Ensuring reliability of the weighting vector: Weak consistent pairwise comparison matrices

In the context of Pairwise Comparison Matrices (PCMs) defined over abelian linearly ordered group, ⊙-consistency and ⊙-transitivity represent a full coherence of the Decision Maker (DM) and the minimal logical requirement that DM's preferences should satisfy, respectively. Moreover, the ⊙-mean vector wm⊙wm⊙ is proposed as weighting vector for the decision elements related to the PCM. In this paper, we investigate the effects of ⊙-inconsistency of a ⊙-transitive PCM on wm⊙wm⊙ and, in order to ensure its reliability as weighting vector, we provide the notion of weak ⊙-consistency; it is weaker than ⊙-consistency and stronger than ⊙-transitivity, and ensures that vectors associated with a PCM, by means of a strictly increasing synthesis functional, are reliable for assigning a preference order on the set of related decision elements. The ⊙-mean vector wm⊙wm⊙ is associated with a PCM by means of one of these functionals. Finally, we introduce an order relation on the rows of the PCM, that is a simple order if and only if the condition of weak ⊙-consistency is satisfied; the simple order allows us to easily determine the actual ranking on the set of related decision elements.