Le problème de tournées de véhicules est un des problèmes d’optimisation combinatoire les plus connus et les plus difficiles. Il s’agit de déterminer les tournées optimales pour une flotte de véhicules afin de servir un ensemble donné de clients. Dans les problèmes classiques de transport, chaque client est normalement servi à partir d’un seul nœud (ou arc). Pour cela, on définit toujours un ensemble donné de nœuds (ou arcs) obligatoires à visiter ou traverser, et on recherche la solution à partir de cet ensemble de nœuds (ou arcs). Mais dans plusieurs applications réelles où un client peut être servi à partir de plus d’un nœud, (ou arc), les problèmes généralisés qui en résultent sont plus complexes. Le but principal de cette thèse est d’étudier trois problèmes généralisés de tournées de véhicules. Le premier problème de la tournée sur arcs suffisamment proche (CEARP), comporte une application réelle intéressante en routage pour le relevé des compteurs à distance ; les deux autres problèmes, problème de tournées couvrantes multi-véhicules (mCTP) et problème généralisé de tournées sur nœuds (GVRP), permettent de modéliser des problèmes de conception des réseaux de transport à deux niveaux. Pour résoudre ces problèmes, nous proposons une approche exacte ainsi que des métaheuristiques. Pour développer la méthode exacte, nous formulons chaque problème comme un programme mathématique, puis nous construisons des algorithmes de type branchement et coupes. Les métaheuristiques sont basées sur le ELS (ou Evolutionary Local Search) et sur le GRASP (ou Greedy Randomized Adaptive Search Procedure). De nombreuses expérimentations montrent la performance de nos méthodes. / The Routing Problem is one of the most popular and challenging combinatorial optimization problems. It involves finding the optimal set of routes for fleet of vehicles in order to serve a given set of customers. In the classic transportation problems, each customer is normally served by only one node (or arc). Therefore, there is always a given set of required nodes (or arcs) that have to be visited or traversed, and we just need to find the solution from this set of nodes (or arcs). But in many real applications where a customer can be served by from more than one node (or arc), the generalized resulting problems are more complex. The primary goal of this thesis is to study three generalized routing problems. The first one, the Close-Enough Arc Routing Problem(CEARP), has an interesting real-life application to routing for meter reading while the others two, the multi-vehicle Covering Tour Problem (mCTP) and the Generalized Vehicle Routing Problem(GVRP), can model problems concerned with the design of bilevel transportation networks. The problems are solved by exact methods as well as metaheuristics. To develop exact methods, we formulate each problem as a mathematical program, and then develop branch-and-cut algorithms. The metaheuristics are based on the evolutionary local search (ELS) method et on the greedy randomized adaptive search procedure (GRASP) method. The extensive computational experiments show the performance of our methods.

Dans cette thèse, nous nous intéressons à un problème de fiabilité dans les réseaux multicouches IP-sur-WDM. Etant donné un ensemble de demandes pour lesquelles on connaît une topologie fiable dans la couche IP, le problème consiste à sécuriser la couche optique WDM en y cherchant une topologie fiable. Nous montrons que le problème est NP-complet même dans le cas d'une seule demande. Ensuite, nous proposons quatre formulations en termes de programmes linéaires en nombres entiers pour le problème. La première est basée sur les contraintes de coupes. Nous considérons le polyèdre associé. Nous identifions de nouvelles familles de contraintes valides et étudions leur aspect facial. Nous proposons également des algorithmes de séparation pour ces contraintes. En utilisant ces résultats, nous développons un algorithme de coupes et branchements pour le problème et présentons une étude expérimentale. La deuxième formulation utilise comme variables des chemins entre des terminaux dans le graphe sous-jacent. Un algorithme de branchements et génération de colonnes est proposé pour cette formulation. Par la suite, nous discutons d'une formulation dite naturelle utilisant uniquement les variables de design. Enfin, nous présentons une formulation étendue compacte qui, en plus des variables naturelles, utilise des variables de routage. Nous montrons que cette formulation fournit une meilleure borne inférieure. / This thesis deals with a problem related to survivability issues in multilayer IP-over-WDM networks. Given a set of traffic demands for which we know a survivable logical routing in the IP layer, the aim is determine the corresponding survivable topology in the WDM layer. We show that the problem is NP-hard even for a single demand. Moreover, we propose four integer linear programming formulations for the problem. The first one is based on the so-called cut inequalities. We consider the polyhedron associated with the formulation. We identify several families of valid inequalities and discuss their facial aspect. We also develop separation routines. Using this, we devise a Branch-and-Cut algorithm and present experimental results. The second formulation uses paths between terminals of the underlying graph as variables. We devise a Branch-and-Price algorithm based on that formulation. In addition, we investigate a natural formulation for the problem which uses only the design variables.  Finally, we propose an extended compact formulation which, in addition to the design variables, uses routing variables. We show that this formulation provides a tighter bound for the problem.

