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3.2.1 Chinese Postman Problem ...30 GAPS: a hybridised framework applied to vehicle routing problems A thesis submitted in partial fulfilment of the requirement for the degree of Doctor of Philosophy Matthew J. W. Morgan December 2008 Cardiff University School of Computer Science UMI Number: U514605 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U514605 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 iii Declaration This work has not previously been accepted in substance for any degree and is not being concurrently submitted in candidature for any degree. Signed ... jna .. c-....................... (candidate) Date ___________ .*\ ...................... Statement 1 This thesis is being submitted in partial fulfillment of the requirements for the degree of PhD. Signed .. (candidate) Date .. .^.'J.l.°.l.°3........................... Statement 2 This thesis is the result of my own independent work/investigation, except where other­ wise stated. Other sources are acknowledged by explicit references. Signed ... (candidate) Date ... 3 J J.v.?. Jp.^..................... Statement 3 I hereby give consent for my thesis, if accepted, to be available for photocopying and for inter-library loan, and for the title and summary to be made available to outside organisa­ tions. Signed ... /V'.erj. ?r> (candidate) Date jp./.q !»....................... iv V Abstract In this thesis we consider two combinatorial optimization problems, the Capacitated Ve­ hicle Routing Problem (CVRP) and the Capacitated Arc Routing Problem (CARP). In the CVRP, the objective is to find a set of routes for a homogenous fleet of vehicles, which must service a set of customers from a central depot. In contrast, the CARP requires a set of routes for a fleet of vehicles to service a set of customers at the street level of an intercity network. After a comprehensive discussion of the existing exact and heuristic algorithmic tech­ niques presented in the literature for these problems, computational experiments to pro­ vide a benchmark comparison of a subset of algorithmic implementations for these meth­ ods are presented for both the CVRP and CARP, run against a series of dataset instances from the literature. All dataset instances are re-catalogued using a standard format to overcome the difficulties of the different naming schemes and duplication of instances that exist between different sources. We then present a framework, which we shall call Genetic Algorithm with Perturbation Scheme (GAPS), to solve a number of combinatorial optimization problems. The idea is to use a genetic algorithm as a container framework in conjunction with a perturbation or weight coding scheme. These schemes make alterations to the underlying input data within a problem instance, after which the changed data is fed into a standard problem specific heuristic and the solution obtained decoded to give a ‘true’ solution cost using the original unaltered instance data. We first present GAPS in a generic context, using the Traveling Salesman Problem (TSP) as an example and then provide details of the specific application of GAPS to both the CVRP and CARP. Computational experiments on a large set of problem instances from the literature are presented and comparisons with the results achieved by the current state of the art algorithmic approaches for both problems are given, highlighting the robustness and effectiveness of the GAPS framework. vi Abstract Acknowledgements Throughout my years of study at Cardiff University, I have been supported and inspired, through interaction with many friends and colleagues within the School of Computer Sci­ ence. I would like to extend my thanks to these individuals, but I particularly wish to acknowledge a number of people who have had a profound impact on my work. First of all I would like to extend my thanks to the School of Computer Science at Cardiff University for their general and financial support. Above all, I extend my heartfelt gratitude to my supervisor Christine Mumford, who has consistently provided me with support, encouragement and friendship during my time at Cardiff University. Thank you for all of your time, patience, interesting discussions and being the sounding board for my ideas, both good and bad!. Finally, I want to thank my family, for all of the unconditional help and support that they have provided me. They have been a continuous inspiration, without whom, the completion of this work would not have been possible. viii Acknowledgements Contents Abstract v Acknowledgements vii Contents ix List of Figures xv List of Tables xix List of Algorithms xxi Notation xxiii Acronyms xxv 1 Introduction 1 1.1 Motivation ................................................................................................... 1 1.2 Real World Vehicle Routing ..................................................................... 2 1.3 Problem Complexity and Solution M ethods ............................................. 3 1.4 Thesis Contribution ..................................................................................... 4 1.5 Algorithmic Implementations ..................................................................... 5 1.6 A Note on Computational Experimentation ............................................. 5 1.7 Thesis Overview ....................................................................................... 6 X Contents 2 Background and Terminology 9 2.1 Introduction .................................................................................................. 9 2.2 Combinatorial O ptim ization ...................................................................... 9 2.3 Graph T heory ............................................................................................... 10 2.4 Computational Complexity ......................................................................... 15 2.5 Algorithmic Classes ..................................................................................... 18 2.6 Routing Problems ......................................................................................... 20 2.7 Chapter Summary ......................................................................................... 21 3 Routing Problem Formulations 23 3.1 Node Routing Problems ................................................................................ 23 3.1.1 Symmetric Traveling Salesman Problem ......................................... 23 3.1.2 Multiple Traveling Salesman Problem ............................................ 25 3.1.3 Capacitated Vehicle Routing P roblem ............................................ 26 3.1.4 Vehicle Routing Problem with B ackhauls ...................................... 28 3.1.5 Vehicle Routing Problem with Time Windows ................................ 28 3.2 Arc Routing Problems ................................................................................ 29 3.2.1 Chinese Postman P ro b lem ............................................................... 30 3.2.2 Directed Chinese Postman P ro b lem ............................................... 32 3.2.3 Mixed Chinese Postman P ro b le m .................................................. 34 3.2.4 Windy Postman Problem .................................................................. 35 3.2.5 Rural Postman Problem .................................................................. 36 3.2.6 Directed Rural Postman Problem ...................................................... 37 3.2.7 Mixed Rural Postman Problem ......................................................... 38 3.2.8 Capacitated Arc Routing P roblem ................................................... 39 3.3 Chapter Summary ......................................................................................... 39 Contents _______________________________________________________________ xi 4 Solving The Capacitated Vehicle Routing Problem 41 4.1 Introduction ................................................................................................. 41 4.2 H euristics.................................................................................................... 41 4.2.1 Single Phase Route-Construction Heuristic Algorithms ................ 41 4.2.2 Two Phase Constructive Heuristic Algorithms ............................... 47 4.2.3 Improvement Heuristic Algorithms .............................................. 56 4.3 Dataset Instances ........................................................................................ 60 4.4 Comparing the CW, Sweep and Petal Heuristics ................... 66 4.4.1 Computational R esults ................................................................... 66 4.5 Chapter Summary ......................................................................................... 72 5 Solving The Capacitated Arc Routing Problem 73 5.1 Introduction .................................................................................................. 73 5.2 Arc Routing Problems ..............................................................................
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