Comparative analysis of mathematical formulations for the two-dimensional guillotine cutting problem

TL;DR

This paper compares various mathematical formulations for the two-dimensional guillotine cutting problem, analyzing their empirical performance, solver efficiency, and proposing hybrid approaches to improve solution times.

Contribution

It provides a comprehensive comparison of existing formulations, adapts them for piece rotation, and introduces hybrid methods that reduce solution times by about 20%.

Findings

Pseudo-polynomial formulations dominate for smaller instances.

Compact formulations maintain good primal solutions for larger instances.

Hybrid formulations reduce branch-and-bound time by approximately 20%.

Abstract

About ten years ago, a paper proposed the first integer linear programming formulation for the constrained two-dimensional guillotine cutting problem (with unlimited cutting stages). Since, six other formulations followed, five of them in the last two years. This spike of interest gave no opportunity for a comprehensive comparison between the formulations. We review each formulation and compare their empirical results over instance datasets of the literature. We adapt most formulations to allow for piece rotation. The possibility of adaptation was already predicted but not realized by the prior work. The results show the dominance of pseudo-polynomial formulations until the point instances become intractable by them, while more compact formulations keep achieving good primal solutions. Our study also reveals a small but consistent advantage of the Gurobi solver over the CPLEX solver in our context; that the choice of solver hardly benefits one formulation over another; and a mistake in the generation of the T instances, which should have the same optima with or without guillotine cuts. Our study also proposes hybridising the most recent formulation with a prior formulation for a restricted version of the problem. The hybridisations show a reduction of about 20% of the branch-and-bound time thanks to the symmetries broken by the hybridisation.