Metaheuristic Optimizationvia Memory and Evolution

Tabu Search and Scatter Search

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Greenberg I A Computer-Assisted Analysis System for Mathematical Programming Models andSolutions: A User's Guide for ANALYZE

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Technologies

Metaheuristic Optimization viaMemory and Evolution

Tabu Search and Scatter Search

edited byCesar Rego and Bahrain Alidaee

Kluwer Academic PublishersBoston/Dordrecht/London

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Rego & Alidaee/ METAHEURISTIC OPTIMIZATION VIA MEMORY AND EVOLUTION:Tabu Search and Scatter Search

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CONTENTS

Foreword vii

Part I: ADVANCES FOR NEW MODEL AND SOLUTION APPROACHES

1. A Scatter Search Tutorial for Graph-Based Permutation Problems 1Cesar Rego and Pedro Ledo

2. A Multistart Scatter Search Heuristic for Smooth NLP and 25MINLP ProblemsZsolt Ugray, Leon Lasdon, John C. Plummer, Fred Glover, Jim Kellyand Rafael Marti

3. Scatter Search Methods for the Covering Tour Problem 59Roberto Baldacci, Marco A. Boschetti, Vittorio Maniezzo and MarcoZamboni

4. Solution of the Sonet Ring Assignment Problem withCapacity Constraints 93Roberto Aringhieri and Mauro DelVAmico

Part II: ADVANCES FOR SOLVING CLASSICAL PROBLEMS

5. A Very Fast Tabu Search Algorithm for Job Shop Problem 117Jozef Grabowski and Mieczyslaw Wodecki

6. Tabu Search Heuristics for the Vehicle Routing Problem 145Jean-Frangois Cordeau and Gilbert Laporte

7. Some New Ideas in TS for Job Shop Scheduling 165Eugeniusz Nowicki and Czeslaw Smutnicki

8. A Tabu Search Heuristic for the Uncapacitated Facility LocationProblem 191Minghe Sun

9. Adaptive Memory Search Guidance for Satisfiability Problems 213Arne L0kketangen and Fred Glover

Part III: EXPERIMENTAL EVALUATIONS

10. Lessons from Applying and Experimenting with Scatter Search 229Manuel Laguna and Vinicius A. Armentano

11. Tabu Search for Mixed-Integer Programming 247Jodo Pedro Pedroso

12. Scatter Search vs. Genetic Algorithms: An Experimental Evaluationwith Permutation Problems 263Rafael Marti, Manuel Laguna and Vicente Campos

Part IV: REVIEW OF RECENT DEVELOPMENTS

13. Parallel Computation, Co-operation, Tabu Search 283Teodor Gabriel Crainic

14. Using Group Theory to Construct and Characterize MetaheuristicSearch Neighborhoods 303Bruce W. Colletti and J. Wesley Barnes

15. Logistics Management: An Opportunity for Metaheuristics 329Helena R. Lourenco

PartV: NEW PROCEDURAL DESIGNS

16. On the Integration of Metaheuristic Strategies in ConstraintProgramming 357Mauro DelVAmico and Andrea Lodi

17. General Purpose Metrics for Solution Variety 373David L Woodruff

18. Controlled Pool Maintenance for Metaheuristics 387Peter Greistorfer and Stefan Vop

19. Adaptive Memory Projection Methods for Integer Programming 425Fred Glover

20. RAMP: A New Metaheuristic Framework for CombinatorialOptimization 441Cesar Rego

Index 461

Foreword

WHAT'S ALL THE COMMOTION ABOUTSCATTER SEARCH AND TABU SEARCHANYWAY?

Fred GloverLeads School of Business, University of Colorado, Boulder, CO 80309-0419, USA,fred.glover@colorado. edu

Given an opportunity to dig into a good book, such as the one youpresently hold in your hands, little can be more of a nuisance than to read apreface. Let me acknowledge at once that I certainly won't be offended if youdon't read this one. (After all, I'll never know whether you did or not!) Asmy grandmother used to say, a preface to a book is like a speech before abanquet: it only delays enjoying what follows.

If you've bothered to read this far, at the least I owe you the courtesy ofletting you know what to expect by reading farther. First, I'll quickly describesome features of the book that I'm delaying you from enjoying. Then I'll seeI can offend as many of our mutual friends as possible (including you, to besure) by suggesting that some popular conceptions of search strategies maybe leading us down the wrong track. If at this point you haven't yet shreddedthe preface in the firm resolve to read the remainder of the book withoutfurther distraction, you will come to a proposal for an experiment to put mycontentions to the test, and thereby to expose their underlying sense ornonsense. Finally, I'll wrap up by commenting on where we are headed in thegrand scheme of things, as a way to remind you once again of what you'vebeen missing by pausing to scan these preliminary pages.

What the book offers.

Tabu search and scatter search have come a long way from their origins,and the papers in this volume disclose some of the significant recentdevelopments that are fueling advances in these areas. New contributions aredocumented for solving a varied collection of challenging problems, boththose representing classical optimization models that have attracted the

attention of researchers for many years, and those representing quite recentmodels, that belong to forefront of modern optimization concerns. Thevolume additionally includes papers devoted to experimentation, exposinglessons learned from evaluating the performance of past and recent proposals.Still another section focuses on identifying currently established andemerging roles for tabu search and scatter search in realms such as paralleloptimization and constraint programming, thereby affording a crossfertilization with other disciplines. And not least, the volume contains acollection of papers offering innovative proposals for new research directions.

This work represents the first time that tabu search and scatter search havebeen examined together in the same volume. Such a side-by-side examinationis long overdue, particularly in view of the fact that the two approaches grewout of common beginnings, and share overlapping principles andperspectives. Today tabu search is typically applied from the perspective ofadaptive memory concepts and strategies that it has introduced into themetaheuristic literature, while scatter search is primarily pursued within thecontext of an evolutionary approach, emphasizing processes concerned withgenerating and managing a population of solutions. Yet tabu search hasalways likewise embraced strategies for managing populations of solutions,notably within the setting of two of its basic types of intensificationprocesses, the first concerned with saving and re-visiting elite solutions toexplore their vicinity more thoroughly and the second concerned withanalyzing these solutions to isolate critical features, such as variables thatstrongly and frequently receive particular values, as a basis for generatingother solutions sharing these features.

