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An integer programming model for two- and three-stage two-dimensional cutting stock problems

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  • Silva, Elsa
  • Alvelos, Filipe
  • Valério de Carvalho, J.M.

Abstract

In this paper, an integer programming model for two-dimensional cutting stock problems is proposed. In the problems addressed, it is intended to cut a set of small rectangular items of given sizes from a set of larger rectangular plates in such a way that the total number of used plates is minimized. The two-stage and three-stage, exact and non-exact, problems are considered. Other issues are also addressed, as the rotation of items, the length of the cuts and the value of the remaining plates. The new integer programming model can be seen as an extension of the "one-cut model" proposed by Dyckhoff for the one-dimensional cutting stock problem. In the proposed model, each decision variable is associated with cutting one item from a plate or from a part of a plate resulting from previous cuts (residual plates). Comparative computational results of the proposed model and of models from the literature are presented and discussed.

Suggested Citation

  • Silva, Elsa & Alvelos, Filipe & Valério de Carvalho, J.M., 2010. "An integer programming model for two- and three-stage two-dimensional cutting stock problems," European Journal of Operational Research, Elsevier, vol. 205(3), pages 699-708, September.
    • Handle: RePEc:eee:ejores:v:205:y:2010:i:3:p:699-708
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    References listed on IDEAS

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    9. Belov, G. & Scheithauer, G., 2006. "A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting," European Journal of Operational Research, Elsevier, vol. 171(1), pages 85-106, May.
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    Cited by:

    1. Wei, Lijun & Tian, Tian & Zhu, Wenbin & Lim, Andrew, 2014. "A block-based layer building approach for the 2D guillotine strip packing problem," European Journal of Operational Research, Elsevier, vol. 239(1), pages 58-69.
    2. Michele Monaci & André Gustavo Santos, 2018. "Minimum tiling of a rectangle by squares," Annals of Operations Research, Springer, vol. 271(2), pages 831-851, December.
    3. François Clautiaux & Ruslan Sadykov & François Vanderbeck & Quentin Viaud, 2019. "Pattern-based diving heuristics for a two-dimensional guillotine cutting-stock problem with leftovers," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(3), pages 265-297, September.
    4. Nestor M Cid-Garcia & Yasmin A Rios-Solis, 2020. "Positions and covering: A two-stage methodology to obtain optimal solutions for the 2d-bin packing problem," PLOS ONE, Public Library of Science, vol. 15(4), pages 1-22, April.
    5. 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).
    6. 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.
    7. Parreño, F. & Alvarez-Valdes, R., 2021. "Mathematical models for a cutting problem in the glass manufacturing industry," Omega, Elsevier, vol. 103(C).
    8. Rapine, Christophe & Pedroso, Joao Pedro & Akbalik, Ayse, 2022. "The two-dimensional knapsack problem with splittable items in stacks," Omega, Elsevier, vol. 112(C).
    9. Iori, Manuel & de Lima, Vinícius L. & Martello, Silvano & Miyazawa, Flávio K. & Monaci, Michele, 2021. "Exact solution techniques for two-dimensional cutting and packing," European Journal of Operational Research, Elsevier, vol. 289(2), pages 399-415.
    10. Martinovic, J. & Scheithauer, G. & Valério de Carvalho, J.M., 2018. "A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems," European Journal of Operational Research, Elsevier, vol. 266(2), pages 458-471.
    11. Parreño, F. & Alonso, M.T. & Alvarez-Valdes, R., 2020. "Solving a large cutting problem in the glass manufacturing industry," European Journal of Operational Research, Elsevier, vol. 287(1), pages 378-388.
    12. Cui, Yaodong & Huang, Baixiong, 2012. "Reducing the number of cuts in generating three-staged cutting patterns," European Journal of Operational Research, Elsevier, vol. 218(2), pages 358-365.
    13. Jean-François Côté & Manuel Iori, 2018. "The Meet-in-the-Middle Principle for Cutting and Packing Problems," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 646-661, November.
    14. de Lima, Vinícius L. & Alves, Cláudio & Clautiaux, François & Iori, Manuel & Valério de Carvalho, José M., 2022. "Arc flow formulations based on dynamic programming: Theoretical foundations and applications," European Journal of Operational Research, Elsevier, vol. 296(1), pages 3-21.
    15. Fabio Furini & Enrico Malaguti & Dimitri Thomopulos, 2016. "Modeling Two-Dimensional Guillotine Cutting Problems via Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 736-751, November.
    16. Jianyu Long & Zhong Zheng & Xiaoqiang Gao & Panos M. Pardalos & Wanzhe Hu, 2020. "An effective heuristic based on column generation for the two-dimensional three-stage steel plate cutting problem," Annals of Operations Research, Springer, vol. 289(2), pages 291-311, June.
    17. 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.
    18. Hadj Salem, Khadija & Silva, Elsa & Oliveira, José Fernando & Carravilla, Maria Antónia, 2023. "Mathematical models for the two-dimensional variable-sized cutting stock problem in the home textile industry," European Journal of Operational Research, Elsevier, vol. 306(2), pages 549-566.
    19. Douglas Nogueira Nascimento & Adriana Cristina Cherri & José Fernando Oliveira, 2022. "The two-dimensional cutting stock problem with usable leftovers: mathematical modelling and heuristic approaches," Operational Research, Springer, vol. 22(5), pages 5363-5403, November.
    20. Yi-Ping Cui & Yongwu Zhou & Yaodong Cui, 2017. "Triple-solution approach for the strip packing problem with two-staged patterns," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 588-604, August.
    21. Pedroso, João Pedro, 2020. "Heuristics for packing semifluids," European Journal of Operational Research, Elsevier, vol. 282(3), pages 823-834.

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