Extension of Saaty's inconsistency index to incomplete comparisons: Approximated thresholds

Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices to analyse such incomplete data sets and even fewer measures have an associated threshold. This paper generalises the inconsistency index proposed by Saaty to incomplete pairwise comparison matrices. The extension is based on the approach of filling the missing elements to minimise the eigenvalue of the incomplete matrix. It means that the well-established values of the random index, a crucial component of the consistency ratio for which the famous threshold of 0.1 provides the condition for the acceptable level of inconsistency, cannot be directly adopted. The inconsistency of random matrices turns out to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly used by practitioners as a statistical criterion for accepting/rejecting an incomplete pairwise comparison matrix.