In a complementary manner, scatter search has made use of adaptivememory, chiefly by virtue of incorporating improvement processes which,curiously enough, have only become recognized as relevant in otherevolutionary approaches in relatively recent years. In the setting of thesecontrasting evolutionary procedures, the slowly dawning appreciation of thevalue of such improvement procedures has spawned the appearance ofmethods that currently go by the names of "hybrid" and "memetic"procedures. Characteristically, however, by its association with tabu search,scatter search is more often disposed to make use of improvement processesthat embody adaptive memory, whereas other evolutionary approaches oftenstill lag in bridging the gap to exploit such strategies.

Curious as it may seem, an eminent body of traditional wisdom inclines usto see little of interest in designs anchored to adaptive memory and associatedstrategies for exploiting it. The TS focus on adaptive memory admittedly

Foreword ix

poses challenges that many researchers may prefer to sidestep. But, grantingsome liberty to speculation, these challenges may also help to define thefrontiers of research into problem-solving processes that seek to incorporateabilities analogous to our own. Perhaps the human impulse to avoidacknowledging that our methods are still somewhat primitive steers us awayfrom ideas that expose the yawning chasms of our ignorance. In any case, theblueprints drawn up for methods that do not venture far into the realms ofmemory and learning seem to capture the devotion of many searchpractitioners. From this standpoint, the papers of this volume represent amore daring collection than those found in many treatments of metaheuristics,by confronting important issues connected with the use of adaptive memoryin search and seeking to draw inferences about experiments that may shedlight on such issues.

Randomization: For Better or Worse1

Apart from the issue of adaptive memory, a topic that conspicuouslyseparates tabu search and scatter search from other approaches concerns theapplication of random choice. It is impossible to look very far in themetaheuristic literature without becoming aware that it is engaged in a loveaffair with randomization. Our "scientific" reports of experiments with naturereflect our fascination with the role of chance. When apparently chaoticfluctuations are brought under control by random perturbations, we seizeupon the random element as the key, while downplaying the importance ofattendant restrictions on the setting in which randomization operates. Thediligently concealed message is that under appropriate controls, perturbationis effective for creating outcomes of desired forms. If the system andattendant controls are sufficiently constrained, perturbation works even whenrandom, but a significant leap is required to conclude that randomization ispreferred to intelligent design. (Instead of accentuating differences betweenworkable and unworkable kinds of perturbation, in our quest to mold theuniverse to match our mystique we often portray the source of usefuloutcomes to be randomization in itself.)2

1 This section and the next draw on portions of the reference F. Glover (2003) "Tabu Search -Uncharted Domains," University of Colorado, Boulder.

The attraction of randomization is particularly evident in the literature of evolutionarymethods, but the theme that randomization is an essential underpinning of modern searchmethods is also echoed throughout many other segments of the metaheuristic literature.Perhaps this orientation relates to our natural disposition to take comfort in the notion thatall can be left up to chance and everything will turn out for the best in the end. Or perhapsrandomization has a seductive charm akin to the appeal of a certain kind of romantic

The orientation behind tabu search and scatter search evidently contrastswith this perspective. Many of the fundamental TS and SS strategies do notrequire randomization for their execution. While tabu search and scattersearch principles do not exclude recourse to randomization, neither do theyembrace it as essential. Occasionally such implementations are found whererandomization is given a highly significant role, a fact that is consonant withthe notion that a probing literature should entertain exceptions. However, as arule, variants of TS and SS that incorporate randomized elements operate in acontext where variation is tightly controlled by strategy.

There is of course a setting where randomization makes good sense. In agame against a shrewd opponent, it can be desirable to make sure one'sbehavior exhibits no demonstrable pattern, in order to avoid the chance thatthe pattern will be detected and turned to the adversary's advantage. In thiscase, recourse to randomization is a prudent policy. On the other hand, if weare entitled to presume that our search spaces are not possessed of a deviousintelligence bent on thwarting our moves, random behavior fulfills nopurpose comparable to its role in game playing. Instead, within the context ofsearch, randomization seems more likely to be a means of outwitting theplayer who uses it. A policy of embarking on a series of random steps wouldseem one of the best ways of confounding the goal of detecting andexploiting structure.

We may grant that systematic search runs the risk of systematic blundering,something we are all susceptible to. But with appropriate monitoring,systematic search also affords an opportunity to isolate and rectify ourblunders, and therefore to carry out more effective explorations. (Of course, ifthe search terrain itself is inherently random, no amount of systematizationwill do any good. The presence or absence of patterned design makes nodifference when all is reduced to blind luck.)3

encounter - which seems somehow more captivating when marked by an element ofcapriciousness.

Even the statement that structured search has little value for a problem whose character is"essentially random" deserves qualification. A sorting problem may involve entirelyrandom data, but may still be handled advantageously by a method that is highlysystematic. (The "P vs. NP" distinction is relevant, but does not remove the need foradditional qualification.)

Foreword xi

An Experiment with Randomization.

I am tempted to pose a research challenge, as a way of testing the relevanceof randomization in search. If a policy of random choice produces a usefulsequence of responses, then we may presume that a different randomlygenerated sequence will likewise prove useful. On the other hand, not allsequences are apt to be equally good if the total number of moves is limited -a relevant consideration if it is important to reach good outcomes before theknell of doomsday, or at least before next summer's vacation. When timematters, we may anticipate that not all randomly generated sequences willprovide the same quality of performance.

This observation prompts the following experiment. To determine therelative utility of randomization, we might examine the outcomes ofexecuting some number, say k, of randomly guided searches (each of limitedduration) with the goal of finding a value of k such that at least one of theunderlying choice sequences will lead to a solution of specified quality. Weseek a value for our parameter that is not too large and that is applicable to alarge portion of problems from a given class.

Now the evident question comes to mind. If we replace the k randomlygenerated choice sequences with k systematically generated sequences, canwe obtain comparable or better results? Or can the replacement allow us toproduce such results for a smaller value of k? In accordance with the theme ofdiversification strategies used in TS and SS, the systematically generatedsequences may be equipped to incorporate feedback, allowing a newsequence may be influenced by the outcomes of preceding sequences. (Thereis an issue of time invested in the systematic approach, so that the value of kmust be adjusted to account for this.)

