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One-Dimensional Cutting Stock

In: Introduction to Cutting and Packing Optimization

Author

Listed:
  • Guntram Scheithauer

    (TU Dresden)

Abstract

In difference to one-dimensional Bin Packing Problems (1BPP) where each item is considered to be a unique one, in a one-dimensional Cutting Stock Problem (1CSP), the number of different piece types is rather small, but their order demands (or availability in case of packing problems) are mostly large. Another differentiator of bin packing and cutting stock problems could be the magnitude of the respective optimal value. If it is small in comparison to the total number of items, then the instance is of BPP type, otherwise the problem type depends on the number of different patterns in a solution. In the beginning of this chapter, we consider the 1CSP with a single type of raw material. We present a solution strategy which is also applicable to higher-dimensional problems as, for instance, in the furniture industry when the production of rectangular pieces has to be optimized. Subsequently, we address generalizations and present alternative models. Finally, we investigate the relation between the standard ILP model and its LP relaxation and observe a small gap for any 1CSP instance.

Suggested Citation

  • Guntram Scheithauer, 2018. "One-Dimensional Cutting Stock," International Series in Operations Research & Management Science, in: Introduction to Cutting and Packing Optimization, chapter 0, pages 73-122, Springer.
    • Handle: RePEc:spr:isochp:978-3-319-64403-5_4
    DOI: 10.1007/978-3-319-64403-5_4
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    Citations

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    Cited by:

    1. John Martinovic & Markus Hähnel & Guntram Scheithauer & Waltenegus Dargie, 2022. "An introduction to stochastic bin packing-based server consolidation with conflicts," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(2), pages 296-331, July.
    2. John Martinovic & Guntram Scheithauer, 2018. "Combinatorial investigations on the maximum gap for skiving stock instances of the divisible case," Annals of Operations Research, Springer, vol. 271(2), pages 811-829, December.
    3. Dell’Amico, Mauro & Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2019. "Mathematical models and decomposition methods for the multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 274(3), pages 886-899.
    4. John Martinovic & Markus Hähnel & Guntram Scheithauer & Waltenegus Dargie & Andreas Fischer, 2019. "Cutting stock problems with nondeterministic item lengths: a new approach to server consolidation," 4OR, Springer, vol. 17(2), pages 173-200, June.
    5. John Martinovic, 2022. "A note on the integrality gap of cutting and skiving stock instances," 4OR, Springer, vol. 20(1), pages 85-104, March.
    6. de Lima, Vinícius L. & Alves, Cláudio & Clautiaux, François & Iori, Manuel & Valério de Carvalho, José M., 2022. "Arc flow formulations based on dynamic programming: Theoretical foundations and applications," European Journal of Operational Research, Elsevier, vol. 296(1), pages 3-21.
    7. Wang, Danni & Xiao, Fan & Zhou, Lei & Liang, Zhe, 2020. "Two-dimensional skiving and cutting stock problem with setup cost based on column-and-row generation," European Journal of Operational Research, Elsevier, vol. 286(2), pages 547-563.
    8. Maxence Delorme & Manuel Iori, 2020. "Enhanced Pseudo-polynomial Formulations for Bin Packing and Cutting Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 101-119, January.

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