A direct consistency test and improvement method for the analytic hierarchy process

The consistency test is a vital component of pairwise comparison matrices if meaningful results are to be guaranteed, and it has been studied extensively since the analytic hierarchy process was developed by Saaty. However, when using the existing methods, it is imperative to carry out matrix operations, which are usually not intuitional. In this paper, a direct method, from the perspective of 3 tuples, is proposed to test and improve the consistency of a pairwise comparison matrix, which is far more intuitional and easier to understand. In the proposed 3 tuples iterative method, only simple mathematical calculations are needed, without the need for matrix operations. Furthermore, the calculation of Saaty's consistency ratio is only conducted to verify the proposed method. Some related theorems and propositions are proved mathematically or verified by random simulations. The proposed method is applied to some published examples to verify its effectiveness and practicability. Finally, we provide an improved iterative method, which is based on the 3 tuples iterative method. The improved iterative method and the 3 tuples iterative method are closely linked and each has its own different perspective. Moreover, they are applied on different occasions and reinforce one another.