Stochastic programming is an optimization technique that incorporates random variables as parameters. Because it better reflects the uncertain real world than its traditional deterministic counterpart, stochastic programming has drawn increasingly more attention among decision-makers, and its applications span many fields including financial engineering, health care, communication systems, and supply chain management. On the flip side, stochastic programs are usually very difficult to solve, which is further compounded by the fact that in many of the aforementioned applications, we also have discrete decisions, thereby rendering these problems even more challenging. In this dissertation, we study the class of two-stage stochastic mixed-integer programs (SMIP), which, as its name suggests, lies at the confluence of two formidable classes of problems. We design a novel algorithm for this class of problems, and also explore specialized approaches for two related real-world applications. Although a number of algorithms have been developed to solve two-stage SMIPs, most of them deal with problems containing purely integer or continuous variables in either or both of the two stages, and frequently require the technology and/or recourse matrices to be deterministic. As a ground-breaking effort, in this work, we address the challenging class of two-stage SMIPs that involve 0-1 mixed-integer variables in both stages. The only earlier work on solving such problems (Carøe and Schultz (1999)) requires the optimization of several non-smooth Lagrangian dual problems using subgradient methods in the bounding process, which turns out to be computationally very expensive. We begin with proposing a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having 0-1 mixed-integer variables in both stages. Since the second-stage problems contain binary variables, their value functions are in general nonconvex and discontinuous; hence, the classical Benders' decomposition approach (or the L-shaped method) for solving two-stage stochastic programs, which requires convex subproblem value functions, cannot be directly applied. This motivates us to relax the second-stage problems and accompany this relaxation with a convexification process. To make this process computationally efficient, we propose to construct a certain partial convex hull representation of the two-stage solution space, using the relaxed second-stage constraints and the restrictions confining the first-stage variables to lie within some hyperrectangle. This partial convex hull is sequentially generated using a convexification scheme, such as the Reformulation-Linearization Technique (RLT), which yields valid inequalities that are functions of the first-stage variables and, of noteworthy importance, are reusable in the subsequent subproblems by updating the values of the first-stage variables. Meanwhile, since the first stage contains continuous variables, whenever we tentatively fix these variables at some given feasible values, the resulting constraints may not be facial with respect to the associated bounding constraints that are used to construct the partial convex hull. As a result, the constructed Benders' subproblems define lower bounds for the second-stage value functions, and likewise, the resulting Benders' master problem provides a lower bound for the original stochastic program defined over the same hyperrectangle. Another difficulty resulting from continuous first-stage variables is that when the given first-stage solution is not extremal with respect to its bounds, the second-stage solution obtained for a Benders' subproblem defined with respect to a partial convex hull representation in the two-stage space may not satisfy the model's binary restrictions. We thus need to be able to detect whether or not a Benders' subproblem is solved by a given fractional second-stage solution. We design a novel procedure to check this situation in the overall algorithmic scheme. A key property established, which ensures global convergence, is that these lower bounds become exact if the given first-stage solution is a vertex of the defining hyperrectangle, or if the second-stage solution satisfies the binary restrictions. Based on these algorithmic constructs, we design a branch-and-bound procedure where the branching process performs a hyperrectangular partitioning of the projected space of the first-stage variables, and lower bounds for the nodal problems are computed by applying the proposed modified Benders' decomposition method. We prove that, when using the least-lower-bound node-selection rule, this algorithm converges to a global optimal solution. We also show that the derived RLT cuts are not only reusable in subsequent Benders iterations at the same node, but are also inheritable by the subproblems of the children nodes. Likewise, the Benders' cuts derived for a given sub-hyperrectangle can also be inherited by the lower bounding master programs solved for its children nodes. Using these cut inheritance properties results in significant savings in the overall computational effort. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach. The sizes of the deterministic equivalent of our test problems range from having 386 continuous variables, 386 binary variables, and 386 constraints, up to 1795 continuous variables, 1539 binary variables, and 1028 constraints. The results reveal an average savings in computational effort by a factor of 9.5 in comparison with using a commercial mixed-integer programming package (CPLEX 8.1) on a deterministic equivalent formulation. We then explore an important application of SMIP to enhance the traditional airline fleet assignment models (FAM). Given a flight schedule network, the fleet assignment problem solved by airline companies is concerned with assigning aircraft to flight legs in order to maximize profit with respect to captured path- or itinerary-based demand. Because certain related crew scheduling regulations require early information regarding the type of aircraft serving each flight leg, the current practice adopted by airlines is to solve the fleet assignment problem using estimated demand data 10-12 weeks in advance of departure. Given the level of uncertainty, deterministic models at this early stage are inadequate to obtain a good match of aircraft capacity with passenger demands, and revisions to the initial fleet assignment become naturally pertinent when the observed demand differs considerably from the assigned aircraft capacities. From this viewpoint, the initial decision should embrace various market scenarios so that it incorporates a sufficient look-ahead feature and provides sufficient flexibility for the subsequent re-fleeting processes to accommodate the inevitable demand fluctuations. With this motivation, we propose a two-stage stochastic programming approach in which the first stage is concerned with the initial