Comparative analysis of pattern-based models for the two-dimensional two-stage guillotine cutting stock problem

Abstract

This study presents a theoretical and computational analysis of integer programs based on the concept of the patterns, the so-called pattern-based models, for a two-dimensional two-stage guillotine cutting stock problem. The full-pattern models and the staged-pattern models with the corresponding column generation problems are analyzed and compared. Those pattern-based models have been actively used, however, their LP-relaxations have not been theoretically analyzed and compared. In this paper, we formally establish a hierarchy of the strength of lower bounds that can be obtained from their LP-relaxations. In addition, through computational tests with benchmark instances, we analyze the trade-off between the strength of the lower bounds and the required computation time to solve the LP-relaxations of the models. The quality of integer solutions obtained by an LP-based branch-and-bound algorithm with the columns generated during the process of solving the LP-relaxations of the models are also compared. The results show that one of the staged-pattern models, which has not been well-studied, shows competitive theoretical and computational performance.