A new method for solving the linear programming problem in an interval-valued intuitionistic fuzzy environment

Abstract

In the last few decades, uncertainty linear programming (ULP) has received important attention among researchers and industrials. In this paper the ULP problem in which all parameters and/or decision variables are specified in terms of interval-valued intuitionistic fuzzy (IVIF) numbers is considered. First, we introduce a new method for solving the LP problem, denoted by IVIFLP, in which all parameters are IVIF numbers. Through using this method, the IVIFLP problem is broken down into nine smaller crisp linear problems (CLPs). Second, we modified the proposed method for solving an LP problem, denoted by FIVIFLP, in which all parameters and decision variables are IVIF numbers. An additional bounded variables constraint is added for the CLPs in which the optimization variables of all lower problems are considered as parameters at the upper problems. According to the reduction technique based on a linear combination between variables, the CLPs in the modified method are reduced to two CLPs, giving the most and the least favorable value of the objective function. The proposed methods are illustrated numerically. Finally, we explored the shortcomings of Bharati and Singh’s method (2018) for solving transportation problems in an IVIF environment and applied the proposed methods to solve such problems.

Keywords

• Fuzzy set theory
• Intuitionistic fuzzy set
• Linear programming
• Uncertainty
• Transportation problem