New neighborhood search structures for the capacitated minimum spanning tree problem

Cover title. "November, 1998."Includes bibliographical references (p. 24-25).by Ravindra K. Ahuja, James B. Orlin, Dushyant Sharma

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

Operations Research in action

Wie der Titel bereits andeutet bezieht sich diese Dissertation auf ein Operations Research Projekt, dass der Ä Osterreichische Telekommunikationsanbieter Telekom Austria in den Jahren 2006 bis 2009 durchfÄuhrte. Die wachsende Zahl von Internet Nutzern, neue Anwendungen im Internet und die zunehmende Konkurrenz von mobilem Internet zwingen Festnetzbetreiber wie Telekom Austria ihre Produkte fÄur den Internet Zugang mit hÄoheren Bandbreiten zu versehen. ZwangslÄau¯g mÄussen die Zugangsnetze verbessert werden, was nur mit hohen Investitionskosten erreichbar ist. Aus diesem Grund kommt der kostenoptimalen Planung solcher Netzwerke eine zentrale Rolle zu. Ein wesentliches Projektziel war es, den Planungsprozess mit Methoden der diskreten Optimie- rung aus dem Bereich Network Design zu unterstÄutzen. Die Ergebnisse, die in dieser Disserta- tion beschrieben werden, beschÄaftigen sich mit Algorithmen aus dem Gebiet Facility Location (Bestimmung von Versorgungsstandorten). Vor der PrÄasentation der dazugehÄorigen Theorie und ihrer Anwendung auf die gestellten Problem werden zweitere grÄundlich analysiert. ZunÄachst wird der Telekommunikationsmarkt bis 2009 mit speziellem Fokus auf den Zeitraum zwischen 2006 und 2009 beschrieben. Die Telekommunikationsindustrie hatte bereits einige Strategien zur Verbesserung der Netzwerkinfrastruktur entwickelt. Ihre Relevanz fÄur die ge- stellten Probleme wird herausgearbeitet Dem folgt eine Au°istung der Problemspezi¯kationen, wie sie in der Evaluierungsphase des Projekts mit den beteiligten Anwendern erstellt wurde. Mit Hilfe eines dynamischen Programmes wird die gestellte Fragestellung unter BerÄucksichtigung aller Spezi¯kationen gelÄost. Eine Au°istung von Bedingungen, wann dieser Algorithmus die optimale LÄosung liefert, und die dazugehÄorigen Beweise beschlie¼en Kapitel 1. In der Folge stellte sich allerdings heraus, dass die Praktiker mit dieser ersten LÄosung nicht zufrieden waren. Die Liste der Spezi¯kationen war nicht vollstÄandig. Sie musste verÄandert und erweitert werden. Mangelnde E±zienz machte die LÄosungen fÄur die Praxis unbrauchbar. Die LÄosungen enthielten Versorgungsstandorte, die minder ausgelastet waren (underutilized), d.h. diesen Standorten waren zu wenige Kunden zugeordnet worden. Solche Lokationen mussten aus den LÄosungen entfernt werden. Dann aber waren die Verbleibenden so zu repositionieren, dass die Versorgung mit einer vorgegebenen MindestÄubertragungsrate fÄur die grÄo¼tmÄogliche Menge an Kunden sichergestellt werden konnte. Diese Strategie wurde mit Hilfe des Konzepts der k-Mediane umgesetzt: Unter der Nebenbedingung, dass die Anzahl der Standorte durch eine Konstante k beschrÄankt ist, wird die optimale Zuordnung von Kunden zu Versorgungs- standorten, d.h. ihre Versorgung, gesucht. Anschlie¼end lÄost man dann k-Median Probleme fÄur verschiedene Werte von k und bestimmt die Mindestauslastungen und Versorgungsraten, die diese LÄosungen erzielen. Dieses Vorgehen versetzt den Anwender in die Lage unter verschie- denen LÄosungen zwischen e±zienter Auslastung der Versorgungsstandorten und der HÄohe der Versorgungsraten balancieren zu kÄonnen. In Kapitel 2 werden zunÄachst die Ereignisse und Diskussionen beschrieben, die eine ÄAnderung der LÄosungsstrategie notwendig machten, und die geÄanderten bzw. neuen Spezi¯kationen wer- den prÄasentiert. Dem folgt die Vorstellung der Theorie der k-Mediane inklusive der Beschrei- bung eines Algorithmus aus der Literatur. Am Ende des zweiten Kapitels wird eine Variante dieses Algorithmus entwickelt, der fÄur die spezi¯schen Anforderungen noch besser geeignet ist: Der Algorithmus aus der Literatur fÄugt Lokationen schrittweise in die LÄosung ein, d.h. pro Iteration erhÄoht sich die Anzahl der Versorgungsstandorte um einen, bis die maximale Anzahl von Lokationen erreicht ist. Im Falle von Zugangsnetzen ist die zu erwartende Anzahl von Standorten aber eher gro¼. Daher ist es vorteilhafter die gewÄunschte Anzahl von oben, durch Reduktion der Anzahl von Versorgungsstandorten in der LÄosung zu erreichen. Kapitel 3 liefert eine extensive empirische Analyse von 106 verschiedenen Zugangsnetzen. Kon- kreter Zweck dieser Demonstration ist es einen Eindruck zu vermittelt, wie man