Enhanced Formulation for Guillotine 2D Cutting Problems

TL;DR

This paper improves MILP formulations for Guillotine 2D Cutting Problems, significantly reducing size and complexity, leading to faster solution times and better bounds on challenging benchmark instances.

Contribution

The authors develop an enhanced MILP formulation that reduces variables and constraints, improves solution efficiency, and extends applicability to harder problem instances.

Findings

Enhanced formulation has only 3.07% of original variables and 8.35% of constraints.

Solves all benchmark instances in about 4 hours, compared to 12 hours for the original.

Finds 17 new optimal solutions and improves bounds on difficult instances.

Abstract

We advance the state of the art in Mixed-Integer Linear Programming (MILP) formulations for Guillotine 2D Cutting Problems by (i) adapting a previously known reduction to our preprocessing phase and by (ii) enhancing a previous formulation by cutting down its size and symmetries. Our focus is the Guillotine 2D Knapsack Problem with orthogonal and unrestricted cuts, constrained demand, unlimited stages, and no rotation -- however, the formulation may be adapted to many related problems. The code is available. Concerning the set of 59 instances used to benchmark the original formulation, and summing the statistics for all models generated, the enhanced formulation has only a small fraction of the variables and constraints of the original model (respectively, 3.07% and 8.35%). The enhanced formulation also takes about 4 hours to solve all instances while the original formulation takes 12 hours to solve 53 of them (the other six runs hit a three-hour time limit each). We integrate, to both formulations, a pricing framework proposed for the original formulation; the enhanced formulation keeps a significant advantage in this situation. Finally, in a recently proposed set of 80 harder instances, the enhanced formulation (with and without the pricing framework) found: 22 optimal solutions for the unrestricted problem (5 already known, 17 new); 22 optimal solutions for the restricted problem (all are new and they are not the same 22 of the optimal unrestricted solutions); better lower bounds for 25 instances; better upper bounds for 58 instances.