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Ant Colony Optimization Algorithms to solve Nonlinear Network Flow PDF

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Ant Colony Optimization Algorithms to solve Nonlinear Network Flow Problems por Marta Sofia Rodrigues Monteiro TesedeDoutoramentoem Ciências Empresariais Orientadapor: Prof. DoutoraDalilaB. M.M. Fontes Prof. DoutorFernando A. C. C. Fontes 2012 Biography Note Marta Monteiro was born and raised in Porto. In 1998 she finishes her graduation in AppliedMathematicstoTechnologyintheFacultyofSciencesoftheUniversityofPorto. After a brief incursion in a business job, in 2001 she decides to apply to a research grantin ImageProcessingat theFacultyofEngineeringoftheUniversityofPorto. This experience was so rewarding that in 2002 she decides to improve her scientific knowledgeby attending a two years Masters Degreein DataAnalysis and Decision Sup- portSystemsat theFacultyofEconomicsoftheUniversityofPorto. MeanwhileshealsoteachesduringtwoyearsattheUniversityofMinho,attheMath- ematicsforScience and TechnologyDepartment. Having finished her Master’s Degree she applies and is accepted to a Doctoral Pro- grammeinBusinessandManagementStudiesattheFacultyofEconomicsoftheUniver- sityofPorto. She has been a full time data analyst at “Confidencial Imobiliário” for the last four years and, atthesametime,ateacherat thePortugueseSchool ofBank Management. i Acknowledgements When we think about all the years that have passed by while we were walking this long road, we come to the conclusion that almost everyone that surrounded us was a helping vehicle for this long journey. Therefore, there is no “first of all” and “second of all”, because all people contributed in the best way they knew and could. Nonetheless, some mustbementionedfortheirutmosteffort. Anenormous“thankyou”isduetomysupervisors,Prof. DalilaFontesandProf. Fer- nando Fontes, but with a special affection for Prof. Dalila Fontes. Without her guidance andwordofadviceIcouldnothavemadeit. SomanytimesIwasdownandtiredbutshe alwaysmanagedtobethereforme,inbothprofessionalandprivatemoments. Morethan asupervisorIconsiderher asa dearfriend. On a more personal level, my thanks go to my husband Miguel and to my parents Blandina and António for their sacrifice. To Miguel I want to thank his patience for a wife not always one hundred percent present. His words of advice, from one who has already travelled this road, and for taking care of our daughter providing me with the necessary timeto work. My motherand father were exceptionalbecause lately they took care ofall my day-life issues so that I could accelerate this process. Rute, my mother-in- law is also to be thanked for also taking care of Alice, which I hopefully believewas not suchabig “sacrifice” forher. Finally, an infinite thank you to my lovely daughter Alice that was born during this journey. She was the one who made the utmost sacrifice because during the months preceding thefinish ofthiswork, shehad tostayaway from hometolet hermotherwork duringweekends. I dohopeshewillforgivemeand beproud ofhermotheroneday. ii Thisthesisisdedicated to Migueland Alice iii Resumo Este trabalho aborda dois problemas de optimização em redes de fluxos: a minimização de custos em redes com uma só fonte e sem restrições na capacidade dos arcos (SSU MCNFP) e a minimização de custos em redes com restrições no número de arcos em cada caminho definido entre a fonte e os nós procura (HMFST). De salientar que para ambososproblemassão consideradasfunções decustonão lineares côncavas. Os modelos matemáticos obtidos podem ser solucionados por software específico, como por exemplo o software de optimização CPLEX, mas apenas para algumas das funções de custo não lineares consideradas. Para além disso, dado que a dimensão do modelo cresce rapidamente com a dimensão do problema, o tempo computacional tor- na-se proibitivo. Propõem-se então duas heurísticas, HACO1 e HACO2, baseadas em optimização por colónias de formigas hibridizadas com pesquisa local, como forma de solucionarosproblemasapontadosrealçando-seaindaoestudoefectuado àsensibilidade dasheurísticasrelativamenteàvariação dosseusparâmetros. As heurísticas desenvolvidas são avaliadas num