A Simple Method to Improve the Multiplicative Consistency of an Interval Fuzzy Preference Relation

In this paper, we propose a method that improves the multiplicative consistency and minimizes indeterminacy (the sum of widths of all interval membership degrees) by using only off-diagonal elements of an interval fuzzy preference relation (IFPR). In addition, this method keeps the initial information as much as possible. To do so, we formulate a concept of the multiplicative consistency that satisfies the additive reciprocity between the related preferences of the IFPR and is invariant under any permutation of objects. Next, the equations which are equivalent to the multiplicative consistency for the IFPR and uses only off-diagonal elements are derived. Based on these equations, the linear models to judge the multiplicative consistency of the IFPR and calculate multiplicatively consistent IFPR minimizing indeterminacy by using only off-diagonal elements are constructed. Based on linear models, we construct an algorithm that calculates the acceptable consistent IFPR keeping the initial information as much as possible and prove that a consistency index of algorithm converges to zero. The proposed method can reduce a large amount of calculations and is correct in judging and improving the multiplicative consistency for the IFPR in comparison with previous results because it uses only off-diagonal elements of the initial IFPR. In addition, a numerical example is provided to show the feasibility and efficiency of the proposed method.