Este trabalho propõe a aplicação de dois diferentes conceitos para a resolução do Problema de Estivagem de Unidades de Celulose - PEUC, que de acordo com Ribeiro e Lorena (2008) pode ser definido como um problema que busca alocar a máxima quantidade de unidades de celulose ao porão de cargas de um dado navio, respeitando as restrições físicas de dimensões, de posicionamento, de não-sobreposição das unidades e de capacidade máxima do porão do navio. Esse tipo de problema se encaixa, no contexto da Pesquisa Operacional, na classe de Corte e Empacotamento (Cutting and Packing - C&P) e pode ser classificado, de acordo com a tipologia de Wäscher, Haußner e Schumann (2007), como sendo um Single Large Object Placement Problem (SLOPP). Em última instância, o objetivo do PEUC é definir o melhor plano de estivagem para o carregamento de unidades de celulose em um dado porão de um navio, maximizando a área ocupada pelas unidades de celulose. Trata-se de um problema NP-Completo (DOWSLAND; DOWSLAND, 1992; BISCHOFF; WÄSCHER, 1995; MALAGUTI; DURáN; TOTH, 2013) e por isso foram propostas duas abordagens para buscar a melhoria das soluções encontradas e/ou redução do tempo computacional necessário. As abordagens propostas, o Modelo Matemático Modificado e o Método Iterativo de Solução, apresentaram bons resultados para instâncias experimentais, confirmando a efetividade de suas aplicações. Os resultados foram melhores tanto na qualidade das soluções (ocupação total do objeto), como no tempo computacional necessário. Também foram avaliadas quatro instâncias reais, com a comparação dos planos de estivagem resultantes da aplicação dos modelos matemáticos com os planos reais, elaborados manualmente por especialistas. Em três dos quatro casos os resultados das abordagens aqui propostas se mostraram melhores que os planos reais. / This work proposes the application of two different concepts to tackle the Woodpulp Stowage Problem - WSP, that according to Ribeiro e Lorena (2008) can be defined as a problem that seeks the allocation of the maximum quantity of woodpulp units inside the hold of a cargo vessel, always respecting the physical constraints, positioning constraints, non-overlapping of units and also the hold capacity. This kind of problem fits, in the context of Operational Research, into the class of Cutting & Packing and can be classified, according to Wäscher, Haußner e Schumann (2007) typology, as a Single Larga Object Placement Problem (SLOPP). Ultimately the objective of the WSP is to define the best stowage plan for the loading of woodpulp units inside a given hold of a given cargo vessel, maximizing the total area occupied by the woodpulp units. As it\'s a NP-Complete problem (DOWSLAND; DOWSLAND, 1992; BISCHOFF; WÄSCHER, 1995; MALAGUTI; DURáN; TOTH, 2013) two approaches were proposed to improve the quality of the resulting solutions and/or the reduction of the computational time needed. The proposed approaches, the Modified Mathematical Model and the Iterative Solution Method, showed good results for experimental instances, confirming the effectiveness of these approaches. The results were better regarding the quality of the solutions (total occupied area of the object) and also regarding the computational time needed. Also, four real instances were evaluated, comparing the results of the mathematical models with the real stowage plans, manually created by specialists. In three of the four instances, the proposed approaches showed better results than the real stowage plans.

