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A Dynamic Programming Operator for Metaheuristics to Solve Vehicle Routing Problems with Optional Visits Leticia Gloria Vargas Suarez

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A dynamic programming operator for metaheuristics to solve vehicle routing problems with optional visits Leticia Gloria Vargas Suarez To cite this version: Leticia Gloria Vargas Suarez. A dynamic programming operator for metaheuristics to solve vehi- cle routing problems with optional visits. Automatic. INSA de Toulouse, 2016. English. NNT : 2016ISAT0027. tel-01355746v2 HAL Id: tel-01355746 https://tel.archives-ouvertes.fr/tel-01355746v2 Submitted on 19 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THTHESEESE`` En vue de l’obtention du DOCTORAT DE L’UNIVERSITEF´ ED´ ERALE´ TOULOUSE MIDI-PYREN´ EES´ D´elivr´e par : l’Institut National des Sciences Appliqu´eesde Toulouse (INSA de Toulouse) Pr´esent´ee et soutenue le 24/06/2016 par : Leticia VARGAS A dynamic programming operator for metaheuristics to solve vehicle routing problems with optional visits JURY Dominique FEILLET Professeur Ecole´ des Mines St-Etienne´ Laetitia JOURDAN Professeur des Universit´es Universit´eLille 1 Nicolas JOZEFOWIEZ Maˆıtrede Conf´erences INSA Toulouse Sandra U. NGUEVEU Maˆıtrede Conf´erences INP Toulouse Caroline PRODHON Maˆıtrede Conf´erences Universit´ede Troyes Christos TARANTILIS Professeur Athens University Ecole´ doctorale et sp´ecialit´e : EDSYS : Informatique 4200018 Unit´e de Recherche : LAAS-CNRS Toulouse Directeur(s) de Th`ese : Nicolas JOZEFOWIEZ et Sandra Ulrich NGUEVEU Rapporteurs : Laetitia JOURDAN et Christos TARANTILIS iii Acknowledgments First and foremost, I thank my thesis advisor M. Nicolas Jozefowiez, Maˆıtrede Conf´erencesINSA-Toulouse, for his relentless support, advice and guidance along these three years. A great thank you also goes to my second thesis supervisor Mlle. Sandra Ulrich Ngueveu, Maˆıtrede Conf´erencesINP-Toulouse, for her continuous encourage- ment. It was a great experience working with both of you. Let me express my sincere gratitude to my jury: Professeur D. Feillet, Professeur de Universit´esL. Jourdan, Maˆıtrede Conf´erencesC. Prodhon, and Professor C. Tarantilis. Their comments, remarks and thought-provoking questions help to improve my work and writings. I am very grateful to our team leader M. Christian Artigues, Directeur de Recherche at LAAS-CNRS, who was always very helpful and supportive aside from being a very charismatic personality always seeking the cohesion of the team and the success of each of its members. I also thank him for always making me feel welcome in a foreign country. I am specially grateful to Mme. Christ`eleMouclier, Assistant ROC team, who helped me out so many times in diverse matters with a very kind and warm demeanour. I would also like to thank all the permanent members of the ROC team as well as all my colleagues for their help in diverse situations, for their patience with my limited French skills, and for making the average work day more fun and interesting. I am specially grateful to Erasmus Mundus Project LAMENITEC which financially supported my studies, fellowship 372362-1-2012-1-ES, and made this endeavor possible. I thank the French government as well for their support. This work was partially supported by the French National Research Agency through the ATHENA project under the grant ANR-13-BS02-0006. I also gratefully acknowledge the economic support received from the Mexican National Council of Science and Technology (CONACyT) as well as from the Instituto de Innovaci´ony Transferencia de Tecnolog´ıade Nuevo Le´on. Last but not least, I would like to thank my friends and family for their support. I especially wish to thank my parents for their inconditional love and support throughout my life. A very special thank you is for my children, Tavo and Tani, and my husband who