1996) Observing logical interdependencies in tabu search. methods and results. In Rayward-Smith V. J., Osman I. H., Reeves C. R., and Smith G. D. (eds) Modern Heuristic Search Methods. John Wiley, Chichester.

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9 Simulated Annealing Solutions for Multi-objective Scheduling and Timetabling Kathryn A. Dowsland Abstract Simulated annealing has been successfully applied to a variety of scheduling and timetabling problems. In cases where multiple objectives are encountered, the traditional approach is to incorporate them all in a single cost function with appropriate weights. If the problem is so highly constrained that feasible solutions are difficult to obtain then the solution space is frequently relaxed and violation of constraints is penalised by further terms in the cost function. This it may well be difficult to find appropriate values for all the weights. This is particularly true if there is a distinct hierarchy of importance. This paper suggests an alternative approach which deals with groups of objectives in distinct phases. The factors which should be taken into account when deciding how to allocate the objectives to phases are considered, and the approach is illustrated using two successful applications. 9.1 Introduction Schedules and timetables cover a broad spectrum of human activity, governing our school days, leisure activities and travel, and affecting our working lives through shift patterns, production plans and delivery schedules. In an industrial setting the main objective when solving such problems is usually some form of cost minimisation. However, where the task is essentially one of scheduling people, personal preferences or concepts of fairness may become major factors in judging the quality or feasibility of a particular schedule. The result is frequently a formulation involving hierarchies of objectives and a mixture of hard and soft constraints, of a size and complexity which renders exact optimisation procedures ineffective. Furthermore the precise nature of the objectives and constraints may vary over time, necessitating a flexible approach. They may also include subjective criteria, making it desirable to produce a number Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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of different schedules so that the final choice can be left to the decision maker. It is, therefore, not surprising that metaheuristics such as simulated annealing and tabu search have proved to be popular and successful solution techniques for problems of this type. Ideally, the search should be conducted over the space of feasible solutions, but for many timetabling and scheduling problems simply finding a feasible solution is often a difficult task and it is usual practice to relax some of the constraints and to use a cost function which is a weighted linear combination of constraint violations and original objectives. As the number of different cost function elements grows it becomes increasingly difficult to balance the weights appropriately and this becomes critical in situations where some terms relate to hard constraints, which must be satisfied, while others are concerned with soft constraints and secondary objectives. In view of these difficulties this paper offers an alternative strategy in which different groups of cost function terms are considered in different phases, so that costs minimised early on are not allowed to increase in subsequent phases. This approach is illustrated with reference to two case studies based on successful simulated annealing solutions to two scheduling problems which arise annually at Swansea University. 9.2 Local Search, Simulated Annealing and Scheduling Problems Simulated annealing (SA) is a local search algorithm which avoids becoming trapped in local optima by accepting some uphill moves in a controlled manner. A full description of the method is given in Figure 9.1. In order to implement SA in the solution of a particular problem it is necessary to make a number of decisions concerning the way in which the problem is to be defined within the local search framework and the details of the parameters which control the algorithm itself. It is well known that both types of decision can have a considerable effect on the efficiency of the search and the quality of solutions produced. The problem specific decisions involve defining a solution space and neighbourhood structure, and associating an appropriate cost with each feasible solution. The generic decisions are mainly concerned with the cooling parameter, t, and define its starting and finishing values and the way it is to be reduced. Scheduling problems are particularly amenable to solution by local search as they frequently give rise to very natural neighbourhoods. These are either based on changing the time, or other attribute, of a single event, or on swapping the attributes of two different events. For example Abramson [Abr91], [AD93] uses the first type of neighbourhood structure in his SA solution to the school timetabling problem. He defines his feasible solutions as the set of all allocations of classes to time-periods, teachers and rooms, and defines a neighbourhood move as the result of changing one of the attributes of a single class. Conversely Wright [Wri91] uses a swap neighbourhood in his application of local search to an umpire scheduling problem, completely swapping the events allocated to any two umpires at each iteration. Once such an algorithm has been implemented with a suitable solution space and neighbourhood structure it is relatively easy to modify the cost function or some of the constraints to deal with any changes or crises which may arise, and this flexibility is frequently cited as a distinct advantage by those electing to solve their problems in this way. It is also worth noting that many problems of this type occur at most once or twice a year and do not require

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Select an initial solution s0; Select an initial temperature t0 > 0; Select a temperature reduction function α; Repeat Repeat Randomly select s ∈ N(s0); δ = f(s) f(s0); if δ < 0 then s0 = s else generate random x ∈ exp(δ/t) then s0 = s; endif endif until iteration-count = nrep; t = α(t); Until stopping condition = true. s0 is the approximation to the optimal solution.

Figure 9.1 Simulated annealing for minimisation immediate solution. Thus the major criticism against local search, that it requires a considerable amount of time to produce high quality solutions, is not an important factor. Many applications allow an overnight or longer run in order to get the best possible results from the search. Many successes have been reported using a cost function which combines all the objectives and constraints into a single weighted sum. For example Abramson [Abr91], [AD93] takes this approach with the school timetabling problem and Forbes et al.[FHKW94] use it successfully to schedule the buses run by Brisbane City Council. However, other researchers suggest that it can be difficult to set the weights appropriately, especially when they involve a mixture of hard and soft constraints and it is unlikely that all terms can simultaneously be reduced to zero. For example, Dige et al. [DLR93] report serious problems in obtaining a feasible solution to their version of the school timetabling problem. Wright[Wri91, Wri92, Wri94] has had more success with a number of different multi-objective problems associated with the English cricket season but concludes that when selecting weights many different factors such as the relative importance of the objectives, their relative difficulty and the way in which they interact with each other and with the solution space should all be considered. In the context of tabu search, he also suggests that if binding constraints are enforced using large weights then some form of intelligent diversification may be necessary at appropriate points, in order to allow the search to accept large increases in the cost

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function. In practice, such an allocation of very high weights to those terms relating to the binding constraints may be the only way of guaranteeing feasibility. In a simulated annealing implementation this will mean that at moderate to low temperatures this part of the cost function will not be increased. Such an approach effectively deals with the heavily weighted terms first and then concentrates on the secondary terms, while ensuring that they are not optimised at the expense of primary terms. At first glance it seems that this is almost equivalent to tackling the problem in distinct phases, considering only the binding constraints in the first phase, and then reducing the solution space so that only feasible solutions are considered while the secondary objectives are optimised. However, in many cases a phased approach can be very much more efficient. The remainder of this paper concentrates on two case studies which serve to illustrate why this is so, and to highlight some of the problems which may be encountered when implementing a phased approach in the solution of a particular problem. 9.3 Examination Scheduling We first consider the problem of producing the annual or biannual examination timetable in higher education establishments. Although the precise rules governing the examination schedule differ from institution to institution the problem is usually one of multiple objectives and a variety of constraints. The most common variant of the problem involves fitting a given set of exams into a fixed number of time-slots, in such a way that no student is required to take more than one exam at a time, and so that there are sufficient desks or rooms to accommodate all the exams scheduled for any one period. In addition, some papers may be subject to time-window constraints, groups of papers may have to be scheduled in a pre-specified order and some pairs of exams may have to be scheduled at the same time or at different times. Other, secondary, requirements which are considered by many institutions involve spreading the exams evenly over the exam period for each individual, placing papers taken by large number of students early in the examination period and incorporating further time-window requests which may be preferred but are not regarded as binding constraints. The problem of scheduling the degree exams at Swansea University is typical and involves around 3000 students taking 600 exams which are to be scheduled within 24 time-slots. The binding constraints are as follows: there are only 24 time-slots available no student clashes to be allowed certain pairs of exams to be scheduled at the same time certain pairs of exams to be scheduled at different times certain groups of exams to be scheduled in order certain exams to be scheduled within time-windows no more than 1200 students to be involved in any one session in addition there are two secondary objectives: minimise the number of exams with over 100 students scheduled after period 10

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minimise the number of occurrences of students having exams in consecutive periods. Finding a feasible solution without considering the secondary objectives is itself a difficult problem. Therefore, the first step in implementing a simulated annealing solution method was to decide which constraints should be incorporated into the definition of the solution space and which should be penalised in the cost function. These decisions were motivated by the underlying graph colouring model for the problem. This involves mapping each exam onto a vertex in a graph and defining an edge between each pair of vertices representing exams which cannot take place in the same time period. A feasible colouring, in which vertices are allocated colours so that no two adjacent vertices have the same colour, then defines a clash-free timetable with the colours representing the different time-slots. Balakrishnan [Bal91] extended this model to include time-windows by adding a set of dummy vertices representing the different time-slots. Using this model the problem at Swansea can be regarded as that of finding a colouring of the underlying graph in 24 colours so that orderings and capacity constraints are obeyed and the secondary objectives are optimised. Chams et al. [CHdW87] report considerable success using simulated annealing to solve the problem of colouring a graph in k colours, and our implementation is based on theirs, with additional restrictions on the solution space and extra terms in the cost function. They used the following problem specific decisions:

Solution space:

partitions of vertices into k colour classes

Neighbourhood move: change of colour of a single vertex Cost:

number of edges between pairs of vertices in the same colour class

The objective is to reduce the cost function to its optimal value of zero. In our case there are four different types of edges in the underlying model: those representing student clashes, those representing exams which cannot go on at the same time for other reasons, those relating to time-window constraints, and those relating to large exams which are effectively time-windowed. As some of these represent binding constraints while the latter represents a secondary objective it is necessary to consider weighting each type differently, giving four separate terms. Of the remaining features, second order conflict is an objective and is therefore included as the fifth term. This leaves the decision as to whether to include the remaining constraints, i.e. desk capacities and orderings, in the definition of the solution space, or to include further terms in the cost function. In any modified local search, such as SA, which aims to be independent of the starting solution, it is important that the solution space remains connected under the chosen neighbourhood structure, in order to ensure that the optimal solutions are reachable from any starting point. This observation led to a decision to include the ordering constraint in the definition of solution space and to include the desk capacity constraint in the cost function, as removing solutions which violate the latter could potentially result in a disconnection. This results in the following definitions.

Solution space:

allocations of exams to 24 time-slots subject to ordering constraints

Neighbourhood move:

change of time-slot for a single exam

Cost:

w1c1 + w2c2 + w3c3 + w4c4 + w5c5 + w6c6

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Page 160 where c1 is the number of student clashes, c2 is the number of exams clashing for other reasons, c3 is the number of time-window violations, c4 is the number of desk capacity violations, c5 is the number of occurrences of second order conflict and c6 is the number of large exams out of place. After considerable experimentation with this formulation using very slow cooling it became apparent that the only way to guarantee that the binding constraints were satisfied was to impose relatively high weights, which effectively meant that through most of the search constraint violations were not reintroduced once they had been eliminated. From this point of view the search is mimicking a two-phase process, converging with respect to the highly weighted elements first and then dealing with the secondary objectives later. However, as the following discussion shows, improved performance can be obtained by strictly dividing the search into two separate phases and running each independently. The initial experiments suggested that the objective of placing large exams early in the exam period was relatively easy to satisfy. Therefore this was incorporated into the time-window constraints for the remaining experiments. In the twophase approach the cost function for the first phase becomes:

and the objective is to reduce this to zero. This problem proved to be relatively easy to solve and setting all the wi, equal to one, and adopting a cooling rate of 0.99 every 1000 iterations produced zero cost solutions quickly and easily. Phase 2 then commenced with the final solution from phase 1 and worked on a reduced solution space from which all solutions with f1 > 0 had been removed. The objective in this phase was to minimise the second order conflict given by cost function:

Very slow cooling was necessary in order to get the best results and the final schedule was chosen as result of both experimentation and theoretical considerations. In this schedule the temperature is initialised with a value of 20 and is reduced according to t → 0.99t every 5000 iterations. In order to compare this method with the single-phase approach, the schedule for the latter was adjusted to allow the same total number of iterations as were used in phases 1 and 2. The results using three different random number streams are shown in the first two rows of Table 9.1. Table 9.1 Units of second-order conflict (c5) in final solutions of three independent runs Run 1

Run 2

Run 3

Mean

Single phase, original neighbourhood:

367

385

365

372.3

Two phase, original neighbourhood:

313

341

336

330.0

Two phase, Kempe chain neighbourhood:

262

265

257

261.3

Note: all other costs were successfully reduced to zero.

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It is clear that using two separate phases gives better results. The reasons for this become apparent when we consider the behaviour of the search in phase 2, and compare it with that at moderate and low temperatures using the single-phase method. In the single-phase approach moves which will cause additional clashes will be generated but will almost certainly be rejected due to the large increase in cost. In the two-phase approach only those moves which do not cause clashes are generated. Further investigation suggested that around 75% of solutions are removed in phase 2. Therefore, if the single-phase approach is allowed 5000 iterations at each temperature, only around 1250 of these will be valid moves in terms of the phase 2 criteria. Thus the two-phase method is effectively cooling four times more slowly. Due to the problem size and complexity slow cooling is important in order to obtain good results, and the two-phase approach allows the search to make maximum use of the time available in generating only promising moves. The above factors are sufficient in themselves to recommend a two-phase approach for this problem. However, dealing with different objectives in different phases has the added advantage that the standard algorithm can be modified to take advantage of specific features of the sub-problems which may not exist when the problem is considered as a whole. Such modifications may be designed to improve the efficiency of the search, or they may improve the quality of the final solution. In phase 1 optimal solutions of zero cost are found relatively easily, but there is still potential for increased efficiency. As the objective is to reduce the cost to zero it is reasonable to assume that the number of exams contributing to the cost function will gradually decrease as the search progresses. Moving an exam which is not making any contribution to cost cannot result in an improvement and does little to help the progress of the search. It is therefore sensible to redefine the neighbourhoods to include only moves relating to exams which are contributing to some element of the cost. This results in a more efficient search. Note that this modification would not be sensible in the single-phase approach where large numbers of exams continue to contribute to second order conflict, even close to optimal solutions. Phase 2 lends itself to modifications which improve solution quality. These were motivated by a concern that removing so many solutions from the phase 1 solution space might cause it to become too sparsely connected for an effective search, and in some circumstances might even disconnect the space entirely. In order to overcome this the neighbourhood structure was changed so that each solution had an increased set of neighbours. The choice of neighbourhood was again motivated by the graph-colouring model. The solution space in phase 2 consists of feasible colourings in 24 colours and any new neighbourhood must maintain this feasibility. This was achieved using a neighbourhood based on Kempe chains. These essentially define subsets of vertices in two colours which can be swapped without rendering the resulting colouring infeasible. Full details can be found in Thompson and Dowsland [TD95]. The results from this neighbourhood are shown in the last row of Table 9.1. The improvement in solution quality is clear, and in view of this, the two-phase approach with the above modifications was chosen as the most appropriate method to produce the annual examination timetable at Swansea University, where it has been in use since 1993.

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9.4 Laboratory Scheduling Our second problem concerns the allocation of students to practical classes. Each year two groups of approximately 150 students must be allocated to a weekly laboratory session. Each group has its own laboratory which is available for around 20 hours each week and has a capacity of 24. Each student has his or her own individual lecture timetable and the primary objective is to find a feasible solution which minimises the total number of sessions required. There are also a number of secondary objectives, some of which are quantifiable, while others are subjective and can only be satisfied by producing a number of alternative solutions. There are inevitably some students who register late for the course or who are forced to change their practical time due to unforseen changes in their lecture timetable. Ideally these changes should be made without affecting the rest of the schedule. This is most likely to be achieved if the spare capacity is evenly spread over all the sessions rather than being concentrated in one or two. This is obviously equivalent to distributing the students evenly among those sessions utilised. Not all students are required to attend every week. Around 80% of weeks will be attended by everyone, but the remaining 20% are used to support a course taken by a subset of around 40% of the students. Ideally this special subgroup should be allocated to a minimum subset of sessions and should be distributed evenly among this set. In practice, if there are ns students the number of sessions required is (where denotes rounding up to the nearest integer). Thus the problem can be formulated as that of finding a solution satisfying the following hard constraints: 1. only n sessions utilised 2. every student allocated to exactly one session 3. every student allocated at a time when he or she is free to attend 4. no more than 24 students per session and as many of the following soft constraints as possible: 5. the number of students in each utilised session is uniform 6. a special subgroup of students are allocated uniformly to a minimum subset of practicals. The set of available sessions will be denoted by Q, the set of students by S and the special subgroup of students involved in the constraint (6) by T. The solution method is again based on simulated annealing, and, as with the examination scheduling problem, the first stage in implementing the procedure was to determine which constraints should be imposed on the solution space and which should be incorporated into the cost function. It was decided that the solution space should consist of all allocations in which each student was placed at a time when he or she was free to attend. The capacity constraints and the number of sessions used were to be included in the cost function, together with additional terms for the soft constraints. The objective is then to reduce the capacity overflow to zero and to minimise the number of sessions required. However this is not a practical definition of cost, as it will give rise to a stepped function with large plateaux which will not be able to guide the search towards good solutions. In reformulating these objectives into a cost function suitable for local search it proved convenient to include soft constraint

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or students, or by zero students. (5). Ideally, if n sessions are to be utilised each will be attended by What is required is a cost function which is zero at any of these extremes, and which will guide the search towards emptying sparsely populated sessions and filling those which are more densely populated. This will be achieved if all moves which move a student into a session with lower occupancy are uphill moves, which implies that the gradient of the cost function must be decreasing as the occupancy increases. As the sine function has the required property between 0 and π was decided to define a penalty for each session, q, as follows. Let U denote (q) denote the number of students allocated to session q. Define penalties:

, L denote

and z

where

The full cost function was made up of these penalties added to a term for capacity overflow, summed over the set of feasible sessions. The problem of satisfying (1) to (5) with student set S and session set Q will be denoted P(S, Q) and the cost function defined above by f(S, Q). It is clear that the problem of satisfying (6) is simply that of satisfying (1) to (5) with the values of n and the parameters of f defined with respect to T, i.e. problem P(T, Q) with cost function f(T, Q). In theory this can be achieved by adding the terms of f(T, Q) to f(S, Q) in a weighted sum. As it was expected that in practice all the constraints could be satisfied simultaneously, finding suitable weights was not anticipated to be too difficult a problem. However, as this implementation is not straightforward, in that the cost function does not directly reflect the objectives, it was decided to test the formulation on problem P(S, Q) before complicating the search with constraint (6). Results from three different historical data sets and some randomly generated variants of these were initially disappointing in that the search frequently failed to find zero cost solutions. It converged readily to low-cost local optima where most students were distributed evenly among n different sessions, but one or two were allocated elsewhere. Examination of the data revealed that these students were not able to attend any of the n sessions to which the majority had been allocated. This problem was overcome by adding an intelligent diversification phase to guide the search towards better solutions. Full details are not important here, and can be found in Dowsland [Dow93]. However the following points are relevant. 1. The diversification phase is called as soon as it is clear that the search is converging irrevocably to one of these local but not global optima. 2. The cost function is adjusted by the addition of further penalty terms relating to the students who cannot attend the selected n practicals. 3. The temperature parameter goes through a gradual heating phase which terminates when a more promising local optimum is approached or when it reaches a threshold value. With this modification the search repeatedly converged to optimal zero cost solutions.

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An attempt was therefore made to satisfy constraint (6) by incorporating f(T, Q) into the cost function. This proved to be unsuccessful, whatever weights were employed. This lack of success was due to three reasons, the first two of which were immediately apparent. First, when the weights were such that the search tended to converge with respect to f(S, Q) before fully considering f(T, Q) all the utilised sessions were filled to their ideal capacity as f(S, Q) converged to zero. Further moves were required in order to optimise with respect to f(T, Q) but any move of a student in T resulted in an imbalance of seat utilisation, resulting in rejection due to the increase in the heavily weighted part of the cost function. Second, the behaviour of the search with respect to f(T, Q) mirrors that with respect to f(s, Q) and thus requires a diversification phase to avoid deep local optima. Both these observations lead to the conclusion that a two-phase approach should be utilised. In this way the neighbourhoods in phase 2 can be redefined to be made up of swaps between a student in T and one in S T. The need for diversification strengthens the argument for separate phases because of the way in which the temperature parameter behaves after diversification. It is unlikely that convergence with respect to both parts of the cost function will occur at the same time. Therefore diversification with respect to one will be initialised while the other is still converging. In order to maintain this convergence the temperature must be cooling, but for the diversification penalties to work effectively it is necessary for the temperature to be increasing. A two-phase approach will avoid this conflict. Such an approach was implemented as follows. The first phase solves P(S, Q) and selects a set of n practicals. This will be denoted by R. In the second phase the solution space is reduced to contain only the n sessions in R, and the neighbourhood is altered to a swap neighbourhood as described above. Problem P(T, R) is then solved using cost function f(T, R). This strategy was not successful, in that the search still frequently failed to reduce the cost to zero in the second phase. Further examination of the data revealed the third reason for failure P(T, Q) is far harder to solve than P(S, Q) because there are very few subgroups of sessions of the correct size which cover all the students in T. Phase 1 severely restricts the solution space, as only a subset of the available sessions are considered. In the previous example this resulted in a sparsely connected space which was overcome by redefining the neighbourhoods. This approach is not suitable here as it possible that all optimal solutions have been removed. We therefore take an alternative approach of reorganising the phases. In this final implementation phase 1 solves P(T, Q), completely ignoring the remaining students. The allocation of students in T is then fixed. The second phase solves P(S, Q) using f(S, Q), but employing a neighbourhood structure which does not allow moves for students in T. This approach was successful in producing solutions satisfying all conditions (1) to (6) and is currently used as the basis of an interactive support system which schedules the practicals at the start of each academic year. Unlike the examination scheduling problem, the difficulty with a single-phase approach is not that the important constraints cannot be satisfied without high weights. The important factor here is the need for a diversification phase which is required in order to optimise both parts of the cost function, and which requires special action with respect to the temperature. This example also illustrates that the allocation of an objective/constraint to a particular phase does not only depend on its importance, but that, in common with Wright's observations, the difficulty of

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satisfying the objective/constraint should also be taken into account. 9.5 Conclusions The above examples illustrate how a phased approach can be superior to using a weighted cost function when solving those highly constrained or multi-objective problems for which feasible solutions are dificult to find. In the first problem the only way of getting a feasible solution was to impose heavy weights on the important terms in the cost function. In these circumstances a phased approach allows the available computation time to be used more efficiently, in that solutions causing large increases in the cost function, due to increases in the heavily weighted terms, are not considered in phase 2. It was also possible to improve both phases using problem specific modifications i.e. a reduced neighbourhood in phase 1, and a new neighbourhood structure in phase 2. In the second example the phased approach allows the search to take advantage of a non-standard enhancement based on a diversification routine which incorporates a 'heating' phase for the temperature function. Here the aim was to completely satisfy all the objectives, and this was best achieved by solving the most difficult, rather than the most important, objective in phase 1. The main argument against such an approach is that the solution space will be more sparsely connected in later phases, so that optimal solutions may no longer be easily reachable from the starting solution. This occurred to some extent in both our examples. In the exam scheduling problem the use of a Kempe chain neighbourhood was sufficient to overcome the problem. In the laboratory scheduling problem the original choice of phases resulted in a phase 2 optimum which was not only disconnected from the starting solution, but was no longer included in the search space. This problem was largely overcome by reversing the phases. This makes it more likely that some optimal solutions will be attainable, but it is still possible to generate phase 1 solutions which exclude all zero cost solutions when the remaining students are added. The current implementation includes a diversification option which returns to phase 1 to produce a new solution should this situation arise. It should, however, be noted that this 'reachability' problem will also be implicit in a single-phase approach with heavy weights, as paths through the solution space which visit heavily weighted solutions are unlikely to be accepted in the middle and late phases of the search. A phased approach has the advantage that by simplifying the objectives in each phase it makes it easier to identify and implement modifications aimed at overcoming reachability problems. Problems in which finding a feasible solution is itself a difficult task pose special difficulties for local search methods. Although many can be solved well using a series of weights, there are others for which this approach has not been able to produce solutions of the required quality. Some of these will belong to the class of problems for which simulated annealing is simply not appropriate, but there are others for which a phased simulated annealing algorithm is able to provide optimal or near optimal solutions within reasonable computational time. The two examples cited in this paper have illustrated two very different reasons for using phases as opposed to weights. Both are currently implemented at Swansea and have been used live for a number of years. In both cases the users are able to obtain good quality solutions in terms of meeting

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constraints and objectives at different levels of importance, while benefiting from the flexibility of simulated annealing in dealing with minor modifications and last minute changes. References [Abr91] Abramson D. (1991) Constructing school timetables using simulated annealing: sequential and parallel algorithms. Management Science 37: 98131. [AD93] Abramson D. and Dang H. (1993) School timetables: a case study in simulated annealing. In Vidal R. (ed) Applied Simulated Annealing, number 396 in Lecture notes in Economics and Mathematical Systems, pages 103124. Springer-Verlag, Berlin. [Bal91] Balakrishnan N. (1991) Examination scheduling: A computerised application. OMEGA 19: 3741. [CHdW87] Chams M., Hertz A., and de Werra D. (1987) Some experiments with simulated annealing for colouring graphs. European Journal of Operational Research 32: 260266. [DLR93] Dige P., Lund C., and Ravn H. (1993) Timetabling by simulated annealing. In Vidal R. (ed) Applied Simulated Annealing, number 396 in Lecture notes in Economics and Mathematical Systems, pages 151174. Springer-Verlag, Berlin. [Dow93] Dowsland K. (1993) Using simulated annealing for the efficient allocation of students to practical classes. In Vidal R. (ed) Applied Simulated Annealing, number 396 in Lecture notes in Economics and Mathematical Systems, pages 125150. Springer-Verlag, Berlin. [FHKW94] Forbes M., Holt J., Kilby P., and Watts A. (1994) BUDI: A software system for bus dispatching. J. Opt. Res. Soc. 45: 497508. [TD95] Thompson J. and Dowsland K. (1995) Variants of simulated annealing for the examination timetabling problem. Annals of O.R 60. [Wri91] Wright M. (1991) Scheduling English cricket umpires. J. Opl. Res. Soc. 42: 447452. [Wri92] Wright M. (1992) A fair allocation of county cricket opponents. J. Opl. Res. Soc. 43: 195202. [Wri94] Wright M. (1994) Timetabling county cricket fixtures using a form of tabu search. J. Opl. Res. Soc. 45: 758770.

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10 A Tabu Search Algorithm for Some Discrete-Continuous Scheduling Problems Joanna Józefowska, Grzegorz Waligóra and Jan Weglarz * Abstract In this paper a tabu search (TS) algorithm for solving some discrete-continuous scheduling problems is proposed. Two inter-related subproblems of a discrete-continuous scheduling problem are distinguished: (i) constructing a feasible sequence of jobs on machines and (ii) allocating the continuous resource among jobs already sequenced. An application of the tabu search algorithm operating on a space of feasible sequences with the schedule length as the objective function is presented. Two methods of tabu list management are compared on a basis of a computational experiment : a method in which solutions on the tabu list are ordered according to the FIFO rule and the reverse elimination method. 10.1 Introduction A problem of scheduling jobs on parallel machines under additional resources is considered. In a classical model considered in the literature (see e.g. Blazewicz* et al.[BESW94]), the amount of the additional resource which can be allotted to a job at a time takes a value from a given finite set. For this model a number of results are known concerning the computational complexity analysis as well as exact and approximation algorithms. However, in many practical situations an additional resource can be allotted to a job in an arbitrary amount from a given interval. Such resources are called continuously divisible or simply continuous ones. Discrete-continuous scheduling problems occur when jobs simultaneously require for their processing both discrete and continuous resources (Józefowska, Weglarz[JW95]). Let us consider the simplest situation of this type, where there is one discrete resource, which is a set of parallel, identical machines, and one continuous, renewable resource. The total amount of the continuous resource available at a time is limited. Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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The processing rate of a job depends on the amount of a continuous resource allotted to this job at a time. Notice that even this simple case already concerns a wide variety of practical situations, for example when jobs are assigned to multiple parallel processors driven by a common (electric, hydraulic or pneumatic) power source. Let us consider, for instance, m refuelling terminals driven by a pump (a common power source) used to refuel a given fleet of n boats. The terminals are the machines and the boats correspond to the jobs. An optimization problem can be formulated as a problem of scheduling the fleet of boats on the terminals with the schedule length as the optimization criterion. This problem was considered in Dror et al.[DSL87] but in a purely discrete way in which job processing times were dependent on the number of jobs in operation. In our model, the delivery of the pump is the continuous resource and the refuelling rate of a boat depends on the part of the delivery of the pump allotted to the terminal refuelling this boat at a time. Notice that the above practical problem is only an example of a fairly large class of real-life situations which can be modelled as discrete-continuous scheduling problems. These are, for example, the problems in which machines (grinding, mixing etc.) or electrolytic tanks are driven by a common electric power source or the problems in which processors of a multiprocessor computer system use a common primary memory treated as a continuous resource (this is well justified in practical cases when the number of memory pages runs into hundreds) (Weglarz *[Weg80]). Using the discrete-continuous approach one can systematically study properties of feasible or optimal schedules as well as algorithms for finding them. 10.2 Problem Formulation In (Józefowska, Weglarz[JW95]) the following model of the discrete-continuous scheduling problem is discussed. Each job i, i = 1, 2, . . .,n requires for its processing at time t a machine j, j = 1,2, . . .,m from a set of parallel, identical machines and, simultaneously, an amount (unknown in advance) of a continuous resource ui(t). Assume that each machine can process at most one job at a time, jobs are nonpre-emptable and independent and all are available at the start of the process. Processing rate of job i is described by the equation:

where xi(t) is the state of job i at time t, ui(t) is the amount of the resource allotted to job i at time t, fi is a continuous, nondecreasing function, fi (0) = 0, Ci is the (unknown in advance) completion time of job i, is the final state (or the processing demand) of job i. Assume that

for every t.