La prise en compte de machines à traitement par batch dans les problèmes d’ordonnancement d’ateliers complexes de type job-shop est particulièrement difficile. La fabrication de semiconducteurs est probablement l’une des applications pratiques les plus importantes pour ce types de problèmes. Nous considérons un problème d’ordonnancement de type job-shop flexible avec « p-batching », des flux rentrants, des temps de préparation dépendant de la séquence et des dates de début au plus tôt. Le but c’est d’optimiser différentes fonctions objectives régulières.Les approches existantes par graphe disjonctif pour ce problème utilise des nœuds dédiés pour représenter explicitement les batches. Afin de faciliter la modification du graphe conjonctif, notre nouvelle modélisation réduit cette complexité en modélisant les décisions de batching à travers les poids des arcs. Une importante contribution de cette thèse est un algorithme original qui prend les décisions de batching lors du parcours du graphe. Cet algorithme est complété par un déplacement (« move ») intégré qui permet de reséquencer ou réaffecter les opérations. Cette combinaison donne un voisinage riche que nous appliquons dans une approche méta-heuristique de type GRASP.Nous étendons cette approche en prenant en compte de nouvelles contraintes qui ont un rôle important dans l’application industrielle considérée. En particulier, nous modélisons de manière explicite les ressources internes des machines, et nous considérons un temps maximum d’attente entre deux opérations quelconques d’une gamme de fabrication. Les résultats numériques sur des instances de la littérature pour des problèmes plus simples ainsi que sur de nouvelles instances montrent la généricité et l’applicabilité de notre approche. Notre nouvelle modélisation permet de faciliter les extensions à d’autres contraintes complexes rencontrées dans les applications industrielles. / The integration of batching machines within a job-shop environment leads to a complex job-shop scheduling problem. Semiconductor manufacturing presumably represents one of the most prominent practical applications for such problems. We consider a flexible job-shop scheduling problem with p-batching, reentrant flows, sequence dependent setup times and release dates while considering different regular objective functions. The scheduling of parallel batching machines and variants of the job-shop scheduling problem are well-studied problems whereas their combination is rarely considered.Existing disjunctive graph approaches for this combined problem rely on dedicated nodes to explicitly represent batches. To facilitate modifications of the graph, our new modeling reduces this complexity by encoding batching decisions into edge weights. An important contribution is an original algorithm that takes batching decisions “on the fly” during graph traversals. This algorithm is complemented by an integrated move to resequence and reassign operations. This combination yields a rich neighborhood that we apply within a GRASP based metaheuristic approach.We extend this approach by taking further constraints into account that are important in the considered industrial application. In particular, we model internal resources of machines in detail and take maximum time lag constraints into account. Numerical results for benchmark instances of different problem types show the generality and applicability of our approach. The conciseness of our idea facilitates extensions towards further complex constraints needed in real-world applications.

The Cooperative Game Theory is a scientific discipline which offers rich mathematical apparatus for describing complex situations in the social reality. Its apparatus includes an extension to hierarchical structures and therefore can be applied to numerous research problems from the International Relations field. However, a cooperative game theoretical approach is very scarcely used. The main goal of the diploma thesis is to demonstrate, on the research problem of decision making in participation in local conflicts, the benefits of results that can be achieved by the application of the Cooperative Game Theory. In the first part of the thesis, theoretical foundations are laid and basic concepts are introduced. The second part is focused on forming a series of models of cooperative games with hierarchical structures from four local conflict situations, which are subsequently restricted in order to describe authoritative relations in structure. Restricted games are solved, the results are interpreted and evaluation of how these results can contribute to addressing the research problem follows.

This report considers an application of mixed-integer disjunctive programming (MIDP)where a theoretical robot can jump from one point to another and where the number ofjumps is to be minimized. The robot is only able to jump to the north, south, east andwest. Furthermore, the robot should also be able to navigate and jump around or across anypotential obstacles on the way. The algorithm for solving this problem is set to terminatewhen the robot has reached a set of end coordinates. The goal of this report is to find amethod for solving this problem and to investigate the time complexity of such a method.The problem is converted to big-M representation and solved numerically. Gurobi is theoptimization solver used in this thesis. The model created and implemented with Gurobiyielded optimal solutions to problems of the form above of varying complexity. For most ofcases tested, the time complexity appeared to be linear, but this is likely due to presolvingperformed by Gurobi before running the optimization. Further tests are needed to determinethe time complexity of Gurobi’s optimization algorithm for this specific type of problem.

Orientador: Rubén Augusto Romero Lázaro / Resumo: Tradicionalmente, o problema de Planejamento da Expansão de Sistemas de Transmissão (PEST) é solucionado considerando apenas um único cenário de geração, embora sistemas elétricos reais operem em diferentes cenários de geração. Nessa pesquisa são propostos modelos matemáticos para resolver o problema de PEST, considerando múltiplos cenários de geração de forma que o plano de expansão obtido permita uma operação adequada do sistema. No modelo proposto, o custo de investimento é maior em relação aos planos de expansão encontrados pelo planejamento tradicional, que considera apenas um cenário de geração. Para reduzir o correspondente custo de investimento são apresentadas estratégias eficientes para encontrar planos de expansão para o problema de PEST considerando múltiplos cenários. As estratégias utilizadas foram: permitir pequenos cortes de carga; permitir o deslocamento do nível de geração em uma pequena faixa de geração mínima e máxima em relação à geração ideal e permitir pequenas sobrecargas nas linhas de transmissão. Adicionalmente, uma combinação entre essas estratégias é apresentada e o problema PEST também foi resolvido para o planejamento multiestágio, considerando múltiplos cenários de geração. O método proposto foi implementado com o uso da linguagem de modelagem algébrica AMPL e resolvido com o uso do solver comercial CPLEX. Os resultados encontrados correspondem à propostas de solução que são válidas para diferentes cenários de geração e apresentam diferentes alt... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: Traditionally, the Transmission Network Expansion Problem is solved considering only a single generation scenario. However, a real power system operates in different generation scenarios. This work presents the disjunctive linear model for the Transmission Network Expansion Problem considering multiple generation scenarios to provide a single expansion plan, which must operate in a appropriate way in each one of the different scenarios. The investment cost of the proposed model is greater in relation to the traditional expansion plans, that consider single generation scenario. In order to reduce the investment costs, efficient strategies are presented to find the expansion plans for multiples scenarios. Therefore those strategies are: allow small load cuts; allow generation level displacement in a narrow generation range in relation to the ideal one; and allow small overload in the transmission lines. Moreover, a combination between those strategies is shown and the Transmission Network Expansion Problem was also solved for multistage planning for multiple generation scenarios. The proposed method was implemented using A Mathematical Programming Language (AMPL) and the commercial solver CPLEX. The results were of optimal quality, considering the characteristics of the used solver, and they were compared with methods found in the specialized literature. / Doutor