conjunto de problemas de referência queestãodisponíveisnainternet. Osresultadoscomputacionaisobtidospermitemmostrar aeficiência das heurísticaspropostastantopara problemasdetamanhopequeno comode tamanhogrande. Em particular, a heurística HACO1 conseguiu melhorar resultados obtidos previa- mentecomumaheurísticaevolutivatantonoquedizrespeitoàqualidadedasoluçãocomo aos tempos computacionais, estes últimos à custa da análise de um número reduzido de soluções comparativamente com outros métodos. A heurística HACO2 conseguiu obter a solução óptima em mais de 75% das vezes em que cada problema foi resolvido, tendo também conseguido melhorar os resultados reportados na literatura e obtidos com uma heurísticabaseadaempopulaçõesdesoluções. Ostemposcomputacionaistambémforam reduzidos tanto em comparação com os tempos obtidos pelo CPLEX como com resulta- dosreportadosnaliteratura. iv Abstract In this work, we address two nonlinear network flow problems: the Single Source Unca- pacitatedMinimumCostNetworkFlowProblem(SSUMCNFP)andtheHop-Constrained MinimumcostFlowSpanningTree(HMFST)problem,bothwithconcavecostfunctions. We propose two hybrid heuristics, HACO1 and HACO2, which are based on Ant Colony Optimization (ACO) and on Local Search (LS), to, respectively, solve SSU MC- NFPs and HMFST problems. Heuristics are proposed due to their combinatorial nature and also to the fact that the total costs are nonlinear concave with a fixed-charge compo- nent. In order to test the performance of our algorithms we solvea set of benchmark prob- lemsavailableonlineandcomparetheresultsobtainedwithotherpopulationbasedheuris- tic methods recently published (HGA and MPGA) and also with a commercial available softwareCPLEX. HACO1 and HACO2 have proven to be very efficient while solving both small and large size problem instances. HACO1 was able to improve further upon some results of HGA, both in terms of computational time and solution quality, proving that ACO based algorithms are a very good alternative approach to solve these problems, with the great advantage of reducing the computational effort by analysing a smaller number of solutions. TheresultsobtainedwithHACO2werecomparedwiththeonesobtainedandreported inliteratureforMPGA.Ouralgorithmprovedtobeabletofindanoptimuminmorethan 75%oftheruns,foreachprobleminstancesolved,andwasalsoabletoimproveonmany of the results reported in the literature. Furthermore, for every single problem instance usedwewerealwaysabletofind afeasiblesolution,whichwasnotthecasefortheother heuristic. Our algorithm improved upon the computational time needed by CPLEX and was alwayslowerthan thatofMPGA. v Contents Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv ListofAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv ListofModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi SummaryofNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Aimsand Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. NetworkFlowProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Classification ofNetworkFlowProblems . . . . . . . . . . . . . . . . . 11 2.3 Problem Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Concaveand NonlinearProgramming . . . . . . . . . . . . . . . . . . . 19 2.5 SolutionMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 vi 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3. AntColonyOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Ant ColonyPrinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 From Biological to Artificial Ants: Ant System and Ant Colony Opti- mization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.1 Ant System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.2 ImprovementstotheAntSystem . . . . . . . . . . . . . . . . . . 36 3.3.3 Ant ColonyOptimization . . . . . . . . . . . . . . . . . . . . . . 38 3.3.4 TheBuildingBlocksofan Ant Algorithm . . . . . . . . . . . . . 39 3.3.4.1 ConstructingaSolution . . . . . . . . . . . . . . . . . 39 3.3.4.2 HeuristicInformation . . . . . . . . . . . . . . . . . . 42 3.3.4.3 Pheromonesand theLaws ofAttraction. . . . . . . . . 46 3.3.4.4 PheromoneUpdate . . . . . . . . . . . . . . . . . . . 49 3.3.4.5 TransitionRuleandProbabilityFunction . . . . . . . . 