**5. Conclusions**

In this study, the role of cardinality of HFE in distance measures on HFSs is analysed. Moreover, a series of parameterized distance measures on HFSs are proposed. The main innovation points of this study are as follows:


However, every method does have its limitations, and it is hoped that these novel distance measures could become perfect step-by-step in practice.

**Author Contributions:** Conceptualization, J.R.L.; F.W.Z. and Q.L.; Methodology, F.W.Z. and J.S.; Software, J.Y.; Validation, S.H.W. and S.J.X.; Formal Analysis, S.J.X. and J.S.; Writing—Review and Editing, S.J.X.; Supervision, F.W.Z.; Project Administration, F.W.Z.; Funding Acquisition, F.W.Z.

**Funding:** This research was funded by National Natural Science Foundation of China (51508319, 61374195, 51409157), the research program of the National Special Authorized Social Science Fund of China (07@ZH005), and the Nature and Science Fund from the Zhejiang Province Ministry of Education (Y201327642).

**Acknowledgments:** The authors are thankful to the editor Ms. Jocelyn He, and the anonymous reviewers for their insightful and constructive comments for improving this paper.

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