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Diskretn. Anal. Issled. Oper., 2020, Volume 27, Issue 1, Pages 127–140 (Mi da947)  

Constructing an instance of the cutting stock problem of minimum size which does not possess the integer round-up property

A. V. Ripatti, V. M. Kartak

Ufa State Aviation Technical University, 12 Karl Marx Street, 450008 Ufa, Russia

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 $L \le 15$ 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 $L=18$. Tab. 2, bibliogr. 14.

Keywords: cutting stock problem, integer round-up property, integrality gap.

Funding Agency Grant Number
Russian Foundation for Basic Research 19-07-00895
This research is supported by Russian Foundation for Basic Research (Project 19–07–00895).


DOI: https://doi.org/10.33048/daio.2020.27.665

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English version:
Journal of Applied and Industrial Mathematics, 2020, 14:1, 196–204

UDC: 519.85
Received: 27.06.2019
Revised: 18.09.2019
Accepted:27.11.2019

Citation: A. V. Ripatti, V. M. Kartak, “Constructing an instance of the cutting stock problem of minimum size which does not possess the integer round-up property”, Diskretn. Anal. Issled. Oper., 27:1 (2020), 127–140; J. Appl. Industr. Math., 14:1 (2020), 196–204

Citation in format AMSBIB
\Bibitem{RipKar20}
\by A.~V.~Ripatti, V.~M.~Kartak
\paper Constructing an instance of the cutting stock problem of~minimum size which does not possess the integer round-up property
\jour Diskretn. Anal. Issled. Oper.
\yr 2020
\vol 27
\issue 1
\pages 127--140
\mathnet{http://mi.mathnet.ru/da947}
\crossref{https://doi.org/10.33048/daio.2020.27.665}
\transl
\jour J. Appl. Industr. Math.
\yr 2020
\vol 14
\issue 1
\pages 196--204
\crossref{https://doi.org/10.1134/S1990478920010184}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85082386230}


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