Decision making with a sequential modeling of pairwise comparison process

The Analytic Hierarchy Process (AHP) is considered to be the natural way to realize decision making with pairwise comparisons of alternatives. Within the framework of the AHP model, the full process of forming comparison matrices is considered as an indivisible whole; and the irrational and uncertain behavior of decision makers has been identified and highlighted. In practice, the process of comparing alternatives is complex, especially when the number of alternatives is large. In this study, we propose a sequential modeling of wise comparison process for alternatives, which subsumes the typical AHP model as a special case. A new mathematical framework based on the leading principal submatrices is formed. The process of quantifying priorities of alternatives and the inconsistency-degree process of judgements are proposed to capture the irrationality and uncertainty experienced by decision makers in pairwise comparisons. It is found that the rank preservation is natural according to the comparison process. The acceptable consistency of multiplicative reciprocal matrices is redefined under the principle of weak rationality. A feedback mechanism is proposed to help decision makers avoid irrational behavior and to produce a better outcome of the decision procedure. Some comparisons with the AHP model demonstrate that the proposed model offers a clear way to capture the irrational and uncertain behavior of decision makers when evaluating the importance of alternatives.