Journal of Global Optimization

, Volume 42, Issue 2, pp 157–175 | Cite as

On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices

  • Sándor Bozóki
  • Tamás RapcsákEmail author


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 case of 3 ×  3 PRM, a differentiable one-to-one correspondence is given between Saaty’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 a ij (i < j) of which were randomly chosen from the ratio scale
$$\dfrac{1}{M}, \dfrac{1}{M-1}, \ldots , \dfrac{1}{2}, 1, 2, \ldots , M - 1, M,$$
with equal probability 1/(2M − 1) and a ji is defined as 1/a ij . 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.


Pairwise comparison matrix Inconsistency Inconsistency index Randomly generated pairwise comparison matrix 


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Copyright information

© Springer Science+Business Media, LLC. 2007

Authors and Affiliations

  1. 1.Computer and Automation Research InstituteHungarian Academy of SciencesBudapestHungary

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