A biased random-key genetic algorithm for the capacitated minimum spanning tree problem

This paper focuses on the capacitated minimum spanning tree(CMST)problem.Given a central processor and a set of remote terminals with specified demands for traffic that must flow between the central processor and terminals,the goal is to design a minimum cost network to carry this demand. Potential links exist between any pair of terminals and between the central processor and the terminals. Each potential link can be included in the design at a given cost.The CMST problem is to design a minimum-cost network connecting the terminals with the central processor so that the flow on any arc of the network is at most Q. A biased random-keygenetic algorithm(BRKGA)is a metaheuristic for combinatorial optimization which evolves a population of random vectors that encode solutions to the combinatorial optimization problem.This paper explores several solution encodings as well as different strategies for some steps of the algorithm and finally proposes a BRKGA heuristic for the CMST problem. Computational experiments are presented showing the effectivenes sof the approach:Seven newbest- known solutions are presented for the set of benchmark instances used in the experiments.Peer ReviewedPostprint (author’s final draft

The capacitated minimum spanning tree problem

In this thesis we focus on the Capacitated Minimum Spanning Tree (CMST), an extension of the minimum spanning tree (MST) which considers a central or root vertex which receives and sends commodities (information, goods, etc) to a group of terminals. Such commodities flow through links which have capacities that limit the total flow they can accommodate. These capacity constraints over the links result of interest because in many applications the capacity limits are inherent. We find the applications of the CMST in the same areas as the applications of the MST; telecommunications network design, facility location planning, and vehicle routing. The CMST arises in telecommunications networks design when the presence of a central server is compulsory and the flow of information is limited by the capacity of either the server or the connection lines. Its study also results specially interesting in the context of the vehicle routing problem, due to the utility that spanning trees can have in constructive methods. By the simple fact of adding capacity constraints to the MST problem we move from a polynomially solvable problem to a non-polynomial one. In the first chapter we describe and define the problem, introduce some notation, and present a review of the existing literature. In such review we include formulations and exact methods as well as the most relevant heuristic approaches. In the second chapter two basic formulations and the most used valid inequalities are presented. In the third chapter we present two new formulations for the CMST which are based on the identification of subroots (vertices directly connected to the root). One way of characterizing CMST solutions is by identifying the subroots and the vertices assigned to them. Both formulations use binary decision variables y to identify the subroots. Additional decision variables x are used to represent the elements (arcs) of the tree. In the second formulation the set of x variables is extended to indicate the depth of the arcs in the tree. For each formulation we present families of valid inequalities and address the separation problem in each case. Also a solution algorithm is proposed. In the fourth chapter we present a biased random-key genetic algorithm (BRKGA) for the CMST. BRKGA is a population-based metaheuristic, that has been used for combinatorial optimization. Decoders, solution representation and exploring strategies are presented and discussed. A final algorithm to obtain upper bounds for the CMST is proposed. Numerical results for the BRKGA and two cutting plane algorithms based on the new formulations are presented in the fifth chapter . The above mentioned results are discussed and analyzed in this same chapter. The conclusion of this thesis are presented in the last chapter, in which we include the opportunity areas suitable for future research.En esta tesis nos enfocamos en el problema del Árbol de Expansión Capacitado de Coste Mínimo (CMST, por sus siglas en inglés), que es una extensión del problema del árbol de expansión de coste mínimo (MST, por sus siglas en inglés). El CMST considera un vértice raíz que funciona como servidor central y que envía y recibe bienes (información, objetos, etc) a un conjunto de vértices llamados terminales. Los bienes solo pueden fluir entre el servidor y las terminales a través de enlaces cuya capacidad es limitada. Dichas restricciones sobre los enlaces dan relevancia al problema, ya que existen muchas aplicaciones en que las restricciones de capacidad son de vital importancia. Dentro de las áreas de aplicación del CMST más importantes se encuentran las relacionadas con el diseño de redes de telecomunicación, el diseño de rutas de vehículos y problemas de localización. Dentro del diseño de redes de telecomunicación, el CMST está presente cuando se considera un servidor central, cuya capacidad de transmisión y envío está limitada por las características de los puertos del servidor o de las líneas de transmisión. Dentro del diseño de rutas de vehículos el CMST resulta relevante debido a la influencia que pueden tener los árboles en el proceso de construcción de soluciones. Por el simple de añadir las restricciones de capacidad, el problema pasa de resolverse de manera exacta en tiempo polinomial usando un algoritmo voraz, a un problema que es muy difícil de resolver de manera exacta. En el primer capítulo se describe y define el problema, se introduce notación y se presenta una revisión bibliográfica de la literatura existente. En dicha revisión bibliográfica se incluyen formulaciones, métodos exactos y los métodos heurísticos utilizados más importantes. En el siguiente capítulo se muestran dos formulaciones binarias existentes, así como las desigualdades válidas más usadas para resolver el CMST. Para cada una de las formulaciones propuestas, se describe un algoritmo de planos de corte. Dos nuevas formulaciones para el CMST se presentan en el tercer capítulo. Dichas formulaciones estás basadas en la identificación de un tipo de vértices especiales llamados subraíces. Los subraíces son aquellos vértices que se encuentran directamente conectados al raíz. Un forma de caracterizar las soluciones del CMST es a través de identificar los nodos subraíces y los nodos dependientes a ellos. Ambas formulaciones utilizan variables para identificar los subraices y variables adicionales para identificar los arcos que forman parte del árbol. Adicionalmente, las variables en la segunda formulación ayudan a identificar la profundidad con respecto al raíz a la que se encuentran dichos arcos. Para cada formulación se presentan desigualdades válidas y se plantean procedimientos para resolver el problema de su separación. En el cuarto capítulo se presenta un algoritmo genético llamado BRKGA para resolver el CMST. El BRKGA está basado en el uso de poblaciones generadas por secuencias de números aleatorios, que posteriormente evolucionan. Diferentes decodificadores, un método de búsqueda local, espacios de búsqueda y estrategias de exploración son presentados y analizados. El capítulo termina presentando un algoritmo final que permite la obtención de cotas superiores para el CMST. Los resultados computacionales para el BRKGA y los dos algoritmos de planos de corte basados en las formulaciones propuestas se muestran en el quinto capítulo. Dichos resultados son analizados y discutidos en dicho capítulo. La tesis termina presentando las conclusiones derivadas del desarrollo del trabajo de investigación, así como las áreas de oportunidad sobre las que es posible realizar futuras investigaciones