The last inequality expresses the fact that the continuous resource is renewable. For every t, the total amount of the resource allotted to all jobs processed at time t must

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not exceed the total available amount of the resource which is assumed equal to one without loss of generality. State xi(t) of job i at time t is a measure of work related to the processing of job i up to time t. For example, in the boat refuelling problem described in the introduction, xi(t) is the volume of fuel bunkers of boat i refuelled up to time t. In the same example is the total volume of fuel bunkers of boat i and ui(t) means the part of the delivery of the pump used for refuelling boat i at time t. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation, which minimize the schedule length M = max{Ci}. Of course, the problem is NP-hard because scheduling nonpre-emptable jobs on parallel machines to minimize the schedule length is NP-hard even without any additional resource (Lenstra et. al.[JKL77]). Notice that the problem defined can be decomposed into two interrelated subproblems: (i) to sequence jobs on machines and (ii) to allocate the continuous resource among jobs already sequenced. It can be proved that for functions fi such that fi(ui) ≤ ciui,ci = fi(1), i = 1,2, . . ., n the problem is trivial because the schedule length is minimized by scheduling all jobs on one machine, where each job is processed using the total available amount of the continuous resource. Let us assume that functions fi,i = 1, 2, . . .,n are concave. Then (see (Józefowska, Weglarz *[JW95])) the parallel configuration is optimal and for n ≤ m (i.e., in fact, n = m) in an optimal schedule jobs are processed using the following constant resource amounts:

where M* is the (unique) positive root of the equation:

In consequence, for concave functions fi and n > m each feasible schedule can be divided into p ≤ n intervals Mk defined by the consecutive completion times of jobs, such that the number of jobs (or rather their parts) performed in Mk, k = 1,2, . . .,p does not exceed m and the continuous resource allocation among jobs in Mk remains constant. Let Zk denote the combination of jobs corresponding to the interval Mk. Thus, with each feasible schedule a feasible sequence S of combinations Zk is associated, where the number of elements in each combination does not exceed m, each job appears in at least one combination and if a job appears in more than one combination, then these combinations are consecutive ones (nonpre-emptability). Example 1 Consider the following feasible schedule of 5 jobs on 3 machines in which it is assumed for simplicity ui(t) = 1/3 for every t (Figure 10.1).

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Figure 10.1 The idea of discrete-continuous scheduling. An example of a schedule. The corresponding feasible sequence of combinations Z1, Z2, Z3 is of the form :

Now, the processing demand of each job can be divided into parts intervals (combinations), as it is shown in Fig. 10.2.

corresponding to particular time

Figure 10.2 The idea of discrete-continuous scheduling. A division of processing demands of jobs. For a given feasible sequence S one can find an optimal division of processing demands of jobs among combinations in S, i.e. such a division which leads to a schedule of the minimal length from among all feasible schedules generated by S. To this end a nonlinear mathematical programming problem can be formulated in which the sum of the minimal-length , is intervals (i.e. parts of a feasible schedule) generated by consecutive combinations in S, as functions of minimized subject to obvious constraints. To define these functions equation (10.3) can be used, written for each Mk, k = 1,2, . . .,p. In this way the properties of optimal resource allocations for n = m are fully utilized, simplifying the resulting optimization problem. It is

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worth mentioning that the optimization problem obtained is always a convex one (see Józefowska and Weglarz * [JW95]). Of course, knowing a division of and (10.3). Formalizing the above idea let

one can easily calculate corresponding resource allocations using (10.2)

be the minimal length of an interval Mk, generated by Zk∈ S, as a

, and let Ki be the set of all indices of Zk's such that i ∈ Zk. Then a solution of the following function of mathematical programming problem defines an optimal demand division (and in consequence an optimal resource allocation) for a given feasible sequence S : Problem P. Minimize

subject to

where

is the unique positive root of the equation

Thus, the problem is to minimize a convex function subject to linear constraints. The importance of the mathematical programming problem P follows from the fact that it finds an optimal demand division for a given feasible sequence S. Based on this formulation one can also prove some properties of optimal resource allocations which simplify the procedure of finding optimal schedules for some classes of fi's (see Józefowska, Weglarz[JW95]). To find an optimal schedule when a feasible sequence is not given one has to answer the question: which feasible sequences should be taken into account ? Let POS (a potentially optimal set) be a set of feasible sequences containing at least one sequence corresponding to an optimal schedule. It is clear that it is desirable to generate the smallest possible POS. However, in general, one has to consider a POS containing all sequences S composed of p = n m + 1 m-element combinations (i.e. so-called general POS) [JW95]. If the number of feasible sequences S in a POS is still not very large one can solve problem P for each S from this POS and choose one (or all) schedule with the minimal length. However, the cardinality of a general POS grows exponentially with the number of jobs. Therefore it is justified to apply local search metaheuristics operating on a space of feasible sequences. A local search heuristic generates a set of feasible sequences S (a subset of a general POS) for a given problem instance and finds an optimal resource allocation for each sequence S in S. Of course, in general, an optimal schedule is found if S=POS. In this paper a heuristic approach based on the tabu search (TS) algorithm is presented. Although the idea is a general one, a computational experiment will be performed for functions fi, i = 1, 2, . . ., n of the form:

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This assumption significantly simplifies the solution of the ''continuous part" of the problem. For fixed values of ci and α in (10.4) an instance of the problem is defined by the number of jobs n, the number of machines m and a vector of processing demands (final states)

.

10.3 An Application of the Tabu Search Algorithm In order to apply a tabu search algorithm [Glo89, Glo90] to a combinatorial optimization problem one has to define the following elements: the representation of a feasible solution, the way of creating a starting solution, the form of the objective function and the method of calculating its values, the mechanism of generating a neighbourhood, the structure and the length of a tabu list. In the following sections the definitions of the above elements for the considered discrete-continuous scheduling problem will be presented. 10.3.1 Representation of a Feasible Solution As it has been mentioned, TS operates on a space of feasible sequences and thus a single feasible solution is simply a feasible sequence S of the form already discussed. Because the general POS is considered, a feasible solution consists of exactly n m + 1 m-element combinations of jobs. It is easy to observe that in this case every combination Zk, k = 2, . . .,p = n m + 1 differs from the previous one by exactly one element. Otherwise repetitions of combinations in a feasible sequence would appear. Such feasible sequences with repeated combinations are redundant and should be excluded from considerations. 10.3.2 Creating a Starting Solution Two methods of creating a starting solution have been applied. In the first method successive jobs are generated randomly but a job is accepted only if it does not violate feasibility of a sequence. It can be done in the following way: the first combination is generated randomly as an m-element combination without repetitions from the n-element set of jobs, every next combination Zk, k = 2, . . ., p = n m + 1 is created from the previous one by generating randomly a position in the combination (from 1 to m) and inserting into that position a randomly generated job which has not appeared in the sequence so far. A sequence of combinations obtained in the above way is random and fulfils the feasibility conditions.

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In the second method a feasible solution generated randomly is transformed according to the vector of processing demands in such a way that the job which occurs the greatest number of times is replaced by the job with the greatest processing demand and so on. In consequence, a job with a larger processing demand appears in a greater number of combinations. It is obvious that for identical fi, i = 1, 2, . . ., n, such a strategy always increases the probability of finding an optimal schedule. 10.3.3 Objective Function The value of the objective function of a feasible solution is defined as the minimal schedule length for the corresponding feasible sequence S. For the class of functions fi considered it is given by the following formula:

where Pi, i = 1, . . ., m denotes the set of jobs assigned to machine i. Formula (10.5) follows from the fact that in this case the minimal schedule length depends on how the jobs are assigned to the machines but is independent of the order of jobs on a particular machine (Józefowska, Weglarz *[JW95]). 10.3.4 Mechanism of Generating a Neighbourhood A neighbour of a feasible solution is obtained from this solution by replacing one job occurring in a position in the sequence by another job. The number of positions is, of course, (n m + 1) · m. It is obvious that a job may be replaced by another one only if it occurs more than once in the feasible solution and only in the first or last combination in that it occurs (non pre-emptability). A neighbourhood consists of feasible solutions obtained by performing all possible replacements of that type. Assume that job i occurs in combination Zk and also in either Zk1 or Zk+1. Then job i from combination Zk is replaced either by 1. job j from Zk1 if (j is not in Zk) and (i is in Zk1) or by 2. job l from Zk+1 if (l is not in Zk) and (i is in Zk+1). Example 2 Let n = 5, m = 3 and S = {1,2,3}{2,3,4}{3,4,5}. The set of the neighbouring solutions is: {1,4,3} {2,3,4} {3,4,5} {1,2,4} {2,3,4} {3,4,5} {1,2,3} {1,3,4} {3,4,5} {1,2,3} {2,3,5} {3,4,5} {1,2,3} {2,3,4} {2,4,5} {1,2,3} {2,3,4} {3,2,5}. It can be proved that the above neighbourhood generation mechanism enables every single feasible solution to be visited. file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_173.html（第 1／2 页）2009-7-7 16:57:42

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10.3.5 Structure of the Tabu List Two strategies of the tabu list management have been tested. The first one is a static method and corresponds to the simplest version of TS. In this case the tabu list contains feasible solutions and works as a queue. A solution stays tabu for a fixed number of iterations. Let us call this version the simple tabu search (STS) algorithm. The second strategy is based on a dynamic method called the reverse elimination method (REM)[Vos93]. In this case the tabu list contains attributes of moves, where a move is a modification leading from a feasible solution to a neighbouring one. Tabu List Management for STS This version corresponds to the simplest and most general form of a TS algorithm. Every visited solution is appended to the tabu list, causing the removal from the list of the oldest existing one (according to the FIFO algorithm). In the worst case checking the status of a solution requires examining the whole list. Because the order of jobs in a combination is not important (it is a set, not a sequence), solutions on the tabu list are sorted within every combination in order to identify them correctly. For example, combinations {1,2,3} and {3,1,2} represent the same part of a schedule. Example 3 Tabu list management for STS where n = 4, m = 3.

visited solutions

tabu list

1. {1,2,3}{2,3,4}

1. {1,2,3}{2,3,4}

2. {1,4,3}{2,3,4}

1. {1,2,3}{2,3,4} 2. {1,3,4}{2,3,4}

3. {1,4,2}{2,3,4}

1. {1,2,3}{2,3,4} 2. {1,3,4}{2,3,4} 3. {1,2,4}{2,3,4}

e.g. {1,3,2}{2,3,4} tabu.

Tabu List Management Based on REM The basic structure for this version is a move. It can be defined as a 3tuple: (index of a combination, job replaced, job introduced). It represents a difference between two neighbouring solutions. In this case a single element of the tabu list is a so-called residual cancellation sequence (RCS) which is, in fact, a set of moves. If the starting solution is denoted by #1 then RCS number j (the j-th element of the tabu list) is the minimal set of moves by which the current solution differs from solution #j (i.e. visited in the j-th iteration). REM uses the fact that any solution can be revisited in the next iteration only if it is a neighbour of the current solution (i.e. if the set of moves by which it differs from the current solution consists of one element only). In consequence, every move occurring on the tabu list as a one-element RCS is tabu in the next iteration. Thus, in order to define the status of a move, it is sufficient to check only those elements of the tabu list which contain exactly one move.

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The way of updating the tabu list using REM is the following. At the start of the algorithm all RCSs are empty. If a move (k, i, j) is performed, a new one-element RCS containing its reverse (i.e. move (k, j, i)) is created and added to the tabu list. Then

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all non-empty RCSs are updated in the following way: if the move (k,i,j) belongs to the RCS considered then it is eliminated (i.e. cancelled) from the RCS otherwise the move (k,j,i) is added to the RCS. It is very important that every RCS should be the minimal set of moves by which the current solution differs from an already visited one. To maintain this condition one has to reduce a RCS every time it is possible. Let v denote a move which is to be added to the tabu list. Then there are two possible kinds of reduction : 1. an added move v and a move already existing in a RCS create a new move and the number of elements of the RCS remains the same; Example: if RCS=(1,4,3) . . . and v = (1,3,2) then RCS=(1,4,2) . . . or if RCS=(1,4,3) . . . and v = (1,5,4) then RCS=(1,5,3) . . . 2. an added move v and two moves already existing in a RCS create a new move and the number of elements of the RCS decreases by 1; Example: if RCS=(1,5,4) . . . (1,3,2) . . . and v = (1,2,5) then RCS=(1,3,2) + (1,2,5) + (1,5,4) . . . = (1,3,4) . . . Of course, the reducing operation is possible only among moves having the same first element. It is easy to observe that in the minimal set the following condition has to be fulfilled: for every two moves (k,i,j) and (k,p,q) in a RCS: (i ≠ p) and (i ≠ q) and (j ≠ p) and (j ≠ q). REM guarantees that solutions will never be revisited. It is obvious, however, that the number of RCSs creating the tabu list increases with the number of iterations. The criterion of removing RCSs from the list (and replacing them by new ones) is based on the fact that the number of elements in the j-th RCS defines the distance between the current solution and solution #j. After the number of iterations equal to the assumed length of the tabu list, an RCS corresponding to the most distant solution from the current one is removed from the list rather than the oldest RCS. It is based on the assumption that going back to the most distant solution should be least likely. Thus, in this case the tabu list does not work as a queue (in contrast to STS). Example 4 Tabu list management based on REM where n=5, m=3.

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visited solutions

tabu list

1. {1,2,3}{2,3,4}{3,4,5} move (2,2,1) 2. {1,2,3}{1,3,4}{3,4,5}

1. (2,1,2)

move (2,4,5) 3. {1,2,3}{1,3,5}{3,4,5}

move (2,1,2)

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4. {1,2,3}{2,3,5}{3,4,5}

move (1,3,5)

1. (2,5,4) 2. (2,5,4)(2,2,1)

3. (2,2,1) 5. {1,2,5}{2,3,5}{3,4,5}

move (1,2,3)

1. (2,5,4)(1,5,3) 2. (2,5,4)(2,2,1)(1,5,3)

3. (2,2,1)(1,5,3) 4. (1,5,3) 6. {1,3,5}{2,3,5}{3,4,5}

1. (2,5,4)(1,5,3)(1,3,2) = (2,5,4)(1,5,2) 2. (2,5,4)(2,2,1)(1,5,2) 3. (2,2,1)(1,5,2) 4. (1,5,2) 5. (1,3,2)

10.4 Computational Experiments A tabu search algorithm has been tested to examine the influence of the type of a starting solution and the strategy of the tabu list management on the performance and the convergence of the algorithm. For each combination of (n, m) (starting with (10,2), finishing with (100,10)) 1000 instances (vectors of processing demands) have been randomly generated and solved by TS using all four combinations of starting solution and tabu list management parameters. The stop criterion was a prespecified computational time limit. The algorithm has been implemented in C++ and the experiment has been carried out on a Silicon Graphics Power Challenge computer with four RISC TFP 75MHz processors in the Poznan * Supercomputing & Networking Center. All the results have been compared with the lower bound calculated from (10.5) under the assumption that nonpreemptability is violated. The minimal schedule length is reached if the particular components in (10.5) are equal to each other. The average relative deviation from the lower bound for each version of the algorithm and the computational time limit for one instance of the problem are presented in Table 10.1.

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The results show that a starting solution dependent on processing demands improves the performance of the algorithm in the overwhelming majority of cases (the only exception is the problem of the biggest size (100,10)). Thus, it seems to be better for TS to start with a good initial solution. It is worth mentioning that in our experiment a good initial solution is a solution with a 'good' value of the objective function (i.e. close to the lower bound). In this sense a good starting solution may be a deep local optimum making the search process more difficult.

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Page 177 Table 10.1 The results of the computational experiment: Average relative deviations from lower bound in percent. r starting solutions generated randomly, d starting solutions dependent on processing demands (n,m)

t [sec.]

STS/r

STS/d

REM/r

REM/d

(10,2)

0.1

0.010005 0.006290 0.004943 0.003763

(10,3)

0.2

0.194713 0.075277 0.114693 0.051590

(20,2)

0.3

0.001241 0.000355 0.000525 0.000192

(20,3)

0.75

0.111323 0.027547 0.086444 0.026085

(50,2)

2

0.000161 0.000012 0.000124 0.000006

(50,3)

6

0.009094 0.003323 0.007371 0.002898

(100,10)

50

0.074322 0.100000 0.070342 0.097967

As far as the tabu list management is concerned, the results obtained for REM appeared to be significantly better than for STS in every case, but especially for small problem sizes. It is not very difficult to give some reasons for such a situation. Firstly, defining the solution status is much simpler using REM. There is no necessity to examine the whole list and the examining process is faster because of much simpler data structures. Secondly, the probability of going back to a solution just removed from the tabu list is much less using REM (in STS it may happen immediately afterwards). This significantly decreases the possibility of cycling in the algorithm. Finally, as a result of computational simplifications, TS using REM is able to perform a greater number of iterations within the same time period. Moreover, the reverse elimination method gives important information about the state of the searching process in the sense of the distance between already visited solutions. Figures 10.3, 10.4 and 10.5 present the convergence curves of both versions of the algorithm for n=10, 20, 50, m=3 and starting solutions dependent on processing demands. Curves for m=2 as well as for random starting solutions are very similar. The figures show the average relative deviation from the lower bound of the best solution found as a function of the iteration number. For iteration numbers not greater than 15 STS produces better results than REM. Although the differences are very small, STS finds a solution relatively close to the lower bound faster than REM. The advantage of REM becomes more significant, however, with the growth of the number of iterations. For both methods there exists an iteration number after which no really vital improvement can be observed. This number is smaller for the STS method. However, for both versions this iteration number decreases with the increase of the problem size. It is worth noticing that for a given number of machines the average distance from the lower bound decreases with the growth of the problem size (the number of jobs) for both STS and REM.

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Figure 10.3 Convergence curves for n=10 and m=3.

Figure 10.4 Convergence curves for n=20 and m=3.

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Figure 10.5 Convergence curves for n=50 and m=3. 10.5 Final Remarks In this paper an application of a tabu search algorithm to a class of discrete- continuous scheduling problems is presented. Particular components of the algorithm are defined, discussed and evaluated in a computational experiment. In further research it is planned to compare all the results obtained with optimal solutions and also to perform a computational experiment for a wider class of functions fi. Moreover, further experiments with the length of the tabu list are needed in order to avoid cycles and lead to good solutions. Also a parallelization of the algorithm is planned to make the searching process more efficient. 10.6 Acknowledgements This research has been supported by the KBN Grant No 8T11F 01008p02. References [BESW94] Blazewicz J., Ecker K. H., Schmidt G., and Weglarz J. (1994) Scheduling in Computer and Manufacturing Systems. Springer Verlag, Berlin,

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2nd edition. [DSL87] Dror M., Stern H. J., and Lenstra J. K. (1987) Parallel machine scheduling with processing rates dependent on number of jobs in operation. Management Science 33(8): 10011009. [Glo89] Glover F. (1989) Tabu search. ORSA J. Computing 1: 190206. [Glo90] Glover F. (1990) Tabu search. ORSA J. Computing 12: 432. [JKL77] J. K. Lenstra A. H. G. Rinnooy Kan P. B. (1977) Complexity of machine scheduling problems. Ann. Discrete Math. 1. [JW95] Józefowska J. and Weglarz * J. (1995) On a methodology for discrete-continuous scheduling. Technical Report RA-95/004, Inst. of Comp. Science, Poznan University of Technology. [Vos93] Voss S. (1993) Concepts for parallel tabu search. In Symposium on Applied Mathematical Programming and Modelling, Volume of Extended Abstracts. Budapest. [Weg80] Weglarz J. (1980) Multiprocessor scheduling with memory allocation a deterministic approach. IEEE Trans. Computers C-29(8): 703710.

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11 The Analysis of Waste Flow Data from Multi-Unit Industrial Complexes Using Genetic Algorithms Hugh M. Cartwright and Ben Jesson Abstract The discharge of liquid waste from industrial chemical sites is heavily regulated. In order to control waste flow within sites, and identify the source if excess pollution is released, a thorough interpretation of data delivered by on-site chemical monitoring stations is essential. This paper discusses how the genetic algorithm, operating upon threedimensional strings, can be used for this task, and can help in the design of waste flow networks. 11.1 Introduction The genetic algorithm (GA) is a search and optimisation technique inspired by the manipulations of natural selection. It operates within a search space consisting of the set of points which define all possible solutions to a problem. At each of these points the value of an 'objective function' can in principle be calculated, which indicates the quality of the solution to which the point corresponds. The task of the algorithm is to move through this search space in an intelligent fashion, looking for points with optimal or near-optimal values of the objective function. There is a trade-off in such a search between the quality of the best solution found and the time required to find it; it is clear that to be of use any searching algorithm whatever its nature must deliver high-quality solutions in reasonable time. The GA is an iterative procedure, which starts from randomly chosen (and, therefore, typically poor) solutions and attempts to improve them progressively. This is, of course, far from a unique approach, but in other respects the algorithm is fundamentally different from more traditional methods. The most far-reaching Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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difference is that the GA manipulates not one solution, but a population of perhaps 50 to 100 solutions. It can therefore take advantage of a high degree of parallelism, since all solutions are manipulated using the same set of operators, and can, during most stages of the algorithm, be operated upon independently. As an intelligent algorithm, the GA creates a 'collective memory' of the characteristics of above-average solutions to a problem. Large numbers of candidate solutions are inspected as the algorithm proceeds, and since the information which the solutions contain may be useful later in the calculation, a portion of this information is passed from one cycle to the next. This memory is of particular value when the search space is large and complex, and is a distinctive and crucial feature of the algorithm. A further distinction from traditional methods is that the GA is stochastically driven, so is less likely to become trapped by local maxima that might pin down more deterministic algorithms. GA's are also generally tolerant about the nature of the objective function; this function need not be continuous, nor have a definable gradient, and this flexibility expands the range of problems to which the GA can be applied. Indeed, the objective function may even generate substantial noise, so that repeated sampling of a point returns different values, without seriously disrupting operation of the algorithm. In a genetic algorithm, the population is composed of a multitude of trial solutions to a problem. Each of these solutions which, in GA terminology, are often known as 'strings', consists of a list of items which may be binary digits, integers or characters from some other alphabet. A GA string completely describes an individual, and therefore contains all the information necessary to specify the trial solution to a problem which the individual represents. The quality of any arbitrary solution can be determined using the objective function, which is usually referred to as the 'fitness function'. The GA attempts to evolve its population of strings using a range of operators, with the aim of locating the string which has the maximum fitness, since this individual is, in principle, the best possible solution. The operation of the algorithm is described in detail in standard texts on the subject (Goldberg[Gol89], Davies[Dav91]). 11.2 Applicability of the Genetic Algorithm The GA has proved to be of value in a wide range of problems (see, for example, Eogarty, Ireson and Bull[FIB95], Reeves[Ree95], Cartwright and Cattell[CC95], and references therein), but is applicable only if the problem satisfies certain criteria. Firstly, an appropriate fitness function must exist, so that the quality of every conceivable solution to the problem which the algorithm could generate can be assessed. Ideally, the global maximum of the fitness function should correspond to the optimum solution, though this is not necessarily a trivial requirement. The fitness function should also yield a fine-grained set of fitness values, so that the fitness of partially correct solutions exceeds that of less desirable ones (this, of course, implies that some meaning can be attached to the idea of a partial solution). This requirement arises because the fitness function provides a sense of direction to the search, so that the algorithm can move in a purposeful way towards a good solution. Without such direction, the algorithm would fiddle aimlessly with the strings which comprise the population. (One should not, however, try to interpret the idea of direction too

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literally; the direction of movement of a genetic algorithm across a search space is rather different from the direction that would be chosen by, say, a hill-climbing algorithm, since the algorithms manoeuvre about the search space in quite different ways). A further criterion for efficient searching is that a suitable recipe must be available, according to which solutions to the problem are encoded as a string (Goldberg[Gol91], Schaffer and Eshelman[SE93]). This is a key requirement, since an unsuitable encoding can destroy much of the power of the algorithm. For example, the operations that 'cut-and-splice' parent strings to form child strings must largely retain the better portions of superior strings, so that those portions can be combined with others in new ways. If the encoding is such that valuable portions are always, or very frequently, split up by the cut-and-splice operation, then the progress of the GA will be severely hampered, and the algorithm is unlikely to converge to acceptable solutions. When these criteria are satisfied, the GA can be a powerful and robust search technique that often out-performs more traditional methods such as exhaustive search, or hill-climbing methods. These methods are by no means made redundant by algorithms such as the GA, however, and much recent research combines the GA with features of these methods to yield powerful hybrid algorithms. 11.3 The Flow of Industrial Waste within Large Complexes Large industrial complexes, comprising as many as 2530 individual factories, are not uncommon within the chemical industry. Clustering of a number of independent synthetic units on a single site leads to economies of scale through the sharing of resources, and minimisation of transportation and storage costs. Among the resources usually shared is the system whose role is to dispose of liquid chemical waste (Figure 11.1). This waste is delivered into and carried away by a pipe network before release into natural waterways. To minimise the impact of chemical waste on the environment, legal restrictions exist on the amounts and types of waste that may be introduced into natural systems from such complexes. The outputs of large sites are monitored, both by site operators and environmental agencies, and substantial financial penalties may be imposed if permitted discharge levels are exceeded.

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Figure 11.1 An industrial waste disposal network. To control effectively discharge from industrial plants, a knowledge of the composition and quantity of waste produced by each factory is crucial information. If permitted discharge levels at the site outfall are exceeded, the source of the pollution within the complex can be traced, provided that sufficiently detailed information is available about the waste output of each factory. This information can be provided by monitoring stations installed at points within the pipe network (Figure 11.2), whose role is to measure and record key data on identity and concentrations of the main pollutants, and additional information such as rate of waste flow or pH. In principle, such a network of monitoring stations yields sufficient information to trace all significant pollution events back to their source. Unfortunately, the cost of installing monitoring stations capable of generating a complete and continuous profile of the waste output for every factory is high, and it is rarely feasible to install a station to monitor the output of each factory. As a consequence, a limited number of monitoring stations are generally installed at what are believed (or guessed) to be key positions in the disposal system (Figure 11.3). A restricted set of stations naturally provides much less information than would be available if monitoring stations were positioned at the outlet of every factory. When a pollution event occurs, it may be difficult, or even impossible to determine which factory was responsible from the reduced information provided by this cut-down network. Without this information one cannot take remedial action to prevent recurrence. We can illustrate the difficulty by considering a very simple case, with two factories and a single monitoring station (Figure 11.4).

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Figure 11.2 The ideal waste monitoring network.

Figure 11.3 A realistic waste monitoring network.

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Figure 11.4 A minimal monitoring network. The monitoring station records concentrations in the effluent of total organic carbon (TOC) and chlorine of, say, 100 ppm and 75 ppm respectively, but the raw measurements provide, in themselves, no clue to how we should apportion chlorine and carbon production between the two factories. Without additional information no algorithm could determine what proportions of the elements came from which factory. However, the outputs of real factories are not infinitely variable; certain parameters are known or can be predicted, such as the types of waste produced at particular points in a production cycle, or the relationships between the amounts of certain pollutants discharged by the factories. In the example above, factory A might perhaps generate purely inorganic waste and therefore produce no organic carbon. By contrast, factory B might produce predominantly chlorocarbon waste, containing a carbon to chlorine ratio of 2:1. Given such additional data we could attribute 25 ppm chlorine to factory A and 100 ppm carbon and 50 ppm chlorine to factory B. A complete or partial interpretation of the waste passing through a particular monitoring station is thus possible, provided that appropriate raw data from the factories are available. Nevertheless, in real complexes the known parameters and the relationships between them are likely to be less clear-cut. The ratio between two pollutants discharged by a factory, for example, might not be fixed, but only known to lie within certain limits. This is a 'fuzzy' relationship, and presents difficulties to many types of algorithm; it is a complex task to work backwards from imprecise monitoring station data to derive the individual outputs of each factory. A more fundamental difficulty also exists. Quite apart from the variability of the monitoring station data, the search space is extremely large; there may be numerous combinations of factory discharge levels which reproduce the monitoring station data with reasonable fidelity. For example, if a plant contained only four factories, each producing waste characterised by four parameters (typically, the rate of flow, pH and the concentration of two elements) recorded over a period of five time steps, a complete solution consists of the specification of 80 parameters (4 factories × 4 parameters × 5 times steps). If each parameter was specified as a percentage to a precision of one part in a hundred, 10160 different solutions to the problem would exist. An exhaustive search across such a space would not be productive and, unless the search space were particularly well-behaved, hill-climbing algorithms and their relations would also

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be inappropriate. The real-world problem is considerably more daunting, involving perhaps twenty synthetic units discharging through monitoring stations which record 6 to 10 parameters on a virtually continuous basis. Solutions consist of many hundreds or thousands of values. The task of identifying high-quality solutions is demanding, and only fairly sophisticated algorithms such as the GA are likely to be successful. 11.4 The Algorithm 11.4.1 String Encoding Each string within the GA population represents a possible solution to the waste-flow analysis problem. The string specifies the amount of waste of each type discharged by each factory at each time step for which readings are available at the monitoring station. If such strings are to be manipulated successfully by the GA, compatibility between the string encoding and the choice of crossover operator is vital. A key consideration in this respect is the relative size of different values in the strings. The raw data generated by the monitoring stations may be in a variety of different units, and therefore vary over different numerical ranges. For example, the volume of discharge may be of the order of thousands of cubic metres per hour, whereas pH values lie within the range 0 to 14. If all values were used without any form of scaling, the fitness of each string would be determined almost entirely by the degree to which large values such as flow rate were fitted, while small values such as the pH would be virtually ignored. In this algorithm all the quantities in the string are therefore scaled to the same range, 0 to 1000. This helps to ensure that all types of quantity are treated with equal weight by the GA, and it allows the string to be internally represented as integers without serious loss of precision, which is computationally desirable. Once scaled, the factory discharges must be collected together to create the string. In most GA applications a linear representation of the solution to a problem is used. For this chemical waste problem we might consider forming such a string in the manner depicted in Figure 11.5.

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Figure 11.5 A possible linear string encoding. When such a linear string is processed by the operators within the algorithm, the crossover operator will cut and splice the string at randomly chosen positions along the length of the string. It is easy to see that this is potentially a very destructive operation. If, for example, an individual arose in a GA run in which all pH data were well fitted, this individual would almost certainly be fitter than others in its generation since it would successfully predict the monitoring station pH measurements. However, in a linear encoding the pH values for different factories are spread throughout the string, so that the crossover operation will generally divide the pH values between different child strings, thereby destroying or at best diluting this valuable information. To circumvent this difficulty one might consider a linear encoding which placed all the pH values together. However, this would in turn separate other related string elements, such as all discharges from a particular factory at a given time. This inevitable disruption of logically connected string elements by crossover will severely affect the performance of the GA and prevent it converging on any high-quality results. To counteract this, a representation of the data must be chosen that does not separate related string elements. This problem has been discussed in the literature[CH93] and can satisfactorily be resolved by using a multidimensional string. In this algorithm, we have used for the first time a three-dimensional string, forming a matrix of data values that, after scaling, directly corresponds to an array of factory discharge data (Figure 11.6).