The outcome of such an experiment of course depends on our skills indesigning systematic choice sequences, and also depends on the types ofproblems we confront. It must be acknowledged that the experiment I amadvocating has been implicit in a variety of TS and SS implementations of thepast. But there may be advantages to bringing the relevant factors into moreexplicit focus. Perhaps this will help us to better understand what is essentialand what is superfluous when we speak of "intelligent strategy," and toidentify better instances of such strategy over time.

Where Are We Headed?

The papers of this book provide a series of landmarks along the way as weinvestigate and seek to better understand the elements of tabu search andscatter search that account for their successes in an astonishingly varied rangeof applications. The contributions of the chapters are diverse in scope, and arenot uniform in the degree that they plumb or take advantage of fundamentalprinciples underlying TS and SS. Collectively, however, they offer a usefulglimpse of issues that deserve to be set in sharper perspective, and that moveus farther along the way toward dealing with problems whose size andcomplexity pose key challenges to the optimization methods of tomorrow.

Foreword xiii

Acknowledgment and Disclaimer

This material is based upon work supported by the National ScienceFoundation under Grant No. 0118305. Any opinions, findings, andconclusions or recommendations expressed in this material are thoseof the author(s) and do not necessarily reflect the views of the NationalScience Foundation.

Chapter 1

A SCATTER SEARCH TUTORIAL FORGRAPH-BASED PERMUTATION PROBLEMS

Cesar Rego1 and Pedro Leao2

lHearin Center for Enterprise Science, School of Business Administration, University ofMississippi, University, MS 38677, USA, crego@bus.olemiss.edu;1 Universidade Portucalense, Departamento de Informdtica, Rua Dr. Antonio Bernardino deAlmeida 541-619, 4200, Porto, Portugal, pedro@upt.pt

Abstract: Scatter search is an evolutionary method that has proved highly effective insolving several classes of non-linear and combinatorial optimization problems.Proposed early 1970s as a primal counterpart to the dual surrogate constraintrelaxation methods, scatter search has recently found a variety of applications ina metaheuristic context. Because both surrogate constraint methods and scattersearch incorporate strategic principles that are shared with certain componentsof tabu search methods, scatter search provides a natural evolutionaryframework for adaptive memory programming. The aim of this paper is toillustrate how scatter search can be effectively used for the solution of generalpermutation problems that involve the determination of optimal cycles (orcircuits) in graph theory and combinatorial optimization. In evidence of thevalue of this method in solving constrained optimization problems, we identifya general design for solving vehicle routing problems that sets our approachapart from other evolutionary algorithms that have been proposed for variousclasses of this problem.

Keywords: Metaheuristics, Scatter Search, Combinatorial Optimization, PermutationProblems, Vehicle Routing Problems

1. Introduction

Scatter search is an evolutionary method proposed by Glover (1977) as aprimal counterpart to the dual approaches called surrogate constraintmethods, which were introduced as mathematical relaxation techniques fordiscrete optimization problems (Glover 1965). As opposed to eliminatingconstraints by plugging them into the objective function as in Lagrangean

relaxations, for example, surrogate relaxations have the goal of generatingnew constraints that may stand in for the original constraints. The method isbased on the principle of capturing relevant information contained inindividual constraints and integrating it into new surrogate constraints as away to generate composite decision rules and new trial solutions. Scattersearch combines vectors of solutions in place of the surrogate constraintapproach of combining vectors of constraints, and likewise is organized tocapture information not contained separately in the original vectors. Also, incommon with surrogate constraint methods it is organized to take advantageof auxiliary heuristic solution methods to evaluate the combinations producedand generate new vectors. As any evolutionary procedure, the methodmaintains a population of solutions that evolves in successive generations.

A number of algorithms based on the scatter search approach haverecently been proposed for various combinatorial problems (see Kelly,Rangaswamy and Xu 1996, Fleurent et al. 1996, Cung et al. 1999, Laguna,Marti and Campos 1999, Campos et al. 1999, Glover, Lokketangen andWoodruff 1999, Atan and Secomandi 1999, Laguna, Lourencpo and Marti2000, Xu, Chiu and Glover 2000).

In this study we provide a general scatter search design for problemsdealing with the optimization of cycles (or circuits) on graphs. The travelingsalesman problem (TSP) which consists of finding the shortest Hamiltoniancycle (or circuit in the asymmetric case) is a typical example of this class ofproblems. It is well-known that the TSP is NP-Complete so that efficientheuristic methods are required to provide high quality solutions for largeproblem instances (see Johnson and McGeoch 1997, Rego 1998). A directextension of the TSP is the vehicle routing problem (VRP) where sideconstraints force the creation of multiple Hamiltonian cycles {routes) startingand ending at a central node (representing a hub or depot). Both TSP andVRP problems provide a general model for a wide range of practicalapplications and are central in the fields of transportation, distribution andlogistics (see Reinelt 1994, Laporte and Osman 1995). Empirical studies haveshown that the VRP is significantly harder to solve than TSPs of similar sizes(see Gendreau, Hertz and Laporte 1994, Rochat and Taillard 1995, Rego1998, 2000, and Toth and Vigo 2003, for some of the current best heuristicalgorithms for the VRP). Because of its theoretical and practical relevance weuse the vehicle routing problem to illustrate the various procedures involvedin the scatter search template.

Also, while other tutorials on scatter search exist (see, e.g. the articles byLaguna 2001, and Glover, Laguna and Marti 2001), there have been noexposition of the approach that disclose its operation within the setting ofrouting.

Scatter Search for Graph-Based Permutation Problems 3

The remainder of this paper is organized as follows. An overview of thescatter search method is presented in Section 2. Section 3 defines the vehiclerouting problem and presents formulations that are relevant for the context ofthe paper. Section 4 illustrates the design of the scatter search procedure foran instance of the problem. Section 5 summarizes and discusses possiblegeneralizations of the proposed template.

