Benders decomposition for Network Design Covering Problems

We consider two covering variants of the network design problem. We are given a set of origin/destination (O/D) pairs and each such O/D pair is covered if there exists a path in the network from the origin to the destination whose length is not larger than a given threshold. In the first problem, called the maximal covering network design problem, one must determine a network that maximizes the total demand of the covered O/D pairs subject to a budget constraint on the design costs of the network. In the second problem, called the partial covering network design problem, the design cost is minimized while a lower bound is set on the total demand covered. After presenting formulations, we develop a Benders decomposition approach to solve the problems. Further, we consider two different stabilization methods to determine the Benders cuts as well as the addition of cut-set inequalities to the master problem. Computational experiments show the efficiency of these different aspects

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

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

Conception et optimisation d'un système de collecte d'énergie éolienne

Resumé : Cette thèse porte sur l'étude du problème de conception d'un réseau de collecte d'énergie éolienne. Celui-ci consiste à installer les capacités sur les liens d'un réseau électrique potentiel pour acheminer au moindre coût l'énergie produite par des éoliennes vers une sous-station de transformation. L'augmentation du nombre d'éoliennes installées dans un parc éolien nécessite la conception des réseaux étendus sur plusieurs kilomètres de lignes électriques et donc des investissements importants. Face à un tel enjeu financier, l'utilisation d'outils d'optimisation de conception de réseaux s'avère capitale. Le problème de conception d'un réseau de collecte d'énergie éolienne fait partie d'une grande classe de problèmes de conception de réseaux. Mise à part sa complexité héritée de cette classe, ce dernier est sujet à une contrainte technique importante à savoir la non-bifurcation de l'énergie. En effet, les concepteurs de réseau exigent que l'énergie produite par un groupe d'éoliennes et qui est combinée sur un lien donné reste combinée jusqu'à ce qu'elle atteigne la sous-station. Dans cette thèse, nous nous sommes concentrés sur deux variantes de ce problème. La première variante consiste à installer des câbles souterrains et des lignes de transmission sur les liens potentiels du réseau électrique tels qu'un seul choix de capacité est disponible. La deuxième variante qui est plus compliquée que la première consiste non seulement à déterminer les liens qui sont utilisés pour acheminer l'énergie vers la sous-station, mais également à choisir la capacité de chaque lien utilisé, ce choix devant être fait parmi un ensemble de possibilités fixé par les concepteurs. Tous ces choix doivent être faits en prenant en considération les pertes engendrées par le transfert de l'énergie dans ce réseau. Dans les deux variantes, nous supposons que l'emplacement des éoliennes est prédéfini et qu'il est possible d'installer plusieurs liens parallèles entre deux sommets du réseau. Face à la complexité de ce problème, nous proposons des approches exactes pour la résolution des deux variantes. Plus précisément, nous commençons d'abord par développer les formulations mathématiques appropriées. Nous nous orientons ensuite vers les techniques basées sur la théorie polyédrale qui, de nos jours, sont qualifiées comme très efficaces pour la résolution de problèmes complexes en optimisation combinatoire. Nous déterminons en particulier des inégalités valides et proposons des techniques pour la génération de plans coupants. Toutes les méthodes proposées sont testées sur des exemplaires concrets fournis par notre partenaire industriel. ---------- Abstract : The aim of this thesis is to study the wind farm collection network design problem which consists of choosing a subset of edges of a potential electrical network and assigning the right capacity to each link so that the energy produced by the wind turbines can be sent to a specific sub-station at minimum cost. Designing such an electrical network incures a significant cost when building a wind farm. Therefore, it will be useful to develop efficient tools to find an optimal design in reasonable computation time. The wind farm collection network design problem belongs to a large class of network design problems which is known to be $NP$-hard. In designing a collection network, one must take an important engineering constraint into account. The energy flowing through a link is unsplittable, i.e., once the energy units produced by wind turbines and bundled on a given link these units have to remain bundled until they reach the substation. In this thesis, we will focus on two versions of this problem. The first one consists of selecting edges to install underground cables and transmission lines such that there is only one capacity for each type of link (cable or transmission line). In the second version of the problem we are allowed to install different types of capacities on the edges of the electrical network. Furthermore, power losses due to energy dissipation are accounted for when calculating the objective function. We assume that the locations of the turbines are already known for the two versions of the problem and several parallel links may be installed between two endpoints. We propose models and algorithms for the two versions of the problem. Then we discuss several ways to strengthen our models by means of valid inequalities. We also develop procedures for generating cutting planes. Computational tests performed on instances provided by our industrial partner confirm the efficiency of the proposed methods

