Abstract
An extension of the concept of the guillotine layout function has been proposed for solving the problem of rectangular orthogonal packing; this extension is a function that assigns a triple of values to the sheet width. In addition to the standard effect for the guillotine layout function (the sheet with a given width has a minimum length), which is sufficient to arrange a given set of rectangles in a guillotine manner, two additional values have been used. They describe the method of cutting this sheet to uniquely form a guillotine cutting card and a guillotine layout card of the set of rectangles. These data involve the characteristics of the first cut of the sheet as well as the partition of the set of rectangles corresponding to the cut into two subsets, which is uniquely determined by the number of one of these subsets. The description of the first cut is modeled by a single numerical value that reflects both the size of the offset from the lower-left corner of the sheet and the orientation of the cut: a cut is required along or transverse of the sheet. It has been shown that this information is sufficient for the recovery of the guillotine cutting card and the guillotine layout card for a set of rectangles. Modifications of the algorithms for calculating the sum of two right-semicontinuous monotonically nonincreasing step functions with a finite number of steps and the minimum of two functions of this type have been proposed to determine additional information about the first cut and calculate the extension of the guillotine layout function. Also, an algorithm for the formation of a guillotine cutting card and a guillotine layout card for rectangles has been proposed that uses the calculated extensions of guillotine layout functions for all subsets of the required set of rectangles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. V. Kantorovich and V. A. Zalgaller, Computation of Rational Cutting of IndustrialMaterials (Lenizdat, Leningrad, 1951) [in Russian].
L. V. Kantorovich and V. A. Zalgaller, Rational Cutting of IndustrialMaterials (Nauka, Novosibirsk, 1971) [in Russian].
E. A. Mukhacheva, Rational Cutting of Industrial Materials. Application in ICS (Nauka, Novosibirsk, 1987) [in Russian].
A. Lodi, S. Martello, and M. Monaci, “Two-dimensional packing problems: A survey,” Eur. J. Oper. Res. 141, 241–252 (2002). doi: 10.1016/S0377-2217(02)00123-6
D. F. Zhang, Y. Kang, and S. Deng, “A new heuristic recursive algorithm for the strip rectangular packing problem,” Comput. Operat. Res. 33, 2209–2217 (2006). doi: 10.1016/j.cor.2005.01.009
M. Chen and W. Huang, “A two-level search algorithm for 2D rectangular packing problem,” Comput. Ind. Eng. 53, 123–136 (2007). doi: 10.1016/j.cie.2007.04.007
E. A. Mukhacheva, A. F. Valeeva, V. M. Kartak, and A. S. Mukhacheva, “Models and methods for solving orthogonal nesting and packaging problems: analytical overview and new technology of block structures,” Inform. Tekhnol., Suppl. 5, 31 (2004).
A. F. Valeeva, “Applying of the constructive heuristic method for cutting-packing problem,” Vestn. Bashkir. Univ., No. 3, 5–6 (2006).
I. V. Romanovskii, “Solution of the guillotine cutting problem by the method of processing a list of states,” Kibernetika, No. 1, 102–103 (1969).
A. B. Gribov, “Algorithm for solving the problem of a plane cutting layout,” Kibernetika, No. 6, 110–115 (1973).
I. V. Romanovskii, Algorithm for Extremal Problems Solving (Nauka, Moscow, 1977) [in Russian].
E. A. Mukhacheva and G. Sh. Rubinshtein, Mathematical Programming (Nauka, Novosibirsk, 1987) [in Russian].
E. Yu. Lerner and V. R. Fazylov, “Guillotine layout function,” Issled. Prikl.Mat. 21, 187–196 (1999).
A. A. Andrianova, T. M. Mukhtarova, and V. R. Fazylov, “Models of the nonguillotine sheet and strip rectangular packing problem,” Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 155 (2), 5–18 (2013).
A. A. Andrianova, T. M. Mukhtarova, and V. R. Fazylov, “Model of the nonguillotine strip rectangular set packing problem,” in Proceedings of the 17th International Conference on Problems of Theoretical Kibernetics (Otechestvo, Kazan’, 2014), pp. 23–26.
A. A. Andrianova, T. M. Mukhtarova, and V. R. Fazylov, “Model of the sheet compact rectangular set packing,” in Proceedings of the 16th Baikal International School-Seminar on Optimization Methods and their Applications (ISEM SO RAN, Irkutsk, 2014), p.33.
E. Yu. Lerner and V. R. Fazylov, “Quasi-inverse functions and their properties,” Issled. Prikl.Mat. 22, 63–74 (1997).
E. Humby, Programs from Decision Tables (Macdonald, Elsevier, London, New York, 1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.A. Andrianova, T.M. Mukhtarova, V.R. Fazylov, 2017, published in Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki, 2017, Vol. 159, No. 2, pp. 161–173.
Rights and permissions
About this article
Cite this article
Andrianova, A.A., Mukhtarova, T.M. & Fazylov, V.R. Formation of the Guillotine Cutting Card of a Sheet by Guillotine Layout Functions. Lobachevskii J Math 39, 439–447 (2018). https://doi.org/10.1134/S1995080218030034
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080218030034