Combinatorial Benders' Cuts for the Strip Packing Problem

Abstract

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. 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 reiterated.We show that both the master and the slave are strongly -hard problems and solve them with tailored preprocessing, 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.

Combinatorial Benders cuts for assembly line balancing problems with setups

“…Constraints (10) ensure that the active workstations are in an ordered sequence. Constraint sets (11), (12) and (13) order tasks and assign setup operations between them if they are assigned to the same workstation. Any two tasks have to be ordered within a workstation due to their precedence relations which is ensured by constraints (14).…”

“…As stated by Côté et al, [11], for a special case of P where c = 0, the slave SP can be used as a feasibility check on the system {Ay + Bx ≥ b, y ≥ 0}. In particular, ifx is not a feasible solution for at least one variable x j causing infeasibility, then this variable must take a different value fromx j .…”

“…and derive a cut from these subsets rather than adding a cut containing all the x variables [11]. Such constraints are called combinatorial Benders cuts by [10], which do not require that SP is continuous.…”

Mathematical models and decomposition algorithms for cutting and packing problems

“…They solved the slave problem by logical deduction methods, such as constraint programming, whose outcome was used to produce valid cuts. Côté et al [83] also used Benders' decomposition where both subproblems were MILPs solved by combinatorial algorithm. To produce effective valid cuts, they used the combinatorial Benders' cuts introduced by Codato and Fischetti [69], in which a third combinatorial algorithm is used to find the cuts that are added to the master.…”

“…The latter also made computational experiments on SPP instances, but the results were not encouraging, as the proposed approaches generally perform worse than the 'old' Martello et al [212] branch-and-bound algorithm. Côté et al [83] found better results by using a Benders' decomposition in which the master problem is a MILP model and the slave is solved through branch-and-bound.…”