Calculating priority weights from interval-valued multiplicative hesitant fuzzy preference relations

To denote the reciprocally uncertain hesitancy of decision makers, this paper introduces a new type of fuzzy sets called interval-valued multiplicative hesitant fuzzy variables (IVMHFVs), which is an extension of multiplicative hesitant fuzzy variables. On the basis of IVMHFVs, interval-valued multiplicative hesitant fuzzy preference relations (IVMHFPRs) are introduced. To rank objects based on IVMHFPRs reasonably, the consistency of IVMHFPRs is studied and a consistency definition is presented. Using this concept, a 0–1 mixed programming model for judging the consistency of IVMHFPRs is built, and a consistency-based programming model for deriving consistent IVMHFPRs is constructed. When the offered IVMHFPRs are incomplete, a model for determining missing values is established. In the setting of group decision making, a consensus index is defined and an interactive method for improving the consensus is offered. After that, an approach to group decision making with inconsistent and incomplete IVMHFPRs is developed. Meanwhile, a practical example is provided to show the application of the new method.