have always been my greatest advocates and cheerleaders. iv Abstract Metaheuristics are problem independent optimisation techniques. As such, they do not take advantage of any specificity of the problem and, therefore, can provide general frameworks that may be applied to many problem classes. These iterative upper level methodologies can furnish a guiding strategy in designing subordinate heuristics to solve specific optimisation problems. Their use in many applications shows their efficiency and effectiveness to solve large and complex problems. Nowadays, metaheuristics applied to the solution of optimisation problems have shifted towards integrating other optimisation techniques, so that solution methods benefit from the advantages each offers. This thesis seeks to contribute to the study of vehicle routing problems with optional visits by providing a dynamic programming-based operator that works embedded into a generic metaheuristic. The operator retrieves the optimal tour of customers to visit, satisfying the side constraints of the problem, while optimising the defined objective. The operator formulates the problem of selecting the best customers to visit as a Resource Constrained Elementary Shortest Path Problem on an auxiliary directed acyclic graph where the side restrictions of the problem considered act as the constraining resource. In vehicle routing problems with optional visits, the customers to serve are not known a priori and this fact leaves a more difficult to solve problem than a classic routing problem, which per se is already NP-hard. Routing problems with optional visits find application in multiple and diverse areas such as bimodal distribution design, humanitarian logistics, health care delivery, tourism, recruitment, hot rolling production, selected collection or delivery, and urban patrolling among others. Keywords: Metaheuristics, dynamic programming, vehicle routing. R´esum´e Les m´etaheuristiques sont des techniques d’optimisation ind´ependantes des probl`emes trait´es.Elles ne profitent pas d’une sp´ecificit´edu probl`emeet, par cons´equent, peuvent fournir des cadres g´en´erauxqui peuvent ˆetreappliqu´es`ade nombreuses classes de probl`emes.Les m´etaheuristiquespeuvent fournir une strat´egiede guidage dans la conception des heuristiques pour r´esoudredes probl`emesd’optimisation sp´ecifiques. Leur utilisation dans de nombreuses applications montre leur efficacit´epour r´esoudredes probl`emesimportants et complexes. De nos jours, les m´etaheuristiquesappliqu´ees`ala solution des probl`emes d’optimisation ont ´evolu´e vers l’int´egrationd’autres techniques d’optimisation, de sorte que les m´ethodes de r´esolution peuvent b´en´eficier des avantages de chacune des composantes. Le travail dans cette th`esevise `acontribuer `al’´etudedes probl`emesde tourn´eesde v´ehicules avec des visites optionnelles en fournissant un op´erateur`abase de programmation dynamique int´egr´edans un processus m´etaheuristiqueg´en´erique.L’op´erateurr´ecup`erele tour optimal de clients `avisiter, r´epondant aux contraintes du probl`eme,tout en optimisant l’objectif d´efini. L’op´erateurpose le probl`eme de la s´electiondes meilleurs clients `avisiter comme un probl`emede plus court chemin avec contraintes de ressources sur un graphe auxiliaire dirig´eacyclique repr´esentant les choix de visite possibles. Dans les probl`emesde tourn´eesde v´ehicules avec des visites optionnelles, les clients `aservir ne sont pas connus a priori et cela rend plus difficile `ar´esoudrele probl`emequ’un probl`emede routage classique qui est lui-mˆeme d´ej`aNP-difficile. Les probl`emes de tourn´ees avec des visites optionnelles trouvent des applications dans des domaines multiples et vari´es tels que la conception de la distribution, la logistique humanitaire, la prestation des soins de sant´e,le tourisme, le recrutement, la collection ou la livraison de marchandises et patrouille en milieu urbain. Mots-cl´es: M´etaheuristiques,programmation dynamique, tourn´eesde v´ehicules. v vi Contents 1 Introduction 1 1.1 Affiliation . .1 1.2 Motivation . .1 1.3 Thesis Proposal . .4 1.4 Overview of the PhD. Thesis . .5 2 Routing Problems and Solution Methods 7 2.1 Introduction . .7 2.2 Routing Problems . .8 2.2.1 The Travelling Salesman Problem . .9 2.2.2 The Capacitated Vehicle Routing Problem . 