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English version:
Journal of Applied and Industrial Mathematics, 2017, 11:4, 472-480

Volume 24, No 4, 2017, P. 5-21

UDC 519.85
V. L. Beresnev and A. A. Melnikov
An upper bound for the competitive location and capacity choice problem with multiple demand scenarios

Abstract:
A new mathematical model is considered related to competitive location problems where two competing parties, the Leader and the Follower, successively open their facilities and try to win customers. In the model, we consider a situation of several alternative demand scenarios which differ by the composition of customers and their preferences. We assume that the costs of opening a facility depend on its capacity; therefore, the Leader, making decisions on the placement of facilities, must determine their capacities taking into account all possible demand scenarios and the response of the Follower. For the bilevel model suggested, a problem of finding an optimistic optimal solution is formulated. We show that this problem can be represented as a problem of maximizing a pseudo-Boolean function with the number of variables equal to the number of possible locations of the Leader’s facilities. We propose a novel system of estimating the subsets that allows us to supplement the estimating problems, used to calculate the upper bounds for the constructed pseudo-Boolean function, with additional constraints which improve the upper bounds.
Bibliogr. 13.

Keywords: competitive facility location, bilevel programming, upper bound.

DOI: 10.17377/daio.2017.24.578

Andrey A. Melnikov 1,2

1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
2. Novosibirsk State University,
2 Pirogov St., 630090 Novosibirsk, Russia
e-mail: beresnev@math.nsc.ru, melnikov@math.nsc.ru

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