The Inverse Spanning Tree of a Fuzzy Graph Based on Credibility Measure

An inverse spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree. In this paper, based on the notion of fuzzy α-minimum spanning tree, the inverse spanning tree problem with fuzzy edge weights is discussed and formulated as a fuzzy programming model with some chance constraints. It shows that when all the edge weights are assumed to be independent fuzzy variables with regular credibility distributions, the proposed model can be reformulated into a traditional nonlinear programming according to the equivalent condition of fuzzy α-minimum spanning tree characterized by a set of constraints on non-tree edges and their tree paths. Moreover, if all the fuzzy weights are triangular fuzzy numbers, a linear programming problem can be obtained and solved efficiently with the help of some well developed software packages.