Geometric mean method for judgement matrices: Formulas for errors
  The geometric mean method (GMM) and the eigenvector method (EM) are well-known approaches to deriving information from pairwise comparison matrices in decision making processes. However, the original algorithms of these methods are logically incomplete and have significant drawbacks: their actual numerical errors are unknown and their reliability is doubted by different rank reversal phenomena including the GMM-EM ones.   Recently (Tomashevskii 2015: Eur J Oper Res 240:774-780), the actual numerical errors were found for EM, and it was shown that all EM rank reversal phenomena have the same cause and are eliminated when the errors are taken into account. In this paper the similar ap-proach is used for GMM: we associate GMM with some standard measuring procedure, ana-lyze this procedure from the viewpoint of measurement theory, and find the actual GMM errors. We show that the GMM and the GMM-EM rank reversal phenomena are eliminated when the GMM and the EM errors are taken into account.   The GMM decision support tool, which has all components of a standard measuring tool, is composed of pairwise comparisons as an initial measuring procedure, GMM as a data processor, and the obtained formulas for GMM errors as an error indicator. This tool is analogous to the EM decision support tool received earlier in the above paper. It is shown that the EM and the GMM versions of the tool are equally suitable to measure and rank any comparable elements with positive numerical values.   We also analyze the Geometric Consistency Index usually used in the original GMM to measure of the inconsistency and to accept or reject an inconsistent pairwise comparison matrix, and show that this index is not an acceptable GMM error indicator. 
I.L. Tomashevskii
math.OC