Fuzzy eigenvector method for obtaining normalized fuzzy weights from fuzzy pairwise comparison matrices

Abstract

Proper formulas for obtaining the fuzzy maximal eigenvalue and the corresponding fuzzy maximal eigenvector of a fuzzy pairwise comparison matrix are proposed in this paper. First, the formulas for obtaining the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by Csutora & Buckley (2001) and by Ishizaka (2014) are reviewed, and the flaws in their formulas regarding the violation of the reciprocity of pairwise comparisons are pointed out. New formulas for obtaining the fuzzy maximal eigenvalue preserving the reciprocity of pairwise comparisons are then proposed. After, a fuzzy extension of Saaty's Consistency Index and Consistency Ratio is introduced in order to verify an acceptable level of inconsistency of a fuzzy pairwise comparison matrix. Further, the methods for obtaining the fuzzy maximal eigenvector corresponding to the fuzzy maximal eigenvalue of a fuzzy pairwise comparison matrix proposed by Wang & Chin (2006) and by Ishizaka (2014) are reviewed. The flaws in Ishizaka's method are pointed out, and Wang & Chin's method is studied and modified in order to preserve the reciprocity of pairwise comparisons. The fuzzy maximal eigenvalues and the corresponding fuzzy maximal eigenvectors obtained by the new formulas are confronted with those obtained by methods proposed by Csutora & Buckley (2001), Wang & Chin (2006) and Ishizaka (2014), and three numerical examples are given for better illustration.