The Steiner Linear Ordering Problem: Application to resource-constrained scheduling problems

When examined through polyhedral study, the resource-constrained scheduling problems have always dealt with processes which have the same priority. With the Steiner Linear Ordering problem, we can address systems where the elements involved have different levels of priority, either high or low. This allows us greater flexibility in modeling different resource-constrained scheduling problems. In this paper, we address both the linear ordering problem and its application to scheduling problems, and provide a polyhedral study of the associated polytopes

Modelos para sequenciação de padrões em problemas de corte de stock

Tese de doutoramento em Engenharia Industrial e de SistemasIn this thesis, we address an optimization problem that appears in cutting stock operations research called the minimization of the maximum number of open stacks (MOSP) and we put forward a new integer programming formulation for the MOSP. By associating the duration of each stack with an interval of time, it is possible to use the rich theory that exists in interval graphs in order to create a model based on the completion of a graph with edges. The structure of this type of graphs admits a linear ordering of the vertices that de nes an ordering of the stacks, and consequently decides a sequence for the cutting patterns. The polytope de ned by this formulation is full-dimensional and the main inequalities in the model are proved to be facets. Additional inequalities are derived based on the properties of chordal graphs and comparability graphs. The maximum number of open stacks is related with the chromatic number of the solution graph; thus the formulation is strengthened by adding the representatives formulation for the vertex coloring problem. The model is applied to the minimization of open stacks, and also to the minimum interval graph completion problem and other pattern sequencing problems such as the minimization of the order spread (MORP) and the minimization of the number of tool switches (MTSP). Computational tests of the model are presented.Nesta tese e abordado um problema de optimização que surge em operações de corte de stock chamado minimização do número máximo de pilhas abertas (MOSP) e e proposta uma nova formulação de programação inteira. Associando a duração de cada pilha a um intervalo de tempo, e possível usar a teoria rica que existe em grafos de intervalos para criar um modelo baseado no completamento de um grafo por arcos. A estrutura deste tipo de grafos admite uma ordenação linear dos vértices que define uma ordenação linear das pilhas e, por sua vez, determina a sequência dos padrões de corte. O politopo definido por esta formulação tem dimensão completa e prova-se que as principais desigualdades do modelo são facetas. São derivadas desigualdades adicionais baseadas nas propriedades de grafos cordais e de grafos de comparabilidades. O número máximo de pilhas abertas está relacionado com o número cromático do grafo solução, pelo que o modelo e reforçado com a formulação por representativos para o problema de coloração de vértices. O modelo e aplicado a minimização de pilhas abertas, e também ao problema de completamento mínimo de um grafo de intervalos e a outros problemas de sequenciação de padrões, tais como a minimização da dispersão de encomendas (MORP) e a minimização do número de trocas de ferramentas (MTSP). São apresentados testes computacionais do modelo.Fundação para a Ciência e a Tecnologia (FCT), programa de financiamento QREN-POPH-Tipologia 4.1-Formação Avançada comparticipado pelo Fundo Social Europeu e por fundos do MCTES (Bolsa individual com a refer^encia SFRH/BD/32151/2006) entre 2006 e 2009, e pela Escola Superior de Estudos Industriais e de Gest~ao do Instituto Polit ecnico do Porto (Bolsa PROTEC com a refer^encia SFRH/BD/49914/2009) entre 2009 e 2010

Linear Orderings of Sparse Graphs

The Linear Ordering problem consists in finding a total ordering of the vertices of a directed graph such that the number of backward arcs, i.e., arcs whose heads precede their tails in the ordering, is minimized. A minimum set of backward arcs corresponds to an optimal solution to the equivalent Feedback Arc Set problem and forms a minimum Cycle Cover. Linear Ordering and Feedback Arc Set are classic NP-hard optimization problems and have a wide range of applications. Whereas both problems have been studied intensively on dense graphs and tournaments, not much is known about their structure and properties on sparser graphs. There are also only few approximative algorithms that give performance guarantees especially for graphs with bounded vertex degree. This thesis fills this gap in multiple respects: We establish necessary conditions for a linear ordering (and thereby also for a feedback arc set) to be optimal, which provide new and fine-grained insights into the combinatorial structure of the problem. From these, we derive a framework for polynomial-time algorithms that construct linear orderings which adhere to one or more of these conditions. The analysis of the linear orderings produced by these algorithms is especially tailored to graphs with bounded vertex degrees of three and four and improves on previously known upper bounds. Furthermore, the set of necessary conditions is used to implement exact and fast algorithms for the Linear Ordering problem on sparse graphs. In an experimental evaluation, we finally show that the property-enforcing algorithms produce linear orderings that are very close to the optimum and that the exact representative delivers solutions in a timely manner also in practice. As an additional benefit, our results can be applied to the Acyclic Subgraph problem, which is the complementary problem to Feedback Arc Set, and provide insights into the dual problem of Feedback Arc Set, the Arc-Disjoint Cycles problem

A note on small linear ordering polytopes

SIGLEAvailable from TIB Hannover: RO7722(329) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman