Preservation of preferences intensity of an inconsistent Pairwise Comparison Matrix

Pairwise comparisons have been a long standing technique for comparing alternatives/criteria and their role has been pivotal in the development of modern multi-criteria decision analysis methods; they represent a useful tool for deriving a weighting vector which establishes a cardinal ranking of the alternatives/criteria. The weighting vector used in this paper gained popularity due to its mathematical properties and satisfies fundamental requirements, such as the immunity to rank reversal and preservation of the algebraic structure of the problem. Under consistency condition, that represents the full decision maker's coherence, this vector is completely reliable because it exactly represents the decision maker's preferences expressed by means of the pairwise comparisons. Unfortunately, the consistency is hard to reach in real situation; thus, in a previous paper, we provided a condition weaker than consistency so that the weighting vector at least preserves the preferences order expressed by means of inconsistent pairwise comparisons. This paper deals with a further crucial step in multi-criteria decision analysis, that is to ensure that the weighting vector preserves the preferences intensity, in addition to the preferences order, expressed by means of inconsistent pairwise comparisons. Thus, firstly, we introduce a necessary condition so that the weighting vector preserves the preferences intensity. Then, we show that this condition is a necessary but not a sufficient condition. Finally, we propose a sufficient condition so that the weighting vector preserves the preferences intensity. In order to provide a general unified framework, we deal with pairwise comparison matrices defined over Abelian linearly ordered groups.