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Constructing an Instance of the Cutting Stock Problem of Minimum Size Which Does Not Possess the Integer Round-Up Property

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Abstract

We consider the well-known one-dimensional cutting stock problem in order to find some integer instances with the minimal length L of a stock material for which the round-up property is not satisfied. The difference between the exact solution of an instance of a cutting stock problem and the solution of its linear relaxation is called the integrality gap. Some instance of a cutting problem has the integer round-up property (IRUP) if its integrality gap is less than 1. We present a new method for exhaustive search over the instances with maximal integrality gap when the values of L, the lengths of demanded pieces, and the optimal integer solution are fixed. This method allows us to prove by computing that all instances with L ≤ 15 have the round-up property. Also some instances are given with L = 16 not-possessing this property, which gives an improvement of the best known result L = 18.

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Funding

The authors were supported by the Russian Foundation for Basic Research (project no. 19-07-00895).

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Authors and Affiliations

  1. Ufa State Aviation Technical University, ul. Karla Marksa 12, Ufa, 450008, Russia

    A. V. Ripatti & V. M. Kartak

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  1. A. V. Ripatti
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  2. V. M. Kartak
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Correspondence to A. V. Ripatti or V. M. Kartak.

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Russian Text © The Author(s), 2020, published in Diskretnyi Analiz i Issledovanie Operatsii, 2020, Vol. 27, No. 1, pp. 127–140.

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Ripatti, A.V., Kartak, V.M. Constructing an Instance of the Cutting Stock Problem of Minimum Size Which Does Not Possess the Integer Round-Up Property. J. Appl. Ind. Math. 14, 196–204 (2020). https://doi.org/10.1134/S1990478920010184

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  • DOI: https://doi.org/10.1134/S1990478920010184

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