The elimination of distribution inefficiencies, occurring due to the timing of customers' orders is an important reason for companies to introduce vendor managed inventory programs. By managing their customers' inventories, suppliers may be able to reduce demand variability and therefore distribution costs. We develop technology to measure the effectiveness of distribution strategies. We develop a methodology that allows the computation of tight lower bounds on the total mileage required to satisfy customer demand over a period of time. As a result, companies will be able to gain insight into the effectiveness of their distribution strategy. This technology can also be used to suggest desirable delivery patterns and to analyze tactical and strategic decisions. Secondly, we study the inventory routing problem with continuous moves (IRP-CM). The typical inventory routing problem deals with the repeated distribution of a single product, from a single facility, with an unlimited supply, to a set of customers that can all be reached with out-and-back trips. Unfortunately, this is not always the reality. We introduce the IRP-CM to study two important real-life complexities: limited product availabilities at facilities and customers that cannot be served using out-and-back tours. We need to design delivery tours spanning several days, covering huge geographic areas, and involving product pickups at different facilities. We develop a heuristic and an optimization algorithm to construct distribution plans. The heuristic is an innovative randomized greedy algorithm, which includes linear programming based postprocessing technology. To solve the IRP-CM to optimality, we give a time-discretized integer programming model and develop a branch-and-cut algorithm. As instances of time-discretized models tend to be large we discuss several possibilities for reducing the problem size. We introduce a set of valid inequalities, called delivery cover inequalities, in order to tighten the bounds given by the LP relaxation of the time-discretized model. We also introduce branching schemes exploiting the underlying structure of the IRP-CM. An extensive computational study demonstrates the effectiveness of the optimization algorithm. Finally, we present an integrated approach using heuristics and optimization algorithms providing effective and efficient technology for solving inventory problems with continuous moves.

La connaissance inter-entreprises permet à chaque société de se renseigner sur ses clients, ses fournisseurs et de développer son activité tout en limitant le risque lié à la solvabilité ou retard de paiement de ses partenaires. Avec les tensions de trésorerie, la nécessité de la croissance et l'augmentation de la concurrence, ce domaine devient plus que jamais stratégique aussi bien pour les PME que pour les grands groupes. La quantité de données traitée dans ce domaine, les exigences de qualité et de fraîcheur, la nécessité de croiser ces données pour déduire des nouvelles informations et indicateurs, posent plusieurs problèmes pour lesquels l'optimisation en général et l'optimisation combinatoire en particulier peuvent apporter des solutions efficaces. Dans cette thèse, nous utilisons l'optimisation combinatoire, l'algorithmique du texte et la théorie des graphes pour résoudre efficacement des problèmes issus du domaine de la connaissance inter-entreprises et posés par Altares D&B. Dans un premier temps, nous nous intéressons à la qualité de la base de données des dirigeants. Ce problème combine la détection et suppression des doublons dans une base de données et la détection d'erreurs dans une chaîne de caractères. Nous proposons une méthode de résolution basée sur la normalisation des données et l'algorithmique de texte et de comparaison syntaxique entre deux chaînes de caractères. Les résultats expérimentaux montrent non seulement que cette méthode est pertinente dans la détection et la suppression des doublons mais aussi qu'elle est efficace de point du vue temps de traitement. Nous nous focalisons par la suite sur les données des liens capitalistiques et nous considérons le problème de calcul des liens indirects et l'identification des têtes des groupes. Nous présentons une méthode de résolution basée sur la théorie des graphes. Nous testons cette méthode sur plusieurs instances réelles. Nous prouvons l'efficacité de cette méthode par son temps de traitement et par l'espace de calcul qu'elle utilise. Enfin, nous remarquons que le temps de calcul de celui-ci augmente de façon logarithmique en fonction de la taille d'instance. Enfin, nous considérons le problème de l'identification des réseaux d'influence. Nous formalisons ce problème en termes de graphes et nous le ramenons à un problème de partitionnement de graphe qui est NP-difficile dans ce cas général. Nous