2. Scatter Search Overview

Scatter search operates on a set of reference solutions to generate newsolutions by weighted linear combinations of structured subsets of solutions.The reference set is required to be made up of high-quality and diversesolutions and the goal is to produce weighted centers of selected subregionsthat project these centers into regions of the solution space to be explored byauxiliary heuristic procedures. Depending on whether convex or nonconvexcombinations are used, the projected regions can be respectively internal orexternal to the selected subregions. For problems where vector componentsare required to be integer a rounding process is used to yield integer valuesfor such components. Rounding can be achieved either by a generalizedrounding method or iteratively, using updating to account for conditionaldependencies that can modify the rounding options. Regardless the type ofcombinations employed, the projected regions are not required to be feasibleso that the auxiliary heuristic procedures are usually designed to incorporate adouble function of mapping an infeasible point to a feasible trial solution andthen to transform this solution into one or more trial solutions. (Although, theauxiliary heuristic commonly includes the function of restoring feasibility,this is not an absolute requirement since scatter search can be allowed tooperate in the infeasible solution space.) From the implementation standpointthe scatter search method can be structured to consist of the followingsubroutines:

Diversification Generation Method - Used to generate diverse trialsolutions from one or more arbitrary seed solutions used to initiate themethod.

Improvement Method - Transform a trial solution into one or moreenhanced trial solutions. (If no improvement occurs for a given trial solution,the enhanced solution is considered to be the same as the one submitted forimprovement.)

Reference Set Update Method - Creates and maintains a set of referencesolutions consisting of the best according to the criteria under consideration.The goal is to ensure diversity while keeping high-quality solutions asmeasured by the objective function.

Subset Generation Method - Generates subsets of the reference set as abasis for creating combined solutions.

Solution Combination Method - Uses weighted structured combinations totransform each subset of solutions produced by the subset generation methodinto one or more combined solutions.

A general template for a scatter search algorithm can be organized in twophases outlined as follows.

• Initial PhaseDiversification Generation MethodImprovement MethodReference Set Update MethodRepeat this initial phase until producing a desirable level of high-quality

and diverse solutions.• Scatter Search Phase

Subset Generation MethodSolution Combination MethodImprovement MethodReference Set Update MethodRepeat this scatter search phase while the reference set converges or until

a specified cutoff limit on the total number of iterations.

3. The Vehicle Routing Problem

The vehicle routing problem is a classic in Combinatorial Optimization.To establish some basic notation, and to set the stage for subsequentillustrations, we provide a formal definition of the vehicle routing problem inthis section. We also present two mathematical formulations which arerelevant for the scatter search design introduced in the next section.

3.1 Problem Definition

In graph theory terms the classical VRP can be defined as follows. LetG = fV,A) be a graph where V = {vo,vl,...,vn} is a vertex set, andA = {(vi}Vj) | vifVj eV,i^ j \ is an arc set. Vertex v0 denotes a depot, where afleet of m identical vehicles of capacity Q are based, and the remainingvertices V'=V\{vQ} represent n cities (or client locations). A nonnegativecost or distance matrix C = (ctj) which satisfies the triangle inequality( Cy <cik +ckj,) is defined on A. When c^-c^ for all (V{,VJ)GA theproblem is said to be symmetric and it is then common to replace A with theedge set E = \(vifVj) | vt,Vj eV,i* j]. It is assumed that m e [rn,m] with

Scatter Search for Graph-Based Permutation Problems 5

m = 1 and m = n - 1 . The value of m can be a decision variable or can befixed depending on the application.

Vehicles make collections or deliveries but not both. With each vertexv{ is associated a quantity q{ (q0 - 0) of some goods to be delivered by avehicle. The VRP consists of determining a set of m vehicle routes ofminimal total cost, starting and ending at a depot v0, such that every vertex inv{ e V is visited exactly once by one vehicle and the total quantity assignedto each route does not exceed the capacity Q of the vehicle which services theroute.

We define a solution for the VRP as a set of m routes, S = {jR1,...,Rm}9

Rk = (vo,vk ,vk ,..-v0) where i ^ i s an ordered set representing consecutivevertices in the route k. Consequently, we denote v{ e Rk if v{ is componentof Rk and similarly (vifVj) e Rk if v{, vk are two consecutive vertices inRk . Finally, the cost of a solution S is defined as C(S) = Xi<jc<mS/f >/? cij •

3.2 Vehicle Flow Formulation

,=0 7=0 *=1

SJ.

(1)

!"? (2)

Z4 = I4=^> f j""'i=0 i=0 " ~ * > • • • > '

i = 0,...,« (4)

In this formulation (Fisher and Jaikumar 1981), variables *•• indicateswhether or not (vifVj) is an edge of route k in the optimal solution, and

similarly y\ specifies whether route k contains vertex v{. Constraints (1)guarantee that vehicle capacity is not exceeded; constraints (2) ensure thateach city is visited by exactly one vehicle and m vehicles visit the depot;constraints (3) ensure that every city is entered and left by the same vehicle;subtour elimination are specified by constraints (4); and equations (5) are theintegrality constraints.

This formulation contains two well-known problems: the generalizedassignment problem (GAP) specified by constraints (1), (2) and (5); and thetraveling salesman problem (TSP). The TSP results when the y\ 's are fixedto satisfy the GAP constraints for a given k, whereupon constraints (3) and (5)define a TSP for vehicle k. Specifically, the GAP is the subproblemresponsible for assigning clients to vehicle routes, and the sequence in whichclients are visited by a vehicle is determined by the TSP solution over eachindividual route.

Laporte, Nobert, and Desrochers (1985) show that a two-indexformulation can be derived from a three-index formulation by aggregating allXij variable into a single variable Xy which does not refer specifically to aparticular route, but simply indicates whether or not an arc (vitVj) is in theoptimal solution. For the symmetrical case the value of the x{j variableindicates the number of times (0,1, or 2) that the corresponding edge is usedin the optimal solution. We will see later that this vehicle-flow formulationand the index-reduction concept proves useful in several parts of our scattersearch model.

