Mathematical models for the minimization of open stacks problem

In this paper, we address the minimization of open stacks problem (MOSP). This problem often appears during production planning of manufacturing industries, such as in the cutting of objects to comply with space constraints around the cutting machine in the glass, furniture, and metallurgical industries. During the processing of the cutting patterns, all the copies of a demanded item are stored in a stack usually placed near the cutting machine. One stack for each type of demanded item, that is, different items do not share the same stack. In this sense, the MOSP consists of finding an optimal sequence of a given set of cutting patterns, while minimizing the maximum number of simultaneously open stacks. To effectively model and solve the problem, we present a novel integer linear programming (ILP) formulation based on a graph representation of the problem. We derive an ILP formulation from the modeling approach of Faggioli and Bentivoglio for the MOSP. Then we develop a simple constraint programming model based on interval variables and renewable resources. We performed computational experiments to evaluate the proposed approaches in comparison with other ILP formulations from the literature. Using a general-purpose solver, the proposed approaches perform well in terms of solution quality and computational time in comparison to the benchmark models for small and moderate-sized problem instances.