The number of circular triads in a pairwise comparison matrix and a consistency test in the AHP

A pairwise comparison matrix in the Analytic Hierarchy Process (AHP), which was proposed by Saaty in 1970s, consists of elements expressed on a numerical scale. The purpose of this paper is to propose a consistency test for ordinality of items in the pairwise comparison matrix. The original of this test is in a sensory test. In a sensory test we use a pick-the-winner ordinal scale to obtain the table of preferences for objects. In 1940 Kendall and Babington Smith proposed a consistency test for the preference table, using the number of circular triads in it. In this paper we show how to apply their test to a pairwise comparison matrix in the binary AHP and to one without a tie for up to nine items in the AHP. This is to test, using a pairwise comparison matrix, whether or not we can accept that items which are factors or alternatives are sufficiently ranked linearly before calculating weights of these items.