The Molds of Intuitionistic Fuzzy Value and Their Applications

In the process of applying Atanassov's intuitionistic fuzzy sets (A-IFSs) to decision-making problems, one key step is to rank Atanassov's intuitionistic fuzzy values (A-IFVs). In this paper, we give four kinds of mold for A-IFVs, accordingly, we give four kinds of admissible order on A-IFVs. It is disclosed that Xu's intuitionistic fuzzy weighted geometric (IFWG) operator and intuitionistic fuzzy weighted averaging (IFWA) operator are monotonic with respect to the third and the fourth kinds of order, respectively. The third kind of mold has some desirable properties, so we define the cosine similarity measures for A-IFSs and the projection formulas based on the third kind of mold. In the end, some applications are given in this paper. We apply the proposed cosine similarity measures and the projection formulas to pattern recognition and intuitionistic fuzzy multi-attribute decision making through two illustrative examples.