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Solving one-dimensional cutting stock problems exactly with a cutting plane algorithm

Author

Listed:
  • G Scheithauer

    (Dresden University of Technology)

  • J Terno

    (Dresden University of Technology)

  • A Müller

    (Dresden University of Technology)

  • G Belov

    (Dresden University of Technology)

Abstract

When solving the one-dimensional cutting stock problem (1D CSP) as an integer linear programming problem one has to overcome computational difficulties arising from the integrality condition and a huge number of variables. In the Gilmore–Gomory approach the corresponding continuous relaxation is solved via column generation techniques followed by an appropriate rounding of the in general non-integer solution. Obviously, there is no guarantee of obtaining an optimal solution in this way but it is extremely effective in practice. However, in two- and three-dimensional cutting stock problems the heuristics are not so good which necessitates the research of effective exact methods. In this paper we present an exact solution approach for the 1D CSP which is based on a combination of the cutting plane method and the column generation technique. Results of extensive computational experiments are reported.

Suggested Citation

  • G Scheithauer & J Terno & A Müller & G Belov, 2001. "Solving one-dimensional cutting stock problems exactly with a cutting plane algorithm," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 52(12), pages 1390-1401, December.
    • Handle: RePEc:pal:jorsoc:v:52:y:2001:i:12:d:10.1057_palgrave.jors.2601242
    DOI: 10.1057/palgrave.jors.2601242
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    Citations

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    Cited by:

    1. Krzysztof C. Kiwiel, 2010. "An Inexact Bundle Approach to Cutting-Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 22(1), pages 131-143, February.
    2. Belov, G. & Scheithauer, G., 2002. "A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths," European Journal of Operational Research, Elsevier, vol. 141(2), pages 274-294, September.
    3. 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.
    4. 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.
    5. Beraldi, P. & Bruni, M.E. & Conforti, D., 2009. "The stochastic trim-loss problem," European Journal of Operational Research, Elsevier, vol. 197(1), pages 42-49, August.
    6. Lijun Wei & Zhixing Luo, & Roberto Baldacci & Andrew Lim, 2020. "A New Branch-and-Price-and-Cut Algorithm for One-Dimensional Bin-Packing Problems," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 428-443, April.
    7. J W Hearne, 2007. "A market-driven approach to the optimal stocking problem on African game ranches," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(4), pages 423-428, April.

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