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Cutting Stock Problems

In: Column Generation

Author

Listed:
  • Hatem Ben Amor

    (École Polytechnique and GERAD)

  • José Valério de Carvalho

    (Universidade do Minho)

Abstract

Column generation has been proposed by Gilmore and Gomory to solve cutting stock problem, independently of Dantzig-Wolfe decomposition. We survey the basic models proposed for cutting stock and the corresponding solution approaches. Extended Dantzig-Wolfe decomposition is surveyed and applied to these models in order to show the links to Gilmore-Gomory model. Branching schemes discussion is based on the subproblem formulation corresponding to each model. Integer solutions are obtained by combining heuristics and branch-and-price schemes. Linear relaxations are solved by column generation. Stabilization techniques such as dual-optimal inequalities and stabilized column generation algorithms that have been proposed to improve the efficiency of this process are briefly discussed.

Suggested Citation

  • Hatem Ben Amor & José Valério de Carvalho, 2005. "Cutting Stock Problems," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 131-161, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-25486-9_5
    DOI: 10.1007/0-387-25486-2_5
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    Citations

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    Cited by:

    1. Raf Jans, 2009. "Solving Lot-Sizing Problems on Parallel Identical Machines Using Symmetry-Breaking Constraints," INFORMS Journal on Computing, INFORMS, vol. 21(1), pages 123-136, February.
    2. Paul A. Chircop & Timothy J. Surendonk & Menkes H. L. van den Briel & Toby Walsh, 2022. "On routing and scheduling a fleet of resource-constrained vessels to provide ongoing continuous patrol coverage," Annals of Operations Research, Springer, vol. 312(2), pages 723-760, May.
    3. Gondzio, Jacek & González-Brevis, Pablo & Munari, Pedro, 2013. "New developments in the primal–dual column generation technique," European Journal of Operational Research, Elsevier, vol. 224(1), pages 41-51.
    4. Timo Gschwind & Stefan Irnich, 2014. "Dual Inequalities for Stabilized Column Generation Revisited," Working Papers 1407, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz, revised 23 Jul 2014.
    5. Krzysztof C. Kiwiel, 2010. "An Inexact Bundle Approach to Cutting-Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 22(1), pages 131-143, February.
    6. Katrin Heßler & Timo Gschwind & Stefan Irnich, 2017. "Stabilized Branch-and-Price Algorithms for Vector Packing Problems," Working Papers 1713, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    7. Silva, Eduardo M. & Melega, Gislaine M. & Akartunalı, Kerem & de Araujo, Silvio A., 2023. "Formulations and theoretical analysis of the one-dimensional multi-period cutting stock problem with setup cost," European Journal of Operational Research, Elsevier, vol. 304(2), pages 443-460.
    8. Dongho Lee & Seunghyun Son & Doyeong Kim & Sunkuk Kim, 2020. "Special-Length-Priority Algorithm to Minimize Reinforcing Bar-Cutting Waste for Sustainable Construction," Sustainability, MDPI, vol. 12(15), pages 1-15, July.
    9. Timo Gschwind & Stefan Irnich, 2016. "Dual Inequalities for Stabilized Column Generation Revisited," INFORMS Journal on Computing, INFORMS, vol. 28(1), pages 175-194, February.
    10. Kallrath, Julia & Rebennack, Steffen & Kallrath, Josef & Kusche, Rüdiger, 2014. "Solving real-world cutting stock-problems in the paper industry: Mathematical approaches, experience and challenges," European Journal of Operational Research, Elsevier, vol. 238(1), pages 374-389.
    11. Heßler, Katrin & Gschwind, Timo & Irnich, Stefan, 2018. "Stabilized branch-and-price algorithms for vector packing problems," European Journal of Operational Research, Elsevier, vol. 271(2), pages 401-419.
    12. Keehoon Kwon & Doyeong Kim & Sunkuk Kim, 2021. "Cutting Waste Minimization of Rebar for Sustainable Structural Work: A Systematic Literature Review," Sustainability, MDPI, vol. 13(11), pages 1-21, May.
    13. Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2016. "Bin packing and cutting stock problems: Mathematical models and exact algorithms," European Journal of Operational Research, Elsevier, vol. 255(1), pages 1-20.
    14. Tao Wu & Kerem Akartunal? & Raf Jans & Zhe Liang, 2017. "Progressive Selection Method for the Coupled Lot-Sizing and Cutting-Stock Problem," INFORMS Journal on Computing, INFORMS, vol. 29(3), pages 523-543, August.

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