A fuzzy shortest path with the highest reliability

  • Authors:

  • Esmaile Keshavarz;Esmaile Khorram

  • Affiliations:

  • Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, 424, Hafez Ave, 15914, Tehran, Iran;Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, 424, Hafez Ave, 15914, Tehran, Iran

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2009

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Abstract

This paper concentrates on a shortest path problem on a network where arc lengths (costs) are not deterministic numbers, but imprecise ones. Here, costs of the shortest path problem are fuzzy intervals with increasing membership functions, whereas the membership function of the total cost of the shortest path is a fuzzy interval with a decreasing linear membership function. By the max-min criterion suggested in [R.E. Bellman, L.A. Zade, Decision-making in a fuzzy environment, Management Science 17B (1970) 141-164], the fuzzy shortest path 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 very solvable. Here, we propose an efficient algorithm, based on the parametric shortest path problem for solving the bi-level programming problem. An illustrative example is given to demonstrate our proposed algorithm.