An uncertainty-induced axiomatic foundation of the analytic hierarchy process and its implication

Abstract Uncertainty is always encountered by decision makers (DMs) when addressing a complex decision-making problem. The existing axiomatic foundation of the analytic hierarchy process (AHP) cannot reflect the uncertainty experienced by DMs. In this paper, it is first to find that the uncertainty experienced by DMs can be characterized by the non-reciprocal property of pairwise comparisons. The novel concept of reciprocal symmetry breaking (RSB) is proposed to capture the situation without reciprocal property. Then an uncertainty-induced axiomatic foundation of the AHP model is reported, where the RSB is one of the proposed axioms. Some interesting results are derived from the new axioms involving the concept of approximate consistency and the method of eliciting priorities. An index is constructed to measure approximate consistency degree of pairwise comparison matrices (PCMs). Some comparisons with the typical AHP model are offered by carrying out some numerical examples. The obtained results reveal that the proposed axiomatic foundation and the derived facts form a novel operational basis of the AHP model under uncertainty.