In this work, we investigate job-shop problems where limited capacity buffers to store jobs in non-processing periods are present. In such a problem setting, after finishing processing on a machine, a job either directly has to be processed on the following machine or it has to be stored in a prespecified buffer. If the buffer is completely occupied the job may wait on its current machine but blocks this machine for other jobs. Besides a general buffer model,also specific configurations are considered.The key issue to develop fast heuristics for the job-shop problem with buffers is to find a compact representation of solutions. In contrast to the classical job-shop problem,where a solution may be given by the sequences of the jobs on the machines, now also the buffers have to be incorporated in the solution representation. In this work, we propose two solution representations for the job-shop problem with buffers. Furthermore, we investigate whether the given solution representations can be simplified for specific buffer configurations. For the general buffer configuration it is shown that an incorporation of the buffers in the solution representation is necessary, whereas for specific buffer configurations possible simplifications are presented. Based on the given solution representations we develop local search heuristics in the second part of this work. Therefore, the well-known block approach for the classical job-shop problem is generalized to the job-shop problem with specific buffer configurations.

Cardinality constrained optimization problems (CCOP) are a new class of optimization problems with many applications. In this thesis, we propose a framework called mathematical programs with disjunctive subspaces constraints (MPDSC), a special case of mathematical programs with disjunctive constraints (MPDC), to investigate CCOP. Our method is different from the relaxed complementarity-type reformulation in the literature. The first contribution of this thesis is that we study various stationarity conditions for MPDSC, and then apply them to CCOP. In particular, we recover disjunctive-type strong (S-) stationarity and Mordukhovich (M-) stationarity for CCOP, and then reveal the relationship between them and those from the relaxed complementarity-type reformulation. The second contribution of this thesis is that we obtain some new results for MPDSC, which do not hold for MPDC in general. We show that many constraint qualifications like the relaxed constant positive linear dependence (RCPLD) coincide with their piecewise versions for MPDSC. Based on such result, we prove that RCPLD implies error bounds for MPDSC. These two results also hold for CCOP. All of these disjunctive-type constraint qualifications for CCOP derived from MPDSC are weaker than those from the relaxed complementarity-type reformulation in some sense. / Graduate

No description available.

This dissertation develops efficient solution techniques for general and problem-specific applications within nonconvex optimization, exploiting the constructs of the Reformulation-Linearization Technique (RLT). We begin by developing a technique to enhance general problems in nonconvex optimization through the use of a new class of RLT cuts, called semidefinite cuts. While these cuts are valid for any general problem for which RLT is applicable, we demonstrate their effectiveness in optimizing a nonconvex quadratic objective function over a simplex. Computational results indicate that on average, the semidefinite cuts have reduced the number of nodes in the branch-and-bound tree by a factor of 37.6, while decreasing solution time by a factor of 3.4. The semidefinite cuts have also led to a significant reduction in the optimality gap at termination, in some cases producing optimal solutions for problems that could not be solved using RLT alone. We then narrow our focus to the class of mixed-integer programming (MIP) problems, and develop a modification of Benders' decomposition method using concepts from RLT and lift-and-project cuts. This method is particularly motivated by the class of two-stage stochastic programs with integer recourse. The key idea is to design an RLT or lift-and-project cutting plane scheme for solving the subproblems where the cuts generated have right-hand sides that are functions of the first-stage variables. An illustrative example is provided to elucidate the proposed approach. The focus is on developing a first comprehensive finitely convergent extension of Benders' methodology for problems having 0-1 mixed-integer subproblems. We next address a specific challenging MIP application known as the facility layout problem, and we significantly improve its formulation through outer-linearization techniques and concepts from disjunctive programming. The enhancements produce a substantial increase in the accuracy of the layout produced, while at the same time, providing a dramatic reduction in computational effort. Overall, the maximum error in department size was reduced from about 6% to nearly zero, while solution time decreased by a factor of 110. Previously unsolved test problems from the literature that had defied even approximate solution methods have been solved to exact optimality using our proposed approach. / Ph. D.