3.3 A Permutation-Based Formulation

In combinatorial optimization it is usually useful to distinguish twoclasses of problems according to whether the objective is to find (1) anoptimal combination of discrete objects, or (2) an optimal permutation ofobjects. In the first case, a solution is a non-ordered set while in the second asolution is a ordered set (or a sequence). For instance, the 0-1 knapsackproblem and the set covering problem are typical examples of combinationproblems where a solution can be interpreted to consist simply of a set ofelements. Permutation problems on the other hand, are exemplified by jobscheduling, traveling salesman, and vehicle routing problems where differentarrangements of the same set represent different solutions (allowing for asequence to be considered equivalent to its reverse ordering in the case ofsymmetric problems). The differentiation of these two types of combinatorialproblems is particularly relevant because the specialized methods that applyto these problems, especially those methods based on graph theoreticrepresentations, differ dramatically in the two cases. We will address to thesefeatures later on when describing the proposed scatter search template.

Scatter Search for Graph-Based Permutation Problems 1

The VRP, which is the primary focus of this tutorial, seeks to determinean optimal permutation according to the following formulation:

mm\=\ i=0

S.t.

n,<Qk, k = l,...,m (1)

k<n (2)k=\

Here, a solution is defined by the permutation yrl,...,^,...^™,...,7r™mj of

vertices {vlf...,vn}. Vertex v0 is implicitly represented by inserting it at the

beginning and end of each subsequence k of the permutation (and may be

expressed by "dummy elements" TTQ and Xnk+i )• The permutation is

partitioned into subsequences by indexes nk and may be interpreted to

consist of m separate routes, where the sequence of nodes on a given route

identifies the order in which clients (vertices) are visited by a vehicle

assigned to that route.

4. Scatter Search Template

Consider the following VRP instance with n=14, Q=30,q = (q.)(i = 0,...,14) = (0,6,20, 8,9,10,8,7,5,5,4,3,4,7) , m e [l,l4], and acost matrix C = (c(j)(i,j = l,...,14,i< j) defined as follows:

0 7.85 10.82 8.18 9.71 4.53 3.13 5.11 6.40 7.54 6.40 9.30 8.51 4.61 7.960 14.40 4.36 4.19 9.41 9.39 2.77 1.57 1.12 2.82 1.73 8.98 12.04 15.74

0 10.89 12.52 15.24 13.62 12.48 13.04 14.96 15.02 16.10 19.19 13.20 13.540

C =

1.84 11.470 12.60

0

10121

.80

.12.710

4.405.307.316.96

0

34881

.78

.30

.46

.22

.300

5.345.308.608.752.721.92

0

6.486.826.857.142.602.701.81

0

5.765.1210.2610.484.343.261.773.40

0

12.5913.055.026.608.278.857.886.248.95

0

12.79 15.8814.3017.544.152.949.4410.7411.509.9813.269.14

0

8.016.91

13.0414.3515.3013.8617.0712.873.97

0

A viewgraph of the clients' spatial distribution and the correspondingvertex coordinates that gave rise to the matrix C is provided in the appendix.We will use this special representation of the problem to illustrate differentprocedures used in the scatter search design.

Now, we can proceed to the illustration of the scatter search design usingthe defined VRP instance as an example of a permutation problem. We firstdescribe in the context of the VRP each of the methods considered in thegeneral scatter search template. Then, we present the various proceduresintegrated in the overall algorithm.

• Initial PhaseDiversification Generation MethodScatter search starts with an initial set of trial solutions which are required

to be different, thus the use of a systematic procedure to generate thesesolutions is appropriate. Regarding the VRP as permutation problem weconsider the generator of combinatorial objects described in Glover (1997),which operates as follows. A trial permutation P is used as a seed to generatesubsequent permutations. Define the subsequence P(h:s) where s is a positiveinteger between 1 and A, so that P(h : s) = (s,s + h,s + 2h,...,s + rh), wherer is the largest nonnegative integer such that s + rh<n. Finally, define thepermutation P(h) for h<n, to be P(h) = (P(h : h), P(h :h-l), ..., P(h : 1)).In the VRP context we consider permutations in n-vectors where componentsare all vertices vx e V\{v0}. Consider for illustration h = 4 , and a seedpermutation P = {1,2,3,4,5,6,7,8,9,10,11,12,13,14] given by the sequence ofclients ordered by their indices. The recursive application of P(4:s) fors = 4,...,l results the subsequences:P = {4,8,12}, P = {3,7,11} , P = {2,6,10,14} and P = {l,5,9,13} , henceP(4) = {4,8,12,3,7,11,2,6,10,14,1,5,9,13} . For illustration purposes weconsider the set of initial vertices assignments A.h. to vehicle routes derivedfrom permutations P(h) (h = l,...,10) where each cluster of vertices in aroute is obtained by successively assigning a vertex vt(ieP(h)) to a routeRhk (initially k=l) until the cumulative quantity Qk = ^v eR q{ does notexceed Q with the insertion of a new vertex vk.. As soon as* such a cutofflimit is attained a new assignment is created by incrementing k by one unit,and the process goes on until all vertices have been assigned. Table 1.1 showsthe assignment A4 derived from permutation P(4) , where shadowed cellsindicate the Qk value associated with the insertion of vertex v{ in a route Rk .

In fact, the result obtained can be viewed as a generalized assignmentprocess which does not rely on the order in which clients are visited, though itensures that all the initial solutions that can be created are feasible anddifferent (since they derive from distinct permutations). Vehicle routes can

Scatter Search for Graph-Based Permutation Problems 9

thus be determined by using any traveling salesman algorithm on thesubgraph defined by the previous assignments.

Table 1P(4)R*.

iq-i-

Ri.

R2.R*R4.

R5.

1. Sequential assignment of clients to vehicle routes488

8715

12318

3927

78

114

12

220

20

610

30

105

5

147

12

16

18

59

27

95

5

134

9

For illustration, we consider the standard 2-exchange procedure (Lin1965) to find a 2-optimal TSP tour for each individual route. The 2-optimalprocedure is a local search improvement method, hence an initial feasibleTSP solution for each route Rk is required to start the method. Astraightforward method to create this solution can be obtained by successivelylinking vertices in the order they appear in the permutation and attaching theinitial and ending vertices to the depot. Then, the method proceeds byreplacing two edges (v{,v{) and (vj,Vj)by two others (vifVj)and (v^Uj)where v denotes the successor of v in a given orientation of the route. Hence,in order to maintain the feasible orientation of the route, the subpath(vif...,Vj) needs to be reversed, so that the subpath (vi,vi,...,Vj,Uj)becomes (viyv^)...>vi>v^) . For convenience, denote the c^ values byc(Ui,Vj) so that the solution cost change produced by a 2-exchange move canbe expressed as A^ = c(v{,Vj) + c(vt,Uj)-c(v{,v{)-c(Uj,Uj) . A 2-optimal(or 2-opt) solution is obtained by iteratively applying 2-exchange moves untilno possible move yield a negative A value.

As the purpose of the diversification generation method is to generatediverse solutions rather than high quality solutions, it is convenient to makethe method fast, therefore we have adopted a first-improvement (as opposedto a best-improvement) strategy for the TSP heuristic.

Table 1.2 shows the four iterations carried out to create the solution &*..Specifically, the first column shows the initial trial solution T4. created fromthe assignment A4 (defined in Table 1.1) and the final solution S+. Figureswithin parenthesis indicate the iteration number. The second column showsvertices v.h vy. that define the move from one iteration to another. (Note that a2-exchange move is completely identified by vertices vt,Vj since the twoother vertices are necessarily their successors, respectively v{ and Uj.) Theremaining columns show the routes in each solution, the solution cost, and thevalue associated with the move that has led to the solution shown in the nextrow. Boldfaced numbers indicate the changes carried out by the

10

corresponding move. The result of these changes are illustrated in Figure 1.1,which depicts the initial trial solution T4. and the final solution S+.

Table 1.2. Application of 2-optSolutionTj".T4<

2>.T <3>1.4.

Tj4).

s4.

Movev.a,v.12.V-lfhV.i

V./,V.j

V.o,Vj.

R,4 812 3128431284312 84312843

procedure to T.4. trial solutionR2.711711711711711

R3.2626262626

R4.10141 510141 510114 5101514110514

R5.913913913913913

Cost163.54151.60139.15136.25135.15

-11.94-12.45-2.90-1.10

Figure 1.1. Solutions TV (left-hand side) and 5* (right-hand side)

The shape of the solutions in the figure clearly show that routes Rj. and R4.were significantly improved. Repeating the process for the rest of thepermutations we obtain the ten solutions produced by the diversificationgeneration method as illustrated in Table 1.3.

Improvement MethodThe improvement method used in the initial phase may or may not be the

same method used in the scatter search phase. This decision usually dependson the context and on the search strategy one might want to implement. Forthe purpose of this tutorial we use the same procedure in both phases, thoughin the scatter search phase the method is required to deal with some additionalfeatures.

Scatter Search for Graph-Based Permutation Problems 11

Table 1.3. Trial solutions (T.h) and final solutions (Ch) produced by the diversificationgeneration method

SolutionTlSIT2S2T3S3T4S4T5S5T6S6T7S7T8S8T9S9T10S10

Routes0 1 2 0 3 4 5 0 6 7 8 9 0 10 11 12 13 14 00 1 2 0 3 4 5 0 6 9 8 7 0 11 10 12 14 13 00 2 4 0 6 8 1 0 1 2 0 1 4 1 3 0 5 7 9 1 1 13 00 2 4 0 6 1 2 1 0 8 0 1 4 3 1 0 7 1 1 9 5 1 3 00 3 6 9 12 0 2 5 0 8 11 14 1 0 4 7 10 13 00 6 1 2 9 3 0 2 5 0 8 1 11 1 4 0 7 4 1 0 1 3 00 4 8 12 3 0 7 1 1 0 2 6 0 10 14 1 5 0 9 13 00 12 8 4 3 0 7 11 0 2 6 0 1 10 5 14 0 9 13 00 5 10 4 9 0 14 3 8 13 0 2 7 0 12 1 6 1 1 00 5 1 0 9 4 0 8 3 14 1 3 0 2 7 0 1 11 12 6 00 6 1 2 5 1 1 0 4 1 0 3 9 0 2 8 0 1 4 1 7 1 3 00 6 5 12 1 1 0 3 4 9 10 0 2 8 0 14 13 1 7 00 7 14 6 13 0 5 12 4 1 1 0 3 10 0 2 9 0 1 8 00 7 6 13 14 0 5 12 1 1 4 0 3 10 0 2 9 0 1 8 00 8 7 6 0 14 5 13 4 0 12 3 1 1 0 2 10 0 1 9 00 7 8 6 0 14 13 5 4 0 12 1 1 3 0 2 10 0 1 9 00 9 8 7 6 0 5 14 4 13 0 3 12 0 2 11 1 0 10 00 7 8 9 6 0 14 1 3 5 4 0 3 12 0 2 1 1 1 0 10 00 10 9 8 7 0 6 5 4 0 14 3 13 0 2 12 1 0 1 1 00 10 9 8 7 0 6 5 4 0 3 14 13 0 2 1 1 2 0 1 1 0

Cost120.90120.83132.28122.74157.24128.27163.54135.15149.08119.50140.97113.74139.83130.47146.83136.29148.42137.16148.65135.92

We consider an iterative improvement method based on local search. Themethod is designed to directly operate on the problem graph using a verystraightforward neighborhood structure consisting of removing a vertex fromits current position and inserting it between two other vertices. Specifically,denoting respectively by v and v the predecessor and successor of a vertexv, the insertion of a vertex vt between vertices vp and vq operates byinserting edges (v{,v{), (vp,v{), (vuvq), and by removing edges (viyv{),(vi)vi) and (vp,vp), where vq=vp. Hence, the solution cost change isgiven by Aip = c(v{,v{) + c(vp>vi) + c(vt,vp)-c(v{,v{)-c(v(,v{)-c(vp,vp) .Applying the procedure on each £,. solution (in Table 1.3) we obtain thecorresponding improved solutions reported in Table 1.4. (A graphicillustration of the method is shown in the scatter search phase where someadditional features of the method are included.)

Reference Set Update MethodThis method is used to create and maintain a set of reference solutions. As

in any evolutionary (population-based) method, a set of solutions (populationof individuals) containing high evaluation combinations of attributes replacesless promising solutions at each iteration (generation) of the method in orderto enhance the quality of the population.

