  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.