Algoritmos RAMP para o Problema de Localização de Instalações com Restrições de Capacidade e um Único Servidor

Os Problemas de Localização de Instalações são problemas de otimização combinatória complexos que têm centrado a atenção da comunidade científica. A importância dada à resolução destes problemas deve-se principalmente à sua relevância nas mais variadas áreas, tais como, economia, indústria, saúde, entre muitas outras. Neste estudo é considerado o Problema de Localização de Instalações com Restrições de Capacidade e um Único Servidor (Single Source Capacitated Facility Location Problem - SSCFLP). No SSCFLP, dado um conjunto de possíveis localizações para a abertura de instalações e um conjunto de clientes a servir, o objetivo é determinar que instalações abrir de forma a satisfazer com custo mínimo a procura dos clientes, garantindo que cada cliente é servido apenas por uma instalação. Neste problema são considerados os custos de abertura das instalações e os custos de afetação dos clientes. O SSCFLP tem várias aplicações práticas, como por exemplo, no planeamento de sistemas de distribuição e na conceção de redes informáticas. Os métodos exatos conseguem garantir a obtenção da solução ótima dos problemas à custa de recursos computacionais elevados, tornando pertinente a investigação de abordagens alternativas, nomeadamente heurísticas/metaheurísticas, que permitam com recursos mais reduzidos, a obtenção de soluções de elevada qualidade. As heurísticas/metaheurísticas têm centrado a sua atenção apenas num dos lados do espaço de soluções dos problemas de otimização combinatória. A dualidade dos problemas tem sido, maioritariamente, utilizada para a criação de soluções iniciais para uma exploração mais intensiva do espaço de soluções por parte de heurísticas primais. A metaheurística RAMP (Relaxation Adaptive Memory Programming), proposta por Rego [1], pretende criar algoritmos que explorem de forma mais eficiente a relação primal-dual dos problemas de otimização combinatória, permitindo, de forma iterativa, a manipulação da informação que é obtida de ambos os lados do espaço de soluções. A aplicação do método RAMP a vários problemas de otimização combinatória, demonstrou a enorme potencialidade desta metaheurística, obtendo algoritmos de estado-da-arte para todos esses problemas. O objetivo deste trabalho é verificar se a aplicação do método RAMP ao SSCFLP também é capaz de rivalizar com outros métodos propostos para a resolução deste problema. Neste trabalho, são apresentados dois novos algoritmos para a resolução do SSCFLP, ambos baseados no método RAMP, que designamos por Dual RAMP e PD-RAMP. O primeiro algoritmo (Dual RAMP) segue a abordagem RAMP na sua versão mais simples. O Dual RAMP baseia-se na resolução do dual lagrangeano do SSCFLP, através de otimização por subgradiente. A solução dual é projetada para o espaço de soluções primal através da aplicação de um método simples de projeção, e a solução primal obtida é sujeita a um método de melhoramento baseado numa abordagem simples da pesquisa tabu. Iterativamente, a informação obtida do lado primal é utilizada para o ajuste dos parâmetros do dual. O segundo algoritmo (PD-RAMP) baseia-se numa versão mais sofisticada da abordagem RAMP. Este algoritmo integra o Dual RAMP com um método evolutivo de forma a fortalecer a relação primal-dual do problema. Na implementação proposta, o método primal do PDRAMP é baseado numa pesquisa por dispersão com um conjunto de referência atualizado por ambos os lados, primal e dual. Os resultados obtidos pelo Dual RAMP e pelo PD-RAMP permitem concluir que a aplicação da metaheurística RAMP ao SSCFLP consegue resultados excelentes, obtendo soluções de elevada qualidade em tempos computacionais reduzidos. Acresce ainda o facto de, ao contrário da maioria das abordagens existentes na literatura, ambos os algoritmos propostos demonstrarem ser extremamente robustos, conseguindo muito bons resultados para todos os conjuntos de testes utilizados.Facility Location Problems are complex combinatorial optimization problems that have been focusing the attention of the scientific community. The importance given to the solution of these problems, is mainly due to their relevance in diversified areas, such as, economics, industry, health, among many others. This study considers the Single Source Capacitated Facility Location Problem (SSCFLP), where, given a set of possible locations for opening facilities and a set of clients to serve, the goal is to determine which facilities to open in order to fulfill with minimum cost the demand of the clients, ensuring that each client is served by only one facility. This problem considers the costs for opening facilities and the client’s assignment costs. SSCFLP has several practical applications, such as, distribution systems planning and computer networks design. Exact methods ensure the achievement of the problem’s optimal solution at the expense of high computational