New Tanino's Additive Consistency Trapezoidal Fuzzy Extension and Optimal Fuzzy Priority Vector Acquisition

Abstract

Multiplication and division of fuzzy arithmetic have brought out theoretical drawbacks to fuzzy analytic hierarchy process based decision making systems. To cope with the drawbacks, trapezoidal fuzzy additive reciprocal preference matrices (TFARPMs) are utilized to characterize preference information and addition and subtraction of fuzzy arithmetic are applied to fuzzy numbers. This paper introduces formulas to calculate left and right spread indices and imprecision indices of cores and supports for fuzzy elements in a TFARPM. Based on the addition of fuzzy arithmetic, a transitivity equation system with a parameterized trapezoidal fuzzy vector is built and computational formulas are devised to identify values of parameters from imprecision indices of cores and supports of the trapezoidal fuzzy elements in a TFARPM. A fuzzy addition based non-parametric transitivity equation is then established to define additive consistency of TFARPMs. Properties of additively consistent TFARPMs are proposed and an index formula is brought forward to compute additive inconsistency degrees of TFARPMs. A novel approach is presented to generate additively consistent TFARPMs from fuzzy vectors and a new framework is put forward to normalize [0, 1]-valued trapezoidal fuzzy vectors. An absolute deviation based minimization model is developed and converted equivalently into a linear program to acquire normalized trapezoidal fuzzy priority vectors from TFARPMs. A closed-form solution of the minimization model is found to calculate normalized optimal trapezoidal fuzzy priority vectors of additively consistent TFARPMs. Two illustrating examples including a multi-criteria decision making problem are provided to validate the proposed models.