KNAPSACK PROBLEMS WITH SETUPS

RÉSUMÉ : We consider two variants of knapsack problems with setups arising as subproblems in a DantzigWolfe decomposition approach to more complex combinatorial optimization problems. In the multiple-class binary knapsack problem with setups, items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. In the continuous knapsack problems with setups, each class holds a single item and a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size

A decomposition approach for multidimensional knapsacks with family-split penalties

The optimization of Multidimensional Knapsacks with Family-Split Penalties has been introduced in the literature as a variant of the more classical Multidimensional Knapsack and Multi-Knapsack problems. This problem deals with a set of items partitioned in families, and when a single item is picked to maximize the utility, then all items in its family must be picked. Items from the same family can be assigned to different knapsacks, and in this situation split penalties are paid. This problem arises in real applications in various fields. This paper proposes a new exact and fast algorithm based on a specific Combinatorial Benders Cuts scheme. An extensive experimental campaign computationally shows the validity of the proposed method and its superior performance compared to both commercial solvers and state-of-the-art approaches. The paper also addresses algorithmic flexibility and scalability issues, investigates challenging cases, and analyzes the impact of problem parameters on the algorithm behavior. Moreover, it shows the applicability of the proposed approach to a wider class of realistic problems, including fixed costs related to each knapsack utilization. Finally, further possible research directions are considered

Symmetric separable convex resource allocation problems with structured disjoint interval bound constraints

Motivated by the problem of scheduling electric vehicle (EV) charging with a minimum charging threshold in smart distribution grids, we introduce the resource allocation problem (RAP) with a symmetric separable convex objective function and disjoint interval bound constraints. In this RAP, the aim is to allocate an amount of resource over a set of $n$ activities, where each individual allocation is restricted to a disjoint collection of $m$ intervals. This is a generalization of classical RAPs studied in the literature where in contrast each allocation is only restricted by simple lower and upper bounds, i.e., $m=1$. We propose an exact algorithm that, for four special cases of the problem, returns an optimal solution in $O \left(\binom{n+m-2}{m-2} (n \log n + nF) \right)$ time, where the term $nF$ represents the number of flops required for one evaluation of the separable objective function. In particular, the algorithm runs in polynomial time when the number of intervals $m$ is fixed. Moreover, we show how this algorithm can be adapted to also output an optimal solution to the problem with integer variables without increasing its time complexity. Computational experiments demonstrate the practical efficiency of the algorithm for small values of $m$ and in particular for solving EV charging problems.Comment: 20 pages, 4 figure

Knapsack Problems with Side Constraints

The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once. The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem. These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts

Modelos e métodos para problemas de dimensionamento de lotes e escalonamento

Tese de doutoramento em Engenharia Industrial e de SistemasO trabalho que se apresenta nesta tese relaciona-se com o desenvolvimento de modelos e de métodos para a resolução de dois problemas de planeamento da produção de médio/curto prazo. A principal motivação consiste na exploração e comparação de diferentes abordagens, baseadas em programação inteira mista, em modelos/métodos de decomposição e em métodos heurísticos, para os problemas em estudo. O primeiro problema, é um problema clássico de dimensionamento de lotes, que está associado às decisões de planeamento da produção de médio-prazo. O problema consiste na determinação de um plano de produção para vários produtos finais ao longo de um determinado horizonte temporal, que minimize todos os custos envolvidos e respeite restrições de procura e de capacidade. Para este problema desenvolve-se um novo modelo exacto, que resulta da aplicação dos princípios da decomposição de Dantzig-Wolfe múltipla a uma formulação de programação inteira mista para o problema. Os princípios gerais de aplicação desta decomposição são também apresentados neste trabalho. A potencial mais valia deste modelo relaciona-se com a obtenção de limites inferiores de boa qualidade. O modelo que resulta da decomposição de Dantzig-Wolfe múltipla é comparado com dois modelos de decomposição alternativos, que se obtêm aplicando directamente os princípios da decomposição de Dantzig-Wolfe, e com o modelo de programação inteira mista, resolvido directamente através de um software de estado-da-arte. Para determinar a solução óptima inteira dos modelos de decomposição aplica-se o método de partição e geração de colunas (branchand- price). São apresentados resultados computacionais partindo de um conjunto de instâncias da literatura, para os vários modelos e métodos. O segundo problema estudado neste trabalho surge associado ao planeamento de curto-prazo e combina as decisões de dimensionamento de lotes, com as decisões de afectação e escalonamento desses lotes. Este estudo foi motivado por um problema real da indústria têxtil, no qual se pretende definir um plano de produção para uma secção de tricotagem, onde os principais componentes dos produtos finais são realizados num conjunto de máquinas paralelas idênticas. Para este problema propõe-se um novo modelo de programação inteira mista, que se resolve através de um software de estadoda- arte. Paralelamente, propõem-se vários métodos heurísticos. Duas das heurísticas propostas são: uma heurística de fluxos em rede e escalonamento e uma heurística de ordenação e escalonamento. Estas heurísticas visam a obtenção de soluções com alguma qualidade em pouco tempo. Propõem-se ainda quatro algoritmos de pesquisa local, que têm em consideração características específicas do problema e que tentam melhorar a qualidade das soluções das heurísticas anteriores. Atendendo ao desempenho dos algoritmos de pesquisa local, estes são combinados através de mudanças sistemáticas das vizinhanças, dando origem a duas meta-heurísticas: uma de descida em vizinhanças variáveis e outra de pesquisa em vizinhanças variáveis. Para avaliar as soluções do modelo de programação inteira mista e dos métodos heurísticos sugere-se uma função de avaliação inovadora, que minimiza os atrasos totais e os níveis em curso de fabrico entre duas etapas sucessivas do processo produtivo. É ainda sugerida uma nova função de avaliação nos métodos heurísticos, também baseada na minimização dos atrasos totais e na minimização dos níveis em curso de fabrico. A principal vantagem desta segunda medida de avaliação é contabilizar de um modo mais rigoroso os níveis em curso de fabrico. Para avaliar o desempenho e a qualidade das soluções do modelo de programação inteira mista e dos métodos heurísticos, desenvolveu-se um gerador de instâncias, que gera instâncias semelhantes às do problema real.This work is associated with the development of models and methods for two medium/short term production planning problems. Our main motivation is the exploration and comparison of different approaches, based on mixed integer programming, on decomposition models and methods and on heuristics, for those two problems. The first one is a classical lot sizing problem associated with the medium-term production planning decisions. The problem consists of finding a production plan for several final items over a given planning horizon that minimizes the overall costs involved, while respecting demand and capacity constraints. An exact model based on a multiple Dantzig-Wolfe decomposition is developed. The general principles of this decomposition are presented in this work too. The potential benefit of this decomposition is the achievement of good quality lower bounds, although our purpose is to obtain integer optimal solutions. The resulting model of multiple Dantzig-Wolfe decomposition is compared with two alternative decomposition models that are obtained when applying directly the Dantzig-Wolfe decomposition principles, and is also compared with an integer programming formulation solved by a state-of-art software. The integer optimal solutions of all the decomposition models are obtained through branch-and-price algorithms. We present computational results for a set of instances from the literature. The second problem studied in this work is a short-term production planning problem that integrates lot sizing, assignment and scheduling decisions. This study was motivated by a real problem from a textile industry. The aim is to define a production plan for a knitting section where the main components of the final items are processed on a set of identical parallel machines. A new mixed integer programming model is proposed for this problem, as well as several heuristics. Two of those heuristics are: a network flow and scheduling heuristic and an ordering and scheduling heuristic. The purpose of these heuristics is to find good quality solutions quickly. Four local search based algorithms that consider specific characteristics of the problem are developed too, in order to try to improve the solutions of the previous heuristics. Taking into account the performance of the four local search heuristics, we combine them through systematic changes of neighborhoods, testing two metaheuristics: variable neighborhood descent and variable neighborhood search. To evaluate the mixed integer programming model solutions and the solutions of all the heuristics, an innovative evaluation function that minimizes a weighted sum of total tardiness and work-in-process levels between two successive production processes is suggested. We study another new evaluation function for the heuristic methods, which is related to the previous one. The main advantage of the second evaluation function over the first one is that it calculates in a more precise way the levels of workin- process inventory. The performance and quality of solutions of all the above presented methods for the second problem are evaluated using a set of instances that are similar to the real ones. Those instances were generated by an instance generator developed by us.Fundação para a Ciência e a Tecnologia (FCT) - SFRH/BD/38582/200

