Mass transportation and the consistency of the empirical optimal conditional locations

We consider the problem of finding the optimal locations of new facilities given the locations of existing facilities and clients. We analyze the general situation where the locations of existing facilities are deterministic while the locations of clients are stochastic with the same unknown marginal distribution. We show how this conditional location-allocation problem can be modeled as a variation of the standard Monge-Kantorovich mass transference problem. We provide a probabilistic formulation of the optimal locations of the new facilities and derive consistent estimators of these theoretical locations from a sample of identically distributed random clients. The integrity of our method is illustrated through some simulation experiments and a real case study.