fleet assignment decisions and, unlike the traditional deterministic methodology, focuses on making only a family-level assignment to each flight leg. The second stage subsequently performs the detailed assignments of fleet types within the allotted family to each leg under each of the multiple potential scenarios that address corresponding path- or itinerary-based demands. In this fashion, the initial decision of what aircraft family should serve each flight leg accomplishes the purpose of facilitating the necessary crew scheduling decisions, while judiciously examining the outcome of future re-fleeting actions based on different possible demand scenarios. Hence, when the actual re-fleeting process is enacted several weeks later, this anticipatory initial family-level assignment will hopefully provide an improved overall fleet type re-allocation that better matches demand. This two-stage stochastic model is complemented with a secondary model that performs adjustments within each family, if necessary, to provide a consistent fleet type-assignment information for accompanying decision processes, such as yield management. We also propose several enhanced fleet assignment models, including a robust optimization model that controls decision variation among scenarios and a stochastic programming model that considers the recapture effect of spilled demand. In addition to the above modeling concepts and framework, we also contribute in developing effective solution approaches for the proposed model, which is a large-scale two-stage stochastic 0-1 mixed-integer program. Because the most pertinent information needed from the initial fleet assignment is at the family level, and the type-level assignment is subject to change at the re-fleeting stage according to future demand realizations, our solution approach focuses on assigning aircraft families to the different legs in the flight network at the first stage, while finding relaxed second-stage solutions under different demand scenarios. Based on a polyhedral study of a subsystem extracted from the original model, we derive certain higher-dimensional convex hull as well as partial convex hull representations for this subsystem. Accordingly, we propose two variants for the primary model, both of which relax the binary restrictions on the second-stage variables, but where the second variant then also accommodates the partial convex hull representations, yielding a tighter, albeit larger, relaxation. For each variant, we design a suitable solution approach predicated on Benders' decomposition methodology. Using certain realistic large-scale flight network test problems having 900 flight legs and 1,814 paths, as obtained from United Airlines, the proposed stochastic modeling approach was demonstrated to increase daily expected profits by about 3% (which translates to about $160 million per year) in comparison with the traditional deterministic model in present usage, which considers only the expected demand. Only 1.6% of the second-stage binary variables turn out to be fractional in the first variant, and this number is further reduced to 1.2% by using the tighter variant. Furthermore, when attempting to solve the deterministic equivalent formulation for these two variants using a commercial mixed-integer programming package (CPLEX 8.1), both the corresponding runs were terminated after reaching a 25-hour cpu time limit. At termination, the software was still processing the initial LP relaxation at the root node for each of these runs, and no feasible basis was found. Using the proposed algorithms, on the other hand, the solution times were significantly reduced to 5 and 19 hours for the two variants, respectively. Considering that the fleet assignment models are solved around three months in advance of departure, this solution time is well acceptable at this early planning stage, and the improved quality in the solution produced by considering the stochasticity in the system is indeed highly desirable. Finally, we address another practical workforce planning problem encountered by a global financial firm that seeks to manage multi-category workforce for functional areas located at different service centers, each having office-space and recruitment-capacity constraints. The workforce demand fluctuates over time due to market uncertainty and dynamic project requirements. To hedge against the demand fluctuations and the inherent uncertainty, we propose a two-stage stochastic programming model where the first stage makes personnel recruiting and allocation decisions, while the second stage, based on the given personnel decision and realized workforce demand, decides on the project implementation assignment. The second stage of the proposed model contains binary variables that are used to compute and also limit the number of changes to the original plan. Since these variables are concerned with only one quality aspect of the resulting workforce plan and do not affect feasibility issues, we replace these binary variables with certain conservative policies regarding workforce assignment change restrictions in order to obtain more manageable subproblems that contain purely continuous variables. Numerical experiments reveal that the stochastic programming approach results in significantly fewer alterations to the original workforce plan. When using a commercial linear programming package CPLEX 9.0 to solve the deterministic equivalent form directly, except for a few small-sized problems, this software failed to produce solutions due to memory limitations, while the proposed Benders' decomposition-based solution approach consistently solved all the practical-sized test problems with reasonable effort. To summarize, this dissertation provides a significant advancement in the algorithmic development for solving two-stage stochastic mixed-integer programs having 0-1 mixed-integer variables in both stages, as well as in its application to two important contemporary real-world applications. The framework for the proposed solution approaches is to formulate tighter relaxations via partial convex hull representations and to exploit the resulting structure using suitable decomposition methods. As decision robustness is becoming increasingly relevant from an economic viewpoint, and as computer technological advances provide decision-makers the ability to explore a wide variety of scenarios, we hope that the proposed algorithms will have a notable positive impact on solving stochastic mixed-integer programs. In particular, the proposed stochastic programming airline fleet assignment and the workforce planning approaches studied herein are well-poised to enhance the profitability and robustness of decisions made in the related industries, and we hope that similar improvements are adapted by more industries where decisions need to be made in the light of data that is shrouded by uncertainty. / Ph. D.

