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

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 biologically inspired network design model

A network design problem is to select a subset of links in a transport network that satisfy passengers or cargo transportation demands while minimizing the overall costs of the transportation. We propose a mathematical model of the foraging behaviour of slime mould P. polycephalum to solve the network design problem and construct optimal transport networks. In our algorithm, a traffic flow between any two cities is estimated using a gravity model. The flow is imitated by the model of the slime mould. The algorithm model converges to a steady state, which represents a solution of the problem. We validate our approach on examples of major transport networks in Mexico and China. By comparing networks developed in our approach with the man-made highways, networks developed by the slime mould, and a cellular automata model inspired by slime mould, we demonstrate the flexibility and efficiency of our approach

A biologically inspired network design model

A network design problem is to select a subset of links in a transport network that satisfy passengers or cargo transportation demands while minimizing the overall costs of the transportation. We propose a mathematical model of the foraging behaviour of slime mould P. polycephalum to solve the network design problem and construct optimal transport networks. In our algorithm, a traffic flow between any two cities is estimated using a gravity model. The flow is imitated by the model of the slime mould. The algorithm model converges to a steady state, which represents a solution of the problem. We validate our approach on examples of major transport networks in Mexico and China. By comparing networks developed in our approach with the man-made highways, networks developed by the slime mould, and a cellular automata model inspired by slime mould, we demonstrate the flexibility and efficiency of our approach

Randomized heuristics for the Capacitated Clustering Problem

In this paper, we investigate the adaptation of the Greedy Randomized Adaptive Search Procedure (GRASP) and Iterated Greedy methodologies to the Capacitated Clustering Problem (CCP). In particular, we focus on the effect of the balance between randomization and greediness on the performance of these multi-start heuristic search methods when solving this NP-hard problem. The former is a memory-less approach that constructs independent solutions, while the latter is a memory-based method that constructs linked solutions, obtained by partially rebuilding previous ones. Both are based on the combination of greediness and randomization in the constructive process, and coupled with a subsequent local search phase. We propose these two multi-start methods and their hybridization and compare their performance on the CCP. Additionally, we propose a heuristic based on the mathematical programming formulation of this problem, which constitutes a so-called matheuristic. We also implement a classical randomized method based on simulated annealing to complete the picture of randomized heuristics. Our extensive experimentation reveals that Iterated Greedy performs better than GRASP in this problem, and improved outcomes are obtained when both methods are hybridized and coupled with the matheuristic. In fact, the hybridization is able to outperform the best approaches previously published for the CCP. This study shows that memory-based construction is an effective mechanism within multi-start heuristic search techniques

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008