IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v48y2000i6p915-926.html
   My bibliography  Save this article

Exact Algorithm for Minimising the Number of Setups in the One-Dimensional Cutting Stock Problem

Author

Listed:
  • François Vanderbeck

    (Mathématiques Appliquées Bordeaux (MAB), Université Bordeaux 1, 351 Cours de la Libération, F-33405 Talence Cedex, France)

Abstract

The cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste. Because changing over from one cutting pattern to another involves significant setups, an auxiliary problem is to minimise the number of different patterns that are used. The pattern minimisation problem is significantly more complex, but it is of great practical importance. In this paper, we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns, each of which corresponds to selecting a fixed number of copies of a specific cutting pattern. The integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems. At each node of the branch-and-bound tree, the linear programming relaxation of our formulation is made tighter by adding super-additive inequalities. Branching rules are presented that yield a balanced tree. Incumbent solutions are obtained using a rounding heuristic. The resulting branch-and-price-and-cut procedure is used to produce optimal or approximately optimal solutions for a set of real-life problems.

Suggested Citation

  • François Vanderbeck, 2000. "Exact Algorithm for Minimising the Number of Setups in the One-Dimensional Cutting Stock Problem," Operations Research, INFORMS, vol. 48(6), pages 915-926, December.
    • Handle: RePEc:inm:oropre:v:48:y:2000:i:6:p:915-926
    DOI: 10.1287/opre.48.6.915.12391
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.48.6.915.12391
    Download Restriction: no
    File URL: https://libkey.io/10.1287/opre.48.6.915.12391?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
    ---><---

    References listed on IDEAS

    as
    1. François Vanderbeck, 2000. "On Dantzig-Wolfe Decomposition in Integer Programming and ways to Perform Branching in a Branch-and-Price Algorithm," Operations Research, INFORMS, vol. 48(1), pages 111-128, February.
    2. A. A. Farley, 1990. "A Note on Bounding a Class of Linear Programming Problems, Including Cutting Stock Problems," Operations Research, INFORMS, vol. 38(5), pages 922-923, October.
    3. Goulimis, Constantine, 1990. "Optimal solutions for the cutting stock problem," European Journal of Operational Research, Elsevier, vol. 44(2), pages 197-208, January.
    4. Vanderbeck, F. & Wolsey, L. A., 1996. "An exact algorithm for IP column generation," LIDAM Reprints CORE 1242, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. 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.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    3. Omid Shahvari & Rasaratnam Logendran & Madjid Tavana, 2022. "An efficient model-based branch-and-price algorithm for unrelated-parallel machine batching and scheduling problems," Journal of Scheduling, Springer, vol. 25(5), pages 589-621, October.
    4. Marc Peeters & Zeger Degraeve, 2004. "The Co-Printing Problem: A Packing Problem with a Color Constraint," Operations Research, INFORMS, vol. 52(4), pages 623-638, August.
    5. Valerio de Carvalho, J. M., 2002. "LP models for bin packing and cutting stock problems," European Journal of Operational Research, Elsevier, vol. 141(2), pages 253-273, September.
    6. Ambros Gleixner & Stephen J. Maher & Benjamin Müller & João Pedro Pedroso, 2020. "Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition," Annals of Operations Research, Springer, vol. 284(2), pages 527-555, January.
    7. Marjan van den Akker & Han Hoogeveen & Steef van de Velde, 2002. "Combining Column Generation and Lagrangean Relaxation to Solve a Single-Machine Common Due Date Problem," INFORMS Journal on Computing, INFORMS, vol. 14(1), pages 37-51, February.
    8. Daniel Villeneuve & Jacques Desrosiers & Marco Lübbecke & François Soumis, 2005. "On Compact Formulations for Integer Programs Solved by Column Generation," Annals of Operations Research, Springer, vol. 139(1), pages 375-388, October.
    9. Zeger Degraeve & Marc Peeters, 2003. "Optimal Integer Solutions to Industrial Cutting-Stock Problems: Part 2, Benchmark Results," INFORMS Journal on Computing, INFORMS, vol. 15(1), pages 58-81, February.
    10. DE WOLF, Daniel, 2003. "Using column generation to solve an industrial mixing problem," LIDAM Discussion Papers CORE 2003042, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. Barry C. Smith & Ellis L. Johnson, 2006. "Robust Airline Fleet Assignment: Imposing Station Purity Using Station Decomposition," Transportation Science, INFORMS, vol. 40(4), pages 497-516, November.
    12. Suliman, S.M.A., 2006. "A sequential heuristic procedure for the two-dimensional cutting-stock problem," International Journal of Production Economics, Elsevier, vol. 99(1-2), pages 177-185, February.
    13. Richard Freling & H. Edwin Romeijn & Dolores Romero Morales & Albert P. M. Wagelmans, 2003. "A Branch-and-Price Algorithm for the Multiperiod Single-Sourcing Problem," Operations Research, INFORMS, vol. 51(6), pages 922-939, December.
    14. Chengbin Chu & Julien Antonio, 1999. "Approximation Algorithms to Solve Real-Life Multicriteria Cutting Stock Problems," Operations Research, INFORMS, vol. 47(4), pages 495-508, August.
    15. Sung, Inkyung & Lee, Taesik, 2016. "Optimal allocation of emergency medical resources in a mass casualty incident: Patient prioritization by column generation," European Journal of Operational Research, Elsevier, vol. 252(2), pages 623-634.
    16. C Alves & J M Valério de Carvalho, 2008. "New integer programming formulations and an exact algorithm for the ordered cutting stock problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(11), pages 1520-1531, November.
    17. Daniel de Wolf & Stéphane Auray & Yves Smeers, 2015. "Using Column Generation To Solve A Coal Blending Problem," Post-Print halshs-02396784, HAL.
    18. Laszlo Ladanyi & Jon Lee & Robin Lougee-Heimer, 2005. "Rapid Prototyping of Optimization Algorithms Using COIN-OR: A Case Study Involving the Cutting-Stock Problem," Annals of Operations Research, Springer, vol. 139(1), pages 243-265, October.
    19. Zeger Degraeve & Linus Schrage, 1999. "Optimal Integer Solutions to Industrial Cutting Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 406-419, November.
    20. Krzysztof C. Kiwiel, 2010. "An Inexact Bundle Approach to Cutting-Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 22(1), pages 131-143, February.

    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:inm:oropre:v:48:y:2000:i:6:p:915-926. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.