Branch-and-price and multicommodity flows

Tese de doutoramento em Engenharia de Produção e Sistemas, área de Investigação OperacionalIn this Thesis, we address column generation based methods for linear and integer programming and apply them to three multicommodity flow problems. For (mixed) integer programming problems, the approach taken consists in reformulating an original model, using the Dantzig-Wolfe decomposition principle, and then combining column generation with branch-and-bound (branch-and-price) in order to obtain optimal solutions. The main issue when developing a branch-and-price algorithm is the branching scheme. The approach explored in this work is to branch on the variables of the original model, keeping the structure of the subproblems of the column generation method unchanged. The incorporation of cuts (branch-and-price-and-cut), again without changing the structure of the subproblem, is also explored. Based on that general methodology, we developed a set of C++ classes (ADDing - Automatic Dantzig-Wolfe Decomposition for INteger column Generation), which implements abranch-and-price algorithm. Its main distinctive feature is that it can be used as a “black-box”: all the user is required to do is to provide the original model. ADDing can also be customised to meet a specific problem, if the user is willing to provide a subproblem solver and/or specific branching schemes. We developed column generation based algorithms for three multicommodity flow problems. In this type of problems, it is desired to route a set of commodities through a capacitated network at a minimum cost. In the linear problem, each unit of each commodity is divisible. By using a model with variables associated with paths and circuits, we obtained significant improvements on the solution times over the standard column generation approach, for instances defined in planar networks (in several instances the relative improvement was greater than 60%). In the integer problem, each unit of each commodity is indivisible; the flow of a commodity can be split between different paths, but the flow on each of those paths must be integer. In general, the proposed branch-and-price algorithm was more efficient than Cplex 6.6 in the sets of instances where each commodity is defined by an origin-destination pair; for some of the other sets of instances, Cplex 6.6 gave better time results. In the binary problem, all the flow of each commodity must be routed along a single path. We developed a branch-and-price algorithm based on a knapsack decomposition and modified (by using a different branching scheme) a previously described branch-and-price-and-cut algorithm based on a path decomposition. The outcome of the computational tests was surprising, given that it is usually assumed that specific methods are more efficient than general ones. For the instances tested, a state-of-the-art general-purpose (Cplex 8.1) gave, in general, much better results than both decomposition approaches.Nesta Tese, abordam-se métodos de geração de colunas para programação linear e inteira. A sua aplicação é feita em três problemas de fluxo multicomodidade. Para problemas de programação inteira (mista), a abordagem seguida é a de reformular um modelo original, utilizando o princípio de decomposição de Dantzig-Wolfe, e combinar geração de colunas com o método de partição e avaliação (partição e geração de colunas) para a obtenção de soluções óptimas. A questão essencial no desenvolvimento de um algoritmo deste tipo é a estratégia de partição. A abordagem seguida neste trabalho é a de realizar a partição nas variáveis do modelo original, mantendo a estrutura do subproblema do método de geração de colunas. A incorporação de cortes, ainda sem alteração da estrutura do subproblema, é também explorada. Com base nesta metodologia geral, foi desenvolvido um conjunto de classes em C++ (ADDing - Automatic Dantzig-Wolfe Decomposition for INteger column Generation), que implementa um algorithmo de partição e geração de colunas. A sua característica fundamental é apenas ser requerido ao utilizador a definição de um modelo original. Num modo mais avançado, o utilizador pode implementar algoritmos para resolver o subproblema e/ou esquemas de partição. Foram desenvolvidos algoritmos baseados em geração de colunas para três problemas de fluxo multicomodidade. Neste tipo de problemas, pretende-se encaminhar um conjunto de comodidades através de uma rede capacitada, minimizando o custo. No problema linear, cada unidade de cada comodidade é divisível. Utilizando um modelo com variáveis associadas a caminhos e a circuitos, obtiveram-se melhorias significativas nos tempos de resolução em relação ao método de geração de colunas usual, para instâncias definidas em redes planares (em várias instâncias a melhoria relativa foi superior a 60%). No problema inteiro, cada unidade de cada comodidade é indivisível; o fluxo de uma comodidade pode ser dividido por diferentes caminhos, mas o fluxo em cada um deles tem de ser inteiro. Em geral, o algoritmo de partição e geração de colunas foi mais eficiente do que o software Cplex 6.6 nos conjuntos de instâncias em que cada comodidade é definida por um par origem-destino; para alguns dos outros conjuntos de instâncias, o software Cplex 6.6 obteve melhores resultados. No problema binário, todo o fluxo de cada comodidade apenas pode utilizar um caminho. Foi desenvolvido um algoritmo de partição e geração de colunas baseado numa decomposição de mochila e modificado (através de um esquema de partição diferente) um algoritmo de partição e geração de colunas com cortes, previamente descrito, baseado numa decomposição por caminhos. Os resultados dos testes computacionais foram surpreendentes, dado que é usualmente assumido que métodos específicos são mais eficientes do que métodos gerais. Para as instâncias testadas, o software Cplex 8.1 obteve, em geral, resultados muito melhores do que as duas decomposições

Design of large scale transportation service networks with consolidation : models, algorithms and applications

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering; and, (S.M.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 1998.Includes bibliographical references (leaves 94-103).by Niranjan Krishnan.S.M

The Multilayer Capacitated Survivable IP Network Design Problem : valid inequalities and Branch-and-Cut

