In this thesis I focus on cutting planes for large Mixed Integer Programming (MIP) problems. More specifically, I focus on two independent cutting planes studies. The first of these deals with cutting planes for the Traveling Salesman Problem (TSP), and the second with cutting planes for general MIPs. In the first study I introduce a new class of cutting planes which I call the Generalized Domino Parity (GDP) inequalities. My main achievements with regard to these are: (1) I show that these are valid for the TSP and for the graphical TSP. (2) I show that they generalize most well-known TSP inequalities (including combs, domino-parity constraints, clique-trees, bipartitions, paths and stars). (3) I show that a sub-class of these (which contains all clique-tree inequalities w/ a fixed number of handles) can be separated in polynomial time, on planar graphs. My second study can be subdivided in two parts. In the first of these I study the Mixed Integer Knapsack Problem (MIKP) and develop a branch-and-bound based algorithm for solving it. The novelty of the approach is that it exploits the notion of "dominance" in order to effectively prune solutions in the branch-and-bound tree. In the second part, I develop a Mixed Integer Rounding (MIR) cut separation heuristic, and embed the MIKP solver in a column generation algorithm in order to assess the performance of said heuristic. The goal of this study is to understand why no other class of inequalities derived from single-row systems has been able to outperform the MIR. Computational results are presented.

This study deals with chocolate production line scheduling. The particular production line allows producing multiple items at the same time. Another distinguishing property affecting the planning methodology is that an item can have different production capacities when produced in different product combinations which are called production patterns in this study. Planning is done on a 12 weeks rolling horizon. There are 21 products and 103 production patterns covering all the production possibilities. The subject of the study is to construct an algorithm that gives 12 weeks&rsquo / production values of each product and to construct the shift based scheduling of the first week of the planning horizon. The first part is Master Production Scheduling (MPS) and the objective is minimizing the shortage and overage costs. A mathematical modeling approach is used to solve the MPS problem. The second part is the scheduling part which aims to arrange the production patterns obtained from the MPS module within the shifts for the first week of the planning horizon considering the setup times. The MPS module is a large integer programming model. The challenge is finding a reasonable lower bound whenever possible. If it is not possible, finding a reasonable upper bound and seeking solutions better than that is the main approach. The scheduling part, after solving MPS, becomes a TSP and the setup times are sequence independent. In this part, the challenge is solving TSP with an appropriate objective function.

In this study, a new selective and time-window routing problem is defined for the first time in the literature, which is called the campaign routing problem (CRP). The two special cases of the CRP correspond to the two real-life problems, namely political campaign routing problem (PCRP) and the experiments on wheels routing problem (EWRP). The PCRP is based on two main decision levels. In the first level, a set of campaign regions is selected according to a given criteria subject to the special time-window constraints. In the second level, a pair of selected regions or a single region is assigned to a campaign day. In the EWRP, a single selected region (school) is assigned to a campaign day. These two problems are modeled using classical mathematical programming and bi-level programming methods, and a two-step heuristic approach is developed for the solution of the problems. Implementation of the solution methods is done using the test instances that are compiled from the real-life data. Computational results show that the solution methods developed generate good solutions in reasonable time.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2009. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed January 5, 2010) Includes bibliographical references (p. 152-159).

The traveling salesperson or salesman problem (TSP) is a well known and important combinatorial optimization problem. The goal is to find the shortest tour that visits each city in a given list exactly once and then returns to the starting city. Despite this simple problem statement, solving the TSP is difficult since it belongs to the class of NP-complete problems. The importance of the TSP arises besides from its theoretical appeal from the variety of its applications. In addition to vehicle routing, many other applications, e.g., computer wiring, cutting wallpaper, job sequencing or several data visualization techniques, require the solution of a TSP. In this paper we introduce the R package TSP which provides a basic infrastructure for handling and solving the traveling salesperson problem. The package features S3 classes for specifying a TSP and its (possibly optimal) solution as well as several heuristics to find good solutions. In addition, it provides an interface to Concorde, one of the best exact TSP solvers currently available. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Thesis (Ph. D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Thomas, Robin; Committee Co-Chair: Cook, William J.; Committee Member: Dvorak, Zdenek; Committee Member: Parker, Robert G.; Committee Member: Yu, Xingxing.