Insolvenzprognosen und Ratings sind wichtige Aufgaben der Finanzbranche und dienen der Kreditwürdigkeitsprüfung von Unternehmen. Eine Möglichkeit dieses Aufgabenfeld anzugehen, ist maschinelles Lernen. Dabei werden Vorhersagemodelle aufgrund von Beispieldaten aufgestellt. Methoden aus diesem Bereich sind aufgrund Ihrer Automatisierbarkeit vorteilhaft. Dies macht menschliche Expertise in den meisten Fällen überflüssig und bietet dadurch einen höheren Grad an Objektivität. Allerdings sind auch diese Ansätze nicht perfekt und können deshalb menschliche Expertise nicht gänzlich ersetzen. Sie bieten sich aber als Entscheidungshilfen an und können als solche von Experten genutzt werden, weshalb interpretierbare Modelle wünschenswert sind. Leider bieten nur wenige Lernalgorithmen interpretierbare Modelle. Darüber hinaus sind einige Aufgaben wie z.B. Rating häufig Mehrklassenprobleme. Mehrklassenklassifikationen werden häufig durch Meta-Algorithmen erreicht, welche mehrere binäre Algorithmen trainieren. Die meisten der üblicherweise verwendeten Meta-Algorithmen eliminieren jedoch eine gegebenenfalls vorhandene Interpretierbarkeit. In dieser Dissertation untersuchen wir die Vorhersagegenauigkeit von interpretierbaren Modellen im Vergleich zu nicht interpretierbaren Modellen für Insolvenzprognosen und Ratings. Wir verwenden disjunktive Normalformen und Entscheidungsbäume mit Schwellwerten von Finanzkennzahlen als interpretierbare Modelle. Als nicht interpretierbare Modelle werden Random Forests, künstliche Neuronale Netze und Support Vector Machines verwendet. Darüber hinaus haben wir einen eigenen Lernalgorithmus Thresholder entwickelt, welcher disjunktive Normalformen und interpretierbare Mehrklassenmodelle generiert. Für die Aufgabe der Insolvenzprognose zeigen wir, dass interpretierbare Modelle den nicht interpretierbaren Modellen nicht unterlegen sind. Dazu wird in einer ersten Fallstudie eine in der Praxis verwendete Datenbank mit Jahresabschlüssen von 5152 Unternehmen verwendet, um die Vorhersagegenauigkeit aller oben genannter Modelle zu messen. In einer zweiten Fallstudie zur Vorhersage von Ratings demonstrieren wir, dass interpretierbare Modelle den nicht interpretierbaren Modellen sogar überlegen sind. Die Vorhersagegenauigkeit aller Modelle wird anhand von drei in der Praxis verwendeten Datensätzen bestimmt, welche jeweils drei Ratingklassen aufweisen. In den Fallstudien vergleichen wir verschiedene interpretierbare Ansätze bezüglich deren Modellgrößen und der Form der Interpretierbarkeit. Wir präsentieren exemplarische Modelle, welche auf den entsprechenden Datensätzen basieren und bieten dafür Interpretationsansätze an. Unsere Ergebnisse zeigen, dass interpretierbare, schwellwertbasierte Modelle den Klassifikationsproblemen in der Finanzbranche angemessen sind. In diesem Bereich sind sie komplexeren Modellen, wie z.B. den Support Vector Machines, nicht unterlegen. Unser Algorithmus Thresholder erzeugt die kleinsten Modelle während seine Vorhersagegenauigkeit vergleichbar mit den anderen interpretierbaren Modellen bleibt. In unserer Fallstudie zu Rating liefern die interpretierbaren Modelle deutlich bessere Ergebnisse als bei der zur Insolvenzprognose (s. o.). Eine mögliche Erklärung dieser Ergebnisse bietet die Tatsache, dass Ratings im Gegensatz zu Insolvenzen menschengemacht sind. Das bedeutet, dass Ratings auf Entscheidungen von Menschen beruhen, welche in interpretierbaren Regeln, z.B. logischen Verknüpfungen von Schwellwerten, denken. Daher gehen wir davon aus, dass interpretierbare Modelle zu den Problemstellungen passen und diese interpretierbaren Regeln erkennen und abbilden.