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An Improved Upper Bound for the Gap of Skiving Stock Instances of the Divisible Case

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Operations Research Proceedings 2017

Part of the book series: Operations Research Proceedings ((ORP))

Abstract

The 1D skiving stock problem (SSP) is a combinatorial optimization problem being of high relevance whenever an efficient utilization of given resources is intended. In the classical formulation, (small) items shall be used to build as many large objects (specified by some required length) as possible. Due to the NP-hardness of the SSP, the performance of the LP relaxation (measured by the additive gap of the related optimal values) and/or heuristics are of scientific interest. In this paper, theoretical properties of the best fit decreasing heuristic (for the SSP) are investigated and shown to provide a new and improved upper bound for the gap of the so-called divisible case.

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Notes

  1. 1.

    An instance \( E=(m,l,L,b) \) of the SSP belongs to the divisible case (for short: \( E \in \mathscr {DC} \)) if \( l_i \mid L \) holds for all \( i \in I \). Then, E can be described by \( E=(m,l,1,b) \) with \(l_i \in \lbrace 1/2,1/3,\ldots \rbrace \) for all \( i \in I \).

  2. 2.

    Each upper bound of the gap that holds for such instances can easily be transferred to unrestricted instances of the divisible case, too (see [6, Theorem 9]).

  3. 3.

    Due to the absence of exact patterns, \( x_1=0 \) has to hold in any feasible solution of (4). This means that, actually, it is not necessary to consider the length \( \kappa _1=1 \) in the corresponding maximization problem. Nevertheless, we will work with the given definition of the vector \( \kappa \) for the sake of an easier presentation, since then \( \kappa _d=1/d \) is true for all \( d \in \lbrace 1,\ldots ,q \rbrace \).

References

  1. Assmann, S. F., Johnson, D. S., Kleitman, D. J., & Leung, J. Y.-T. (1984). On a dual version of the one-dimensional bin packing problem. Journal of Algorithms, 5, 502–525.

    Article  Google Scholar 

  2. Chen, Y., Song, X., Ouelhadj, D., & Cui, Y. (2017). A heuristic for the skiving and cutting stock problem in paper and plastic film industries. To appear in: International Transactions in Operational Research, https://doi.org/10.1111/itor.12390.

  3. Johnson, M. P., Rennick, C., & Zak, E. J. (1997). Skiving addition to the cutting stock problem in the paper industry. SIAM Review, 39(3), 472–483.

    Article  Google Scholar 

  4. Martinovic, J., Jorswieck, E., Scheithauer, G., & Fischer, A. (2017). Integer linear programming formulations for cognitive radio resource allocation. IEEE Wireless Communication Letters, 6(4), 494–497.

    Article  Google Scholar 

  5. Martinovic, J., & Scheithauer, G. (2016). Integer linear programming models for the skiving stock problem. European Journal of Operational Research, 251(2), 356–368.

    Article  Google Scholar 

  6. Martinovic, J., & Scheithauer, G. (2016). Integer rounding and modified integer rounding for the skiving stock problem. Discrete Optimization, 21, 118–130.

    Article  Google Scholar 

  7. Martinovic, J., & Scheithauer, G. (2016). New Theoretical Investigations on the Gap of the Skiving Stock Problem. Preprint MATH-NM-03-2016, Technische Universität Dresden

    Google Scholar 

  8. Martinovic, J., & Scheithauer, G. (2017). An upper bound of \(\Delta (E) < 3/2\) for skiving stock instances of the divisible case. Discrete Applied Mathematics, 229, 161–167.

    Article  Google Scholar 

  9. Zak, E. J. (2003). The skiving stock problem as a counterpart of the cutting stock problem. International Transactions in Operational Research, 10, 637–650.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Numerische Mathematik, Technische Universität Dresden, 01062, Dresden, Germany

    John Martinovic & Guntram Scheithauer

Authors
  1. John Martinovic
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  2. Guntram Scheithauer
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Corresponding author

Correspondence to John Martinovic .

Editor information

Editors and Affiliations

  1. Department of Information Systems, Freie Universität Berlin, Berlin, Germany

    Natalia Kliewer

  2. Management Science Group, Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany

    Jan Fabian Ehmke

  3. Department of Mathematics, Freie Universität Berlin, Berlin, Germany

    Ralf Borndörfer

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Martinovic, J., Scheithauer, G. (2018). An Improved Upper Bound for the Gap of Skiving Stock Instances of the Divisible Case. In: Kliewer, N., Ehmke, J., Borndörfer, R. (eds) Operations Research Proceedings 2017. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-89920-6_25

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