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Figure 11.6 Three-dimensional string encoding. In this form of string, every item of data in the string is located near to the items to which it most closely relates. Therefore crossover does not unavoidably disrupt strings and the GA evolution can now proceed with a good chance of success. 11.4.2 The Fitness Function Two factors are used to assess the quality of a string. The more important factor relates to the deviation between the data recorded at the monitoring station and the equivalent data predicted by the string. A high-quality string codes for a solution which faithfully reproduces the monitoring data. Each string is therefore initially decoded to give factory discharge data. From these data the corresponding flow of waste within the pipe network is found, and the consequent monitoring station readings are calculated. These calculated monitoring station measurements are then compared to the actual measurements. The poorer the agreement, the lower the fitness of the individual. In this way the GA is encouraged to search for a set of factory discharges that accurately mimic the recorded monitoring station data. Every string is also assessed in terms of the extent to which it meets known constraints on the discharges from each factory. Information such as the maximum possible range over which discharge of a particular pollutant may vary, the typical short-term variation of each pollutant and ratios between different quantities may all be built into the algorithm. If the output of a factory, as encoded in a string, then deviates from some specified bounds, the fitness of the individual is reduced by an amount related to the extent of the deviation. The GA is thus gently guided towards solutions that not only reproduce the monitoring station data but that also give realistic factory emissions.

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11.4.3 Crossover Operator The GA operators used to select parent strings and to establish the next generation (tournament selection with replacement and elitist selection respectively) have been chosen for simplicity and speed of execution, and their refinement will be the subject of future work on this algorithm. The crossover and mutation operators are less standard due the unusual nature of the problem. The new crossover operator that has been developed is an extension of Harris' UNBLOX operator (Harris[Har91]) which was proposed to perform an unbiased two-point crossover on a two-dimensional string. In this method, two points are chosen within the string matrix, and define a rectangular section of that matrix. This area is then swapped between individuals. In three dimensions the requirement that crossover be unbiased leads to a variety of crossover operators (Figures 11.7, 11.8). To ensure that the operator is unbiased with respect to each of the string elements, the crossover area is allowed to 'wrap around' the string matrix. All locations within the string are then selected for crossover with equal probability, and therefore can be fitted with equal weight by the GA.

Figure 11.7 A simple three-dimensional crossover operation.

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Figure 11.8 Crossover volumes wrapping around in 0, 1, 2 or 3 directions. 11.4.4 Mutation Operator In traditional binary GA representations, individuals are mutated with a very low probability, commonly 0.01 or lower. However, when the items in the string are not binary but chosen from a large range of integers (or from the infinite range of real numbers), mutation is considerably more important. For example, suppose each position in a population of 50 strings contains a random integer in the range 0 to 1000. For any given position, a maximum of 50 of the possible 1000 integers will occur in the population. Since crossover does not introduce new values into the population but merely reshuffles existing ones, without mutation at least 950 integers would be unavailable to each location in the string. Mutation ensures that the entire search space is accessible to the GA, but if the mutation rate is low, the time required to cover the range of permissible values within a string becomes extremely large. Therefore in this algorithm, the mutation probability is high at 0.5 per string. Moreover, it is tested five times for each individual, so that each new child individual produced will, on average receive 2.5 mutations. The form of the mutation used is one in which an existing string element is displaced by a random amount, chosen from within a maximum mutation range that varies during the GA run (Bramlette[Bra91]). Initially the mutation can generate a new value selected arbitrarily from within the complete range of integers, so that the whole search space is available to the GA. The range decreases exponentially to a small value and then remains constant for the remainder of the run. This allows the GA to investigate the search space very widely initially, but later gives the GA more local search behaviour, so that it refines the solutions it has found. There are obvious

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similarities here with simulated annealing. 11.5 Results 11.5.1 Input Data Generation To evaluate the performance of the algorithm, and investigate its behaviour, a simulated chemical plant has been used. Each monitoring station can measure any combination of waste flow parameters to any chosen degree of precision. The characteristics of each factory are also specified, including any bounds and relationships that are required. The factories are then simulated over a period of time and their outputs used to calculate the 'recorded' monitoring station data. Finally, these recorded data are overlaid with Gaussian noise. There are several advantages to using simulated data; in particular it is possible to determine the performance of the algorithm by direct comparison of the results obtained with the original factory discharges used in creating the data set. 11.5.2 Results for a Four Factory System The first data set for which results are reported is a modest one, involving four factories and three monitoring stations (Figure 11.9), though even for a system of this size 100 variables are required within the GA string.

Figure 11.9 The initial waste flow network. A portion of typical factory discharge data produced by the simulation is given in Table 11.1. Flow rates are measured in m3 per hour, and concentrations in ppm. Repeated GA runs were made using the monitoring station data produced by these discharge data, with a population size of 50 and a run length of 10 000 generations. The GA converged to solutions of similar fitness within 4 000 to 5 000 generations. Table 11.2 shows a portion of the best string generated in one run. For ease of comparison

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Page 193 Table 11.1 Time Step

Factory 1

Factory 2

Flow

pH

conc 1

conc 2

conc 3

Flow

pH

conc 1

conc 2

conc 3

1

1396

3.66

207

525

54.7

5572

12.36

677

358

9

2

1840

3.68

18

46

77.1

5589

12.57

665

366

7.7

3

1353

3.55

270

697

54.9

5635

12.17

658

343

9.7

4

1447

3.37

49

111

43.8

5834

11.84

641

325

7.6

5

1319

2.83

48

114

60.2

5604

11.97

618

304

9.1

Table 11.2 Time Step

Factory 1

Factory 1

Flow

pH

conc 1

conc 2

conc 3

Flow

pH

conc 1

conc 2

conc 3

1

1380

3.65

207

525

54.7

5600

12.33

673

362

9.1

2

1830

3.68

18

46

77.1

5560

12.59

668

364

7.4

3

1350

3.56

270

697

54.9

5640

12.18

657

347

9.4

4

1450

3.37

49

111

43.8

5830

11.79

642

329

7.3

5

1330

2.83

48

114

60.2

5570

11.97

623

311

9.4

with the known 'theoretical' results, the string is plotted in Figure 11.10 as a scatter graph. It is evident from the figure that agreement is generally good, but that three values for factory 3 are poorly fitted by the algorithm. A similar plot for the emissions of factory 3 alone indicates that the most badly fitted points all correspond to pH measurements. We can understand why agreement with the simulation is poorer for factory 3 by considering the geometry of the flow network and the flow rate data for the four factories (Table 11.3). Factory 1 is the sole input to the first monitoring station (Figure 11.9), and, as shown in Figure 11.10, the data from this factory are well fitted by the GA. Similarly the outputs of factory 4 are available to the GA as the differences between the data of the second and third monitoring stations. The second monitoring station records the combined outputs of the first three factories. Of these, factory 2 has by far the greatest output volume, and therefore has the greatest influence on the combined waste, which is measured in terms of pH's and concentrations, 'swamping' the much smaller output of factory 3. The GA therefore has correspondingly little information about it, and is unable to fit it effectively. Of all the parameters used to characterise the waste, the pH is the most susceptible to this swamping effect. For example at time step 1 in the data set, factory 2 produces Table 11.3 Time Step

1

2

3

4

5

Factory 1

1396

1840

1353

1447

1319

Factory 2

5572

5589

5635

5834

5604

Factory 3

901

952

927

902

959

Factory 4

7566

7556

7585

7595

7603

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Figure 11.10 A scatter graph of calculated versus actual data. waste at 5572 m3 hour1 at a pH of 12.36, whereas factory 3 produces waste at 901 m3 hour1 at a pH of 4.84. Combining these amounts of waste gives a total waste flow of 6473 m3 hour1 at a pH of 12.29 (pH readings are calculated on a logarithmic scale, so cannot simply be summed). This figure clearly gives much more information about factory 2 than it does about factory 3. From this example of the application of the algorithm, a general conclusion can be drawn about how it might handle other, more complex, problems. The GA unavoidably concentrates on those discharges that have the most significant effect on the eventual output of the waste network. It is clear, therefore, that the GA, in general, will be able to accurately determine the parameters describing high-output factories, whilst those factories with smaller outputs will be relatively poorly fitted. This is an inherent feature of the GA fitness function, which is primarily driven by the monitoring station data. Whilst it would be possible to override this tendency through the introduction of different weights into the fitness function, it is not necessarily in itself undesirable. The speed of the algorithm on this model problem is encouraging, in view of the need to scale up the calculation to handle real-world complexes. Increasing the number of factories or the number of monitored quantities would of course increase the size of the string used and dramatically increase the size and complexity of the search space. However, early results on much larger systems have provided good evidence that the algorithm continues to perform well (Cartwright, Jesson and Sztandera[CJSon]). 11.6 Applications of the Algorithm In its current form the algorithm works on a batch of monitoring station data, but it can be recast into a form suitable for continuous analysis. In this form a continuous stream of data from the monitoring network is analysed to reveal the outputs of factories at any given time. The algorithm could then provide rapid feedback to the

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plant and factory managers. The algorithm also has potential as an analytical tool for the design and maintenance of waste disposal networks. The GA can be applied to the analysis of simulated data within waste networks of various geometries, in order to assess the optimum points within that network to locate monitoring stations. It could equally well provide guidance on the types of pollutant it would be most productive to monitor at those stations. If the GA, when applied to the data from a given network, provided solutions with substantial uncertainty in one or more parameters, this would indicate that extra monitoring data will be required to pin down the solution more precisely. By examining how the consistency of the results depends upon the geometry of the network, high-quality geometries can be developed. 11.7 Acknowledgements We are pleased to acknowledge the support of Hewlett-Packard for this research. References [Bra91] Bramlette M. F. (1991) Initialization, mutation and selection methods in genetic algorithms for function optimization. In Proceedings of Fourth International Conference on Genetic Algorithms, page 101. San Mateo. [CC95] Cartwright H. M. and Cattell J. R. (1995) Studies of continuous-flow chemical synthesis using genetic algorithms. In Rayward-Smith V. J. (ed) Applications of Modern Heuristic Methods, pages 129144. Alfred Waller. [CH93] Cartwright H. M. and Harris S. P. (1993) Analysis of the distribution of airborne pollution using genetic algorithms. Atmospheric Environment 27A: 17831791. [CJSon] Cartwright H. M., Jesson B., and Sztandera L. M. (In preparation) Intelligent algorithmic interpretation of waste flow data from multi-unit industrial complexes. Information Sciences. [Dav91] Davies L. D. (1991) Handbook of Genetic Algorithms. Van Nostrand, New York. [FIB95] Fogarty T. C., Ireson N. S., and Bull L. (1995) Genetics based machine learning applications in industry and commerce. In Rayward-Smith V. J. (ed) Applications of Modern Heuristic Methods, pages 91110. Alfred Waller. [Go189] Goldberg D. E. (1989) Genetic Algorithms in Search, Optimisation and Machine Learning. Addison Wesley, Reading, Mass. [Go191] Goldberg D. E. (1991) The theory of virtual alphabets. In Schwefel H. P. and Manner R. (eds) Parallel Problem Solving from Nature. Springer-Verlag, Berlin. [Har91] Harris S. P. (1991) Chemical mass balance calculations using genetic algorithms. PhD thesis, Chemistry Part II, Oxford University. [Ree95] Reeves C. R. (1995) Genetic algorithms and combinatorial optimisation. In Rayward-Smith V. J. (ed) Applications of Modern Heuristic Methods, pages 111126. Alfred Waller. [SE93] Schaffer J. D. and Eshelman L. J. (1993) Real-coded genetic algorithms and interval schemata. In Whitley L. D. (ed) Foundations of Genetic Algorithms 2. Morgan Kaufmann, San Mateo.

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12 The Evolution of Solid Object Designs Using Genetic Algorithms P. J. Bentley and J. P. Wakefield Abstract This paper describes an attempt to enable computers to generate truly novel conceptual designs of three-dimensional solid objects by using genetic algorithms (GAs). These designs are represented using spatial partitions of 'stretched cubes' with optional intersecting planes[BW94]. Each individual three-dimensional solid object has its functionality specified by an explicit objective function, which is utilised by GAs to evolve candidate designs. 12.1 Introduction The genetic algorithm (GA) is a highly flexible and powerful search algorithm that uses principles of evolution observed in nature to direct its search[Hol75, Gol89]. GAs can tackle optimisation problems if these problems are formulated in terms of search, where the search space is the space containing all possible combinations of parameter values, and the optimal solution is the point in that space where the parameters take optimal values. Indeed, GAs are commonly used to optimise designs, with some remarkable results[PD94]. However, in this paper we aim to demonstrate that GAs are able to do more than just optimise existing designs we propose that they can create entirely new designs from scratch. The area of design creation using genetic algorithms is a relatively unexplored area. Using GAs to create new designs has the potentially great benefit of new conceptual designs being automatically created in addition to optimal designs. Research has so far been performed in limited ways, such as the use of GAs to create new conceptual designs from highlevel building blocks[PY93] although this often seems to be little more than the optimisation of connections between existing designs rather than true design by the GA. Genetic art is becoming more popular, with various voting systems now being on-line on the Internet (for example John Mount's 'Interactive Genetic Art' at http://robocop.modmath.cs.cmu. edu.8001). Other art-evolution systems using Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith© 1996 John Wiley & Sons Ltd.

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humans as design evaluators have been created[Daw86, TL92], but as yet very few, if any, systems exist that can evolve a design from scratch with no human input during the evolution process. This paper describes an early prototype of an evolutionary design system, capable of doing just that. 12.2 Representation Evolving designs from scratch rather than optimising existing designs requires a very different approach to the representation of designs. When optimising an existing design, only selected parameters need have their values optimised (e.g. for a jet-turbine blade, such parameters could be the angle and rotation speed of the blade). To allow a GA to create a new design, the GA must be able to modify more than a small selected part of that design it must be able to modify every part of the design. This means that a design representation is required, which is suitable for manipulation by GAs. Many such representations exist, and some are in use by the evolutionary-art systems: a variant of constructive solid geometry (CSG) is used by Todd and Latham[TL92], others use fractal equations (e.g. John Mount see the WWW address given above), and Dawkins[Daw86] uses tree-like structures. For a system capable of designing a wide variety of different solid object designs, however, a more generic representation is needed. A variant of spatial-partitioning representation has been developed for this work [BW94]. This representation uses spatial partitions of variable sizes, each one being a 'cuboid' with variable width, height, depth and position in space. Each primitive shape can also be intersected by a plane of variable orientation, but this refinement to the representation will not be considered in this paper. The main benefit of using such a variable-sized primitive shape as a spatialpartition is that few partitions are required to represent designs. Significantly, the fewer the partitions in a design, the fewer the number of parameters that need to be considered by the GA. The main disadvantage, however, is that it is possible to represent illegal designs using the representation, i.e. it is possible for two spatial-partitions to overlap each other causing redundancy and ambiguity. Thus some form of correction is needed, to convert overlapping primitives into non-overlapping primitives. Since this correction process needs to be executed for every new candidate design in the population, every generation, it must be fast. 12.3 Correcting Illegal Designs Various alternative methods for correcting illegal designs were examined: 12.3.1 Method 1 Intersecting primitives are located by detecting whether any corner of one primitive lies within another primitive, Figure 12.1. If they are found to overlap, the positions of the primitives are examined relative to one another. If the difference in position

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between the two primitives is greatest in the x direction, the widths of the primitives are reduced, if the difference is greatest in the y direction, the heights are reduced, and for a greatest difference in the z direction, the depths are reduced, Figure 12.2. Using the positions to determine which sides to move ensures that the primitives are changed by the minimum amount possible. When reducing the dimensions, they are reduced until instead of overlapping, the appropriate sides are touching. To prevent too much unnecessary squashing, all other sides of the primitives are kept in the same place by moving the positions of the two primitives slightly.

Figure 12.1 Two primitive shapes are detected as overlapping: a corner of one primitive lies within the other.

Figure 12.2 Distance between overlapping primitives is greatest along y-axis: heights (and ypositions) are changed. This method of squashing any overlapping primitives until they touch was found to be very successful. However, a flaw in the overlapping detection method became apparent: it is possible for two primitives to overlap without any of the corners of one being within the other, Figure 12.3. This meant an alternative detection method was required.

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Figure 12.3 Primitives not detected as intersecting: no corner of one lies within the other. 12.3.2 Method 2 In this method, primitives begin 'life' as single points (specified by the centre positions of the primitives as described in the genotype). They are then grown in size by a small amount for a specific number of iterations. When two primitives grow to touch each other, the touching sides stop growing, otherwise the sides continue to grow until they reach the size specified by the genotype, Figure 12.4.

Figure 12.4 Growing the primitives. This method was found to work most of the time, and indeed mirrors nature attractively in that genotypes are mapped to phenotypes by a growth process. Another attraction was the way it was implemented the time complexity was reduced from O(np2) for the first method to O(npc), where n is the number of individuals in the population, p is the number of primitives in a design, and c is the constant (maximum) number of iterations the primitives were grown for.

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Figure 12.5 Two primitives begin too close together and grow to overlap each other. Unfortunately, the method did not work all of the time: it is unable to detect whether two primitives starting too close to one another and growing by too large an amount are touching or not, and so they grow to overlap each other, Figure 12.5. A more serious problem though, was one of speed despite being a potentially faster method for very high numbers of primitives in a design, it was actually a very slow method for usable numbers of primitives. Another method was needed. 12.3.3 Method 3 This method of squashing overlapping primitives is really a combination of parts of the first two methods. Primitives, when detected as overlapping, are squashed as described in the first method, Figure 12.2. To detect whether the primitives were overlapping, an idea was taken and extended from the second method. This method examines the sides of the two primitives and if at least one side of one primitive is between two sides of the other primitive for all three axes, the two primitives are overlapping. The time complexity returns to O(np2), but because the detection is nothing more than a conditional check, (see Figure 12.6) the process runs very quickly indeed.

Figure 12.6 If side 1 or 3 lies between a and c, or if side a or c lies between 1 and 3, and if side 2 or 4 lies between b and d or if side b or d lies between 2 and 4, the two primitives are overlapping. So the third method provides an effective way to correct illegal designs. However, this creates the question: where should designs be corrected?

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12.3.4 Where to Correct the Designs Each candidate design (or phenotype) is defined by the representation, and coded as the genotype which is directly manipulated by the GA. The question is then, should incorrect designs be corrected in the genotype, in the phenotype, or should they be prevented from existing in the population at all? To deal with the last, first: if every time the GA creates an illegal design, that design is discarded, the GA will take substantially longer to evolve designs, since illegal designs are often evolved. If the illegal designs are allowed to exist in a population, but are corrected in the genotype, the population of candidate designs becomes very static, with little evolution occurring. This is because a high proportion of all new designs evolved by the GA are illegal designs if these are corrected every time, evolution is effectively being undone so the population stagnates. So the solution must be to correct the phenotypes. This is achieved by allowing an illegal design (a design with spatial partitions overlapping) in the genotype, but correcting the design when it is mapped to the phenotype. In this way, the genotype no longer directly specifies the phenotype (or design), but instead gives directions that should be carried out if permissible. This method corresponds well with nature. For example, suppose that there is a single gene within a person's genotype that specifies how tall he should be (in reality it is probably a collection of genes). It might 'say' his ideal height should be 6 ft 5 in but in real life he might only be 6 ft. Why? Because whilst growing, he was restricted by gravity. If he had grown up in space without the restriction of gravity, he may well have become 6 ft 5 in. Equally, the designs are restricted by the rules of the representation. Even though in the genotype, a primitive shape may be given a large width, when the genotype is mapped to the phenotype, that primitive may be squashed to a smaller size and new position by its neighbours. It is apparent that in nature, evolution takes into account such restrictions our genes compensate for gravity when specifying our heights, so although the human genes may be asking for a height of 6 ft. 5 in. they 'know' that, because of gravity, this equates to 6 ft. Equally, as is shown by the experiments below, the evolution of the GA takes into account the restriction of the design representation, and compensates.

Figure 12.7 Operation of the prototype evolutionary design system. So, the GA evolves new designs by manipulating coded indirect representations in the genotypes, which are mapped to the direct representation of the phenotypes,

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Figure 12.7. Some form of guidance is obviously necessary to direct the evolution of the GA. This is provided by simple evaluation software effectively a software version of the design specification. There follows a series of experiments to demonstrate the way this evolutionary design system can evolve new designs from scratch, guided only by such evaluation software. 12.4 The Evolution of a Table The genetic algorithm used for the following series of experiments remains unchanged throughout, apart from varying the population size and the number of generations it runs for. A basic canonical GA was used[Müh92], with primitive 'roulette wheel' partner selection[Gol89], real coding[Gol92] and no special multi-objective function handling capabilities. Designs were all limited to five primitives and the population was initialised with primitives of random size and location at the beginning of every test run (i.e. starting from scratch). Evaluation software was utilised to describe the functionality of a table by the use of an objective function consisting of a weighted sum of a number of design objectives. Five experiments were performed, with the evaluation software being made more sophisticated for each experiment by the addition of an extra design objective. 12.4.1 Size Perhaps the most basic requirement for a table is that of appropriate size: a table the size of a house or a matchbox is of no great use. The size of the design is specified by minimum and maximum extents for the left, right, back, front, top and bottom of the design. The fitness of a candidate design decreases the further it is from the ideal size. Using a population of 50 individuals, designs of the required size had evolved after 25 generations, Figure 12.8.

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Figure 12.8 Evaluation of size only. 12.4.2 Mass Another basic, but vital requirement is that of mass: a table weighing a ton would not be very practical. An ideal mass is defined, the fitness of the design decreasing the more the actual mass differs from the ideal mass. The ideal mass was defined as being a very low value. Designs with good fitnesses (the right size and very close to the ideal mass) had evolved after 100 generations, with a population of 50, Figure 12.9.

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Figure 12.9 Evaluation of size and low mass. The easiest way for the GA to create designs of low mass is to reduce the dimensions of the primitive shapes that make up the designs. This can produce fragmented designs however, where primitives become disconnected from the main part of the design. Since this work concentrates on the evolution of a single object at a time, such fragmented designs are heavily penalised. The fitness of fragmented designs only based on the sum of the distance of each primitive from the origin (normally the centre of the design). This addition to the evaluation software means that in a population of fragmented designs, evolution will first create whole designs by drawing in the primitives towards the origin. In a population of whole designs, a fragmented design is very unfit and normally never reproduces. 12.4.3 Stability A more complex requirement is stability under gravity the table must stand upright, and not fall over under its own weight. The stability of a candidate design is determined by calculating whether the centre of mass of the design falls within the outer extents of the primitive(s) touching the ground. (The ground is assumed to be at the lowest part of the design.) The more unstable the design is, the worse its fitness becomes. To help evolution, the population size was increased to 80 [GDC92]. After 130 generations, some very fit designs had evolved (the right size, stable, and almost the right mass), Figure 12.10 and 12.11. Strangely, evolution to good designs seemed remarkably improved with the addition of this new constraint.

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Figure 12.10 Evaluation of size, low mass and stability.

Figure 12.11 Evaluation of size, low mass and stability.

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12.4.4 Flat Surface Perhaps the most important requirement of a table is its flat, upper surface the table top. A good table should have a flat surface at an appropriate height from the ground, and covering an appropriate area. These criteria were evaluated by randomly sampling, five times, the top of the design within the required area of the table top, and checking how much the height at these points differed from the required height. (Analogous to five objects of negligible mass, e.g. feathers, being dropped onto the design within the area for which the flat surface is required. Should a feather be supported at too high or too low a height, or not be supported at all, the fitness of the design decreases proportionately.) Since this evaluation is of a random nature, a design may be given a different fitness value every time it is evaluated (unless it fully satisfies the constraint). Not surprisingly, the GA found evolution to good designs a little harder to perform, so the population size was increased to 100, and the GA was allowed to run for 300 generations. The resulting designs were usually very good, but compromises were being made by the GA. It was not always able to fully satisfy all of the constraints since, in order to have a large flat surface, the size of the primitives must be increased, but to have a low mass, the size of the primitives must be decreased. Results were often designs with excellent table tops, but too massive (Figure 12.12), or designs with unusual multi-level table tops with the right mass (Figure 12.13). All designs were still the right size and stable, however.

Figure 12.12 Evaluation of size, low mass, stability and flat top.

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Figure 12.13 Evaluation of size, low mass, stability and flat top. 12.4.5 Supportiveness and Stability One final requirement for a table is that it must be able to support objects on its flat upper surface. Although the tables evolved so far stand up on their own, if an object was to be placed on them, most would still topple over. Greater stability is therefore required. This is achieved by a simple extension to the 'stability under gravity' constraint, which requires the table to be able to support a heavy object placed at the very edges of its extremities without falling over. The mass of the object was deliberately set very high (twice the required mass of the table itself). The GA was allowed to run for 500 generations, and produced some good designs. Again compromises were made the GA quickly 'discovered' that an easy way to support a heavy object is to make the table even heavier, so it was necessary to increase the weighting of the 'low mass' constraint. Once this was done, the GA produced some remarkably fit designs, coming up with two main methods for increased stability: a very low centre of mass (Figure 12.14) or a very wide base (Figure 12.15), as well as producing designs the right size, nearly perfect flat surfaces, and almost the right mass.

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Figure 12.14 Evaluation of size, low mass, flat top and greater stability.

Figure 12.15 Evaluation of size, low mass, flat top and greater stability.

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12.4.6 What the Tables are Not Although it should be apparent that the GA can indeed create novel, usable designs for tables, it is also clear that the tables do not resemble the sort of four-legged table most of us are used to. This should not really be surprising, since the GA has no world knowledge and so cannot be influenced by existing designs. (Also the small number of primitives combined with the initial random seeding of the population is a factor in producing some of the more unusual designs.) As the software stands, it is theoretically possible for the GA to evolve a four-legged table, but it is highly unlikely. It would require four primitive shapes to be precisely the same length so that they all touch the ground at once. Any primitive which was a fraction too long or too short would no longer behave as a leg, thus resulting in a poor design, so a mutation or crossover would have to produce all four legs in one unlikely jump. However, it is not at all difficult to make the GA produce four-legged tables. By enforcing symmetry about the x and z axis, more traditional, symmetrical table designs can be evolved. (An alternative approach would be to seed the initial population with existing table designs and allow the GA to combine the best features of these, but this would no longer be designing from scratch.) The tables produced are not evaluated for aesthetics, which would require human input, and no stress analysis takes place, so deficiencies in these areas are occasionally apparent. 12.5 Conclusions The genetic algorithm is capable of more than design optimisation it can evolve entirely new designs. To do this, a flexible representation is required, to allow the GA to modify every part of the design. As shown in this paper, the GA can automatically take into account the background correction of illegal designs (i.e. those not following the rules of the representation). It can cope with partially conflicting constraints (e.g. the low mass against the large table-top and greater stability), and randomly varying evaluations (e.g. the random sampling of the table-top evaluation) and still produce some excellent results. In the 'evolution of a table' example, the multiobjective function defining the table was treated as a single objective function by weighting and summing the separate functions. Although the GA coped surprisingly well, this is not an ideal solution[Go189] and in the future the GA will be modified to evolve Pareto optimal solutions to such multicriteria design problems[HN93]. Future work will also be directed towards the creation of a system capable of evolving shapes other than 'cuboid' in appearance by allowing the GA to use the refinement to the representation, i.e. primitive shapes will be permitted to have portions 'sliced' off them. It is also anticipated that the GA can be made to evolve not only the position and dimensions of the primitive shapes, but also the number of primitives making up a design.

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References [BW94] Bentley P. J. and Wakefield J. P. (1994) Generic representation of solids for genetic search. Microcomputers in Civil Engineering 11(3): to appear. [Daw86] Dawkins R. (1986) The Blind Watchmaker. Longman Scientific & Technical Publishers, Harlow. [GDC92] Goldberg D. E., Deb K., and Clark J. H. (1992) Genetic algorithms, noise, and the sizing of populations. Complex Systems 6: 333362. [Gol89] Goldberg D. E. (1989) Genetic Algorithms in Search, Optimization & Machine Learning. Addison- Wesley, Reading, MA. [Gol92] Goldberg D. E. (1992) Real-coded genetic algorithms, virtual alphabets and blocking. Complex Systems 5: 139167. [HN93] Horn J. and Nafpliotis N. (1993) Multiobjective optimisation using the niched pareto genetic algorithm. Technical Report 93005, Illinois Genetic Algorithms Laboratory. [Hol75] Holland J. H. (1975) Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor. [Müh92] Mühlenbein H. (1992) Darwin's continent cycle theory and its simulation by the prisoner's dilemma. In Conf. Proc. 1st European Conference on Artificial Life Towards a Practice of Autonomous Devices, Paris, pages 236244. [PD94] Parmee I. C. and Denham M. J. (1994) The integration of adaptive search techniques with current engineering design practice. In Proceedings from Adaptive Computing in Engineering Design and Control -'94, Plymouth, pages 113. [PY93] Pham D. T. and Yang Y. (1993) A genetic algorithm based preliminary design system. Journal of Automobile Engineers 207(D2): 127133. [TL92] Todd S. and Latham W. (1992) Evolutionary Art and Computers. Academic Press, New York.

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13 The Convoy Movement Problem with Initial Delays Y. N. Lee, G. P. McKeown and V. J. Rayward-Smith Abstract The convoy movement problem (CMP) with initial delays is presented and three methods for its solution are discussed and compared. These methods are a pure branch-and-bound (B&B) algorithm, a hybrid genetic algorithm / B&B algorithm and a pure genetic algorithm. Each approach is shown to have its advantages and the hybrid method also provides an illustration of a genetic algorithm in which the fitness of a solution depends upon the time when it is evaluated. 13.1 Introduction In [YNLRS94], a combinatorial optimization problem called the convoy movement problem (CMP) is introduced and described. A branch-and-bound algorithm for solving a basic version of the CMP, and a description of its implementation on a network of transputers, is given in [LMRS94]. In this paper, we describe work carried out to incorporate delays into the model and, in particular, the use of genetic algorithms within solution techniques. In this section, we give a brief formal description of the CMP with delays and in subsequent sections, three different approaches to solving this problem are discussed, two of which use genetic algorithms. Results are presented for experiments on test data provided by the DRA (Malvern) for each of these approaches. Finally, we give our conclusions and raise a number of issues relating to further work. In the CMP, we are given a collection of m military units (convoys),

Convoy i is to move from a given source position, si, to a given target position, ti, over a limited route network. The objective is to find a set of paths with the minimal completion time for all of the convoys subject to certain constraints. Different types of convoy may move at different speeds along the same parts of Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith© 1996 John Wiley & Sons Ltd.

