Combinatorial Benders’ Cuts for the Strip Packing Problem

We study the strip packing problem, in which a set of two-dimensional rectangular items has to be packed in a rectangular strip of fixed width and infinite height, with the aim of minimizing the height used. The problem is important because it models a large number of real-world applications, including cutting operations where stocks of materials such as paper or wood come in large rolls and have to be cut with minimum waste, scheduling problems in which tasks require a contiguous subset of identical resources, and container loading problems arising in the transportation of items that cannot be stacked one over the other. The strip packing problem has been attacked in the literature with several heuristic and exact algorithms, nevertheless, benchmark instances of small size remain unsolved to proven optimality since many years. In this paper we propose a new exact method, that solves a large number of the open benchmark instances within a limited computational effort. Our method is based on a Benders’ decomposition, in which in the master we cut items into unit-width slices and pack them contiguously in the strip, and in the slave we attempt to reconstruct the rectangular items by fixing the vertical positions of their unit-width slices. If the slave proves that the reconstruction of the items is not possible, then a cut is added to the master, and the algorithm is re-iterated. We show that both the master and the slave are strongly NP-hard problems, and solve them with tailored pre-processing, lower and upper bounding techniques, and exact algorithms. We also propose several new techniques to improve the standard Benders’ cuts, using the so-called combinatorial Benders’ cuts, and an additional lifting procedure. Extensive computational tests show that the proposed algorithm provides a substantial breakthrough with respect to previously published algorithms.