Fuzzy Number with Nonlinear Membership Functions to Provide Flexibility in a Multi Objective Travelling Salesman Problem

Abstract

AbstractTravelling Salesman Problem (TSP), as extensively discussed in literature is an NP hard problem and among the most challenging problems in operations research, industrial engineering and computational mathematics, which has been deciphered and scrutinized under different headings and using different approaches e.g. Artificial Intelligence techniques, evolutionary algorithms and linear programming models under deterministic conditions. However, the information about real life processes is not always crisp but is often available as vague, uncertain and imprecise data. Fuzzy numbers finds application in handling vague terms, and therefore they can be suitably used to model real life scenarios involving vague parameters so as to obtain optimal solutions. Fuzzy multi-objective linear programming usually deals with flexible aspiration levels that are indicative of optimality when considering all objectives or goals simultaneously with possible deviation in objectives or constraints. Therefore in this study we develop a fuzzy multi-objective linear programming model with nonlinear membership functions for solving a multi objective TSP in order to simultaneously minimize the three parameters cost, distance and time. The importance of these parameters is assigned as weights to these objectives in the final model using AHP. The proposed model will give a compromised solution for best optimality and higher satisfaction level for the three parameters being considered in uncertain environment. The primary contribution of this study is a fuzzy mathematical model using nonlinear membership functions, more precisely the exponential functions to ensure an optimal solution in vague, imprecise and uncertain environment.KeywordsTravelling Salesman Problem (TSP)nonlinear fuzzy numbersexponential fuzzy numbersfuzzy multi objective linear programmingvague parametersAHP