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the network. Indeed, some types of convoy may not be able to use some parts of the network at all. Furthermore, a convoy of a given type may travel at a different speed when moving in one direction along part of the network than it does when moving in the opposite direction. Let K = {0,1, . . .,p 1} denote the set of convoy types and let V = {0, . . ., n 1} denote the set of intersection points in the route network. For each type of convoy, k ∈ K, a matrix [Cuvk] is specified. Each Cuvk value is either a positive integer or ∞. If Cuvk is ∞ then no convoy of type k may travel directly from vertex u to vertex v. Otherwise, Cuvk represents the number of time units required by a convoy of type k to travel directly from vertex v to vertex v. A convoy travels 'directly' from vertex u to vertex v if and only if it passes through no other intersection point en-route from u to v. We will refer to Cuvk as the 'cost' for a convoy of type k travelling along the edge from u to v. Paths followed by different convoys do not have to be disjoint but two convoys may not occupy the same part of the network at the same time. With each convoy, i, we associate a time window, wi∈ Z+, representing the number of time units that the entire convoy takes to pass a point on the slowest part of the network for this type of convoy. wi is so defined that if the head of convoy i arrives at a point at time t, then the tail of i arrives at that point at time t + wi,. Thus convoy i occupies this point from the start of the tth time unit up to the end of the (t + wi)th time unit. Associated with each convoy, i, is its earliest ready time, ei. This is the earliest time at which convoy i can start its movement. Without loss of generality, we may assume that ei = 0 for at least one convoy. However, convoy i does not have to start at time ei but may be delayed by di time units. Convoy i would then start at time ei + di. Although ei is a fixed parameter for a particular problem instance, di is a variable whose value is to be determined. We define a path of length l by π = vo . . . vl, where vg∈ V, g = 0, . . ., l. The cost of path π for a convoy of type k is defined by

and its length is denoted by lπ Let IIuv denote the set of all possible paths, uv1 . . .uh v from u ∈ V to v ∈ V via zero or more intermediate vertices vg∈ V, and let πi, denote a path taken by convoy i. Then πi is a feasible path for convoy i and cost (πi,ki) < ∞. If πi = vo . . . vg . . . vl then πig denotes vo . . . vg, i.e. πig denotes the first g legs of πi. To refer to individual vertices in a path, we use vig to denote the (g + 1)th vertex in πi thus

The solution space, S, for the CMP is an m-tuple of paths, one for each of the convoys. We define S to include partial paths for convoys as well as complete paths. Thus,

A feasible solution to the CMP is then a solution (π0, . . ., πm1) ∈ S such that vi = ti, for each i and for which constraints relating to the unique occupancy of parts of the network at any time are satisfied.

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The cost of the movement of a convoy is equal to the sum of the path cost, the actual starting time and the time window for the convoy. The time window is included here to allow time for the tail of the convoy to arrive at the target position. Thus the movement cost of convoy i is defined as

Figure 13.1 Constraints for the CMP. The primary objective of the CMP is to find an optimal solution having the least completion time over all feasible movements, i.e. to minimize

where the minimum is taken over all feasible (π0, . . ., πm1) ∈ S. To avoid generating paths with unnecessarily high costs, we include a secondary objective to minimize the total completion time of all the movements,

Again, we seek to minimize objective function (02) over all feasible movements in S. The CMP constraints are specified formally in Figure 13.1.

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Constraints (C1) ensure that each convoy finishes its movement on or before its specified finishing time. Constraints (C2) prevent more than one convoy using the same vertex at the same time. In these constraints, qig denotes the set of time units during which convoy i occupies vertex vig on its path πi. If convoy i is of type k, then the time, aig, at which the head of convoy i arrives at vig is given by

The time at which the tail of convoy i arrives at vig is

It then follows that

The qualification in (C2) that g and h are not both simultaneously zero means that two convoys may share the same starting vertex. Constraints (C3) ensure that no convoy overtakes any other convoy on any edge at any time. We also prohibit any convoy from passing any other convoy moving in the opposite direction along the same edge (C4). Allowing delayed starts means that a convoy can wait at its source location for an arbitrary length of time if this leads to a better overall solution. When a convoy, i, waits at its source location, si, other convoys are allowed to use s, until convoy i starts to move. For simplicity, we assume that the initial delay for a convoy must be a multiple of a prescribed waiting interval, or waiting cost. The given waiting interval may be different for different source vertices and may also be different at a given source vertex for different types of convoy. For convenience, we assume that the waiting intervals associated with vertices are represented by diagonal entries in the [cuvk] matrices. The starting time of a , where ri≥ 0 is a variable to be determined and represents the convoy, i, with source vertex, v, is thus number of waiting intervals that convoy i waits at its source vertex, v. Other convoys can use vertex v from time 0 to time

.

13.2 A Branch-and-Bound Approach We have developed a program implementing a branch-and-bound (B&B) algorithm for solving the CMP with delays. This program is based on the program described in [MRSLL94b] and details are given in [MRSLL94a]. The programs have been developed within the general B&B framework described in [MRST91] and build on the associated kernel software. The CMP has a number of features in common with multi-commodity problems (see, for example, [KH80, MW84, GC94]). All experiments reported in this report were performed on a DEC Alpha 3000/400 running OSF/1 and using the DEC cc compiler with -02 optimization turned on. The timings reported are measured in CPU seconds. We have used four test problems for our experiments. Problems 1 and 2 were supplied by the DRA and problem 3 is our own generated problem. A summary of the DRA problems is shown in Table 13.1. All of the edges in these examples are bi-directional.

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Table 13.1 DRA test problems No. of vertices

No. of edges

No. of convoys

Problem 1

160

212

17

Problem 2

530

724

25

Test problem 2a is a variation of test problem 2 in which the input order of the convoys is different. Test problem 3 has been specially generated to test the pathological situation where every vertex in the network is connected to several vertices which form a bottleneck. Figure 13.2 shows an example of such a network. Test problem 3 has 15 vertices on each side of the bottleneck and a bottleneck comprising just three nodes. The 15 vertices either side of the bottleneck form the source and target locations for 15 convoys. These source and target vertices are connected directly to the bottleneck vertices and do not have any other connections. All connections are bi-directional and the cost of each edge is generated randomly between 1 and 19. Note that the cost of travelling in one direction is not necessarily the same as the cost of travelling in the other direction. All 15 convoys are of the same type and have time windows set to 1. This means that any convoy will take 2 time units to pass any point on the network.

Figure 13.2 Test problem 3. Various parameters of the B&B code can be adjusted easily by altering the input data files. However, for the results we report here, we used a depth-first search method; both 01 and 02 were used in the bounding and priority functions, with 01 having the primary role. We took a time window for each convoy of 10 for test problems 1 and 2 and of 1 for test problem 3, our initial bounds for the two objective functions

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Page 218 were (∞, ∞) and the path length limit was 70. The maximum size of the active pool was 3000. For each problem, a maximum of 4 experiments were conducted. For experiment 1, no delay was allowed. For experiments 2, 3 and 4, delayed starts were allowed. The waiting intervals at all the source locations are set to the same value for each experiment. For all problems, the waiting intervals used in experiments 2 and 3 are prescribed as 1 and 2, respectively. For experiment 4, the value 11 (the average of all time windows + 1) is used for problems 1 and 2 and the value 3 is used for problem 3. Note that the waiting intervals at all source locations are set to the same value for simplicity only. They can take different values if desired. Table 13.2 shows the results. The optimal value found in experiment 1 is (1339, 16 064) when delays are disallowed. However, when delayed starts are allowed, better solutions are found. The optimal values for experiments 2, 3 and 4 are (1339, 15792), (1339, 15 799) and (1339, 15 819), respectively. As we can see from Table 13.2, the price we pay for an improved solution is an increased execution time. This is because the size of search space is significantly increased when delayed starts are permitted. However, we should also note that in all of the three cases, a solution that is better than the optimal solution of experiment 1 is found in a shorter time. Table 13.2 Results of test problem 1 B&B approach Experiment

1

2

3

4

Waiting Costs

0

1

2

11

O1, O2

1339,16 201

1339,16 201

1339,16 201

1339,16 201

Time (secs)

0.83

0.87

0.85

0.86

O1, O2

1339,16 149

1339,16 060

1339,16 060

1339,16 057

Time (secs)

0.95

2.56

3.13

3.46

O1, O2

1339,16 064

1339,15 792

1339,15 799

1339,15 819

Time (secs)

3.77

242.00

135.20

21.29

5.87

279.09

152.85

24.64

First Solution

Intermediate Solution

Optimal Solution

Termination Time Time (secs)

If we interpret no delayed starts as having very high waiting costs at all source locations, then in general, the total execution time of the B&B increases with decreasing waiting costs while the quality of the optimal solutions improves with decreasing waiting costs. We also performed some experiments using best-first search. For experiment 1, the algorithm terminates after about 5 seconds, which is faster than the time for the depth-first search. However, for experiments 2 and 3, the active pool overflowed. This may be expected because the waiting costs are set to very low values. This means that nodes with delayed starts are likely to have very good bounds. With a best-first search, these nodes will be expanded first, which in turn generates more nodes with delayed

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Page 219 starts. Thus, the active pool can very quickly become flooded with these nodes. Table 13.3 shows the results of test problem 2. As problem 2 is much larger than problem 1, with a very large search space, we have not managed to solve this problem to optimality. Despite the fact that the program has not terminated, we have been able to find some very good solutions within a relatively short time. The best values found for experiments 1, 2 and 3 are (2377, 35648), (2377, 35644) and (2377, 35644), respectively. The value for the primary objective, 01, in these solutions is optimal whilst the value for the secondary objective, 02, is within 4.7% of the lower bound of the optimal value. We force the termination of all three experiments after resolving another 500 to 1000 clashes without producing a better solution. The percentage deviation shows how close a solution is from a lower bound for the optimal value. Table 13.3 Results of test problem 2 B&B approach Experiment

1

2

3

Waiting Cost

0

1

2

Lower Bound of Optimal Objectives : 2 377,34 070 First Solution O1, O2

2377,35 701

2 377,35 701

2 377,35 701

% Deviation

0, 4.78

0, 4.78

0, 4.78

Time (secs)

11.86

11.95

12.05

O1, O2

2377, 35 654

2 377, 35 648

2 377, 35 648

% Deviation

0, 4.65

0, 4.63

0, 4.63

Time (secs)

56.16

417.45.76

82.74

O1, O2

2377,35 648

2377,35 644

2377,35644

% Deviation

0, 4.63

0, 4.62

0, 4.62

Time (secs)

81.68

454.78

170.41

139.13

579.43

322.79

Intermediate Solution

Best Solution

Forced Termination Time (secs)

Table 13.4 shows the results of test problem 2a. For each experiment in this set, we have left the program running for 5000 seconds. We can see from the table that, in all our cases, a longer time is needed to generate a good solution with this ordering than with the ordering of problem 2. For problems 1 and 2, we obtained the optimal 01 value in the first solution produced. However, for problem 3, in all four experiments reported in Table 13.5, the 01 value of the first solution found is not optimal. This is due to the bottleneck in the road network as a result of which the critical convoy is not so easily identifiable. Delayed starts improve the quality of the solution significantly. Both 01 and 02 values are improved with delayed starts. As expected, experiment 2 takes longer to terminate than experiment 3 but generates better values. Experiment 4, surprisingly, did not terminate as quickly as we expected. It had not terminated after 1700 seconds, which is longer than the time for experiment 2. This may be due to the high waiting cost used (greater than wi + 1), which makes the scheduling around the bottleneck even

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Page 220 Table 13.4 Results of test problem 2A B&B approach Experiment

1

2

3

Waiting Cost

0

1

2

Lower Bound of Optimal Objectives : 2 377,34 070 First Solution O1, O2

2 377,36 508

2 377,36 508

2 377,36 508

% Deviation

0, 7.16

0, 7.16

0, 7.16

Time (secs)

13.68

13.69

14.01

O1, O2

2 377,36 070

2 377,36 070

2 377,36 070

% Deviation

0, 5.87

0, 5.87

0, 5.87

Time (secs)

2 225.33

2 219.09

2 216.93

5000

5000

5000

Best Solution

Forced Termination Time (secs) harder. Table 13.5 Results of test problem 3 B&B approach Experiment

1

Waiting Costs

2

3

4

1

2

3

First Solution O1, O2

30, 278

30, 278

30, 278

28, 254

Time (secs)

0.54

0.56

0.54

0.51

O1, O2

26,238

23, 210

23, 215

24, 220

Time (secs)

2.51

1 551.20

315.20

4.70

71.02

1 616.57

349.12

Optimal Solution

Termination Time Time (secs) We can see from our results that: 1. Delayed starts do in general produce better solutions. The trade-off for this is longer execution times. 2. The finer the waiting costs, the better the solutions. The trade-off is again greater computational time. 3. The ordering of the convoys affects the positions of individual solutions in the search tree, which in turn affects the total execution time. file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_220.html（第 1／2 页）2009-7-7 16:58:06

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One of the problems with our approach is that the B&B search tree is potentially very large and it is not physically possible to store the whole tree. Therefore, it is important to use a search strategy which minimizes the proportion of the search tree

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which must be held in memory. This effectively rules out both breadth-first and best-first search strategies as they require far too much memory to be practical. While the use of a depth-first search can keep a check on the storage requirements, we have very little control over the execution time of the algorithm. Indeed, if we start off on the wrong branch, the algorithm may not terminate within a reasonable time. This is because the depth-first search chooses a node for expansion based on its position in the tree rather than the quality (bound) of the node. Thus, with a large search tree, the execution time is heavily dependent on finding a good solution at an early stage in order to allow early pruning of the tree. The structure of the search tree is determined by the branching strategy used. We have described and discussed various branching strategies and their effects on the structure of the search tree in [MRSLL94b]. All the branching strategies we have considered may be strongly affected by the ordering in which the convoys are considered, although the extent of the impact varies from strategy to strategy. This impact is clearly shown by the difference between the results of problem 2 and 2A. In some cases, the difference is much more significant. Unfortunately, we have not currently been able to identify characteristics for good and bad orderings. Although we suspect there is no single best ordering for all problems, we expect that there may be good orderings for problems with certain characteristics. As the performance may vary significantly with different orderings, this may be worthy of further research. A node in the search tree represents an m-tuple of convoy paths, one for each convoy, from its source location, si, to the current location, ui. In our current implementation, we use a fairly simple bound calculation based on the current cost (from si to ui) and the minimum cost path from ui to ti. 13.3 A Hybrid Approach : Using GAs for Delays and B&B for Paths In this section, we present a hybrid approach for solving the CMP with delays. We use a GA to compute the delays and a B&B algorithm to compute the paths. We will refer to this as the hybrid system. As we can see from the results presented above, allowing delays does in general produce better solutions. However, by incorporating delays into the B&B implementation, the total search space increases significantly. For large problems, our current B&B implementation may take a very long time to terminate due to the potentially large search space. For example, for test problem 2, we do not know how long our B&B program would take to terminate naturally, either with or without delays. We have left a version without delays running for more than a week without it terminating. By adding delayed starts, the program can take considerably longer to terminate naturally because of the increased search space. Thus, using a GA to compute delays may be more effective than a pure B&B approach when dealing with large problems. For smaller problems, pure B&B can find the optimal solution very quickly, and we would not expect the hybrid approach to be worthwhile in such cases. The hybrid system is implemented on a DEC Alpha workstation using the X-GAmeter toolkit which provides an Xwindow interface. X-GAmeter has been

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developed at UEA and provides an easy to use framework and interface for designing and experimenting with GAs for optimization problems. General GA functions are included in the toolkit. Details of X-GAmeter can be found in [KRSS93]. For any GA, the most important (problem dependent) issues are the representation of the bit string and the definition of the fitness function. We now discuss both of these issues for the GA used in our hybrid system. For simplicity, let the delay time, di, associated with convoy, i, be of the form ri × wi, where wi is a prescribed waiting interval. The bit string is then used to represent the m-tuple (ro, ri, . . ., rm1). Figure 13.3 shows the structure of such a bit string. Each ri, is represented by b bits which means that ri can take a value between 0 and 2b 1, inclusively.

Figure 13.3 Bit string hybrid approach. In our system, we allow the values of b and wi to vary from problem to problem, by including them as part of the problem data. This allows us to perform various experiments on these two values. The default values are b = 8 and . In order to allow experimentation, we have implemented both the binary and Gray code representation of the bit string. In our hybrid system, each evaluation of the fitness function involves an execution of the B&B algorithm to determine the paths for the convoys. The bit string is decoded to give a set of delayed starting times for the convoys, and this set is then passed to the B&B algorithm for determining the paths. The values of the objective functions 01 and 02 are output from the B&B algorithm to the GA. X-GAmeter restricts the fitness value to be a single number, double or long, so we must encode the pair of objective values output from the B&B algorithm as a single number representing the quality of the solution found. In our implementation, the fitness value is represented by a long number in which we use the lower 32 bits to represent the 02 value, β say, and the rest of the higher order bits to represent the 01 value, α say. The fitness value is then given by

This is thus equivalent to giving the 01 value a weighting of 232.

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In the context of a GA, it may be argued that a smaller weighting for α is to be preferred. This is because α is the cost of a single convoy (the critical convoy) and so depends on only b bits of the bit string, whilst β, which is the total cost of all convoys, depends on all of the bits in the string. A large weighting on α may lead to quick convergence to a local optimum. Furthermore, as α is based on the b bits that correspond to the critical convoy, other 'good' bits in the bit string may not be preserved. This is because the contributions of these bits to the fitness value is so trivial when compared with the bits corresponding to the critical convoy. However, when comparing the quality of two solutions for the CMP, only if the O1 values are the same for both solutions do we need to consider the O2 values. Thus, we need a representation that can differentiate the quality of individual solutions in the GA. Obviously, we could reduce the value of the weighting for problems with smaller O2 values. We could also employ some fitness scaling techniques to re-scale the raw fitness value. Generally, the evaluation of the fitness function of a GA is relatively simple. Typically, we can measure the number of evaluations performed per second. However, in our case, each evaluation invokes the B&B algorithm for generating the paths, which may take a very long time (possibly days or even weeks) before the algorithm terminates. In the implementation of our hybrid system, we impose a time limit, max_exec_time, on the execution of the B&B algorithm. If the algorithm has not terminated naturally by the end of max_exec_time seconds, we force its termination. The value of max_exec_time should be problem dependent. For smaller and/or easier problems, the algorithm may terminate after several seconds while for others, it may take tens or hundreds of seconds before a feasible solution is found. Thus, for our system, we input the value of max_exec_time as part of the problem data so that the value can be adjusted easily. Our hybrid system possesses a special feature which makes it rather different from a typical GA system. In our system, the evaluation of two identical bit strings at different times may yield different fitness values. This is because the initial bound used by the B&B evaluation function may change as the GA progresses. The use of an initial bound gives a lower bound for any solution in the search space. Branches of the search tree which do not contain a solution that has better objective values than the initial bound can be pruned immediately. Thus, the use of different initial bounds on the same search tree may produce different results. We use the best O1 and O2 values found as the initial bound for each evaluation. Assuming the B&B algorithm is allowed to execute to natural termination, it would terminate with either a better overall solution or no new solution. However, with the introduction of a time limit on the execution of the B&B algorithm, a number of outcomes may occur. 1. The B&B algorithm terminates with an optimal solution. In this case, we return the values of the optimal solution. 2. The B&B algorithm terminates and no new solution is found. This means that the search space corresponding to the current bit string does not contain a better solution. To penalize this, we return the current best O1 and O2 values with the O1 value increased by 1. As the O1 value has a weighting of 232, this is equivalent to increasing the fitness value

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by 232. 3. The B&B algorithm is forced to terminate because of the time limit but a better solution is found. In this case, the values of the best solution found are returned. 4. The B&B algorithm is forced to terminate because of the time limit and no new solution is found. This means that the search space may or may not contain a better solution. In this case, we return the current best O1 and O2 values with the O2 value increased by 1. As the GA progresses, the initial bound for use in the B&B algorithm typically gets tighter. For larger problems when natural termination is unlikely, it is then harder for the B&B algorithm to find a better solution. We would need to relax the time constraint to increase the chance of finding a better solution. This is achieved by an automatic increase in the time limit after a fixed number of evaluations. Again, this parameter is very problem dependent. Thus, we allow the user to input it as part of the problem data. By default, we increase the execution time by 1 CPU second after every 100 evaluations. Due to the length of time required to run each of the experiments, only a very limited number of experiments have been performed. Unlike most GA experiments where the same experiment is run many times, and from the results of which statistics are then compiled, we have only been able to run each of the test problems 3 to 4 times each. For all the experiments, we use the GA settings shown in Table 13.6. For a general introduction and survey of GA terminology, see [BBM93a, BBM93b, Whi94]. Table 13.6 GA settings Parameters

Setting

Selection

Roulette

Crossover

One Point (60%)

Merge

New Solution

Mutation

1%

Pool Size

10

Parent Chromosome

2 to 10

Termination

No change for 50 generations

Table 13.7 shows the results for test problem 1 using the hybrid approach. For comparison purposes, we also present the timings (Table 13.8) at which solutions of a similar quality are found using a pure B&B approach. As the waiting intervals for these experiments are set to 1, we use the timings from the second experiments for the pure B&B method, in which the same waiting interval setting is used. As we can seen from the tables, the hybrid approach does not perform very well for problem 1 in comparison with the pure B&B approach in terms of both the quality of the solution and computation time. For test problem 2, the results are given in Table 13.9 and 13.10. In this case, a better solution is found using the hybrid approach than was found using pure B&B.

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Page 225 Table 13.7 Results of test problem 1 hybrid approach Run

Best Solution

Average Solution

1

2

3

O1

1339

1339

1339

1339

1339

O2

15 793

15801

15 820

15 793

15 804.67

Individual

172.79

428.88

685.68

172.79

429.12

Accumulated

172.79

601.67

1287.35

172.79

Solution Found

170

258

62

170

163.33

Terminate

220

308

112

220

213.33

Solution Found

1019

1498

383

1019

967.33

Terminate

1294

1789

657

1294

1246.66

Time:

Generations:

Evaluations:

Table 13.8 Extracted results of problem 1 using B&B approach O1,O2

Time

Intermediate Solution

1339, 15 848

14.77

Intermediate Solution

1339, 15 822

14.91

Optimal Solution

1339, 15 792

242.00

Termination

279.09

The best O2 value found using the hybrid approach is 35 631, 13 less than that found by the B&B approach. However, the execution time of the hybrid approach is much longer. As different termination conditions are used for the two approaches, we cannot rule out the possibility that a similar quality solution may be found using the B&B approach if the same amount of execution time is allowed. The particular ordering of the convoys in test problem 2A seems to favour the hybrid approach over the B&B approach. On comparing the results for problems 2 and 2A, we can see that better results are obtained for problem 2A than for problem 2 using the hybrid approach. When using the B&B approach, the results are better for problem 2 than for problem 2A. The results presented in Tables 13.11 and 13.12 show that for problem 2A better solutions are obtained using the hybrid approach than using the B&B approach. The best O2 value obtained by the hybrid approach is 35 554, 516 less than the value obtained from the B&B approach. In fact, the results of all the runs are better. However, we should note that the execution time taken by the hybrid approach far exceeds the execution time used by the B&B approach.

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Table 13.13 shows the results of problem 3. When compared with the results obtained using the pure B&B approach (Table 13.14), we can see that although the hybrid approach does not find the optimal solution, in each of the three runs very

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Page 226 Table 13.9 Results of test problem 2 hybrid approach Run

Best Solution

Average Solution

1

2

3

O1

2 377

2 377

2 377

2 377

2 377

O2

35 665

35 631

35 676

35 631

35 657.33

Individual

4794.56

14 509.81

3840.88

14 509.81

7715.08

Accumulated

4794.56

19 304.37

23 145.25

19 304.37

Solution Found

23

99

16

99

Terminate

73

149

66

149

Solution Found

145

592

87

592

274.67

Terminate

446

883

388

883

572.33

Time:

Generations: 46

Evaluations:

Table 13.10 Extracted Results of Problem 2 using B&B Approach Waiting Costs = 1 Intermediate Solution Best Solution Found

O1, O2

Time

2377, 35 656

265.38

2377, 35 644

454.78

good solutions were found in a much shorter time than that needed by the pure B&B approach. From our experiments, we draw the following conclusions. 1. The hybrid approach is computationally very intensive since each evaluation of the fitness function involves the execution of a B&B algorithm. 2. The approach is only worthwhile for larger problems or those involving significant bottlenecks. For smaller and simpler problems, the amount of time taken to evaluate the initial pool may exceed the time taken for the pure B&B algorithm to solve the problem. 3. The length of time required for each evaluation of the fitness function is nowhere near constant. The B&B algorithm may terminate immediately or take the whole max_time_limit seconds. This is due to the use of a progressively improving initial bound. In our current implementation, a very simple fitness function is used and the method for penalising bit strings that do not produce better solutions is also rather crude. Research is being undertaken to develop better schemes.

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The lower bound on the optimal value is calculated using the minimum cost path, with a zero delay being assumed for each convoy. If a solution is found with the same O1 value, then we have found the critical convoy. We also know that the delay of

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Page 227 Table 13.11 Results of test problem 2A hybrid approach Run

Best Solution

Average Solution

1

2

3

4

O1

2377

2377

2377

2377

2377

2377

O2

35 586

35 912

35 77

35 554

35 554

35 707.75

Individual

16 984.06

16 245.38

3402.31

32 850.21

32 850.21

12 210.58

Accumulated

16 984.06

33 229.44

36 631.75

69 481.96

69 481.96

Solution Found

149

69

7

224

224

112.25

Terminate

199

119

57

274

274

162.25

Solution Found

939

422

42

1396

1396

699.75

Terminate

1218

717

352

1686

1686

993.25

Time (secs):

Generations:

Evaluations:

Table 13.12 Extracted results of problem 2A using B&B approach Waiting Costs = 1

O1, O2

Time

Best Solution Found

2377, 36 070

2219.09

this critical convoy has to be 0. Thus, we could shorten the bit strings by removing the b bits corresponding to the critical convoy just determined. This should make the problem easier to solve. This has yet to be implemented. 13.4 A Pure GA Approach In this section, we describe a direct GA approach for solving the CMP with delays. We use a GA for computing the delays and for selecting paths. At the preprocessing stage, a fixed number (specified in the data file) of minimum cost paths is generated for each convoy. We have implemented the loopless k-shortest path algorithm presented by Yen [Yen71]. These paths and their corresponding costs are stored for later use. As in the hybrid system, for each convoy, i, we use a fixed number of bits (b, say) to encode ri, where di = ri = wi. In addition, for each convoy, i, we encode the number corresponding to the selected path. As in the hybrid approach, the number of bits used to represent delays is input from the data file. The number of bits used to represent the path number depends on the number of (near) minimum cost paths generated during the preprocessing phase. If 16 minimum cost paths are generated, then 4 bits are used to represent the path number for each convoy. Figure 13.4 shows the structure of such

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Page 228 Table 13.13 Results of test problem 3 hybrid approach Run 1

2

3

Best Solution

Average Solution

O1

24

23

23

23

23.33

O2

217

219

222

219

219.33 136.28

Time: Individual

116.93

155.79

136.13

155.79

Accumulated

116.93

272.72

408.85

272.72

Generations: Solution Found

134

36

35

36

68.33

Terminate

184

86

65

86

118.33

Solution Found

749

211

183

211

1143

Terminate

1000

508

430

508

646

Evaluations:

Table 13.14 Extracted results of problem 3 using B&B approach Waiting Costs = 1

O1, O2

Time

First Solution Found

30, 278

0.56

Intermediate Solution Found

23, 222

1331.48

Optimal Solution

23, 210

1551.20

Termination

1616.57

a bit string. Both binary and Gray code representations have been implemented.

