abstract: A good production schedule in a semiconductor back-end facility is critical for the on time delivery of customer orders. Compared to the front-end process that is dominated by re-entrant product flows, the back-end process is linear and therefore more suitable for scheduling. However, the production scheduling of the back-end process is still very difficult due to the wide product mix, large number of parallel machines, product family related setups, machine-product qualification, and weekly demand consisting of thousands of lots. In this research, a novel mixed-integer-linear-programming (MILP) model is proposed for the batch production scheduling of a semiconductor back-end facility. In the MILP formulation, the manufacturing process is modeled as a flexible flow line with bottleneck stages, unrelated parallel machines, product family related sequence-independent setups, and product-machine qualification considerations. However, this MILP formulation is difficult to solve for real size problem instances. In a semiconductor back-end facility, production scheduling usually needs to be done every day while considering updated demand forecast for a medium term planning horizon. Due to the limitation on the solvable size of the MILP model, a deterministic scheduling system (DSS), consisting of an optimizer and a scheduler, is proposed to provide sub-optimal solutions in a short time for real size problem instances. The optimizer generates a tentative production plan. Then the scheduler sequences each lot on each individual machine according to the tentative production plan and scheduling rules. Customized factory rules and additional resource constraints are included in the DSS, such as preventive maintenance schedule, setup crew availability, and carrier limitations. Small problem instances are randomly generated to compare the performances of the MILP model and the deterministic scheduling system. Then experimental design is applied to understand the behavior of the DSS and identify the best configuration of the DSS under different demand scenarios. Product-machine qualification decisions have long-term and significant impact on production scheduling. A robust product-machine qualification matrix is critical for meeting demand when demand quantity or mix varies. In the second part of this research, a stochastic mixed integer programming model is proposed to balance the tradeoff between current machine qualification costs and future backorder costs with uncertain demand. The L-shaped method and acceleration techniques are proposed to solve the stochastic model. Computational results are provided to compare the performance of different solution methods. / Dissertation/Thesis / Ph.D. Industrial Engineering 2011

[pt] A década de 90 foi marcante para a indústria de eletricidade, com a introdução de mercados competitivos em que os agentes geradores são livres para tomar suas decisões de produção e investimento, assumindo integralmente os riscos decorrentes de suas estratégias. O despacho e o preço spot neste tipo de mercado são definidos através de leilões diários, onde os agentes fornecem seus lances de preços/quantidades que expressam suas disposições em vender ou comprar energia. Os lances aceitos nos leilões, que estabelecem compromissos de geração, são definidos um dia antes da energia ser fisicamente gerada e injetada na rede. A ocorrência de eventos improváveis, como quebra de máquinas ou alterações nas condições meteorológicas, gera a necessidade de ajustes para compensar os desequilíbrios entre geração e carga, e para isso criaram-se mercados de ajustes. A base experimental do trabalho foi o Nord Pool, o mercado livre de energia dos países nórdicos que possui um mercado de ajustes chamado Elbas. Neste trabalho foi desenvolvido um modelo computacional que otimiza a estratégia de oferta de um agente hidrelétrico price-taker atuando no Nord Pool, que além de representar de forma detalhada as características operativas das usinas, leva em conta as negociações no mercado Elbas e o nível de aversão a risco do agente gerador, através da função objetivo que maximiza uma combinação convexa do valor esperado e do CVaR (Conditional Value at Risk) da renda líquida obtida da venda de energia. Cenários de preços spot e do mercado Elbas foram gerados baseados em modelos de séries temporais ARMA e GARCH, e para reduzir o esforço computacional e viabilizar o uso de um número adequado de cenários foram utilizadas técnicas de decomposição de Benders e Benders Multicut. O modelo desenvolvido possibilitou estudar a atuação dos agentes nos mercados spot e Elbas sob dois pontos de vista distintos: sob a ótica dos geradores, que buscam maximizar suas margens operacionais; e sob a ótica do regulador, cujo foco é investigar