Inconsistent and contradictory judgements in pairwise comparison method in the AHP

Abstract

The aim of this paper is to show the relationship between inconsistent and contradictory matrices of data obtained as a result of the pairwise comparison of factors in the sense of the analytic hierarchy process. The consistency check is performed to ensure that judgements are neither random nor illogical. This paper shows that even if a matrix will pass a consistency test successfully, it can be contradictory. Moreover an algorithm of checking contradictions is proposed. Scope and purpose A consistency of judgements in the pairwise comparison method in the analytic hierarchy process can be measured using the consistency index of judgement matrices. The consistency index is based on a judgement matrix rank. Depending on its value judgement matrices can be classified into two sets, namely a set of consistent and a set of inconsistent matrices. The paper introduces a new class of judgement matrices which is defined as contradictory and shows that a judgement matrix can be very closed to fully consistent in sense of the assumed consistency index but still to be contradictory.