On Vargas's proof of consistency test for 3x3 comparison matrices in AHP

In this paper we have raised, discussed, and solved the problems relating the Vargas's verification of consistency test for 3 x 3 comparison matrices in the Analytic Hierarchy Process (AHP). In the AHP community, assessing attribute weights by using a consistency ratio of 0.1 or less has been widely practiced. Though Vargas did study the problem, the method he adopted might lead to certain questionable results. By trying to clarify such a critical situation, we pointed out the problems arose form the solutions of two inequality systems within the proof process of Vargas. We proved that the solution for the first system was an interval, not a point as he had indicated. After we solved the first inequality system, we found that the criterion for the solution interval of the first system would satisfy the second inequality system. We have then justified the inherent problim in Vargas's verification procedure.