Profit maximising single competitive facility location in the plane

A single facility has to be located in the plane in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, which fully patronise the facility to which it is most attracted. Attraction by a facility is expressed by some general attractiveness of the facility divided by a power of its euclidean distance to demand. For existing facilities attractiveness is fixed, while the costs connected with the new facility are an increasing function of its attractiveness. Each demand point attracted by the new facility generates a given amount of income. The aim is to find that location for the new facility which maximises the resulting profits. It is shown that this problem is well posed under the additional assumption that consumers are novelty oriented, i.e. attraction ties are resolved in favor of the new facility. The problem then reduces to a parametric maxcovering problem with inflated euclidean distances, which is solvable in polynomial time.