52 3.3.4.6 ParameterValues . . . . . . . . . . . . . . . . . . . . 54 3.3.5 MultipleAnt Colonies . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.6 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.7 Books andSurveys . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4. MinimumCostNetworkFlowProblems. . . . . . . . . . . . . . . . . . . . . 66 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 ConcaveMinimumCostNetwork FlowProblems . . . . . . . . . . . . . 67 4.2.1 Characteristics ofConcaveMCNFPs Solutions . . . . . . . . . . 68 4.2.2 ComplexityIssues . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 TheSingleSourceUncapacitatedConcaveMinimumCostNetworkFlow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 vii 4.3.1 Problem DefinitionandFormulation . . . . . . . . . . . . . . . . 73 4.3.2 SolutionCharacteristics . . . . . . . . . . . . . . . . . . . . . . 74 4.3.3 TransformingMCNFPs intoSSU ConcaveMCNFPs . . . . . . . 75 4.4 SolutionMethodsforMCNFPs . . . . . . . . . . . . . . . . . . . . . . . 77 4.4.1 Exact SolutionMethods . . . . . . . . . . . . . . . . . . . . . . 78 4.4.2 Heuristicand ApproximateSolutionMethods . . . . . . . . . . . 82 4.4.3 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.5 An Ant Colony Optimization Algorithm Outline for the SSU Concave MCNFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5.1 Representation oftheSolution . . . . . . . . . . . . . . . . . . . 90 4.5.2 ConstructionoftheSolution . . . . . . . . . . . . . . . . . . . . 91 4.5.3 PheromoneUpdate . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5.4 PheromoneBounds . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5.6 Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.6 ComputationalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.6.1 Parameters Setting . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.6.1.1 SettingtheHeuristicInformation . . . . . . . . . . . . 104 4.6.1.2 Settingα and β ParameterValues . . . . . . . . . . . . 104 4.6.1.3 SettingthePheromoneEvaporationRateValue . . . . . 107 4.6.1.4 OtherParameters . . . . . . . . . . . . . . . . . . . . 109 4.6.1.5 FinalParameters . . . . . . . . . . . . . . . . . . . . . 109 4.6.2 ComparingOurResults withtheOnes inLiterature . . . . . . . . 110 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5. Hop-ConstrainedMinimumCostFlowSpanningTrees . . . . . . . . . . . . . 117 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 MinimumSpanningTree: Problemand Extensions . . . . . . . . . . . . 118 viii 5.2.1 DiameterConstrainedMinimumSpanningTreeProblem . . . . . 119 5.2.2 Capacitated MinimalSpanningTreeProblem . . . . . . . . . . . 122 5.2.3 TheHop-ConstrainedMinimumSpanningTreeProblem . . . . . 124 5.2.4 FormulationsfortheHMSTProblem . . . . . . . . . . . . . . . 125 5.2.5 TheHop-ConstrainedMinimumcostSpanningTreeProblemwith Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2.5.1 Definingand FormulatingtheHMFST Problem . . . . 129 5.2.5.2 CostFunctions . . . . . . . . . . . . . . . . . . . . . . 131 5.2.5.3 TestProblems . . . . . . . . . . . . . . . . . . . . . . 133 5.3 SolutionMethodsfortheHFMST and SomeRelated Problems . . . . . . 133 5.4 SolvingHop-ConstrainedMinimumCost FlowSpanningTree Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.4.1 Representation oftheSolution . . . . . . . . . . . . . . . . . . . 149 5.4.2 SolutionConstruction . . . . . . . . . . . . . . . . . . . . . . . 149 5.4.3 Updatingand BoundingPheromoneValues . . . . . . . . . . . . 150 5.4.4 TheAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.4.5 Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.4.6 Sets ofTested Parameters . . . . . . . . . . . . . . . . . . . . . 156 5.5 ComputationalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6. ConclusionsandFutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 ix

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This experience was so rewarding that in 2002 she decides to improve her scientific knowledge by attending a two years Masters Degree in .. 4.5 An Ant Colony Optimization Algorithm Outline for the SSU Concave. MCNFP . its route at the same distribution center. The authors solve the problem in
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