Air transportation offers both passenger and freight services that are essential for economic growth and development. In a highly competitive environment, airline companies have to control their operating costs by managing their flights, aircraft, and crews effectively. This motivates the extensive use of analytical techniques to solve complex problems related to airline operations planning, which includes schedule design, fleet assignment, aircraft routing, and crew scheduling. The initial problem addressed by airlines is that of schedule design, whereby a set of flights having specific origin and destination cities as well as departure and arrival times is determined. Then, a fleet assignment problem is solved to assign an aircraft type to each flight so as to maximize anticipated profits. This enables a decomposition of subsequent problems according to the different aircraft types belonging to a common family, for each of which an aircraft routing problem and a crew scheduling or pairing problem are solved. Here, in the aircraft routing problem, a flight sequence or route is built for each individual aircraft so as to cover each flight exactly once at a minimum cost while satisfying maintenance requirements. Finally, in the crew scheduling or pairing optimization problem, a minimum cost set of crew rotations or pairings is constructed such that every flight is assigned a qualified crew and that work rules and collective agreements are satisfied. In practice, most airline companies solve these problems in a sequential manner to plan their operations, although recently, an increasing effort is being made to develop novel approaches for integrating some of the airline operations planning problems while retaining tractability. This dissertation formulates and analyzes three different models, each of which examines a composition of certain pertinent airline operational planning problems. A comprehensive fourth model is also proposed, but is relegated for future research. In the first model, we integrate fleet assignment and schedule design by simultaneously considering optional flight legs to select along with the assignment of aircraft types to all scheduled legs. In addition, we consider itinerary-based demands pertaining to multiple fare-classes. A polyhedral analysis of the proposed mixed-integer programming model is used to derive several classes of valid inequalities for tightening its representation. Solution approaches are developed by applying Benders decomposition method to the resulting lifted model, and computational experiments are conducted using real data obtained from a major U.S. airline (United Airlines) to demonstrate the efficacy of the proposed procedures as well as the benefits of integration. A comparison of the experimental results obtained for the basic integrated model and for its different enhanced representations reveals that the best modeling strategy among those tested is the one that utilizes a variety of five types of valid inequalities for moderately sized problems, and further implements a Benders decomposition approach for relatively larger problems. In addition, when a heuristic sequential fixing step is incorporated within the algorithm for even larger sized problems, the computational results demonstrate a less than 2% deterioration in solution quality, while reducing the effort by about 21%. We also performed an experiment to assess the impact of integration by comparing the proposed integrated model with a sequential implementation in which the schedule design is implemented separately before the fleet assignment stage based on two alternative profit maximizing submodels. The results obtained demonstrate a clear advantage of utilizing the integrated model, yielding an 11.4% and 5.5% increase in profits in comparison with using the latter two sequential models, which translates to an increase in annual profits by about $28.3 million and $13.7 million, respectively. The second proposed model augments the first model with additional features such as flexible flight times (i.e., departure time-windows), schedule balance, and demand recapture considerations. Optional flight legs are incorporated to facilitate the construction of a profitable schedule by optimally selecting among such alternatives in concert with assigning the available aircraft fleet to all the scheduled legs. Moreover, network effects and realistic demand patterns are effectively represented by examining itinerary-based demands as well as multiple fare-classes. Allowing flexibility on the departure times of scheduled flight legs within the framework of an integrated model increases connection opportunities for passengers, hence yielding robust schedules while saving fleet assignment costs. A provision is also made for airlines to capture an adequate market share by balancing flight schedules throughout the day. Furthermore, demand recapture considerations are modeled to more realistically represent revenue realizations. For this proposed mixed-integer programming model, which integrates the schedule design and fleet assignment processes while considering flexible flight times, schedule balance, and recapture issues, along with optional legs, itinerary-based demands, and multiple fare-classes, we perform a polyhedral analysis and utilize the Reformulation-Linearization Technique in concert with suitable separation routines to generate valid inequalities for tightening the model representation. Effective solution approaches are designed by applying Benders decomposition method to the resulting tightened model, and computational results are presented to demonstrate the efficacy of the proposed procedures. Using real data obtained from United Airlines, when flight times were permitted to shift by up to 10 minutes, the estimated increase in profits was about $14.9M/year over the baseline case where only original flight legs were used. Also, the computational results indicated a 1.52% and 0.49% increase in profits, respectively, over the baseline case, while considering two levels of schedule balance restrictions, which can evidently also enhance market shares. In addition, we measured the effect of recaptured demand with respect to the parameter that penalizes switches in itineraries. Using values of the parameter that reflect 1, 50, 100, or 200 dollars per switched passenger, this yielded increases in recaptured demand that induced additional profits of 2.10%, 2.09%, 2.02%, and 1.92%, respectively, over the baseline case. Overall, the results obtained from the two schedule balance variants of the proposed integrated model that accommodate all the features of flight retiming, schedule balance, and demand recapture simultaneously, demonstrated a clear advantage by way of $35.1 and $31.8 million increases in annual profits, respectively, over the baseline case in which none of these additional features is considered. In the third model, we integrate the schedule design, fleet assignment, and aircraft maintenance routing decisions, while considering optional legs, itinerary-based demands, flexible flight retimings, recapture, and multiple fare-classes. Instead of utilizing the traditional time-space network (TSN), we formulate this model based on a flight network (FN) that provides greater flexibility in accommodating integrated operational considerations. In order to consider through-flights (i.e., a sequence of flight legs served by the same aircraft), we append a set of constraints that matches aircraft assignments on certain inbound legs into a station with that on appropriate outbound legs at the same station. Through-flights can generate greater revenue because passengers are willing to pay a premium for not having to change aircraft on connecting flights, thereby reducing the possibility of delays and missed baggage. In order to tighten the model representation and reduce its complexity, we apply the Reformulation-Linearization Technique (RLT) and also generate other classes of valid inequalities. In addition, since the model possesses many equivalent feasible solutions that can be obtained by simply reindexing the aircraft of the same type that depart from the same station, we introduce a set of suitable hierarchical symmetry-breaking constraints to enhance the model solvability by distinguishing among aircraft of the same type. For the resulting large-scale augmented model formulation, we design a Benders decomposition-based solution methodology and present extensive computational results to demonstrate the efficacy of the proposed approach. We explored four different algorithmic variants, among which the best performing procedure (Algorithm A1) adopted two sequential levels of Benders partitioning method. We then applied Algorithm A1 to perform several experiments to study the effects of different modeling features and algorithmic strategies. A summary of the results obtained is as follows. First, the case that accommodated both mandatory and optional through-flight leg pairs in the model based on their relative effects on demands and enhanced revenues achieved the most profitable strategy, with an estimated increase in expected annual profits of $2.4 million over the baseline case. Second, utilizing symmetry-breaking constraints in concert with compatible objective perturbation terms greatly enhanced problem solvability and thus promoted the detection of improved solutions, resulting in a $5.8 million increase in estimated annual profits over the baseline case. Third, in the experiment that considers recapture of spilled demand from primary itineraries to other compatible itineraries, the different penalty parameter values (100, 50, and 1 dollars per re-routed passenger) induced average respective proportions of 3.2%, 3.4%, and 3.7% in recaptured demand, resulting in additional estimated annual profits of $3.7 million, $3.8 million, and $4.0 million over the baseline case. Finally, incorporating the proposed valid inequalities within the model to tighten its representation helped reduce the computational effort by 11% on average, while achieving better solutions that yielded on average an increase in estimated annual profits of $1.4 million. In closing, we propose a fourth more comprehensive model in which the crew scheduling problem is additionally integrated with fleet assignment and aircraft routing. This integration is important for airlines because crew costs are the second largest component of airline operating expenses (after fuel costs), and the assignment and routing of aircraft plus the assignment of crews are two closely interacting components of the planning process. Since crews are qualified to typically serve a single aircraft family that is comprised of aircraft types having a common cockpit configuration and crew rating, the aircraft fleeting and routing decisions significantly impact the ensuing assignment of cockpit crews to flights. Therefore it is worthwhile to investigate new models and solution approaches for the integrated fleeting, aircraft routing, and crew scheduling problem, where all of these important inter-dependent processes are handled simultaneously, and where the model can directly accommodate various work rules such as imposing a specified minimum and maximum number of flying hours for crews on any given pairing, and a minimum number of departures at a given crew base for each fleet group. However, given that the crew scheduling problem itself is highly complex because of the restrictive work rules that must be heeded while constructing viable duties and pairings, the formulated integrated model would require further manipulation and enhancements along with the design of sophisticated algorithms to render it solvable. We therefore recommend this study for future research, and we hope that the modeling, analysis, and algorithmic development and implementation work performed in this dissertation will lend methodological insights into achieving further advances along these lines. / Ph. D.

