A study of column generation embedded in scalarization methods for the bi-objective cutting stock problem

TL;DR

This paper investigates a bi-objective cutting stock problem using embedded column generation within scalarization methods, showing improved Pareto front approximation and offering practical insights for industry applications.

Contribution

It introduces a novel combination of dynamic column generation and scalarization methods for bi-objective CSP, enhancing Pareto front approximation.

Findings

Dynamic column generation improves Pareto front approximation.

Scalarization methods are complementary, not superior individually.

Union of solutions from three methods yields better Pareto front metrics.

Abstract

Research on multi-objective combinatorial optimization and on the Cutting Stock Problem (CSP) has been widely developed over the years. In contrast, the multi-objective Cutting Stock Problem has received limited attention and has been explored in only a small number of studies. In this paper a bi-objective study of the one-dimensional and the two-dimensional CSP is presented. It distinguishes itself from other research in the literature in two key aspects, among others. The first regards the model used to represent the problem and the second is the solution strategy based on dynamic column generation embedded into scalarization methods. Three methods adapted from the literature to analyse the trade-off between the minimization of the total number of objects and the total number of saw cycles are implemented. The computational results show that the use of dynamic column generation provides a better approximation of the Pareto front. As for the scalarization methods, none stood out. In fact, they can be viewed as complementary. The cardinality and hypervolume of an approximation of the Pareto front built from the union of the points generated by the three methods is always greater than the ones generated by each one. Dealing with the multi-objective nature of CSP, this study provides insights and computational tools to help industry obtain pragmatic solutions.