Generating unconstrained two-dimensional non-guillotine cutting patterns by a Recursive Partitioning Algorithm

Abstract

In this study, a dynamic programming approach to deal with the unconstrained twodimensional non-guillotine cutting problem is presented. The method extends the recently introduced recursive partitioning approach for the manufacturer's pallet loading problem. The approach involves two phases and uses bounds based on unconstrained two-staged and non-staged guillotine cutting. The method is able to find the optimal cutting pattern of a large number of problem instances of moderate sizes known in the literature and a counterexample for which the approach fails to find known optimal solutions was not found. For the instances that the required computer runtime is excessive, the approach is combined with simple heuristics to reduce its running time. Detailed numerical experiments show the reliability of the method.

Una revisión literaria del Problema de Carga del Pallet

“…In other cases, programming languages such as C+ in its different versions have been the chosen language for the development of the algorithms as referred in [3], [8], [10], [12], [14], [23], [25], [26], [31], [37] and [39].…”

“…It could be observed that the majority of them use exact algorithms with the incorporation of heuristics. The conformation of blocks (G4, G5), patterns or groups the heuristic method more widely employed when solving the PLP in either of its two variations as in the works of [1], [6], [7], [10], [14], [17], [31], [34], [37], [38] and [39]. Heuristics are normally used within these solution methods because they allowed the possibility to reduce computational time for this NPhard problem.…”

Symmetry-breaking constraints for packing identical rectangles within polyhedra

“…All the models considered in the numerical experiments were implemented in C/C++ (gcc version 4.4.3) and solved with CPLEX (IBM ILOG AMPL/CPLEX 12.1.0) using its default parameters. It means that a solution is reported as optimal by the solver when absolute gap = best feasible solution − best lower bound ≤ 10 In order to test the proposed models, values for constants α, β, and M o in (10), M w and M h in (4,12), andĉ x andĉ y in (5,13) must be given. For α and β we considered random numbers with uniform distribution within the interval [−1, 1].…”

“…See also [8,10,11]. When the convex region is a polyhedra -as opposed to a region with curved boundariesthe problem can be modeled as a mixed integer programming (MIP) problem and solved to optimality (see [1,3,4,6] for the particular case of packing rectangles within rectangles). When all the rectangles are identical, it is possible to define equivalent solutions in which the roles of a pair of rectangles R i and R j are reversed.…”

“…If, for example, Ω were an equilateral triangle of side L, it would be enough to locate it with its bottom-left corner at the origin of the Cartesian axes and to considerĉ x ≡ L andĉ y ≡ ( √ 3/2)L. Constraints (1,3,4) and (1,3,5) define MIP (feasibility) models, named P LGDP . In P k,[Ω,h,w] BM there are two additional binary variables (q ij , q ji , j = i + 1, .…”