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.

一家进入公司希望通过开设一些在有限的潜在地点（离散空间）中选择的新设施来争夺该地区的市场份额。客户在空间上是分开的，并且该地区已经有其他公司在运营。在本文中，我们使用了众所周知的霍夫（比例）客户选择规则的变体，即所谓的帕累托-霍夫（Pareto-Huff），由于其非线性公式，因此对其文献关注很少。该未经测试的规则认为，客户会根据其吸引力在质量（最大化）和距离（最小化）方面对帕累托最优的设施进行需求分配，即，一个遥远的设施将仅捕获客户的需求如果它比其他更接近的设施具有更高的质量。首先提出了一个非线性规划问题的公式，然后提出了一个线性规划问题的等效公式，这使我们能够获得中等大小问题的精确解。对于大问题，还提出了一种启发式方法以获得最佳近似解决方案。检查其性能是否存在小尺寸问题，并将其解决方案与标准优化程序Xpress给出的最佳解决方案进行比较，并使用西班牙城市的真实地理坐标和人口数据。