Additive consistency analysis and normalized optimal utility vector derivation for triangular fuzzy additive reciprocal preference relations

Abstract

This paper focuses on extending Tanino’s additive consistency to triangular fuzzy additive reciprocal preference relations (TFARPRs) and deriving normalized optimal fuzzy utility vectors from TFARPRs. Spread based indices are introduced to measure vagueness of [0, 1]-valued triangular fuzzy judgments and TFARPRs. A transitivity equation with parameters is devised to define additive consistency of TFARPRs. Values of parameters are then determined to introduce an equivalent definition of additively consistent TFARPRs. Pivotal properties of additively consistent TFARPRs are put forward and a computational formula is designed to obtain additive consistency indices of TFARPRs. A novel method is proposed to generate additively consistent TFARPRs from [0, 1]-valued triangular fuzzy utility vectors. By examining the equivalency of [0, 1]-valued triangular fuzzy utility vectors, a framework is presented to normalize triangular fuzzy utility vectors. A goal programming model is subsequently built to derive normalized triangular fuzzy utility vectors from TFARPRs. This optimization model is transformed into a linear program and its analytical solution is achieved for additively consistent TFARPRs. The performance and validity of the proposed models are demonstrated by three illustrations with comparative analysis.