Lopez Ibanez MOACO

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Lopez Ibanez MOACO

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Multi-ObjectiveAnt Colony OptimizationTECHNI SCHEUNI VERSI TTDARMSTADTniversidad deranadaUGManuel Lpez-Ibez InfanteOctober 2003 July 2004Supervisors:Luis PaqueteThomas SttzleLicenseMulti-objectiveAntColonyOptimization.DiplomaThesisbyManuelLopez-Iba nezCopyright c _ 2004 Manuel Lopez-Iba nez. All rights reserved.This work is licensed under the Creative Commons AttributionLicense.To view a copy of this license, visit http://creativecommons.org/licenses/by/2.0/orsendalettertoCreativeCommons, 559NathanAbbottWay,Stanford, California 94305, USA.Youarefree:to copy, distribute, display, and perform the workto make derivative worksto make commercial use of the workUnderthefollowingconditions:Attribution. You must give the original author credit.For any reuse or distribution, you must make clear to others the licenseterms of this work.Any of these conditions can be waived if you get permission from thecopyright holder.Your fair use and other rights are in no way aected by the above.This is a human-readable summary of the Legal Code(http://creativecommons.org/licenses/by/2.0/legalcode).PrefaceRealworldproblemsusuallyinvolveseveralandoftenconictiveobjectivesthat must be simultaneously optimized in order to achieve a satisfactory solu-tion. Multi-objective optimization has its roots at the end of the nineteenthcentury, in the studies of Edgeworth and Pareto, but had not experimentedagreatdevelopmentuntil thefties. Sincelastdecade, thereisanongo-ing research eort to develop approximate algorithms and metaheuristic ap-proaches to solve multi-objective problems. As a result, various metaheuris-tic approaches have been applied to multi-objective optimization,includingsimulatedannealing(SA), taboosearch(TS)andevolutionaryalgorithms.Nevertheless, therearemanyimportantopenquestionsinmulti-objectiveoptimization, e.g., theadequateperformanceassessmentof multi-objectiveoptimization algorithms.Ant Colony Optimization (ACO) is a metaheuristic approach applied suc-cessfully to single objective combinatorial problems, but few work has beendone to apply ACO principles to multi-objective combinatorial optimization(MOCO) and most of the proposed algorithms are only applicable to problemswheretheobjectivescanbeorderedaccordingtotheirimportance. There-fore, itisstill unclearhowACOalgorithmsfortheseproblemsshouldbedesigned.Thisworkexaminestheconceptsinvolvedonthedesignof ACOalgo-rithmstotackleMOCOproblems, e.g., denitionof multi-objectivepher-omone information, multiple cooperative ant colonies, pheromone updatestrategiesinthemulti-objectivecontextandlocalsearchmethodsformul-tipleobjectives. Mostof theseconceptshavebeenpreviouslyusedintheliterature. Nevertheless, weexaminethemindependentlyofanyparticularproblem and we propose alternative concepts still not considered. Moreover,we study these concepts in relation to the available knowledge about the bestperforming ACO algorithms for single objective optimization and the searchstrategies currently used in multi-objective optimization. As well, these con-cepts are studied as components of a general multi-objective ACO (MO-ACO)algorithm.Next, weapplytheseconceptstodesignanACOalgorithmforthebi-objective case of a recently proposed multi-objective variant of the quadraticiv Prefaceassignment problem (QAP). Therefore, we consider various congurations ofthe MO-ACO algorithm and compare their performance when applied to thebi-objective QAP. These congurations use the //A-/1^Ant System asthe underlying algorithm because it is considered the best performing ACOalgorithmforthesingleobjectiveQAP. Inaddition, theparticularcompo-nents of each conguration are chosen in such a way that the congurationreects a particular search strategy. This search strategy is either based ondominance criteria or based on scalarizations of the objective vector. To ad-dress the performance assessment of these experiments, we use a methodologythat combines up-to-date ideas from the research area of multi-objective op-timization with well-known statistical techniques from experimental design.The experimental results show that the use of local search with MO-ACOisessential forthebi-objectiveQAP. Moreover, otherparametersbecomeless signicant when local search is applied. The underlying search strategyfollowedbyeachcongurationplaysanimportantroleintheshapeofthePareto set obtained by the algorithm.The organization of concepts of MO-ACO into modular components whichcanbecombinedintovariouscongurationsallowsapplyingMO-ACOtootherMOCOproblems. TheavailableknowledgeofsingleobjectiveACO,multi-objective optimization and about particular problems may be used todesignnewcomponents, e.g., dierentlocal searchmethods, whichcanbecombined into congurations of MO-ACO in order to tackle dierent MOCOproblems.vThis document is structured as follows:Chapter1introduces the basic concepts of multi-objective optimization interms of Pareto