Discretization results for the Huff and Pareto–Huff competitive location models on networks

In this paper we prove that there always exists a finite set that includes an optimal solution for the Huff and the Pareto–Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto–Huff model, we prove that there is always an optimal solution at, or an ε–optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.