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 non-convex 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.

在从三维材料中提取最大项目/对象的问题中，我们获得了一块固体材料和一个具有预定比例的对象。目的是从固体材料中提取最大可能的凸形或非凸形对象。该问题有两种变体：固体材料为非凸面的一般情况和特殊情况，其由凸形固体材料组成。我们提出了一种数学方法来解决这两个问题。对于特殊情况，该问题也被建模为非线性规划公式。同时，在一般情况下，基于非凸材料分解为有限数量的凸区，提供了混合整数线性规划公式。计算实验评估了两种模型在小规模实例中的性能，分别考虑了特殊情况和一般情况下分别具有最多1400和370个顶点的对象。我们观察到，所需的计算量随对象的顶点数量而增加，因此，还考虑了具有更多顶点数量的对象，还针对较复杂的实例评估了所提出的数学方法的效率。为了建立用于将来研究的基准实例，使用的所有实例都是公开可用的。还考虑了顶点数量更多的对象，评估了所提出的数学方法在更复杂的情况下的效率。为了建立用于将来研究的基准实例，使用的所有实例都是公开可用的。还考虑了顶点数量更多的对象，评估了所提出的数学方法在更复杂的情况下的效率。为了建立用于将来研究的基准实例，使用的所有实例都是公开可用的。