se o mercado Elbas cumpre seu papel de equilibrar a oferta e a demanda, e não fazendo com que os geradores especulem através de estratégias conjuntas nos dois mercados. Todos esses efeitos foram estudados e analisados para diferentes perfis de risco dos agentes e diferentes condições de mercado, ou seja, considerando períodos de diferentes volatilidades dos preços praticados no mercado Elbas e diferentes valores (ou custos de oportunidade) da água armazenada nos reservatórios das usinas hidrelétricas. Sob a ótica do agente, o trabalho mostrou que há um incentivo para o agente neutro a tentar usufruir de possíveis preços mais altos no mercado Elbas que os praticados no spot. Sob a ótica do regulador, os resultados mostram que o agente menos avesso a risco, dependendo das condições de mercado, opta por deslocar parte de sua energia do mercado spot para o Elbas, mostrando seu apetite por ganhos maiores independentemente do risco associado às suas decisões. O agente avesso a risco opta por transacionar menos energia no Elbas, principalmente em períodos mais voláteis, evitando com isso os piores cenários. Finalmente, considerando que normalmente empresas de energia são avessas a risco, o modelo de ajustes através do mercado Elbas se mostrou adequado, cumprindo naturalmente seu papel sem a necessidade de interferência do regulador. / [en] The widespread introduction of competitive mechanisms during the 1990s changed the panorama of the electricity industry around the world. Vertically integrated and centrally operated systems were replaced by market environments in which generators became free to make their production and investment decisions and, at the same time, assume the risk of their chosen strategies. Both the dispatch and the energy spot price in such markets result from two-sided auctions in which producing and consuming agents submit their price-quantity bids, expressing how much energy they are willing to buy or sell. The accepted bids, which commit agents to either deliver or consume power, are set a day before the energy delivery. However, since unexpected events may occur - such as changes in weather conditions or breakdowns of generation turbines - some adjustments might have to be done in order to compensate for the unbalances between total generation and load. These adjustments usually take place in the balancing markets. In the present work, we propose an optimization model for a price-taking hydropower producer who trades energy in the Nord Pool – the competitive electricity market encompassing the Nordic countries that comprises a balancing market called Elbas. The proposed model represents in details the operating aspects of the plants and takes into account the possibility of trading energy in the Elbas market. The model represents the level of risk aversion of the agent in its objective function, by maximizing a convex combination of the expected value and the CVaR (Conditional Value at Risk) of the net income obtained. Scenarios of spot and Elbas prices were generated based on time series models ARMA and GARCH and, in order to reduce the computational effort and enable the use of an adequate number of scenarios, Benders decomposition and Benders Multicut methods were applied. The developed model allowed us to study the behavior of agents in the spot and Elbas markets under two different viewpoints: from the perspective of the generators, which aim at maximizing its operating income; and from the viewpoint of the regulator, whose focus is on analyzing whether the Elbas market meets its role of balancing supply and demand, rather than leading generators to speculate through combined strategies in both markets. All these effects were studied and analyzed for different risk-averse profiles of the agents, and for different market conditions, i.e., considering periods of different volatilities of Elbas market prices and different water values (or opportunity costs) stored in the reservoirs of the hydroelectric power plants. From the perspective of the agent, the study showed that there are incentives for the risk- neutral agent to try to take advantage of possible higher prices in the Elbas. From the regulator’s viewpoint, the results show that the risk-neutral agents, depending on market conditions, choose to shift some of its energy generation to the Elbas market, showing their desire for higher incomes regardless of the risk associated with their decisions. The risk-averse agent chooses to trade less energy in Elbas, especially in volatile periods, thereby avoiding the worst scenarios. Finally, considering that energy companies are usually risk-averse, the adjustments made in the Elbas market were shown to be adequate, naturally meeting its role without requiring interventions from the regulator.

