Fast neighborhood search for two- and three-dimensional nesting problems

In this paper we present a new heuristic solution method for two-dimensional nesting problems. It is based on a simple local search scheme in which the neighborhood is any horizontal or vertical translation of a given polygon from its current position. To escape local minima we apply the meta-heuristic method Guided Local Search. The strength of our solution method comes from a new algorithm which is capable of searching the neighborhood in polynomial time. More precisely, given a single polygon with m edges and a set of polygons with n edges the algorithm can find a translation with minimum overlap in time O(mn log(mn)). Solutions for standard test instances are generated by an implementation and a comparison is done with recent results from the literature. The solution method is very robust and most of the best solutions found are also the currently best results published. Our approach to the problem is very flexible regarding problem variations and special constraints, and as an example we describe how it can handle materials with quality regions. Finally, we generalize the algorithm for the fast neighborhood search and present a solution method for three-dimensional nesting problems. 2006 Elsevier B.V. All rights reserved.