An And-or-graph Approach For Two-dimensional Cutting Problems

The problem of generating guillotine cutting patterns for a rectangular plate is studied and a type of structure is proposed for representing the solution of the problem, called and-or graph, which is commonly used in the Artificial Intelligence environment. To search the graph we combined two classical strategies: depth-first and hill-climbing. Further, some heuristics are considered and computational results are presented, relating their performance on examples from both literature as well as randomly generated. © 1992.582263271Beasley, Algorithms for unconstrained two-dimensional guillotine cutting (1985) Journal of the Operational Research Society, 4, pp. 297-306Christofides, Whitlock, An Algorithm for Two-Dimensional Cutting Problems (1977) Operations Research, 25, pp. 30-44Gilmore, Gomory, Multistage Cutting Stock Problems of Two and More Dimensions (1965) Operations Research, 13, pp. 94-120Herz, Recursive Computational Procedure for Two-dimensional Stock Cutting (1972) IBM Journal of Research and Development, 16, pp. 462-469Morabito, Corte de Estoque Bidimensional (1989) Dissertação de Mestrado, , Instituto de Ciências Matemáticas de São Carlos, Universidade de São PauloMorabito, Arenales, Arcaro, An and—or-graph representation to generate cutting patterns for the two-dimensional cutting problem (1989) Workshop on Combinatorial Optimization, , Rio de JaneiroPearl, (1984) Heuristics: Intelligent Search Strategies for Computer Problem Solving, , Addison-Wesley, Reading, MARich, (1983) Artificial Intelligence, , McGraw-Hill, New Yor

Reordenamento eficiente das colunas básicas na programação de lotes e cortes

Neste trabalho consideramos o problema combinado, que acopla os problemas de dimensionamento de lotes e de corte de estoque, incluindo uma formulação matemática deste problema. Consideramos algumas propriedades da matriz de restrições deste modelo e como construir uma base esparsa para ela, utilizando um reordenamento estático das colunas. Resultados numéricos de uma implementação que realiza trocas de colunas básicas e verifica sua esparsidade, simulando o método simplex são apresentados. Experimentos numéricos também comprovam a robustez desta abordagem. Concluímos que a proposta de construção da base estática esparsa leva a bons resultados computacionais com relação à velocidade e robustez em comparação com abordagens que não consideram a estrutura esparsa da matriz.<br>In this work the combined problem is considered, which solves simultaneously the lot sizing and the cutting stock problems. We study some properties of the matrix of constraints and how to factorize the base without losing sparsity in the simplex method context, by a static reordering of the basic columns. Numerical results simulating simplex iterations and verify the sparsity of the factorizations are presented. Numerical experiments had also proven the robustness of this strategy. We conclude that the approach of constructing of the static sparse base reordering leads to very good computational results for both: speed and robustness, in comparison with approaches which do not consider the sparse structure of the matrix of constraints