12

Table 1.4. Improved SolutionsSolution

SIS2S3S4S5S6S7S8S9S10

Routes0 2 0 3 4 5 0 6 9 8 7 0 1 1 1 10 12 14 13 00 2 0 3 4 1 8 0 6 5 13 14 0 7 1 1 9 10 12 00 6 5 12 9 0 2 0 8 1 11 13 14 0 7 3 4 10 00 8 4 3 0 7 1 119 10 0 2 0 12 5 6 0 13 14 00 5 12 10 9 4 0 7 3 14 13 0 2 0 8 1 1 1 6 00 6 5 0 3 4 9 10 12 0 2 8 0 14 13 11 1 7 0

0 6 5 13 14 0 3 4 1 1 10 12 0 2 0 9 1 8 7 00 7 8 1 9 0 14 13 5 6 0 12 10 1 1 4 3 0 2 00 7 8 0 14 13 5 6 0 3 4 9 10 12 0 2 11 100 10 911 8 7 0 6 5 0 4 3 14 13 0 2 1 12 0

Cost

109.6792.51106.3796.84111.52106.3092.4892.48102.11107.74

In genetic algorithms, for example, the updating process relies onrandomized selection rules which select individuals according to their relativefitness value. In scatter search the updating process relies on the use ofmemory and is confined to maintain a good balance between intensificationand diversification of the solution process. In advanced forms of scattersearch reference solutions are selected based on the use of memory whichoperates by reference to different dimensions as defined in tabu search.Depending on the context and the search strategy, different types of memorycan be used. The way they are integrated to achieve both intensification anddiversification is generically called adaptive memory programming. (SeeGlover and Laguna 1997 for a detailed explanation of various forms and usesof memory within search processes.) For the purpose of this tutorial we use asimple rule to update the set of reference solutions, where intensification isachieved by the selection of high-quality solutions (in terms of the objectivefunction value) and diversification is basically induced by including diversesolutions from the current candidate set CS. Thus the reference set RS can bedefined by two distinct subsets B and D, representing respectively the subsetsof high-quality and diverse solutions, hence RS = B u D . Also, in the contextof our example the terms "highest evaluated solution" and "best solution" areinterchangeable and refer to the solution that best fits the evaluation criterionunder consideration.

Consider the candidate set CS = {S1,...,S1O} defined by the improvedsolutions, and a reference set of size \RS\ = 6 where |#| = \D\ = 3.1.. Before

} . The cardinality of B and D does not need to be identical and can vary duringthe search. For instance, relatively higher values of B (D) can beappropriate during a phase that is more strongly characterized by anintensification (diversification) emphasis. Also different schemes can bechosen to implement these variations. A dynamic variation of these valuescan be implemented by a perturbation scheme conducted strategicallyrather than randomly. For example, a strategic oscillation can be

Scatter Search for Graph-Based Permutation Problems 13

proceeding to the creation of the reference set we first need to eliminatepossible repeated solutions in the current set of trial solutions of Table 1.4.We can see that S.7. and S.8. represent the same solution, so we drop S.8. from thesolution candidate set, making CS = CS\ {s8}. Now, to create RS we start byselecting the three best solutions S!7, S.2. and S4. in CS to generate B, then theset D of diverse solutions is generated by successively selecting the solutionwhich mostly differs from the ones currently belonging to RS. As a diversitymeasure we define di}f = (S{\JSJ)\(S{-.r\Sjj^ as the distance betweensolutions S.j. and £,., which gives the number of edges by which the twosolutions differ from each other. For example, solution S.3. contains 6 edges(1,8), (3,7), (4,10), (9,0), (9,12), (11,13), which are not in solution S4, andsolution S4 has 7 edges (1,7), (3,0), (4,8), (9,10), (9,11), (12,0), (13,0), whichare not in the solution S3.. Hence the distance between the two solutions is 13.Table 1.5 shows devalues for each pair of solutions Sf e RS and Sj e CS.

Candidate solutions are included in RS according to the Maxmin criterionwhich maximizes the minimum distance of each candidate solution to all thesolutions currently in the reference set. The method starts with RS - B and ateach step RS is extended with a solution S • e CS, to be RS = RSv [Sj\, andconsequently CS is reduced to CS = CS\ {Sj j . Then the distance of solutionSj to every solution currently in the reference set is computed to makepossible the selection of a new candidate solution according to the Maxmincriterion. More formally, the selection of a candidate solution is given by

min \dij:j = l>...\CS\).—l \RS\ "

Table 1.5. Distances between solutions

Reference set (AS)S.7.S.2.s+Minimum distanceS.5.Minimum distanceS,Minimum distance

Candidate solutions (CS)S,.162019161816

S.3.

1616131316132213

s,22181717

s«1612111116111811

S.9.1210131018101610

S-io2018151518151815

The process is repeated until \RS\ is achieved. In the table, boldfaced figuresrepresent the maxmin values obtained at each step of the method.

implemented by using critical event memory as an indicator of the order ofmagnitude of the relative variations. Note that since the cardinality ofsubsets B and D are complementary in relation to the RS size and thedecision can be uniquely based on the variation of either one.

14

This results in an initial reference set formed by solutions S.7., S.2., S.4., S.5.,S.j., S.1O.. For convenience, we reorder the solution indexes by settingRS = {Sl9S29S39S49S59S6} as illustrated in Table 1.6.

Table 1.6. Initial reference set.Solution

SIS2S3S4S5S6

Routes0 6 5 13 14 0 3 4 1 1 10 12 0 2 0 9 1 8 7 00 2 0 3 4 1 8 0 6 5 13 14 0 7 1 1 9 10 12 00 8 4 3 0 7 1 119 10 0 2 0 12 5 6 0 13 14 00 5 12 10 9 4 0 7 3 14 13 0 2 0 8 1 1 1 6 00 2 0 3 4 5 0 6 9 8 7 0 1 1 1 10 12 14 13 00 10 9 1 1 8 7 0 6 5 0 4 3 14 13 0 2 1 1 2 0

Cost

92.4892.5196.84

111.52109.67107.74

It is important to note that a balance between intensification anddiversification is achieved by an evaluation criterion that causes the highest-evaluated solutions considered for the reference to include qualities other thana "good" (small) objective function value. Solution S.g. with a cost of 102.11 isby-passed in favor of other solutions that will add more diversity to the set.As indicated in Table 1.5, the distance between S.9 to solutions in thereference set are relatively lower, making this solution unattractive from thestandpoint of diversification.

