Upper Bound for the Competitive Facility Location Problem with Demand Uncertainty

We consider a competitive facility location problem with two competing parties operating in a situation of uncertain demand scenarios. The problem of finding the best solutions for the parties is formulated as a discrete bilevel mathematical programming problem. A procedure for computing an upper bound for the objective function on solution subsets is suggested. The procedure could be employed in implicit enumeration schemes capable of computing an optimal solution for the problem under study. Within the procedure, additional constraints (cuts) iteratively augment the high-point relaxation of the initial bilevel problem, which strengthens the relaxation and improves the upper bound's quality. A new procedure for generating such cuts is proposed, which allows us to construct the strongest cuts without enumerating the parameters encoding them.