Cette thèse relate certains aspects liés à l'analyse post-Pareto issue de Problèmes d'Optimisation Vectorielle Stochastique. Un problème d'optimisation Vectorielle Stochastique consiste à optimiser l'espérance d'une fonction vectorielle aléatoire définie sur un ensemble arbitraire et à valeurs dans un espace sectoriel ordonné. L'ensemble des solutions de ce problème (appelé ensemble de Pareto) est composé des solutions admissibles qui assurent un certain équilibre entre les objectifs : il est impossible d'améliorer la valeur d'un objectif sans détériorer celle d'un autre. D'un point de vue technique, chaque solution de Pareto est acceptable. Nous nous posons alors le problème de la sélection de l'une d'entre elles : en supposant l'existence d'un décideur qui aurait son propre critère de décision, nous considérons le problème post-Pareto Stochastique qui vise à minimiser cette fonctionnelle sur l'ensemble de Pareto associé à un Problème d'Optimisation Vectorielle Stochastique. / This thesis explore related aspects to post-Pareto analysis arising from Stochastic Vector Optimization Problem. A Stochastic Vector Optimization Problem is to optimize a random vector objective function defined on an arbitrary set, and taking values in a partially ordered set. Its solution set (called Pareto set) consists of the feasible solutions which ensure some sort of equilibrium amongst the objectives. That is to say, Pareto solutions are such that noneof the objectives values can be improved further without deterioring another. Technically speaking, each Pareto solution is acceptable. The natural question that arises is : how to choose one solution ? One possible answer is to optimize an other objective over the Pareto set. Considering the existence of a decision-maker with its own criteria, we deal with the post-Pareto Stochastic Optimization Problem of minimizing its real-valued criteria over the Pareto set.

Le but de cette thèse est de proposer des méthodes générales afin de contourner l’intractabilité de problèmes d’optimisation multi-objectifs.Dans un premier temps, nous essayons d’apprécier la portée de cette intractabilité en déterminant une borne supérieure, facilement calculable, sur le nombre de points non dominés, connaissant le nombre de valeurs prises par chaque critère.Nous nous attachons ensuite à produire des représentations discrètes et tractables de l’ensemble des points non dominés de toute instance de problèmes d’optimisation multi-objectifs. Ces représentations doivent satisfaire des conditions de couverture, i.e. fournir une bonne approximation, de cardinalité, i.e. ne pas contenir trop de points, et si possible de stabilité, i.e. ne pas contenir de redondances. En s’inspirant de travaux visant à produire des ensembles ε-Pareto de petite taille, nous proposons tout d’abord une extension directe de ces travaux, puis nous axons notre recherche sur des ensembles ε-Pareto satisfaisant une condition supplémentaire de stabilité. Formellement, nous considérons des ensembles ε-Pareto particuliers, appelés (ε, ε′)-noyaux, qui satisfont une propriété de stabilité liée à ε′. Nous établissons des résultats généraux sur les (ε, ε′)-noyaux puis nous proposons des algorithmes polynomiaux qui produisent des (ε, ε′)-noyaux de petite taille pour le cas bi-objectif et nous donnons des résultats négatifs pour plus de deux objectifs. / The goal of this thesis is to propose new general methods to get around the intractability of multi-objective optimization problems.First, we try to give some insight on this intractability by determining an, easily computable, upper bound on the number of nondominated points, knowing the number of values taken on each criterion. Then, we are interested in producingsome discrete and tractable representations of the set of nondominated points for each instance of multi-objective optimization problems. These representations must satisfy some conditions of coverage, i.e. providing a good approximation, cardinality, i.e. it does not contain too many points, and if possible spacing, i.e. it does not include any redundancies. Starting from works aiming to produce ε-Pareto sets of small size, we first propose a direct extension of these works then we focus our research on ε-Pareto sets satisfying an additional condition of stability. Formally, we consider special ε-Pareto sets, called (ε, ε′)-kernels, which satisfy a property of stability related to ε′. We give some general results on (ε, ε′)-kernels and propose some polynomial time algorithms that produce small (ε, ε′)-kernels for the bicriteria case and we give some negative results for the tricriteria case and beyond.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-05-07T12:48:18Z No. of bitstreams: 2 Dissertação - Paulo Sérgio Zanin Júnior - 2018.pdf: 9799514 bytes, checksum: 91a3707c80f8aa8c8ee94301653047f9 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-05-07T13:03:18Z (GMT) No. of bitstreams: 2 Dissertação - Paulo Sérgio Zanin Júnior - 2018.pdf: 9799514 bytes, checksum: 91a3707c80f8aa8c8ee94301653047f9 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-05-07T13:03:18Z (GMT). No. of bitstreams: 2 Dissertação - Paulo Sérgio Zanin Júnior - 2018.pdf: 9799514 bytes, checksum: 91a3707c80f8aa8c8ee94301653047f9 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-04-20 / Distributed generation represents a new paradigm to the electricity offer expansion in distribution networks, which has been studied in several aspects. This dissertation aims to propose a multi-objective sizing and siting analysis model to assist the distributed generation units planning, based on three perspectives of concern: technical aspect, from the distribution company viewpoint, financial aspect, from the distributed generation owner’s viewpoint, and social-environmental aspect, from the society viewpoint. In this analysis, a genetic algorithm and the Maximin metric are implemented to obtain a Pareto optimal solutions set. As to decision-making analysis, the present value distributed generators net income; minimum losses set; and the Max-Min approximation are used to choose a final solution from the Pareto set. Study cases are accomplished using distribution networks by the proposed model and Pareto-sets are obtained, which allow the view of the best solutions. The mentioned decision-making analysis is applied to capture one of the Pareto-set solutions. Therefore, the results enable the choice of a distributed generation siting and sizing set and also allow complementary analysis referring to generation sources and the model itself. / A geração distribuída representa um novo paradigma para expansão da oferta de eletricidade em redes de distribuição e tem sido estudada em diversos aspectos. Este trabalho propõe um modelo multiobjetivo de alocação e dimensionamento para auxiliar o planejamento de unidades de geração distribuída, baseado em três perspectivas de interesse: aspecto técnico, do ponto de vista da companhia de distribuição; aspecto financeiro, do ponto do empreendedor de geração distribuída; e aspecto socioambiental, do ponto de vista da sociedade como um todo. Nesta análise, um algoritmo genético e a métrica Maximin são implementados para obter um conjunto de soluções ótimas de Pareto. Para a análise de tomada de decisão, de forma a escolher uma solução final do conjunto de Pareto, são utilizados: o valor presente da receita líquida total dos geradores distribuídos; configuração de perdas mínimas; e a aproximação Max-Min. São realizados estudos de caso em redes elétricas de distribuição por meio do modelo proposto e são obtidos conjuntos de Pareto, que permitem visualização das melhores soluções e então a análise de tomada de decisão é aplicada para apreender uma das soluções do conjunto-Pareto. Dessa forma, os resultados permitem a escolha de uma das possíveis configurações de alocação e dimensionamento de geração distribuída e também permitem análises complementares referentes às fontes de geração e ao próprio modelo proposto.