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Integer linear programming models for the skiving stock problem

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  • Martinovic, J.
  • Scheithauer, G.

Abstract

We consider the one-dimensional skiving stock problem which is strongly related to the dual bin packing problem: find the maximum number of items with minimum length L that can be constructed by connecting a given supply of m∈N smaller item lengths l1,…,lm with availabilities b1,…,bm. For this optimization problem, we present three new models (the arcflow model, the onestick model, and a model of Kantorovich-type) and investigate their relationships, especially regarding their respective continuous relaxations. To this end, numerical computations are provided. As a main result, we prove the equivalence between the arcflow model, the onestick approach and the existing pattern-oriented standard model. In particular, this equivalence is shown to hold for the corresponding continuous relaxations, too.

Suggested Citation

  • Martinovic, J. & Scheithauer, G., 2016. "Integer linear programming models for the skiving stock problem," European Journal of Operational Research, Elsevier, vol. 251(2), pages 356-368.
    • Handle: RePEc:eee:ejores:v:251:y:2016:i:2:p:356-368
    DOI: 10.1016/j.ejor.2015.11.005
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. 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.
    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. Martinovic, J. & Scheithauer, G. & Valério de Carvalho, J.M., 2018. "A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems," European Journal of Operational Research, Elsevier, vol. 266(2), pages 458-471.
    4. 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.
    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. Mathijs Barkel & Maxence Delorme, 2023. "Arcflow Formulations and Constraint Generation Frameworks for the Two Bar Charts Packing Problem," INFORMS Journal on Computing, INFORMS, vol. 35(2), pages 475-494, March.

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