Consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations

• A positive triangular fuzzy number is expressed by a cross-ratio-expressed triplet.
• Operational laws are established for cross-ratio-expressed triangular fuzzy numbers (CRETFNs).
• Multiplicative consistency is analyzed for triangular fuzzy additive reciprocal preference relations.
• A triangular fuzzy weighted geometric operator is developed to aggregate CRETFNs
• A novel method is put forward for comparing two CRETFNs.

Triangular fuzzy numbers are effective in modeling imprecise and uncertain information, and have been widely applied in decision making. This paper uses a cross-ratio-expressed triplet to characterize a positive triangular fuzzy number, and introduces notions of cross-ratio-expressed triangular fuzzy numbers (CRETFNs) and triangular fuzzy additive reciprocal preference relations (TFARPRs). We present transformation methods between TFARPRs and triangular fuzzy multiplicative reciprocal preference relations, and develop operational laws of CRETFNs, such as complement, addition, multiplication and power. A cross-ratio-expressed triangular fuzzy multiplication based transitivity equation is established to define multiplicative consistency of TFARPRs. The new consistency captures Tanino's multiplicative consistency among the cross-ratio-expressed modal values, and geometric consistency of the interval fuzzy preference relation constructed from lower and upper support values of cross-ratio-expressed triangular fuzzy judgments. Some desirable properties are furnished for multiplicatively consistent TFARPRs. We propose a cross-ratio-expressed triangular fuzzy weighted geometric operator to aggregate CRETFNs, and extend it to fuse TFARPRs. Score and uncertainty index functions are defined and employed to devise a novel comparison method for CRETFNs. A detailed procedure is put forward to solve group decision making problems with TFARPRs. Six numerical examples are provided to illustrate the validity and applicability of the proposed models.