IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v179y2010i1p297-31510.1007-s10479-008-0457-4.html
   My bibliography  Save this article

A heuristic approach based on dynamic programming and and/or-graph search for the constrained two-dimensional guillotine cutting problem

Author

Listed:
  • Reinaldo Morabito
  • Vitória Pureza

Abstract

In this paper we present a heuristic method to generate constrained two-dimensional guillotine cutting patterns. This problem appears in different industrial processes of cutting rectangular plates to produce ordered items, such as in the glass, furniture and circuit board business. The method uses a state space relaxation of a dynamic programming formulation of the problem and a state space ascent procedure of subgradient optimization type. We propose the combination of this existing approach with an and/or-graph search and an inner heuristic that turns infeasible solutions provided in each step of the ascent procedure into feasible solutions. Results for benchmark and randomly generated instances indicate that the method’s performance is competitive compared to other methods proposed in the literature. One of its advantages is that it often produces a relatively tight upper bound to the optimal value. Moreover, in most cases for which an optimal solution is obtained, it also provides a certificate of optimality. Copyright Springer Science+Business Media, LLC 2010

Suggested Citation

  • Reinaldo Morabito & Vitória Pureza, 2010. "A heuristic approach based on dynamic programming and and/or-graph search for the constrained two-dimensional guillotine cutting problem," Annals of Operations Research, Springer, vol. 179(1), pages 297-315, September.
    • Handle: RePEc:spr:annopr:v:179:y:2010:i:1:p:297-315:10.1007/s10479-008-0457-4
    DOI: 10.1007/s10479-008-0457-4
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-008-0457-4
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10479-008-0457-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. P. C. Gilmore & R. E. Gomory, 1966. "The Theory and Computation of Knapsack Functions," Operations Research, INFORMS, vol. 14(6), pages 1045-1074, December.
    2. Wang, P. Y. & Wascher, G., 2002. "Cutting and packing," European Journal of Operational Research, Elsevier, vol. 141(2), pages 239-240, September.
    3. Oliveira, Jose Fernando & Wascher, Gerhard, 2007. "Cutting and Packing," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1106-1108, December.
    4. Oliveira, JoseFernando & Ferreira, JoseSoeiro, 1990. "An improved version of Wang's algorithm for two-dimensional cutting problems," European Journal of Operational Research, Elsevier, vol. 44(2), pages 256-266, January.
    5. K. V. Viswanathan & A. Bagchi, 1993. "Best-First Search Methods for Constrained Two-Dimensional Cutting Stock Problems," Operations Research, INFORMS, vol. 41(4), pages 768-776, August.
    6. Victor Parada & Rodrigo Palma & Daniel Sales & Arlindo Gómes, 2000. "A comparative numerical analysis for the guillotine two-dimensional cutting problem," Annals of Operations Research, Springer, vol. 96(1), pages 245-254, November.
    7. Lodi, Andrea & Martello, Silvano & Monaci, Michele, 2002. "Two-dimensional packing problems: A survey," European Journal of Operational Research, Elsevier, vol. 141(2), pages 241-252, September.
    8. Dowsland, Kathryn A. & Herbert, Edward A. & Kendall, Graham & Burke, Edmund, 2006. "Using tree search bounds to enhance a genetic algorithm approach to two rectangle packing problems," European Journal of Operational Research, Elsevier, vol. 168(2), pages 390-402, January.
    9. Mhand Hifi, 2004. "Dynamic Programming and Hill-Climbing Techniques for Constrained Two-Dimensional Cutting Stock Problems," Journal of Combinatorial Optimization, Springer, vol. 8(1), pages 65-84, March.
    10. E. K. Burke & G. Kendall & G. Whitwell, 2004. "A New Placement Heuristic for the Orthogonal Stock-Cutting Problem," Operations Research, INFORMS, vol. 52(4), pages 655-671, August.
    11. Christofides, Nicos & Hadjiconstantinou, Eleni, 1995. "An exact algorithm for orthogonal 2-D cutting problems using guillotine cuts," European Journal of Operational Research, Elsevier, vol. 83(1), pages 21-38, May.
    12. Morabito, R. N. & Arenales, M. N. & Arcaro, V. F., 1992. "An and--or-graph approach for two-dimensional cutting problems," European Journal of Operational Research, Elsevier, vol. 58(2), pages 263-271, April.
    13. Nicos Christofides & Charles Whitlock, 1977. "An Algorithm for Two-Dimensional Cutting Problems," Operations Research, INFORMS, vol. 25(1), pages 30-44, February.
    14. Wascher, Gerhard & Hau[ss]ner, Heike & Schumann, Holger, 2007. "An improved typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1109-1130, December.
    15. 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.
    16. P. Y. Wang, 1983. "Two Algorithms for Constrained Two-Dimensional Cutting Stock Problems," Operations Research, INFORMS, vol. 31(3), pages 573-586, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Carlos V. G. C. Lima & Fábio Protti & Dieter Rautenbach & Uéverton S. Souza & Jayme L. Szwarcfiter, 2018. "And/or-convexity: a graph convexity based on processes and deadlock models," Annals of Operations Research, Springer, vol. 264(1), pages 267-286, May.