This dissertation focuses on selected issues in regard to the mathematical modeling of electricity markets. In a first step the interrelations of electric power market modeling are highlighted a crossroad between operations research, applied economics, and engineering. In a second step the development of a large-scale continental European economic engineering model named ELMOD is described and the model is applied to the issue of wind integration. It is concluded that enabling the integration of low-carbon technologies appears feasible for wind energy. In a third step algorithmic work is carried out regarding a game theoretic model. Two approaches in order to solve a discretely-constrained mathematical program with equilibrium constraints using disjunctive constraints are presented. The first one reformulates the problem as a mixed-integer linear program and the second one applies the Benders decomposition technique. Selected numerical results are reported.

[pt] Portfólios de fontes renováveis de energia elétrica são mecanismos de gerenciamento de risco interessantes para comercialização de energia em mercados de negociação bilateral. Quando formados por agentes que pertencem a diferentes companhias sua estabilidade depende da maneira com que os benefícios de mitigação de risco gerados pelo portfólio são alocados individualmente entre os participantes. O problema de se encontrar uma solução estável pode ser matematicamente formulado através da busca de um vetor de alocação de quotas que pertença ao núcleo do jogo cooperativo, que por sua vez pode ser formulado como um conjunto de restrições lineares que aumenta exponencialmente com o número de participantes. Adicionalmente, o lado direito de cada restrição que define o núcleo do jogo cooperativo define o valor de uma determinada coalisão que, no presente trabalho, é obtido através de um modelo de otimização estocástica de dois estágios. Este trabalho compara diferentes metodologias computacionalmente eficientes baseadas em programação linear inteira mista e na técnica de decomposição de Benders para encontrar vetores de alocação de quotas que pertençam ao núcleo de portfólios de larga escala de geradores de energia renovável. São apresentados estudos de casos que utilizam dados reais do sistema elétrico brasileiro. / [en] Portfolios of renewable electricity sources are interesting risk-management mechanisms for trading in electricity contract markets. When they are formed by players belonging to different companies, their stability relies on the way the riskmitigation benefit generated by the optimal portfolio is allocated through individual participants. The problem of reaching a stable allocation can be mathematically formulated in terms of finding a quota-sharing vector belonging to the Core of a cooperative game, which can be formulated as a set of linear constraints that exponentially grows with the number of participants. Moreover, the right-hand-side of each constraint defining the Core relies on a given coalition value which, in the present work, is obtained by a two-stage stochastic optimization model. This work presents and compares efficient methodologies mainly based on mixed integer linear programming and Benders decomposition to find quota allocation vectors that belongs to the Core of large-scale renewable energy portfolios. Case studies are presented with realistic data from the Brazilian power system.

[pt] Recentes apagões em todo o mundo fazem da confiabilidade de sistemas de potência, no tocante a contingências múltiplas, um tema de pesquisa mundial. Dentro desse contexo, se faz importante investigar métodos eficientes de proteger o sistema contra falhas de alguns de seus componentes, sejam elas dependentes e/ou independentes de outras falhas. Nesse sentido, se tornou crucial a incorporação de critérios de segurança mais rigorosos na operação e planejamento de sistemas de potência. Contingências múltiplas são mais comuns e desastrosas do que falhas naturais e independentes. A principal razão para isso reside na complexidade da estabilidade dinâmica de sistemas de potência. Além disso, o sistema de proteção que opera em paralelo ao sistema de distribuição não é livre de falhas. Portanto, interrupções naturais podem causar contingências em cascata em decorrência do mau funcionamento de mecanismos de proteção ou da instabilidade do sistema elétrico como um todo. Nesse contexto, se dá a motivação pela busca de critérios de segurança mais severos como, por exemplo, o n - K, onde K pode ser maior do que 2. Nesse trabalho, o principal objetivo é incorporar o crtitério de segurança geral n-K para geração e transmissão em modelos de operação e planejamento de sistemas de potência. Além de interrupções em geradores, restrições de rede, bem como falhas em linhas de transmiss˜ao também são modeladas. Esse avanço leva a novos desafios computacionais, para os quais formulamos metodologias de solução eficientes baseadas em decomposição de Benders. Considerando operação, duas abordagens são apresentadas. A primeira propõe um modelo de otimização trinível para decidir o despacho ótimo de energia e reservas sob um critério de segurançaa n - K. Nessa abordagem, a alta dimensionalidade do problema, por contemplar restrições de rede, bem como falhas de geradores e de linhas de transmissão, é contornada por meio da implícita consideração do conjunto de possíveis contingências. No mesmo contexto, a segunda abordagem leva em conta a incerteza da carga a ser suprida e a correlação entre demandas de diferentes barras. Considerando planejamento de expansão da transmissão, outro modelo de otimização trinível é apresentado no intuito de decidir quais linhas de transmissão, dentro de um conjunto de candidatas, devem ser construídas para atender a um critério de segurança n - K e, consequentemente, aumentar a confiabilidade do sistema como um todo. Portanto, as principais contribuições do presente trabalho são as seguintes: 1) modelos de otimização trinível para considerar o critério de segurança n - K em operação e planejamento de sistemas de potência, 2) consideração implícita de todo o conjunto de contingências por meio de uma abordagem de otimização robusta ajustável, 3) otimização conjunta de energia e reserva para operação de sistemas de potência, considerando restrições de rede e garantindo a entregabilidade das reservas em todos os estados pós-contingência considerados, 4) metodologias de solução eficientes baseadas em decomposição de Benders que convergem em passos finitos para o ótimo global e 5) desenvolvimento de restrições válidas que alavancam a eficiência computacional. Estudos de caso ressaltam a eficácia das metodologias propostas em capturar os efeitos econômicos de demanda nodal correlacionada sob um critério de segurançaa n - 1, em reduzir o esfor¸co computacional para considerar os critérios de seguran¸ca convencionais n-1 e n-2 e em considerar critérios de segurança mais rigorosos do que o n - 2, um problema intratável até então. / [en] Recent major blackouts all over the world have been a driving force to make power system reliability, regarding multiple contingencies, a subject of worldwide research. Within this context, it is important to investigate efficient methods of protecting the system against dependent and/or independent failures. In this sense, the incorporation of tighter security criteria in power systems operation and planning became crucial. Multiple contingencies are more common and dangerous than natural independent faults. The main reason for this lies in the complexity of the dynamic stability of power systems. In addition, the protection system, that operates in parallel to the supply system, is not free of failures. Thus, natural faults can cause subsequent contingencies (dependent on earlier contingencies) due to the malfunction of the protection mechanisms or the instability of the overall system. These facts drive the search for more stringent safety criteria, for example, n - K, where K can be greater than 2. In the present work, the main objective is to incorporate the joint generation and transmission general security criteria in power systems operation and planning models. Here, in addition to generators outages, network constraints and transmission lines failures are also accounted for. Such improvement leads to new computational challenges, for which we design efficient solution methodologies based on Benders decomposition. Regarding operation, two approaches are presented. The first one proposes a trilevel optimization model to decide the optimal scheduling of energy and reserve under an n - K security criterion. In such approach, the high dimensionality curse of considering network constraints as well as outages of generators and transmission assets is withstood by implicitly taking into account the set of possible contingencies. The second approach includes correlated nodal demand uncertainty in the same framework. Regarding transmission expansion planning, another trilevel optimization model is proposed to decide which transmission assets should be built within a set of candidates in order to meet an n - K security criterion, and, consequently, boost the power system reliability. Therefore, the main contributions of this work are the following: 1) trilevel models to consider general n - K security criteria in power systems operation and planning, 2) implicit consideration of the whole contingency set by means of an adjustable robust optimization approach, 3) co-optimization of energy and reserves for power systems operation, regarding network constraints and ensuring the deliverability of reserves in all considered post-contingency states, 4) efficient solution methodologies based on Benders decomposition that finitely converges to the global optimal solution, and 5) development of valid constraints to boost computational efficiency. Case studies highlight the effectiveness of the proposed methodologies in capturing the economic effect of nodal demand correlation on power system operation under an n - 1 security criterion, in reducing the computational effort to consider conventional n-1 and n-2 security criteria, and in considering security criteria tighter than n - 2, an intractable problem heretofore.