Notre travail de recherche traite la problématique de la conception d’une chaîne logistique multi-niveaux tout en tenant compte du cycle de vie du produit. Par cycle de vie du produit, nous voulons dire la succession des quatre phases de commercialisation que traverse un produit à travers le temps, à savoir : l’introduction, la croissance, la maturité et le déclin. L’objectif est de mette en place un modèle mathématique qui soit fondé sur une analyse approfondie des différents acteurs de la chaîne, selon la phase du cycle de vie du produit.Trois principaux modèles ont été développés dans cette thèse. Chacun fait l’objet d’un chapitre à part entière.Le premier modèle développé vise à concevoir une chaîne logistique de coût minimum, tout en prenant en considération l’efficacité des différents acteurs potentiels calculée selon plusieurs critères (coût, qualité, innovation, qualité du service, délais de livraisons, …), ainsi que sa variation au cours du cycle de vie du produit. Un deuxième modèle a été mis en place pour la conception d’une chaîne logistique durable, tout en prenant en considération le cycle de vie du produit. Dans ce modèle, trois objectifs différents ont été pris en compte à la fois, à savoir, un objectif économique, un objectif environnemental et un objectif social. Dans les deux premiers modèles, nous avons supposé que le produit aura un cycle de vie classique. Cependant, dans la réalité, ceci n’est pas toujours le cas. En effet, quelques produits connaissent des cycles de vie très atypiques et donc très éloignés de la courbe d’un cycle de vie théorique. Pour ce faire, un troisième modèle stochastique a été proposé pour la conception d’une chaîne logistique robuste, tenant compte des différents scénarios du cycle de vie du produit. / Our research addresses the problem of designing a multi-level supply chain, while taking into consideration the product life cycle. By product life cycle, we mean the succession of the four marketing stages that a product goes through since its introduction to the market and until it will be removed from. All products have a life cycle which can be classified into four discrete stages: introduction, growth, maturity and decline.Depending on the product life cycle phases, and based on a thorough analysis of the different supply chain potential actors, this study aims to establish mathematical models to design an efficient supply chain network. Three main models have been developed in this thesis. The first proposed model aims to design a product-driven supply chain with a minimal total cost, taking into consideration the evaluation of the different potential actors effectiveness, according to several criteria (cost, quality, innovation, quality service, timely delivery, ...).A second model was developed to design of a sustainable supply chain network, taking into account the product life cycle. In this model, three different objectives at the time were considered, namely, an economic objective, an environmental objective and a social objective.In the two previous models, we have assumed that the product has a classical life cycle. However, in the reality this is not always the case. Indeed, some products have very atypical life cycles, whose curves are very different from the classical one. To tackle this problem, in the third part of this thesis, we propose a stochastic model to design a robust supply chain network, taking into account the different product life cycle scenarios.

A incerteza é um elemento fundamental da realidade. Então, torna-se natural a busca por métodos que nos permitam representar o desconhecido em termos matemáticos. Esses problemas originam uma grande classe de programas probabilísticos reconhecidos como modelos de programação estocástica. Eles são mais realísticos que os modelos determinísticos, e tem por objetivo incorporar a incerteza em suas definições. Essa tese aborda os problemas probabilísticos da classe de problemas de multi-estágio com incerteza e com restrições probabilísticas e com restrições probabilísticas conjuntas. Inicialmente, nós propomos um modelo de administração de ativos e passivos multi-estágio estocástico para a indústria de fundos de pensão brasileira. Nosso modelo é formalizado em conformidade com a leis e políticas brasileiras. A seguir, dada a relevância dos dados de entrada para esses modelos de otimização, tornamos nossa atenção às diferentes técnicas de amostragem. Elas compõem o processo de discretização desses modelos estocásticos Nós verificamos como as diferentes metodologias de amostragem impactam a solução final e a alocação do portfólio, destacando boas opções para modelos de administração de ativos e passivos. Finalmente, nós propomos um “framework” para a geração de árvores de cenário e otimização de modelos com incerteza multi-estágio. Baseados na tranformação de Knuth, nós geramos a árvore de cenários considerando a representação filho-esqueda, irmão-direita o que torna a simulação mais eficiente em termos de tempo e de número de cenários. Nós também formalizamos uma reformulação do modelo de administração de ativos e passivos baseada na