New lower bounds based on column generation and constraint programming for the pattern minimization problem

The pattern minimization problem is a cutting and packing problem that consists in finding a cutting plan with the minimum number of different patterns. This objective may be relevant when changing from one pattern to another involves a cost for setting up the cutting machine. When the minimization of the number of different patterns is done by assuming that no more than the minimum number of rolls can be used, the problem is also referred to as the cutting stock problem with setup costs. Most of the approaches described in the literature are based on heuristics. Solving the problem exactly has been a real challenge, and only very few exact solution methods have been reported so far in the literature. In this paper, we intend to contribute to the resolution of the pattern minimization problem with new results. We explore a different integer programming model that can be solved using column generation, and we describe different strategies to strengthen it, among which are constraint programming and new families of valid inequalities. Lower bounds for the pattern minimization problem are derived from the new integer programming model, and also from a constraint programming model. Our approaches were tested on a set of real instances, and on a set of random instances from the literature. For these instances, the computational experiments show a clear improvement on the quality of the lower bounds.