A Branch and Bound Algorithm to Solve Travelling Salesman Problem (TSP) with Uncertain Parameters

Abstract

The core of the theoretical Computing science and mathematics is computational complexity theory. It is usually concerned with the classification of computational problems in to P and NP problems by using their inherent challenges. There is no efficient algorithms for these problems. Travelling Salesman Problem is one of the most discussed problems in Combinatorial Mathematics. To deduct a Hamiltonian cycle in which the cost or time is minimum is the main objective of the TSP. There exist many algorithms to solve it. Since all the existing algorithms are not efficient to solve it, still many researchers are working to produce efficient algorithms. If the description of the parameters is vague, then fuzzy notions which include membership value are applied to model the parameters. Still the modeling does not give the exact representation of the vagueness. The Intuitionistic fuzzy set which includes non-membership value along with membership values in its domain is applied to model the parameters. The decision variables in the TSP, the cost, time or distance are modeled as intuitionistic fuzzy numbers, then the TSP is named as Intuitionistic fuzzy TSP (InFTSP). We develop the intuitionistic fuzzified version of littlewood's formula or branch and bound method to solve the Intuitionistic fuzzy TSP. This method is effective because it involves the simple arithmetic operation of Intuitionistic fuzzy numbers and ranking of intuitionistic fuzzy numbers. Ordering of intuitionistic fuzzy numbers is vital in optimization problems since it is equivalent to the ordering of alternatives. In this article, we used weighted arithmetic mean method to order the fuzzy numbers. Weighted arithmetic mean method satisfies linear property which is a very important characteristic of ranking function. Numerical examples are solved to validate the given algorithm and the results are discussed.