proposons alors une formulation en programme linéaire en nombre entier pour ce problème. Nous étudions le polyèdre associé et décrivons plusieurs classes de contraintes valides. Nous donnons des conditions nécessaires pour que ces contraintes définissent des facettes et discutons des algorithmes de séparations de ces contraintes. En utilisant les résultats polyédraux obtenus, nous développons un algorithme de coupes et branchements. Enfin, nous donnons quelques résultats expérimentaux qui montrent l'efficacité de notre algorithme de coupes et branchements / The inter-companies knowledge allows to every partner to learn about its customers, its suppliers and to develop its activity. Also this permits to limit the risk related to the creditworthiness, or the late payment of its partners. With the cash flow pressures, the need for growth and increased competition, this area becomes more strategic than ever, for both small (PME) and large groups. The amount of data processed in this domain, the requirements of quality and freshness, the need to cross these data to obtain new information and indicators, yield several optimization problems for which the recent techniques and computational tools can bring effective solutions. In this thesis, we use combinatorial optimization, text algorithms as well as graph theory to solve efficiently problems arising in the field of inter-companies knowledge. In particular, such problems was encountered in Altares D&B. First, we focus on the quality of the managers database. This problem combines the detection and removal of duplicates in a database, as well as the error detection in a string. We propose a method for solving this problem, based on data normalization, text algorithms and syntactic comparison between two strings. Our experimental results show that this method is relevant for the detection and removal of duplicates, and it is also very efficient in terms of processing time. In a second part of the thesis, we address a problem related to the data of ownership links. We compute the indirect links, and identify the group heads. We propose a method for solving this problem using graph theory and combinatorial optimization. We then perform a set of experiments on several real-world instances. The computational results show the effectiveness of our method in terms of CPU-time and resource allocation. In fact, the CPU time for computation increases logarithmically with the size of the instances. Finally, we consider the problem of identifying influence networks. We give a description of this problem in terms of graphs, and show that it can reduce to a graph partitioning problem. The latter is NP-hard. We then propose an integer linear programming formulation to model the problem. We investigate the associated polyhedron and describe several classes of valid inequalities. We give some necessaryand sufficient conditions for these inequalities to define facets of the considered polyhedron, and we discuss the related separation problems. Based on the obtained polyhedral results, we devise a Branch-and-Cut algorithm to solve the problem. Some numerical results are presented to show the efficiency of our algorithm.

L’Optimisation Combinatoire (OC) est un domaine de recherche qui est en perpétuel changement. Résoudre un problème d’optimisation combinatoire (POC) consiste essentiellement à trouver la ou les meilleures solutions dans un ensemble des solutions réalisables appelé espace de recherche qui est généralement de cardinalité exponentielle en la taille du problème. Pour résoudre des POC, plusieurs méthodes ont été proposées dans la littérature. On distingue principalement les méthodes exactes et les méthodes d’approximation. Ne pouvant pas viser une résolution exacte de problèmes NP-Complets lorsque la taille du problème dépasse une certain seuil, les chercheurs on eu de plus en plus recours, depuis quelques décennies, aux algorithmes dits hybrides (AH) ou encore à au calcul parallèle. Dans cette thèse, nous considérons la classe POC des problèmes de conception d'un réseau fiable. Nous présentons un algorithme hybride parallèle d'approximation basé sur un algorithme glouton, un algorithme de relaxation Lagrangienne et un algorithme génétique, qui produit des bornes inférieure et supérieure pour les formulations à base de flows. Afin de valider l'approche proposée, une série d'expérimentations est menée sur plusieurs applications: le Problème de conception d'un réseau k-arête-connexe avec contrainte de borne (kHNDP) avec L=2,3, le problème de conception d'un réseau fiable Steiner k-arête-connexe (SkESNDP) et ensuite deux problèmes plus généraux, à savoir le kHNDP avec L >= 2 et le problème de conception d'un réseau fiable k-arête-connexe (kESNDP). L'étude expérimentale de la parallélisation est présentée après cela. Dans la dernière