Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process

Intuitionistic fuzzy preference relations (IFPRs), which are based on Atanassov's intuitionistic fuzzy sets (A-IFS), have turned out to be a useful structure in expressing the experts' uncertain judgments, and the intuitionistic fuzzy analytic hierarchy process (IFAHP) is a method for solving multiple criteria decision making problems. To provide a theoretical support for group decision making with IFAHP, this paper presents some straightforward and useful results regarding to the aggregation of IFPRs. Firstly, a new type of aggregation operator, namely, simple intuitionistic fuzzy weighted geometric (SIFWG) operator, is developed to synthesize individual IFPRs. It is well known that for traditional comparison matrices, if all individual comparison matrices are of acceptable consistency, then their weighted geometric mean complex judgment matrix is of acceptable consistency. In this paper, we prove that this property holds for IFPRs as well if we use the SIFWG operator to synthesize the individual IFPRs. A numerical example is given to verify the theorems. It is pointed out that the well-known simple intuitionistic fuzzy weighted averaging (SIFWA) operator, the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy weighted geometric (IFWG) operator and the symmetric intuitionistic fuzzy weighted geometric (SYIFWG) operator do not have this property. Finally, the group IFAHP (GIFAHP) procedure is developed to aid group decision making process with IFPRs.