Abstract
We consider the well-known cutting stock problem (CSP). The gap of a CSP instance is the difference between its optimal function value and optimal value of its continuous relaxation. For most instances of CSP the gap is less than 1 and the maximal known gap \(6/5=1.2\) was found by Rietz and Dempe [11]. Their method is based on constructing instances with large gaps from so-called sensitive instances with some additional constraints, which are hard to fulfill. We adapt our method presented in [15] to search for sensitive instances with required properties and construct a CSP instance with gap \(77/64=1.203125\). We also present several instances with large gaps much smaller than previously known.
Supported by RFBR, project 19-07-00895.
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References
Baum, S., Trotter Jr., L.: Integer rounding for polymatroid and branching optimization problems. SIAM J. Algebraic Discrete Methods 2(4), 416–425 (1981)
Delorme, M., Iori, M., Martello, S.: Bin packing and cutting stock problems: mathematical models and exact algorithms. Eur. J. Oper. Res. 255(1), 1–20 (2016)
Dyckhoff, H., Kruse, H.J., Abel, D., Gal, T.: Trim loss and related problems. Omega 13(1), 59–72 (1985)
Fieldhouse, M.: The duality gap in trim problems. SICUP Bull. 5(4), 4–5 (1990)
Gilmore, P., Gomory, R.: A linear programming approach to the cutting-stock problem. Oper. Res. 9(6), 849–859 (1961)
Kantorovich, L.V.: Mathematical methods of organizing and planning production. Manage. Sci. 6(4), 366–422 (1960)
Kartak, V.M., Ripatti, A.V.: Large proper gaps in bin packing and dual bin packing problems. J. Global Optim. 74(3), 467–476 (2018). https://doi.org/10.1007/s10898-018-0696-0
Kartak, V.M., Ripatti, A.V., Scheithauer, G., Kurz, S.: Minimal proper non-IRUP instances of the one-dimensional cutting stock problem. Discrete Appl. Math. 187(Complete), 120–129 (2015)
Marcotte, O.: An instance of the cutting stock problem for which the rounding property does not hold. Oper. Res. Lett. 4(5), 239–243 (1986)
Rietz, J.: Untersuchungen zu MIRUP für Vektorpackprobleme. Ph.D. thesis, Technischen Universität Bergakademie Freiberg (2003)
Rietz, J., Dempe, S.: Large gaps in one-dimensional cutting stock problems. Discrete Appl. Math. 156(10), 1929–1935 (2008)
Rietz, J., Scheithauer, G., Terno, J.: Families of non-IRUP instances of the one-dimensional cutting stock problem. Discrete Appl. Math. 121(1), 229–245 (2002)
Rietz, J., Scheithauer, G., Terno, J.: Tighter bounds for the gap and non-IRUP constructions in the one-dimensional cutting stock problem. Optimization 51(6), 927–963 (2002)
Ripatti, A.V., Kartak, V.M.: Bounds for non-IRUP instances of cutting stock problem with minimal capacity. In: Bykadorov, I., Strusevich, V., Tchemisova, T. (eds.) MOTOR 2019. CCIS, vol. 1090, pp. 79–85. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-33394-2_7
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(1), 196–204 (2020). https://doi.org/10.1134/S1990478920010184
Scheithauer, G., Terno, J.: About the gap between the optimal values of the integer and continuous relaxation one-dimensional cutting stock problem. In: Gaul, W., Bachem, A., Habenicht, W., Runge, W., Stahl, W.W. (eds.) Operations Research Proceedings. Operations Research Proceedings 1991, vol. 1991. Springer, Heidelberg (1992). https://doi.org/10.1007/978-3-642-46773-8_111
Scheithauer, G., Terno, J.: The modified integer round-up property of the one-dimensional cutting stock problem. Eur. J. Oper. Res. 84(3), 562–571 (1995)
Sweeney, P.E., Paternoster, E.R.: Cutting and packing problems: a categorized, application-orientated research bibliography. J. Oper. Res. Soc. 43(7), 691–706 (1992)
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The authors would like to thank the anonymous referees for their valuable remarks.
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Ripatti, A.V., Kartak, V.M. (2020). Sensitive Instances of the Cutting Stock Problem. In: Kochetov, Y., Bykadorov, I., Gruzdeva, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Communications in Computer and Information Science, vol 1275. Springer, Cham. https://doi.org/10.1007/978-3-030-58657-7_9
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