Research on the consistency of additive trapezoidal fuzzy preference relations

Abstract

Trapezoidal fuzzy numbers offer a suitable and flexible way to express vague judgments of decision makers. This paper studies decision making with additive trapezoidal fuzzy preference relations. It first analyzes the issues encountered in previous consistency concepts for additive trapezoidal fuzzy preference relations. Then, two new consistency concepts, namely, an additive consistency concept and a multiplicative consistency concept, are introduced. To verify the rationality, several of their properties are discussed. To efficiently assess the consistency of additive trapezoidal fuzzy preference relations, several corresponding optimization models are formed. When the objective function values of the built models are zero, the associated additive trapezoidal fuzzy preference relations are consistent. Considering the case where additive trapezoidal fuzzy preference relations are incomplete, optimization models based on the consistency analysis are built, by which identified unknown judgments have the highest consistency level with the known ones. In general, additive trapezoidal fuzzy preference relations are unacceptably consistent. To rank objects from additive trapezoidal fuzzy preference relations, several optimization models for improving the consistency level are built, which consider the self-confidence of the decision makers, the total adjustment and the number of adjusted elements. According to the above discussion, two frameworks for ranking objects from additive trapezoidal fuzzy preference relations are provided that are based on the additive and multiplicative consistency analysis, respectively. Finally, numerical examples are provided to highlight the concrete application and to deliver the comparative analysis.