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A General Framework for Bounds for Higher-Dimensional Orthogonal Packing Problems

Author

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  • Sándor P. Fekete
  • Jörg Schepers

Abstract

Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. In the context of a branch-and-bound framework for solving these packing problems to optimality, it is of crucial importance to have good and easy bounds for an optimal solution. Previous efforts have produced a number of special classes of such bounds. Unfortunately, some of these bounds are somewhat complicated and hard to generalize. We present a new approach for obtaining classes of lower bounds for higher-dimensional packing problems; our bounds improve and simplify several well-known bounds from previous literature. In addition, our approach provides an easy framework for proving correctness of new bounds. This is the second in a series of four articles describing new approaches to higher-dimensional packing. Copyright Springer-Verlag 2004

Suggested Citation

  • Sándor P. Fekete & Jörg Schepers, 2004. "A General Framework for Bounds for Higher-Dimensional Orthogonal Packing Problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(2), pages 311-329, October.
    • Handle: RePEc:spr:mathme:v:60:y:2004:i:2:p:311-329
    DOI: 10.1007/s001860400376
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    Cited by:

    1. Sebastian Kraul & Markus Seizinger & Jens O. Brunner, 2023. "Machine Learning–Supported Prediction of Dual Variables for the Cutting Stock Problem with an Application in Stabilized Column Generation," INFORMS Journal on Computing, INFORMS, vol. 35(3), pages 692-709, May.
    2. Manuel Iori & Juan-José Salazar-González & Daniele Vigo, 2007. "An Exact Approach for the Vehicle Routing Problem with Two-Dimensional Loading Constraints," Transportation Science, INFORMS, vol. 41(2), pages 253-264, May.
    3. François Clautiaux & Cláudio Alves & José Valério de Carvalho, 2010. "A survey of dual-feasible and superadditive functions," Annals of Operations Research, Springer, vol. 179(1), pages 317-342, September.
    4. Wei, Lijun & Oon, Wee-Chong & Zhu, Wenbin & Lim, Andrew, 2011. "A skyline heuristic for the 2D rectangular packing and strip packing problems," European Journal of Operational Research, Elsevier, vol. 215(2), pages 337-346, December.
    5. 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.
    6. Alberto Caprara, 2008. "Packing d -Dimensional Bins in d Stages," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 203-215, February.
    7. Torsten Buchwald & Kirsten Hoffmann & Guntram Scheithauer, 2014. "Relations between capacity utilization, minimal bin size and bin number," Annals of Operations Research, Springer, vol. 217(1), pages 55-76, June.
    8. Polyakovsky, Sergey & M'Hallah, Rym, 2009. "An agent-based approach to the two-dimensional guillotine bin packing problem," European Journal of Operational Research, Elsevier, vol. 192(3), pages 767-781, February.
    9. Selma Khebbache-Hadji & Christian Prins & Alice Yalaoui & Mohamed Reghioui, 2013. "Heuristics and memetic algorithm for the two-dimensional loading capacitated vehicle routing problem with time windows," 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. 21(2), pages 307-336, March.
    10. Polyakovskiy, Sergey & M’Hallah, Rym, 2021. "Just-in-time two-dimensional bin packing," Omega, Elsevier, vol. 102(C).
    11. Clautiaux, Francois & Carlier, Jacques & Moukrim, Aziz, 2007. "A new exact method for the two-dimensional orthogonal packing problem," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1196-1211, December.
    12. Liao, Chung-Shou & Hsu, Chia-Hong, 2013. "New lower bounds for the three-dimensional orthogonal bin packing problem," European Journal of Operational Research, Elsevier, vol. 225(2), pages 244-252.
    13. Polyakovskiy, Sergey & M’Hallah, Rym, 2022. "A lookahead matheuristic for the unweighed variable-sized two-dimensional bin packing problem," European Journal of Operational Research, Elsevier, vol. 299(1), pages 104-117.
    14. Richard Korf & Michael Moffitt & Martha Pollack, 2010. "Optimal rectangle packing," Annals of Operations Research, Springer, vol. 179(1), pages 261-295, September.
    15. Polyakovskiy, Sergey & M’Hallah, Rym, 2018. "A hybrid feasibility constraints-guided search to the two-dimensional bin packing problem with due dates," European Journal of Operational Research, Elsevier, vol. 266(3), pages 819-839.
    16. Nikolaus Furian & Siegfried Vössner, 2014. "A hybrid algorithm for constrained order packing," 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(1), pages 157-186, March.
    17. Kartak, Vadim M. & Ripatti, Artem V., 2018. "The minimum raster set problem and its application to the d-dimensional orthogonal packing problem," European Journal of Operational Research, Elsevier, vol. 271(1), pages 33-39.
    18. Khanafer, Ali & Clautiaux, François & Talbi, El-Ghazali, 2010. "New lower bounds for bin packing problems with conflicts," European Journal of Operational Research, Elsevier, vol. 206(2), pages 281-288, October.
    19. F. Parreño & R. Alvarez-Valdes & J. Oliveira & J. Tamarit, 2010. "A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing," Annals of Operations Research, Springer, vol. 179(1), pages 203-220, September.
    20. Bayliss, Christopher & Currie, Christine S.M. & Bennell, Julia A. & Martinez-Sykora, Antonio, 2021. "Queue-constrained packing: A vehicle ferry case study," European Journal of Operational Research, Elsevier, vol. 289(2), pages 727-741.
    21. 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.

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