On the order of the preference intensities in fuzzy Analytic Hierarchy Process

We show that a recently discovered fundamental problem with the Analytic Hierarchy Process (AHP) concerning the meaning of the resultant preference intensities is also evident for the fuzzy AHP. We prove that if there is a judgmental inconsistency in the fuzzy pair-wise comparisons, it is impossible to ensure the preservation of the order regarding preference intensities in the resultant priority vector. Further, it is shown with an example from the published literature that the order of the preference intensities may not be preserved even if there is no inconsistency in the judgment set, albeit it is possible to comply with this order via using fuzzy preference programming (FPP) methodology. Finally, it is proved that if the interval judgments regarding the decompositions of original judgments to α-level sets are consistent, FPP guarantees the preservation of the order of the preference intensities at those levels.