A multi-criteria group decision-making method based on consistency of interval-valued distributed preference relations

In multi-criteria group decision making (MCGDM) with interval-valued distributed preference relations (IDPRs), the consistency of IDPRs is important to guarantee the rationality of IDPRs and the rationality of group-satisfactory solutions. In existing studies, the consistency of the score intervals of IDPRs was considered to ensure the consistency of IDPRs. Meanwhile, a comparison of the score intervals of IDPRs was considered to approximately compare the IDPRs. The transformation from IDPRs into score intervals involved in the consistency and comparison of IDPRs results in information loss and unexpected group solutions. To avoid the influence of information loss on the group solutions, this study proposes an MCGDM method based on the consistency of IDPRs. Six transitive rules were constructed and used to define the distributed transitivity of IDPRs, which differs from existing types of consistency because IDPRs are on a set of discrete grades. With the grade scores, an abstract function with two variables was defined, in which ten conditions were satisfied. Using an abstract function with the scores of the two grades as two parameters, the mechanism of combining the two grades in the two IDPRs was designed by following the principles of mutual independence and probabilistic operation. The consistency of IDPRs is then constructed based on the combination mechanism, which is theoretically proven to satisfy distributed transitivity. In addition, the mean of the odd-order origin moment intervals of an IDPR was used to compare the alternatives. Information loss was avoided from two perspectives. The ultrasonic machine supplier selection problem of a tertiary hospital located in Hefei, Anhui, China was solved using the proposed method to demonstrate its applicability and effectiveness.