Dans cette thèse, nous étudions le problème d’optimisation des effectifs dans les centres d’appels, dans lequel nous visons à minimiser les coûts d’exploitation tout en offrant aux clients une qualité de service (QoS) élevée. Nous introduisons également l'utilisation de contraintes probabilistes qui exigent que la qualité de service soit satisfaite avec une probabilité donnée. Ces contraintes sont adéquates dans le cas où la performance est mesurée sur un court intervalle de temps, car les mesures de QoS sont des variables aléatoires sur une période donnée. Les problèmes de personnel proposés sont difficiles en raison de l'absence de forme analytique pour les contraintes probabilistes et doivent être approximées par simulation. En outre, les fonctions QoS sont généralement non linéaires et non convexes. Nous considérons les problèmes d’affectation personnel dans différents contextes et étudions les modèles proposés tant du point de vue théorique que pratique. Les méthodologies développées sont générales, en ce sens qu'elles peuvent être adaptées et appliquées à d'autres problèmes de décision dans les systèmes de files d'attente. La thèse comprend trois articles traitant de différents défis en matière de modélisation et de résolution de problèmes d'optimisation d’affectation personnel dans les centres d'appels à compétences multiples. Les premier et deuxième article concernent un problème d'optimisation d'affectation de personnel en deux étapes sous l'incertitude. Alors que dans le second, nous étudions un modèle général de programmation stochastique discrète en deux étapes pour fournir une garantie théorique de la consistance de l'approximation par moyenne échantillonnale (SAA) lorsque la taille des échantillons tend vers l'infini, le troisième applique l'approche du SAA pour résoudre le problème d’optimisation d'affectation de personnel en deux étapes avec les taux d’arrivée incertain. Les deux articles indiquent la viabilité de l'approche SAA dans notre contexte, tant du point de vue théorique que pratique. Pour être plus précis, dans le premier article, nous considérons un problème stochastique discret général en deux étapes avec des contraintes en espérance. Nous formulons un problème SAA avec échantillonnage imbriqué et nous montrons que, sous certaines hypothèses satisfaites dans les exemples de centres d'appels, il est possible d'obtenir les solutions optimales du problème initial en résolvant son SAA avec des échantillons suffisamment grands. De plus, nous montrons que la probabilité que la solution optimale du problème de l’échantillon soit une solution optimale du problème initial tend vers un de manière exponentielle au fur et à mesure que nous augmentons la taille des échantillons. Ces résultats théoriques sont importants, non seulement pour les applications de centre d'appels, mais également pour d'autres problèmes de prise de décision avec des variables de décision discrètes. Le deuxième article concerne les méthodes de résolution d'un problème d'affectation en personnel en deux étapes sous incertitude du taux d'arrivée. Le problème SAA étant coûteux à résoudre lorsque le nombre de scénarios est important. En effet, pour chaque scénario, il est nécessaire d'effectuer une simulation pour estimer les contraintes de QoS. Nous développons un algorithme combinant simulation, génération de coupes, renforcement de coupes et décomposition de Benders pour résoudre le problème SAA. Nous montrons l'efficacité de l'approche, en particulier lorsque le nombre de scénarios est grand. Dans le dernier article, nous examinons les problèmes de contraintes en probabilité sur les mesures de niveau de service. Notre méthodologie proposée dans cet article est motivée par le fait que les fonctions de QoS affichent généralement des courbes en S et peuvent être bien approximées par des fonctions sigmoïdes appropriées. Sur la base de cette idée, nous avons développé une nouvelle approche combinant la régression non linéaire, la simulation et la recherche locale par région de confiance pour résoudre efficacement les problèmes de personnel à grande échelle de manière viable. L’avantage principal de cette approche est que la procédure d’optimisation peut être formulée comme une séquence de simulations et de résolutions de problèmes de programmation linéaire. Les résultats numériques basés sur des exemples réels de centres d'appels montrent l'efficacité pratique de notre approche. Les méthodologies développées dans cette thèse peuvent être appliquées dans de nombreux autres contextes, par exemple les problèmes de personnel et de planification dans d'autres systèmes basés sur des files d'attente avec d'autres types de contraintes de QoS. Celles-ci soulèvent également plusieurs axes de recherche qu'il pourrait être intéressant d'étudier. Par exemple, une approche de regroupement de scénarios pour atténuer le coût des modèles d'affectation en deux étapes, ou une version d'optimisation robuste en distribution pour mieux gérer l'incertitude des données. / In this thesis, we study the staffing optimization problem in multiskill call centers, in which we aim at minimizing the operating cost while delivering a high quality of service (QoS) to customers. We also introduce the use of chance constraints which require that the QoSs are met with a given probability. These constraints are adequate in the case when the performance is measured over a short time interval as QoS measures are random variables in a given time period. The proposed staffing problems are challenging in the sense that the stochastic constraints have no-closed forms and need to be approximated by simulation. In addition, the QoS functions are typically non-linear and non-convex. We consider staffing optimization problems in different settings and study the proposed models in both theoretical and practical aspects. The methodologies developed are general, in the sense that they can be adapted and applied to other staffing/scheduling problems in queuing-based systems. The thesis consists of three articles dealing with different challenges in modeling and solving staffing optimization problems in multiskill call centers. The first and second articles concern a two-stage staffing optimization problem under uncertainty. While in the first one, we study a general two-stage discrete stochastic programming model to provide a theoretical guarantee for the consistency of the sample average approximation (SAA) when the sample sizes go to infinity, the second one applies the SAA approach to solve the two-stage staffing optimization problem under arrival rate uncertainty. Both papers indicate the viability of the SAA approach in our context, in both theoretical and practical aspects. To be more precise, in the first article, we consider a general two-stage discrete stochastic problem with expected value constraints. We formulate its SAA with nested sampling. We show that under some assumptions that hold in call center examples, one can obtain the optimal solutions of the original problem by solving its SAA with large enough sample sizes. Moreover, we show that the probability that the optimal solution of the sample problem is an optimal solution of the original problem, approaches one exponentially fast as we increase the sample sizes. These theoretical findings are important, not only for call center applications, but also for other decision-making problems with discrete decision variables. The second article concerns solution methods to solve a two-stage staffing problem under arrival rate uncertainty. It is motivated by the fact that the SAA version of the two-stage staffing problem becomes expensive to solve with a large number of scenarios, as for each scenario, one needs to use simulation to approximate the QoS constraints. We develop an algorithm that combines simulation, cut generation, cut strengthening and Benders decomposition to solve the SAA problem. We show the efficiency of the approach, especially when the number of scenarios is large. In the last article, we consider problems with chance constraints on the service level measures. Our methodology proposed in this article is motivated by the fact that the QoS functions generally display ``S-shape'' curves and might be well approximated by appropriate sigmoid functions. Based on this idea, we develop a novel approach that combines non-linear regression, simulation and trust region local search to efficiently solve large-scale staffing problems in a viable way. The main advantage of the approach is that the optimization procedure can be formulated as a sequence of steps of performing simulation and solving linear programming models. Numerical results based on real-life call center examples show the practical viability of our approach. The methodologies developed in this thesis can be applied in many other settings, e.g., staffing and scheduling problems in other queuing-based systems with other types of QoS constraints. These also raise several research directions that might be interesting to investigate. For examples, a clustering approach to mitigate the expensiveness of the two-stage staffing models, or a distributionally robust optimization version to better deal with data uncertainty.