• Scatter Search PhaseSubset Generation MethodThis method consists of generating subsets of reference solutions to

create structured combinations in the next step. The method is typicallydesigned to organize subsets of solutions to cover different promising regionsof the solution space. In a spatial representation, the convex-hull of eachsubset delimits the solution space in subregions containing all possibleconvex combinations of solutions in the subset. In order to achieve a suitableintensification and diversification of the solution space, three types of subsetsare required to be organized:1. subsets containing only solutions in B,2. subsets with only solutions in D, and3. subsets mixing in solutions in B and D in different proportions.

Subsets defined by solutions of type 1 are conceived to intensify thesearch in regions of high-quality solutions while subsets of type 2 are createdto diversify the search to unexplored regions. Finally, subsets of type 3integrate both high-quality and diverse solutions with the aim of exploitingsolutions across these two types of subregions.

Again, adaptive memory can be used to appropriately define combinedrules for clustering elements in the various types of subsets. This has the

Scatter Search for Graph-Based Permutation Problems 15

advantage of incorporating additional information about the search space andproblem context.

Since the use of sophisticated memory features is beyond the scope of thisstudy, we consider a simpler yet systematic procedure to generate thefollowing types of subsets:1. All 2-element subsets.2. 3-element subsets derived from two element subsets by augmenting each

2-element subset to include the best solution (as measured by the objectivefunction value) not in this subset.

3. 4-element subsets derived from the 3-elelement subsets by augmentingeach 3-element subset to include the best solution (as measured by theobjective function value) not in this subset.

4. The subsets consisting of the best b elements (as measured by theobjective function value), for b = 5,...,\BTable 1.7 shows the subset generated using the current reference set.

Table 1.7. Subset generationType1

2

3

4

Subsets(SI, S2), (SI, S3), (SI, S4), (SI, S5), (SI, S6), (S2, S3), (S2, S4), (S2, S5), (S2, S6),(S3, S4), (S3, S5), (S3, S6), (S4, S5), (S4, S6), (S5, S6)(SI, S2, S3), (SI, S2, S4), (SI, S2, S5), (SI, S2, S6), (SI, S3, S4), (SI, S3, S5), (SI,S3, S6), (SI, S4, S5), (SI, S4, S6), (SI, S5, S6)(SI, S2, S3, S4), (SI, S2, S3, S5), (SI, S2, S3, S6), (SI, S2, S4, S5), (SI, S2, S4,S6), (SI, S2, S5, S6);(SI, S2, S3, S5, S6), (SI, S2, S3, S4, S5, S6)

Subsets of type 1 are the fifteen possible combinations of two solutions.For type 2 subsets, we start by adding solution S.3. to the subset (S.j, S.2). Thensince adding solution S.2. to the subset (S.2, S.3) leads to a repeated subset, theprocedure jumps to the next subset, adding S.2. to the subset (S.h S.4), and soforth. Type 3 subsets are created in a similar way, now starting by addingsolution S+ to the subset (S.i, S.2, S.3). Finally, subsets of type 3 consist ofsuccessively adding solutions S.5. and S4 to subsets (S.i,S.2,S.3?S.6) and

Solution Combination MethodThis method is designed to explore subregions within the convex-hull of

the reference set. We consider solutions encoded as vectors of variablesXyrepresenting edges (vuVj) . New solutions are generated by weightedlinear combinations which are structured by the subsets defined in the laststep. In order to restrict the number of solutions only one solution isgenerated in each subset by a convex linear combination defined as follows.

16

Let E be a subset defined in RS, \E\ = r, and let H(E) denote the convex-hullof E. We generate solutions S E H(E) represented as

t=i

t=i

At>0 j = l,...,r

where the multiplier At represents the weight assigned to solution £.,.. Wecompute these multipliers by

so that the better (lower cost) solutions receive higher weight than lessattractive (higher cost) solutions. Then, we calculate the score of eachvariable Xy relative to the solutions in E by computing

t=i

where x\j means that x^ is an edge in the solution St. Finally as variables

are required to be binary, the value is obtained by rounding its score to givex{j =\score(Xy) +.5\. The computation of the value for each variable in E

results in the linear combination of the solutions in E. Table 1.8 shows thecomputation of the linear combination of solutions Si, S2, S3, and S4 in theinitial reference set illustrated in Table 1.7. For the sake of simplicity onlyedges that appear in at least one solution are represented in the table, since theremaining variables correspond to xtj = 0 .

Scatter Search for Graph-Based Permutation Problems 17

Table LSolution

A,Edges

0,4)(1,7)

(1,8)

(1,9)

(1,11)

(2,0)

(3,0)

(3,4)

(3,7)

(3,14)

(4,0)

(4,8)

(4,9)

(4,11)

(5,0)

(5,6)

(5,12)

(5,13)

(6,0)

(6,11)

(7,0)

(7,8)

(7,11)

(8,0)

(9,0)

(9,10)

(9,11)

(10,0)

(10,11)

(10,12)

(12,0)

(13,0)

(13,14)

(14,0)

8. Linear combinations of solution vectors

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

0.2643

S.2.0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

0.2642

s*.0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

0.2524

S4.0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

0.2191

score(Xij)

0.2642

0.2524

0.7476

0.2643

0.4715

1.0000

0.7809

0.7809

0.2191

0.2191

0.2191

0.2524

0.2191

0.2643

0.2191

0.7809

0.4715

0.5285

1.0000

0.2191

1.0000

0.2643

0.2642

0.7357

0.2643

0.7357

0.5166

0.2524

0.2643

1.0000

0.5285

0.4715

1.0000

0.5285

*ij

0010011100000001011010010110011011

It is important to observe the effect of this weighting on the resultingsolutions. As previously intimated, the least cost solution S.j. has the largestweight in the combination. Also, note that both edges (5,12) and (5,13)appear in two solutions, though only the variable x513 is considered for theresulting solution.