Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;

Design of robust networks. Application to the design of wind farm cabling networks.

RÉSUMÉ: Aujourd’hui, la conception de réseaux est une problématique cruciale qui se pose dans beaucoup de domaines tels que le transport ou l’énergie. En particulier, il est devenu nécessaire d’optimiser la façon dont sont conçus les réseaux permettant de produire de l’énergie. On se concentre ici sur la production électrique produite à travers des parcs éoliens. Cette énergie apparait plus que jamais comme une bonne alternative à la production d’électricité via des centrales thermiques ou nucléaires. Nous nous intéressons dans cette thèse à la conception du câblage collectant l’énergie dans les parcs éoliens. On connaît alors la position de l’ensemble des éoliennes appartenant au parc ainsi que celle du site central collecteur vers laquelle l’énergie doit être acheminée. On connaît également la position des câbles que l’on peut construire, leurs capacités, et la position des noeuds d’interconnexion possibles. Il s’agit de déterminer un câblage de coût minimal permettant de relier l’ensemble des éoliennes à la sous-station, tel que celui-ci soit résistant à un certain nombre de pannes sur le réseau. Mots clés: Recherche opérationnelle, Optimisation combinatoire, Conception de réseaux robustes, Théorie des graphes, Programmation en nombres entiers, Câblage de parcs éoliens.----------ABSTRACT: Nowadays, the design of networks has become a decisive problematic which appears in many fields such as transport or energy. In particular, it has become necessary and important to optimize the way in which networks used to produce, collect or transport energy are designed. We focus in this thesis on electricity produced through wind farms. The production of energy by wind turbines appears more than ever like a good alternative to the electrical production of thermal or nuclear power plants, giving that both of those production can have harmful consequences on the environment. It has then become necessary to optimize the design and construction of such networks. We focus in this thesis on the design of the cabling network which allows to collect and route the energy from the wind turbines to a sub-station, linking the wind farm to the electrical network. In this problem, we know the location of each wind turbine of the farm and the one of the sub-station. We also know the location of possible inter-connection nodes which allow to connect different cables between them. Each wind turbine produces a known quantity of energy and with each cable are associated a cost and a capacity (the maximum amount of energy that can be routed through this cable). The optimization problem that we consider is to select a set of cables of minimum cost such that the energy produced from the wind turbines can be routed to the sub-station in the network induced by this set of cables, without exceeding the capacity of each cable. We focus on cabling networks resilient to breakdowns. Keywords : Operations Research, Combinatorial optimization, Robust networks design, Graph theory, Mixed integer programming, Wind farm cabling networks

Problema da árvore de suporte de custo mínimo com restrição de grau e custos associados aos nodos

Tese de doutoramento, Estatística e Investigação Operacional (Optimização), 2009, Universidade de Lisboa, Faculdade de CiênciasNesta dissertação aborda-se uma variante de Problemas de Árvores em Grafos onde, para além de custos associados às ligações entre os nodos, existem também custos associados ao grau dos nodos. Esta variante é motivada no contexto de redes de telecomunicações, onde estes custos se encontram associados a equipamento de routing que é necessário instalar em todo os nodos que estejam ligados a mais do que um nodo na rede. Neste tipo de redes é ainda usual restringir o número máximo de ligações de cada vértice de forma a reduzir interferências de sinal. Esta variante é aplicada a dois problemas clássicos de Árvores em Grafos: o problema da Árvore de Suporte de Custo Mínimo e o problema da Árvore de Steiner com Recolha de Prémios. Incorporando esta variante nas formulações tradicionalmente utilizadas para estes problemas chega-se a modelos não lineares, devido à presença dos custos associados ao grau dos vértices. Duas técnicas de reformulação de modelos s aoent ao utilizadas: a técnica de Reformulação por Discretização e a técnica de Reformulação por Caminhos. Ao utilizar qualquer uma destas duas técnicas no modelos tradicionais, obt em-se modelos lineares. Além disso, as duas técnicas permitem construir conjuntos de desigualdades válidas que ao ser adicionadas a um modelo fortalecem-no no que diz respeito à respectiva relaxação linear. A segunda técnica só pode ser aplicada ao primeiro problema devido à existência de uma estrutura de Saco Mochila, presente neste problema. Os modelos apresentados são comparados utilizando um conjunto de instâncias com 25 e 50 nodos.In this dissertation a new variant of Tree Problems in Graphs is considered.In this variant, besides the costs associated to the links between the nodes,there also exist costs associated with the degree of the nodes. This variantis motivated in the context of telecommunications networks where this typeof costs is associated with routing equipment that has to be installed inevery node that is connected with more than one node in the network. Inthis kind of networks it is usual to limit the number of links in any node toprevent signal interferences. This variant is applied to two classical problems:the Spanning Tree Problem and the Prize-collecting Steiner Tree Problem.By integrating this variant in traditional formulations for these problem oneobtains non-linear models, due to the presence of the costs associated withthe degree of the nodes. Two reformulations techniques are then used: theReformulation by Discretization technique and the Reformulation by Pathstechnique. Linear models are obtained by using any of these two techniquesin the traditional models. In addition, different sets of valid inequalities canbe constructed with the use of these techniques which, when added to amodel, strengthen it in terms of the respective linear relaxation. The secondtechnique can only be applied to the first problem due to the existence of aKnapsack structure present in this problem.The presented models are compared using a set of instances with 25 and 50nodes