VRP with Time Windows

Abstract This paper presents a survey of the research on the Vehicle Routing Problem with Time Windows (VRPTW), an extension of the Capacitated Vehicle Routing Problem. In the VRPTW, the service at each customer must start within an associated time window and the vehicle must remain at the customer location during service. Soft time windows can be violated at a cost while hard time windows do not allow for a vehicle to arrive at a customer after the latest time to begin service. We first present a multi-commodity network flow formulation with time and capacity constraints for the VRPTW. Approximation methods proposed in the literature to derive upper bounds are then reviewed. Then we explain how lower bounds can be obtained using optimal approaches, namely, Lagrangean relaxation and column generation. Next, we provide branching and cutting strategies that can be embedded within these optimal approaches to produce integer solutions. Special cases and extensions to the VRPTW follow as well as our conclusions. Résumé Cet article synthèse porte sur les récents développements concernant le problème du routage de véhicules sous des contraintes de fenêtres de temps. Dans ce problème, le serviceà un client doit débuterà l'intérieur d'un intervalle de temps. Celui-ci peutêtre, soit relaché au prix d'une certaine pénalité, soit rigide, auquel cas, il n'est pas permis de dépasser la limite supérieure. Nous présentons un modèle de réseau multi-flots avec des contraintes de temps et de capacité. Les méthodes heuristiques permettant de calculer des bornes supérieures sont d'abord présentées. Suivent les modèles d'optimisation basés sur la relaxation lagrangienne et la génération de colonnes pourévaluer des bornes inférieures. Enfin, on présente les stratégies de coupes et de branchements liéesà ces méthodes afin de déterminer des solutions entières. L'article se termine par l'étude de cas particuliers et d'extensions ainsi que nos conclusions

Matheuristics: using mathematics for heuristic design

Matheuristics are heuristic algorithms based on mathematical tools such as the ones provided by mathematical programming, that are structurally general enough to be applied to different problems with little adaptations to their abstract structure. The result can be metaheuristic hybrids having components derived from the mathematical model of the problems of interest, but the mathematical techniques themselves can define general heuristic solution frameworks. In this paper, we focus our attention on mathematical programming and its contributions to developing effective heuristics. We briefly describe the mathematical tools available and then some matheuristic approaches, reporting some representative examples from the literature. We also take the opportunity to provide some ideas for possible future development

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

Model-Based Heuristics for Combinatorial Optimization

Many problems arising in several and different areas of human knowledge share the characteristic of being intractable in real cases. The relevance of the solution of these problems, linked to their domain of action, has given birth to many frameworks of algorithms for solving them. Traditional solution paradigms are represented by exact and heuristic algorithms. In order to overcome limitations of both approaches and obtain better performances, tailored combinations of exact and heuristic methods have been studied, giving birth to a new paradigm for solving hard combinatorial optimization problems, constituted by model-based metaheuristics. In the present thesis, we deepen the issue of model-based metaheuristics, and present some methods, belonging to this class, applied to the solution of combinatorial optimization problems

Comparison of heuristic approaches for the multiple depot vehicle scheduling problem

Given a set of timetabled tasks, the multi-depot vehicle scheduling problemis a well-known problem that consists of determining least-cost schedulesfor vehicles assigned to several depots such that each task is accomplishedexactly once by a vehicle. In this paper, we propose to compare theperformance of five different heuristic approaches for this problem,namely, a heuristic \\mip solver, a Lagrangian heuristic, a columngeneration heuristic, a large neighborhood search heuristic using columngeneration for neighborhood evaluation, and a tabu search heuristic. Thefirst three methods are adaptations of existing methods, while the last twoare novel approaches for this problem. Computational results on randomlygenerated instances show that the column generation heuristic performs thebest when enough computational time is available and stability is required,while the large neighborhood search method is the best alternative whenlooking for a compromise between computational time and solution quality.tabu search;column generation;vehicle scheduling;heuristics;Lagrangian heuristic;large neighborhood search;multiple depot

