Hesitant Fuzzy Monotonic Dependent OWA Operator and Its Application in Symmetric Group Decision-Making
Abstract
:1. Introduction
2. Preliminaries
2.1. Hesitant Fuzzy Sets
- (1)
- If , then ;
- (2)
- If , then .
- (1)
- .
- (2)
- .
- (3)
- (4)
- Given k HFNs , then we have [40]:
- (5)
- (6)
2.2. Dependent OWA Operators
- (1)
- is continuous with respect to .
- (2)
- is monotonic decreasing on .
- (3)
- if .
- (2’)
- is monotonic increasing on .
- (3’)
- if .
3. Hesitant Fuzzy Monotonic Dependent OWA Operators
3.1. Hesitant Fuzzy Monotonic Dependent OWA Operators with Identical Variable Weight Vectors
3.2. Hesitant Fuzzy Monotonic Dependent OWA Operators with Different Variable Weight Vectors
4. Hesitant Fuzzy Hybrid Monotonic Dependent OWA Operators
4.1. The First Class of Hesitant Fuzzy Hybrid Monotonic Dependent OWA Operators
4.2. The Second Class of Hesitant Fuzzy Hybrid Monotonic Dependent OWA Operators
5. Algorithm of Group Decision Making Based on HFMDOWA Operators
6. Numerical Example
6.1. Description of the Decision Problem
6.2. Evaluation Process
6.3. Comparison with HFWA Operator Developed by Xia and Xu [10]
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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0.8 | 0.6 | 0.9 | 0.7 | 0.6 | 0.8 | 0.7 | 0.9 | 0.8 | 0.7 | 0.6 | |
0.7 | 0.9 | 0.6 | 0.8 | 0.7 | 0.5 | 0.6 | 0.9 | 0.4 | 0.7 | 0.6 | |
0.9 | 0.6 | 0.5 | 0.8 | 0.7 | 0.4 | 0.6 | 0.5 | 0.7 | 0.8 | 0.9 | |
0.8 | 0.5 | 0.6 | 0.9 | 0.7 | 0.9 | 0.8 | 0.4 | 0.6 | 0.7 | 0.5 | |
0.6 | 0.8 | 0.5 | 0.9 | 0.9 | 0.8 | 0.6 | 0.5 | 0.7 | 0.5 | 0.8 |
0.8 | 0.5 | 0.7 | 0.8 | 0.9 | 0.4 | 0.8 | 0.6 | 0.7 | 0.9 | 0.5 | |
0.9 | 0.8 | 0.9 | 0.5 | 0.4 | 0.6 | 0.7 | 0.5 | 0.9 | 0.8 | 0.7 | |
0.8 | 0.6 | 0.5 | 0.7 | 0.9 | 0.6 | 0.8 | 0.7 | 0.8 | 0.6 | 0.7 | |
0.9 | 0.6 | 0.5 | 0.7 | 0.8 | 0.6 | 0.5 | 0.5 | 0.8 | 0.6 | 0.8 | |
0.7 | 0.8 | 0.6 | 0.5 | 0.9 | 0.7 | 0.6 | 0.8 | 0.7 | 0.8 | 0.6 |
0.8 | 0.6 | 0.9 | 0.7 | 0.8 | 0.6 | 0.5 | 0.9 | 0.6 | 0.8 | 0.7 | |
0.8 | 0.7 | 0.6 | 0.6 | 0.8 | 0.9 | 0.7 | 0.6 | 0.8 | 0.6 | 0.8 | |
0.9 | 0.5 | 0.6 | 0.8 | 0.7 | 0.6 | 0.8 | 0.9 | 0.5 | 0.7 | 0.6 | |
0.8 | 0.6 | 0.9 | 0.7 | 0.7 | 0.8 | 0.6 | 0.5 | 0.8 | 0.6 | 0.7 | |
0.8 | 0.6 | 0.9 | 0.7 | 0.8 | 0.6 | 0.8 | 0.9 | 0.6 | 0.7 | 0.6 |
0.9 | 0.8 | 0.6 | 0.7 | 0.9 | 0.6 | 0.5 | 0.8 | 0.7 | 0.9 | 0.7 | |
0.8 | 0.5 | 0.7 | 0.6 | 0.9 | 0.8 | 0.5 | 0.7 | 0.9 | 0.5 | 0.8 | |
0.8 | 0.6 | 0.7 | 0.5 | 0.9 | 0.5 | 0.7 | 0.6 | 0.9 | 0.8 | 0.7 | |
0.9 | 0.5 | 0.7 | 0.6 | 0.8 | 0.7 | 0.9 | 0.5 | 0.7 | 0.8 | 0.6 | |
