Interval multiplicative pairwise comparison matrix: Consistency, indeterminacy and normality

Abstract To manifest human judgments, a long-established method called Pairwise Comparison (PC) has been successfully applied in the Analytic Hierarchy Process (AHP). In practice, human judgments are often made with uncertainty, and can be characterized by an Interval Multiplicative Pairwise Comparison Matrix (IMPCM). Since consistency is a key issue that has plagued decision makers and researchers for a long time, it is useful to propose a transformation that can effectively convert an inconsistent IMPCM into a consistent one, especially in group decision-making. However, a consistent IMPCM is not sufficient to be acceptable, indeterminacy should also be considered. Moreover, the interval priority weights should be normalized. To consider consistency, indeterminacy, and normality simultaneously, we put forward a new definition of acceptable IMPCM. To obtain such an acceptable IMPCM, we propose a theorem of consistency, a consistent transformation, and a normalized prioritization scheme. As a result, the proposed methods guarantee an inconsistent IMPCM can be directly converted into an acceptable IMPCM. Five theorems are proved to corroborate the proposed methods. A numerical example is presented to illustrate the validity and superiority of the proposed methods. Finally, discussion and conclusions are given.