Two-stage two-dimensional guillotine cutting stock problems with usable leftover

In this study, we solve the nonexact two-stage two-dimensional guillotine cutting problem considering usable leftovers, in which stock plates remainders of the cutting patterns (nonused material or trim loss) can be used in the future, if they are large enough to fulfill future demands for items (ordered smaller plates). This cutting problem can be characterized as a residual bin-packing problem because of the possibility of putting back into stock residual pieces, as the trim loss of each cutting/packing pattern does not necessarily represent waste of material depending on its size. Two bilevel mathematical programming models to represent this nonexact two-stage two-dimensional residual bin-packing problem are presented. The models basically consist of cutting/packing the ordered items using a set of plates of minimum cost and, among all possible solutions of minimum cost, choosing one that maximizes the value of the generated usable leftovers. Because of special characteristics of these bilevel models, they can be reformulated as one-level mixed integer programming models. Results of some numerical experiments are presented to show that the models represent appropriately the problem and illustrate their performance.