partie de ce travail, nous présentons deux algorithmes parallèles exactes: un Branch-and-Bound distribué et un Branch-and-Cut distribué. Une série d'expérimentation a été menée sur une grappe de 128 processeurs, et des accélération intéressantes ont été atteintes pour la résolution du problèmes kHNDP avec k=3 et L=3. / Combinatorial Optimization (CO) is an area of research that is in a constant progress. Solving a Combinatorial Optimization Problem (COP) consists essentially in finding the best solution (s) in a set of feasible solutions called a search space that is usually exponential in cardinality in the size of the problem. To solve COPs, several methods have been proposed in the literature. A distinction is made mainly between exact methods and approximation methods. Since it is not possible to aim for an exact resolution of NP-Complete problems when the size of the problem exceeds a certain threshold, researchers have increasingly used Hybrid (HA) or parallel computing algorithms in recent decades. In this thesis we consider the COP class of Survivability Network Design Problems. We present an approximation parallel hybrid algorithm based on a greedy algorithm, a Lagrangian relaxation algorithm and a genetic algorithm which produces both lower and upper bounds for flow-based formulations. In order to validate the proposed approach, a series of experiments is carried out on several applications: the k-Edge-Connected Hop-Constrained Network Design Problem (kHNDP) when L = 2,3, The problem of the Steiner k-Edge-Connected Network Design Problem (SkESNDP) and then, two more general problems namely the kHNDP when L >= 2 and the k-Edge-Connected Network Design Problem (kESNDP). The experimental study of the parallelisation is presented after that. In the last part of this work, we present a two parallel exact algorithms: a distributed Branch-and-Bound and a distributed Branch-and-Cut. A series of experiments has been made on a cluster of 128 processors and interesting speedups has been reached in kHNDP resolution when k=3 and L=3.

Die Tourenbildung beschäftigt sich mit der Konstruktion kostengünstiger Transportrouten zur Belieferung von Verbrauchern. Sie ist eine der weitreichensten Erfolgsgeschichten des Operations Research. Das starke Interesse an diesen Problemen durch Industrie und Forschung liegt zum einen am wirtschaftlichen Potenzial der Tourenbildung und -optimierung, zum anderen macht ihr Reichtum an Struktur sie zu einem faszinierenden Forschungsgebiet. In der vorliegenden Arbeit soll ein Überblick über einige, u. a. auch neuere mathematische Modell- und Lösungsansätze gegeben werden. Auf Grund der hohen Anzahl der Veröffentlichungen auf diesem Gebiet wird nicht zwingend ein Anspruch auf die vollständige Darlegung aller möglichen Problemstellungen im Zusammenhang mit dem TSP sowie dem VRP und deren Lösungsansätze erhoben. An den gegebenen Stellen wird statt dessen auf weiterführende Literatur verwiesen.

The work presented in this dissertation has been focused on exploiting the branch-and-price (BNP) method for the solution of various stochastic mixed integer programming problems (MIPs). In particular, we address the stochastic generalized assignment problem (SGAP), a hospital staff scheduling problem (HSSP), a stochastic hospital staff scheduling problem (SHSSP), and a stochastic short-term personnel planning problem (SSTPP). The BNP method has been developed in concert with the dual stabilization technique and other enhancements of this method for each of these problems. In view of an excessive number of scenarios that arise for these problems, we also implement the Monte Carlo method within the BNP scheme. The superiority of the BNP-based method over the branch-and-cut (BNC) method is demonstrated for all of these problems. The first problem that we address is the SGAP for which the processing time of a job on a machine is assumed to be stochastic. Even though the generalized assignment problem (GAP) has been solved using the BNP method, yet no study has been reported in the literature on the use of the BNP method for the solution of the SGAP. Our work has been motivated by the desire to fill this gap. We begin by showing that it is better to solve the SGAP as a stochastic program in contrast to solving it by using the expected values of the times required to process the jobs on the machines. Then, we show that the stochastic model of the SGAP is a complete recourse model — a useful property which permits the first stage decisions to produce feasible solutions for the recourse problems. We develop three BNP-based methods for the solution of the SGAP. The first of these is BNP-SGAP, which is a combination of branch-and-bound and column generation methods. The pricing problem of BNP-SGAP is separable with regard to each machine, and it is a