Three-dimensional guillotine cutting problems with constrained patterns: MILP formulations and a bottom-up algorithm

In this paper, we address the Constrained Three-dimensional Guillotine Cutting Problem (C3GCP), which consists of cutting a larger cuboid block (object) to produce a limited number of smaller cuboid pieces (items) using orthogonal guillotine cuts only. This way, all cuts must be parallel to the object’s walls and generate two cuboid sub-blocks, and there is a maximum number of copies that can be manufactured for each item type. The C3GCP arises in industrial manufacturing settings, such as the cutting of steel and foam for mattresses. To model this problem, we propose a new compact mixed-integer non-linear programming (MINLP) formulation by extending its two-dimensional version, and develop a mixed-integer linear programming (MILP) version. We also propose a new model for a particular case of the problem which considers 3-staged patterns. As a solution method, we extend the algorithm of Wang (1983) to the three-dimensional case. We emphasise that the C3GCP is different from 3D packing problems, namely from the Container Loading Problem, because of the guillotine cut constraints. All proposed approaches are evaluated through computational experiments using benchmark instances. The results show that the approaches are effective on different types of instances, mainly when the maximum number of copies per item type is small, a situation typically encountered in practical settings with low demand for each item type. These approaches can be easily embedded into existing expert systems for supporting the decision-making process.