Fuzzy α-minimum spanning tree problem: definition and solutions

In this paper, the minimum spanning tree problem is investigated on the graph with fuzzy edge weights. The notion of fuzzy α-minimum spanning tree is presented based on the credibility measure, and then the solutions of the fuzzy α-minimum spanning tree problem are discussed under different assumptions. First, we respectively assume that all the edge weights are triangular fuzzy numbers and trapezoidal fuzzy numbers, and prove that the fuzzy α-minimum spanning tree problem can be transformed to a classical problem on a crisp graph in these two cases, which can be solved by classical algorithms such as the Kruskal algorithm and the Prim algorithm in polynomial time. Subsequently, as for the case that the edge weights are general fuzzy numbers, a fuzzy simulation-based genetic algorithm using Prüfer number representation is designed for solving the fuzzy α-minimum spanning tree problem. Some numerical examples are also provided for illustrating the effectiveness of the proposed solutions.