resources, justifying the exploration of alternative approaches, such as heuristics/metaheuristics, that can obtain high quality solutions with lower resources. Heuristics/metaheuristics have focused their attention on only one side of the combinatorial optimization problems solution space. The problems duality has been mostly used for creating initial solutions for a more intensive exploration of the solution space by primal heuristics. The RAMP (Relaxation Adaptive Memory Programming) metaheuristic proposed by Rego [1] aims to create algorithms that exploit more efficiently the primal-dual relationship of combinatorial optimization problems, allowing iteratively, the manipulation of information that is obtained by both sides of the solutions space. The RAMP application to several combinatorial optimization problems, demonstrated the great potential of this metaheuristic, obtaining state-of-the-art algorithms for all of those problems. With this study we intend to verify if the application of the RAMP method to the SSCFLP is also capable of competing with other proposed methods for the solution of this problem. In this work, we present two new algorithms for solving the SSCFLP, both based on the RAMP method, designated by Dual RAMP and PD-RAMP. The first algorithm (Dual RAMP) follows the RAMP approach in its simplest version. The Dual RAMP is based on the solution of the Lagrangean dual through subgradient optimization. The dual solution is projected to the primal solution space through the application of a simple projection method, and the obtained solution is subjected to an improvement method based on a simple tabu search approach. Iteratively, the information obtained from the primal side is used to adjust the dual parameters. The second algorithm (PD-RAMP) is based on a more sophisticated version of the RAMP approach. This algorithm integrates the Dual RAMP algorithm with an evolutionary method in order to strengthen the primal-dual relationship of the problem. In the proposed implementation, the PD-RAMP primal method is based on Scatter Search with a reference set updated by both sides, primal and dual. The results obtained by the Dual RAMP and the PD-RAMP algorithms showed that the application of the RAMP metaheuristic to the SSCFLP attains excellent results, obtaining high quality solutions in reduced computational times. Moreover, unlike most of the existing approaches in the literature, both proposed algorithms proved to be extremely robust, achieving very good results for all sets of tests

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

Logistics service network design : models, algorithms, and applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2004.Includes bibliographical references (leaves 177-186).Service network design is critical to the profitability of express shipment carriers. In this thesis, we consider two challenging problems associated with designing networks for express shipment service. The first problem is to design an integrated network for premium and deferred services simultaneously. Related existing models adapted to this problem are intractable for realistic instances of this problem: computer memory requirements and solution times are excessive. We introduce a disaggregate information-enhanced column generation approach for this problem that reduces the number of variables to be considered in the integer program from hundreds of thousands to only thousands, allowing us to solve previously unsolvable problem instances. The second problem is to determine the express package service network design in its entirety, including aircraft routings, fleet assignments, and package flow routings, including hub assignments. Existing models applied to this problem have weak associated linear programming bounds and hence, fail to produce quality feasible solutions. For example, for a small network design problem instance it takes days to produce a feasible solution that is provably near- optimal using the best performing existing model. To overcome these tractability challenges, we introduce a new model, referred to as the gateway cover and flow formulation. Applying our new formulation to the same network design instance, it takes only minutes to find an optimal solution.(cont.) Applying our disaggregate information-enhanced column generation approach and gateway cover and flow formulation and solution approach to the network design problems of a large express package service provider, we demonstrate tens of millions of dollars in potential annual operating cost savings and reductions in the numbers of aircraft needed to perform the service. Moreover, we illustrate that, though designed for tactical planning, our new model and solution approach can provide insights for strategic decision-making, such as hub opening/closure, hub capacity expansion, and fleet composition and size.by Su Shen.Ph.D

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