Extracting Maximal Objects From Three-Dimensional Solid Materials

In the problem of extracting maximal items/objects from three-dimensional materials, we are given a piece of solid material and an object with predefined ratios. The goal is to extract the largest possible convex or nonconvex object from the solid material. There are two variants of the problem: the general case where the solid material is non-convex and its special case consisting of a convex solid material. We propose a matheuristic approach for solving both problems. For the special case, the problem is also modeled as a non-linear programming formulation. Meanwhile, for the general case a mixed integer linear programming formulation is provided based on the decomposition of the non-convex material into a finite number of convex regions. Computational experiments evaluate the performance of both models for small-scale instances considering objects with at most 1400 and 370 vertices for the special and general case, respectively. We observe that the required computational effort increases with the number of vertices of the objects, therefore, it is also evaluated the efficiency of the proposed matheuristic approach for more complicated instances, considering objects with a higher number of vertices. In order to establish benchmark instances for future research, all instances used are publicly available.