geometric mean / Additive Normalization Method  The Eigenvector Method has a consistency test in AHP method. But the geometric or Additive Normalization Method  didn't have any measure of consistency. This is correct ?
Can i use the consistency test with geometric mean or Additive Normalization Method ? basically, find the consistency ratio (CR)
 A: Before entering into the discussion of consistency ratios, I must say that I personally have very strong reservation to the use of AHP. I think this method is mathematically unsound and it is more like a tool to disguise meaningless figures as meaningful than a useful and practical tool to prioritize things reasonably. (However, don't ask me which method I think is OK. I don't know.)
As to your question, recall that when the eigenvector method (EV) is used, the consistency ratio is defined as
$$\textrm{consistency ratio} = CI/RCI
= \left.\underbrace{\frac{\lambda_\textrm{max}-n}{n-1}}_{CI}\right/RCI,$$
where CI is the so-called consistency index, and RCI, the random consistency index, is the average CI computed using some 500 randomly generated reciprocal matrices.
This framework for constructing a consistency ratio is applicable to the geometric mean method (GM) and the additive normalization method (AN), provided that appropriate definitions of CIs exist. In EV, the above definiton of CI can be rewritten as
$$
CI = \frac{\textrm{max. eigenvalue}-\textrm{theoretical min. of the max. eigenvalue}}{n-1}.
$$
To devise a consistency ratio for AN or GM, one may replace the $\lambda_\textrm{max}$ and its lower bound by some other notions, and compute RCI using the new definition of CI. [Edit] Yet, to my knowledge, no one has done this for AN. For GM, though, someone has devised a measure called "geometric consistency index" (GCI). I have googled this term and found a downloadable version of the original paper. The authors show that GCI is somehow has an almost linear relationship with CR (and also with CI, because RCI is constant). So, changing EV to GM and CR to GCI does not really bring something new to the table. [End edit]
Some people do try to apply the consistency ratio test to AN (and/or GM), however. For examples, they may do the followings:


*

*Apply the original consistency ratio test. In other words, they perform both EV and AN/GM, but EV is used to justify consistency, and AN/GM is used to generate priorities.

*Replace $\lambda_\textrm{max}$ by some weird quantity that has superficial similarity to eigenvalue in appearance. For instance, some books or papers (which I will not name here) simply replace the definition of $\lambda_\textrm{max}$ by $\textrm{mean}\left[(Av)/v\right]$, where $A$ is the reciprocal matrix, $v$ is the priority vector obtained by AN or GM, the division "/" is performed entrywise and $\textrm{mean}(x)$ means the average of the entries of a vector $x$. The consistency ratio is then computed as
$$
\textrm{consistency ratio} = \left.\underbrace{\frac{\textrm{new definition of}\ \lambda_\textrm{max}-n}{n-1}}_{\textrm{new } CI}\right/\textrm{RCI for EV}.
$$
This practice is very questionable, because people are mixing quantities from AN/GM (namely, the new definition of $\lambda_\textrm{max}$) with benchmark values for EV (namely, the old RCI value for EV and the $n$ in the numerator of the definition of CI --- the question of whether $n$ is still a theoretical lower bound for $\lambda_\textrm{max}$ is ignored).

