Consistency of distributed preference relations

Distributed preference relation (DPR) is a way to simultaneously characterize the preferred, nonpreferred, indifferent, and global ignorance degrees of one alternative over another. To make DPRs applicable in decision-making, the consistency of DPRs is addressed in this paper. Under the assumption that any two DPRs on a set of alternatives are independent of each other, four rules regarding the relationship between alternatives are designed to develop the transitivity of three DPRs. By considering eight cases of relationship between two grades from two DPRs, an abstract function is defined to combine the two grades. With the abstract function, the consistency of three DPRs is developed. Three consistent DPRs are theoretically proven to be transitive. A multiple criteria group decision-making (MCGDM) process is developed based on the proposed consistency of the DPRs to analyze group decision-making problems, in which DPRs between neighboring alternatives and those between a specific pair of nonadjacent alternatives are required to be provided. The problem of selecting the associate head of the ultrasonic department of a tertiary hospital is analyzed using the MCGDM process to demonstrate the real application of the consistency of DPRs.