Étant donné un graphe G = (V U T, E), tel que V U T représente l'ensemble des sommets où T est un ensemble de terminaux, et une fonction poids w associée aux sommets non terminaux, le problème du séparateur de poids minimum consiste à partitionner V U T en k + 1 sous-ensembles {S, V1,..., Vk} tel qu'il n'y a aucune arête entre deux sous-ensembles différents Vi et Vj, chaque Vi contient exactement un terminal et le poids de S est minimum. Dans cette thèse, nous étudions le problème d'un point de vue polyèdral. Nous donnons deux formulations en nombres entiers pour le problème, et pour une de ces formulations, nous étudions le polyèdre associé. Nous présentons plusieurs inégalités valides, et décrivons des conditions de facette. En utilisant ces résultats, nous développons un algorithme de coupes et branchement pour le problème. Nous étudions également le polytope des séparateurs dans les graphes décomposables par sommets d'articulation. Si G est un graphe qui se décompose en G1 et G2, alors nous montrons que le polytope des séparateurs dans G peut être décrit à partir de deux systèmes linéaires liés à G1 et G2. Ceci donne lieu à une technique permettant de caractériser le polytope des séparateurs dans les graphes qui sont récursivement décomposables. Ensuite, nous étudions des formulations étendues pour le problème et proposons des algorithmes de génération de colonnes et branchement ainsi que des algorithmes de génération de colonnes, de coupes et branchement. Pour chaque formulation, nous présentons un algorithme de génération de colonnes, une procedure pour le calcul de la borne duale ainsi qu'une règle de branchement. De plus, nous présentons quatre variantes du problème du séparateur. Nous montrons que celles-ci sont NP-difficiles, et pour chacune d'elles nous donnons une formulation en nombres entiers et présentons certaines classes d'inégalités valides. / Given a graph G = (V U T, E) with V U T the set of vertices, where T is a set of terminals, and a weight function w, associated with the nonterminal nodes, the multi-terminal vertex separator problem consists in partitioning V U T into k + 1 subsets {S, V1,..., Vk} such that there is no edge between two different subsets Vi and Vj, each Vi contains exactly one terminal and the weight of S is minimum. In this thesis, we consider the problem from a polyhedral point of view. We give two integer programming formulations for the problem, for one of them, we investigate the related polyhedron. We describe some valid inequalities and characterize when these inequalities define facets. Using these results, we develop a Branch-and-Cut algorithm for the problem. We also study the multi-terminal vertex separator polytope in the graphs decomposable by one node cutsets. If G is a graph that decomposes into G1 and G2, we show that the multi-terminal vertex separator polytope in G can be described from two linear systems related to G1 and G2. This gives rise to a technique for characterizing the multi-terminal vertex separator polytope in the graphs that are recursively decomposable. Moreover, we propose three extended formulations for the problem and derive Branch-and-Price and Branch-and-Cut-and-Price algorithms. For each formulation we present a column generation scheme, the way to compute the dual bound, and the branching scheme. Finally, we discuss four variants of the multi-terminal vertex separator problem. We show that all these variants are NP-hard and for each one we give an integer programming formulation and present some class of valid inequalities.

Dans un graphe non orienté, le Problème de Coloration de Graphe (PCG) consiste à assigner à chaque sommet du graphe une couleur de telle sorte qu'aucune paire de sommets adjacents n'aient la même couleur et le nombre total de couleurs est minimisé. DSATUR est un algorithme exact efficace pour résoudre le PCG. Un de ses défauts est qu'une borne inférieure est calculée une seule fois au noeud racine de l'algorithme de branchement, et n'est jamais mise à jour. Notre nouvelle version de DSATUR surpasse l'état de l'art pour un ensemble d'instances aléatoires à haute densité, augmentant significativement la taille des instances résolues. Nous étudions trois formulations PLNE pour le Problème de la Somme Chromatique Minimale (PSCM). Chaque couleur est représentée par un entier naturel. Le PSCM cherche à minimiser la somme des cardinalités des sous-ensembles des sommets recevant la même couleur, pondérés par l'entier correspondant à la couleur, de telle sorte que toute paire de sommets adjacents reçoive des couleurs différentes. Nous nous concentrons sur l'étude d'une formulation étendue et proposons un algorithme de Branch-and-Price. / Given an undirected graph, the Vertex Coloring Problem (VCP) consists of assigning a color to each vertex of the graph such that two adjacent vertices do not share the same color and the total number of colors is minimized. DSATUR is an effective exact algorithm for the VCP. We introduce new lower bounding techniques enabling the computing of a lower bound at each node of the branching scheme. Our new DSATUR outperforms the state of the art for random VCP instances with high density, significantly increasing the size of solvable instances. Similar results can be achieved for a subset of high density DIMACS instances. We study three ILP formulations for the Minimum Sum Coloring Problem (MSCP). The problem is an extension of the classical Vertex Coloring Problem in which each color is represented by a positive natural number. The MSCP asks to minimize the sum of the cardinality of subsets of vertices receiving the same color, weighted by the index of the color, while ensuring that vertices linked by an edge receive different colors. We focus on studying an extended formulation and devise a complete Branch-and-Price algorithm.

