Ant Colony Optimization Algorithms to solve Nonlinear Network Flow

211 Pages·2012·1.83 MB·English

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.
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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
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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.
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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.
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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
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