Telecommunication networks can be seen as the stacking of several layers like, for instance, IP-over-Optical networks. This infrastructure has to be sufficiently survivable to restore the traffic in the event of a failure. Moreover, it should have adequate capacities so that the demands can be routed between the origin-destinations. In this paper we consider the Multilayer Capacitated Survivable IP Network Design problem. We study two variants of this problem with simple and multiple capacities. We give two multicommodity flow formulations for each variant of this problem and describe some valid inequalities. In particular, we characterize valid inequalities obtained using Chvatal-Gomory procedure from the well known Cutset inequalities. We show that some of these inequalities are facet defining. We discuss separation routines for all the valid inequalities. Using these results, we develop a Branch-and-Cut algorithm and a Branch-and-Cut-and-Price algorithm for each variant and present extensive computational results

Lagrangian-based methods for single and multi-layer multicommodity capacitated network design

Le problème de conception de réseau avec coûts fixes et capacités (MCFND) et le problème de conception de réseau multicouches (MLND) sont parmi les problèmes de conception de réseau les plus importants. Dans le problème MCFND monocouche, plusieurs produits doivent être acheminés entre des paires origine-destination différentes d’un réseau potentiel donné. Des liaisons doivent être ouvertes pour acheminer les produits, chaque liaison ayant une capacité donnée. Le problème est de trouver la conception du réseau à coût minimum de sorte que les demandes soient satisfaites et que les capacités soient respectées. Dans le problème MLND, il existe plusieurs réseaux potentiels, chacun correspondant à une couche donnée. Dans chaque couche, les demandes pour un ensemble de produits doivent être satisfaites. Pour ouvrir un lien dans une couche particulière, une chaîne de liens de support dans une autre couche doit être ouverte. Nous abordons le problème de conception de réseau multiproduits multicouches à flot unique avec coûts fixes et capacités (MSMCFND), où les produits doivent être acheminés uniquement dans l’une des couches. Les algorithmes basés sur la relaxation lagrangienne sont l’une des méthodes de résolution les plus efficaces pour résoudre les problèmes de conception de réseau. Nous présentons de nouvelles relaxations à base de noeuds, où le sous-problème résultant se décompose par noeud. Nous montrons que la décomposition lagrangienne améliore significativement les limites des relaxations traditionnelles. Les problèmes de conception du réseau ont été étudiés dans la littérature. Cependant, ces dernières années, des applications intéressantes des problèmes MLND sont apparues, qui ne sont pas couvertes dans ces études. Nous présentons un examen des problèmes de MLND et proposons une formulation générale pour le MLND. Nous proposons également une formulation générale et une méthodologie de relaxation lagrangienne efficace pour le problème MMCFND. La méthode est compétitive avec un logiciel commercial de programmation en nombres entiers, et donne généralement de meilleurs résultats.The multicommodity capacitated fixed-charge network design problem (MCFND) and the multilayer network design problem (MLND) are among the most important network design problems. In the single-layer MCFND problem, several commodities have to be routed between different origin-destination pairs of a given potential network. Appropriate capacitated links have to be opened to route the commodities. The problem is to find the minimum cost design and routing such that the demands are satisfied and the capacities are respected. In the MLND, there are several potential networks, each at a given layer. In each network, the flow requirements for a set of commodities must be satisfied. However, the selection of the links is interdependent. To open a link in a particular layer, a chain of supporting links in another layer has to be opened. We address the multilayer single flow-type multicommodity capacitated fixed-charge network design problem (MSMCFND), where commodities are routed only in one of the layers. Lagrangian-based algorithms are one of the most effective solution methods to solve network design problems. The traditional Lagrangian relaxations for the MCFND problem are the flow and knapsack relaxations, where the resulting Lagrangian subproblems decompose by commodity and by arc, respectively. We present new node-based relaxations, where the resulting subproblem decomposes by node. We show that the Lagrangian dual bound improves significantly upon the bounds of the traditional relaxations. We also propose a Lagrangian-based algorithm to obtain upper bounds. Network design problems have been the object of extensive literature reviews. However, in recent years, interesting applications of multilayer problems have appeared that are not covered in these surveys. We present a review of multilayer problems and propose a general formulation for the MLND. We also propose a general formulation and an efficient Lagrangian-based solution methodology for the MMCFND problem. The method is competitive with (and often significantly better than) a state-of-the-art mixedinteger programming solver on a large set of randomly generated instances

Dynamic unsplittable flows with path-change penalties: new formulations and solution schemes for large instances

In this work, we consider the dynamic unsplittable flow problem. This variation of the unsplittable flow problem has received little attention so far. The unsplittable flow problem is an NP-hard extension of the multi-commodity flow problem where each commodity sends its flow on only one path. In its dynamic version, this problem features several time steps and a penalty is paid when a commodity changes its path from one time step to the next. We present several mixed-integer linear programming formulations for this problem and compare the strength of their linear relaxation. These formulations are embedded in several solvers which are extensively compared on small to large instances. One of these formulations must be solved through a column generation process whose pricing problem is more difficult than those used in classical flow problems. We present limitations of the pricing schemes proposed in earlier works and describe two new schemes with a better worst-case complexity. Overall, this work lays a strong algorithmic baseline for the resolution of the dynamic unsplittable flow problem, proposes original formulations, and discusses the compared advantages of each, thus hopefully contributing a step towards a better understanding of this problem for both OR researchers and practical applications