Role of Consistency and Random Index in Analytic Hierarchy Process—A New Measure

Abstract

This paper explains about the role of consistency and Random Index in analytic hierarchy process [2], which is one of the popular and most effectively used decision-making techniques developed by professor Saaty in the 1970s. The techniques of AHP have been applied in many areas and several decision-making issues. Consistency plays a vital role in AHP [2], Saaty introduced consistency test to accept or reject the decision-maker’s pairwise comparison matrix, i.e., \( {\text{Consistency Ratio}} = \frac{\text{Consistency Index}}{\text{Random Index}} \) should less than 0.1 [3]. Here Consistency Index \( \left( {{\text{C}}.{\text{I}}} \right) = \frac{{\lambda_{\max } - x}}{x - 1} \) and Random Index (R.I) decide the consistency of comparison matrix obtained by decision-maker. Various authors have proposed different R.I values based on the size of the comparison matrix. This paper explains a statistical technique to get R.I values for various sizes of pairwise comparison matrices by using \( {\text{R}}.{\text{I }}\left( x \right) = \frac{{\bar{\lambda }_{\max } - x}}{x - 1} \). The effectiveness of the new technique was also verified with illustrations, and the results were analyzed [1].