An axiomatic property based triangular fuzzy extension of Saaty’s consistency

Abstract

The fuzzy analytic hierarchy process (FAHP) is a widely applied fuzzy multi-criteria decision making method. In triangular FAHP, consistency of triangular fuzzy multiplicative preference relations (TFMPRs) plays a very important role for checking the quality of human fuzzy judgments. This paper introduces some indices to measure fuzziness of triangular fuzzy judgments and row fuzziness proportionalities in a TFMPR. An expected interval based possibility degree formula is developed to compare two triangular fuzzy judgments and used to judge whether the average-based intensity of a triangular fuzzy judgment is changed. The paper proposes four axiomatic properties aimed at characterizing multiplicative consistency of TFMPRs, and illustrates that four known consistency definitions fail to meet all the four axiomatic properties. By analyzing the axiomatic property of restriction on ambiguity of single judgments, a constrained triangular fuzzy multiplication based transitivity equation is devised and used to construct a triangular fuzzy matrix from a TFMPR. Based on the constructed fuzzy matrix, the paper establishes an ordinary triangular fuzzy multiplication based transitivity equation to define consistency of TFMPRs. Important properties of consistent TFMPRs are offered to demonstrate that all the four axiomatic properties are satisfied by the proposed consistency definition. Three numerical examples are given to illustrate the use of the proposed consistency model and the obtained properties for judging consistency of TFMPRs and determining unknown values of incomplete TFMPRs.