abordagem extensiva implícita para o modelo de otimização. Essa técnica é projetada pela definição de um processo de filtragem com “bundles”; e codifciada com o auxílio de uma linguagem de modelagem algébrica. A eficiência dessa metodologia é testada em um modelo de administração de ativos e passivos com incerteza com restrições probabilísticas conjuntas. Nosso framework torna possível encontrar a solução ótima para árvores com um número razoável de cenários. / Uncertainty is a key element of reality. Thus, it becomes natural that the search for methods allows us to represent the unknown in mathematical terms. These problems originate a large class of probabilistic programs recognized as stochastic programming models. They are more realistic than deterministic ones, and their aim is to incorporate uncertainty into their definitions. This dissertation approaches the probabilistic problem class of multistage stochastic problems with chance constraints and joint-chance constraints. Initially, we propose a multistage stochastic asset liability management (ALM) model for a Brazilian pension fund industry. Our model is formalized in compliance with the Brazilian laws and policies. Next, given the relevance of the input parameters for these optimization models, we turn our attention to different sampling models, which compose the discretization process of these stochastic models. We check how these different sampling methodologies impact on the final solution and the portfolio allocation, outlining good options for ALM models. Finally, we propose a framework for the scenario-tree generation and optimization of multistage stochastic programming problems. Relying on the Knuth transform, we generate the scenario trees, taking advantage of the left-child, right-sibling representation, which makes the simulation more efficient in terms of time and the number of scenarios. We also formalize an ALM model reformulation based on implicit extensive form for the optimization model. This technique is designed by the definition of a filtration process with bundles, and coded with the support of an algebraic modeling language. The efficiency of this methodology is tested in a multistage stochastic ALM model with joint-chance constraints. Our framework makes it possible to reach the optimal solution for trees with a reasonable number of scenarios.

abstract: This research develops decision support mechanisms for power system operation and planning practices. Contemporary industry practices rely on deterministic approaches to approximate system conditions and handle growing uncertainties from renewable resources. The primary purpose of this research is to identify soft spots of the contemporary industry practices and propose innovative algorithms, methodologies, and tools to improve economics and reliability in power systems. First, this dissertation focuses on transmission thermal constraint relaxation practices. Most system operators employ constraint relaxation practices, which allow certain constraints to be relaxed for penalty prices, in their market models. A proper selection of penalty prices is imperative due to the influence that penalty prices have on generation scheduling and market settlements. However, penalty prices are primarily decided today based on stakeholder negotiations or system operator’s judgments. There is little to no methodology or engineered approach around the determination of these penalty prices. This work proposes new methods that determine the penalty prices for thermal constraint relaxations based on the impact overloading can have on the residual life of the line. This study evaluates the effectiveness of the proposed methods in the short-term operational planning and long-term transmission expansion planning studies. The second part of this dissertation investigates an advanced methodology to handle uncertainties associated with high penetration of renewable resources, which poses new challenges to power system reliability and calls attention to include stochastic modeling within resource scheduling applications. However, the inclusion of stochastic modeling within mathematical programs has been a challenge due to computational complexities. Moreover, market design issues due to the stochastic market environment make it more challenging. Given the importance of reliable and affordable electric power, such a challenge to advance existing deterministic resource scheduling applications is critical. This ongoing and joint research attempts to overcome these hurdles by developing a stochastic look-ahead commitment tool, which is a stand-alone advisory tool. This dissertation contributes to the derivation of a mathematical formulation for the extensive form two-stage stochastic programming model, the utilization of Progressive Hedging decomposition algorithm, and the initial implementation of the Progressive Hedging subproblem along with various heuristic strategies to enhance the computational performance. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2017

