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Classical and inverse median location problems under uncertain environment. (English) Zbl 1436.90074
Summary: 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 \(\alpha\)-\(p\)-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.
90B80 Discrete location and assignment
90C27 Combinatorial optimization
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