0.9 | 0.6 | 0.8 | 0.7 | 0.5 | 0.6 | 0.8 | 0.9 | 0.7 | 0.6 | 0.8 |
0.1772 | 0.1472 | 0.1356 | 0.1310 | 0.1473 | 0.0954 | 0.0566 | 0.1473 | 0.0602 | 0.1559 | 0.0458 | |
0.1559 | 0.1926 | 0.1356 | 0.1011 | 0.1294 | 0.1273 | 0.0458 | 0.1184 | 0.0704 | 0.1042 | 0.0602 | |
0.1772 | 0.1135 | 0.0893 | 0.1097 | 0.1595 | 0.0681 | 0.0602 | 0.1184 | 0.0696 | 0.1166 | 0.0706 | |
0.1772 | 0.1135 | 0.1184 | 0.1356 | 0.1310 | 0.1389 | 0.0662 | 0.0584 | 0.0602 | 0.1166 | 0.0537 | |
0.1389 | 0.1714 | 0.1273 | 0.1287 | 0.1405 | 0.1166 | 0.0610 | 0.1405 | 0.0516 | 0.1042 | 0.0610 |
Ranking Results | ||||||
---|---|---|---|---|---|---|
0.10904 | 0.10554 | 0.09906 | 0.09939 | 0.10648 | ||
0.11100 | 0.10721 | 0.10025 | 0.10088 | 0.10806 | ||
0.11299 | 0.10888 | 0.10149 | 0.10239 | 0.10949 | ||
0.11499 | 0.11057 | 0.10275 | 0.10390 | 0.11098 | ||
0.11709 | 0.11227 | 0.10405 | 0.10542 | 0.11243 | ||
0.11896 | 0.11398 | 0.10537 | 0.10694 | 0.11396 | ||
0.11956 | 0.11449 | 0.10578 | 0.10740 | 0.11424 | ||
0.11968 | 0.11467 | 0.10591 | 0.10755 | 0.11442 |
Ranking Results | ||||||
---|---|---|---|---|---|---|
0.10350 | 0.09756 | 0.09508 | 0.09531 | 0.09945 | ||
0.10634 | 0.10036 | 0.09690 | 0.09740 | 0.10198 | ||
0.10926 | 0.10331 | 0.09880 | 0.09957 | 0.10462 | ||
0.11225 | 0.10641 | 0.10078 | 0.10181 | 0.10736 | ||
0.11528 | 0.10967 | 0.10283 | 0.10412 | 0.11018 | ||
0.11836 | 0.11308 | 0.10496 | 0.10650 | 0.11309 | ||
0.11929 | 0.11413 | 0.10561 | 0.10722 | 0.11398 | ||
0.11960 | 0.11448 | 0.10583 | 0.10746 | 0.11427 |
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Li, D.; Bian, H.; Wang, H.; Ma, R.; Zeng, W. Hesitant Fuzzy Monotonic Dependent OWA Operator and Its Application in Symmetric Group Decision-Making. Symmetry 2024, 16, 1450. https://doi.org/10.3390/sym16111450
Li D, Bian H, Wang H, Ma R, Zeng W. Hesitant Fuzzy Monotonic Dependent OWA Operator and Its Application in Symmetric Group Decision-Making. Symmetry. 2024; 16(11):1450. https://doi.org/10.3390/sym16111450
Chicago/Turabian StyleLi, Deqing, Hongya Bian, Hongjian Wang, Rong Ma, and Wenyi Zeng. 2024. "Hesitant Fuzzy Monotonic Dependent OWA Operator and Its Application in Symmetric Group Decision-Making" Symmetry 16, no. 11: 1450. https://doi.org/10.3390/sym16111450
APA StyleLi, D., Bian, H., Wang, H., Ma, R., & Zeng, W. (2024). Hesitant Fuzzy Monotonic Dependent OWA Operator and Its Application in Symmetric Group Decision-Making. Symmetry, 16(11), 1450. https://doi.org/10.3390/sym16111450