Measuring Weak Consistency and Weak Transitivity of Pairwise Comparison Matrices

Under the assumption of rational economics, the opinions of decision makers should exhibit some transitivity properties. It is an important issue on how to measure the transitivity properties of the provided preference relations over a set of alternatives. In this study, we report the methods for measuring weak consistency (w-consistency) and weak transitivity (w-transitivity) of pairwise comparison matrices (PCMs) originating from the analytic hierarchy process (AHP). First, some interesting properties of PCMs with w-consistency and w-transitivity are studied. Second, novel methods are proposed to construct the quantification indices of w-consistency and w-transitivity of PCMs, respectively. Some comparisons with the existing methods are offered to illustrate the novelty of the proposed ones. Third, an optimization model is put forward to modify a PCM without any transitivity property to a new one with w-consistency and w-transitivity, respectively. The particle swarm optimization (PSO) algorithm is adopted to solve the nonlinear optimization problems. A novel decision-making model is established by considering the w-transitivity as the minimum requirement. Some numerical examples are carried out to illustrate the developed methods and models. It is observed that the proposed indices can be computed efficiently and reflect the inherent relations of the entries in a PCM with w-consistency and w-transitivity, respectively.