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.

In this thesis, we study a series of closely related multi-stage stochastic programming models in production planning, from both a modeling and an algorithmic point of view. We first consider a very simple multi-stage stochastic lot-sizing problem, involving a single item with no fixed charge and capacity constraint. Although a multi-stage stochastic integer program, this problem can be shown to have a totally unimodular constraint matrix. We develop primal and dual algorithms by exploiting the problem structure. Both algorithms are strongly polynomial, and therefore much more efficient than the Simplex method. Next, motivated by applications in semiconductor tool planning, we develop a general capacity planning problem under uncertainty. Using a scenario tree to model the evolution of the uncertainties, we present a multi-stage stochastic integer programming formulation for the problem. In contrast to earlier two-stage approaches, the multi-stage model allows for revision of the capacity expansion plan as more information regarding the uncertainties is revealed. We provide analytical bounds for the value of multi-stage stochastic programming over the two-stage approach. By exploiting the special simple stochastic lot-sizing substructure inherent in the problem, we design an efficient approximation scheme and show that the proposed scheme is asymptotically optimal. We conduct a computational study with respect to a semiconductor-tool-planning problem. Numerical results indicate that even an approximate solution to the multi-stage model is far superior to any optimal solution to the two-stage model. These results show that the value of multi-stage stochastic programming for this class of problem is extremely high. Next, we extend the simple stochastic lot-sizing model to an infinite horizon problem to study the planning horizon of this problem. We show that an optimal solution of the infinite horizon problem can be approximated by optimal solutions of a series of finite horizon problems, which implies the existence of a planning horizon. We also provide a useful upper bound for the planning horizon.

Electric energy constitutes one of the most crucial elements to almost every aspect of life of people. The modern electric power systems face several challenges such as efficiency, economics, sustainability, and reliability. Increase in electrical energy demand, distributed generations, integration of uncertain renewable energy resources, and demand side management are among the main underlying reasons of such growing complexity. Additionally, the elements of power systems are often vulnerable to failures because of many reasons, such as system limits, weak conditions, unexpected events, hidden failures, human errors, terrorist attacks, and natural disasters. One common factor complicating the operation of electrical power systems is the underlying uncertainties from the demands, supplies and failures of system components. Stochastic programming provides a mathematical framework for decision making under uncertainty. It enables a decision maker to incorporate some knowledge of the intrinsic uncertainty into the decision making process. In this dissertation, we focus on application of two-stage and multistage stochastic programming approaches to electric energy systems modeling and optimization. Particularly, we develop models and algorithms addressing the sustainability and reliability issues in power systems. First, we consider how to improve the reliability of power systems under severe failures or contingencies prone to cascading blackouts by so called islanding operations. We present a two-stage stochastic mixed-integer model to find optimal islanding operations as a powerful preventive action against cascading failures in case of extreme contingencies. Further, we study the properties of this problem and propose efficient solution methods to solve this problem for large-scale power systems. We present the numerical results showing the effectiveness of the model and investigate the performance of the solution methods. Next, we address the sustainability issue considering the integration of renewable energy resources into production planning of energy-intensive manufacturing industries. Recently, a growing number of manufacturing companies are considering renewable energies to meet their energy requirements to move towards green manufacturing as well as decreasing their energy costs. However, the intermittent nature of renewable energies imposes several difficulties in long term planning of how to efficiently exploit renewables. In this study, we propose a scheme for manufacturing companies to use onsite and grid renewable energies provided by their own investments and energy utilities as well as conventional grid energy to satisfy their energy requirements. We propose a multistage stochastic programming model and study an efficient solution method to solve this problem. We examine the proposed framework on a test case simulated based on a real-world semiconductor company. Moreover, we evaluate long-term profitability of such scheme via so called value of multistage stochastic programming.

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.

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.

This dissertation addresses the modeling and solution of mixed-integer linear multistage stochastic programming problems involving both endogenous and exogenous uncertain parameters. We propose a composite scenario tree that captures both types of uncertainty, and we exploit its unique structure to derive new theoretical properties that can drastically reduce the number of non-anticipativity constraints (NACs). Since the reduced model is often still intractable, we discuss two special solution approaches. The first is a sequential scenario decomposition heuristic in which we sequentially solve endogenous MILP subproblems to determine the binary investment decisions, fix these decisions to satisfy the first-period and exogenous NACs, and then solve the resulting model to obtain a feasible solution. The second approach is Lagrangean decomposition. We present numerical results for a process network planning problem and an oilfield development planning problem. The results clearly demonstrate the efficiency of the special solution methods over solving the reduced model directly. To further generalize this work, we also propose a graph-theory algorithm for non-anticipativity constraint reduction in problems with arbitrary scenario sets. Finally, in a break from the rest of the thesis, we present the basics of stochastic programming for non-expert users.

This thesis deals with multi-stage stochastic programming in the context of random process representation. Basic structure for random process is a scenario tree. The thesis introduces general and stage-independent scenario tree and their properties. Scenario trees can be also combined with Markov chains which describe the state of the system and determine which scenario tree should be used. Another structure which enables reduce the complexity of the problem is a scenario lattice. Scenario generation is performed using moment method. Scenario trees are used for representation of random returns as the input to the investment problem.

