Triangular Intuitionistic Fuzzy Triple Bonferroni Harmonic Mean Operators and Application to Multi-attribute Group Decision Making

As an special intuitionistic fuzzy set defined on the real number set, triangular intuitionistic fuzzy number (TIFN) is a fundamental tool for quantifying an ill-known quantity. In order to model the decision maker's overall preference with mandatory requirements, it is necessary to develop some Bonferroni harmonic mean operators for TIFNs which can be used to effectively intergrate the information of attribute values for multi-attribute group decision making (MAGDM) with TIFNs. The purpose of this paper is to develop some Bonferroni harmonic operators of TIFNs and apply to the MAGDM problems with TIFNs. The weighted possibility means of TIFN are firstly defined. Hereby, a new lexicographic approach is presented to rank TIFNs sufficiently considering the risk preference of decision maker. The sensitivity analysis on the risk preference parameter is made. Then, three kinds of triangular intuitionistic fuzzy Bonferroni harmonic aggregation operators are defined, including a triangular intuitionistic fuzzy triple weighted Bonferroni harmonic mean operator (TIFTWBHM) operator, a triangular intuitionistic fuzzy triple ordered weighted Bonferroni harmonic mean (TIFTOWBHM) operator and a triangular intuitionistic fuzzy triple hybrid Bonferroni harmonic mean (TIFTHBHM) operator. Some desirable properties for these operators are discussed in detail. By using the TIFTWBHM operator, we can obtain the individual overall attribute values of alternatives, which are further integrated into the collective ones by the TIFTHBHM operator. The ranking order of alternatives is generated according to the collective overall attribute values of alternatives. A real investment selection case study verifies the validity and applicability of the proposed method.