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The proper relaxation and the proper gap of the skiving stock problem

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  • John Martinovic

    (Technical University of Dresden)

  • Guntram Scheithauer

    (Technical University of Dresden)

Abstract

We consider the 1D skiving stock problem (SSP) which is strongly related to the dual bin packing problem: find the maximum number of products with minimum length L that can be constructed by connecting a given supply of $$ m \in {\mathbb {N}} $$ m ∈ N smaller item lengths $$ l_1,\ldots ,l_m $$ l 1 , … , l m with availabilities $$ b_1,\ldots , b_m $$ b 1 , … , b m . For this NP-hard optimization problem, we focus on the proper relaxation and introduce a modeling approach based on graph theory. Additionally, we investigate the quality of the proper gap, i.e., the difference between the optimal objective values of the proper relaxation and the SSP itself. As an introductorily motivation, we prove that the SSP does not possess the integer round down property (IRDP) with respect to the proper relaxation. The main contribution of this paper is given by a construction principle for an infinite number of non-equivalent non-proper-IRDP instances and an enumerative approach that leads to the currently largest known (proper) gap.

Suggested Citation

  • John Martinovic & Guntram Scheithauer, 2016. "The proper relaxation and the proper gap of the skiving stock problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(3), pages 527-548, December.
    • Handle: RePEc:spr:mathme:v:84:y:2016:i:3:d:10.1007_s00186-016-0552-2
    DOI: 10.1007/s00186-016-0552-2
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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. 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.
    2. 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.
    3. 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.
    4. Vadim M. Kartak & Artem V. Ripatti, 2019. "Large proper gaps in bin packing and dual bin packing problems," Journal of Global Optimization, Springer, vol. 74(3), pages 467-476, July.

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