optimality.Chapter2reviewsthesingleobjectiveQAP,describingtheproblemde-nition and briey the current knowledge about types of QAP instancesand related measures. Afterward, this chapter introduces the proposalson multi-objective QAP, and nally it describes the bi-objective QAPinstances used as test instances.Chapter3gives a brief introduction to Ant Colony Optimization, the ap-plicationofACOtothesingleobjectiveQAPandthe //A-/1^Ant System.Chapter4examines the design of multi-objective ACO algorithms. Partic-ularly, we discuss the use of multiple colonies, several pheromone updatestrategies and dierent search strategies. Finally, several congurationsare considered in order to tackle the bQAP.Chapter5describestheassessmentmethodologyfollowedtoevaluatethecongurations considered to tackle the bQAP.Chapter6analyzes the results of the experiments performed with respectto the assessment methodology.Chapter7concludes and indicates future research directions.ContentsPreface iii1 Multi-objectiveOptimization 11.1 Multi-objective Optimization Problems . . . . . . . . . . . . . 11.2 Pareto Optimality . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Relations between Pareto Sets . . . . . . . . . . . . . . . . . . 31.4 Weighted Sum Scalarization. . . . . . . . . . . . . . . . . . . 41.5 Multi-objective Combinatorial Optimization Problems . . . . 42 TheQuadraticAssignmentProblem 72.1 The Single Objective QAP . . . . . . . . . . . . . . . . . . . . 72.1.1 Types of QAP Instances. . . . . . . . . . . . . . . . . 82.1.2 Measures of QAP Instances . . . . . . . . . . . . . . . 92.2 The Multi-objective QAP (mQAP). . . . . . . . . . . . . . . 92.2.1 Bi-objective QAP (bQAP) Instances . . . . . . . . . . 103 AntColonyOptimization 133.1 ACO Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Ant System Applied to the QAP . . . . . . . . . . . . . . . . 153.3 Improvements on AS. . . . . . . . . . . . . . . . . . . . . . . 163.4 //A-/1^Ant System. . . . . . . . . . . . . . . . . . . . 174 Multi-objectiveACO 194.1 ACO Algorithms for MOCO Problems . . . . . . . . . . . . . 194.2 Multi-objective Pheromone Information . . . . . . . . . . . . 214.2.1 Multiple Pheromone Information . . . . . . . . . . . . 224.2.2 Single Pheromone Information . . . . . . . . . . . . . 254.2.3 Computational Eciency . . . . . . . . . . . . . . . . 264.3 Pheromone Update Strategies . . . . . . . . . . . . . . . . . . 274.4 Multiple Colonies. . . . . . . . . . . . . . . . . . . . . . . . . 294.4.1 Weight Vectors in the Multi-colony Approach. . . . . 304.4.2 Candidate Set in the Multi-colony Approach . . . . . 314.4.3 Pheromone Update Strategies with Multiple Colonies 314.5 Local Search Methods for MO-ACO . . . . . . . . . . . . . . 33viii Contents4.5.1 Local Search for Single Objective Problems . . . . . . 344.5.2 Pareto Local Search (PLS) . . . . . . . . . . . . . . . . 344.5.3 Bounded Pareto Local Search (BPLSA) . . . . . . . . 344.6 MO-ACO Applied to the bQAP . . . . . . . . . . . . . . . . . 355 PerformanceAssessment 395.1 Binary-measure . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Unary-measure . . . . . . . . . . . . . . . . . . . . . . . . . 415.2.1 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . 415.2.2 Analysis of Variance (ANOVA) . . . . . . . . . . . . . 425.3 Median Attainment Surface . . . . . . . . . . . . . . . . . . . 435.4 Reference Solutions . . . . . . . . . . . . . . . . . . . . . . . . 446 Experiments 456.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . 456.2 Analysis of Experimental Results . . . . . . . . . . . . . . . . 466.2.1 Analysis Based on Binary-measure . . . . . . . . . . 466.2.2 Analysis Using Unary-measure and ANOVA . . . . . 526.3 Median Attainment Surfaces . . . . . . . . . . . . . . . . . . 577 Conclusions 637.1 Multi-objective ACO. . . . . . . . . . . . . . . . . . . . . . . 637.2 MO-ACO Applied to the bQAP . . . . . . . . . . . . . . . . . 647.3 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 65AANOVAAssumptions 67Bibliography 73ListofFigures3.1 Algorithmic schema of ACO algorithms . . . . . . . . . . . . 155.1 Pictorial view of the lower bound for the bQAP. . . . . . . . 426.1 Lower bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Interactions: unstructured ( = 0.75) instance, using LS . . . 556.3 Main factors: unstructured ( = 0.00) instance, using LS . . . 566.4 Interactions: unstructured ( = 0.00) instance, using LS . . . 566.5 Interactions: unstructured ( = 0.75) instance, using LS . . 576.6 Median attainment surfaces: unstructured instances . . . . . 606.7 Median attainment surfaces: structured instances. . . . . . . 61A.1 ANOVA assumptions: unstructured instances, using LS . . . 69A.2 ANOVA assumptions: unstructured instances, not using LS . 70A.3 ANOVA assumptions: structured instances, using LS. . . . . 71ListofTables2.1 bQAP unstructured instances . . . . . . . . . . . . . . . . . . 112.2 bQAP structured instances . . . . . . . . . . . . . . . . . . . 114.1 Search strategies when applying MO-ACO to the bQAP. . . 384.2 Congurations of