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Multi-Objective Tools for the Vehicle Routing Problem with Time Windows

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Multi-Objective Tools for the Vehicle Routing Problem with Time Windows Multi-Objective Tools for the Vehicle Routing Problem with Time Windows Juan Castro-Gutierrez Thesis submitted to The University of Nottingham for the degree of Doctor of Philosophy March 2012 Dedicated to my parents Juan Luis and Nieves i Abstract Most real-life problems involve the simultaneous optimisation of two or more, usu- ally conflicting, objectives. Researchers have put a continuous effort into solving these problems in many different areas, such as engineering, finance and computer science. Over time, thanks to the increase in processing power, researchers have created meth- ods which have become increasingly sophisticated. Most of these methods have been based on the notion of Pareto dominance, which assumes, sometimes erroneously, that the objectives have no known ranking of importance. The Vehicle Routing Problem with Time Windows (VRPTW) is a logistics problem which in real-life applications appears to be multi-objective. This problem consists of designing the optimal set of routes to serve a number of customers within certain time slots. Despite this problem’s high applicability to real-life domains (e.g. waste collec- tion, fast-food delivery), most research in this area has been conducted with hand-made datasets. These datasets sometimes have a number of unrealistic features (e.g. the as- sumption that one unit of travel time corresponds to one unit of travel distance) and are therefore not adequate for the assessment of optimisers. Furthermore, very few studies have focused on the multi-objective nature of the VRPTW. That is, very few have studied how the optimisation of one objective affects the others. This thesis proposes a number of novel tools (methods + dataset) to address the above- mentioned challenges: 1) an agent-based framework for cooperative search, 2) a novel multi-objective ranking approach, 3) a new dataset for the VRPTW, 4) a study of the pair-wise relationships between five common objectives in VRPTW, and 5) a simplified Multi-objective Discrete Particle Swarm Optimisation for the VRPTW. ii Acknowledgements The research presented in this thesis was undertaken under the supervision of Dr. Dario Landa-Silva to whom I am deeply grateful for his invaluable input and guid- ance. I am also profoundly thankful to Professor José Andrés Moreno Pérez for his patience, continuous encouragement, support and friendship as co-advisor. I wish to express my sincere gratitude to the members of the Group of Intelligent Com- puting (GCI) at the University of La Laguna (Spain). Thanks to Marcos Moreno for introducing me to field of Heuristics, and to him and Belén Melián to give me the opportunity to work with them as a research assistant. Thanks also to Julio Brito for sharing his knowledge with me. I want to thank all my beloved friends in Tenerife. Thanks to Miguel Ángel (Miki) for being always there for me, for being like a brother. Thanks to Jonatan (Jony) for sharing dreams, happiness, sadness, films, ... Thanks to Santi for his friendship and encouraging conversations. Thanks also to Carlos and Javi for their friendship and the collections of good memories I have from our undergraduate times. Thanks to Inés, Elsa and Eva for their unconditional friendship and support. I want to express my gratitude to all the friends I have made through these Ph.D. years. Thanks to Elkin for being my brother in the UK, for those long and inspiring conversa- tions, for being there in both good and bad moments. Thanks to those special house- mates and friends, such as Tiago and Daniel Muller to whom I am deeply grateful for sharing their life with me. Thanks to Daniel Soria for his sincere friendship and continuous support. Thanks also to Pawel, Claudio, María, Fran, Madda, Ben, Pilar, Javier, Alberto, and a long etcetera of good friends who have shared pieces of their iii lives with me. I would also like to thank ’mijn geliefde en lieve vrouw’ Hilde for her support and love, for her advice and for proof-reading this work, for helping me to look always at the bright side and for cheering me up in those hard moments. This work would have not been possible without the continuous love, support and encouragement of my parents Juan Luis and Nieves, my brothers Javier and Sergio, and my uncle José Castro. iv Contents List of figuresx List of tables xiv 1 Introduction2 1.1 Motivation....................................2 1.2 Thesis Structure.................................4 1.3 Contribution of this thesis...........................5 1.4 Articles resulting from this thesis.......................7 2 Literature Review9 2.1 Introduction...................................9 2.2 Vehicle Routing Problem............................ 