A strong integer linear optimization model to the compartmentalized knapsack problem

Abstract

The compartmentalized knapsack problem (CKP) is a relatively new type of problem with a wide application in industrial processes, arising, for instance, in the case of cutting steel coils in two phases in the metallurgicalindustry. In the literature, there are two mathematical formulations for the CKP: a classical formulation, which is a nonlinear integer programming (IP) model, and a recent (linear) IP formulation, obtained by discretizing the compartments that can be built for each class of items; the latter is an important contribution, because it makes the problem amenable to solution by mixed-integer linear programming tools. Combinatorial enumeration algorithms and several pseudo-polynomial decomposition heuristics were also developed for theCKP. This paper presents a new model for the exact solution of the CKP, denoted as the strong integer linear model, derived from the (linear) IP formulation by strengthening data, reducing symmetry, and lifting, and also a new pseudo-polynomial heuristic, the heuristic of the pk strong capacities. Computational experiments are presented with a large set of instances that show the advantage of the new approaches. The strong model solves the CKP exactly more than seven times faster, and the new heuristic is more efficient, presenting a good balance in the terms ofeffectiveness.