Figure 13.4 Structure of Bit String GA Approach

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The set of convoy paths represented within a bit string corresponds to a potential solution. The evaluation of the fitness function involves the validation of this solution against all the constraints specified in our model. We check each convoy path against

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all other paths to make sure that constraints C1, C2, C3 and C4 are not violated, taking into consideration the delays for each convoy. The result of the evaluation is a triple which is composed of the infeasibility value of this solution, the O1 value and the O2 value. The infeasibility value is equal to the total number of jointly infeasible pairs of paths. A jointly infeasible pair of paths is defined to be a pair of paths which, if taken by their corresponding convoys, causes one or more of the constraints to be violated. As the evaluation function returns three values, we need to encode them into a single fitness value. As before, let α represent the O1 value and let β represent the O2 value. Let ival denote the infeasibility value. The fitness value is then calculated in the following way.

where

and

As the evaluation of the fitness function in this case is fairly simple, a large number of experiments have been performed to test out the effect of various settings. We have concentrated mainly on testing problem dependent parameters such as the number of bits used to represent delays, the number of minimum cost paths generated for each convoy, the value of the waiting interval and the bit string representation. We have also tested two different population pool sizes. As the ordering of convoys is not significant in this approach, we have not used test problem 2a. For all of the experiments, the GA settings given in Table 13.15 are used. Table 13.16 shows the settings for various test sets. Table 13.15 GA settings Parameters

Setting

Selection

Roulette

Crossover

One Point (60%)

Merge

New Solution

Mutation

(1%)

Parent Chromosome

2 to 10

Termination

No change for 500 generations

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Page 230 Table 13.16 Settings for various test sets Test Sets

Pool Size

Bit String Encoding

Number of Delay Bits Per Convoy

Waiting Interval

Convoy Time Window

2

10

Binary

8

1

10

3

10

Gray Code

8

2

10

4

10

Gray Code

6

11

10

5

10

Gray Code

6

1

10

6

10

Gray Code

6

2

10

7

20

Gray Code

6

1

10

1 10

Gray Code

8 1 10

We should note that the objective values of the test sets may be different for different waiting costs. Thus, there may be three different optimal values for the test sets, one for each choice of waiting cost. We should also note that the optimal values obtained using this approach may not be as good as the pure B&B approach and the hybrid approach when the same waiting costs are used. This is because loops are allowed in paths generated using the B&B algorithm, while the kshortest path algorithm only allows loopless paths. Thus, the paths used by the pure GA approach may not be as good as those used by the other two approaches. For test problem 1, experiments were performed on each of the test sets using different numbers of minimum cost paths. The results for 10 runs of each experiment were recorded. Table 13.17 summarizes the best solution values recorded and Table 13.18 shows the corresponding timings for that particular run. We have omitted the O1 value as the optimal O1 value was obtained for all of the runs. For this test problem, we observe the following. Comparing the results for test sets 1 and 2, we can see that on the whole the better results were obtained using the Gray code representation. The results for test sets 1 and 3 indicate that using a waiting interval of 1 produces the best results when more than 2 paths are used. The worst set of results is obtained with test set 4 when the waiting interval is set to 11; this is expected and is consistent with the results of the pure B&B approach. In general, using 6 bits to represent delays produces better results as the total search space is reduced. A large population pool does not seem to make much difference. For test problem 2, our pure GA method failed to find a feasible solution. For our third test problem, however, the pure GA does extremely well, consistently finding quite good solutions in very reasonable computational times, as shown in Tables 13.19 and 13.20. 13.5 Conclusions We have fully developed and implemented three methods of solution of the convoy movement problem with initial delays:

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Page 231 Table 13.17 Best results obtained for test problem 1 Test Sets

Number of Minimum Cost Paths Used 1

2

4

8

16

32

64

1

15 810

15 810

15 810

15 810

15 792

15 792

15 817

2

15 794

15 816

15 822

15 815

15 857

15 815

15 823

3

15 800

15 800

15 828

15 814

15 827

15 828

15 820

4

15 881

15 837

15 837

15 863

15 848

15 870

15 841

5

15 792

15 792

15 792

15 792

15 801

15 810

15 792

6

15 800

15 800

15 820

15 824

15 800

15 820

15 800

7

15 792

15 792

15 810

15 814

15 792

15 810

15 794

Table 13.18 Timings for best results obtained for test problem 1 Test Sets

Number of Minimum Cost Paths Used 1

2

4

8

16

32

64

1

209.63

197.55

333.02

186.28

376.66

254.28

398.55

2

130.87

90.22

121.51

197.72

191.65

113.65

226.67

3

177.55

210.61

222.43

288.22

377.12

374.22

273.76

4

109.20

130.57

183.01

213.59

296.24

346.91

593.25

5

111.85

167.57

121.54

143.34

181.70

171.23

305.29

6

108.04

148.94

176.05

247.31

225.79

163.92

218.74

7

166.12

199.87

155.75

354.52

190.77

295.65

271.67

MA pure branch-and-bound algorithm A hybrid genetic algorithm using a branch-and-bound algorithm as the evaluation function A pure genetic algorithm We have tested these algorithms on DRA data and on limited test data generated by ourselves. Further testing is necessary but, even with the limited experiments we have performed, we have been able to show the strength of each approach. The pure branch-and-bound approach produces very satisfactory results especially for test problem 1. On occasions, it can prove far too slow and we realise that examples are likely to arise where the use of this method will be unrealistic. This will remain the case even if we accept that the pure branch-and-bound does not run to optimality but terminates after a fixed time, reporting the best solution that it has found. Heuristics will need to be developed. The hybrid GA uses a much simpler branch-and-bound algorithm but this code is called many, many times. From a research point of view, the hybrid offers the challenge of a changing environment, whereby the fitness of a solution depends on the time when it is evaluated. This is an unusual phenomenon in GA research and has yet to be fully explored. The hybrid GA appears to produce satisfactory solutions especially for test problem 2. However, the costs of running even a simple

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Page 232 Table 13.19 Best results obtained for test problem 3 Test Sets

Number of Minimum Cost Paths Used 1

2

4

8

16

32

64

5

25 231

23 221

23 230

23 232

23 245

23 248

24 226

6

26 236

26 230

24 249

23 200

23 227

25 228

26 257

Table 13.20 Timings for Best Results Obtained for Test Problem 3 Test Sets

Number of Minimum Cost Paths Used 1

2

4

8

16

32

64

5

10.76

14.12

18.91

19.67

28.56

37.54

61.23

6

9.38

11.33

15.04

27.49

24.03

38.33

45.64

branch-and-bound might mean that it will not provide a viable solution to certain problems. As far as the pure GA approach is concerned, we have shown that that too has potential, especially for applications involving considerable bottlenecks. It produced good quality results for our contrived problem 3, in very reasonable computational times. Incorporating start delays has given us scope for improving the quality of the convoy routes. However, delays are currently limited to delays at the start only. Delays at certain intermediate nodes may be militarily feasible and such delays could enhance still further the quality of the solutions. Such delays are currently being incorporated into the model and into the suite of algorithms. References [BBM93a] Beasley D., Bull D. R., and Martin R. R. (1993) An overview of genetic algorithms : Part 1, fundamentals. University Computing 15(2): 5869. [BBM93b] Beasley D., Bull D. R., and Martin R. R. (1993) An overview of genetic algorithms : Part 2, research topics. University Computing 15(4): 170181. [GC94] Gendron B. and Crainic T. G. (1994) Bounding procedures for multicommodity capacitated fixed charge network design problems. Technical Report CRT-95-12, Centre de Recherche sur les Transports, University of Montreal. [KH80] Kennington J. L. and Helgason R. V. (1980) Algorithms for Network Programming. John Wiley, New York. [KRSS93] Kapsalis A., Rayward-Smith V. J., and Smith G. D. (1993) Fast sequential and parallel implementation of genetic algorithms using the gameter toolkit. In Proc. Int. Conference on Neural Networks and Genetic Algorithms. Innsbruck. [LMRS94] Lee Y. N., McKeown G. P., and Rayward-Smith V. J. (1994) Solving

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the convoy movement problem using branch-and bound on a network of transputers. In A. De Gloria M. R. J. and Marini D. (eds) Transputer Applications and Systems '94, pages 786796. IOS Press. [MRSLL94a] McKeown G. P., Rayward-Smith V. J., Lee Y. N., and Ling P. D. (1994) The convoy movement problem with delays. Technical report, University of East Anglia, Norwich. [MRSLL94b] McKeown G. P., Rayward-Smith V. J., Lee Y. N., and Ling P. D. (1994) Massively parallel integer programming. Technical Report CB/RAE/9/4/2162/025, University of East Anglia, Norwich. [MRST91] McKeown G. P., Rayward-Smith V. J., and Turpin H. J. (1991) Branch-and-bound as a higher order function. Annals of Operations Research 33: 379401. [MW84] Magnanti T. L. and Wong R. T. (1984) Network design and transportation planning: Models and algorithms. Transportation Science 18: 155. [Whi94] Whitley D. (1994) A genetic algorithm tutorial. Journal of Statistics and Computing 4: 6585. [Yen71] Yen J. Y. (1971) Finding the k shortest loopless paths in a network. Manag. Sci. 17: 712716. [YNLRS94] Y. N. Lee G. P. M. and Rayward-Smith V. J. (1994) The convoy movement problem. Technical Report SYS-C94-01, University of East Anglia, Norwich.

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14 A Comparison of Heuristics for Telecommunications Traffic Routing Jason W. Mann and George D. Smith Abstract In the routing of traffic across telecommunications networks, shortest path routing is often used on a call by call basis. Although this is the most efficient solution for each particular call, it can lead to ineffective use of the global network and even congestion. The approach adopted here is to search for routing strategies that attempt to spread the expected traffic across the network by forcing the utilisation for each link below a specified threshold. We compare the quality of solutions obtained and efficiency of two approaches to the search for such strategies, namely genetic algorithms and simulated annealing. 14.1 Introduction The telecommunications industry is one of the most important industries in the modern world and its continuing growth relies, among other things, on the solution of many differing and difficult optimisation problems. In order to cope with an ever increasing demand and rapidly changing market conditions, as well as the incorporation of new technologies, the industry needs to apply the whole spectrum of optimisation techniques to the solution of decision problems in network design, network management, resource allocation, fault management, fraud management and many other complex problem areas. These techniques range from problem specific, exact algorithms such as those based on shortest path routines for routing, through linear and integer programming to the generic heuristics such as simulated annealing, genetic algorithms, tabu search and constraint satisfaction techniques. In this paper, we describe a particular case study which the authors have carried out as part of a long-term project with Nortel Technology, formerly BNR Europe Ltd. In the following section, we describe the particular flavour of the heuristic techniques Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith© 1996 John Wiley & Sons Ltd.

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used in this study, namely genetic algorithms (GAs) and simulated annealing (SA), and review some relevant applications of GAs and SA within the telecommunications domain. In Section 14.3, we introduce the particular routing problem studied and describe the representation used for both the GA and SA application. It may be argued that in some applications, a different representation may be more effective for one paradigm than another, but in this case the representation chosen was extremely effective for both GA and SA approaches and is similar to that successfully adopted in [SHY93] and [LMRS94] for the representation of routes over a network. The resulting routing strategies obtained using the particular GA and SA applications are compared, not only with each other, but also with the routing strategy based on a shortest path routine to route the traffic on a set of specified networks. These results are presented in Section 14.4. The work was carried out on a UNIX HP 9000 712/80 workstation using the GA toolkit GAmeter and the SA toolkit SAmson, both developed at UEA to aid comparative studies using GAs and SA. The fitness function developed for the problem can be used in either of the toolkits, provided the same representation is used. 14.2 Background Genetic algorithms and simulated annealing algorithms are members of a family of heuristics commonly known as modern heuristic techniques, see [Ree93, RS95]. We give a brief introduction to each technique in order to explain the terminology used in the description of the parameters. 14.2.1 The Genetic Algorithm GA {objective is to minimise value(p) such that p ∈ U } P, Q, R : multi-set of solutions ⊂ U; initialise(P); while not finish(P) do begin Q := select(P); R := create(Q); P := merge(P, Q, R) end end GA

Figure 14.1 The genetic algorithm paradigm. Genetic algorithms are general search paradigms that simulate the mechanisms of natural evolution, see [Gol89]. The GA starts with a population of solutions

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to the problem in hand and, using selection of the fittest mechanisms and genetic recombination processes, evolves populations of solutions which converge to (near) optimal solutions to the problem. A typical version of the basic algorithm, as used in function and combinatorial optimisation, is shown in Figure 14.1. Here, P is the population of chromosomes, each one representing a solution to the problem and each evaluated according to some objective function, Q represents the gene pool containing the parents selected from P and R is the pool of offspring created from Q using genetic operators. In applying our GA to the problem described in Section 14.3, we must fix parameters such as population size, crossover mechanism, crossover rate, etc. As part of this study, a pre-trial phase is used to choose the select, create and merge mechanisms (see Figure 14.1) from a limited choice described below. The remaining parameters are fixed at values that are now used by many GA practitioners to be typical values acceptable for a wide range of applications. This includes setting the crossover rate to 60% and mutation rate to 0.1%, for example. In addition, we choose to use a small population size (20) but with selection and replacement mechanisms designed to help retain diversity. The stopping criterion used is to finish if there is no improvement in the best solution after 500 evaluations. This final point is discussed in more detail in Section 14.4. The function select in Figure 14.1 is used to sample from the (initial) population P in order to generate a gene pool, Q. The desired quality in any effective select mechanism is that it should simulate selection of the fittest. Thus, good solutions should have a better than average chance of being selected while bad solutions should have a worse than average chance. The toolkit GAmeter has many choices of select built in, but, for the purposes of this study, we limit the choice of select to two commonly used mechanisms: Roulette Select with a probability based on the relative fitness of each solution. Tournament Randomly select two solutions (2-party tournament) and accept the better solution into Q. Repeat this the required number of times until Q is filled. Similarly, GAmeter includes many genetic crossover operators that can be called from the create function. For the purposes of this study, we limit the crossover operator to one of the following commonly used operators: 1-point crossover see [Gol89], Uniform crossover see [Sys89]. Finally, the merge process determines the makeup of the next generation. Once again, many different mechanisms have been proposed and implemented for this process and the most effective of these have been incorporated into GAmeter. We limit the choice of merge here to two such mechanisms: Best fit With this mechanism, the best solution in R replaces the worst solution in P until either all solutions in P are better than the remaining solutions in R or R is empty. New Solutions Here, only the solutions in R that do not already exist in P and are better than the current members of P are accepted

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into P, replacing solutions in P that are worse. This has the effect of retaining the diversity of the population throughout the search and the added bonus of providing a multitude of effective routing strategies when the search is complete. Genetic algorithms have been used extensively in the design and management of telecommunications networks. One of the first 'real-world' applications of GAs was in the determination of network link sizes [CD87, DC87]. This process is the final stage in the design of wide-area packet switching networks that satisfy customer performance criteria while minimising network cost. The chromosome used in these studies was a set of n link speeds, where n is the number of links in the network. Each link speed was chosen from a set of possible link speeds for the associated link. The evaluation ran a simulator routing algorithm to assess the performance of the resulting network against a set of criteria, for example, link utilisation should not exceed 90%, or average link utilisation should not exceed 60%, etc. Compared with a greedy heuristic, the GA found better, and in fact, near optimal solutions, but used more processing power to do so. More recently, Davis has worked on dynamic anticipatory routing [CDQ91] and survivable network design [DOCQ93]. Several heuristics, including GAs and GA-hybrids, were used in approaches to the solution of this problem. The work by [DOCQ93] compared a GA with a mixed integer programming approach and a greedy algorithm, with the GA finding better solutions in a reasonably fast time for large networks. A new dynamic routing method based on genetic algorithms has been proposed by [SHY93] for the real-time management of dynamic traffic changes in broadband networks. The authors applied their model to a simple network and periodically searched for a new solution when the traffic changed such that the maximum call loss-rate exceeded a pre-set target value. They concluded that a solution can be obtained by a genetic algorithm, even after only a few generations, and thus within a very short span of real time. One of the techniques they used to speed up the search was to seed the population of a new search with solutions that had been used previously. In other words, past routing patterns were injected into the initial population in order to improve convergence. This memory of past solutions may be useful in itself as a collection of 'optimal' network configurations for different scenarios. See also [PW91] for an application of GAs to route traffic on an ATM network. There are many other applications of genetic algorithms to network management and network design problems in telecommunications and the reader is referred to [Man95a] for a full discussion. 14.2.2 Simulated Annealing Simulated annealing, as a mechanism for searching the (feasible) search space of an optimisation problem, was first proposed by [KGV83] and, independently, by [Cer85], Their algorithms were based on that of [MRR×53] which simulated the controlled cooling of a material by a process now known as annealing. The simulated annealing technique is essentially local search in which a move to an inferior solution is allowed with a probability that decreases, as the search progresses, according to some, typically Boltzmann-type, distribution.

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A basic, sequential version of the SA paradigm for optimisation problems is described in Figure 14.2 and a detailed overview of SA can be found in [Dow93]. SA {objective is to minimise value(p) such that p ∈ U} p := p0 where p0 is some initial solution T := T0 where T0 is some initial temperature while not finish(p, T) do while not reached equilibrium do begin randomly select p′ ∈ N(p); ∆ := value(p) value(p′); if ∆ > 0 then p := p′ else begin generate x ∈ uniform[0,1]; if x < exp(∆/T) then p := p′ end {else} end {while}; T := cool(T) end {while} accept p as the solution. end {SA}

Figure 14.2 A basic version of the simulated annealing paradigm. As with the GA, all parameters in the SA algorithm must be set prior to its use. We fix certain parameters and use a pretrial phase to select others from a limited selection. The fixed parameters for the SA approach are the type of neighbourhood, in this case fixed as a simple random 'flip' of bits, the initial temperature, fixed at 20, which is around the best value obtained for all networks using an initial 'heating' process to determine the best starting temperature, and the number of iterations per temperature step, here fixed at 30 for the Geometric Cooling, and at 1 for the Lundy & Mees Cooling, see below. As with the GA, the stopping criterion is to finish if there is no improvement after 500 evaluations. The parameter settings determined by a pre-trial phase are the cool function and the reached-equilibrium condition, see Figure 14.2. The function cool determines the cooling schedule, i.e. the sequence of values of the temperature parameter, T, used during a run of the algorithm. Typically, this consists of a sequence of monotonically decreasing values of T in order to reduce the probability exp(∆/T) of moving to a worse solution, the rate of decrease determined by the cooling rate function. A large number of variations in the choice of the cooling rate function exists, see, for example,

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[Dow93]. We choose to select from two mechanisms commonly used: Geometric Cooling Here the temperature is reduced according to the formula

Here the value of α is chosen close to one, typically in the range 0.85 to 0.95. (The value 0.9 is used in this study.) Lundy & Mees Cooling This is an example of a study [LM86] using inhomogeneous Markov chains, i.e. the temperature is decreased in gradual steps, according to the formula

where β is a constant very close to 0. (The value used in this study is β = 0.01.) Normally, when this cooling rate is used, the number of iterations at each temperature step is set to 1. Moreover, the condition reached-equilibrium determines the number of iterations, i.e. neighbourhood moves, at each temperature step, basically the length of the (homogeneous) Markov chain at each temperature step. Once again, there are many ways of choosing this condition and we limit the choice available to two effective mechanisms: Constant number of iterations Here, the number of iterations is simply governed by the number of proposed moves1, i.e. the user specifies a value for this, usually large for homogeneous systems and 1 for the Lundy & Mees cooling function, and the temperature is reduced after this has been reached and the counter reset. Variable number of iterations (feedback) Arguably, a better way of ensuring sufficient searching at lower temperatures is to only allow a temperature reduction after a prescribed number of accepted moves. An additional feature which is occasionally used in SA is to periodically increase the temperature, T, to reduce the likelihood of becoming trapped in a local minimum. The term tempering is used to describe this process. If we define cooling mechanism to mean the general behaviour of the temperature parameter, T, then the choices considered for the cooling mechanism are: Monotonic The temperature is reduced according to the chosen cooling rate and the process stops when some final temperature is reached, or some other stopping criterion is achieved. Tempering The temperature is reduced, according to the chosen cooling rate, until some value, or threshold, is achieved; it is then increased significantly and allowed to continue reducing. This occurs periodically. Within SAmson, we have adopted the following scheme to simulate the tempering process: 1 A proposed move is the process of generating a neighbour p′ of p, whether this new solution is accepted or not. An accepted move is when the proposed solution p′ becomes the current solution p.

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1. Select Tempering and enter a threshold value η, say 10%. 2. Start with a temperature T = Ts (= T0, initially). 3. Reduce temperature according to chosen cooling rate until T = η % of Ts. 4. Reset the start temperature: Ts := Ts/2 and go to step 2. Once again, with two options for each of the mechanisms cooling rate, reached-equilibrium and cooling mechanism, we have eight possible combinations of these parameter settings alone. By applying each of these to a pre-trial network, we are able to determine an effective combination to be used for the full trial. The details are described in Section 14.4. Although simulated annealing has been applied to solve problems in the telecommunications domain, there are certainly fewer such studies than with GAs, particularly in routing problems. One such study, [PCR95], compares an SA approach with a classifier system, hill climbing and random search on a routing and scheduling problem. Within a given telecommunications network, requests for services will arrive at indeterminate times. Those requests will need to be routed and possibly scheduled within a certain time frame. The various search techniques were applied to the scheduling element of this problem, a standard router being used to complete the evaluation of a schedule. The SA technique consistently outperformed the other techniques in terms of quality of solution obtained. [CM94] used SA to determine the most effective upgrade to existing networks, in terms of customer satisfaction and cost-effectiveness. Upgrading all switches, for example, would achieve maximum customer satisfaction but at a prohibitive cost. The question is, which subset of switches to upgrade to maximise some weighted sum of the two objectives. The SA implementation provided very satisfactory results, but no comparison was made with other techniques. 14.3 Problem Specification A common problem in the field of telecommunications is the routing of traffic through a given network. Automatic route finding is a central part of any telecommunications network system. Whatever the underlying network technology, there is the need to send a signal from one node to another across a route. Although the use of shortest path routing for each single call request ensures the best possible route for that particular request, it can be shown to lead to suboptimal routing or even highly congested network routing solutions when one considers the network utilisation as a whole [Wan92]. Use of global information, including not only the available link capacities but also the expected traffic profile for the period in question, can lead to routing strategies designed to minimise congestion and make better use of available network resources. This is the particular question posed for this study: what routing strategies should be adopted for a particular network to ensure that no link is over-utilised and, if possible, that all links are evenly loaded, i.e. a balanced network is achieved, for the predicted traffic profile. The work presented in this study centres around the design of optimal routing

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strategies for synchronous digital hierarchy (SDH) networks, but the approach used can be adopted for any (telecommunications) network. An SDH network uses three transmission modes: STM-1 (155 Mbits/s), STM-4 (622 Mbits/s) and STM-16 (2.4 Gbits/s). We are given a telecommunications network over which we must route multiple traffic requests in such a manner as to achieve a feasible routing assignment, i.e. no link is over-capacitated (hard constraint). This is the primary objective. In addition, we impose a secondary objective that link utilisations should all be below a specified, fixed target utilisation. Finally, the routing assignment attempts to minimise the communications costs, assuming that costs are assigned to link usage. Specifically, we are given a network G = (N,E), where N is the set of n nodes and E is the set of m edges. Associated with each edge e ∈ E is a bandwidth, b(e), and a cost, c(e). For the applications we have been considering, the network is always bidirectional, there is only a single type of traffic, n is in the range 7 to 50 and m is between 9 and 125. The bandwidths, {b(e) | e ∈ E}, lie in {63,4 × 63,16 × 63}, which is determined by the number of VC12s (virtual containers) which fit onto an STM1, STM4 and STM16, respectively, see [New93]. We thus introduce a bandwidth factor which is a number in {1,4,16}. This is multiplied by 63 to give the actual bandwidth. For each distinct v, w ∈ N, there is an amount of network traffic, t(v, w), which must be routed from v to w in the network. This traffic must all be routed on the same path, P(v, w), which has to be determined. The total traffic, f(e), on any edge e ∈ E is given by

This must not exceed the bandwidth of that edge, i.e.

This can be achieved by minimising the deviation f(e) b(e), so long as f (e) > b(e), i.e.

A second objective is to find a minimum cost allocation of traffic through the network, satisfying constraint (02). The cost of routing all traffic, t(v, w), on the path, P(v, w), between v and w is given by

This is summed over all possible source/destination pairs to yield the second objective

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bearing in mind the bidirectional constraint so we are summing only over n(n 1)/2 possible pairs. The third and final objective specified is to minimise the deviation from a target utilisation, u, for each link on the network, i.e.

Thus, so long as f(e) > (u × b(e)/100), for at least one link e ∈ E, there will exist some pressure on the optimisation process to find a more balanced solution. For our experiments, we set u = 50. The full and final objective is therefore to

Here, p, w1 and w2 are weights applied to the various terms in the evaluation function. These can be adjusted as necessary to prioritise any of the objectives. Obviously, attaining objective (O2) means that we have automatically attained objective (O2). However, the inclusion of both terms in the evaluation function means that we can concentrate on feasible solutions while still retaining the flexibility to choose between low cost or more balanced solutions. Ideally, the values for the weights p, w1 and w2 would be determined for each problem studied in order to achieve the desired characteristics in the solution. After a large number of trials on different networks, concentrating on wellbalanced, feasible routing strategies, the preferred values for these parameters, and the values used throughout the experiments, are p = 10, w1 = 1 and w2 = 5. The value of the objective function (O2) can be determined once the routing allocation has been done. This brings us to the representation of the routing decisions. The objective of the exercise is to determine the path, P(v, w), for each v, w ∈ N, to minimise the expression (O2). If there are n nodes in the network, then there are n(n 1)/2 possible pairs of connections. Thus the representation is simply a string of n(n 1)/2 paths, one for each source-destination pair. For the genetic algorithm and the simulated annealing algorithm, therefore, the representation contains n(n 1)/2 'genes' or parameters, each representing a path for a unique pair. In order to avoid explicitly representing each path, each gene holds a pointer to a look-up table held in memory, one for each connection pair. The question is: which paths should we hold in this look-up table? The obvious choice is the kth shortest paths [Yen71], where k has to be decided. By limiting the size of k, one can ensure that only reasonable routes are considered for each connection. For our purposes, we found that k = 8 was acceptable, thus ensuring that excessively long paths were removed from consideration without limiting the search space sufficiently to preclude an effective search. For larger networks, or networks with multiple links

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between nodes, this figure may need to be adjusted accordingly. Indeed, there is an argument for having different values of k for different node pairs, depending on 'distance' apart. There is a danger that, for small values of k, we may be precluding the optimal solution from our search. However, we argue that we are basing the search on the estimated traffic profile, and that effective (sub)-optimal solutions found within a reasonable time frame are acceptable. A more effective strategy, and one that we are currently pursuing, is to hold the k shortest diverse paths, where diverse can mean node or link diverse. As previously stated, we limit the choice of the path, P(v,w), to be one of the eight shortest paths. In this case, we can represent this choice using three bits for each source/destination pair. This can clearly be generalised to obtain a k bit representation should we wish to consider the 2k shortest paths. When the problem is initially loaded, a pre-processing stage determines, for each node pair, the eight shortest possible paths plus associated costs, and stores them in a look-up table. A routing solution is thus represented as a concatenation of n(n 1)/2 three-bit strings. This is then fed into the 'fitness' function (of the GA or SA application) to determine an overall fitness for that routing strategy, depending on the values of the weights p, w1 and w2. The results reported in Section 14.4 were obtained using the GA toolkit GAmeter and the SA toolkit SAmson. The coding of the evaluation function, equation (O2), once tested, can be used in either toolkits, allowing ready comparison of the two techniques for a given problem, see [MKS95, Man95b]. Each of these toolkits has a Windows-based interface developed on X/Motif on Unix. In addition to the many interactive problem-independent features available, the user can develop problem-dependent windows to allow easy input of parameter choices and to produce visual displays of the results. The resulting GA and SA applications for this problem have two problem-dependent windows in common. The first of these is a Problem Parameters dialog box, in which the user can set the values of the three parameters, p, w1 and w2, appearing in the objective equation (O2), see [Man95a]. The second dialog box which is common to both toolkits is the Utilisation Table of the best solution found so far, see Figure 14.3. This displays, for each link in the network, the nodes adjoining the link, the bandwidth (or capacity), the load (i.e. the amount of traffic on the link) and the utilisation (= load/bandwidth × 100). Additionally, the GA application, GAmeter, offers the Utilisation Table for any selected member of the current population, thus presenting a wider choice of strategies. 14.4 Results To compare the performances of GAs and SA applied to this problem, it is necessary to compare related attributes. We note here that the SA application can be enhanced by incorporating some 'delta evaluations'. This is due to the fact that any new solution is generated from an existing solution through a relatively small change, based on some neighbourhood operator. It is possible to derive the 'fitness' of the new solution from

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Figure 14.3 An example Utilisation Table in this case for the best solution in the current population in the GAmeter application. that of the current solution and the cost of the change. This makes the SA approach much more effective in terms of processing time. Recent studies [SKRSK95] comparing the number of evaluations per second achieved using both a GA and SA approach for the Radio Link Frequency Assignment Problem showed that SA can achieve at least 50 times the number of evaluations per second achieved by the corresponding GA. Building 'delta evaluation' into the GA to a limited extent reduced this advantage to around 20 times the number. For this reason, we do not use the CPU time as one of the measures of comparison, although we note in passing that this is a major advantage of SA over GAs. Instead we compare the number of function evaluations in both algorithms. Thus for the GA, we present the results as Fitness versus Number of evaluations while for the SA, we use Fitness versus Number of proposed moves2. As described previously, we fix many of the parameters used within each algorithm and use a pre-trial study on a moderately sized network (12 nodes) to determine the remaining parameter settings. 2 Each proposed move represents an evaluation of a new solution.

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Table 14.1 The parameter options for the GA and SA applications tested during the pre-trial phase GA

Options

select mechanism

Roulette (R)

Tournament (T)

create mechanism

One-point (O)

Uniform (U)

merge mechanism

Best Fit (B)

New Solutions (N)

SA

Options

cooling rate

Geometric (G)

Lundy and Mees (L)

reached-equilibrium

Constant (C)

Feedback (F)

cooling mechanism

Monotonic (M)

Tempering (T)

For the GA, we chose the select, create and merge functions from one of two options, see Section 14.2.1 and Table 14.1. Similarly, for the SA application, we selected the cooling rate, reached-equilibrium condition and cooling mechanism from one of two options, see Section 14.2.2 and Table 14.1. Thus there were a total of 8 parameter settings for each heuristic. Each experiment on this pre-trial network was repeated 10 times for each setting. This set of experiments was designed, not so much to determine 'optimal' settings for these parameters, but to allow a fair comparison of the techniques through moderate tuning. Many more combinations would have to be tested, and many more runs would be required, to achieve the most effective GA or SA parameters for this problem. Even then, we cannot guarantee that this would be the best set for a wide range of problems. The results of this 'tuning' phase are shown in Table 14.2, where the letters in columns 2, 3 and 4 refer to the entries in Table 14.1. For each heuristic, the choice of parameter setting which gave a minimum value to a weighted sum of the minimum value, the average and the standard deviation was chosen as the representative setting. This turned out to be parameter set 3 for the GA (Roulette, One-point crossover and New Solutions) and parameter set 3 for the SA (Geometric, Constant and Tempering). The case for the final choice of the SA settings is not so clear cut as that for the GA.

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Page 247 Table 14.2 The results of the 'tuning' stage of the heuristics. The letters in columns 2, 3 and 4 refer to the entries in Table 14.1 GA trial

Select

Create

Merger

Min.

1

R

O

B

718.5 734.35 18.0

2

R

U

B

722.0 751.05 29.9

3

R

O

N

714.5 731.85 15.2

4

R

U

N

714.5 733.10 18.3

5

T

O

B

717.5 740.90 30.5

6

T

U

B

719.5 740.50 19.2

7

T

O

N

718.0 730.00 17.6

8

T

U

N

720.5 747.95 40.25

SA trial

Cooling Rate

Temp. Equil.

Cooling Mech.

1

G

C

M

716.5 748.15 29.0

2

G

F

M

721.0 731.95 15.5

3

G

C

T

718.0 737.95 18.2

4

G

F

T

749.0 783.45 25.8

5

L

C

M

717.0 751.25 38.3

6

L

F

M

717.0 743.00 28.5

7

L

C

T

718.5 744.25 38.3

8

L

F

T

717.0 747.05 32.6

Min.

Ave.

Ave.

SD

SD

The question remains: for how long do we run each experiment? There are many options open to us, including running for the same number of evaluations or for the same time. The choice we made was to stop the run when there was very little likelihood of further improvement. In fact, as noted earlier, each run ceased when no improvement on the best solution occurred after 500 evaluations. Charts showing the convergence of the fitness function for both approaches showed a striking similarity for any given network, indicating that we were comparing the GA and SA at a similar state of convergence, see [Man95a]. We note in passing, however, that the SA achieves this state in an order of magnitude fewer evaluations than the GA for all problems studied. Now that we have set the parameters for the GA and the SA applications, the final step is to apply the resulting algorithms to a range of test networks with known traffic profiles. The networks chosen are shown in Figure 14.4.