Thesis (M.S.)--Texas State University--San Marcos, 2008. / Vita. Includes bibliographical references (leaves 50-51). Also available on microfilm.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-07-01Bitstream added on 2014-06-13T18:30:54Z : No. of bitstreams: 1 carretero_mm_me_sjrp.pdf: 795127 bytes, checksum: 64b07e80db6689945e91fc1c317deb3c (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Em alguns setores, o planejamento da produção envolve dois aspectos: o dimensionamento do tamanho dos lotes e a programação da produção (sequenciamento dos lotes). O primeiro problema consiste em determinar o tamanho dos lotes de produção de cada item a ser produzido em uma ou mais máquinas em cada período ao longo de um horizonte de planejamento finito. O segundo problema consiste em encontrar a ordem em que os lotes devem ser produzidos em um dado conjunto de máquinas. Estes dois aspectos do planejamento da produção podem ser tratados de forma independente: em um estágio é resolvido o problema de dimensionamento dos lotes e no outro, realizado antes ou depois, é resolvido o problema de seqüenciamento. No entanto, uma tendência recente na literatura são trabalhos que apresentam modelos matemáticos que capturam simultaneamente as relações entre os dois problemas. Na literatura pode-se encontrar modelos integrados que incluem restrições de eliminação de subrotas, propostas para o Problema do Caixeiro Viajante (PCV), para formular as restrições de sequenciamento. No entanto, alguns dos modelos propostos usam restrições de ordem polinomial que fornecem uma relaxação linear fraca. O objetivo desse trabalho é avaliar o uso de inequações válidas, propostas na literatura, para obtenção de formulações mais fortes para o problema integrado de dimensionamento e sequenciamento da produção. Resultados computacionais usando exemplares aleatórios e exemplares da literatura mostram que as reformulações propostas são eficientes para cenários em que o modelo original não é eficiente. / Often, the production planning involves the lot sizing and scheduling of items. The first problem is to determine the lot size of each item to be produced in one or more machines in each period over a finite planning horizon. The second problem is to find the order in which the items will be produced. These two aspects of the production planning can be treated independently: in one stage the lot sizing problem is solved, and in the other, that can be executed before or after, the scheduling problem is solved. A recent trend in the literature is to propose mathematical models that capture the relationships between these two problems. In the literature one can find integrated models that include subtour elimination constraints, proposed for the Traveling Salesman Problem, to formulate the scheduling decisions. However, in some of these models, constraints of polynomial order, that provides a weak linear relaxation, are used.The purpose of this study is to evaluate the use of valid inequalities proposed in the literature to obtain stronger formulations to the lot and scheduling problem. Computational results using random instances and instances from the literature show that the proposed formulations have a better performance in scenarios where the original model is not efficient.

Orientador: Luziane Ferreira de Mendonça / Dissertação ( mestrado profissional) - Universidade Estadual de Campionas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T15:32:08Z (GMT). No. of bitstreams: 1 Tavora_RogerioCarvalhoMendes_M.pdf: 4616708 bytes, checksum: 3a393a830c9e1127f5694b0a56d9fcd9 (MD5) Previous issue date: 2011 / Resumo: Comemorando os 200 anos de Academia Militar no Brasil a partir de março de 2011, estão previstas várias implementações e melhorias na estrutura de visitação da AMAN que, consequentemente, vão gerar um aumento substancial no número de grupos visitantes no ano de seu bicentenário. Diante dos fatos percebe-se a necessidade de um modelo matemático eficiente cuja finalidade seja permitir aos grupos visitantes percorrerem trajetos otimizados, ou seja, que passem pelos pontos principais de visitação no menor tempo e distância possíveis. O modelo matemático a ser adotado neste trabalho é o Problema do Caixeiro Viajante (Traveling Salesman Problem - TSP), um clássico da Otimização Combinatória pertencente 'a classe de problemas NP - difícil, que já possui eficientes algoritmos desenvolvidos. Serão utilizadas heurísticas próprias para a resolução do TSP com o intuito de se obter numericamente itinerários ótimos de visitação, considerando os diferentes grupos visitantes e suas dificuldades de acesso, dentre outras particularidades. / Abstract: Celebrating 200 years of the Military Academy in Brazil from March 2011, provides a lot of implementations and improvements in the structure of visitation of AMAN, consequently, will generate substantial growth in the number of visiting groups in the year of its bicentennial. Given the facts we see the need for an efficient mathematical model whose purpose is to allow visitors to wander paths optimized groups, ie passing through the main points of visitation in the shortest possible time and distance. The mathematical model to be adopted in this work is TSP (Traveling Salesman Problem - TSP), a classic combinatorial optimization class of problems NP - hard, that have already efficient algorithms. We will use own heuristics for solving the TSP in order to obtain numerically optimal routes for visitors, considering the various visiting groups and their difficulties of access, among other features. / Mestrado / Mestre em Matemática