This thesis deals with multi-stage stochastic programming in the context of random process representation. Basic structure for random process is a scenario tree. The thesis introduces general and stage-independent scenario tree and their properties. Scenario trees combined with Markov chains are also introduced. Markov chains states determine if there is a crisis period or not. Information about historical number of crises helps us to construct a scenario lattice. Scenario generation is performed using moment method. Scenario trees are used as an input to the investment problem.

The thesis deals with a multistage stochastic model and its application to a number of practical problems. Special attention is devoted to the case where a random element follows an autoregressive sequence and the constraint sets correspond to the individual probability constraints. For this case conditions under which is the problem well-defined are specified. Further, the approximation of the problem and its convergence rate under the empirical estimate of the distribution function is analyzed. Finally, an example of the investment in financial instruments is solved, which is defined as a two-stage stochastic programming problem with the probability constraint and a random element following an autoregressive sequence. Powered by TCPDF (www.tcpdf.org)

The main objective of this thesis is to investigate risk neutral and risk averse approaches to multistage stochastic programming with applications to hydrothermal operation planning problems. The purpose of hydrothermal system operation planning is to define an operation strategy which, for each stage of the planning period, given the system state at the beginning of the stage, produces generation targets for each plant. This problem can be formulated as a large scale multistage stochastic linear programming problem. The energy rationing that took place in Brazil in the period 2001/2002 raised the question of whether a policy that is based on a criterion of minimizing the expected cost (i.e. risk neutral approach) is a valid one when it comes to meet the day-to-day supply requirements and taking into account severe weather conditions that may occur. The risk averse methodology provides a suitable framework to remedy these deficiencies. This thesis attempts to provide a better understanding of the risk averse methodology from the practice perspective and suggests further possible alternatives using robust optimization techniques. The questions investigated and the contributions of this thesis are as follows. First, we suggest a multiplicative autoregressive time series model for the energy inflows that can be embedded into the optimization problem that we investigate. Then, computational aspects related to the stochastic dual dynamic programming (SDDP) algorithm are discussed. We investigate the stopping criteria of the algorithm and provide a framework for assessing the quality of the policy. The SDDP method works reasonably well when the number of state variables is relatively small while the number of stages can be large. However, as the number of state variables increases the convergence of the SDDP algorithm can become very slow. Afterwards, performance improvement techniques of the algorithm are discussed. We suggest a subroutine to eliminate the redundant cutting planes in the future cost functions description which allows a considerable speed up factor. Also, a design using high performance computing techniques is discussed. Moreover, an analysis of the obtained policy is outlined with focus on specific aspects of the long term operation planning problem. In the risk neutral framework, extreme events can occur and might cause considerable social costs. These costs can translate into blackouts or forced rationing similarly to what happened in 2001/2002 crisis. Finally, issues related to variability of the SAA problems and sensitivity to initial conditions are studied. No significant variability of the SAA problems is observed. Second, we analyze the risk averse approach and its application to the hydrothermal operation planning problem. A review of the methodology is suggested and a generic description of the SDDP method for coherent risk measures is presented. A detailed study of the risk averse policy is outlined for the hydrothermal operation planning problem using different risk measures. The adaptive risk averse approach is discussed under two different perspectives: one through the mean-$avr$ and the other through the mean-upper-semideviation risk measures. Computational aspects for the hydrothermal system operation planning problem of the Brazilian interconnected power system are discussed and the contributions of the risk averse methodology when compared to the risk neutral approach are presented. We have seen that the risk averse approach ensures a reduction in the high quantile values of the individual stage costs. This protection comes with an increase of the average policy value - the price of risk aversion. Furthermore, both of the risk averse approaches come with practically no extra computational effort and, similarly to the risk neutral method, there was no significant variability of the SAA problems. Finally, a methodology that combines robust and stochastic programming approaches is investigated. In many situations, such as the operation planning problem, the involved uncertain parameters can be naturally divided into two groups, for one group the robust approach makes sense while for the other the stochastic programming approach is more appropriate. The basic ideas are discussed in the multistage setting and a formulation with the corresponding dynamic programming equations is presented. A variant of the SDDP algorithm for solving this class of problems is suggested. The contributions of this methodology are illustrated with computational experiments of the hydrothermal operation planning problem and a comparison with the risk neutral and risk averse approaches is presented. The worst-case-expectation approach constructs a policy that is less sensitive to unexpected demand increase with a reasonable loss on average when compared to the risk neutral method. Also, we comp are the suggested method with a risk averse approach based on coherent risk measures. On the one hand, the idea behind the risk averse method is to allow a trade off between loss on average and immunity against unexpected extreme scenarios. On the other hand, the worst-case-expectation approach consists in a trade off between a loss on average and immunity against unanticipated demand increase. In some sense, there is a certain equivalence between the policies constructed using each of these methods.