TOPSIS-Based Nonlinear-Programming Methodology for Multiattribute Decision Making With Interval-Valued Intuitionistic Fuzzy Sets

Interval-valued intuitionistic fuzzy (IVIF) sets are useful to deal with fuzziness inherent in decision data and decision-making processes. The aim of this paper is to develop a nonlinear-programming methodology that is based on the technique for order preference by similarity to ideal solution to solve multiattribute decision-making (MADM) problems with both ratings of alternatives on attributes and weights of attributes expressed with IVIF sets. In this methodology, nonlinear-programming models are constructed on the basis of the concepts of the relative-closeness coefficient and the weighted-Euclidean distance. Simpler auxiliary nonlinear-programming models are further deduced to calculate relative-closeness of IF sets of alternatives to the IVIF-positive ideal solution, which can be used to generate the ranking order of alternatives. The proposed methodology is validated and compared with other similar methods. A real example is examined to demonstrate the applicability and validity of the methodology proposed in this paper.

I. Introduction

Atanassov [1] introduced the concept of the intuitionistic fuzzy (IF) set, which is characterized by a degree of membership and a degree of nonmembership. The IF set gives us the possibility to model hesitation and uncertainty by using an additional degree [2]–[9]. The IF set is a generalization of the fuzzy set [10]. Other extensions of the fuzzy set are the vague set [11] and the interval-valued fuzzy (IVF) set [12]. The vague set, which is characterized by a truth function and a false function, was proven to be the same as the IF set [13]. The IVF set is defined by an interval-valued membership function, which approximates the “real,” but unknown membership degree of an element to the given set. It was proven that the IVF set is isomorphic to the IF set in the sense of Atanassov [14]–[17], i.e., the IVF set is mathematically equivalent to the IF set, even if their interpretive settings and motivation are quite different [16]. The former captures the idea of the ill-known membership degree, while the latter starts from the idea of evaluating degrees of membership and nonmembership independently.