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 facilitylocation and discriminatory pricing is considered. When solving such a problem, it is necessary tofind the threshold stability radius and a semifeasible solution of the original bilevel problem suchthat the leader’s revenue is not less than a predetermined value (threshold) for any deviation ofbudgets 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 withwhich these conditions are satisfied.

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