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

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...

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Vydáno v:European journal of operational research Ročník 191; číslo 1; s. 61 - 85
Hlavní autoři: Cintra, G.F., Miyazawa, F.K., Wakabayashi, Y., Xavier, E.C.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 16.11.2008
Elsevier
Edice:European Journal of Operational Research
Témata:
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
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Popis
Shrnutí: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.
ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2007.08.007