A branch-and-price-and-cut algorithm for the pattern minimization problem

n cutting stock problems, after an optimal (minimal stockusage) cutting plan has been devised, one might want to further reducethe operational costs by minimizing the number of setups. A setupoperation occurs each time a different cutting pattern begins to beproduced. The related optimization problem is known as the PatternMinimization Problem, and it is particularly hard to solve exactly. Inthis paper, we present different techniques to strengthen a formulationproposed in the literature. Dual feasible functions are used for thefirst time to derive valid inequalities from different constraints of themodel, and from linear combinations of constraints. A new arc flowformulation is also proposed. This formulation is used to define thebranching scheme of our branch-and-price-and-cut algorithm, and itallows the generation of even stronger cuts by combining the branchingconstraints with other constraints of the model. The computationalexperiments conducted on instances from the literature show that ouralgorithm finds optimal integer solutions faster than other approaches.A set of computational results on random instances is also reported.info:eu-repo/semantics/publishedVersio

Un problema no lineal de archivo de corte

In this work we introduce a new method to minimize the numberof processed objects and the setup number in a unidimensional cutting stockproblem. A nonlinear integer programming problem can be used to representthe problem studied here. The term related to the minimization of the setupnumber is a nonlinear discontinuous function, we smooth it and generate thecutting patterns using a modified Gilmore-Gomory strategy. Numerical testson a wide range of test problems are very encouraging and the new methodcompares favorably with other methods in the literature.  En este trabajo presentamos un nuevo método para reducir almínimo el número de objetos elaborados y el número de patrones de corte enun problema de corte unidimensional. Un problema de programación enterano lineal se puede utilizar para representar el problema estudiado. El términorelacionado con la reducción al mínimo del número de patrones de corte esuna función discontinua no lineal, la cual suavizamos y genera los patronesde corte utilizando una estrategia de modificación Gilmore-Gomory. Pruebasnuméricas en una amplia gama de problemas fueron muy alentadores y elnuevo método se compara favorablemente con otros métodos en la literatura

Un problema no lineal de archivo de corte

En este trabajo presentamos un nuevo método para reducir almínimo el número de objetos elaborados y el número de patrones de corte enun problema de corte unidimensional. Un problema de programación enterano lineal se puede utilizar para representar el problema estudiado. El términorelacionado con la reducción al mínimo del número de patrones de corte esuna función discontinua no lineal, la cual suavizamos y genera los patronesde corte utilizando una estrategia de modificación Gilmore-Gomory. Pruebasnuméricas en una amplia gama de problemas fueron muy alentadores y elnuevo método se compara favorablemente con otros métodos en la literatura

Facets of a mixed-integer bilinear covering set with bounds on variables

We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a certain way. This description does not introduce any new variables, but consists of exponentially many inequalities. An extended formulation with a few extra variables and much smaller number of constraints is presented. We also derive a linear time separation algorithm for finding the facet defining inequalities of this convex hull. We study the effectiveness of the new inequalities and the extended formulation using some examples

Nonlinear cutting stock problem model to minimize the number of different patterns and objects

In this article we solve a nonlinear cutting stock problem which represents a cutting stock problem that considers the minimization of, both, the number of objects used and setup. We use a linearization of the nonlinear objective function to make possible the generation of good columns with the Gilmore and Gomory procedure. Each time a new column is added to the problem, we solve the original nonlinear problem by an Augmented Lagrangian method. This process is repeated until no more profitable columns is generated by Gilmore and Gomory technique. Finally, we apply a simple heuristic to obtain an integral solution for the original nonlinear integer problem.617

Progressive selection method for the coupled lot-sizing and cutting-stock problem

The coupled lot-sizing and cutting-stock problem has been a challenging and significant problem for industry, and has therefore received sustained research attention. The quality of the solution is a major determinant of cost performance in related production and inventory management systems, and therefore there is intense pressure to develop effective practical solutions. In the literature, a number of heuristics have been proposed for solving the problem. However, the heuristics are limited in obtaining high solution qualities. This paper proposes a new progressive selection algorithm that hybridizes heuristic search and extended reformulation into a single framework. The method has the advantage of generating a strong bound using the extended reformulation, which can provide good guidelines on partitioning and sampling in the heuristic search procedure so as to ensure an efficient solution process. We also analyze per-item and per-period Dantzig-Wolfe decompositions of the problem and present theoretical comparisons. The master problem of the per-period Dantzig-Wolfe decomposition is often degenerate, which results in a tailing-off effect for column generation. We apply a hybridization of Lagrangian relaxation and stabilization techniques to improve the convergence. The discussion is followed by extensive computational tests, where we also perform detailed statistical analyses on various parameters. Comparisons with other methods indicate that our approach is computationally tractable and is able to obtain improved results