Title: Reinsurance optimization using stochastic programming and risk measures Author: Jan Došel Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Martin Branda, Ph.D., Department of Probability and Mathe- matical Statistics Abstract: The diploma thesis deals with an application of a stochastic progra- mming in a reinsurance optimization problem in terms of a present regulatory framework of the insurance companies within the European Union, i.e. Solvency II. In this context, the reinsurance does not only transfer a portion of the risk to the reinsurer but also reduces an amout of required capital. The thesis utilizes certain risk measures and their properties, premium principles and non-linear in- teger programming. In the theoretical part, there are basic terms from Solvency II, reinsurance, risk measures and the comonotonicity of random variables descri- bed and the optimization problem itself is derived. The approach is then applied in the practical part on data of Czech Insurers' Bureau using the GAMS software. Finally, a stability of the solution is tested depending on several parameters. Keywords: reinsurance optimization, stochastic programming, Solvency II, risk measures 1

A incerteza é um elemento fundamental da realidade. Então, torna-se natural a busca por métodos que nos permitam representar o desconhecido em termos matemáticos. Esses problemas originam uma grande classe de programas probabilísticos reconhecidos como modelos de programação estocástica. Eles são mais realísticos que os modelos determinísticos, e tem por objetivo incorporar a incerteza em suas definições. Essa tese aborda os problemas probabilísticos da classe de problemas de multi-estágio com incerteza e com restrições probabilísticas e com restrições probabilísticas conjuntas. Inicialmente, nós propomos um modelo de administração de ativos e passivos multi-estágio estocástico para a indústria de fundos de pensão brasileira. Nosso modelo é formalizado em conformidade com a leis e políticas brasileiras. A seguir, dada a relevância dos dados de entrada para esses modelos de otimização, tornamos nossa atenção às diferentes técnicas de amostragem. Elas compõem o processo de discretização desses modelos estocásticos Nós verificamos como as diferentes metodologias de amostragem impactam a solução final e a alocação do portfólio, destacando boas opções para modelos de administração de ativos e passivos. Finalmente, nós propomos um “framework” para a geração de árvores de cenário e otimização de modelos com incerteza multi-estágio. Baseados na tranformação de Knuth, nós geramos a árvore de cenários considerando a representação filho-esqueda, irmão-direita o que torna a simulação mais eficiente em termos de tempo e de número de cenários. Nós também formalizamos uma reformulação do modelo de administração de ativos e passivos baseada na abordagem extensiva implícita para o modelo de otimização. Essa técnica é projetada pela definição de um processo de filtragem com “bundles”; e codifciada com o auxílio de uma linguagem de modelagem algébrica. A eficiência dessa metodologia é testada em um modelo de administração de ativos e passivos com incerteza com restrições probabilísticas conjuntas. Nosso framework torna possível encontrar a solução ótima para árvores com um número razoável de cenários. / Uncertainty is a key element of reality. Thus, it becomes natural that the search for methods allows us to represent the unknown in mathematical terms. These problems originate a large class of probabilistic programs recognized as stochastic programming models. They are more realistic than deterministic ones, and their aim is to incorporate uncertainty into their definitions. This dissertation approaches the probabilistic problem class of multistage stochastic problems with chance constraints and joint-chance constraints. Initially, we propose a multistage stochastic asset liability management (ALM) model for a Brazilian pension fund industry. Our model is formalized in compliance with the Brazilian laws and policies. Next, given the relevance of the input parameters for these optimization models, we turn our attention to different sampling models, which compose the discretization process of these stochastic models. We check how these different sampling methodologies impact on the final solution and the portfolio allocation, outlining good options for ALM models. Finally, we propose a framework for the scenario-tree generation and optimization of multistage stochastic programming problems. Relying on the Knuth transform, we generate the scenario trees, taking advantage of the left-child, right-sibling representation, which makes the simulation more efficient in terms of time and the number of scenarios. We also formalize an ALM model reformulation based on implicit extensive form for the optimization model. This technique is designed by the definition of a filtration process with bundles, and coded with the support of an algebraic modeling language. The efficiency of this methodology is tested in a multistage stochastic ALM model with joint-chance constraints. Our framework makes it possible to reach the optimal solution for trees with a reasonable number of scenarios.