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.

There are many real cases where a company needs to determine the price of its products so as to maximise its revenue or profit.<p>To do so, the company must consider customers' reactions to these prices, as they may refuse to buy a given product or service if its price is too high. This is commonly known in literature as a pricing problem.<p>This class of problems, which is typically bilevel, was first studied in the 1990s and is NP-hard, although polynomial algorithms do exist for some particular cases. Many questions are still open on this subject.<p><p>The aim of this thesis is to investigate mathematical properties of pricing problems, in order to find structural properties, formulations and solution methods that are as efficient as possible. In particular, we focus our attention on pricing problems over a network. In this framework, an authority owns a subset of arcs and imposes tolls on them, in an attempt to maximise his/her revenue, while users travel on the network, seeking for their minimum cost path.<p><p>First, we provide a detailed review of the state of the art on bilevel pricing problems. <p>Then, we consider a particular case where the authority is using an unit toll scheme on his/her subset of arcs, imposing either the same toll on all of them, or a toll proportional to a given parameter particular to each arc (for instance a per kilometre toll). We show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo-polynomial.<p>We then address a robust approach taking into account uncertainty on parameters. We solve some polynomial cases of the pricing problem where uncertainty is considered using an interval representation.<p><p>Finally, we focus on another particular case where toll arcs are connected such that they constitute a path, as occurs on highways. We develop a Dantzig-Wolfe reformulation and present a Branch-and-Cut-and-Price algorithm to solve it. Several improvements are proposed, both for the column generation algorithm used to solve the linear relaxation and for the branching part used to find integer solutions. Numerical results are also presented to highlight the efficiency of the proposed strategies. This problem is proved to be APX-hard and a theoretical comparison between our model and another one from the literature is carried out. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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.

The air transportation market has been growing steadily for the past three decades since the airline deregulation in 1978. With competition also becoming more intense, airline companies have been trying to enhance their market shares and profit margins by composing favorable flight schedules and by efficiently allocating their resources of aircraft and crews so as to reduce operational costs. In practice, this is achieved based on demand forecasts and resource availabilities through a structured airline scheduling process that is comprised of four decision stages: schedule planning, fleet assignment, aircraft routing, and crew scheduling. The outputs of this process are flight schedules along with associated assignments of aircraft and crews that maximize the total expected profit. Traditionally, airlines deal with these four operational scheduling stages in a sequential manner. However, there exist obvious interdependencies among these stages so that restrictive solutions from preceding stages are likely to limit the scope of decisions for succeeding stages, thus leading to suboptimal results and even infeasibilities. To overcome this drawback, we first study the aircraft routing problem, and develop some novel modeling foundations based on which we construct and analyze an integrated model that incorporates fleet assignment, aircraft routing, and crew pairing within a single framework. Given a set of flights to be covered by a specific fleet type, the aircraft routing problem (ARP) determines a flight sequence for each individual aircraft in this fleet, while incorporating specific considerations of minimum turn-time and maintenance checks, as well as restrictions on the total accumulated flying time, the total number of takeoffs, and the total number of days between two consecutive maintenance operations. This stage is significant to airline companies as it directly assigns routes and maintenance breaks for each aircraft in service. Most approaches for solving this problem adopt set partitioning formulations that include exponentially many variables, thus requiring the design of specialized column generation or branch-and-price algorithms. In this dissertation, however, we present a novel compact polynomially sized representation for the ARP, which is then linearized and lifted using the Reformulation-Linearization Technique (RLT). The resulting formulation remains polynomial in size, and we show that it can be solved very efficiently by commercial software without complicated algorithmic implementations. Our numerical experiments using real data obtained from United Airlines demonstrate significant savings in computational effort; for example, for a daily network involving 344 flights, our approach required only about 10 CPU seconds for deriving an optimal solution. We next extend Model ARP to incorporate its preceding and succeeding decision stages, i.e., fleet assignment and crew pairing, within an integrated framework. We formulate a suitable representation for the integrated fleeting, routing, and crew pairing problem (FRC), which accommodates a set of fleet types in a compact manner similar to that used for constructing the aforementioned aircraft routing model, and we generate eligible crew pairings on-the-fly within a set partitioning framework. Furthermore, to better represent industrial practice, we incorporate itinerary-based passenger demands for different fare-classes. The large size of the resulting model obviates a direct solution using off-the-shelf software; hence, we design a solution approach based on Benders decomposition and column generation using several acceleration techniques along with a branch-and-price heuristic for effectively deriving a solution to this model. In order to demonstrate the efficacy of the proposed model and solution approach and to provide insights for the airline industry, we generated several test instances using historical data obtained from United Airlines. Computational results reveal that the massively-sized integrated model can be effectively solved in reasonable times ranging from several minutes to about ten hours, depending on the size and structure of the instance. Moreover, our benchmark results demonstrate an average of 2.73% improvement in total profit (which translates to about 43 million dollars per year) over a partially integrated approach that combines the fleeting and routing decisions, but solves the crew pairing problem sequentially. This improvement is observed to accrue due to the fact that the fully integrated model effectively explores alternative fleet assignment decisions that better utilize available resources and yield significantly lower crew costs. / Ph. D.