The leader–follower facility location problem consists of a leader and a follower who are competitors that locate new facilities sequentially. Traditional studies have generally assumed that the leader has partial or full advance information of the follower’s response when making a decision. However, this assumption might be invalid or impracticable in practice. In this paper, we consider that the leader needs to locate a predetermined number of new facilities without knowing anything about the follower’s response. By separating the scenarios in which the follower responds with different numbers of new facilities, a minimax regret model is proposed for the leader to minimise its maximum possible loss. Based on the structural characteristics of the proposed model, a set of solving procedures is provided that transforms the follower’s nonlinear (fraction) programming model into a linear model. In the numerical experiments, the proposed model is compared with two other location models, a deterministic model and a risk model, and the efficiency of the linearisation in decreasing the computation time is verified. The results show that the proposed model is more applicable to the leader when there is no information about the number or probability distribution of the follower’s new facilities.

中文翻译：

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领导者-跟随者设施位置问题的极小极大后悔模型

领导者 - 追随者设施定位问题由作为竞争者的领导者和追随者组成，他们是依次定位新设施的竞争者。传统研究通常假设领导者在做出决定时拥有追随者反应的部分或全部预先信息。然而，这种假设在实践中可能无效或不切实际。在本文中，我们认为领导者需要在不知道跟随者响应的情况下定位预定数量的新设施。通过分离跟随者响应不同数量的新设施的场景，为领导者提出了一个极小极大后悔模型，以最小化其最大可能的损失。基于所提出模型的结构特点，提供了一组求解程序，将跟随者的非线性（分数）规划模型转换为线性模型。在数值实验中，将所提出的模型与其他两个位置模型、确定性模型和风险模型进行了比较，并验证了线性化在减少计算时间方面的效率。结果表明，当没有追随者新设施的数量或概率分布信息时，所提出的模型更适用于领导者。