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

Robust Design of Single-Commodity Networks

The results in the present work were obtained in a collaboration with Eduardo Álvarez- Miranda, Valentina Cacchiani, Tim Dorneth, Michael Jünger, Frauke Liers, Andrea Lodi and Tiziano Parriani. The subject of this thesis is a robust network design problem, i.e., a problem of the type “dimension a network such that it has sufficient capacity in all likely scenarios.” In our case, we model the network with an undirected graph in which each scenario defines a supply or demand for each node. We say that a flow in the network is feasible for a scenario if it can balance out its supplies and demands. A scenario polytope B defines which scenarios are relevant. The task is now to find integer capacities that minimize the total installation costs while allowing for a feasible flow in each scenario. This problem is called Single-Commodity Robust Network Design Problem (sRND) and was introduced by Buchheim, Liers and Sanità (INOC 2011). The problem contains the Steiner Tree Problem (given an undirected graph and a terminal set, find a minimum cost subtree that connects all terminals) and therefore is N P-hard. The problem is also a natural extension of minimum cost flows. The network design literature treats the case that the scenario polytope B is given as the finite set of its extreme points (finite case) and that it is given as the feasible region of finitely many linear inequalities (polyhedral case). Both descriptions are equivalent, however, an efficient transformation is not possible in general. Buchheim, Liers and Sanità (INOC 2011) propose a Branch-and-Cut algorithm for the finite case. In this case, there exists a canonical problem formulation as a mixed integer linear program (MIP). It contains a set of flow variables for every scenario. Buchheim, Liers and Sanità enhance the formulation with general cutting planes that are called target cuts. The first part of the dissertation considers the problem variant where every scenario has exactly two terminal nodes. If the underlying network is a complete, unweighted graph, then this problem is the Network Synthesis Problem as defined by Chien (IBM Journal of R&D 1960). There exist polynomial time algorithms by Gomory and Hu (SIAM J. of Appl. Math 1961) and by Kabadi, Yan, Du and Nair (SIAM J. on Discr. Math.) for this special case. However, these algorithms are based on the fact that complete graphs are Hamiltonian. The result of this part is a similar algorithm for hypercube graphs that assumes a special distribution of the supplies and demands. These graphs are also Hamiltonian. The second part of the thesis discusses the structure of the polyhedron of feasible sRND solutions. Here, the first result is a new MIP-based capacity formulation for the sRND problem. The size of this formulation is independent of the number of extreme points of B and therefore, it is also suited for the polyhedral case. The formulation uses so-called cut-set inequalities that are known in similar form from other network design problems. By adapting a proof by Mattia (Computational Optimization and Applications 2013), we show that cut-set inequalities induce facets of the sRND polyhedron. To obtain a better linear programming relaxation of the capacity formulation, we interpret certain general mixed integer cuts as 3-partition inequalities and show that these inequalities induce facets as well. The capacity formulation has exponential size and we therefore need a separation algorithm for cut-set inequalities. In the finite case, we reduce the cut-set separation problem to a minimum cut problem that can be solved in polynomial time. In the polyhedral case, however, the separation problem is N P-hard, even if we assume that the scenario polytope is basically a cube. Such a scenario polytope is called Hose polytope. Nonetheless, we can solve the separation problem in practice: We show a MIP based separation procedure for the Hose scenario polytope. Additionally, the thesis presents two separation methods for 3-partition inequalities. These methods are independent of the encoding of the scenario polytope. Additionally, we present several rounding heuristics. The result is a Branch-and-Cut algorithm for the capacity formulation. We analyze the algorithm in the last part of the thesis. There, we show experimentally that the algorithm works in practice, both in the finite and in the polyhedral case. As a reference point, we use a CPLEX implementation of the flow based formulation and the computational results by Buchheim, Liers and Sanità. Our experiments show that the new Branch-and-Cut algorithm is an improvement over the existing approach. Here, the algorithm excels on problem instances with many scenarios. In particular, we can show that the MIP separation of the cut-set inequalities is practical

Algorithms for vehicle routing problems with heterogeneous fleet, flexible time windows and stochastic travel times

Orientador: Vinícius Amaral ArmentanoTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Este trabalho aborda três variantes multiatributo do problema de roteamento de veículos. A primeira apresenta frota heterogênea, janelas de tempo invioláveis e tempos de viagem determinísticos. Para resolvê-la, são propostos algoritmos ótimos baseados na decomposição de Benders. Estes algoritmos exploram a estrutura do problema em uma formulação de programação inteira mista, e três diferentes técnicas são desenvolvidas para acelerá-los. A segunda variante contempla os atributos de frota heterogênea, janelas de tempo flexíveis e tempos de viagem determinísticos. As janelas de tempo flexíveis permitem o início do serviço nos clientes com antecipação ou atraso limitados em relação às janelas de tempo invioláveis, com custos de penalidade. Este problema é resolvido por extensões dos algoritmos de Benders, que incluem novos algoritmos de programação dinâmica para a resolução de subproblemas com a estrutura do problema do caixeiro viajante com janelas de tempo flexíveis. A terceira variante apresenta frota heterogênea, janelas de tempo flexíveis e tempos de viagem estocásticos, sendo representada por uma formulação de programação estocástica inteira mista de dois estágios com recurso. Os tempos de viagem estocásticos são aproximados por um conjunto finito de cenários, gerados por um algoritmo que os descreve por meio da distribuição de probabilidade Burr tipo XII, e uma matheurística de busca local granular é sugerida para a resolução do problema. Extensivos testes computacionais são realizados em instâncias da literatura, e as vantagens das janelas de tempo flexíveis e dos tempos de viagem estocásticos são enfatizadasAbstract: This work addresses three multi-attribute variants of the vehicle routing problem. The first one presents a heterogeneous fleet, hard time windows and deterministic travel times. To solve this problem, optimal algorithms based on the Benders decomposition are proposed. Such algorithms exploit the structure of the problem in a mixed-integer programming formulation, and three algorithmic enhancements are developed to accelerate them. The second variant comprises a heterogeneous fleet, flexible time windows and deterministic travel times. The flexible time windows allow limited early and late servicing at customers with respect to their hard time windows, at the expense of penalty costs. This problem is solved by extensions of the Benders algorithms, which include novel dynamic programming algorithms for the subproblems with the special structure of the traveling salesman problem with flexible time windows. The third variant presents a heterogeneous fleet, flexible time windows and stochastic travel times, and is represented by a two-stage stochastic mixed-integer programming formulation with recourse. The stochastic travel times are approximated by a finite set of scenarios generated by an algorithm which describes them using the Burr type XII distribution, and a granular local search matheuristic is suggested to solve the problem. Extensive computational tests are performed on instances from the literature, and the advantages of flexible windows and stochastic travel times are stressed.DoutoradoAutomaçãoDoutor em Engenharia Elétrica141064/2015-3CNP