Heuristic Approach for Packing Identical Rectangular Tiles in an Irregular Marble Slab

Packing identical rectangular tiles in a large marble slab is a necessary task. However, when the region of the marble slab is a concave set, the characteristics of the concave set make it difficult to determine whether the rectangular tiles are in the region. For this purpose, this article proposes a heuristic approach to the problem of packing identical rectangular items with orthogonal constraints in a concave irregular region. The heuristic approach uses a tree-search structure, and the final layout is built by inserting a new item in an irregular polygon. The newly inserted item must satisfy the containment and the nonoverlapping constraints associated with the irregular polygon. Regarding the containment constraint, we propose an improved slope algorithm to obtain the inner-fit polygon (IFP) of irregular polygons and rectangular items and apply IFP to determine the relative position of a rectangular item and an irregular polygon to guarantee containment constraint. Compared with existing methods, this approach can not only be used in convex regions but also in concave regions. Numerical experiments and application examples illustrate the effectiveness of the approach.