An extended multifacility minimax location problem revisited

We consider the following model of Drezner for the location of several facilities. The weighted sum of distances to all facilities plus a set-up cost is calculated for each separate demand point. The maximal value among these sums is to be minimized. In this note we show that if the weights used in the model decompose into a product of two factors, one depending only on the demand point, the other only on the new facilities, there exists at least one optimal solution such that all new facilities coincide. We also investigate when a unique optimal solution of coincidence type exists, and obtain a full description of the set of optimal solutions when the weights have this multiplicative structure and the norm is round. An example shows that this kind of coincidence does not necessarily happen when the weights may have any value.