Iterative restricted space search : a solving approach based on hybridization

Face à la complexité qui caractérise les problèmes d'optimisation de grande taille l'exploration complète de l'espace des solutions devient rapidement un objectif inaccessible. En effet, à mesure que la taille des problèmes augmente, des méthodes de solution de plus en plus sophistiquées sont exigées afin d'assurer un certain niveau d 'efficacité. Ceci a amené une grande partie de la communauté scientifique vers le développement d'outils spécifiques pour la résolution de problèmes de grande taille tels que les méthodes hybrides. Cependant, malgré les efforts consentis dans le développement d'approches hybrides, la majorité des travaux se sont concentrés sur l'adaptation de deux ou plusieurs méthodes spécifiques, en compensant les points faibles des unes par les points forts des autres ou bien en les adaptant afin de collaborer ensemble. Au meilleur de notre connaissance, aucun travail à date n'à été effectué pour développer un cadre conceptuel pour la résolution efficace de problèmes d'optimisation de grande taille, qui soit à la fois flexible, basé sur l'échange d'information et indépendant des méthodes qui le composent. L'objectif de cette thèse est d'explorer cette avenue de recherche en proposant un cadre conceptuel pour les méthodes hybrides, intitulé la recherche itérative de l'espace restreint, ±Iterative Restricted Space Search (IRSS)>>, dont, la principale idée est la définition et l'exploration successives de régions restreintes de l'espace de solutions. Ces régions, qui contiennent de bonnes solutions et qui sont assez petites pour être complètement explorées, sont appelées espaces restreints "Restricted Spaces (RS)". Ainsi, l'IRSS est une approche de solution générique, basée sur l'interaction de deux phases algorithmiques ayant des objectifs complémentaires. La première phase consiste à identifier une région restreinte intéressante et la deuxième phase consiste à l'explorer. Le schéma hybride de l'approche de solution permet d'alterner entre les deux phases pour un nombre fixe d'itérations ou jusqu'à l'atteinte d'une certaine limite de temps. Les concepts clés associées au développement de ce cadre conceptuel et leur validation seront introduits et validés graduellement dans cette thèse. Ils sont présentés de manière à permettre au lecteur de comprendre les problèmes que nous avons rencontrés en cours de développement et comment les solutions ont été conçues et implémentées. À cette fin, la thèse a été divisée en quatre parties. La première est consacrée à la synthèse de l'état de l'art dans le domaine de recherche sur les méthodes hybrides. Elle présente les principales approches hybrides développées et leurs applications. Une brève description des approches utilisant le concept de restriction d'espace est aussi présentée dans cette partie. La deuxième partie présente les concepts clés de ce cadre conceptuel. Il s'agit du processus d'identification des régions restreintes et des deux phases de recherche. Ces concepts sont mis en oeuvre dans un schéma hybride heuristique et méthode exacte. L'approche a été appliquée à un problème d'ordonnancement avec deux niveaux de décision, relié au contexte des pâtes et papier: "Pulp Production Scheduling Problem". La troisième partie a permit d'approfondir les concepts développés et ajuster les limitations identifiées dans la deuxième partie, en proposant une recherche itérative appliquée pour l'exploration de RS de grande taille et une structure en arbre binaire pour l'exploration de plusieurs RS. Cette structure a l'avantage d'éviter l'exploration d 'un espace déjà exploré précédemment tout en assurant une diversification naturelle à la méthode. Cette extension de la méthode a été testée sur un problème de localisation et d'allocation en utilisant un schéma d'hybridation heuristique-exact de manière itérative. La quatrième partie généralise les concepts préalablement développés et conçoit un cadre général qui est flexible, indépendant des méthodes utilisées et basé sur un échange d'informations entre les phases. Ce cadre a l'avantage d'être général et pourrait être appliqué à une large gamme de problèmes

Matheuristics:survey and synthesis

In integer programming and combinatorial optimisation, people use the term matheuristics to refer to methods that are heuristic in nature, but draw on concepts from the literature on exact methods. We survey the literature on this topic, with a particular emphasis on matheuristics that yield both primal and dual bounds (i.e., upper and lower bounds in the case of a minimisation problem). We also make some comments about possible future developments

Optimised search heuristics: combining metaheuristics and exact methods to solve scheduling problems

Tese dout., Matemática, Investigação Operacional, Universidade do Algarve, 2009Scheduling problems have many real life applications, from automotive industry to air traffic control. These problems are defined by the need of processing a set of jobs on a shared set of resources. For most scheduling problems there is no known deterministic procedure that can solve them in polynomial time. This is the reason why researchers study methods that can provide a good solution in a reasonable amount of time. Much attention was given to the mathematical formulation of scheduling problems and the algebraic characterisation of the space of feasible solutions when exact algorithms were being developed; but exact methods proved inefficient to solve real sized instances. Local search based heuristics were developed that managed to quickly find good solutions, starting from feasible solutions produced by constructive heuristics. Local search algorithms have the disadvantage of stopping at the first local optimum they find when searching the feasible region. Research evolved to the design of metaheuristics, procedures that guide the search beyond the entrapment of local optima. Recently a new class of hybrid procedures, that combine local search based (meta) heuristics and exact algorithms of the operations research field, have been designed to find solutions for combinatorial optimisation problems, scheduling problems included. In this thesis we study the algebraic structure of scheduling problems; we address the existent hybrid procedures that combine exact methods with metaheuristics and produce a mapping of type of combination versus application and finally we develop new innovative metaheuristics and apply them to solve scheduling problems. These new methods developed include some combinatorial optimisation algorithms as components to guide the search in the solution space using the knowledge of the algebraic structure of the problem being solved. Namely we develop two new methods: a simple method that combines a GRASP procedure with a branch-and-bound algorithm; and a more elaborated procedure that combines the verification of the violation of valid inequalities with a tabu search. We focus on the job-shop scheduling problem