Au cours des dernières années, les centres d'appels ont été introduits avec succès par de nombreuses entreprises axées sur les services comme les banques et les compagnies d'assurance. Ils deviennent le principal point de contact avec les clients, et une partie intégrante de la majorité des sociétés. L'émergence à grande échelle des centres d'appels a créé une source féconde de problèmes de gestion des opérations. Dans cette thèse, nous nous concentrons sur la question de dimensionnement et définition des emplois du temps dans les centres d'appels. L'objectif de notre travail consiste à développer des analyses qualitatives ainsi que quantitatives, afin de déduire des recommandations utiles aux managers.Nous analysons quatre problèmes qui tiennent compte de l'incertitude sur les paramètres d'arrivée des appels. Le processus d'arrivée des appels est supposé suivre un processus non stationnaire et doublement stochastique avec un taux moyen d'arrivée aléatoire.Dans le premier modèle, nous considérons un centre d’appels avec une seule vacation possible. Les agents traitent en même temps des appels entrants et des tâches de back-office. Ceci permet d’avoir une certaine souplesse pour modifier en temps réel la capacité instantanée de traitement des appels entrants. Nous analysons l'impact de la flexibilité offerte par les charges de travail de back-office.Dans le deuxième modèle, nous considérons un centre d'appels avec plusieurs vacations possibles. Les agents traitent seulement des appels entrants. Dans ce modèle, le dimensionnement initialement établi peut être corrigé au cours de la journée de travail. Nous proposons une approche de programmation stochastique en deux étapes et une approche de programmation réglable robuste pour résoudre le problème d’optimisation. En particulier, nous analysons et montrons l'avantage supplémentaire d'utiliser le réglage dynamique sur les coûts de dimensionnement du centre d’appels. Dans le troisième modèle, nous considérons un autre type d'incertitude supplémentaire, qui est l'incertitude sur la distribution de probabilité d'un paramètre aléatoire. Nous proposons une approche combinant la programmation stochastique et la programmation distributionnellement robuste, et nous évaluons son rendement. Le dernier problème de dimensionnement d’un centre d'appels pour lequel le manager se propose de satisfaire un niveau de service global pour toute la journée au lieu d’un niveau de service objectif par période. Nous permettons également la mise à jour du dimensionnement au cours de la journée. Dans notre analyse, nous montrons en particulier les avantages de l'ajout de la flexibilité de mise à jour, et soulignons l'impact d'avoir une contrainte de service niveau globale sur les performances. / In the past few years, call centers have been introduced with great success by many service-oriented companies such as banks and insurance companies. They become the main point of contact with the customers, and an integral part of the majority of corporations. The large-scale emergence of call centers has created a fertile source of management issues. In this thesis, we focus on the issue of staffing and scheduling of call centers. The objective of our work is to derive both qualitative and quantitative results for practical management.We specifically address the analysis of four problems that take into account the important feature of uncertainty in the call arrival parameters. The call arrival process is assumed to follow a doubly non-stationary stochastic process with a random mean arrival rate.In the first model, we consider a single-shift call center blending inbound calls and back-office jobs. By allowing the possibility of real-time changes in the capacity dealing with inbound calls, we analyze the impact of the flexibility offered by back-office jobs.In the second model, we consider a multi-shift call center with single type of inbound calls, in which the scheduling update is allowed. We propose a two-stage stochastic programming approach and an adjustable robust programming approach to efficiently solve the problem. We also analyze the benefits of using dynamic adjustment on scheduling.In the third model, we consider an additional type of uncertainty, namely the uncertainty on the probability distribution of a random parameter. We propose an approach combining stochastic programming and distributionally robust programming, and evaluate its performance.The last model deals with a call center optimization under a global service level constraint instead of period by period constraints. We again allow scheduling decisions to be updated during the middle of the day. We show the advantages of adding the update flexibility, and point out the impact of having a global service level constraint on performance