10 2.2.3 The Vehicle Routing-Allocation Problem . 10 Covering Problems . 12 Profit Problems . 15 2.3 Overview of Solution Methods . 18 2.3.1 Exact Methods . 18 2.3.2 Approximate Methods . 22 2.4 Classical Heuristics in Vehicle Routing . 23 2.4.1 Construction Heuristics . 24 2.4.2 Improvement Heuristics . 27 2.5 Large Neighborhood Search Metaheuristics . 28 2.5.1 Neighborhoods . 28 2.5.2 Neighborhood Search . 29 2.5.3 Large Neighborhood Search . 29 2.5.4 Adaptive Large Neighborhood Search . 31 2.6 Hybrid Metaheuristics . 34 2.7 Conclusions . 35 3 Solution Methodology Developed 37 3.1 Introduction . 37 3.2 Foundations of Selector ........................... 39 3.2.1 Split Operator . 39 3.2.2 Resource-Constrained Elementary Shortest Path Problem . 41 3.3 Selector Operator . 43 3.3.1 Auxiliary Graph . 43 3.3.2 Labels . 44 3.3.3 Algorithm . 45 3.4 Generating a Feasible Solution . 49 3.5 Performance Improvements . 50 3.5.1 Restricting the Number of Labels Extended . 50 viii Contents 3.5.2 Computing a Lower/Upper Bound . 51 3.5.3 Bidirectional Search . 53 3.6 Multi-Vehicle Selector ............................ 54 3.7 Metaheuristic Framework . 55 3.7.1 Destroy Sub-heuristics . 57 Shaw Removal Heuristic . 57 Worst Removal Heuristic . 58 Random Removal Heuristic . 58 How Many to Remove . 58 3.7.2 Repair Sub-heuristics . 58 Best Greedy Heuristic . 58 First Greedy Heuristic . 58 Regret Heuristic . 59 Choosing a Destroy-Repair Heuristic Pair . 59 Adaptive Weight Adjustment . 59 3.7.3 Simulated Annealing Guides the Search . 60 3.8 Conclusions . 61 4 Solving the Covering Tour Problem with Selector 63 4.1 Introduction .
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  • Notes on Dynamic Programming Algorithms & Data Structures 1

    Notes on Dynamic Programming Algorithms & Data Structures 1

    Notes on Dynamic Programming Algorithms & Data Structures Dr Mary Cryan These notes are to accompany lectures 10 and 11 of ADS. 1 Introduction The technique of Dynamic Programming (DP) could be described “recursion turned upside-down”. However, it is not usually used as an alternative to recursion. Rather, dynamic programming is used (if possible) for cases when a recurrence for an algorithmic problem will not run in polynomial-time if it is implemented recursively. So in fact Dynamic Programming is a more- powerful technique than basic Divide-and-Conquer. Designing, Analysing and Implementing a dynamic programming algorithm is (like Divide- and-Conquer) highly problem specific. However, there are particular features shared by most dynamic programming algorithms, and we describe them below on page 2 (dp1(a), dp1(b), dp2, dp3). It will be helpful to carry along an introductory example-problem to illustrate these fea- tures. The introductory problem will be the problem of computing the nth Fibonacci number, where F(n) is defined as follows: F0 = 0; F1 = 1; Fn = Fn-1 + Fn-2 (for n ≥ 2). Since Fibonacci numbers are defined recursively, the definition suggests a very natural recursive algorithm to compute F(n): Algorithm REC-FIB(n) 1. if n = 0 then 2. return 0 3. else if n = 1 then 4. return 1 5. else 6. return REC-FIB(n - 1) + REC-FIB(n - 2) First note that were we to implement REC-FIB, we would not be able to use the Master Theo- rem to analyse its running-time. The recurrence for the running-time TREC-FIB(n) that we would get would be TREC-FIB(n) = TREC-FIB(n - 1) + TREC-FIB(n - 2) + Θ(1); (1) where the Θ(1) comes from the fact that, at most, we have to do enough work to add two values together (on line 6).