Cette thèse porte sur la modélisation du trafic dans les réseaux routiers et comment celle-ci est intégrée dans des modèles d'optimisation. Ces deux sujets ont évolué de manière plutôt disjointe: le trafic est prédit par des modèles mathématiques de plus en plus complexes, mais ce progrès n'a pas été incorporé dans les modèles de design de réseau dans lesquels les usagers de la route jouent un rôle crucial. Le but de cet ouvrage est d'intégrer des modèles d'utilités aléatoires calibrés avec de vraies données dans certains modèles biniveaux d'optimisation et ce, par une décomposition de Benders efficace. Cette décomposition particulière s'avère être généralisable par rapport à une grande classe de problèmes communs dans la litérature et permet d'en résoudre des exemples de grande taille. Le premier article présente une méthodologie générale pour utiliser des données GPS d'une flotte de véhicules afin d'estimer les paramètres d'un modèle de demande dit recursive logit. Les traces GPS sont d'abord associées aux liens d'un réseau à l'aide d'un algorithme tenant compte de plusieurs facteurs. Les chemins formés par ces suites de liens et leurs caractéristiques sont utilisés afin d'estimer les paramètres d'un modèle de choix. Ces paramètres représentent la perception qu'ont les usagers de chacune de ces caractéristiques par rapport au choix de leur chemin. Les données utilisées dans cet article proviennent des véhicules appartenant à plusieurs compagnies de transport opérant principalement dans la région de Montréal. Le deuxième article aborde l'intégration d'un modèle de choix de chemin avec utilités aléatoires dans une nouvelle formulation biniveau pour le problème de capture de flot de trafic. Le modèle proposé permet de représenter différents comportements des usagers par rapport à leur choix de chemin en définissant les utilités d'arcs appropriées. Ces utilités sont stochastiques ce qui contribue d'autant plus à capturer un comportement réaliste des usagers. Le modèle biniveau est rendu linéaire à travers l'ajout d'un terme lagrangien basé sur la dualité forte et ceci mène à une décomposition de Benders particulièrement efficace. Les expériences numériques sont principalement menés sur un réseau représentant la ville de Winnipeg ce qui démontre la possibilité de résoudre des problèmes de taille relativement grande. Le troisième article démontre que l'approche du second article peut s'appliquer à une forme particulière de modèles biniveaux qui comprennent plusieurs problèmes différents. La décomposition est d'abord présentée dans un cadre général, puis dans un contexte où le second niveau du modèle biniveau est un problème de plus courts chemins. Afin d'établir que ce contexte inclut plusieurs applications, deux applications distinctes sont adaptées à la forme requise: le transport de matières dangeureuses et la capture de flot de trafic déterministe. Une troisième application, la conception et l'établissement de prix de réseau simultanés, est aussi présentée de manière similaire à l'Annexe B de cette thèse. / The subject of this thesis is the modeling of traffic in road networks and its integration in optimization models. In the literature, these two topics have to a large extent evolved independently: traffic is predicted more accurately by increasingly complex mathematical models, but this progress has not been incorporated in network design models where road users play a crucial role. The goal of this work is to integrate random utility models calibrated with real data into bilevel optimization models through an efficient Benders decomposition. This particular decomposition generalizes to a wide class of problems commonly found in the literature and can be used to solved large-scale instances. The first article presents a general methodology to use GPS data gathered from a fleet of vehicles to estimate the parameters of a recursive logit demand model. The GPS traces are first matched to the arcs of a network through an algorithm taking into account various factors. The paths resulting from these sequences of arcs, along with their characteristics, are used to estimate parameters of a choice model. The parameters represent users' perception of each of these characteristics in regards to their path choice behaviour. The data used in this article comes from trucks used by a number of transportation companies operating mainly in the Montreal region. The second article addresses the integration of a random utility maximization model in a new bilevel formulation for the general flow capture problem. The proposed model allows for a representation of different user behaviors in regards to their path choice by defining appropriate arc utilities. These arc utilities are stochastic which further contributes in capturing real user behavior. This bilevel model is linearized through the inclusion of a Lagrangian term based on strong duality which paves the way for a particularly efficient Benders decomposition. The numerical experiments are mostly conducted on a network representing the city of Winnipeg which demonstrates the ability to solve problems of a relatively large size. The third article illustrates how the approach used in the second article can be generalized to a particular form of bilevel models which encompasses many different problems. The decomposition is first presented in a general setting and subsequently in a context where the lower level of the bilevel model is a shortest path problem. In order to demonstrate that this form is general, two distinct applications are adapted to fit the required form: hazmat transportation network design and general flow capture. A third application, joint network design and pricing, is also similarly explored in Appendix B of this thesis.

This dissertation focuses on selected issues in regard to the mathematical modeling of electricity markets. In a first step the interrelations of electric power market modeling are highlighted a crossroad between operations research, applied economics, and engineering. In a second step the development of a large-scale continental European economic engineering model named ELMOD is described and the model is applied to the issue of wind integration. It is concluded that enabling the integration of low-carbon technologies appears feasible for wind energy. In a third step algorithmic work is carried out regarding a game theoretic model. Two approaches in order to solve a discretely-constrained mathematical program with equilibrium constraints using disjunctive constraints are presented. The first one reformulates the problem as a mixed-integer linear program and the second one applies the Benders decomposition technique. Selected numerical results are reported.

