A Study of the Threshold Stability of the Bilevel Problem of Facility Location and Discriminatory Pricing

The problem of threshold stability for a bilevel problem with a median type of facility location and discriminatory pricing is considered. When solving such a problem, it is necessary to find the threshold stability radius and a semifeasible solution of the original bilevel problem such that the leader’s revenue is not less than a predetermined value (threshold) for any deviation of budgets that does not exceed the threshold stability radius and which preserves its semifeasibility. Thus, the threshold stability radius determines the limit of disturbances of consumer budgets with which these conditions are satisfied.

Two approximate algorithms for solving the threshold stability problem based on the heuristic of descent with alternating neighborhoods are developed. These algorithms are based on finding a good approximate location of facilities as well as on calculating the optimal set of prices for the found location of facilities. The algorithms differ in the way they compare various locations of facilities; this ultimately leads to different estimates of threshold stability radius. A numerical experiment has shown the efficiency of the chosen approach both in terms of the running time of the algorithms and the quality of the solutions obtained.