Fuzzy eigenvector method for deriving normalized fuzzy priorities from fuzzy multiplicative pairwise comparison matrices

The eigenvector method is one of the most used methods for deriving priorities of objects from multiplicative pairwise comparison matrices in Analytic Hierarchy Process (AHP). Fuzzy extension of AHP has been of much attention in order to capture uncertainty stemming from subjectivity of human thinking and from incompleteness of information that are integral to multi-criteria decision-making problems. Various fuzzy extensions of the eigenvector method have been introduced in order to derive fuzzy priorities of objects from fuzzy multiplicative pairwise comparison matrices. These fuzzy extensions are critically reviewed in this paper, and it is showed that (i) they violate multiplicative reciprocity of the related pairwise comparisons, (ii) they are not invariant under permutation of objects, (iii) the fuzzy maximal eigenvectors are not normalized, or (iv) a given normalized fuzzy maximal eigenvector does not consist of normalized maximal eigenvectors obtainable from multiplicative pairwise comparison matrices forming the fuzzy multiplicative pairwise comparison matrices. Afterwards, a new fuzzy extension of the eigenvector method based on the constrained fuzzy arithmetic is introduced and it is shown that it satisfies all four desirable properties.

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