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An Exact Algorithm for Constrained Two-Dimensional Two-Staged Cutting Problems

Author

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  • Mhand Hifi

    (LaRIA, Laboratoire de Recherche en Informatique d'Amiens, 5 rue du Moulin Neuf, 80000 Amiens, France; CERMSEM-CNRS UMR 8095, Maison des Sciences Economiques, Université Paris 1 Panthéon-Sorbonne, France; and LRIA-LPBD, EA 3387, EPHE, Paris, France)

  • Rym M'Hallah

    (Department of Statistics and Operations Research, Kuwait University, P.O. Box 5969, Safat 13060, State of Kuwait)

Abstract

The constrained two-dimensional cutting (C_TDC) problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate S with length L and width W , to maximize the sum of the profits of the pieces to be cut. Each piece type i , i =1,…, n , is characterized by a length l i , a width w i , a profit (or weight) c i , and an upper demand value b i . The upper demand value is the maximum number of pieces of type i that can be cut on S . In this paper, we study the two-staged C_TDC problem, noted C_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two cuts. We solve the C_2TDC problem using an exact algorithm that is mainly based on a bottom-up strategy. We introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns. We evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm.

Suggested Citation

  • Mhand Hifi & Rym M'Hallah, 2005. "An Exact Algorithm for Constrained Two-Dimensional Two-Staged Cutting Problems," Operations Research, INFORMS, vol. 53(1), pages 140-150, February.
    • Handle: RePEc:inm:oropre:v:53:y:2005:i:1:p:140-150
    DOI: 10.1287/opre.1040.0154
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    References listed on IDEAS

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    1. Mhand Hifi & Catherine Roucairol, 2001. "Approximate and Exact Algorithms for Constrained (Un) Weighted Two-dimensional Two-staged Cutting Stock Problems," Journal of Combinatorial Optimization, Springer, vol. 5(4), pages 465-494, December.
    2. P. C. Gilmore & R. E. Gomory, 1961. "A Linear Programming Approach to the Cutting-Stock Problem," Operations Research, INFORMS, vol. 9(6), pages 849-859, December.
    3. L. V. Kantorovich, 1960. "Mathematical Methods of Organizing and Planning Production," Management Science, INFORMS, vol. 6(4), pages 366-422, July.
    4. Silvano Martello & Paolo Toth, 1997. "Upper Bounds and Algorithms for Hard 0-1 Knapsack Problems," Operations Research, INFORMS, vol. 45(5), pages 768-778, October.
    5. Morabito, Reinaldo & Arenales, Marcos N., 1996. "Staged and constrained two-dimensional guillotine cutting problems: An AND/OR-graph approach," European Journal of Operational Research, Elsevier, vol. 94(3), pages 548-560, November.
    6. David Pisinger, 1997. "A Minimal Algorithm for the 0-1 Knapsack Problem," Operations Research, INFORMS, vol. 45(5), pages 758-767, October.
    7. Dyckhoff, Harald, 1990. "A typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 44(2), pages 145-159, January.
    8. P. C. Gilmore & R. E. Gomory, 1965. "Multistage Cutting Stock Problems of Two and More Dimensions," Operations Research, INFORMS, vol. 13(1), pages 94-120, February.
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    Cited by:

    1. Mhand Hifi & Rym M'Hallah & Toufik Saadi, 2008. "Algorithms for the Constrained Two-Staged Two-Dimensional Cutting Problem," INFORMS Journal on Computing, INFORMS, vol. 20(2), pages 212-221, May.
    2. Cui, Yaodong & Yang, Liu & Zhao, Zhigang & Tang, Tianbing & Yin, Mengxiao, 2013. "Sequential grouping heuristic for the two-dimensional cutting stock problem with pattern reduction," International Journal of Production Economics, Elsevier, vol. 144(2), pages 432-439.
    3. Song, X. & Chu, C.B. & Lewis, R. & Nie, Y.Y. & Thompson, J., 2010. "A worst case analysis of a dynamic programming-based heuristic algorithm for 2D unconstrained guillotine cutting," European Journal of Operational Research, Elsevier, vol. 202(2), pages 368-378, April.
    4. Leung, Stephen C.H. & Zhang, Defu & Sim, Kwang Mong, 2011. "A two-stage intelligent search algorithm for the two-dimensional strip packing problem," European Journal of Operational Research, Elsevier, vol. 215(1), pages 57-69, November.
    5. Celia Glass & Jeroen Oostrum, 2010. "Bun splitting: a practical cutting stock problem," Annals of Operations Research, Springer, vol. 179(1), pages 15-33, September.
    6. Cui, Yaodong & Zhao, Zhigang, 2013. "Heuristic for the rectangular two-dimensional single stock size cutting stock problem with two-staged patterns," European Journal of Operational Research, Elsevier, vol. 231(2), pages 288-298.
    7. Silva, Elsa & Oliveira, José Fernando & Silveira, Tiago & Mundim, Leandro & Carravilla, Maria Antónia, 2023. "The Floating-Cuts model: a general and flexible mixed-integer programming model for non-guillotine and guillotine rectangular cutting problems," Omega, Elsevier, vol. 114(C).
    8. Mhand Hifi & Toufik Saadi, 2012. "A parallel algorithm for two-staged two-dimensional fixed-orientation cutting problems," Computational Optimization and Applications, Springer, vol. 51(2), pages 783-807, March.
    9. Rapine, Christophe & Pedroso, Joao Pedro & Akbalik, Ayse, 2022. "The two-dimensional knapsack problem with splittable items in stacks," Omega, Elsevier, vol. 112(C).

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