Mathematical models for the two-dimensional variable-sized cutting stock problem in the home textile industry

Abstract

In this paper, we consider the two-dimensional Variable-Sized Cutting Stock Problem (2D-VSCSP) with guillotine constraint, applied to the home textile industry. This is a challenging class of real-world problems where, given a set of predefined widths of fabric rolls and a set of piece types, the goal is to decide the widths and lengths of the fabric rolls to be produced, and to generate the cutting patterns to cut all demanded pieces. Each piece type considered has a rectangular shape with a specific width and length and a fixed demand to be respected. The main objective function is to minimize the total amount of the textile materials produced/cut to satisfy the demand. According to Wäscher, Haußner, & Schumann (2007), the addressed problem is a Cutting Stock Problem (CSP), as the demand for each item is greater than one. However, in the real-world application at stake, the demand for each item type is not very high (below ten for all item types). Therefore, addressing the problem as a Bin-Packing Problem (BPP), in which all items are considered to be different and have a unitary demand, was a possibility. For this reason, two approaches to solve the problems were devised, implemented, and tested: (1) a CSP model, based on the well-known Lodi and Monaci (2003) model (3 variants), and (2) an original BPP-based model. Our research shows that, for this level of demand, the new BPP model is more competitive than CSP models. We analyzed these different models and described their characteristics, namely the size and the quality of the linear programming relaxation bound for solving the basic mono-objective variant of the problem. We also propose an ϵ−constraint approach to deal with a bi-objective extension of the problem, in which the number of cutting patterns used must also be minimized. The quality of the models was evaluated through computational experiments on randomly generated instances, yielding promising results.