Abstract
We consider a mathematical model of decision making by a company attempting to win a market share. We assume that the company releases its products to the market under the competitive conditions that another company is making similar products. Both companies can vary the kinds of their products on the market as well as the prices in accordance with consumer preferences. Each company aims to maximize its profit. A mathematical statement of the decision-making problem for the market players is a bilevel mathematical programming problem that reduces to a competitive facility location problem. As regards the latter, we propose a method for finding an upper bound for the optimal value of the objective function and an algorithm for constructing an approximate solution. The algorithm amounts to local ascent search in a neighborhood of a particular form, which starts with an initial approximate solution obtained simultaneously with an upper bound. We give a computational example of the problem under study which demonstrates the output of the algorithm.
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References
H. von Stackelberg, The Theory of the Market Economy (Springer, Berlin, 1934; Univ. Press, Oxford, 1952).
L. A. Stole, “Price Discrimination and Competition,” in Handbook of Industrial Organization, Vol. 3, Ed. by M. Armstrong and R. H. Porter (Amsterdam, Elsevier, 2007), pp. 2221–2299.
C. M. Campos Rodriguez and J. A. Moreno Perez, “Multiple Voting Location Problems,” European J. Oper. Res. 191(2), 436–452 (2008).
H. A. Eiselt and G. Laporte, “Sequential Location Problems,” European J. Oper. Res. 96, 217–231 (1996).
G. Dobson and U. Karmarkar, “Competitive Location on Network,” Oper. Res. 35, 565–574 (1987).
Facility Location: Applications and Theory, Ed. by Z. Drezner and H. W. Hamacher (Springer, Berlin, 2002).
Discrete Location Theory, Ed. by P. B. Mirchandani and R. L. Francis (John Wiley and Sons, New York, 1990).
F. Plastria, “Static Competitive Facility Location: An Overvew of Optimization Approaches,” European. J. Oper. Res. 129, 461–470 (2001).
F. Plastria and L. Vanhaverbeke, “DiscreteModels for Competitive Location with Foresight,” Comput. Oper. Res. 35, 683–700 (2008).
D. R. Santos Penate, R. Suarez Vega, and P. Dorta Gonzalez, “The Leader-Follower Location Model,” Networks and Spatial Economics 7(1), 45–61 (2007).
V. L. Beresnev, “Upper Bounds for Objective Functions of Discrete Competitive Facility Location Problems,” Diskret. Anal. Issled. Oper. 15(4), 3–24 (2008) [J. Appl. Indust. Math. 3 (4), 419–432 (2009)].
S. Dempem, Foundations of Bilevel Programming (Kluwer Acad. Publ., Dordrecht, 2002).
V. L. Beresnev, Discrete Facility Location Problems and Polynomials of Boolean Variables (Inst.Math., Novosibirsk, 2005) [in Russian].
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Original Russian Text © V.L. Beresnev, V.I. Suslov, 2009, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2009, Vol. XII, No. 1, pp. 11–24.
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Beresnev, V.L., Suslov, V.I. A mathematical model of market competition. J. Appl. Ind. Math. 4, 147–157 (2010). https://doi.org/10.1134/S199047891002002X
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DOI: https://doi.org/10.1134/S199047891002002X