This paper presents a novel framework for the study of hesitant fuzzy rough sets by integrating rough sets with hesitant fuzzy sets. Lower and upper approximations of hesitant fuzzy sets with respect to a hesitant fuzzy approximation space are first defined. Properties of hesitant fuzzy approximation operators are examined. Relationships between hesitant fuzzy approximation spaces and hesitant fuzzy topological spaces are then established. It is proved that the set of all lower approximation sets based on a hesitant fuzzy reflexive and transitive approximation space forms a hesitant fuzzy topology. And conversely, for a hesitant fuzzy rough topological space, there exists a hesitant fuzzy reflexive and transitive approximation space such that the topology in the hesitant fuzzy rough topological space is exactly the set of all lower approximation sets in the hesitant fuzzy reflexive and transitive approximation space. That is to say, there exists a one-to-one correspondence between the set of all hesitant fuzzy reflexive and transitive approximation spaces and the set of all hesitant fuzzy rough topological spaces.
KeywordsApproximation operators Hesitant fuzzy rough sets Hesitant fuzzy sets Hesitant fuzzy topological spaces Rough sets
The authors would like to thank the anonymous referees for their valuable comments and suggestions.
This study is funded by the National Natural Science Foundation of China (Nos. 11461082, 11601474), the Natural Science Foundation of Gansu Province (No. 17JR5RA284), the Research Project Funds for Higher Education Institutions of Gansu Province (No. 2016B-005), the Fundamental Research Funds for the Central Universities of Northwest MinZu University (No. 31920190055) and the first-class discipline program of Northwest Minzu University.
Compliance with ethical standards
Conflict of interest
The author declares that there is no conflict of interests.
This article does not contain any studies with human participants or animals performed by any of the authors.
- Chakrabarty K, Gedeon T, Koczy L (1998) Intuitionistic fuzzy rough set. In: Proceedings of fourth joint conference on information sciences (JCIS), Durham, NC, pp 211–214Google Scholar
- Cornelis C, Cock MD, Kerre EE (2003) Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge. Expert Syst Appl 20:260–270Google Scholar
- Liang DC, Liu D (2015) A novel risk decision-making based on decision-theoretic rough sets under hesitant fuzzy information. IEEE Trans Fuzzy Syst 23:237–247Google Scholar
- Liu HF, Xu ZS, Liao HC (2016) The multiplicative consistency index of hesitant fuzzy preference relation. IEEE Trans Fuzzy Syst 24:82–93Google Scholar
- Rizvi S, Naqvi HJ, Nadeem D (2002) Rough intuitionistic fuzzy set. In: Proceedings of the sixth joint conference on information sciences (JCIS), Durham, NC, pp 101–104Google Scholar
- Thiele H (2001) On aximatic characterisation of fuzzy approximation operators: II. The rough fuzzy set based case. In: Proceeding of the 31st IEEE international symposium on multiple-valued logic, pp 330–335Google Scholar
- Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy systems, Jeju Island, Korea, pp 1378–1382Google Scholar
- Wu QE, Wang T, Huang YX et al (2008) Topology theory on rough sets. IEEE Trans Syst Man Cybern Part B Cybern 38:68–77Google Scholar
- Yeung DS, Chen DG, Tsang ECC et al (2005) On the generalization of fuzzy rough sets. IEEE Trans Fuzzy Syst 13:343–361Google Scholar
- Zhan JM, Liu Q, Herawan T (2017) A novel soft rough set: soft rough hemirings and its multicriteria group decision making. Appl Soft Comput 54:393–402Google Scholar