An Intuitionistic Fuzzy Approach to the Travelling Salesman Problem

Abstract

The travelling salesman problem (TSP) is a classical problem in the combinatorial optimization. Its objective is to find the cheapest route of a salesman starting from a given city, visiting all other cities only once and finally come to the same city where he started. There are different approaches for solving travelling salesman problems with clear data. In real life in one situation there may be not possible to get the delivery costs as a certain quantity. To overcome this Zadeh introduce fuzzy set concepts to deal with an imprecision. There exist algorithms for solution of this problem based on fuzzy or triangular intuitionistic fuzzy numbers (private case of intuitionistic fuzzy sets (IFSs)). But many times the degrees of membership and non-membership for certain element are not defined in exact numbers. Atanassov and Gargov in 1989 first identified it in the concept of interval-valued intuitionist fuzzy sets (IVIFS) which is characterized by sub-intervals of unit interval. In this paper, a new type of TSP is formulated, in which the travelling cost from one city to another is interval-valued intuitionistic fuzzy number (IVIFN), depending on the availability of the conveyance, condition of the roads, etc. We propose for the first time the Hungarian algorithm for finding of an optimal solution of TSP using the apparatuses of index matrices (IMs), introduced in 1984 by Atanassov, and of IVIFSs. The example shown in this paper guarantees the effectiveness of the algorithm. The presented approach for solving a new type of TSP can be applied to problems with imprecise parameters and can be extended in order to obtain the optimal solution for other types of multidimensional TSPs.