This paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices. Statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a Monte–Carlo simulation, which confirms that relatively high percentage of comparison matrices, satisfying Saaty’s CR criterion are ordinarily inconsistent. It is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process, in contrast with cardinal inconsistency. A heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices. The proposed algorithm generates near-optimal solutions and outperforms other tested approaches with respect to computation time.