An axiom system of the analytic hierarchy process based on the completeness of preferences

Decision outcomes rely on the preferences of the decision maker (DM) over alternatives. The completeness axiom is a fundamental assumption of preference theory. This paper reports an axiom system of the analytic hierarchy process (AHP) under the theory of preferences. First, the existing axiomatic foundations of AHP are analyzed. It is found that the Saaty's one is limited by strict mathematical intuition for the reciprocal property of comparison ratios. The uncertainty-induced one could lead to the confusion of preferences owing to the breaking of reciprocal property. Second, the constrained relation of comparison ratios is constructed according to the completeness of preferences, which is used to form a novel axiom system of AHP. Third, the concepts of ordinal consistency and acceptable ordinal consistency are developed for pairwise comparison matrices (PCMs). The index of measuring ordinal consistency is constructed using Spearman rank correlation coefficient to overcome the shortcoming in the existing one. The prioritization method is discussed to reveal that the geometric mean method is suitable for PCMs with ordinal consistency. It is revealed that the developed axiomatic foundation of AHP reasonably embeds the preference information of DMs. The results help to identify how to flexibly apply AHP in practice by considering the complete preferences of DMs under uncertainty.