In this paper, we introduce a new variant of the $p$-median facility location problem in which it is assumed that the exact location of the potential facilities is unknown. Instead, each of the facilities must be located in a region around their initially assigned location (the neighborhood). In this problem, two main decisions have to be made simultaneously: the determination of the potential facilities that must be open to serve the demands of the customers and the location of the open facilities in their neighborhoods, at global minimum cost. We present several mixed integer non-linear programming formulations for a wide family of objective functions which are common in Location Analysis: ordered median functions. We also develop two math-heuristic approaches for solving the problem. We report the results of extensive computational experiments.

29 pages; 6 tables; 7 figures