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Figure 14.4 The problem networks. The first network is a small network, with 7 nodes and 9 links. However, the size of the network is only half the story, the other half being the predicted traffic profile. We assume for this network that the link capacities are uniform (STM1s). Network 2 is a typically meshed network containing 15 nodes and 21 links. Once again, we assume that the link capacities are uniform (STM1s). The main characteristic to note is the 'bridge', i.e. the single link to one node. Although telecommunications networks are generally designed to avoid this situation, this may be the result of a link failure. Apart from this, the network possesses a fair degree of survivability under link failure. Network 3 is a larger, highly meshed network, typical of telecommunications networks that have had to grow to meet a growing demand. Not only does the number of nodes and links increase, but also the quantity of traffic on the network. Once again, we assume that the link capacities are uniform (STM1s).

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Finally, we consider a network consisting of a 'ring of rings'. Ring structures are desirable properties in telecommunications networks as they offer a high degree of survivability. If a link fails, the traffic can still be routed through the network. Such ring-based structures are very rarely symmetrical, and the network used in this final example is skewed accordingly. An additional feature in this final example is the use of STM4s forming the inner ring, shown as bolder lines, the remaining links being STM1s. Recall that an STM4 can carry 4 times the traffic of an STM1. Any node transmitting to a node in another ring must use the inner ring to do so, thus the requirement for more capacity for the inner ring. Note also that, should there only be two levels of rings, with no crossover links, then there are at most only eight unique paths between any two nodes, thus simplifying the task of generating the eight shortest paths. The results of applying the GA and the SA algorithm, with parameter settings prescribed by the pre-trial phase, are shown in Table 14.3. Each algorithm was applied 10 times to each network and the minimum, average and standard deviation of the number of evaluations (Ev) and the resulting fitness values (Fv) are shown for each. If we restrict attention initially to the quality of solutions obtained, i.e. the fitness values obtained, there is very little to separate the two algorithms. For network 1, SA finds the best solution 7 times out of 10, while the GA manages it only 4 times. Thus the SA approach has a more consistent performance for this network. However, the GA is marginally better for network 2 and slightly better for network 3, while SA performs more consistently for network 4. The main difference between the two approaches is in the number of evaluations required to attain this level of quality. Reference to the Ev columns in Table 14.3 shows us that the GA approach requires anything between 5 and 15 times as many evaluations as SA to achieve the same level of quality. For completeness, we note that, for network 1, a typical, randomly created initial solution has a fitness value in the range 800 to 900, and the traffic matrix used in this network, when routed using a shortest path strategy, yields one overutilised link and has a fitness of 369.0. For network 2, a typical initial solution has a fitness of approximately 5800 and a shortest path routing strategy yields two over-utilised links and a fitness of 3245.0. For network 3, a typical initial solution has a fitness around 40 029, while the shortest path routing generates 5 over-utilised links and a fitness of 30 915. Finally, the shortest path routing strategy for network 4 yields a solution that has 4 overloaded links, with a fitness of 3007.5. This last result is depicted in Figure 14.5, where the link utilisation obtained using the shortest path strategy is compared with the best found using either the GA or the SA approach (Table 14.3).

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Page 250 Table 14.3 GA/SA results for all networks. The figures in brackets are the number of times the minimum was found in 10 runs, if greater than 1 Genetic Algorithm Ev Network 1

Min Average St. Dev.

Network 2

Min Average St. Dev.

Network 3

Min Average St. Dev.

Network 4

Min Average St. Dev.

2 207 6 916 4 502 23 090 25 315 1 706 325 649 360 867 36 353 16 715 22 533 6104

Simulated Annealing

Fv

Ev

225.5 (4)

Min

236.9

Average

13.5

St. Dev.

2790 (10)

Min

2790

Average

0

St. Dev.

28 257.5

Min

28 323.8

Average

44.2

St. Dev.

2343.5

Min

2366.4

Average

14.5

St. Dev.

571 1475 477 4913 5783 735 22 070 24 788 1725 3377 4860 1234

Fv 225.5 (7) 230.35 7.9 2790 (4) 2791.6 1.7 28 364.0 28 519.6 89.7 2343.5 2365.0 10.6

For comparison, we report in Table 14.4 the approximate execution times for the various approaches on each problem network. Note that the shortest path execution time is the preprocessing time taken to determine all the 8 shortest paths for each problem. Table 14.4 Approximate execution times for each algorithm Shortest Path Times (secs)

GA Time (secs)

SA Time (secs)

Network 1

0.02

10

1

Network 2

0.2

200

20

Network 3

0.35

17 000

700

Network 4

17

350

25

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The times for the GA and SA applications are the times to finish the run, i.e. no improvement after 500 evaluations. It is obvious from Network 3, for instance, that the GA is continuing to improve the solution very gradually even late on in a run.

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Figure 14.5 Comparison of the link utilisation of different routing strategies for network 4. The top chart shows the link utilisation based on the shortest path routing, with 4 violated links. The bottom chart shows the best routing solution found by both GAs and SA. 14.5 Current and Future Work The work presented in this paper is currently being extended to incorporate: Routing of different traffic types For instance, data traffic may incur different costs and require a different priority objective than video or voice. Minimising delay for delay-sensitive traffic (voice, video) This is an additional objective which is built into equation (14.7).

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Splitting of traffic between multiple routes In many networks, the capability to split traffic across multiple links is an important factor with respect to reducing network overload and increasing network survivability. The actual percentage split may be user-controlled or built in to the optimisation process. Multiple destination routing For video-on-demand services, for instance, there is a requirement to route the same traffic to multiple destinations. The problem then extends to locating the distribution nodes, i.e. nodes at which one copy of a traffic item arrives and from which multiple copies are sent on different links to different destinations. This problem is a variant of the Steiner Tree Problem in Graphs (see [KRSS93]). Miscellaneous constraints These include the deliberate inclusion or exclusion of particular nodes and/or links. For example, some (emergency) services may be routed only over the shortest paths. Provision of alternative routes This feature returns multiple routes for each service request, allowing the network operator or real-time router to choose the most appropriate. Link and node diverse routing To improve survivability, multiple routes can be generated in such a manner that no node or link appears in more than one route. 14.6 Conclusions A genetic algorithm and a simulated annealing algorithm have been applied to a general routing problem on a small set of sample telecommunications networks. The aim of the exercise is to route the traffic across the network in an attempt to balance the load. The results obtained have been compared with those using a traditional shortest path strategy. For all the sample networks, the solutions obtained using either the GA or the SA were feasible, that is to say that no link was over utilised. In many of the solutions, the secondary constraint of pushing the utilisation below 50% was achieved on the majority of links. In all cases, the shortest path algorithm failed to find a feasible routing strategy. Although both GA and SA are stochastic techniques, there is evidence that for this problem, both methods provide consistently good results, with relatively small standard deviations in the fitness values found. There was little dispersion in the 10 run plots of fitness versus number of evaluations for the GA and similarly for the SA, see [Man95a]. Although the GA and the SA application perform equally well, the GA displays a slight edge for the larger and more complex network. However, in terms of the quality of solution obtained generally, there is very little to separate the techniques. With more intuitive tuning of either algorithm, it is likely that comparable performance will be attained even for larger networks.

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The SA application is, however, much more effective in terms of processor time. Typically, the SA converges within 10% to 25% of the number of function evaluations required by the GA to converge. Added to that the possibility of building in 'delta evaluations' to the SA application would argue in favour of this approach. In conclusion, we can say that using global information about the state of the network can lead to better utilisation of the resources, and to a more survivable network as a result. This study has shown that both genetic algorithms and simulated annealing can be used to provide routing strategies to meet multi-objective requirements. 14.7 Acknowledgements The research is being undertaken within a Teaching Company Scheme programme, grant reference GR/K40086, between Nortel Technology (formerly BNR Europe Ltd) and the University of East Anglia. References [CD87] Coombs S. and Davis L. (1987) Genetic algorithms and communication link speed design: Constraints and operators. In Grefenstette J. (ed) Proceedings of the Second International Conference on Genetic Algorithms and their Applications, pages 257260. Lawrence Erlbaum Associates, Hillsdale, NJ. [CDQ91] Cox L. A., Davis L., and Qiu Y. (1991) Dynamic anticipatory routing in circuit-switched telecommunications networks. In Davis L. (ed) Handbook of Genetic Algorithms. Van Nostrand, New York. [Cer85] Cerny V. (1985) A thermodynamical approach to the travelling salesman problem: an efficient simulation algorithm. J. of Optimization Theory and Applic. 45: 4155. [CM94] Crabtree I. B. and Munaf D. (1994) Planning beneficial and profitable network upgrade paths. BT Technology Journal 12(4). [DC87] Davis L. and Coombs S. (1987) Genetic algorithms and communication link speed design: Theoretical considerations. In Grefenstette J. (ed) Proceedings of the Second International Conference on Genetic Algorithms and their Applications, pages 252256. Lawrence Erlbaum Associates, Hillsdale, NJ. [DOCQ93] Davis L., Orvosh D., Cox A., and Qiu Y. (1993) A genetic algorithm for survivable network design. In Forrest S. (ed) Proc. Fifth Int. Conf. on Genetic Algorithms, pages 14081415. Morgan Kaufmann, San Mateo, CA. [Dow93] Dowsland K. (1993) Simulated annealing. In Reeves C. (ed) Modern Heuristic Techniques, chapter 2, pages 2069. Blackwell Scientific, Oxford. [Gol89] Goldberg D. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading, Mass. [KGV83] Kirkpatrick S., Gelatt C. D., and Vecchi M. P. (1983) Optimization by simulated annealing. Science 220 (4598): 671680. [KRSS93] Kapsalis A., Rayward-Smith V. J., and Smith G. D. (1993) Solving the graphical Steiner tree problem using genetic algorithms. Journal of the Operational Research Society 44(4): 397406. [LM86] Lundy M. and Mees A. (1986) Convergence of an annealing algorithm. Math. Prog. 34: 111124. [LMRS94] Lee Y. N., McKeown G. P., and Rayward-Smith V. J. (1994) The convoy movement problem with delays. DRA Project Final Report; completed

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December 1994. [Man95a] Mann J. W. (1995) Applications of genetic algorithms in telecommunications. Master's thesis, University of East Anglia. [Man95b] Mann J. W. (1995) X-SAmson v1.0 User Manual. Technical report, University of East Anglia. [MKS95] Mann J. W., Kapsalis A., and Smith G. D. (1995) The GAmeter toolkit. In Rayward-Smith V. (ed) Applications of Modern Heuristic Methods, chapter 12, pages 195209. Alfred Waller. [MRR+53] Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H., and Teller E. (1953) Equation of state calculation by fast computing machines. J. of Chem. Phys. 21: 10871091. [New93] Newall C. (ed) (1993) Synchronous Transmission Systems. Northern Telecom Europe Ltd, London. [PCR95] Purohit B., Clark T., and Richards T. (1995) Techniques for routing and scheduling services on a transmissions network. BT Technology Journal 13(1): 164172. [PW91] Pan H. and Wang I. Y. (1991) The bandwidth allocation of ATM through genetic algorithms. In Proceedings of GLOBECOM'91, pages 4.4.14.4.5. IEEE Press. [Ree93] Reeves C. (ed) (1993) Modern Heuristic Techniques for Combinatorial Problems. Blackwell Scientific. [RS95] Rayward-Smith V. (1995) Applications of Modern Heuristic Methods. Alfred Waller. [SHY93] Shimamoto N., Hiramatsu A., and Yamasaki K. (1993) Dynamic routing control based on a genetic algorithm. In Proceedings of IEEE International Conference on Neural Networks, pages 11231128. IEEE Computer Science Press, San Francisco, CA. [SKRSK95] Smith G. D., Kapsalis A., Rayward-Smith V. J., and Kolen A. (1995) Radio link frequency assignment problem: Implementation and testing of genetic algorithms. Technical Report 2.1, EUCLID CALMA. [Sys89] Syswerda G. (1989) Uniform crossover in genetic algorithms. In Schaffer J. (ed) Proceedings of the Third International Conference on Genetic Algorithms, pages 29. Morgan Kaufmann, Los Altos, CA. [Wan92] Wang (1992) dRouting Control in Networks. PhD thesis, Imperial College, London. [Yen71] Yen J. (1971) Finding the k shortest loopless paths in a network. Management Science 17(11): 712716.

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15 A Brief Comparison of Some Evolutionary Optimization Methods A.J. Keane Abstract The subject of evolutionary computing is a rapidly developing one where many new search methods are being proposed all the time. Inevitably, some of these new methods will not still be in current use in a few years as a small subset becomes the preferred choice of the optimization community. A key part in this process is the back-to-back testing of competing methods on test problems that simulate real problems. This brief paper presents results for such a test, focusing on the robustness of four methods when applied to a pair of high dimensional test functions. The work presented shows that some modern search methods can be highly sensitive to mistuning of their control parameters and, moreover, that different problems may well require radically different settings of such parameters if the searches are to be effective. 15.1 Introduction This paper briefly reviews the behaviour of four different evolutionary optimization methods when applied to a pair of difficult, high dimension test functions. The methods considered are the genetic algorithm (GA) [Gol89], evolutionary programming (EP) [Fog93], evolution strategies (ES) [BHS91] and simulated annealing (SA) [KGV83]. The two problems considered are the royal road function proposed by Holland, here called 'jhrr' [MFH92] and the fifty dimensional 'bump' problem introduced by Keane [Kea95]. These two problems are very hard for most optimizers to deal with: 'jhrr' is highly discontinuous and very misleading for most methods. Conversely, 'bump' is quite smooth but contains tens of thousands of peaks, all of similar heights. Moreover, its optimal value is defined by the presence of a constraint boundary. The purpose of the work presented is to see how well the different methods cope with these tasks and, more importantly, to investigate what parameter settings are required Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith© 1996 John Wiley & Sons Ltd.

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for robust performance on these two very different problems. It should be stated at the outset, however, that in terms of the sophistication with which each method is implemented, the GA used is the most complex and to some extent should therefore be expected to give the best results. Nonetheless, the comparisons are representative of the kind of tests typically set for new optimization methods, i.e., comparison with the best available alternatives. Still, the reader should be warned that more sophisticated implementations of the other methods could be expected to improve their overall best performance. Finally, in this introductory session, it must be remembered that for any given problem it is always possible to produce a dedicated optimizer that will work better than any other, more general purpose approach: thus comparisons between methods should be treated with care. 15.2 The Optimization Problems The optimization problems to be tackled here have already been discussed in the literature and so will be only briefly outlined. The 'bump' problem is defined as

for

subject to

starting from

where the xi are the variables (expressed in radians) and n is the number of dimensions. This function gives a highly bumpy surface (Figure 15.1 shows the surface for n = 2) where the true global optimum is usually defined by the product constraint. The 'jhrr' problem takes a binary string as input and produces a real value which must be maximized. There are no constraints to be satisfied. The string is composed of 2k non-overlapping contiguous regions, each of length b + g. With Holland's defaults, k = 4, b = 8, g = 7, there are 16 regions of length 15, giving an overall string length of 240. Each region is divided into two non-overlapping pieces. The first, of length b, is called the block and the second of length g the gap. In the fitness calculation only the bits in the block part of each region are considered. The calculation consists of two steps. The part calculation adds to the block's fitness by v for every 1 it contains up to a limit of m*. If a block contains more than m* 1's but less than b 1's it receives v for each 1 over the limit. The default settings are v = 0.02 and m* = 4, so a block with six 1's is assigned a fitness of (6 4) × 0.02 = 0.04. Lastly, if a block consists

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entirely of 1's it receives nothing from the part calculation but it is then considered complete. This then leads to the bonus calculation which rewards complete blocks. The first complete block receives an additional u*, default 1.0, and each subsequent block u, default 0.3. Next, adjacent pairs of complete blocks are rewarded in the same way and then four complete blocks in a row and so on until all 16 blocks are complete. This leads to the maximum objective value which is 1.0 + (1.0 + 0.3) + (1.0 + 3 × 0.3) +(1.0 + 7 × 0.3) + (1.0 + 15 × 0.3) = 12.8. The presence of the gap regions, which do not affect the calculation, ensures that there are a considerable number of strings that have this maximal value. Although the royal road function was designed to be easy for a GA to deal with and hard for other methods, it turns out that even GA's find it difficult to solve [FM93]. It is therefore useful as an extreme example of a case where the relationship between the variables in the function and its value are extremely non-linear, exhibiting many step-like changes. 15.3 The Optimizers Optimization problems of the sort discussed here are characterized by having many variables, highly non-linear relationships between the variables, and an objective function that has many peaks and troughs: in short they are difficult to deal with. The search for methods that can cope with such problems has led to the subject of evolutionary computation. Techniques in this field are characterized by a stochastic approach to the search for improved solutions, guided by some kind of evolutionary control strategy. There are four main methods in use today: (i) genetic algorithms (GA) [Gol89], where the methods of Darwinian evolution are applied to the selection of 'fitter' designs; (ii) evolutionary programming (EP) [Fog93], which is a more heuristic approach to the problem based on ranked mutations; (iii) evolution strategies (ES) [BHS91], where individual designs are mutated using adaptive mutation rates, individually tuned to each variable in the problem and (iv) simulated annealing (SA) [KGV83], where the control strategy is based on an understanding of the kinetics of solidifying crystals and where changes are sometimes allowed even if they make the solution worse. The first three of these methods work on groups of designs called populations while the last deals with only one solution at a time. The versions of the methods used here are fairly standard and as discussed in the references but with the exceptions detailed below. All the methods are set up to use either a one-pass external penalty function or an evolving FiaccoMcCormick internal and external function that is made more severe as the search progresses [Kea94]. These are, of course, not used on the 'jhrr' problem, which is unconstrained. The methods are also all set up to work with a binary encoding of the real valued design variables using up to 16 bit accuracy (default 12 bit) except the ES which works directly on the real valued quantities. This encoding is quite normal with a GA and also allows a simple method for effectively randomizing the variables for the SA and EP. Such a scheme helps the methods deal with the 'jhrr' problem, as there is then a direct mapping between mutations and function value. The GA used encompasses a number of new ideas that have proven well suited to engineering design problems [Kea93] [Kea95]. It uses an elitist survival strategy which ensures that the best of

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each generation always enters the next generation and has optional niche forming to prevent dominance by a few moderately successful designs preventing wide ranging searches. The main parameters used to control the method may be summarized as: 1. Ngen the number of generations allowed (default 10); 2. Npop the population size or number of trials used per generation which is therefore inversely related to the number of generations given a fixed number of trials in total (default 100); 3. P[best] the proportion of the population that survive to the next generation (default 0.8); 4. P [cross] the proportion of the surviving population that are allowed to breed (default 0.8); 5. P[invert] the proportion of this population that have their genetic material re-ordered (default 0.5); 6. P[mutation] the proportion of the new generation's genetic material that is randomly changed (default 0.005); 7. a proportionality flag which selects whether the new generation is biased in favour of the most successful members of the previous generation or alternatively if all P[best] survivors are propagated equally (default TRUE) and 8. the penalty function choice. The niche forming method used here is based on MacQueen's Adaptive KMEAN algorithm [And75] which has recently been applied with some success to multi-peak problems [YG93]. This algorithm subdivides the population into clusters that have similar properties. The members of each cluster are then penalized according to how many members the cluster has and how far it lies from the cluster centre. It also, optionally, restricts the crossover process that forms the heart of the GA, so that large successful clusters mix solely with themselves. This aids convergence of the method, since radical new ideas are prevented from contaminating such sub-pools. The version of the algorithm used here is controlled by: 1. Dmin the minimum non-dimensional Euclidean distance between cluster centres, with clusters closer than this being collapsed (default 0.1); 2. Dmax the maximum non-dimensional Euclidean radius of a cluster, beyond which clusters sub-divide (default 0.2); 3. Nclust the initial number of clusters into which a generation is divided (default 25); 4. Nbreed the minimum number of members in a cluster before exclusive inbreeding within the cluster takes place (default 5) and 5. α the penalising index for cluster members, which determines how severely members sharing an over-crowded niche will suffer, with small numbers giving less penalty (default 0.2), i.e., the objective functions of members of a cluster of m solutions are scaled by mmin(α,1)[1 (E/Dmax)α] + (E/Dmax)α, where E is the Euclidean distance of the member from its cluster centre (which is always less than Dmax). The EP routine is exactly as per the references, except that, as already noted, it works with a binary encoding and uses a choice of penalty function. It works by

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forming a given number of random guesses and then attempts to improve on them, maintaining a number of best guesses as the process continues (the population). In this method evolution is carried out by forming a mutated child from each parent in the population where mutation is related to the objective function so that successful ideas are mutated less. The objective functions of the children are then formed and a stochastic ranking process used to select the next parents from the combined set of parents and children. The best of the solutions is kept unchanged at each pass to ensure that only improvements are allowed. The mutation is controlled by a variable which sets the order of the mutation process with ranking. In all cases the best parent is not mutated and almost all bits in the worst are changed. The stochastic process for deciding which elements survive involves jousting each member against a tournament of other members selected at random (including possibly itself) with a score being allocated for the number in the tournament worse than the case being examined. Having scored all members of both parent and child generations the best half are kept to form the next set of parents. The average number of solutions in the tournament is set by a control variable. The ES routine used allows either the (µ = λ) or the (µ, λ) variants but does not use correlated mutations. It allows for either discrete or intermediate recombination on both design variables and the mutation control vectors, with the mutations being applied to non-dimensionalized versions of the design vectors. The algorithm works by forming a given number of random guesses and then attempts to improve on them, maintaining a number of best guesses as the process continues (the population). Evolution is carried out by forming a child population from mating pairs of parents, followed by mutation. The next generation is then made up from good members of the old and child populations. Two methods can be used: either the best from the combined and child populations are kept (µ = λ) or, alternatively, only the best from the child population are used (µ, λ), with any shortfall being made up from the best of the parents. The mutations are controlled by vectors of standard deviations which also evolve with the various population members. The rate at which these S.D.'s change is controlled by a parameter of the method. Additionally, the breeding of new children can be discrete and then individual design vector values are taken from either parent randomly, or intermediate when the values are the average of those of the parents. A similar interchange takes place between the vectors of S.D.'s of the parent designs. The best of the solutions is remembered at each pass to ensure that the final result is the best solution of all. The SA used has a logarithmic cooling schedule which is based on the following strategy: 1. choose the total number of trials N; 2. set NT = NP; 3. then for i = 1 to NT let

and do N/NT tests at this temperature.

Here there are three parameters: P, W and C : P lies between 0 and 1 with 0 implying a random search at one temperature and 1 a single trial at each temperature. W is set to lie in the range 1 to 10 and C 0.1 to 10, thus large values of W give a wide range of temperatures, while large values of C bias this range to lower, colder values (W of 1.0 fixes all temperatures to be the same while C of 2.0 biases the temperatures equally between hot and cold). Here the default values are 0.3333, 5.0 and 2.0 respectively. Finally the 'Boltzmann' test is applied as follows:

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1. always keep a better result; 2. keep a worse result if exp((Unew Uold)/(dUfirstTi)) > R where Uold is the previous objective function value, Unew is the new (worse) value, R is a random number between 0 and 1 and dUfirst is the magnitude of the difference between the objective function of the starting point in the search and the first trial at the highest temperature (and takes the place of the Boltzmann constant). The use of dUfirst effectively non-dimensionalizes the annealing temperature and is vital for the schedule to be general purpose. The method used here also allows the user to control how big the changes are between the current point and the next point. This is done by using a binary discretization of the design vector and then allowing a given probability of a bit flip to get to the next trial (by default a 10% chance of each and every bit being flipped). This method of choosing neighbours allows quite radical changes to be proposed at all times and seems to help the search. 15.4 Initial Results Application of the various optimizers to the two test problems using 150,000 trials leads to the results of Tables 15.1 and 15.2, which are illustrated in Figures 15.2 to 15.9. For these runs a population size of 250 was used for the GA, while for the EP and ES methods populations of 50 were taken. The (305, λ) version of the ES was used together with a child population size of 100, i.e., twice that of the parent population. The large GA population size reflects the results of earlier studies using the clustering method which show that it works better with bigger populations. Similar increases for the EP and ES methods seem to reduce their efficiency on the functions optimized here. The one pass penalty function was adopted for 'bump' in all cases. The figures show the optimization traces for five runs in each case and it is seen that all the methods find the two problems very difficult to deal with. It is notable that the searches all tend to get stuck while dealing with the 'jhrr' problem. This is no doubt due to the discrete, step-like nature of its objective function. The EP method in particular seems to suffer in this respect. Also clearly visible are the 'liquid', 'freezing' and 'frozen' stages in the SA runs, indicating that the generic schedules used have spanned the correct temperature ranges. It is also clear that the GA seems to work best and the SA worst with the other two methods somewhere in between. It is, of course, to be expected that all methods could be tuned to produce improved results (it is known, for example, that a hand-coded annealing schedule allows the SA to perform at least as well as the EP and ES methods on the problems studied here). A good optimizer should, however, need little attention in this regard if it is to be widely useful. The remainder of this paper addresses this aspect of optimization.

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Page 261 Table 15.1 Initial optimization results for 'bump', 150 000 trials method

1

2

3

4

5

avg.

GA

0.778

0.777

0.785

0.780

0.776

0.779

EP

0.706

0.706

0.634

0.617

0.703

0.673

ES

0.610

0.597

0.595

0.570

0.516

0.578

SA

0.389

0.423

0.357

0.423

0.384

0.395

Table 15.2 Initial optimization results for 'jhrr', 150 000 trials. method

1

2

3

4

5

avg.

GA

7.30

8.02

7.08

10.68

10.66

8.75

EP

3.42

3.86

7.08

5.60

4.08

4.81

ES

7.48

5.60

7.40

5.24

5.74

6.29

SA

4.82

3.84

3.84

3.40

3.62

3.90

15.5 Optimizer Tuning Having illustrated the difficulties the four methods find with the two test problems, attention is next turned to tuning the parameters that control the actions of the optimizers. Selecting these parameters represents an optimization problem in its own right and a code has been developed to tackle this essentially recursive problem [Kea95]. It is, as might be expected, a difficult and time consuming problem to deal with and is not pursued further here. Nonetheless, the GA used does reflect the results of this earlier study on the 'bump' problem, which may go some way to explain its good performance. It has not, however, been tuned to 'jhrr' and so its robustness with this very different problem with the same parameter settings is encouraging (if the mutation control rate P[mutation] is increased, the GA often achieves the true optimum of 12.8 on this problem). Having noted that a considerable amount of effort has been devoted in the past to tuning the variant of the GA used, attention is next focused on the EP and ES methods: is it possible to improve their performance by careful selection of their most important control parameters ? The EP method has two such parameters (aside from the choice of penalty function, run length, population size and number of bits in the encoding used here). These are the order of the mutation process with ranking (Imutnt = 1 for linear, 2 for quadratic, etc.,) and the tournament size used in the selection process as a fraction of the population size (ftourn). The ES method has slightly more

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controls, these being the size of the child population compared to the main or parent population (fchild), the mutation vector evolution control rate (∆σ), the choice of intermediate or discrete crossover for both the design and mutation vectors and lastly the distinction between the (µ + λ) and (µ, λ) forms of the method. Figures 15.10 and 15.11 show contour maps of the variation in the average objective function for 'bump' and 'jhrr', respectively. In both cases these are averaged over five runs and are plotted for changes in Imutnt and ftourn using the EP method. 20 000 evaluations have been allowed per run and all other parameters kept as per the previous figures. Figures 15.12 and 15.13 show similar plots using the ES method for variations in ∆σ and fchild, using discrete variable crossover, intermediate mutation vector crossover and the (µ, λ) form of the method, as before and which trials have shown best suit both these problems (n.b., as has already been noted, when the child population is smaller than the parent population the shortfall is made up from the best of the parents). In all cases the small square marker on the plots indicates the default values used in producing Figures 15.2 to 15.9. The contour plots clearly illustrate how strongly the control parameters can affect the performance of the two optimizers. Consider first the EP method and Figures 15.10 and 15.11. These both show that the tournament size does not strongly influence the method, with values in the range 0.1 to 0.3 being best for these test problems. The class of mutation has, by comparison, a much greater effect. Moreover, linear mutation variations seem to suit the 'jhrr' problem while cubic changes are more suitable for 'bump'. Thus no single setting of this parameter can be proposed as providing a general purpose value: the quadratic mutation variations used by default being clearly a compromise with variations around this value resulting in ±20% changes in performance on the test problems. Turning next to the ES method and Figures 15.12 and 15.13, it is apparent that both plots show a valley of poor performance when the child population is equal in size to the parent population. Moreover, diametrically opposed values of both ∆σ and fchild give the best results for the two test problems studied. Worse is the fact that the range of performances spanned by the plots vary from 50% to +67% when compared to the default values, indicating that the performance of the method is very sensitive to these controls. It does, however, appear that, when correctly set up, the ES method can give good performance on the 'jhrr' problem (i.e., with a high value of ∆σ and small child populations). Finally, consider SA: the method is tuned by modifying the cooling schedule. As has already been set out, the SA schedule used here is a general purpose one that can be controlled via three parameters affecting the ratio of trials per temperature to number of temperatures, the width of the range of temperatures considered and the bias of this range between high and low temperatures. Figures 15.5 and 15.9 demonstrate that the temperatures where most improvements take place lie in the middle of those considered and so the parameters P and W were varied with C fixed. This leads to the contour maps of Figures 15.14 and 15.15, again for 'bump' and 'jhrr', respectively. Both figures indicate that, using the generic cooling schedule adopted here, SA performs significantly worse than any of the three other methods, even for quite broad changes in the two key control parameters of the schedule. Nonetheless, the method seems to offer relatively consistent performance over a wide range of controls. Such variation as there is does, however, tend in different directions for the temperature range parameter W, with 'bump' being best handled with W = 1.2 and

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'jhrr' with W = 7. Both methods seem to perform best with low values of P, i.e., few distinct annealing temperatures, although variations in this parameter do not give such significant changes in performance. Finally, it should be noted, as has been mentioned earlier, that hand-coded schedules allow the SA to perform at least as well as EP and ES on these two problems. 15.6 Conclusion This brief paper has shown that, although they have some advantages, basic implementations of evolutionary programming, evolution strategies and simulated annealing are still all outperformed by an up-to-date GA on the test functions considered here. No doubt further refinements of these methods would allow their performance to be improved. Such improvements would need, however, to go beyond simple parameter tuning and address new mechanisms, such as the niche forming adopted in the GA here. It should also be noted, however, that simple public domain GA's [Gre84] are not so robust against such problems [Kea94]. Perhaps less dependent on the sophistication of the implementations is the robustness of the various methods. The GA seems to be less sensitive to its control parameters than the other methods used here and it is difficult to see how such robustness can be achieved with these methods. So, in summary, it may be said that when comparing new optimizers with existing methods described in the literature, care should be taken not to test simply against readily available but rather simplistic public domain versions of existing methods and then only to compare best results. Nonetheless, it remains true that one of the difficulties of working in this field remains that of getting reliable test results for up-to-date methods on functions of interest.

