Consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean

• Existing consistency notions of triangular fuzzy preference relations (TFPRs) are illustrated to be non-robust.
• New consistency and acceptable consistency definitions are put forward for TFPRs.
• We propose a notion of normalized triangular fuzzy multiplicative weights (NTFMWs).
• Transformation formulae are provided to convert NTFMWs into consistent TFPRs.
• A logarithmic least square model is developed to derive a NTFMW vector from a TFPR.

Triangular fuzzy preference relation (TFPR) is an effective framework to model pairwise estimations with imprecision and vagueness. In order to obtain a reliable and rational decision result, it is important to investigate consistency and priority derivation of TFPRs. The paper analyzes existing definitions and properties of consistent TFPRs, and illustrates that they have no invariance with respect to permutations of decision alternatives. A new triangular fuzzy arithmetic based transitivity equation is introduced to define consistent TFPRs. The new transitivity equation reflects multiplicative consistency of modal values and multiplicative consistency of geometric means of triangular fuzzy estimations. Some properties are presented for consistent TFPRs, and a notion of acceptable consistency is put forward for TFPRs. Geometric mean and uncertainty ratio based transformation formulae are devised to convert normalized triangular fuzzy multiplicative weights into consistent TFPRs. A logarithmic least square model is further established for deriving a normalized triangular fuzzy multiplicative weight vector from a TFPR with acceptable consistency. A geometric mean based method is developed to compare and rank triangular fuzzy multiplicative weights. Three numerical examples including a group decision making problem are examined to demonstrate validity and advantages of the proposed models.