A new method for group decision making with hesitant fuzzy preference relations based on multiplicative consistency

This paper develops a new method for group decision making with hesitant fuzzy preference relations (HFPRs) considering the multiplicative consistency and consensus simultaneously. A consistency index of HFPR is introduced and the acceptable consistent HFPR is defined. For improving the unacceptable consistent HFPR, a mathematical program is constructed to derive an acceptable consistent HFPR. Thus, an algorithm for checking and improving consistency of HFPR is proposed. A consensus index for the group is defined to measure the agreement among decision makers (DMs). To improve the consistency and consensus simultaneously, a new goal program is established to obtain a group of HFPRs with acceptable consistency and consensus. Subsequently, an interval-valued hesitant fuzzy group decision matrix (IVHFGDM) is elicited from the individual HFPRs. A positive ideal matrix, a left negative ideal matrix, and a right negative matrix are derived from the IVHFGDM. According to the relative closeness degrees, DMs' weights are determined objectively. Afterward, the individual HFPRs are integrated into a normalized collective HFPR. It is proven that the normalized collective HFPR is acceptable consistent if all individual HFPRs are acceptable consistent. The final ranking is generated by the collective overall values of alternatives. Some examples of fuzzy decision-making applications are provided to validate effectiveness of the method.