Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes
Abstract
:1. Introduction
2. Preliminaries
2.1. Several Types of NS
2.2. The Distance and Similarity Measures for SVNHFSs
- (1)
- ;
- (2)
- iff ;
- (3)
- ;
- (4)
- If , then , .
- (1)
- ;
- (2)
- iff ;
- (3)
- ;
- (4)
- If , then , .
- (1)
- if A is a crisp set;
- (2)
- iff ;
- (3)
- if A is more crisper than B;
- (4)
- , where is the complement of A.
3. The Distance and Similarity Measures of PSVNHFS
- (1)
- If the probability values are equal for the same type of hesitant membership function, i.e.,Then, the normal PNHFS is reduced to the SVNHFS.
- (2)
- If and , then the normal PNHFS reduces to the SVNS.
- (3)
- If (there is also ), , then the normal PNHFS reduces to the PDHFS, which can be expressed by .
- (4)
- If the normal PNHFS satisfies the conditions in (3), and , then the normal PNHFS reduces to the DHFS, denoted by
- (5)
- If (there is also ), then the normal PNHFS reduces to the PHFS, the mathematical symbol is .
- (6)
- If the normal PNHFS satisfies the conditions in (5), and , the normal PNHFS reduces to the HFS, denoted by .
- (7)
- If (there is also ), , ,, then the normal NHFS reduces to the IFS, denoted by .
- (8)
- If (there is also ), , , and , then the normal NHFS reduces to the FS.
3.1. The Method of Comparing PNHFSs
- (1)
- If , then ;
- (2)
- If , then ;
- (3)
- If , then (i) If , then ; (ii) If , then ;
- (4)
- If , then (i) If , then ; (ii) If , then .
- (1)
- If , then ;
- (2)
- If , then ;
- (3)
- If , then .
3.2. Distance and Similarity Measures of PNHFSs
- (1)
- iff ;
- (2)
- ;
- (3)
- , when .
- (1)
- iff ;
- (2)
- ;
- (3)
- If , then and .
- (1)
- iff ;
- (2)
- ;
- (3)
- If , then , .
- (1)
- iff ;
- (2)
- ;
- (3)
- If , then and .
3.3. The Interrelations among Distance, Similarity and Entropy Measures
- (1)
- if oror ;
- (2)
- if ;
- (3)
- iff holds the requirement that , in which is the complement of A.
- (4)
- when or , in which.
- Let , or , thus the corresponding ISs of A are shown:Next, the entropy measure of A is calculated as follows:
- Let , then the complementary of A is obtained: . By Definition 9, the following equality is obtained: . Obviously, .
- Suppose that B and C are two PNHFS of X, . Thus, the corresponding IS of A is . By Theorem 5, the following similarity measures can be obtained:
4. Method Analysis Based on Illustrations and Applications
4.1. Comparative Evaluations
4.2. Streamlining the Talent Selection Process
5. Conclusions and Future Research
Author Contributions
Funding
Conflicts of Interest
References
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Parameter | Ranking | ||||
---|---|---|---|---|---|
Method | Ranking | The Best Result | The Worst Result |
---|---|---|---|
Xu and Xia’s Method | |||
Singh’s Method | |||
Sahin’s Method |
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Shao, S.; Zhang, X. Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes. Mathematics 2019, 7, 649. https://doi.org/10.3390/math7070649
Shao S, Zhang X. Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes. Mathematics. 2019; 7(7):649. https://doi.org/10.3390/math7070649
Chicago/Turabian StyleShao, Songtao, and Xiaohong Zhang. 2019. "Measures of Probabilistic Neutrosophic Hesitant Fuzzy Sets and the Application in Reducing Unnecessary Evaluation Processes" Mathematics 7, no. 7: 649. https://doi.org/10.3390/math7070649