A Representable Uninorm-Based Intuitionistic Fuzzy Analytic Hierarchy Process

Intuitionistic fuzzy preference relations (IFPRs) have been exposed to be an appropriate and effective preference representation framework in an analytic hierarchy process (AHP) with vagueness and hesitancy. This article focuses mainly on obtaining an intuitionistic fuzzy extension of Tanino's multiplicative consistency and deriving an analytic solution of normalized intuitionistic fuzzy weights (NIFWs) from IFPRs as well as checking acceptability of IFPRs. This article first introduces two indices to, respectively, measure hesitancy of intuitionistic fuzzy judgments and hesitancy of IFPRs, and illustrates that any existing multiplicative consistency model of IFPRs is not an actual intuitionistic fuzzy extension of Tanino's multiplicative consistency. A conjunctive-representable cross-ratio uninorm-based functional equation is then developed to define multiplicative consistency of IFPRs and a consistency index is devised to measure the inconsistency degree of an IFPR. This article establishes a representable uninorm-based transformation method for consistent IFPRs and intuitionistic fuzzy weights, and proposes a new framework of NIFWs. Based on the transformation method and the row hesitancy distribution of an IFPR, a logarithmic least square model is constructed and its analytic solution is found by applying the Lagrange multiplier method to its equivalent least square model. This article puts forward a novel acceptability checking method by taking both acceptable consistency and acceptable hesitancy into consideration. A representable uninorm-based fusion method is presented to aggregate local NIFWs into global intuitionistic fuzzy weights and a representable uninorm-based likelihood formula is given and used to compare and rank intuitionistic fuzzy weights in the proposed intuitionistic fuzzy AHP. Six numerical examples including an outstanding Ph.D. student selection problem are provided to illustrate and validate the obtained results.