Les contraintes en probabilité constituent un modèle pertinent pour gérer les incertitudes dans les problèmes de décision. En management d’énergie de nombreux problèmes d’optimisation ont des incertitudes sous-jacentes. En particulier c’est le cas des problèmes de gestion de la production au court-terme. Dans cette Thèse, nous investiguons les contraintes probabilistes sous l’angle théorique, algorithmique et applicative. Nous donnons quelques nouveaux résultats de différentiabilité des contraintes en probabilité et de convexité des ensembles admissibles. Des nouvelles variantes des méthodes de faisceaux « proximales » et « de niveaux » sont spécialement mises au point pour traiter des problèmes d’optimisation convexe sous contrainte en probabilité. Ces algorithmes gèrent en particulier, les erreurs d’évaluation de la contrainte en probabilité, ainsi que son gradient. La convergence vers une solution du problème est montrée. Enfin, nous examinons deux applications : l’optimisation d’une vallée hydraulique sous incertitude sur les apports et l’optimisation d’un planning de production sous incertitude sur la demande. Dans les deux cas nous utilisons une contrainte en probabilité pour gérer les incertitudes. Les résultats numériques présentés semblent montrer la faisabilité de résoudre des problèmes d’optimisation avec une contrainte en probabilité jointe portant sur un système de environ 200 contraintes. Il s’agit de l’ordre de grandeur nécessaire pour les applications. Les nouveaux résultats de différentiabilité concernent à la fois des contraintes en probabilité portant sur des systèmes linéaires et non-linéaires. Dans le deuxième cas, la convexité dans l’argument représentant le vecteur incertain est requise. Ce vecteur est supposé suivre une loi Gaussienne ou Student multi-variée. Les formules de gradient permettent l’application directe d’un schéma d’évaluation numérique efficient. Pour les contraintes en probabilité qui peuvent se réécrire à l’aide d’une Copule, nous donnons de nouveau résultats de convexité pour l’ensemble admissibles. Ces résultats requirent la concavité généralisée de la Copule, les distributions marginales sous-jacents et du système d’incertitude. Il est suffisant que ces propriétés de concavité généralisée tiennent sur un ensemble spécifique. / In optimization problems involving uncertainty, probabilistic constraints are an important tool for defining safety of decisions. In Energy management, many optimization problems have some underlying uncertainty. In particular this is the case of unit commitment problems. In this Thesis, we will investigate probabilistic constraints from a theoretical, algorithmic and applicative point of view. We provide new insights on differentiability of probabilistic constraints and on convexity results of feasible sets. New variants of bundle methods, both of proximal and level type, specially tailored for convex optimization under probabilistic constraints, are given and convergence shown. Both methods explicitly deal with evaluation errors in both the gradient and value of the probabilistic constraint. We also look at two applications from energy management: cascaded reservoir management with uncertainty on inflows and unit commitment with uncertainty on customer load. In both applications uncertainty is dealt with through the use of probabilistic constraints. The presented numerical results seem to indicate the feasibility of solving an optimization problem with a joint probabilistic constraint on a system having up to 200 constraints. This is roughly the order of magnitude needed in the applications. The differentiability results involve probabilistic constraints on uncertain linear and nonlinear inequality systems. In the latter case a convexity structure in the underlying uncertainty vector is required. The uncertainty vector is assumed to have a multivariate Gaussian or Student law. The provided gradient formulae allow for efficient numerical sampling schemes. For probabilistic constraints that can be rewritten through the use of Copulae, we provide new insights on convexity of the feasible set. These results require a generalized concavity structure of the Copulae, the marginal distribution functions of the underlying random vector and of the underlying inequality system. These generalized concavity properties may hold only on specific sets.

Este trabalho apresenta uma abordagem via programação estocástica linear para a tomada de decisão de investimento em um fundo de pensão com plano de benefícios do tipo benefício definido. Propõe-se uma nova metodologia para a definição da alocação da carteira do fundo no instante inicial baseada na média de vários cenários econômicos gerados aleatoriamente. Como exemplo de aplicação, essa metodologia é utilizada para resolver o problema da alocação inicial da carteira de um grande fundo de pensão brasileiro e a alocação inicial obtida é avaliada em termos da probabilidade de insolvência e VaR, valor em risco, do fundo no instante final do horizonte de planejamento de investimento. / This paper presents an approach via linear stochastic programming for investment decision making in a defined benefit pension fund plan. It proposes a new methodology for defining the allocation of the portfolio at the initial time based on the average of several randomly generated economic scenarios. As an illustrative example, this methodology is used to solve the problem of portfolio initial allocation of a large Brazilian pension fund and the obtained initial allocation is evaluated in terms of funds probability of default and VaR, Value-at-Risk, at the final time of the investment planning horizon.