Tree of fuzzy shortest paths with the highest quality

In this paper we present a network with a finite set of nodes and a set of imprecise arc lengths (costs) instead of real numbers. The imprecise lengths (costs) are modeled as fuzzy intervals with increasing membership functions (based on the quality), whereas the total cost of the shortest paths is a fuzzy interval with a decreasing linear membership function. To obtain a tree of fuzzy shortest paths from a source node to all other nodes, an algorithm is developed. By the max-min criterion suggested by Bellman and Zadeh, the fuzzy shortest path (with highest quality) problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can be simplified into a bi-level programming problem that is easily solvable. An efficient algorithm, based on the parametric shortest path, is proposed for solving the bi-level programming problem. An illustrative example is also included to demonstrate our proposed algorithm.