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Figure 15.1 Contour map of the two-dimensional 'bump' problem.

Figure 15.2 Optimization traces for 'bump' using the GA.

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Figure 15.3 Optimization traces for'bump' using EP.

Figure 15.4 Optimization traces for'bump' using ES.

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Figure 15.5 Optimization traces for 'bump' using SA.

Figure 15.6 Optimization traces for 'jhrr' using the GA.

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Figure 15.7 Optimization traces for 'jhrr' using EP.

Figure 15.8 Optimization traces for 'jhrr' using ES.

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Figure 15.9 Optimization traces for 'jhrr' using SA.

Figure 15.10 Effect on 'bump' optimizations of variations in EP control parameters.

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Figure 15.11 Effect on 'jhrr' optimizations of variations in EP control parameters.

Figure 15.12 Effect on 'bump' optimizations of variations in ES control parameters.

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Figure 15.13 Effect on 'jhrr' optimizations of variations in ES control parameters.

Figure 15.14 Effect on 'bump' optimizations of variations in SA control parameters.

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Figure 15.15 Effect on 'jhrr' optimizations of variations in SA control parameters. References [And75] Anderberg M. (1975) Cluster Analysis for Applications. Academic Press, New York. [BHS91] Back T., Hoffmeister F., and Schwefel H.-P. (1991) A survey of evolution strategies. In Belew R. and Booker L. (eds) Proceedings of the 4th International Conference on Genetic Algorithms (ICGA IV), pages 29. Morgan Kaufman Publishers, Inc., San Diego. [Fog93] Fogel D. (1993) Applying evolutionary programming to selected traveling salesman problems. Cybernetics and Systems 24(1): 2736. [Gol89] Goldberg D. (1989) Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading, MA. [Gre84] Grefenstette J. (1984) A user's guide to GENESIS. [Kea93] Keane A. (1993) Structural design for enhanced noise performance using genetic algorithm and other optimization techniques. In Albrecht R., Reeves C., and Steele N. (eds) Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms, pages 536543. Springer-Verlag, Innsbruck. [Kea94] Keane A. (1994) Experiences with optimizers in structural design. In Parmee I. (ed) Proceedings of the Conference on Adaptive Computing in Engineering Design and Control 94, pages 1427. P.E.D.C., Plymouth. [Kea95] Keane A. (1995) Genetic algorithm optimization of multi-peak problems: studies in convergence and robustness. Artificial Intelligence in Engineering 9(2): 7583. [KGV83] Kirkpatrick S., Gelatt C., and Vecchi M. (1983) Optimization by simulated annealing. Science 220(4598): 671680.

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[MPH92] Mitchell M., Forrest S., and Holland J. (1992) The royal road for genetic algorithms: Fitness landscapes and ga performance. In Proceedings of the First

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European Conference on Artificial Life. MIT Press, Cambridge, MA. [FM93] Forrest S. and Mitchell M. (1993) Relative building-block fitness and the building-block hypothesis. In Whitley L. (ed) Foundations of Genetic Algorithms 2. Morgan Kaufman Publishers, Inc., San Mateo. [YG93] Yin X. and Germay N. (1993) A fast genetic algorithm with sharing scheme using cluster methods in multimodal function optimization. In Albrecht R., Reeves C., and Steele N. (eds) Proceedings of the International Conference on Artificial Neural Nets and Genetic Algorithms, pages 450457. Springer-Verlag, Innsbruck.

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16 When 'Herby' met 'ElViS' Experiments with Genetics Based Learning Systems Paul Devine, Geoff Kendal and Ray Paton Abstract A number of mechanisms for learning and adaptation in computational systems are based on biologically motivated ideas. Two major approaches are selectionism and connectionism. One technique which variously combines these two is the classifier system. The paper discusses the development of two such systems which address practical problems in different scientific disciplines: Herby a system for investigating simulations of complex natural world ecologies and ElViS a vision system which learns to process the event it perceives. Some ecological phenomena do not lend themselves to conventional analytical modelling techniques, one area being the relationship between the behaviour of the animals within a population and the overall demography of that population. Herby allows us to apply evolutionary computational techniques to the investigation of the effects of evolution and behavioural adaptation on population dynamics. The focus of our studies with Herby is a computational ecology, that is, a software simulation or model (derived from some observed biological system) comprising an environment and inhabited by autonomous agents. The design and implementation of such a system based on plant-herbivore interactions is described, this utilises connectionist and selectionist techniques on a number of conceptual levels in what is essentially a multi-animal approach. A highly object oriented system architecture has been designed to provide considerable flexibility in both the nature of the environment and the design of the agent. In addition, it provides a simple mechanism which facilitates the possible development and transfer of a suitable level of agent functionality while retaining the scope for large populations, and presents some results obtained from this system. Some of the results of the simulation experiments are presented and discussed. ElViS (Ecological Vision System) seeks to apply ecological ideas to the development of computer vision. Biological vision invariably exists only as part of a larger system a creature situated within a diverse and dynamic environment. A creature has needs Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith© 1996 John Wiley & Sons Ltd.

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and desires which must be fulfilled if it is to survive. In this ecological context, vision (and indeed sensory perception in general) is just one of the faculties a creature uses to achieve its goals. A theoretical basis for the incremental development of adaptable, robust vision systems from an initially simple level is presented. It may be the case that general, real world artificial vision will not be possible unless an ecological perspective is taken. This is the thesis underlying the development of ElViS. In order to achieve this goal of ecological vision, a learning system is required and ElViS uses a classifier system to do this. Both Herby and ElViS apply ideas from ecology in various ways. The common theme, together with the commonalities of the learning systems, are compared and contrasted. This paper concludes with a discussion of some of the general principles that have been learned through this work. 16.1 Introduction In a previous paper one of the authors discussed the design of adaptable systems through the study and application of biological source models [Pat95]. The purpose of this paper is to extend some of the ideas that were developed there and also examine a number of issues associated with the design and implementation of complex learning systems which in various ways make use of ideas from ecology. Two systems which we are developing will be described Herby and ELViS. The former system system seeks to simulate the evolution of behaviour in a computational ecology and the latter applies ecological ideas about perception to the design of a computer vision system. Both systems are highly adaptive and rely on the capabilities of their agents to learn. We will discuss some of the motivations behind these designs. A number of mechanisms for learning and adaptation in computational systems are based on biologically motivated ideas. Two major approaches are selectionism and connectionism. One technique which variously combines these two is the classifier system [LDJ90]. 16.2 Genetics Based Learning The mechanism and application of genetic algorithms are well known and in widespread use. However, they do not in themselves constitute entities which learn. Holland [Hol92] proposed the learning classifier system (LCS) (Figure 16.2) as a vehicle for the application of genetic algorithms to the problem of machine learning [LDJ90]; here we are specifically concerned with a subset of classifier systems known as animats [Wil90]. In this context we define these as autonomous entities (usually pure software constructs) existing in corresponding environments which present them with one or more learning tasks. This approach effectively isolates the investigation of machine learning from the engineering and computing problems of locomotion, gathering of sensory input etc. Briefly, a classifier system consists of a large population of rules (condition-action productions with a numerical weighting), these are encoded in the form of binary and trinary strings (Figure 16.2) and compete to be acted upon. Rules bid to be placed on the message board when their condition parts match a message already posted;

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these messages are environmental input and the action parts of rules that have bid successfully in previous rounds. Bids are calculated on the basis of the strength of a rule and its specificity, i.e. the number of non-wildcard characters in the condition string. Successful bidders 'pay' their bid to the rules whose postings have triggered them rules which succeed in having their action parts acted upon receive any environmental payoff, be that positive, negative or nil. This 'market economy' ensures that payoffs are percolated down through the chains of triggering rules and is a form of credit assignment known as the bucket brigade algorithm. A genetic algorithm is applied to the rule population and it is the operation of this selectionist mechanism in rule discovery that moves the system towards a (local) behavioural optimum in the problem space. Classifier systems may possess a more complex structure than the basic one illustrated including multiple conditions, tag mediated look-ahead [Hol90] and fuzzy logic. One of the great attractions of classifier systems over conventional symbolic AI techniques in this area of work is that the emphasis is on the development of a learning mechanism, this is environment and problem independent due to the nature of the encoding of input, rules and output. The two systems described in this article utilise variants on the basic LCS outlined above but differ in purpose and emphasis. Herby uses a simple LCS as the basis of a large number autonomous agents in a computational ecology, it is system in which machine learning techniques are applied to the investigation of ecological phenomena. Conversely, Elvis possesses a more complex LCS in a more conventional animat environment, Elvis represents the application of ecological concepts to the problem of machine learning, specifically computer vision. 16.3 Herby This work is a study of the use of adaptive computational approaches as a mechanism for individual agent based modelling of adaptation in ecological systems; it is particularly appropriate to situations which are unsuited to conventional analytical techniques (i.e. using continuous mathematics). In this context we define the term computational ecology to mean the software simulation or model (derived from some observed biological system) comprising an environment and inhabited by autonomous agents. Behavioural ecology is generally concerned with the behaviour of animals both individually and in groups. Population biology, on the other hand, concerns itself with with the effects on the population as a whole of environmental and biological changes. Traditionally, neither address the demographic impact of behavioural changes. Recently there has been a move towards investigating the effects of individual behaviour on population dynamics [HM85]; much of this work employs analytical mathematical methods. The rationale behind Herby is based on the premise that the non-linear behaviour of ecologies cannot be ignored when investigating behavioural and demographic phenomena. The importance of individual variation within a population on long-term outcomes at both the population and individual levels has already been established by individual based field work; the following examples [HDP84] are not strictly behavioural but provide straightforward illustrations of analysis at this level of the

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Figure 16.1 A generalised animat. population. The fish experiment could be interpreted as relying on behavioural effects. The development of size distribution in plant populations is very sensitive to the individual organisms involved and slight differences in initial conditions. Light is an essential resource for plants and they compete for it, consequently relative differences in plant height are critical to the outcome of this competition. If there is a low initial variation in height then the final population will be relatively homogeneous in size. Conversely, a high initial variation confers a competitive edge on the taller individuals which will capture an increasingly large portion of the available light and as a result grow at a higher rate. This positive feedback will accelerate the divergence of sizes and the final population will consist of a few very large individuals and a number of much smaller ones. Another instance of the importance of slight differences in initial sizes of individuals is a series of experiments involving populations of 250 bass in 50 gallon aquaria. Cannibalism is common in bass and it has been observed that a bass is capable of eating another bass up to half of its own size. The corollary is that in cases where there is a large initial variation the larger bass tended to eat the smaller bass and therefore grew rapidly; a low initial variation led to a larger population of smaller individuals as a consequence of the lack of intraspecies predation. These examples would readily lend themselves to computer simulation. Herby is designed to go beyond this and investigate the effects of both variation at the behavioural level and the development of behaviour itself. The proposed system would out of necessity involve a degree of abstraction from the biological reality but it was essential that the essential components and relationships of a real ecology were preserved. A plant-herbivore arrangement was chosen and

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Figure 16.2 A learning classifier system.

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Figure 16.3 Herbivore classification. developed in accordance with the classification scheme shown in Figure 16.3 [CL81] which derived its primary division from Monro's dichotomy [Mon67]. In this context 'herbivore' refers not only to cows, sheep etc. but to all creatures feeding on vegetable matter and 'grazing' encompasses all eating of plants by animals. Non-interactive systems are typified by the herbivore's inability to influence the rate at which its resources are renewed. Further subdivision produces: reactive systems where the rate of increase of the herbivores reacts to the rate at which the resources are renewed; non-reactive systems where the rate of increase of the herbivores is independent of the parameters of resource renewal. In interactive systems the animals do influence the rate of plant renewal, and this in turn influences the rate of increase of the animals. Further subdivision here gives: laissez-faire systems where the herbivores do not interfere with each other's feeding activities; interferential systems where the animals do interfere with each other's ability to obtain food. 16.3.1 System Components The system has been designed and implemented in accordance with the object oriented paradigm. However, for the purposes of this description it is best to focus on it in terms of the environment and the agent. These two components constitute the ecology.

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16.3.2 The Environment The environment comprises a 2D lattice of discrete areas; the boundaries of this lattice are manipulated to produce an effectively toroidal world. This format was chosen because the inclusion of boundaries would in itself impose further, unnecessary learning tasks on the agents that really lie beyond the scope of this investigation. This toroidal approach has been used by other workers [Dew84] and contrasts with simulations where the environment is explicitly bounded. The individual areas have two primary attributes, the type of flora growing there (the resource) and its density; inter area connectivity exists only along edges and not at vertices. The nature of the environment may be determined in a number of ways, in its present state the system allows for four different types of flora (three if the existence of barren areas is selected) the distribution of which may be manipulated by the user. The rate of growth and nutritional value of the individual species are also user set parameters. 16.3.3 The Agent As stated above, the agent employs a classifier system as the mechanism of adaptive change; this operates both in respect of 'learning' during the agent's lifetime and long-term evolutionary change over the generations. The agent also contains state information: The accumulated somatic investment (ASI) a measure of the amount of food invested in the agent's own physical state; residual reproductive value (RRV) a measure of the agent's reproductive success; age. A simple form of life history may be imposed on the agent by the use of a predetermined age threshold for maturity at which reproduction is possible; this can also be tied into the agent's ASI (see Figure 16.4). The inputs for the classifier system consist of the following: type of flora in the current area; density of flora in the current area; the presence of other agents in the current area; the type of flora in adjoining areas; its own fitness level (determined by somatic investment and reproductive value; its own state of maturation and has the following options for action: eat; move (with a choice of four adjacent squares this amounts to four options); reproduce; do nothing.

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Figure 16.4 The agent and its world view

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Figure 16.5 Adaptive Agent Population Over Time

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Figure 16.6 Random Agent Population Over Time

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16.3.4 The Ecology Naturally, the combination of the environment and a population of agents constitute the ecology. In a typical run the geography of the environment is specified and the attributes of the areas are individually set. The starting population of agents is introduced into the environment and the simulation is then left to run for either a predetermined number of timesteps or the extinction of the population, whichever arrives first. The setting of various flags and parameters allows various herbivore type systems to be simulated in accordance with the previous discussion. In order to deduce anything about the behaviour of the agents in an adaptive context it was necessary to provide some sort of control simulation as a point of reference in addition to comparisons with real biological systems. A considerably simplified version of the agent also exists; this has no adaptive abilities whatsoever and its decisions are made on a purely pseudo-random basis. Environmental simulations may be performed with this non-adaptive agent but otherwise identical parameters; this provides a yardstick for the behaviour of the adaptive agents and illustrates the non-linearity introduced to the system by the adaptive agents. 16.3.5 Interim and Prospectus The simulations conducted so far are encouraging. One of the main problems with the system is currently being addressed in a new version. This relates specifically to the starting conditions of the agents. We feel that our current practice of generation a random rule base (this is heavily constrained in size by the necessity of running over one thousand animats simultaneously) and letting the agents loose in the environment is unrealistic. We are now moving to a system of 'selectively breeding' animats for certain traits prior to simulations. The two plots (Figures 16.5 and 16.6) illustrate the marked difference in the behaviour observed in the population dynamics of the random and adaptive agents. Though the random agents peak at higher population levels in this interferential arrangement the striking periodicity that the population exhibits does not truly reflect biological observation. The adaptive agents exhibit a more complex dynamic which is reminiscent of real systems. 16.3.6 Corollary Herby relies upon the distributed, parallel search heuristic exemplified by genetic algorithms and classifier systems in general. In addition to this it embodies a modelling and simulation paradigm based on explicitly representing or implementing the individual elements of a system rather than assuming that the aggregate behaviour of the system may be described treating collections of discrete entities as discrete entities in themselves. This approach is particularly limited where individual variation can be a crucial factor, as is often the case in evolutionary and ecological systems.

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16.4 Exploring Ecological Vision with ElViS In this section we consider how a complex adaptable system which processes visual information has been developed. We show how an appreciation of a number of biologically inspired aspects of perceptual systems can help in the specification of a design. This biomimetic approach provides us with insights into how adaptable systems interact with their environments and why learning is a key to general purpose computer vision. The learning strategy adopted is based on an enhanced classifier system architecture. We contend that consideration of ecological issues is a significant factor in the design of any autonomous system (see also [Hor94]), and it may be the case that general, real world artificial vision will not be possible unless such a perspective is taken. This is the hypothesis underlying the development of ElViS (Ecological Vision System). An assumption underlying much artificial vision research is that the fundamental problem is to discover, from an image of a scene, a description of the objects in that scene in terms of their appearance, location, and so on [Ros88]. Within this framework, the task of a 'vision system' is essentially to recognize and label visible objects. In contrast, biological vision invariably exists only as part of a larger system a creature situated within a diverse and dynamic environment [Wil90]. A creature has needs and desires which must be fulfilled if it is to survive. In this ecological context, vision (and indeed sensory perception in general) is just one of the faculties an agent uses to achieve its goals: others include the ability to move in, adapt to, and manipulate the environment. An important point becomes apparent if vision is considered from an ecological viewpoint, that being the complementary nature of agent and environment. Arbib notes: 'the job of the visual system is to provide the [agent] not with a representation of the world in the abstract, but rather with the information it needs to interact with the world' [Arb89] Rather than following a set sequence of actions, a visual system's activity is dependent on the situation. Examples include: searching for a specific object; supplying contextual information based on what has already been discovered; or providing more detailed analysis of potentially interesting regions of the environment [Arb89]. The problems of recognising objects in images of arbitrary scenes are well documented in the literature [Ros88]. However, within an ecological framework object recognition is not the sole task of the visual processor. Ideally vision should provide an agent with as much information about its environment as is necessary given its current needs and situation. For instance, to move through a crowd one does not require a detailed description of the appearance of each individual encountered; however, the latter may become necessary when searching the crowd for a specific person. Survival in all but the simplest static environments and certainly in the real world requires that an agent has some ability to adapt to unexpected situations. Therefore on encountering an unknown object, a visual system should not fail utterly, but rather provide as much information as possible, since the survival of the agent may depend on a quick but rudimentary assessment of the situation. A system concerned solely with labelling objects would not be able to provide this level of functionality. Survival

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needs greatly influence behaviour, and as a consequence have a powerful effect on the formation of an agent's percepts [Wil90]: this fundamental point can only be acknowledged in computer vision research if an ecological approach is taken. Research along the lines of the recently emerging 'active vision' paradigm (e.g. [Baj88] and [Cli90]) embodies the above ideas to varying degrees. A major goal of the ElViS project is to develop a sound theoretical framework incorporating all of the above considerations upon which future artificial vision research can build. Gibson was a strong proponent of the ecological approach to perception [Gib79]. He described an environment as consisting of substances, a medium, and the surfaces that separate the two. The medium permits locomotion and the sensing of substances at all places. Locomotion and behaviour are continually regulated by the senses. The substances of the environment need to be distinguished, and a powerful way of doing so is by seeing their surfaces. Gibson argued that the fundamental ways in which surfaces are laid out in the environment have an intrinsic meaning for behaviour, unlike the abstract, formal concepts of mathematical space. The theory states that 'affordances' embody this behavioural significance these being what the substances in the environment offer or provide the creature, for good or ill [Gib79]. Consequently one does not have to classify and label things in order to discover what they afford. Affordances are perceived through the detection of variations over time (so called 'ecological events') in the local structure of the 'ambient optic array' the many-times-reflected light within the environment arriving at a point of observation. The implicit notion that vision is continuous is perhaps the most important aspect of Gibson's work with respect to artificial vision research: vision is concerned with detecting events in the environment, rather than some arbitrary idea of regularity (edges, homogeneous regions, etc..) in an image. In order to exploit Gibsonian ideas computationally, it is necessary to investigate what constitutes an 'event' in terms of changes in an agent's environment which warrant changes in its actions and/or its view of the world. Two recent papers ([MA93] and [SM94]) maintain that the discovery and analysis of unanticipated changes in the environment is both a necessary and fundamental requirement of vision. The ElViS project takes the notion of unexpectedness as the basis for ecological events. Given this, the question arises: how can detection of unexpected changes in the optic array be physically realised given a sequence of discrete images? 'Stimulus profusion' the sheer volume of sensory input which must be considered is a significant problem in the perception of real environments [Wil90]. Attempts to reduce the data to manageable levels generally rely upon finding tokens in the visual field (such as high contrast edges or grey-level homogeneous regions) which are then utilised by further processes. However, such techniques are invariably 'brittle', being heuristically chosen for their ability in very limited domains (e.g. [Oht83]). When one considers that most methods of token extraction are dependent upon image characteristics rather than environmental ones, perhaps this is not surprising. Animat theory maintains that to overcome the problems of stimulus profusion and brittleness, the generality and simplicity of percepts must be ensured [Wil90]: it is the interaction of such percepts which leads to more specific information about the environment. The ElViS approach looks towards human vision research in an attempt to fulfil these criteria. Surfaces are perceived by humans from range images, photographs, line drawings, and so on where the only common theme is the preservation of spatial

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structure [WT83]. Human vision immediately detects spatial organisation such as symmetry, proximity, and repetition phenomena the Gestalt psychologists described (but did little to explain) in detail [Zus70]. However, most computer vision approaches either ignore this [Low85] or attempt ad hoc mimicry of such abilities with no theoretical grounding [Pop94]. At a computational level it is difficult to define what perceived structure actually is [Lig91]. One attempt at an explanation relies on the concept of 'non-accidentalness' [WT83]: that regular structural relationships in images (such as parallelism or collinearity) are unlikely to arise by chance, and therefore they imply possible underlying causal relationships in the scene. Non-accidentalness was proposed as a means to aid object recognition, but we extend the idea to ecological vision, the theory being that spatiotemporal relationships in the optic array reflect likely spatiotemporal relationships in the environment. Since the environment's structure greatly influences an agent's possible and required movement and other actions, the ability to perceive it is of great importance. It is hypothesised that spatiotemportal relationships such as parallelism, collinearity and constant velocity embody important information regarding the structure of the environment, and are hence worthy of attention. It should be noted that as candidate percepts, spatiotemporal relationships adhere to the animat recommendations (simplicity and generality), are independent of scale, and apply just as well to sequences of images as to the optic array, yet do not rely on arbitrary definitions of tokens. Hence they form an ideal basis for developing ecological vision within a camera-based system. The key to this approach is that structure in the optic array is not likely to arise by chance. It follows that when spatiotemporal relationships are discovered in the optic array, to ascertain whether they have arisen by chance (i.e.: whether their existence is dependent on the agent's position and velocity) they must be continually monitored for changes. A basic requirement of ElViS is the discovery and tracking of spatiotemporal relationships in the structure of the optic array. These two activities cannot be considered in isolation, since they are interdependent. For example, if when monitoring two discovered instances of collinearity their motion appears to be related, there may be sufficient evidence to deduce the existence of a composite relationship embodying the two. In the real world, such an occurrence might be due to the detection of two sides of a single object. Variation in the structure of the optic array is dependent upon two factors: changes in the point of observation; and changes in the surroundings. The former can be predicted to a certain extent, since it is a reasonable ecological premise that an agent is party to information regarding its movement. Hence such changes are not wholly unexpected. The latter usually requires an adjustment of the agent's world view, so these changes can be classed as unexpected in an immediate sense. Consideration of spatiotemporal relationships within the ecological framework discussed earlier leads to the following definition: Unexpected variations in spatiotemporal relationships discovered in the optic array form the basis of what is here termed an 'optic event'. An optic event indicates the need to focus activity on specific aspects of the optic array (and hence, likely aspects of the environment) which the agent has been hitherto unaware of. On analysing an optic event, the creature's view of the world should be readjusted to accommodate the new information and its actions altered as necessary.

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Figure 16.7 Example of an optic event. One important issue arises when vision is considered in this way. Any event which occurs may significantly affect the agent's view of the world, even though the event itself may appear relatively trivial when considered alone. An informal example demonstrates this. If the circle in Figure 16.7(a) is moving to the right, suppose in time the arrangement resembles 16.7(b), and later 16.7(c). Considered in isolation (ie: locally), the circle in 16.7(c) has become slightly deformed, and following 16.7(a) and 16.7(b) this could be considered an unexpected change: an optic event has occurred. Looking at the whole picture results in a strong perception that the circle has moved behind the rectangle: this is a global consequence of the event. This simple example illustrates how high level changes in the world view can emerge from relatively simple variations in the optic array. This effect is compounded in a real world environment, where many types of optic event may occur over a range of scales (both spatial and temporal). The emergent nature of the event-based approach is clear. There is no single executive control mechanism, but rather a collection of many interacting 'optic event detectors' which may all influence the behaviour of the visual system and the resulting view of the world. The ElViS design draws on Arbib's schema theory [Arb89] in an attempt to model such interactive behaviour. A schema is an active modular entity embodying both perceptual structures and motor control, which is defined by its interaction with a physical environment (via sensors, actuators, and the exchange of information with other schemas). A schemabased system's representation of its environment is embodied in 'the pattern of relationships between all its partial representations' [AC90]p.143, those being its component schemata. During execution, the use, recall and representation of the system's knowledge is embodied in a network of schema 'instances'. This network is dynamic: schema instantiation may be triggered by requests from other instances or by events in the environment; deinstantiation occurs if an instance has fulfilled its purpose or if it becomes obsolete given the current situation. The notion of an 'abstract conformation' (AC) has been developed as a schema-inspired means of implementing an optic event detector as described above. Each type of AC has two separate components. The first the 'discovery' component constantly scans the optic array for new instances of a given type of spatiotemporal relationship (such as parallelism). Discovery of such a relationship triggers the creation of an

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Figure 16.8 An abstract conformation system. instance of the AC's second part the 'tracking' component. This latter comprises the following: 1. Properties Physical attributes specific to the instance (e.g.: distance from agent) Confidences that the instance is representative of underlying structure in the environment (and therefore that it is of use) 2. Behavioural specification How the spatiotemporal relationship embodied by the AC is expected to evolve given the possible actions of the agent What new optic events should be posted and how properties should be affected when expectancies are refuted What effect optic events posted by other ACs have on the instance's properties What property values/variations warrant the posting of new optic events What property values/variations affect the instance's existence (i.e.: its relevance to the current world view) An abstract conformation based visual system is shown in Figure 16.8. Each AC

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receives notification of the agent's actions, and may influence future actions depending on its specific role in the system. An extension to the system (described elsewhere) incorporates higher level goal-oriented visual activity through the application of 'ecological context' (see [Ken94]). Since a schema's interaction is the crucial aspect of its operation, the actual means of implementation is not generally a primary issue [AC90]. However, there are several reasons why the ElViS project employs an adaptive approach. Firstly, animat theory maintains that adaptive techniques should be utilised wherever possible to minimise the introduction of new mechanisms [Wil90]. Secondly, in real world environments spatiotemporal relationships are unlikely to be precise. For example: parallelism undergoes distortion due to perspective effects; a body with constant velocity appears to speed up on approaching the agent; and all relationships are subject to sensor noise. Mathematically, it is difficult to specify such aberrations, but it may be possible to learn common ones by example. Finally, it may be relatively easy to describe roughly what optic events an abstract conformation should respond to, and how its properties should be altered accordingly. However, following the previous point, specifying such behaviour precisely is a problem. Ideally, it would be preferable to describe an AC's behaviour in loose terms, and through adaptation, fine tune these to improve performance within the environment. A closer look at an AC's behaviour shows that it examines input messages in the context of internal property values, and as a result updates the internal properties and produces output messages. A classifier system [BGH90] (described previously in this paper) seems appropriate for this kind of task, provided two extensions are made, which follow directly from the above points. The first is the necessity of handling internal properties as well as input and output messages. The second and most important point is that it is necessary to handle rough or vague rules and properties. Some work along the lines of this second point has already been done, through the combination of classifier systems and fuzzy theory [VR91]. Briefly, fuzzy theory provides a way to mathematically describe vague rules such as 'if temperature is hot then turn fan up' and use such rules in complex inference strategies (see [Kos94] for an in depth study). The current ElViS prototype is following a research strategy in which ACs are implemented using an enhanced fuzzy classifier system (EFCS). The researcher using ElViS defines the fuzzy properties and the input and output messages relevant to the AC, and describes the fuzzy rule base in broad terms. An example of such a rule is 'if distance is faraway and visual-angle is small and movement-type is step-forward then change-in-visual-angle is small'. A simulated environment has been completed which enables the automatic generation of various forms of training data. Selected rules from an AC's rule base are then refined through the application of this training data, and new rules are produced through the application of a genetic algorithm. References [AC90] Arbib M. and Cobas A. (1990) Schemas for prey-catching in frog and toad. In From Animals to Animats Proc 1st Int Conf on Simulation of Adaptive

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Behaviour. [Arb89] Arbib M. (1989) The Metaphorical Brain 2: Neural Networks & Beyond. Wiley. [Baj88] Bajcsy R. (1988) Active perception. Proc IEEE Special Issue on Computer Vision 76(8). [BGH90] Booker L., Goldberg D., and Holland J. (1990) Classifier systems and genetic algorihms. Artificial Intelligence. [CL81] Coughley G. and Laughton J. (1981) Plant-herbivore systems. In May R. (ed) Theoretical Ecology. Blackwell. [Cli90] Cliff D. (1990) The computational hoverfly: A study in computational neuroethology. In From Animals to Animats Proc 1st Int Conf on Simulation of Adaptive Behaviour. [Dew84] Dewdney A. (1984) Sharks and fish wage war on the planet wa-tor. Scientific American 251(6): 1422. [Gib79] Gibson J. (1979) The ecological approach to visual perception. Boston: Houghton-Mifflin. [HDP84] Huston M., DeAngelis D., and Post W. (1984) New computer models unify ecological theory. Bioscience 38 (10): 682691. [HM85] Hassell M. and May R. (1985) From individual behaviour to population dynamics. In Silby R. and Smith R. (eds) Behavioural Ecology Ecological Consequences of Adaptive Behaviour. Blackwell. [Hol90] Holland J. (1990) Concerning the emergence of tag-mediated lookahead in classifier systems. In Forrest S. (ed) Emergent Computation, volume 42, pages 188201. MA:MIT. [Hol92] Holland J. H. (1992) Adaptation in Natural and Artificial Systems. MIT Press. [Hor94] Horswill I. (1994) An ecological analysis of a system for detecting nods of the head. In Paton R. C. (ed) Computing with Biological Metaphors. Chapman & Hall. [Ken94] Kendal G. (December 1994) Towards artificial ecological vision. Technical report, University of Liverpool. [Kos94] Kosko B. (1994) Fuzzy Thinking : The new science of Fuzzy Logic. Flamingo. [LDJ90] L.Booker, D.E.Goldberg, and J.H.Holland (1990) Classifier systems and genetic algorithms. Artificial Intelligence. [Lig91] Ligomenides P. A. (1991) Early perception and structural identity: neural implementation. In Proceedings of the SPIE Intelligent Robots and Computer Vision X, volume 1608. [Low85] Lowe D. G. (1985) Perceptual Organisation and Visual Recognition. Kluwer Academic. [MA93] Marshall J. A. and Alley R. K. (October 1993) A self-organizing neural network that learns to detect and represent visual depth from occlusion events. In Bowyer K. W. and Hall L. (eds) Proceedings of the AAAI Fall Symposium on Machine Learning and Computer Vision, pages 7074. Research Triangle Park, NC. [Mon67] Monro J. (1967) The exploitation and conservation of resources by populations of insects. Journal of Animal Ecology 36: 53147. [Oht83] Ohta Y. (1983) Knowledge-based interpretation of outdoor natural colour scenes. Pitman Advanced Publishing Program, London.

