Finding the Most Vital Arc in the Shortest Path Problem with Fuzzy Arc Lengths

The shortest path problem is to find the shortest distance between two specified nodes in a network. An arc is called a single most vital arc in the network, if its removal from the network results in the greatest increase in the shortest distance. The most vital arcs problems provide a means by which the importance of arc’s availability can be measured. In the traditional most vital arcs problems, the arc lengths are assumed to be crisp numbers. In this paper, we consider the case that the arc lengths are fuzzy numbers. We first show that the membership function of the shortest distance can be found by using a fuzzy linear programming approach. Based on this result, we give a theorem which may be used to reduce the effort required for finding the membership function of the shortest distance, when an arc is removed. Moreover, we may also reduce the number of candidates for the single most vital arc by using the theorem.

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