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Bozóki, S., & Rapcsák, T. (2007). On Saaty's and Koczkodaj's inconsistencies of pairwise comparison matrices. Journal of Global Optimization, 42(2), 157–175. 
Added by: Klaus D. Goepel (07 Jun 2019 12:42:03 Asia/Singapore)   Last edited by: Klaus-admin (08 Jun 2019 02:27:49 Asia/Singapore)
Resource type: Journal Article
DOI: 10.1007/s10898-007-9236-z
BibTeX citation key: 2007
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Categories: AHP/ANP
Keywords: Analytic Hierarchy Process (AHP), consistency index, inconsistency, weighting methods
Creators: Bozóki, Rapcsák
Publisher: Springer
Collection: Journal of Global Optimization
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Abstract
The aim of the paper is to obtain some theoretical and numerical properties of Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices (PRM). In the caseof 3×3PRM, a differentiable one-to-one correspondence is given betweenSaaty’s inconsistency ratio and Koczkodaj’s inconsistency index based on the elements of PRM. In order to make a comparison of Saaty’s and Koczkodaj’s inconsistencies for 4×4 pairwise comparison matrices, the average value of the maximal eigenvalues of randomly generated n×n PRM is formulated, the elements aij(i<j) of which were randomly chosen from the ratio scale 1/M, 1/(M-1), ... 1/2, 1, 2, M-1, M with equal probability 1/(2M−1) and aji is defined as 1/aij . By statistical analysis, the empirical distributions of the maximal eigenvalues of the PRM depending on the dimension number are obtained. As the dimension number increases, the shape of distributions gets similar to that of the normal ones. Finally, the inconsistency of asymmetry is dealt with, showing a different type of inconsistency.

 


  
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