Exponential membership function and duality gaps for i-fuzzy linear programming problems

Abstract

Fuzziness is ever presented in real life decision making problems. In this paper, we adapt the pessimistic approach tostudy a pair of linear primal-dual problem under intuitionistic fuzzy (I-fuzzy) environment and prove certain dualityresults. We generate the duality results using exponential membership and non-membership functions to represent thedecision maker’s satisfaction and dissatisfaction level. Further, two numerical examples have been given. In each ofthese illustrations, varying the values of the shape variables in the exponential membership functions, various nonlinearoptimization problems have been constructed, analyzed and solved. The duality gaps for all these optimization problemshave been computed and compared with the duality gap under the linear membership function. We found that thesegaps for the I-fuzzy linear primal-dual pair under the exponential membership functions are smaller as compared withthe linear membership functions. The main advantage of using the exponential membership function is that it has theflexibility of altering the values of the shape parameters (as per the decision-maker’s satisfaction). Finally, with thehelp of a suitable ranking defuzzification function, we have extended our approach to the I-fuzzy linear problem withfuzzy parameters and fuzzy constraints.