Weak Consistency for Ensuring Priority Vectors Reliability

Consistency and transitivity are important and leading research topics in the study of decision‐making in terms of pairwise comparison matrices. In this paper, we search for conditions that, in case of inconsistency, guarantee ordinal compatibility between ordinal ranking (actual ranking) derived from a transitive matrix and cardinal rankings provided by the most usual priority vectors proposed in the scientific literature. We provide the notion of weak consistency; it is a condition weaker than consistency and stronger than transitivity and ensures that vectors associated with a matrix, by means of a strictly increasing synthesis functional, provide a preference order, on the related set of decision elements, equal to the actual ranking. This notion extends, to the case in which the decision‐maker can be indifferent between two or more alternatives/criteria, weak consistency introduced in previous papers under constraint of no indifference. Finally, we introduce an order relation on the rows of the matrix, that is, a simple order if and only if weak consistency is satisfied; this simple order allows us to easily determine the actual ranking on the set of decision elements. Copyright 2015 John Wiley & Sons, Ltd.