A model for the inverse 1-median problem on trees under uncertain costs

Abstract

Abstract.We consider the problem of justifying vertex weights of a tree under uncertain costs so that a prespecified vertex become optimal and the total cost should be optimal in the uncertainty scenario. We propose a model which delivers the information about the optimal cost which respect to each confidence level α ∈ [0, 1]. To obtain this goal, we first define an uncertain variable with respect to the minimum cost in each confidence level. If all costs are independently linear distributed, we present the inverse distribution function of this uncertain variable in O(n 2 log n) time, where n is the number of vertices in the tree.

A HYBRID ALGORITHM FOR THE UNCERTAIN INVERSE p-MEDIAN LOCATION PROBLEM

“…Note that the goal is to change w(v) and ℓ e with respect to modification bounds so that {v 2 , v 3 } becomes a 2-median at minimum total cost under the new vertex weights and edge lengths. (14,34,25,7,22,10,8,20,12,7,10,26,12,6,10,23,31,21,22) u + e (5, 4, 5, 4, 7, 2, 5, 6, 3, 9, 2, 13, 1, 3, 5, 1, 7, 4, 1) u − e (10,30,15,3,17,8,4,10,8,4,5,20,6,1,8,13,2,3,13) w(v) (34,18,14,13,21,11,13,20,40,22,9,17,…”

“…Some researchers applied the uncertainty theory to deal with the location problems,for example Gao [14] modeled the single facility location problems with uncertain demands. Wen et al [43] investigated the capacitated facility location-allocation problem with uncertain demands and also Nguyen and Chi [31] studied inverse 1-median problem on a tree with uncertain costs and showed that the inverse distribution function of the minimum cost can be obtained at O(n 2 log n) time. For a survey on uncertain location problems, we refer the interested reader to [15,22,27,34,40,46].…”

A SOLUTION ALGORITHM FOR p-MEDIAN LOCATION PROBLEM ON UNCERTAIN RANDOM NETWORKS

“…When we do not have enough samples to estimate the probability distributions of the vertex weights and the distances between vertices, we have to invite experts to give the belief degrees about the vertex weights and distances between vertices. Some researchers applied the uncertainty theory to deal with the location problems: for example the uncertain models for single facility location problems were investigated by Yuan Gao [12], the hierarchical facility location for the reverse logistics network design under uncertainty was studied by Wang and Yang [38], the capacitated facility location-allocation problem under uncertain environment was investigated by Wen et al [39], the inverse 1-median problem on a tree under uncertain cost coefficients was solved by Nguyen and Chi [29] and the classical p-center location problem on a network with the uncertain vertex weights and the uncertain distances was studied by Soltanpour et al [35].…”