Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation

• Published in: European journal of operational research Vol. 191; no. 1; pp. 61 - 85
• Main Authors: Cintra, G.F., Miyazawa, F.K., Wakabayashi, Y., Xavier, E.C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 16-11-2008; Elsevier
• Series: European Journal of Operational Research
• Subjects: Applied sciences; Column generation; Cutting stock; Dynamic programming; Exact sciences and technology; Flows in networks. Combinatorial problems; Guillotine cutting; Operational research and scientific management; Operational research. Management science; Strip packing; Two-dimensional packing; Two-dimensional packing; Cutting stock; Column generation; Dynamic programming; Strip packing; Guillotine cutting; Recurrence; Cutting stock problem; Bin packing problem; Polynomial method; Rotation; Knapsack problem; Polynomial time; Packing
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2007.08.007

We investigate several two-dimensional guillotine cutting stock problems and their variants in which orthogonal rotations are allowed. We first present two dynamic programming based algorithms for the
Rectangular Knapsack (RK) problem and its variants in which the patterns must be staged. The first algorithm solves the recurrence formula proposed by Beasley; the second algorithm – for staged patterns – also uses a recurrence formula. We show that if the items are not so small compared to the dimensions of the bin, then these algorithms require polynomial time. Using these algorithms we solved all instances of the RK problem found at the OR-LIBRARY, including one for which no optimal solution was known. We also consider the
Two-dimensional Cutting Stock problem. We present a column generation based algorithm for this problem that uses the first algorithm above mentioned to generate the columns. We propose two strategies to tackle the residual instances. We also investigate a variant of this problem where the bins have different sizes. At last, we study the
Two-dimensional Strip Packing problem. We also present a column generation based algorithm for this problem that uses the second algorithm above mentioned where staged patterns are imposed. In this case we solve instances for two-, three- and four-staged patterns. We report on some computational experiments with the various algorithms we propose in this paper. The results indicate that these algorithms seem to be suitable for solving real-world instances. We give a detailed description (a pseudo-code) of all the algorithms presented here, so that the reader may easily implement these algorithms.