die entwickel- ten und adaptierten Methoden bei der Vorbereitung des Planungsprozesses einsetzen kann. So ist es einerseits mÄoglich strategischen Fragestellungen vorab zu analysieren (z.B. E®ekt der Erzwingung des HV Kreises, Balance zwischen Auslastung der Versorgungsstandorte und der Versorgungsrate), und andererseits VorschlÄage fÄur passende Planungsprozesse fÄur die An- wender zu entwickeln (z.B. durch Laufzeitanalysen). ZusÄatzlich werden die beiden Methoden zur LÄosung des k-Median Problems, die in dieser Abreit vorgestellt werden, noch bzgl. ihres Laufzeitverhaltens verglichen.As indicated by the title this thesis is based on an Operations Research project which was conducted at the Austrian telecommunications provider Telekom Austria between 2006 and 2009. An increasing number of internet users, new internet applications and the growing competition of mobile internet access force ¯xed line providers like Telekom Austria to o®er higher rates for data transmission via their access networks. As a consequence access nets have to be improved which leads to investments of signi¯cant size. Therefore, minimizing such investments by a cost optimal planning of networks becomes a key issue. The main goal of the project was to support the planning process by utilizing discrete opti- mization methods from the ¯eld of network design. The key results which are presented in this thesis are algorithms for facility location. However, before dealing with the theory and the solutions | in practice as well as in this thesis | a thorough analysis of the stated problem is undertaken. To begin with the telecommunication market before 2006 and especially between 2006 and 2009 is reviewed to provide some background information. The industry had already developed di®erent strategies to improve ¯xed line infrastructure. Their relevance for the stated problem is presented. Furthermore, the most important problem speci¯cations as they were gathered in cooperation with the practitioners are listed and discussed in detail. A ¯rst solution was based on a dynamic program for solving the facility location problem which was derived from the speci¯cations. The statement of conditions for the optimality of this algorithm and their proofs conclude Chapter 1. It turned out that this ¯rst solution did not provide the desired result. It rather fostered the discussion process between operations researches and practitioners. New speci¯cations were added to the existing list. The planners dismissed these ¯rst solutions because they were not e±cient enough. These solutions contained facilities which were underutilized, i.e. too few customers were assigned to such facilities. To overcome this problem facilities of low utilization had to be removed from the solutions. The remaining facilities were rearranged in a way to maximize the coverage with a certain minimum transmission rate. This strategy was realized by adapting the concept of the k-median problem: The number of facilities is bounded whereas simultaneously the number of optimally supplied customers is maximized. Then for di®erent bounds the minimum facility utilization is reported. That way the practitioner is enabled to ¯nd the optimal balance between e±cient facility utilization and coverage of customer demands. After sketching the events and discussions which made further development necessary and listing the additional speci¯cations, the theory of the k-median problem is presented and a basic algorithm from the literature is cited. For the speci¯c requirements of the given problem a variant of the algorithm is developed and described at the end of Chapter 2: The algorithm from the literature inserts facilities one by one into the solution that way approaching the bound in an ascending manner. However, since the expected number of facilities is usually large it is more advantageous to approach the bound from above in a descending manner. Finally, an extensive empirical study of 106 di®erent local access areas is presented. The main purpose of this demonstration is to give a concrete impression of how the adapted and developed methods can be utilized in preparation of the planning process by studying strategic questions (e.g. CO circle enforcement, balancing between facility utilization and coverage) and by providing information (runtime) which is useful to set up an appropriate working environment for the future users. Additionally, the two variants of the k-median algorithm | the ascending and the descending method | can be compared

