**8. Conclusions**

In this manuscript, we have utilized the concept of PDHFS to handle the uncertainty in the data so as to capture the information with some more degree of freedom. For it, we have defined some new distance measures based on the size of two PDHFSs. Further, by focussing on the advantages of the aggregation operators into the decision-making process, we propose some series of weighted averaging and geometric aggregation operators by using Einstein norm operations. The major advantages of the proposed operators are that it considers the probability information to each dual hesitant membership degrees which give more information and help for the decision maker to take a decision more clearly. Further, since the decision makers are more sensitive to the loss and their bounded rationality, so there is a need for the probabilistic information into the analysis to solve the related MCDM problems. Also, its prominent characteristic is that it can consider the decision makers psychological behavior. The primary contribution of this paper is summarized as follows:


(4) Finally, the presented group decision-making approach is explained with the help of numerical example and an extensive comparative analysis has been conducted with the existing decision making theories [42,48,50] to show the advantages of the proposed approach.

Thus, we can conclude that the proposed notion about the PDHFSs is widely used in the different scenarios such as when a person provides the information about the fact that 'how much he/she sure about the uncertain information evaluated by him/her?'; in the situations, when the evaluators have no knowledge of the importance of their decision as well the considered criteria. Thus, the proposed concepts are efficaciously applicable to the situation under uncertainties and expected to have wide applications in complex DM problems. In the future, there is a scope of extending the proposed method to some different environment and its application in the various fields related to decision-theory [53–63].

**Author Contributions:** Conceptualization, H.G. and G.K.; Methodology, G.K.; Validation, H.G.; Formal analysis, H.G. and G.K.; Investigation, H.G. and G.K.; Writing-original draft preparation, H.G. and G.K.; Writing-review and editing, H.G.; Visualization, H.G.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors wish to thank the anonymous reviewers for their valuable suggestions.

**Conflicts of Interest:** The authors declare no conflict of interest.