Ant Colony Optimisation – A Proposed Solution Framework for the Capacitated Facility Location Problem

This thesis is a critical investigation into the development, application and evaluation of ant colony optimisation metaheuristics, with a view to solving a class of capacitated facility location problems. The study is comprised of three phases. The first sets the scene and motivation for research, which includes; key concepts of ant colony optimisation, a review of published academic materials and a research philosophy which provides a justification for a deductive empirical mode of study. This phase reveals that published results for existing facility location metaheuristics are often ambiguous or incomplete and there is no clear evidence of a dominant method. This clearly represents a gap in the current knowledge base and provides a rationale for a study that will contribute to existing knowledge, by determining if ant colony optimisation is a suitable solution technique for solving capacitated facility location problems. The second phase is concerned with the research, development and application of a variety of ant colony optimisation algorithms. Solution methods presented include combinations of approximate and exact techniques. The study identifies a previously untried ant hybrid scheme, which incorporates an exact method within it, as the most promising of techniques that were tested. Also a novel local search initialisation which relies on memory is presented. These hybridisations successfully solve all of the capacitated facility location test problems available in the OR-Library. The third phase of this study conducts an extensive series of run-time analyses, to determine the prowess of the derived ant colony optimisation algorithms against a contemporary cross-entropy technique. This type of analysis for measuring metaheuristic performance for the capacitated facility location problem is not evident within published materials. Analyses of empirical run-time distributions reveal that ant colony optimisation is superior to its contemporary opponent. All three phases of this thesis provide their own individual contributions to existing knowledge bases: the production of a series of run-time distributions will be a valuable resource for future researchers; results demonstrate that hybridisation of metaheuristics with exact solution methods is an area not to be ignored; the hybrid methods employed in this study ten years ago would have been impractical or infeasible; ant colony optimisation is shown to be a very flexible metaheuristic that can easily be adapted to solving mixed integer problems using hybridisation techniques

REVISIÓN DE LA LITERATURA DEL PROBLEMA DE RUTEO DE VEHÍCULOS EN UN CONTEXTO DE TRANSPORTE VERDE

In the efficient management of the supply chain the optimal management of transport of consumables and finished products appears. The costs associated with transport have direct impact on the final value consumers must pay, which in addition to requiring competitive products also demand that they are generated in environmentally friendly organizations. Aware of this reality, this document is intended to be a starting point for Master's and Doctoral degree students who want to work in a line of research recently proposed: green routing. The state of the art of the vehicle routing problem is presented in this paper, listing its variants, models and methodologies for solution. Furthermore, the proposed interaction between variants of classical routing problems and environmental effects of its operations, known in the literature as Green-VRP is presented. The goal is to generate a discussion in which mathematical models and solution strategies that can be applied within organizations that consider within their objectives an efficient and sustainable operation are posed. En el gerenciamiento eficiente de la cadena de suministro aparece la gestión óptima del transporte de insumos y productos terminados. Los costos asociados al transporte tienen impacto directo sobre el valor final que deben pagar los consumidores, que además de requerir productos competitivos también exigen que los mismos sean generados en organizaciones amigables con el medioambiente. Consientes de esa realidad este documento pretende ser un punto de partida para estudiantes de maestría y doctorado que quieran trabajar en una línea de investigación propuesta recientemente: el ruteo verde. En este trabajo se muestra un estado del arte del problema de ruteo de vehículos, enumerando sus variantes, modelos y metodologías de solución. Además, se presenta la interacción que se ha propuesto entre variantes clásicas de los problemas de ruteo y los efectos ambientales de su operación, denominados en la literatura como Green-VRP. El objetivo es generar una discusión donde se planteen modelos matemáticos y estrategias de solución que puedan ser aplicadas en organizaciones que consideren dentro de sus objetivos una operación eficiente y sustentable. Document type: Articl