Recently, there has been a surge in the research and application of "Green energy" in the United States. This has been driven by the following three objectives: (1) to reduce the nation's reliance on foreign oil, (2) to mitigate emission of greenhouse gas, and (3) to create an economic stimulus within the United States. Switchgrass is the biomass of choice for the Southeastern United States. In this dissertation, we address a feedstock logistics problem associated with the delivery of switchgrass for conversion into biofuel. In order to satisfy the continual demand of biomass at a bioenergy plant, production fields within a 48-km radius of its location are assumed to be attracted into production. The bioenergy plant is expected to receive as many as 50-400 loads of biomass per day. As a result, an industrialized transportation system must be introduced as early as possible in order to remove bottlenecks and reduce the total system cost. Additionally, we assume locating multiple bioenergy plants within a given region for the production of biofuel. We develop mixed integer programming formulations for the feedstock logistics problem that we address and for some related problems, and we solve them either through the use of decomposition-based methods or directly through the use of CPLEX 12.1.0. The feedstock logistics problem that we address spans the entire system-from the growing of switchgrass to the transporting of bio-crude oil, a high energy density intermediate product, to a refinery for conversion into a final product. To facilitate understanding, we present the reader with a case study that includes a preliminary cost analysis of a real-life-based instance in order to provide the reader appropriate insights of the logistics system before applying optimization techniques for its solution. First, we consider the benefits of active versus passive ownership of the production fields. This is followed by a discussion on the selection of baler type, and then, a discussion of contracts between various business entities. The advantages of storing biomass at a satellite storage location (SSL) and interactions between the operations performed at the production field with those performed at the storage locations are then established. We also provide a detailed description of the operations performed at a SSL. Three potential equipment options are presented for transporting biomass from the SSLs to a utilization point, defined in this study as a Bio-crude Plant (BcP). The details of the entire logistics chain are presented in order to highlight the need for making decisions in view of the entire chain rather than basing them on its segments. We model the feedstock logistics problem as a combination of a 2-level facility location-allocation problem and a multiple traveling salesmen problem (mATSP). The 2-level facility location-allocation problem pertains to the allocation of production fields to SSLs and SSLs to one of the multiple bioenergy plants. The mATSP arises because of the need for scheduling unloading operations at the SSLs. To this end, we provide a detailed study of 13 formulations of the mATSP and their reformulations as ATSPs. First, we assume that the SSLs are always full, regardless of when they are scheduled to be unloaded. We, then, relax this assumption by providing precedence constraints on the availability of the SSLs. This precedence is defined in two different ways and, is then, effectively modeled utilizing all the formulations for the mATSP and ATSP. Given the location of a BcP for the conversion of biomass to bio-crude oil, we develop a feedstock logistics system that relies on the use of SSLs for temporary storage and loading of round bales. Three equipment systems are considered for handling biomass at the SSLs, and they are either placed permanently or are mobile, and thereby, travel from one SSL to another. We use a mathematical programming-based approach to determine SSLs and equipment routes in order to minimize the total cost incurred. The mathematical program is applied to a real-life production region in South-central Virginia (Gretna, VA), and it clearly reveals the benefits of using SSLs as a part of the logistics system. Finally, we provide a sensitivity analysis on the input parameters that we used. This analysis highlights the key cost factors in the model, and it emphasizes areas where biggest gains can be achieved for further cost reduction. For a more general scenario, where multiple BcPs have to be located, we use a nested Benders' decomposition-based method. First, we prove the validity of using this method. We, then, employ this method for the solution of a potential real-life instance. Moreover, we successfully solve problems that are more than an order of magnitude larger than those solved directly by CPLEX 12.1.0. Finally, we develop a Benders' decomposition-based method for the solution of a problem that gives rise to a binary sub-problem. The difficulty arises because of the sub-problem being an integer program for which the dual solution is not readily available. Our approach consists of first solving the integer sub-problem, and then, generating the convex hull at the optimal integer point. We illustrate this approach for an instance for which such a convex hull is readily available, but otherwise, it is too expensive to generate for the entire problem. This special instance is the solution of the mATSP (using Benders' decomposition) for which each of the sub-problems is an ATSP. The convex hull for the ATSP is given by the Dantzig, Fulkerson, and Johnson constraints. These constraints at a given integer solution point are only polynomial in number. With the inclusion of these constraints, a linear programming solution and its corresponding dual solution can now be obtained at the optimal integer points. We have proven the validity of using this method. However, the success of our algorithm is limited because of a large number of integer problems that must be solved at every iteration. While the algorithm is theoretically promising, the advantages of the decomposition do not seem to outweigh the additional cost resulting from solving a larger number of decomposed problems. / Ph. D.

This dissertation studies models for locating facilities in time varying demand environments. We describe the characteristics of the time varying demand that motivate the analysis of our location models in terms of total demand and the change in value and location of the demand of each customer. The first part of the dissertation is devoted to the dynamic location model, which determines the optimal time and location for establishing capacitated facilities when demand and cost parameters are time varying. This model minimizes the total cost over a discrete and finite time horizon for establishing, operating, and closing facilities, including the transportation costs for shipping demand from facilities to customers. The model is solved using Lagrangian relaxation and Benders? decomposition. Computational results from different time varying total demand structures demonstrate, empirically, the performance of these solution methods. The second part of the dissertation studies two location models where relocation of facilities is not allowed and the objective is to determine the optimal location of capacitated facilities that will have a good performance when demand and cost parameters are time varying. The first model minimizes the total cost for opening and operating facilities and the associated transportation costs when demand and cost parameters are time varying. The model is solved using Benders? decomposition. We show that in the presence of high relocation costs of facilities (opening and closing costs), this model can be solved as a special case by the dynamic location model. The second model minimizes the maximum regret or opportunity loss between a robust configuration of facilities and the optimal configuration for each time period. We implement local search and simulated annealing metaheuristics to efficiently obtain near optimal solutions for this model.