Anuário Científico – 2009 & 2010 Resumos de Artigos, Comunicações, Teses, Patentes, Livros e Monografias de Mestrado

O Conselho Técnico-Científico do Instituto Superior de Engenharia de Lisboa (ISEL), na senda da consolidação da divulgação do conhecimento e da ciência desenvolvidos pelo nosso corpo docente, propõe-se publicar mais uma edição do Anuário Científico, relativa à produção científica de 2009 e 2010. A investigação, enquanto vertente estratégica do Instituto Superior de Engenharia de Lisboa (ISEL), tem concorrido para o seu reconhecimento nacional e internacional como instituição de referência e de qualidade na área do ensino das engenharias. É também nesta vertente que o ISEL consubstancia a sua ligação à sociedade portuguesa e internacional através da transferência de tecnologia e de conhecimento, resultantes da sua atividade científica e pedagógica, contribuindo para o seu desenvolvimento e crescimento de forma sustentada. São parte integrante do Anuário Científico todos os conteúdos com afiliação ISEL resultantes de resumos de artigos publicados em livros, revistas e atas de congressos que os docentes do ISEL apresentaram em fóruns e congressos nacionais e internacionais, bem como teses e patentes. Desde 2002, ano da publicação da primeira edição, temos assistido a uma evolução crescente do número de publicações de conteúdos científicos, fruto do trabalho desenvolvido pelos docentes que se têm empenhado com afinco e perseverança. Contudo, nestes dois anos (2009 e 2010) constatou-se um decréscimo no número de publicações, principalmente em 2010. Uma das causas poderá estar diretamente relacionada com a redução do financiamento ao ensino superior uma vez que limita toda a investigação no âmbito da atividade de I&D e da produção científica. Na sequência da implementação do Processo de Bolonha em 2006, o ISEL promoveu a criação de cursos de Mestrado disponibilizando uma oferta educativa mais completa e diversificada aos seus alunos, mas também de outras instituições, dotando-os de competências inovadoras apropriadas ao mercado de trabalho que hoje se carateriza mais competitivo e dinâmico. Terminados os períodos escolar e de execução das monografias dos alunos, os resumos destas são igualmente parte integrante deste Anuário, no que concerne à conclusão dos Mestrados em 2009 e 2010.A fim de permitir uma maior acessibilidade à comunidade científica e à sociedade civil, o Anuário Científico será editado de ora avante em formato eletrónico. Excecionalmente esta edição contempla publicações referentes a dois anos – 2009 e 2010

Knapsack Problems with Setups

Knapsack problems with setups find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig-Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently. Here, we consider the multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size

Knapsack problems with setups

Knapsack problems with setups find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig-Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently. Here, we consider the multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size.Knapsack problem Fixed cost Setup Variable upper bound Branch-and-bound