Extended Intuitionistic Fuzzy Sets Based on the Hesitant Fuzzy Membership and their Application in Decision Making with Risk Preference

On the basis of the hesitant fuzzy membership, this study proposes the extended intuitionistic fuzzy set (EIFS) and the extended intuitionistic fuzzy number (EIFN) to synthesize the characters of the intuitionistic fuzzy set and the hesitant fuzzy set. We further develop two simplified and applied EIFSs, namely the credible EIFS (C-EIFS) and the possible EIFS (P-EIFS), to comprehensively mine the hesitant fuzzy membership information and to avoid the logical difficulty of simultaneously providing the membership and non-membership in each EIFS or EIFN. Then we investigate the foundations of C-EIFS and P-EIFS, including their expressions, operations, functions, differences, and selection rules. The corresponding aggregation operators are also proposed, and the calculation and relationships of these operators are proven. The prominent properties of C-EIFNs and P-EIFNs are focused on the boundary and average values, respectively; that is, the C-EIFN tends to aggregate the extreme information, whereas the P-EIFN prefers aggregating complete information. Therefore, applying them to decision making with risk preference is suitable, and two risk preference investment cases are provided to demonstrate the applications of these concepts and approaches.