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

Abstract

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.