multiple-constraint knapsack problem. The second method is BNP-SGAP implemented in concert with the dual stabilization technique (DST), and it is designated as BNPDST-SGAP. We have introduced a new DST by modifying the Boxstep method of Pigatti et al. [76]. We have shown that our method performs better than the method of Pigatti et al. [76] resulting in over two-fold savings in cpu times on average. The third method that we develop for the solution of the SGAP is BNPDST-SGAP implemented with an advanced start to obtain an initial feasible solution. We use a greedy heuristic to obtain this solution, and this heuristic is a modification of a similar method used for the knapsack problem. It relies on the information available at a node of the underlying branch-and-bound tree. We have shown that this procedure obtains an initial feasible solution, if it exists at that node. We designate this method as BNPDSTKP-SGAP. We have also developed a BNC method to solve the SGAP using CPLEX 9.0. We have compared the performances of the BNP and BNC methods on various problem instances obtained by varying the number of machines, the ratio of the number of machines to the number of jobs, the machine capacity, and the penalty cost per unit of extra resource required at each machine. Our results show that all BNP-based methods perform better than the BNC method, with the best performance obtained for BNPDSTKP-SGAP. An issue with the use of the scenario-based methods that we have employed for the solution of the SGAP is that the number of scenarios generally grows exponentially in problem parameters, which gives rise to a large-size problem. To overcome the complexity caused by the presence of a large number of scenarios for the solution of the SGAP, we introduce the use of the Monte Carlo method (MCM) within the BNP scheme. We designate this method as BNPDSTKP-SGAP with MCM. It affords the use of a small subset of scenarios at a time to estimate the "true" optimal objective function value. Replications of the subsets of scenarios are carried out until the objective function value satisfies a stopping criterion. We have established theoretical results for the use of the MCM. These pertain to determining unbiased estimates of: (i) lower and upper bounds of the "true" optimal objective function value, (ii) the "true" optimal solution, and (iii) the optimality gap. We have also provided the 100(1-ï ¡) confidence interval on the optimality gap. Our experimental investigation has shown the efficacy of using this method. It obtains almost optimal solutions, with the objective function value lying within 5% of the "true" optimal objective function value, while giving almost ten-fold savings in cpu time. Our experimentation has also revealed that an increment in the number of scenarios in each replication makes a greater impact on the quality of the solution obtained than an increment in the number of replications. We have also observed the impact of a change in the variance of a processing time distribution on cpu time. As expected, the optimal objective function value increases with increment in processing time variability. Also, by comparing the results with the expected value solution, it is observed that the greater the variability in the data, the better it is to use the stochastic program. The second problem that we study is the hospital staff scheduling problem. We address the following three versions of this problem: HSSP (General): Implementation of schedule incorporating the four principal elements, namely, surgeons, operations, operating rooms, and operation times; HSSP (Priority): Inclusion of priority for some surgeons over the other surgeons regarding the use of the facility in HSSP (General); HSSP (Pre-arranged): Implementation of a completely pre-fixed schedule for some surgeons. The consideration of priority among the surgeons mimics the reality. Our BNP method for the solution of these problems is similar to that for the SGAP except for the following: (i) a feasible solution at a node is obtained with no additional assignment, i.e., it consists of the assignments made in the preceding nodes of that node in the branch-and-bound tree; (ii) the columns with positive reduced cost are candidates for augmentation in the CGM; and (iii) a new branching variable selection strategy is introduced, which selects a fractional variable as a branching variable by fixing a value of which we enforce the largest number of variables to either 0 or 1. The priority problem is separable in surgeons. The results of our experimentation have shown the efficacy of using the BNP-based method for the solution of each HSSP as it takes advantage of the inherent structure of each of these problems. We have also compared their performances with that of the BNC method developed using CPLEX. For the formulations