We deal with the location-quantity problem for competing firms when they locate multiple facilities and offer the same type of product. Competition is performed under delivered quantities that are sent from the facilities to the customers. This problem is reduced to a location game when the competing firms deliver the Cournot equilibrium quantities. While existence conditions for a Nash equilibrium of the location game have been discussed in many contributions in the literature, computing an equilibrium on a network when multiple facilities are to be located by each firm is a problem not previously addressed. We propose an integer linear programming formulation to fill this gap. The formulation solves the profit maximization problem for a firm, assuming that the other firms have fixed their facility locations. This allows us to compute location Nash equilibria by the best response procedure. A study with data of Spanish municipalities under different scenarios is presented and conclusions are drawn from a sensitivity analysis.

中文翻译：

---

数量竞争网络上多设施场所纳什均衡的计算

当竞争性公司找到多个设施并提供相同类型的产品时，我们将为其解决位置数量问题。竞争是根据从工厂发送给客户的交货数量进行的。当竞争公司交付古诺均衡量时，这个问题被简化为位置博弈。尽管在文献中的许多文章中都讨论了位置博弈的纳什均衡的存在条件，但是当每个公司要定位多个设施时，在网络上计算均衡是先前未解决的问题。我们提出一种整数线性规划公式来填补这一空白。该公式解决了一家公司的利润最大化问题，并假设其他公司已固定其设施位置。这使我们能够通过最佳响应程序来计算位置纳什均衡。提出了在不同情况下使用西班牙市政数据进行的研究，并从敏感性分析中得出了结论。