www.tjprc.org editor@tjprc.org TYPE REDUCTION ON FUZZY SHORTEST PATH PROBLEM V. ANUSUYA 1 & R. SATHYA 2 1 PG and Research, Department of Mathematics, Seethalakshmi Ramaswami College, Tiruchirappalli, Tamil Nadu, India 2 Department of Mathematics, K. S. Rangasamy College of Arts & Science, Tiruchengode, Tamil Nadu, India ABSTRACT In this paper we have developed an algorithm for finding shortest path in a fuzzy network by edge type reduction method with centroid and centre of gravity of fuzzy set. A proposed algorithm gives the fuzzy shortest path in which type-2 discrete fuzzy number is assigned to each arc length. An illustrative example also included to demonstrate our proposed algorithm. KEYWORDS: Type-2 Fuzzy Set, Centroid of a Type-2 Fuzzy Set, Extension Principle, Centre of Gravity of Fuzzy Set 1. INTRODUCTION The shortest path problem is a classical and important network optimization problem appearing in many real life applications. Dubois and Prade[4] first analyzed this problem and proposed an algorithm to find the shortest path. Klein[10] presented new models based on fuzzy shortest paths and also given a general algorithm based on dynamic programming to solve the new models. Chuang and Kung[2] proposed a fuzzy shortest path length procedure that can find a fuzzy shortest path length among all possible paths in a network. Yao and Lin[16] presented two new types of fuzzy shortest path network problems. Thus, numerous papers have been published on the fuzzy shortest path problem. According to Hisdal[5], “Increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner. The concept of type-2 fuzzy sets was introduced by Zadeh as an extension of the concept of an ordinary fuzzy set. The output of a type-1 fuzzy logic system is a type-1 fuzzy set. This set is usually defuzzified and as is well known, many of the most useful defuzzification methods involve a centroid calculation [12,3]. Recently a type-2 fuzzy logic system [6, 7, 8] has been developed, and its output is a type-2 fuzzy set. A major calculation in a type-2 fuzzy logic system is type-reduction[9], which is an extension[18] of a type-1 defuzzification procedure. In this paper, we focus on the centroid of Gaussian type-2 fuzzy sets and centre of gravity of fuzzy sets for finding fuzzy shortest path. The rest of the paper is structured as follows: Section 2, discusses background theory of type-2 fuzzy set and type reduction method. We developed an algorithm to find the shortest path with type-2 fuzzy number is section 3. Section 4 gives the network terminology. Finally to illustrate the proposed algorithm the numerical example is solved in section 5. 2. CONCEPTS Type-2 Fuzzy Set A fuzzy set A in X is defined as a set of ordered pairs A = {(x,μ A (x)/x ∈ X } where is called the membership function maps each element of X to a membership grade between 0 and 1. International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 4, Issue 6, Dec 2014, 53-60 © TJPRC Pvt. Ltd.