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

Distance-constrained vehicle routing problem: exact and approximate solution (mathematical programming)

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The asymmetric distance-constrained vehicle routing problem (ADVRP) looks at finding vehicle tours to connect all customers with a depot, such that the total distance is minimised; each customer is visited once by one vehicle; every tour starts and ends at a depot; and the travelled distance by each vehicle is less than or equal to the given maximum value. We present three basic results in this thesis. In the first one, we present a general flow-based formulation to ADVRP. It is suitable for symmetric and asymmetric instances. It has been compared with the adapted Bus School Routing formulation and appears to solve the ADVRP faster. Comparisons are performed on random test instances with up to 200 customers. We reach a conclusion that our general formulation outperforms the adapted one. Moreover, it finds the optimal solution for small test instances quickly. For large instances, there is a high probability that an optimal solution can be found or at least improve upon the value of the best feasible solution found so far, compared to the other formulation which stops because of the time condition. This formulation is more general than Kara formulation since it does not require the distance matrix to satisfy the triangle inequality. The second result improves and modifies an old branch-and-bound method suggested by Laporte et al. in 1987. It is based on reformulating a distance-constrained vehicle routing problem into a travelling salesman problem and uses the assignment problem as a lower bounding procedure. In addition, its algorithm uses the best-first strategy and new branching rules. Since this method was fast but memory consuming, it would stop before optimality is proven. Therefore, we introduce randomness in choosing the node of the search tree in case we have more than one choice (usually we choose the smallest objective function). If an optimal solution is not found, then restart is required due to memory issues, so we restart our procedure. In that way, we get a multistart branch and bound method. Computational experiments show that we are able to exactly solve large test instances with up to 1000 customers. As far as we know, those instances are much larger than instances considered for other VRP models and exact solution approaches from recent literature. So, despite its simplicity, this proposed algorithm is capable of solving the largest instances ever solved in literature. Moreover, this approach is general and may be used in solving other types of vehicle routing problems. In the third result, we use VNS as a heuristic to find the best feasible solution for groups of instances. We wanted to determine how far the difference is between the best feasible solution obtained by VNS and the value of optimal solution in order to use the output of VNS as an initial feasible solution (upper bound procedure) to improve our multistart method. Unfortunately, based on the search strategy (best first search), using a heuristic to find an initial feasible solution is not useful. The reason for this is because the branch and bound is able to find the first feasible solution quickly. In other words, in our method using a good initial feasible solution as an upper bound will not increase the speed of the search. However, this would be different for the depth first search. However, we found a big gap between VNS feasible solution and an optimal solution, so VNS can not be used alone unless for large test instances when other exact methods are not able to find any feasible solution because of memory or stopping conditions

Lagrangian Relaxation for q-Hub Arc Location Problems

The topic of this Master thesis is an in-depth research study on a specific type of network systems known as hub-and-spoke networks. In particular, we study q-Hub Arc Location Problems that consist, at a strategical level, of selecting q hub arcs and at most p hub nodes, and of the routing of commodities through the so called hub level network. We propose strong formulations to two variants of the problem, namely the q-hub arc location problem and the $q$-hub arc location problem with isolated hub nodes. We present a Lagrangian relaxation that exploits the structure of these problems by decomposing them into |K|+2 independent easy-to-solve subproblems and develop Lagrangian heuristics that yield high quality feasible solutions to both models. We, further, provide some insights on the structure of the optimal solutions to both models and investigate the cost benefit of incomplete hub networks with and without isolated hub nodes. Finally, computational results on a set of benchmark instances with up to 100 nodes are reported to assess the performance of the proposed MIP formulations and of our algorithmic approach

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

Fiber to the home

In den letzten Jahren gab es zunehmenden Bedarf für breitbandige Telekommunikations Netzwerke. Eine von Telekommunikationsunternehmen angewandte Strategie um die Bandbreite entlang der last-mile des Netzwerks zu erhöhen ist, Glasfaserkabel direkt bis zum Endkunden zu verlegen. Diese Strategie wird fiber to the home (FTTH) genannt. In der vorliegenden Arbeit wird das local access network design problem (LAN) und die Variante mit prize-collecting (PC-LAN) verwendet, um das Problem der FTTH Planung zu modellieren. Das LAN Problem zielt darauf ab eine kostenminimale Lösung zu finden und gestattet es sowohl verschiedene Kabeltechnologien und existierende Infrastruktur, als auch die Zusatzkosten zu modellieren, die anfallen wenn neue Verbindungen hergestellt werden. Darüber hinaus, erlaubt das PC-LAN Problem den Aspekt zu modellieren, dass nicht unbedingt alle Kunden mit FTTH versorgt werden müssen. Stattdessen wird eine Teilmenge der Kunden versorgt mit dem Ziel den Profit zu maximieren. Um LAN und PC-LAN Problem Instanzen zu lösen, werden folgende Methoden des Operations Research angewandt: Preprocessing, ganzzahlige Programmierung, Stärkung der mathematischen Modelle durch Disaggregation der Variablen, Benders' Dekomposition und adaptive Multi-Start-Heuristiken. In einem Projekt von Universität Wien und Telekom Austria wurden große FTTH Datensätze untersucht und die hier vorgestellten Methoden entworfen. Diese Lösungsansätze wurden als Computerprogramme implementiert und ihre Tauglichkeit zur Behandlung von FTTH Planungsfragen konnte gezeigt werden.Within recent years the request for broadband telecommunication networks has been constantly increasing. A strategy employed by telecommunication companies to increase the bandwidth on the last mile of the network is to lay optical fiber directly to the end customer. This strategy is denoted as fiber to the home (FTTH). In this thesis the local access network design problem (LAN) and its prize-collecting variant (PC-LAN) are used to formalize the planning of FTTH networks. The LAN problem asks for a cost minimal solution and allows to model different cable technologies, existing infrastructure and the overhead cost incurred by building new connections. In addition, the PC-LAN problem covers the aspect, that not all customers must necessarily be connected with FTTH, but instead we search for a subset of customers in order to maximize profits. To solve LAN and PC-LAN instances, the following operations research methods are employed: Preprocessing, mixed integer programming, model strengthening by variable disaggregation, Benders' decomposition and adaptive multi-start heuristics. In a project between University of Vienna and Telekom Austria, large real world data sets for FTTH planning were investigated and the methods presented in this thesis have been designed. These solution methods have been implemented as computer programs and empirically verified to be reasonable approaches to FTTH network design problems

The Optimum Communication Spanning Tree Problem : properties, models and algorithms

For a given cost matrix and a given communication requirement matrix, the OCSTP is defined as finding a spanning tree that minimizes the operational cost of the network. OCST can be used to design of more efficient communication and transportation networks, but appear also, as a subproblem, in hub location and sequence alignment problems. This thesis studies several mixed integer linear optimization formulations of the OCSTP and proposes a new one. Then, an efficient Branch & Cut algorithm derived from the Benders decomposition of one of such formulations is used to successfully solve medium-sized instances of the OCSTP. Additionally, two new combinatorial lower bounds, two new heuristic algorithms and a new family of spanning tree neighborhoods based on the Dandelion Code are presented and tested.Postprint (published version