Using Trapezoidal Intuitionistic Fuzzy Number to Find Optimized Path in a Network

In real life, information available on situations/issues/problems is vague, inexact, or insufficient and so the parameters involved therein are grasped in an uncertain way by the decision maker. But in real life such uncertainty is unavoidable. One possible way out is to consider the knowledge of experts about the parameters involved as fuzzy data. In a network, the arc length may represent time or cost. In Relevant literature reports there are several methods to solve such problems in network-flow. This paper proposes an optimized path for use in networks, using trapezoidal intuitionistic fuzzy numbers, assigned to each arc length in a fuzzy environment. It proposes a new algorithm to find the optimized path and implied distance from source node to destination node. 1. Introduction Fuzzy network problems have appeared in literature for quite some time, of which the simplest and most often researched mode is the fuzzy shortest path problem. The main objective of the optimized path problem is to find a path with minimum distance. The classical fuzzy shortest path problem seems to have been first introduced by Dubois and Prade [1]. They employed the fuzzy minimum operator to find the shortest path length, but they did not develop any method to decide the shortest path. They have used a fuzzy number instead of a real number assigned to each of the edges. Okada and Soper [2] concentrated on an optimized path problem and introduced the concept of degree of possibility in which an arc is on the optimized path. Takahashi and Yamakami [3] discussed the optimized path from a specified node to every other node on a network. Another algorithm for this problem was presented by Okada and Gen [4, 5], where there is a generalization of Dijkstra’s algorithm. In this algorithm the weight of the arcs is considered to be interval numbers and is defined from a partial order between interval numbers. Klein [6] proposed an improved algorithm that can get the shortest path length as well as the shortest path. However, the assumption made in the algorithm, that each arc with the length between “1” and a fixed integer “M” was not reasonable and practical. Szmidt and Kacprzyk [7] found distance between intuitionistic fuzzy sets using Hamming distance, the normalized Hamming distance, the Euclidean distance, and the normalized Euclidean distance. Kung and Chuang [8] pointed out that there are several methods to solve this kind of problem in the available literature. Przemys？aw Grzegorzewski [9] discussed two families of metrics in space of intuitionistic fuzzy numbers. Nagoor