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Ant Colony Optimization Algorithms to Solve Nonlinear Network Flow Problems

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Ant Colony Optimization Algorithms to Solve Nonlinear Network Flow Problems Ant Colony Optimization Algorithms to solve Nonlinear Network Flow Problems por Marta Sofia Rodrigues Monteiro Tese de Doutoramento em Ciências Empresariais Orientada por: Prof. Doutora Dalila B. M. M. Fontes Prof. Doutor Fernando A. C. C. Fontes 2012 Biography Note Marta Monteiro was born and raised in Porto. In 1998 she finishes her graduation in Applied Mathematics to Technology in the Faculty of Sciences of the University of Porto. After a brief incursion in a business job, in 2001 she decides to apply to a research grant in Image Processing at the Faculty of Engineering of the University of Porto. This experience was so rewarding that in 2002 she decides to improve her scientific knowledge by attending a two years Masters Degree in Data Analysis and Decision Sup- port Systems at the Faculty of Economics of the University of Porto. Meanwhile she also teaches during two years at the University of Minho, at the Math- ematics for Science and Technology Department. Having finished her Master’s Degree she applies and is accepted to a Doctoral Pro- gramme in Business and Management Studies at the Faculty of Economics of the Univer- sity of Porto. She has been a full time data analyst at “Confidencial Imobiliário” for the last four years and, at the same time, a teacher at the Portuguese School of Bank 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 must be mentioned for their utmost effort. An enormous “thank you” is due to my supervisors, Prof. Dalila Fontes and Prof. Fer- nando Fontes, but with a special affection for Prof. Dalila Fontes. Without her guidance and word of advice I could not have made it. So many times I was down and tired but she always managed to be there for me, in both professional and private moments. More than a supervisor I consider her as a dear friend. 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 time to work. My mother and father were exceptional because lately they took care of all 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 believe was not such a big “sacrifice” for her. 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 the finish of this work, she had to stay away from home to let her mother work during weekends. I do hope she will forgive me and be proud of her mother one day. ii This thesis is dedicated to Miguel and 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 ambos os problemas são consideradas funções de custo não 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 solucionar os problemas apontados realçando-se ainda o estudo efectuado à sensibilidade das heurísticas relativamente à variação dos seus parâmetros. As heurísticas desenvolvidas são avaliadas num conjunto de problemas de referência que estão disponíveis na internet. Os resultados computacionais obtidos permitem mostrar a eficiência das heurísticas propostas tanto para problemas de tamanho pequeno como de tamanho grande. Em particular, a heurística HACO1 conseguiu melhorar resultados obtidos previa- mente com uma heurística evolutivatanto no que diz respeito à qualidade da solução como 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 heurística baseada em populações de soluções. Os tempos computacionais também foram reduzidos tanto em comparação com os tempos obtidos pelo CPLEX como com resulta- dos reportados na literatura. iv Abstract In this work, we address two nonlinear network flow problems: the Single Source Unca- pacitated Minimum Cost Network Flow Problem (SSU MCNFP) and the Hop-Constrained Minimum cost Flow Spanning Tree (HMFST) problem, both with concave cost functions. 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 solve a set of benchmark prob- lems available online and compare the results obtained with other population based heuris- tic methods recently published (HGA and MPGA) and also with a commercial available software CPLEX. 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. The results obtained with HACO2 were compared with the ones obtained and reported in literature for MPGA. Our algorithm proved to be able to find an optimum in more than 75% of the runs, for each problem instance solved, and was also able to improve on many of the results reported in the literature. Furthermore, for every single problem instance used we were always able to find a feasible solution, which was not the case for the other heuristic. Our algorithm improved upon the computational time needed by CPLEX and was always lower than that of MPGA. v Contents Contents ....................................... ix List of Figures .................................... xii List of Tables .................................... xiv List of Algorithms .................................. xv List of Models .................................... xvi Summary of Notation ................................ xvi 1. Introduction ................................... 1 1.1 Motivation.................................. 2 1.2 AimsandScope............................... 4 1.3 Contribution................................. 5 1.4 ThesisOutline................................ 6 2. Network Flow Problems ............................. 8 2.1 Terminology................................. 9 2.2 ClassificationofNetworkFlowProblems . ... 11 2.3 ProblemComplexity ............................ 16 2.4 ConcaveandNonlinearProgramming . 19 2.5 SolutionMethods .............................. 20 vi 2.6 Summary .................................. 28 3. Ant Colony Optimization ............................ 29 3.1 Introduction................................. 29 3.2 AntColonyPrinciples............................ 31 3.3 From Biological to Artificial Ants: Ant System and Ant Colony Opti- mization................................... 32 3.3.1 AntSystem............................. 32 3.3.2 ImprovementstotheAntSystem. 36 3.3.3 AntColonyOptimization. 38 3.3.4 TheBuildingBlocksofanAntAlgorithm . 39 3.3.4.1 ConstructingaSolution . 39 3.3.4.2 HeuristicInformation . 42 3.3.4.3 PheromonesandtheLawsofAttraction. 46 3.3.4.4 PheromoneUpdate . 49 3.3.4.5 TransitionRuleandProbabilityFunction . 52 3.3.4.6 ParameterValues . 54 3.3.5 MultipleAntColonies . 55 3.3.6 Hybridization............................ 58 3.3.7 BooksandSurveys ......................... 64 3.4 Summary .................................. 65 4. Minimum Cost Network Flow Problems ..................... 66 4.1 Introduction................................. 66 4.2 ConcaveMinimumCostNetworkFlowProblems . .. 67 4.2.1 CharacteristicsofConcaveMCNFPsSolutions . ... 68 4.2.2 ComplexityIssues . .. .. .. .. .. .. .. 69 4.3 The Single Source Uncapacitated Concave Minimum Cost Network Flow Problem................................... 71 vii 4.3.1 ProblemDefinitionandFormulation . 73 4.3.2 SolutionCharacteristics . 74 4.3.3 Transforming MCNFPs into SSU Concave MCNFPs . 75 4.4 SolutionMethodsforMCNFPs. 77 4.4.1 ExactSolutionMethods . 78 4.4.2 HeuristicandApproximateSolutionMethods . ... 82 4.4.3 TestProblems............................ 88 4.5 An Ant Colony Optimization Algorithm Outline for the SSU Concave MCNFP..................................
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