Weak and Strong Consistency of an Interval Comparison Matrix

Milan Hladík, Martin Cerný

Abstract: We consider interval-valued pairwise comparison matrices and two types of consistency – weak (consistency for at least one realization) and strong (acceptable consistency for all realizations). Regarding weak consistency, we comment on the paper [Y. Dong and E. Herrera-Viedma, Consistency-Driven Automatic Methodology to Set Interval Numerical Scales of 2-Tuple Linguistic Term Sets and Its Use in the Linguistic GDM With Preference Relation, IEEE Trans. Cybern., 45(4):780–792, 2015], where, among other results, a characterization of weak consistency was proposed. We show by a counterexample that in general the presented condition is not sufficient for weak consistency. It provides a full characterization only for matrices up to size of three. We also show that the problem of having a closed form expression for weak consistency is closely related with P-completeness theory and that an optimization version of the problem is indeed P-complete. Regarding strong consistency, we present a sufficient condition and a necessary condition, supplemented by a small numerical study on their efficiency. We leave a complete characterization as an open problem.