Deriving a ranking from hesitant fuzzy preference relations under group decision making

In this paper, we explore the ranking methods with hesitant fuzzy preference relations (HFPRs) in the group decision making environments. As basic elements of hesitant fuzzy sets, hesitant fuzzy elements (HFEs) usually have different numbers of possible values. In order to compute or compare HFEs, we have two principles to normalize them, i.e., the α-normalization and the β-normalization. Based on the α-normalization, we develop a new hesitant goal programming model to derive priorities from HFPRs. On the basis of the β-normalization, we develop the consistency measures of HFPRs, establish the consistency thresholds to measure whether or not an HFPR is of acceptable consistency, and then use the hesitant aggregation operators to aggregate preferences in HFPRs to obtain the ranking results.