10 2.2.1 Vehicle Routing Problem with Time Windows........... 11 2.2.2 Solution Techniques for VRP and VRPTW.............. 12 2.3 Multi-objective Optimisation......................... 25 2.3.1 Pareto Efficiency............................ 25 2.3.2 Multi-objective approaches...................... 26 2.3.3 Methodologies for Multi-Objective Optimisation.......... 27 2.3.4 Quality Metrics for Multi-Objective Optimisation......... 33 v CONTENTS 2.4 Cooperation................................... 35 2.5 Particle Swarm Optimisation......................... 37 2.5.1 Discrete Particle Swarm Optimisation................ 38 2.5.2 Multi-Objective Particle Swarm Optimisation........... 44 3 CODEA - An Agent Based Multi-Objective Optimisation Framework 51 3.1 Introduction................................... 51 3.2 Frameworks for Multi-objective Optimisation................ 52 3.2.1 JMetal.................................. 54 3.2.2 ECJ.................................... 55 3.2.3 HeuristicLab............................... 56 3.2.4 ParadisEO................................ 57 3.3 CODEA...................................... 59 3.3.1 CODEA v2................................ 60 3.3.2 Neighbourhood............................. 61 3.3.3 Core................................... 61 3.3.4 Phases.................................. 64 3.3.5 CODEA v3................................ 65 3.4 Other uses of CODEA............................. 69 3.4.1 Exploring feasible and infeasible regions in the Vehicle Routing Problem with Time Windows using a Multi-Objective Particle Swarm Optimisation Approach.................... 70 3.4.2 Particle Swarm Optimisation for the Steiner Tree in Graph and Delay Constrained Multicast Routing Problems.......... 71 3.4.3 CODEA - An Agent Based Multi-Objective Optimisation Frame- work................................... 73 vi CONTENTS 3.4.4 Developing Asynchronous Cooperative Multi-agent Search... 74 3.5 Conclusions................................... 75 4 Dynamic Lexicographic Approach for Multi-objective Optimisation 77 4.1 Introduction and motivation.......................... 78 4.2 Dynamic Lexicographic Approach...................... 79 4.3 Experiments................................... 82 4.4 Results...................................... 85 4.4.1 Qualitative analysis.......................... 87 4.4.2 Quantitative analysis.......................... 88 4.5 Conclusions................................... 90 5 Overview of Solomon’s dataset 93 5.1 Introduction................................... 93 5.2 Structure of the data files............................ 95 5.3 Geographical distribution of customers................... 96 5.4 Time Windows and Service Times...................... 100 5.5 Demands and Vehicles Capacities....................... 102 5.6 Multi-objective Suitability........................... 103 5.7 Difficulty..................................... 104 5.8 Conclusions................................... 105 6 A Dataset for the Multi-Objective Vehicle Routing Problem with Time Win- dows 106 6.1 Introduction................................... 107 6.2 Structure of data files.............................. 108 6.3 Geographical distribution of customers................... 109 vii CONTENTS 6.4 Characterisation of Time Windows...................... 111 6.5 Characterisation of Demands......................... 113 6.6 Dataset Settings................................. 113 6.7 Multi-objective Suitability........................... 114 6.8 Difficulty..................................... 115 6.9 Conclusions................................... 115 7 Investigating the Multi-objective Suitability of VRPTW datasets 117 7.1 Introduction................................... 118 7.2 Experimental Design.............................. 120 7.2.1 VNS settings............................... 120 7.2.2 NSGA-II settings............................ 121 7.3 Discussion of Results.............................. 123 7.3.1 Correlation Between Objectives throughout the Optimisation Pro- cess................................... 123 7.3.2 Correlation Between Objectives in Pareto Approximation Sets.. 129 7.4 Conclusions................................... 135 8 A simplified MODPSO for VRPTW 138 8.1 Introduction................................... 139 8.2 Discrete Particle Swarm Optimisation (DPSO)............... 142 8.3 Proposed MODPSO Algorithm........................ 143 8.3.1 Solution Representation and Initialisation.............. 144 8.3.2 Constraints and Objectives...................... 145 8.3.3 Operators: Mutation and Crossover................. 145 8.4 Experimental Design.............................. 146 8.5 Discussion of Results.............................. 148 viii CONTENTS 8.5.1 Hypervolume.............................. 149 8.5.2 Coverage................................. 150 8.5.3 Overall Non-Dominated Vector Generation............. 151 8.6 Speed
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