In this paper, we first consider the classical p -median location problem on a network in which the vertex weights and the distances between vertices are uncertain variables. The uncertainty distribution of the optimal objective value of the p -median problem is given and the concepts of the α-p -median, the most p -median and the expected p -median are introduced. Then, it is shown that the uncertain p -median problem is NP-hard on general networks. However, if the underlying network is a tree, an efficient algorithm for the uncertain 1-median problem with linear time complexity is proposed. Finally, we investigate the inverse 1-median problem on a tree with uncertain vertex weights and present a programming model for the problem. Then, it is shown that the proposed model can be reformulated into a deterministic programming model.

中文翻译：

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不确定环境下的经典和逆中值位置问题

在本文中，我们首先考虑网络上的经典 p 中值位置问题，其中顶点权重和顶点之间的距离是不确定变量。给出了p-中值问题的最优目标值的不确定性分布，并引入了α-p-中值、最大p-中值和期望p-中值的概念。然后，证明了不确定 p 中值问题在一般网络上是 NP-hard 问题。然而，如果底层网络是一棵树，则提出了一种具有线性时间复杂度的不确定1-中值问题的有效算法。最后，我们研究了具有不确定顶点权重的树上的逆 1 中值问题，并提出了该问题的编程模型。然后，表明所提出的模型可以重新表述为确定性编程模型。