Orientadores: Fernando Jose Von Zuben, Leandro Nunes de Castro / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-15T01:42:44Z (GMT). No. of bitstreams: 1 Gomes_LalinkadeCamposTeixeira_D.pdf: 3303378 bytes, checksum: 65adc8d5ec20cd1f431eaca2fe3765cc (MD5) Previous issue date: 2006 / Resumo: Problemas de otimização combinatória apresentam grande relevância prática e surgem em uma ampla gama de aplicações. Em geral, a otimização combinatória está associada a uma explosão de candidatos à solução, inviabilizando a aplicação de métodos exatos. Frente à intratabilidade desta classe de problemas via métodos exatos, nos últimos anos tem havido um crescente interesse por métodos heurísticos capazes de encontrar soluções de alta qualidade, não necessariamente ótimas. Considerando o notório sucesso empírico de meta-heurísticas concebidas através da inspiração biológica e na natureza, essas abordagens vêm ganhando cada vez mais atenção por parte de pesquisadores. É fato conhecido que não existe uma única metodologia capaz de sempre produzir os melhores resultados para todas as classes de problemas, ou mesmo para todas as instâncias de uma mesma classe. Assim, a busca de solução para problemas de natureza combinatória constitui uma linha de pesquisa desafiadora. Nesta tese são considerados problemas de otimização combinatória multicritério e multimodal. Como principal contribuição, destaca-se a concepção de novas meta-heurísticas para a solução de problemas combinatórios de elevada complexidade, tendo sido propostas duas classes de ferramentas computacionais. A primeira envolve um método híbrido fundamentado em mapas auto-organizáveis de Kohonen e inferência nebulosa, em que um conjunto de regras guia o processo de treinamento do mapa de modo a permitir o tratamento de problemas com restrições e múltiplos objetivos. A segunda abordagem baseia-se em sistemas imunológicos artificiais. Em particular, a abordagem imunológica levou à proposição de meta-heurísticas capazes de encontrar e manter diversas soluções de alta qualidade, viabilizando o tratamento de problemas multimodais. Como casos de estudo, foram consideradas duas classes de problemas de otimização combinatória multimodal: o problema de roteamento de veículos capacitados e o problema do caixeiro viajante simétrico. As técnicas propostas foram também adaptadas para a solução de problemas de bioinformática, em particular ao problema de análise de dados de expressão gênica, produzindo resultados diferenciados e indicando um elevado potencial para aplicações práticas. / Abstract: Combinatorial optimization problems possess a high practical relevance and emerge on a wide range of applications. Usually, combinatorial optimization is associated with an explosion of candidates to the solution, making exact methods unfeasible. Before the unfeasibility of exact methods when dealing with this class of problems, lately there has been an increasing interest in heuristic methods capable of finding high-quality solutions, not necessarily the optimal one. Considering the widely known empirical success of metaheuristics conceived with inspiration on biological systems and on the nature itself, such approaches are receiving more and more attention from the scientific community. Evidently, there is no single methodology able to always produce the best results for all classes of problems, or even for all instances of one specific class. That is why the search for solutions to combinatorial problems remains a challenging task. This thesis considers multicriteria and multimodal combinatorial optimization problems. As the main contribution, one can emphasize the conception of new metaheuristics designed to the solution of high-complexity combinatorial optimization problems, and two classes of computational tools have been proposed. The first one involves hybrid method based on Kohonen self-organizing maps and fuzzy inference, in which a set of rules guides the training of the self-organizing maps in order to allow the handling of problems with constraints and multiple objectives. The second approach is based on artificial immune systems. Particularly, the immune-inspired approach leads to the proposal of metaheuristics capable of finding out and maintaining multiple high-quality solutions, making it possible to deal with multimodal problems. As case studies, the capacitated vehicle routing problem and the symmetric traveling salesman problem are considered, giving rise to combinatorial and multimodal problems. The proposed techniques were also adapted to the solution of problems in the field of bioinformatics, specifically the analysis of gene expression data, leading to distinguished results and indicating a high potential for practical applications. / Doutorado / Engenharia de Computação / Doutor em Engenharia Elétrica