Saaty’s consistency trapezoidal fuzzy extension and optimized fuzzy utility vector acquisition

Abstract

Fuzzy extensions of Saaty’s consistency play an important role in developing fuzzy analytic hierarchy process based decision-making methods. This paper proposes formulas to calculate ratio-based increasing and decreasing spread indices and to obtain ratio-based indeterminacy indices of modal and support intervals for fuzzy judgments of trapezoidal fuzzy reciprocal preference matrices (TFRPMs). A transitivity equation with parametric fuzzy elements is introduced and combined with the existence of parameter values to define consistency of TFRPMs. To cope with the challenge of identifying the existence, computational formulas are developed to directly obtain values of parameters from the fuzzy judgments in a TFRPM, and an equivalent consistency concept is proposed for TFRPMs. Properties of consistent TFRPMs are presented and a logarithmic deviation based formula is provided to compute consistency indices of TFRPMs. A new framework of normalized vectors with trapezoidal fuzzy elements is introduced. A logarithmic goal programming model is established and transformed into a linear program for acquiring normalized trapezoidal fuzzy utility vectors from TFRPMs. An analytical solution is found for optimized trapezoidal fuzzy utility vectors of consistent TFRPMs. The developed models are validated by three numerical illustrations possessing comparative studies.