HSSP (General), HSSP (Priority), and HSSP (Pre-arranged), the BNP method gives better results for 22 out of 30, 29 out of 34, and 20 out 32 experiments over the BNC method, respectively. Furthermore, while the BNC method fails to obtain an optimal solution for 15 experiments, the BNP method obtains optimal solutions for all 96 experiments conducted. Thus, the BNP method consistently outperforms the BNC method for all of these problems. The third problem that we have investigated in this study is the stochastic version of the HSSP, designated as the Stochastic HSSP (SHSSP), in which the operation times are assumed to be stochastic. We have introduced a formulation for this formulation, designated as SHSSP2 (General), which allows for overlapping of schedules for surgeons and operating rooms, and also, allows for an assignment of a surgeon to perform an operation that takes less than a pre-arranged operation time, but all incurring appropriate penalty costs. A comparison of the solution of SHSSP2 (General) and its value with those obtained by using expected values (the corresponding problem is designated as Expected-SHSSP2 (General)) reveals that Expected-SHSSP2 (General) may end up with inferior and infeasible schedules. We show that the recourse model for SHSSP2 (General) is a relatively complete recourse model. Consequently, we use the Monte Carlo method (MCM) to reduce the complexity of solving SHSSP2 (General) by considering fewer scenarios. We employ the branch-and-cut (BNC) method in concert with the MCM for solving SHSSP2 (General). The solution obtained is evaluated using tolerance ratio, closeness to optimality, length of confidence interval, and cpu time. The MCM substantially reduces computational effort while producing almost optimal solutions and small confidence intervals. We have also considered a special case of SHSSP2 (General), which considers no overlapping schedules for surgeons and operating rooms and assigns exactly the same operation time for each assignment under each scenario, and designate it as SHSSP2 (Special). With this, we consider another formulation that relies on the longest operation time among all scenarios for each assignment of a surgeon to an operation in order to avoid scheduling conflicts, and we designate this problem as SHSSP (Longest). We show SHSSP (Longest) to be equivalent to deterministic HSSP, designated as HSSP (Equivalent), and we further prove it to be equivalent to SHSSP (General) in terms of the optimal objective function value and the optimal assignments of operations to surgeons. The schedule produced by HSSP (Equivalent) does not allow any overlap among the operations performed in an operating room. That is, a new operation cannot be performed if a previous operation scheduled in that room takes longer than expected. However, the schedule generated by HSSP (Equivalent) may turn out to be a conservative one, and may end up with voids due to unused resources in case an operation in an operating room is completed earlier than the longest time allowed. Nevertheless, the schedule is still a feasible one. In such a case, the schedule can be left-shifted, if possible, because the scenarios are now revealed. Moreover, such voids could be used to perform other procedures (e.g., emergency operations) that have not been considered within the scope of the SHSSP addressed here. Besides, such a schedule can provide useful guidelines to plan for resources ahead of time. The fourth problem that we have addressed in this dissertation is the stochastic short-term personnel planning problem, designated as Stochastic STPP (SSTPP). This problem arises due to the need for finding appropriate temporary contractors (workers) to perform requisite jobs. We incorporate uncertainty in processing time or amount of resource required by a contractor to perform a job. Contrary to the SGAP, the recourse model for this problem is not a relatively complete recourse model. As a result, we cannot employ a MCM method for the solution of this problem as it may give rise to an infeasible solution. The BNP method for the SSTPP employs the DST and the advanced start procedure developed for the SGAP, and due to extra constraints and presence of binary decision variables, we use the branching variable selection strategy developed for the HSSP models. Because of the distinctive properties of the SSTPP, we have introduced a new node selection strategy. We have compared the performances of the BNC-based and BNP-based methods based on the cpu time required. The BNP method outperforms the BNC method in 75% of the experiments conducted, and the BNP method is found to be quite stable with smaller variance in cpu times than those for the BNC method. It affords solution of difficult problems in smaller cpu times than those required for the BNC method. / Ph. D.

