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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.
    2. Csirik, J. & Frenk, J.B.G. & Galambos, G. & Rinnooy Kan, A.H.G., 1991. "Probabilistic analysis of algorithms for dual bin packing problems," Econometric Institute Research Papers 11733, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. Harald Dyckhoff, 1981. "A New Linear Programming Approach to the Cutting Stock Problem," Operations Research, INFORMS, vol. 29(6), pages 1092-1104, December.
    4. Hatem Ben Amor & Jacques Desrosiers & José Manuel Valério de Carvalho, 2006. "Dual-Optimal Inequalities for Stabilized Column Generation," Operations Research, INFORMS, vol. 54(3), pages 454-463, June.
    5. 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.
    6. Vijayakumar, Bharathwaj & Parikh, Pratik J. & Scott, Rosalyn & Barnes, April & Gallimore, Jennie, 2013. "A dual bin-packing approach to scheduling surgical cases at a publicly-funded hospital," European Journal of Operational Research, Elsevier, vol. 224(3), pages 583-591.
    7. Gau, T. & Wascher, G., 1995. "CUTGEN1: A problem generator for the standard one-dimensional cutting stock problem," European Journal of Operational Research, Elsevier, vol. 84(3), pages 572-579, August.
    8. Peeters, Marc & Degraeve, Zeger, 2006. "Branch-and-price algorithms for the dual bin packing and maximum cardinality bin packing problem," European Journal of Operational Research, Elsevier, vol. 170(2), pages 416-439, April.
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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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