Spatial point process models for location-allocation problems

The problem of finding an optimal location frequently occurs in geomarketing, economics and other fields: positioning a new branch of a bank, a supermarket, a fire station, a plant, designing a traffic network, etc. The optimal location of the source facility is the argument-minimum of an optimization problem parametrized by some characteristics of the clients. The random nature of some of these characteristics has already been recognized, but few stochastic models for location-allocation problems address the issue of uncertainty of the locations of the clients, and even then they do it with very naive tools. It is proposed to recognize uncertainty in the spatial positions of the clients, and possible spatial autocorrelation as well, by considering the random inputs of the optimization as one realization of a spatial marked point process. The method, called SPP location-allocation, involves fitting a point process model, simulating from the adjusted process, and solving a family of optimization problems for each simulated set of observations. The advantage of this approach over the deterministic one is twofold: it gives an indication of the spatial variability of the optimal solution, and it allows one to solve larger problems. Finally an application to the optimal positioning of a new fire station in the Toulouse area (France) is presented with some heuristic algorithms.