In this thesis, we consider the k-edge connected L-hop-constrained network design problem. Given a weighted graph G=(N,E), a set D of pairs of terminal nodes, and two integers k,L > 1, it consists in finding in G the minimum cost subgraph containing at least k edge-disjoint paths of at most L edges between each pair in D. This problem is of great interest in today's telecommunication industry, where highly survivable networks need to be constructed.<p><p>We first study the particular case where the set of demands D is reduced to a single pair {s,t}. We propose an integer programming formulation for the problem, which consists in the st-cut and trivial inequalities, along with the so-called L-st-path-cut inequalities. We show that these three classes of inequalities completely describe the associated polytope when k=2 and L=2 or 3, and give necessary and sufficient conditions for them to be facet-defining. We also consider the dominant of the associated polytope, and discuss how the previous inequalities can be separated in polynomial time.<p><p>We then extend the complete and minimal description obtained above to any number k of required edge-disjoint L-st-paths, but when L=2 only. We devise a cutting plane algorithm to solve the problem, using the previous polynomial separations, and present some computational results.<p><p>After that, we consider the case where there is more than one demand in D. We first show that the problem is strongly NP-hard, for all L fixed, even when all the demands in D have one root node in common. For k=2 and L=2,3, we give an integer programming formulation, based on the previous constraints written for all pairs {s,t} in D. We then proceed by giving several new classes of facet-defining inequalities, valid for the problem in general, but more adapted to the rooted case. We propose separation procedures for these inequalities, which are embedded within a Branch-and-Cut algorithm to solve the problem when L=2,3. Extensive computational results from it are given and analyzed for both random and real instances.<p><p>Since those results appear less satisfactory in the case of arbitrary demands (non necessarily rooted), we present additional families of valid inequalites in that situation. Again, separation procedures are devised for them, and added to our previous Branch-and-Cut algorithm, in order to see the practical improvement granted by them.<p><p>Finally, we study the problem for greater values of L. In particular, when L=4, we propose new families of constraints for the problem of finding a subgraph that contains at least two L-st-paths either node-disjoint, or edge-disjoint. Using these, we obtain an integer programming formulation in the space of the design variables for each case.<p><p>------------------------------------------------<p><p>Dans cette thèse, nous considérons le problème de conception de réseau k-arete connexe à chemins L-bornés. Etant donné un graphe pondéré G=(N,E), un ensemble D de paires de noeuds terminaux, et deux entiers k,L > 1, ce problème consiste à trouver, dans G, un sous-graphe de cout minimum tel que, entre chaque paire dans D, il existe au moins k chemins arete-disjoints de longueur au plus L. Ce problème est d'un grand intéret dans l'industrie des télécommunications, où des réseaux hautement fiables doivent etre construits.<p><p>Nous étudions tout d'abord le cas particulier où l'ensemble des demandes D est réduit à une seule paire de noeuds. Nous proposons une formulation du problème sous forme de programme linéaire en nombres entiers, laquelle consiste en les inégalités triviales et de coupe, ainsi que les inégalités dites de L-chemin-coupe. Nous montrons que ces trois types d'inégalités décrivent complètement le polytope associé lorsque k=2 et L=2,3, et donnons des conditions nécessaires et suffisantes pour que celles-ci en définissent des facettes. Nous considérons également le dominant du polytope associé et discutons de la séparation polynomiale des trois classes précédentes.<p><p>Nous étendons alors cette description complète et minimale à tout nombre k de chemins arete-disjoints de longueur au plus 2. De plus, nous proposons un algorithme de plans coupants utilisant les précédentes séparations polynomiales, et en présentons quelques résultats calculatoires, pour tout k>1 et L=2,3.<p><p>Nous considérons ensuite le cas où plusieurs demandes se trouvent dans D. Nous montrons d'abord que le problème est fortement NP-dur, pour tout L fixé et ce, meme si les demandes sont toutes enracinées en un noeud. Pour k=2 et L=2,3, nous donnons une formulation du problème sous forme de programme linéaire en nombres entiers. Nous proposons également de nouvelles classes d'inégalités valides, pour lesquelles nous réalisons une étude faciale. Celles-ci sont alors séparées dans le cadre d'un algorithme de coupes et branchements pour résoudre des instances aléatoires et réelles du problème.<p><p>Enfin, nous étudions le problème pour de plus grandes valeurs de L. En particulier, lorsque L=4, nous donnons de nouvelles familles de contraintes pour le problème consistant à déterminer un sous-graphe contenant entre deux noeuds fixés au moins deux chemins de longueur au plus 4, que ceux-ci doivent etre arete-disjoints ou noeud-disjoints. Grace à ces dernières, nous parvenons à donner une formulation naturelle du problème dans chacun de ces deux cas. <p> / Doctorat en sciences, Spécialisation Informatique / info:eu-repo/semantics/nonPublished