Group consistency and group decision making under uncertain probabilistic hesitant fuzzy preference environment

With regard to element uncertainty in a hesitant fuzzy element (HFE), the probabilistic hesitant fuzzy element (PHFE) was developed. It has been noted that the occurrence probabilities of elements in PHFE are difficult to obtain accurately and sufficiently through subjective evaluation. Therefore, in this study, we propose the uncertain probabilistic hesitant fuzzy element (UPHFE), a generalized fuzzy number, which includes four types of HFEs. Compared with other fuzzy numbers, subjective preference information can be described more properly by the UPHFE. Furthermore, the UPHFE is extended to the uncertain probabilistic hesitant fuzzy preference relations (UPHFPRs). Besides, investigations about the expected consistency, acceptable expected consistency, probability-obtaining approach, and consistency-improving iterative algorithm for reasonable application of UPHFPRs are illustrated respectively. Then, we introduce the UPHFPRs and these methods into a group decision-making process, for which two operators are proposed to aggregate the UPHFPRs and ensure that the aggregated preference relations can remain UPHFPRs. Because the aggregated UPHFPRs may be inconsistent, we further design an acceptable group consistency test. Subsequently, we summarize the group decision-making process under the UPHFPR environment. Finally, an example that selects the optimal objective from four newly listed stocks on the Growth Enterprises Market board in China is provided to demonstrate the proposed approaches.