    2. 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.
    3. Russo, Mauro & Sforza, Antonio & Sterle, Claudio, 2013. "An improvement of the knapsack function based algorithm of Gilmore and Gomory for the unconstrained two-dimensional guillotine cutting problem," International Journal of Production Economics, Elsevier, vol. 145(2), pages 451-462.
    4. Krzysztof Fleszar, 2016. "An Exact Algorithm for the Two-Dimensional Stage-Unrestricted Guillotine Cutting/Packing Decision Problem," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 703-720, November.
    5. Oliviana Xavier Nascimento & Thiago Alves Queiroz & Leonardo Junqueira, 2022. "A MIP-CP based approach for two- and three-dimensional cutting problems with staged guillotine cuts," Annals of Operations Research, Springer, vol. 316(2), pages 805-835, September.
    6. Velasco, André Soares & Uchoa, Eduardo, 2019. "Improved state space relaxation for constrained two-dimensional guillotine cutting problems," European Journal of Operational Research, Elsevier, vol. 272(1), pages 106-120.
    7. Wei, Lijun & Lim, Andrew, 2015. "A bidirectional building approach for the 2D constrained guillotine knapsack packing problem," European Journal of Operational Research, Elsevier, vol. 242(1), pages 63-71.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Krzysztof Fleszar, 2016. "An Exact Algorithm for the Two-Dimensional Stage-Unrestricted Guillotine Cutting/Packing Decision Problem," INFORMS Journal on Computing, INFORMS, vol. 28(4), pages 703-720, November.
    2. 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.
    3. Wei, Lijun & Lim, Andrew, 2015. "A bidirectional building approach for the 2D constrained guillotine knapsack packing problem," European Journal of Operational Research, Elsevier, vol. 242(1), pages 63-71.
    4. Mhand Hifi, 2004. "Dynamic Programming and Hill-Climbing Techniques for Constrained Two-Dimensional Cutting Stock Problems," Journal of Combinatorial Optimization, Springer, vol. 8(1), pages 65-84, March.
    5. Russo, Mauro & Sforza, Antonio & Sterle, Claudio, 2013. "An improvement of the knapsack function based algorithm of Gilmore and Gomory for the unconstrained two-dimensional guillotine cutting problem," International Journal of Production Economics, Elsevier, vol. 145(2), pages 451-462.
    6. 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.
    7. 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.
    8. Hifi, Mhand, 1997. "The DH/KD algorithm: a hybrid approach for unconstrained two-dimensional cutting problems," European Journal of Operational Research, Elsevier, vol. 97(1), pages 41-52, February.
    9. de Armas, Jesica & Miranda, Gara & León, Coromoto, 2012. "Improving the efficiency of a best-first bottom-up approach for the Constrained 2D Cutting Problem," European Journal of Operational Research, Elsevier, vol. 219(2), pages 201-213.
    10. 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.
    11. 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).
    12. Yanasse, Horacio Hideki & Pinto Lamosa, Maria Jose, 2007. "An integrated cutting stock and sequencing problem," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1353-1370, December.
    13. Sławomir Bąk & Jacek Błażewicz & Grzegorz Pawlak & Maciej Płaza & Edmund K. Burke & Graham Kendall, 2011. "A Parallel Branch-and-Bound Approach to the Rectangular Guillotine Strip Cutting Problem," INFORMS Journal on Computing, INFORMS, vol. 23(1), pages 15-25, February.
    14. Parada Daza, Victor & Gomes de Alvarenga, Arlindo & de Diego, Jose, 1995. "Exact solutions for constrained two-dimensional cutting problems," European Journal of Operational Research, Elsevier, vol. 84(3), pages 633-644, August.
    15. Igor Kierkosz & Maciej Luczak, 2014. "A hybrid evolutionary algorithm for the two-dimensional packing problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 22(4), pages 729-753, December.
    16. 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.
    17. Hadjiconstantinou, Eleni & Iori, Manuel, 2007. "A hybrid genetic algorithm for the two-dimensional single large object placement problem," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1150-1166, December.
    18. István Borgulya, 2019. "An EDA for the 2D knapsack problem with guillotine constraint," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 329-356, June.
    19. Ortmann, Frank G. & Ntene, Nthabiseng & van Vuuren, Jan H., 2010. "New and improved level heuristics for the rectangular strip packing and variable-sized bin packing problems," European Journal of Operational Research, Elsevier, vol. 203(2), pages 306-315, June.
    20. Vera Neidlein & Andrèa C. G. Vianna & Marcos N. Arenales & Gerhard Wäscher, 2008. "The Two-Dimensional, Rectangular, Guillotineable-Layout Cutting Problem with a Single Defect," FEMM Working Papers 08035, Otto-von-Guericke University Magdeburg, Faculty of Economics and Management.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:179:y:2010:i:1:p:297-315:10.1007/s10479-008-0457-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.