Mixed integer programming approaches to problems combining network design and facility location

Viele heutzutage über das Internet angebotene Dienstleistungen benötigen wesentlich höhere Bandbreiten als von bestehenden lokalen Zugangsnetzen bereitgestellt werden. Telekommunikationsanbieter sind daher seit einigen Jahren bestrebt, ihre zum Großteil auf Kupferkabeln basierenden Zugangsnetze entsprechend zu modernisieren. Die gewünschte Erweiterung der bereitgestellten Bandbreiten wird oftmals erzielt, indem ein Teil des Kupfernetzes durch Glasfaser ersetzt wird. Dafür sind Versorgungsstandorte notwendig, an welchen die optischen und elektrischen Signale jeweils in einander umgewandelt werden. In der Praxis gibt es mehrere Strategien für die Installation von optischen Zugangsnetzen. Fiber-to-the-Home bezeichnet Netze, in denen jeder Haushalt direkt per Glasfaser angebunden wird. Wird je Wohngebäude eine optische Verbindung bereitgestellt, nennt man dies Fiber-to-the-Building. Endet die Glasfaserverbindung an einem Versorgungsstandort, welcher die Haushalte eines ganzen Wohnviertels durch Kupferkabel versorgt, bezeichnet man dies als Fiber-to-the-Curb. Inhalt dieser Dissertation sind mathematische Optimierungsmodelle für die kosteneffiziente Planung von auf Glasfaser basierenden lokalen Zugangsnetzen. Diese Modelle decken mehrere Aspekte der Planung ab, darunter die Fiber-to-the-Curb-Strategie mit zusätzlichen Restriktionen betreffend Ausfallssicherheit, gemischte Fiber-to-the-Home und Fiber-to-the-Curb-Netze sowie die Kapazitätenplanung von Fiber-to-the-Curb-Netzen. Ergebnis dieser Dissertation sind die theoretische Analyse der beschriebenen Modelle sowie effiziente Lösungsalgorithmen. Es kommen Methoden der kombinatorischen Optimierung zum Einsatz, darunter Umformulierungen auf erweiterten Graphen, zulässige Ungleichungen und Branch-and-Cut-Verfahren.In recent years, telecommunication service providers started to adapt their local access networks to the steadily growing demand for bandwidth of internet-based services. Most existing local access networks are based on copper cable and offer a limited bandwidth to customers. A common approach to increase this bandwidth is to replace parts of the network by fiber-optic cable. This requires the installation of facilities, where the optical signal is transformed into an electrical one and vice versa. Several strategies are commonly used to deploy fiber-optic networks. Connecting each customer via a fiber-optic link is referred to as Fiber-to-the-Home. If there is a fiber-optic connection for every building this is commonly referred to as Fiber-to-the-Building. If a fiber-optic connection leads to each facility that serves an entire neighborhood, this is referred to as Fiber-to-the-Curb. In this thesis we propose mathematical optimization models for the cost-efficient design of local access networks based on fiber-optic cable. These models cover several aspects, including the Fiber-to-the-Curb strategy under additional reliability constraints, mixed Fiber-to-the-Home and Fiber-to-the-Curb strategies and capacity planning of links and facilities for Fiber-to-the-Curb networks. We provide a theoretical analysis of the proposed models and develop efficient solution algorithms. We use state-of-the-art methods from combinatorial optimization including polyhedral comparisons, reformulations on extended graphs, valid inequalities and branch-and-cut procedures

Models for planning the evolution of local telecommunication networks

Includes bibliographical references.Research initiated through a grant from GTE Laboratories, Inc. Supported in part by an AT&T research award. Supported in part by the Systems Theory and Operations Research Program of the National Science Foundation. ECS-8316224 Supported in part by ONR. N0000-14-86-0689A. Balakrishnan ... [et al.]

Models for planning the evolution of local telecommunication networks

Includes bibliographical references.Research initiated through a grant from GTE Laboratories, Inc. Supported in part by an AT&T research award. Supported in part by the Systems Theory and Operations Research Program of the National Science Foundation. ECS-8316224 Supported in part by ONR. N0000-14-86-0689A. Balakrishnan ... [et al.]

Optimal Trees

Not Availabl

ROLAND : a tool for the realistic optimisation of local access network design

Bibliography: p. 141-147.Investment in the local access network represents between 50% and 70% of capital investment of a telecommunications company. This thesis investigates algorithms that can be used to design economical access networks and presents ROLAND: a tool that incorporates several of these algorithms into an interactive environment. The software allows a network designer to explore different approaches to solving the problem, before adopting a particular one. The family of problems that are tackled by the algorithms included in ROLAND involve determining the most economical way of installing concentrators in an access network and connecting demand nodes such as distribution points to these concentrators. The Centre-of-Mass (COM) Algorithm identifies clusters of demand in the network and suggests good locations for concentrators to be installed. The problem of determining which concentrators in a set of potential sites to install is known as the concentrator location problem (CPL) and is an instance of the classical capacitated plant location problem. Linear programming techniques such as branch-and-bound can be used to find an optimal solution to this problem, but soon becomes infeasible as the network size increases. Some form of heuristic approach is needed, and ROLAND includes two such heuristics, namely the Add and Drop Heuristic. Determining the layout of multi-drop lines, which allow a number of demand nodes to share the same connection to a concentrator, is analogous to finding minimal spanning trees in a graph. Greedy approaches such as Kruskal's algorithm are not ideal however, and heuristics such as Esau-William's algorithm achieve better results. Kruskal's algorithm and Kershenbaum's Unified Algorithm (which encapsulates a number of heuristics) have been implemented and come bundled with ROLAND. ROLAND also includes an optimal terminal assignment algorithm for associating distribution points to concentrators. A description of ROLAND's architecture and GUI are provided. The graphical elements are kept separate from the algorithm implementations, and an interface class provides common data structures and routines for use by new algorithm implementations. A test data generator, able to create random or localized data, is also included. A new hybrid concentrator location algorithm, known as the Cluster-Add Heuristic is presented. The implementation of this algorithm is included in ROLAND, and demonstrates the ease with which new solution methods can be integrated into the tool's framework. Experimentation with the concentrator location algorithms is conducted to show the Cluster-Add Heuristic's relative performance