Abstract
We consider a mathematical model similar in a sense to competitive location problems. There are two competing parties that sequentially open their facilities aiming to “capture” customers and maximize profit. In our model, we assume that facilities’ capacities are bounded. The model is formulated as a bilevel integer mathematical program, and we study the problem of obtaining its optimal (cooperative) solution. It is shown that the problem can be reformulated as that of maximization of a pseudo-Boolean function with the number of arguments equal to the number of places available for facility opening. We propose an algorithm for calculating an upper bound for values that the function takes on subsets which are specified by partial (0, 1)-vectors.
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References
V. L. Beresnev, Discrete Location Problems and Polynomials of Boolean Variables (Inst. Mat., Novosibirsk, 2005) [in Russian].
V. L. Beresnev, “Local Search Algorithms for the Problem of Competitive Location of Enterprises,” Avtomat. i Telemekh. No. 3, 12–27 (2012) [Automat. Remote Control 73 (3), 425–439 2012].
V. L. Beresnev, “On the Competitive Facility Location Problem with a Free Choice of Suppliers,” Avtomat. i Telemekh. No. 4, 94–105 (2014) [Automat. Remote Control 75 (4), 668–676 2014].
V. L. Beresnev, E. N. Goncharov, and A. A. Mel’nikov, “Local Search with a Generalized Neighborhood in the Optimization Problem for Pseudo-Boolean Functions,” Diskret. Anal. Issled. Oper. 18 (4), 3–16 (2011) [J. Appl. Indust. Math. 6 (1), 22–30 2012].
V. L.Beresnev and A. A. Mel’nikov, “ApproximateAlgorithms for the Competitive Facility Location Problem,” Diskret. Anal. Issled. Oper. 17 (6), 3–19 (2010) [J. Appl. Indust. Math. 5 (2), 180–190 2011].
V. L. Beresnev and A. A. Mel’nikov, “The Branch-and-Bound Algorithm for a Competitive Facility Location Problem with the Prescribed Choice of Suppliers,” Diskret. Anal. Issled. Oper. 21 (2), 3–23 (2014) [J. Appl. Indust. Math. 8 (2), 177–189 2014].
A. V. Kononov, Yu. A. Kochetov, and A. V. Plyasunov, “Competitive Facility Location Models,” Zh. Vychisl. Mat. Mat. Fiz. 49 (6), 1037–1054 (2009) [Comput. Math. Math. Phys. 49 (6), 994–1009 2009].
A. A. Mel’nikov, “Randomized Local Search for the Discrete Competitive Facility Location Problem,” Avtomat. i Telemekh. No. 4, 134–152 (2014) [Automat. Remote Control 75 (4), 700–714 2014].
A.V. Plyasunov and A. A. Panin, “The Pricing Problem. Part I: Exact and Approximate Algorithms,” Diskret. Anal. Issled. Oper. 19 (5), 83–100 (2012) [J. Appl. Indust. Math. 7 (2), 241-251 (2013)].
A. V. Plyasunov and A. A. Panin, “The Pricing Problem. Part II: Computational Complexity,” Diskret. Anal. Issled. Oper. 19 (6), 56–71 (2012) [J. Appl. Indust. Math. 7 (3), 420-430 (2013)].
V. L. Beresnev, “Branch-and-Bound Algorithm for a Competitive Facility Location Problem,” Comput. Oper. Res. 40 (8), 2062–2070 (2013).
S. Dempe, Foundations of Bilevel Programming (Kluwer Acad. Publ., Dordrecht, 2002).
P. L. Hammer and S. Rudeanu, “Pseudo-Boolean Programming,” Oper. Res. 17 (2), 233–261 (1969).
J. Krarup and P.M. Pruzan, “The Simple Plant Location Problem: Survey and Synthesis,” European J.Oper. Res. 12 (1), 36–81 (1983).
H. von Stackelberg, The Theory of the Market Economy (Hedge, London, 1952).
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Original Russian Text © V.L. Beresnev, A.A. Melnikov, 2016, published in Diskretnyi Analiz i Issledovanie Operatsii, 2016, Vol. 23, No. 1, pp. 35–48.
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Beresnev, V.L., Melnikov, A.A. A capacitated competitive facility location problem. J. Appl. Ind. Math. 10, 61–68 (2016). https://doi.org/10.1134/S1990478916010075
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DOI: https://doi.org/10.1134/S1990478916010075