Exact and heuristic solutions of a discrete competitive location model with Pareto-Huff customer choice rule

An entering firm wants to compete for market share of an area by opening some new facilities selected among a finite set of potential locations (discrete space). Customers are spatially separated and there already are other firms operating in that area. In this paper, we use a variant of the well known Huff (proportional) customer choice rule, the so called Pareto-Huff, which have had little attention on the literature because of its nonlinear formulation. This untested rule considers that customers split their demand among the facilities that are Pareto optimal with respect to quality (to be maximized) and distance (to be minimized), proportionally to their attractions, i.e., a distant facility will capture demand of a customer only if it has higher quality than any other closer facility. A first formulation as a nonlinear programming problem is proposed, and then an equivalent formulation as a linear programming problem is presented, which allows us to obtain exact solutions for medium size problems. For large size problems, a heuristic procedure is also proposed to obtain the best approximate solutions. Its performance is checked for small size problems and its solutions are compared with the optimal solutions given by a standard optimizer, Xpress, using real geographical coordinates and population data of municipalities in Spain.