A Direct Consistency Improvement Method for the Probability-Hesitant 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, how to deal with the consistency of a probability-hesitant analytic hierarchy process with different probability values is a specific problem which needs to be studied further. This paper provides a direct consistency test and an improvement method for the probability-hesitant analytic hierarchy process. A paradox in Saaty's consistency test is first proposed, and then a direct consistency improvement method is suggested for dealing with the consistency of a probability-hesitant analytic hierarchy process with different probability values, from the perspective of three tuples, rather than carrying out matrix operations. In our method, only simple mathematical calculations are needed and there is no need for matrix operations. By using a partial correction, the proposed method can retain most of the information provided by the original comparison matrix, and simultaneously indicate the modification direction and provide the optimal values. At the same time, using the proposed method, the inconsistent elements in the probability-hesitant pairwise comparison matrix can be found rapidly and with a high degree of accuracy. Moreover, our proposed iterative algorithm is a method that is both very quick and leads to good results being obtained.