This study considers the \( (r{\mid }p) \) hub–centroid problem under the price war, which was recently proposed in the literature. The objective is profit maximization by choosing the best hub and spoke topology, with the corresponding price structure, in a leader–follower setting. Because this bi–level optimization problem is NP–hard, the use of metaheuristics is a natural choice for solving real–size instances. A variable neighborhood search algorithm is designed as a solution approach for the leader. The characterization of optimal routes under the price equilibrium is given in order to simplify and improve the algorithm. When it comes to the follower, we have shown how to reformulate in a linear fashion the initial non–linear model. The computational experiments are conducted on the CAB instances. The results of these experiments are thoroughly discussed, highlighting the effects of different parameters and providing some interesting managerial insights.

本研究考虑\（（r {\ mid} p）\）最近在文献中提出的价格战下的中心-质心问题。目的是通过在领导者-从属者设置中选择最佳的轮毂和轮辐拓扑结构以及相应的价格结构来实现利润最大化。由于此双层优化问题是NP难题，因此使用元启发式方法是解决实际大小实例的自然选择。可变邻域搜索算法被设计为领导者的一种解决方法。为了简化和改进算法，给出了价格均衡下最优路径的刻画。对于跟随者，我们已经展示了如何以线性方式重新构造初始非线性模型。计算实验是在CAB实例上进行的。这些实验的结果经过了详尽的讨论，