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[Pat95] Paton R. (1995) Designing adaptable systems through the study of biological sources. In Rayward-Smith V. (ed) Applications of Modern Heuristic Methods, pages 3954. Alfred Waller/Unicom. [Pop94] Pope A. R. (January 1994) Model-based object recognition: A survey of recent research. Technical Report TR9404, University of British Columbia, Computer Science Dept. [Ros88] Rosenfeld A. (1988) Computer vision: Basic principles. Proc IEEE

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Special Issue on Computer Vision 76(8). [SM94] Sommerhoff G. and MacDorman K. (1994) An account of consciousness in physical and functional terms. Integrative Physiology and Behavioural Science 29(2): 151181. [VR91] Valenzuela-Rendón M. (1991) The fuzzy classifier system: motivations and first results. In Schwefel H. and Männer R. (eds) Parallel Problem Solving from Nature, pages 330334. Springer (Verlag) Berlin. [Wil90] Wilson S. W. (1990) The Animat path to AI. In From Animals to Animats Proc 1st Int Conf on Simulation of Adaptive Behaviour. [WT83] Witkin A. P. and Tenenbaum J. M. (1983) On the Role of Structure in Vision. Academic Press (New York). [Zus70] Zusne L. (1970) Visual Perception of Form. Academic Press (New York).

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Index (MGT+OrOPT)5 heuristic, 132, 146, 147 A acceptance function, 10 adaptable system, 274, 283 adaptive memory, vi, 41, 55 animat, 274, 275, 282, 284, 285, 287 assignment problem, 2, 4 Bi-Quadratic (Bi-QAP), 76 Quadratic (QAP), 40, 47, 53, 54, 67, 76 Quadratic Semi (QSAP), 53, 54 Radio Link Frequency, 246 ATM network, 239 attraction basin, 60, 69, 71-73 autonomous agents, 273, 275 B Boltzmann distribution, 239 Branch-and-Bound (B&B), 4, 61, 213, 216, 218-229, 231, 232 best-first search, 218-220 branching strategy, 220, 221 depth-first search, 218-220 search tree, 220, 221, 224 bucket brigade algorithm, 275 bump problem, 256, 261, 264 C

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chaotic attractors, 68, 69 chromosome, 17, 18, 20, 21, 87, 92, 93, 237, 238 classifier system, 241, 273-275, 279, 282, 283, 288 Enhanced Fuzzy Classifier System (EFCS), 288 Learning Classifier System (LCS), 274, 275, 277 combinatorial optimization, 2, 4, 5, 28, 39, 40, 48, 62, 77, 97, 98, 115, 116, 124, 130, 135, 174, 213, 237 computational ecology, 273-275 computer vision, 274, 275, 283-285 constraint satisfaction, v, 25, 26 constraint satisfaction problem (CSOP), 26 crossover, 17-21, 98, 99, 101, 102, 190-193, 211, 237, 249, 258, 262 1-point, 238, 247 complementary, 99, 101, 103 1-Point (1CX), 102, 104-110 1-point (1CX), 106 2-point (2CX), 102-110 crossover operator, vi, 99, 189, 190, 192, 238 partially mapped (PMX), 21, 99, 100 UNBLOX operator, 192 uniform, 20, 22, 238 cutting planes, 40, 61 D delta evaluations, 245, 246, 253 descent algorithm, 130 methods, 5, 6 designs, 199, 200, 202-206, 208, 209, 211 candidate, 199, 200, 203, 205, 206 conceptual, 199

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creation, 199 evolutionary design system, 200, 204 fragmented, 206 illegal, 200, 203, 204, 211 objectives, 205 optimisation, 211 representation, 200, 203, 204 solid object, 200 divide-and-conquer, 87, 95 dynamic programming, 4 E elite solutions, 46, 53, 54 ElViS, 273-275, 283-288 evolution, 64, 71-75, 191, 199, 200, 203, 204, 206, 208, 211, 236, 255, 257, 259, 262, 273, 274, 279, 283 Evolution Strategies (ES), 61, 255, 257-263, 265, 269, 270 evolutionary optimisation, 255 Evolutionary Programming (EP), 255, 257-263, 265, 267-269 F Fast Component Mounter (FCM), 85, 86, 89, 93, 95 feasible solution, 3, 5, 7, 29, 35, 37, 39, 48, 49, 87, 93, 116, 118, 132, 155, 156, 158,

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159, 162, 165, 166, 174-176, 215, 224, 231, 243 feedback-based parameter tuning, 60 fitness function, 184, 185, 191, 196, 222-224, 226, 227, 229, 230, 236, 244, 247 flight scheduling problem, 25, 27, 34, 35, 37 freight train regrouping problem (FTRP), 25, 27, 36, 37 fuzzy theory, 288 G GAmeter, 222, 223, 236-238, 244, 245 genes, 17, 18, 95, 204, 244 Genetic Algorithms (GAs), v, vi, 1, 2, 17-22, 83, 87, 89, 92, 93, 95, 97-101, 106-111, 183-185, 189-197, 199, 200, 203, 204, 206, 208, 209, 211, 222, 224, 225, 227, 229-233, 236-241, 244-247, 249-253, 255-258, 260, 261, 263, 264, 266, 274, 282 alleles, 17, 18 breeding, 17 population, 17-22, 98, 106, 184, 185, 189, 193, 194, 230, 231, 236-238, 245, 258, 275 replacement, 21, 22, 192, 237 schema theorem, 18, 19, 83, 84 selection, 19, 20, 97, 98, 111, 183, 192, 236, 237, 240 roulette wheel partner selection, 20, 204 stochastic universal, 20 tournament, 20, 192 genetic art, 199 genotype, 18, 201-204 global optimum, 25 graph colouring, 159, 162 graph partitioning, 97, 100, 101, 103, 104, 111 greedy algorithm, 238 H file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_292.html（第 1／3 页）2009-7-7 16:58:52

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Hamming distance, 66, 69-75, 104 Herby, 273-276, 278, 282 Heuristic Layout Generator (HLG), 87, 89, 90, 92, 93, 95 hill climbing, 101, 103, 111, 185, 188, 241 1-bit Hamming hill climber (HC), 98, 102-105, 107-110 hybridisation, 97 I integer programming (IP), 1, 3, 41, 236, 238 interchange process, 137, 138 iterative improvement algorithm, 116-118, 123, 124 K k-shortest path algorithm, 228, 231 Kempe chains, 161, 162, 165 knapsack problem 0-1 Knapsack Problem, 2, 3, 76 multi-constraint (MCKP), 53 Multi-Knapsack Problem, 76 L LB5+VRP heuristic, 133, 146, 147 Line Balancing Problem, 84, 85 linear programming (LP), v, 1-4, 61, 236 local optimisation, 97-99, 111 local search, v, 1, 26, 39-41, 49, 55, 59-62, 69, 97, 98, 108, 111, 115, 116, 118, 122-124, 129, 130, 135, 156, 158, 160, 163, 166, 173, 193, 239 LPK heuristic, 101-111 M MacQueen's Adaptive KMEAN algorithm, 258 Markov chain, 7, 8, 61, 240

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MBSA heuristic, 133, 146, 147 meshed network, 248 minimum spanning tree, 49, 117 capacitated minimum spanning tree (CMST), 53, 54 modified route perturbation procedure (MRPERT), 133, 134, 143-147, 149-151 Monro's dichotomy, 278 mutation, 17-19, 21, 22, 103, 105, 107-110, 193, 211, 237, 257-259, 261, 262 mutation operator, 192, 193 N neighbourhood, v, vi, 5, 7, 9-12, 14, 17, 22, 25, 28, 30, 33-35, 42, 49, 51, 55, 59-62, 64, 68, 69, 77, 98, 99, 101, 102, 104, 106, 108, 111, 115-124, 129, 130, 134, 135, 137, 138, 140, 141, 151, 156, 159-162, 164, 165, 174, 175, 240, 246 restricted, 10 neighbourhood search (NS), v, 5-7, 11, 12, 17, 98-100, 111 neural networks, 76, 77 No-wait Flowshop Problem, 97, 99 O objective function, vi, 1, 3, 5, 12, 16, 18, 26, 36, 41, 43, 53, 99, 104, 130, 135, 139, 169, 174, 175, 178, 183, 184, 199, 204, 205, 211, 216, 218, 223, 237, 244, 257, 259, 260, 262 Operational Research (OR), vi, 1, 4, 20, 22 Optic Event, 285-287 Or-OPT exchange procedure, 132 P packet switching networks, 238 parameter optimization problem, 83, 87, 95 Pareto optimal solutions, 211 phenotype, 18, 202-204 Potentially Optimal Set (POS), 173, 174

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R Radio Link Frequency Assignment Problem, 246 random search, 28, 62, 241, 260 Reactive Search (RS), 46, 59-61, 76, 79 recombination, 17, 20, 22, 97-101, 104, 106, 108-111, 236, 259 representation, 21, 49, 83, 84, 124, 131, 174, 189, 190, 193, 200, 211, 222, 223, 228, 230, 231, 236, 244, 283, 286 reproduction, 18, 97 route perturbation procedure (RPERT), 130, 133, 134, 143-146, 149, 151 routing, 97, 129, 130, 133, 134, 150, 151, 235, 236, 238, 241-244, 249, 251-253 dynamic anticipatory, 238 shortest path, 235, 236, 242, 249, 251 royal road function, 255, 257 S SAmson, 236, 241, 244 scheduling, 1, 25, 27, 3437, 155, 156, 158, 159, 162, 163, 165, 169171, 220, 241 discrete-continuous, 169, 170, 172, 174, 181 multi-objective, 155 schema, 286, 287 set covering problem, 3, 86 shift process, 137, 138 Simplex algorithm, 4 Simulated Annealing (SA), v, 1, 2, 68, 1013, 17, 22, 2530, 3237, 62, 97, 103, 116, 155, 156, 158160, 163, 166, 194, 235, 236, 239241, 244247, 249, 250, 252, 253, 255, 257260, 262, 263, 266, 268, 270, 271 annealing schedules, 2530, 32, 33, 35 cooling schedule, 7, 8, 10, 62, 240, 259, 262 Lundy & Mees Cooling, 11, 240 geometric cooling, 240

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Localised Simulated Annealing (LSA), 25, 2737 reannealing, 10 temperature, 711, 2530, 32, 33, 35, 37, 158, 161, 164, 165, 239241, 259, 260, 262, 263 tempering, 241, 246, 247 solid geometry constructive solid geometry (CSG), 200 spatial-partitioning, 199, 200, 204 starting solution, 5, 45, 148, 149, 174, 176, 178, 179 Steiner Tree Problem, vi, 49, 115, 121, 124 directed graphs (SPD), 48, 49, 52, 54 graphs (SP), 4850, 54, 115118, 121, 124, 252 rectilinear, 4850, 52 Steiner tree, 48, 115, 117124 strings, 1721, 61, 62, 6467, 69, 70, 77, 83, 98, 104, 183185, 189193, 244, 257, 275 bit, 70, 83, 104, 224, 227 encoding, 189191 multi-dimensional, 190 T Tabu Search (TS), v, vi, 1, 2, 11, 12, 14, 16, 17, 22, 3943, 6466, 116, 118, 120, 129, 130, 135137, 140, 141, 143, 144, 148, 151, 156, 158, 169, 173, 174, 176, 178, 179, 181, 236 aspiration, 12, 15, 17, 43, 44, 53, 135, 142, 143 cancellation sequence, 4246, 5255 residual cancellation sequence (RCS), 4446, 176, 177 candidate list strategies, 17, 42, 135 diversification, 12, 16, 41, 46, 5355, 6062, 6669, 73, 151 hashing, 13, 46, 65, 66, 68 hashing function, 46, 65, 69, 142 influence, 12, 15, 16, 59, 178 intensification, 12, 16, 41, 46, 55, 60, 62, 151 memory, 11, 13, 41, 43, 54, 55, 6568, 135, 140 adaptive, 41, 55 file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_293.html（第 2／3 页）2009-7-7 16:58:53

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attribute based, 41 explicit, 41 frequency, 12, 15, 16 frequency based, 41 long term, 15, 41, 42, 55 short term, 15, 41, 42, 55 move, 5, 6, 4148, 55, 6167, 70, 73, 135, 137, 176, 177 1-interchange (1IM), 138142 complementary, 41, 44, 46 multi-attribute, 44, 45 moving gap, 47, 48, 54 prohibition, 12, 43, 59, 61, 63, 64, 66, 67, 7377 quality, 12, 15, 16, 22, 140, 142, 148, 149 reactive (RTS), 46, 54, 59, 64, 66, 68, 73, 7679, 148 recency, 12, 13, 15, 17, 41 Reverse Elimination Method (REM), 4246, 49, 51, 5355, 65, 148, 169, 176, 177, 179 Simple Tabu Search (STS), 65, 176, 177, 179 tabu list, 11, 4147, 65, 135, 141, 142, 147, 148, 151, 169, 174, 176, 177, 179, 181 management, 39, 42, 43, 169, 176179 tabu tenure, 12, 15, 41, 45, 53 telecommunications networks, 48, 235, 238, 241, 242, 248, 249, 252

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time complexity, 117119, 121123, 202, 203 timetabling, 155, 156, 158 Travelling Salesman Problem (TSP), 4, 5, 9, 10, 20, 21, 3335, 87, 98, 99, 131, 132, 138 Euclidean, 124 V vehicle fleet mix problem (VFM), 129133, 137, 141, 149 Tabu Search vehicle fleet mix problem (TSVFM), 143147, 149, 150 video-on-demand, 252 virtual containers, 242 W warehouse location problem (WLP), 48, 5254

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Modern Heuristic Search Methods Edited by V. J. Rayward-Smith University of East Anglia, Norwich I. H. Osman University of Kent, Canterbury C. R. Reeves Coventry University G. D. Smith University of East Anglia, Norwich

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Copyright © 1996 by John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex PO19 1UD, England National 01243 779777 International (+44) 1243 779777 All rights reserved. No part of this book may be reproduced by any means, or transmitted, or translated into a machine language without the written permission of the publisher. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA Jacaranda Wiley Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1L1, Canada John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 0512 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 471 96280 5 Produced from camera-ready copy supplied by the editors Printed and bound in Great Britain by Bookcraft (Bath) Ltd This book is printed on acid-free paper responsibly manufactured from sustainable forestation, for which at least two trees are planted for each one used for paper production.

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Preface The last few decades have been an exciting time for those of us working in optimisation. The theoretical foundations of complexity have been laid starting with Cook's theorem in the seventies and followed by a plethora of results on NPCompleteness and the definition of yet more complexity classes. Added to this, we have had remarkable progress in mathematical programming. In Linear Programming, we have had Khachiyan's polynomial time algorithm and a whole spectrum of more practical, interior point algorithms. Together with techniques such as branch-and-cut and polyhedral combinatorics and the very significant advances in computer technology, these greatly enhance the prospect of finding exact solutions to difficult optimisation problems. Nevertheless, the vast majority of large scale, commercial optimisation problems pose a real challenge and their exact solution usually remains computationally intractable. The Artificial Intelligence community has also made considerable advances and many of the methods that they espouse are now coming to fruition and can be used for practical applications. For example, expert systems, case-based reasoning, constraint satisfaction programming and genetic algorithms have all come from the AI stable and are now widely used by industry to solve large real-world problems. Local search is a more mundane technique, at heart quite naive, but nevertheless, for many of the most difficult realworld problems, it provides one of the very best ways of obtaining a near optimal solution with reasonable computational effort. During the last three decades, local search has also been developed and powerful variants of the simple neighbourhood search paradigm have emerged. The problem with neighbourhood search is its propensity to prematurely converge to a local optimum. One way of avoiding this is to use variable depth search which allows the exploration of a more complex neighbourhood in an attempt to find an improved solution. Other recent methods, sometimes known as 'modern heuristic techniques', are also designed to escape from the local optima traps. In general, they still do not guarantee an optimal solution nor even to meet given performance bounds. However, as this book shows, they often work exceptionally well in practice and have many advantages including versatility, efficiency and simplicity. Three of the main modern heuristic methods are simulated annealing, tabu search and genetic algorithms but, with the wider use of hybrid methods, the distinctions will become increasingly blurred. Simulated annealing uses a probability function which Modern Heuristic Search Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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List of Contributors E. H. L. Arts Department of Mathematics & Computer Science Eindhoven University of Technology Eindhoven, Netherlands N. A. S. Azarmi IC-Parc Imperial College London R. Battiti Laboratorio di Informatica e Calculo (LAICA) Dipartimento di Matematica Universita Segli Studi di Trento Trento, Italy P. J. Bentley Dept. of Electrical Engineering University of Huddersfield Huddersfield H. M. Cartwright Physical Chemistry Laboratory Oxford University Oxford P. Devine The Liverpool Biocomputation Group Department of Computer Science University of Liverpool Liverpool K. A. Dowsland European Business Management School Swansea University Swansea L. J. Eshelman Philips Laboratories Briarcliff Manor New York, USA C. Höhn Institut fur Grundlagen Elektrotechnik/Electronik Technische Universitat Dresden, Germany

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B. Jesson Physical Chemistry Laboratory Oxford University Oxford Modern HeuristicSearch Methods Editor V. J. Rayward-Smith, I. H. Osman, C. R. Reeves and G. D. Smith©1996 John Wiley & Sons Ltd.

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Y. J. Jiang IC-Parc Imperial College London J. Józefowska Institute of Computing Science Poznan University of Technology Poznan, Poland A. J. Keane Dept. of Engineering Science University of Oxford Oxford G. Kendall The Liverpool Biocomputation Group Department of Computer Science University of Liverpool Liverpool Y. N. Lee School of Information Systems University of East Anglia Norwich Y. H. Li IC-Parc Imperial College London G. P. McKeown School of Information Systems University of East Anglia Norwich J. W. Mann Nortel Technology Harlow I. H. Osman Institute of Mathematics & Statistics University of Kent Canterbury R. Paton The Liverpool Biocomputation Group Department of Computer Science University of Liverpool Liverpool V. J. Rayward-Smith School of Information Systems University of East Anglia Norwich file:///C|/Documents and Settings/Red Apple/桌面/0471962805/files/page_vi.html（第 1／2 页）2009-7-7 16:58:55

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C. R. Reeves School of Mathematical & Information Sciences Coventry University Coventry

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E. B. Richards IC-Parc Imperial College London S. Salhi School of Mathematics & Statistics University of Birmingham Birmingham J. D. Schaffer Philips Laboratories Briarcliff Manor New York, USA M. E. M. Severens Dept. of Mathematics & Computer Science Eindhoven University of Technology Eindhoven, Netherlands G. D. Smith School of Information Systems University of East Anglia Norwich M. G. A. Verhoeven Dept. of Mathematics & Computer Science Eindhoven University of Technology Eindhoven Netherlands S. Voß Institut für Wirtschaftswissenschaften Abteilung Allgemeine BWL Wirtschaftsinformatik und Informationsmanagement Technische Universitat Carolo-Wilhelmina Braunschweig, Germany J. P. Wakefield Department of Electrical Engineering University of Huddersfield Huddersfield G. Waligóra Institute of Computing Science Poznan University of Technology Poznan, Poland J. Weglarz * Institute of Computing Science Poznan University of Technology Poznan, Poland

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allows a move from a solution to a worse solution with a decreasing probability as the search progresses. Tabu Search uses a deterministic rather than stochastic search. Tabu Search moves to the best admissible solution even if this causes the objective function to deteriorate but, to avoid cycling, a short term adaptive memory is used. With genetic algorithms, a pool of solutions is used and the neighbourhood function is extended to act on pairs of solutions (using the so-called crossover operator. The pool is then maintained according to the 'survival of the fittest' principle. These methods can be hybridised with each other or with more traditional Operational Research techniques. Moreover, they have considerable scope for parallel or distributed implementation. Anyone working in practical operational research, artificial intelligence or any area of decision support needs to be a polymath. Modern heuristic techniques have to be part of his or her weaponry. To meet the need for training in this area, UNICOM have run a series of conferences, workshops and seminars on this and related topics. These conferences have brought together academic researchers and those working in commerce and industry. The result has been most fruitful. In the Spring of 1995, a particularly successful meeting on 'Adaptive Decision Technologies' was held at Brunel University. One of the three main streams was on modern heuristic methods. This book grew out of this meeting but it is not simply the conference proceedings. It comprises one specially written introductory chapter and a carefully refereed selection of extended versions of the best papers presented at that meeting. The book is divided into three sections. The introductory chapter provides a clear overview of the basic techniques and useful pointers to further reading and to current research. The second section concerns some of the most recent and exciting developments of the basic techniques. There are suggestions not only for extending and improving the techniques but also for hybridising and incorporating automatic adaptation. The third section contains a number of case studies, surveys and comparative studies. These span a wide range of application areas ranging from the classic Steiner tree problem to more practical problems arising in telecommunications and data analysis. In the preparation of this text, a large number of people have made significant contributions. Firstly, the staff of UNICOM, in particular Julie Valentine, Mark Salter and Professor Gautam Mitra were instrumental in organising the Conference and in encouraging us to produce this book. The four of us who edited this book called upon a large number of referees who, as usual, we do not name individually but to whom we owe a considerable debt of gratitude. They gave invaluable advice and, in many cases, detailed and always constructive comments. The main credit though must go to the authors who have researched their chapters and worked hard to produce manuscripts in the limited time available. During the final preparation phase, we have had essential administrative and technical help from staff at the University of East Anglia, in particular from Grace Granger and from Romek Szczesniak. V. J. RAYWARD-SMITH, CHAIRMAN, ADT95 CONFERENCE.

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Contents

List of Contributors

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Preface

ix

Introduction 1 Modern Heuristic Techniques

Abstract

1.1 Introduction

1.2 Combinatorial Optimization

1.3 The Case for Heuristics

1.4 Neighbourhood Search

1.5 Simulated Annealing

1.5.1 A Simple Example

1.5.2 Generic Decisions

1.5.3 Problem Specific Decisions

1.5.4 Other Factors

1.5.5 Summary

1.6 Tabu Search

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1.6.1 Basic Concepts

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1.6.2 Recency

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1.6.3 Frequency

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1.6.4 Summary

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1.7 Genetic Algorithms

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1.7.1 Crossover

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1.7.2 Mutation

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1.7.3 Reproduction

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1.7.4 A Simple Example

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1.7.5 Modifications and Enhancements

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1.7.6 Summary

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1.8 Conclusion

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References

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Techniques 2 Localized Simulated Annealing in Constraint Satisfaction and Optimization

Abstract

2.1 Introduction

2.2 Subspaces and Their Evaluation

2.3 Localized Simulated Annealing (LSA)

2.4 Analysis and Justification

2.5 Empirical Results

2.5.1 The Flight Scheduling Problem

2.5.2 The Freight Train Regrouping Problem

2.6 Conclusion

References 3 Observing Logical Interdependencies in Tabu Search: Methods and Results

Abstract

3.1 Introduction

3.2 Tabu Search

3.2.1 Historical Development

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3.2.2 Guiding Process Versus Application Process

3.2.3 Tabu Navigation Method Static Tabu Search

3.2.4 Reverse Elimination Method and Cancellation Sequence Method

3.2.5 Logical Interdependencies

3.2.6 Moving Gap

3.3 Applications in Location Planning

3.3.1 Network Design and Steiner Trees

3.3.2 Warehouse Location Problems

3.3.3 Further Location Problems

3.4 Conclusions

References 4 Reactive Search: Toward Self-tuning Heuristics

Abstract

4.1 Reactive Search: Feedback Applied to Heuristics

4.2 Tabu Search: Beyond Local Search

4.3 Some Forms of Tabu Search

4.3.1 Dynamical Systems

4.3.2 Implementations

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4.4 Reactive Tabu Search (RTS)

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4.4.1 Self-adjusted Prohibition Period

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4.4.2 The Escape Mechanism

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4.4.3 Fast Algorithms for Using the Search History

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4.5 Different Ways of Escaping from a Local Attractor

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4.5.1 Strict-TS

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4.5.2 Fixed-TS

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4.5.3 Reactive-TS

4.6 Applications of Reactive Search: A Review

4.6.1 Combinatorial Tasks

4.6.2 Continuous Optimization

4.6.3 Sub-symbolic Machine Learning (Neural Networks)

4.6.4 VLSI Systems with Learning Capabilities

4.7 Software

4.8 Acknowledgements

References 5 Combinatorial Optimization by Genetic Algorithms: The Value of the Genotype/Phenotype Distinction

Abstract

5.1 Background

5.2 The Line Balancing Problem

5.3 An Inadequate Approach

5.4 A GA Approach

5.5 How Well Does it Work?

5.6 Conclusions

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References 6 Integrating Local Search into Genetic Algorithms

Abstract

6.1 Introduction

6.2 GAs and NS

6.3 No-wait Flowshop Problem

6.4 Graph Partitioning

6.4.1 The LPK Heuristic

6.4.2 Complementary Crossover

6.4.3 1-bit Hamming Hill Climber (HC)

6.4.4 Mutation (MUT)

6.5 Evaluation of the Operators

6.6 Recombination

6.7 Conclusion

References

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Abstract

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7.1 Introduction

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7.2 Local Search

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7.2.1 The Steiner Tree Problem in Graphs (SP)

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7.3 Neighborhoods for the SP

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7.4 Computational Results

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References

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8 Local Search Strategies for the Vehicle Fleet Mix Problem

Abstract

8.1 Introduction

8.2 The Vehicle Fleet Mix (VFM) Problem

8.2.1 Representation of the VFM

8.2.2 Literature Review of the VFM

8.3 Modified RPERT Procedure (MRPERT)

8.4 Tabu Search

8.5 Computational Experience

8.6 Conclusion and Future Directions

8.7 Acknowledgement

References 9 SA Solutions for Timetabling

Abstract

9.1 Introduction

9.2 Local Search, Simulated Annealing and Scheduling Problems

9.3 Examination Scheduling

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9.5 Conclusions

References 10 A Tabu Search Algorithm for Some Discrete-Continuous Scheduling Problems

Abstract

10.1 Introduction

10.2 Problem Formulation

10.3 An Application of the Tabu Search Algorithm

10.3.1 Representation of a Feasible Solution

10.3.2 Creating a Starting Solution

10.3.3 Objective Function

10.3.4 Mechanism of Generating a Neighbourhood

10.3.5 Structure of the Tabu List

10.4 Computational Experiments

10.5 Final Remarks

10.6 Acknowledgements

References 11 The Analysis of Waste Flow Data from Multi-Unit Industrial Complexes Using Genetic Algorithms

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Abstract

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11.2 Applicability of the Genetic Algorithm

11.3 The Flow of Industrial Waste within Large Complexes

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11.4 The Algorithm

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11.4.1 String Encoding

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11.4.2 The Fitness Function

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11.4.3 Crossover Operator

11.4.4 Mutation Operator

11.5 Results

11.5.1 Input Data Generation

11.5.2 Results for a Four Factory System

11.6 Applications of the Algorithm

11.7 Acknowledgements

References 12 The Evolution of Solid Object Designs Using Genetic Algorithms

Abstract

12.1 Introduction

12.2 Representation

12.3 Correcting Illegal Designs

12.3.1 Method 1

12.3.2 Method 2

12.3.3 Method 3

12.3.4 Where to Correct the Designs

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12.4 The Evolution of a Table

12.4.1 Size

12.4.2 Mass

12.4.3 Stability

12.4.4 Flat Surface

12.4.5 Supportiveness and Stability

12.4.6 What the Tables are Not

12.5 Conclusions

References 13 The Convoy Movement Problem with Initial Delays

Abstract

13.1 Introduction

13.2 A Branch-and-Bound Approach

13.3 A Hybrid Approach : Using GAs for Delays and B&B for Paths

13.4 A Pure GA Approach

13.5 Conclusions

References 14 A Comparison of Heuristics for Telecommunications Traffic Routing

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14.2.1 The Genetic Algorithm

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14.2.2 Simulated Annealing

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14.3 Problem Specification

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14.4 Results

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14.5 Current and Future Work

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14.6 Conclusions

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14.7 Acknowledgements

References 15 A Brief Comparison of Some Evolutionary Optimization Methods

Abstract

15.1 Introduction

15.2 The Optimization Problems

15.3 The Optimizers

15.4 Initial Results

15.5 Optimizer Tuning

15.6 Conclusion

References 16 When 'Herby' met 'ElViS' Experiments with Genetics Based Learning Systems

Abstract

16.1 Introduction

16.2 Genetics Based Learning

16.3 Herby

16.3.1 System Components

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16.3.2 The Environment

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16.3.3 The Agent

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16.3.4 The Ecology

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16.3.5 Interim and Prospectus

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16.3.6 Corollary

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16.4 Exploring Ecological Vision with ElViS

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References Index

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