On the similarity between ranking vectors in the pairwise comparison method

Abstract

There are many priority deriving methods for pairwise comparison (PC) matrices. It is known that when these matrices are consistent all these methods result in the same priority vector. However, when they are inconsistent, the results may vary. The presented work formulates an estimation of the difference between priority vectors in the two most popular ranking methods: the eigenvalue method and the geometric mean method. The estimation provided refers to the inconsistency of the PC matrix. Theoretical considerations are accompanied by Monte Carlo experiments showing the discrepancy between the values of both methods.

Keywords:

• Pairwise comparisons
• analytic hierarchy process
• eigenvalue method
• geometric mean method
• AHP
• EVM
• GMM

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The existence of ${\lambda }_{\mathit{max}}$ is guaranteed by the Perron-Frobenius theorem (Gantmaher, 2000, vol. 2, p. 53).

2 $\Vert ·{\Vert }_m$ is the Manhattan norm.

3 In fact, $CR$ also depends on the measurement scale, however, in most of the cases the scale $1/9,\dots ,1,\dots ,9$ is used.