Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods
Abstract
:1. Introduction
2. Some Concepts of LNNs and BM
2.1. Linguistic Neutrosophic Numbers and Their Operational Laws
- If E() > E(), then ;
- If E() = E() then
- If H() > H(), then ;
- If H() = H(), then ;
- If H() < H(), then .
2.2. Bonferroni Mean Operators
3. Two BM Aggregation Operators of LNNs
3.1. Normalized Weighted BM Operators of LNNs
- (1)
- ;
- (2)
- ;
- (3)
- (4)
- (5)
- (6)
3.2. Normalized Weighted Geometric BM Operators of LNNs
4. MAGDM Methods Based on the LNNNWBM or LNNNWGBM Operator
5. Illustrative Examples
5.1. The Decision-Making Process Based on the LNNNWBM Operator or LNNNWGBM Operator
5.2. Analysis the Influence of the Parameters p and q on Decision Results
6. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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C1 | C2 | C3 | |
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A1 | |||
A2 | |||
A3 | |||
A4 |
C1 | C2 | C3 | |
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A1 | |||
A2 | |||
A3 | |||
A4 |
C1 | C2 | C3 | |
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A1 | |||
A2 | |||
A3 | |||
A4 |
C1 | C2 | C3 | |
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p, q | LNNNWBM Operator | Ranking |
---|---|---|
p = 1, q = 0 | E(a1) = 0.7528, E(a2) = 0.7777, E(a3) = 0.7613, E(a4) = 0.8060 | |
p = 1, q = 0.5 | E(a1) = 0.7311, E(a2) = 0.7534, E(a3) = 0.7435, E(a4) = 0.7886 | |
p = 1, q = 2 | E(a1) = 0.7329, E(a2) = 0.7545, E(a3) = 0.7453, E(a4) = 0.7897 | |
p = 0, q = 0 | E(a1) = 0.7573, E(a2) = 0.7766, E(a3) = 0.7656, E(a4) = 0.8046 | |
p = 0.5, q = 1 | E(a1) = 0.7326, E(a2) = 0.7530, E(a3) = 0.7449, E(a4) = 0.7879 | |
p = 2, q = 1 | E(a1) = 0.7349, E(a2) = 0.7562, E(a3) = 0.7463, E(a4) = 0.7902 | |
p = 2, q = 2 | E(a1) = 0.7343, E(a2) = 0.7537, E(a3) = 0.7458, E(a4) = 0.7884 |
p, q | LNNNWGBM Operator | Ranking |
---|---|---|
p = 1, q = 0 | E(a1) = 0.7397, E(a2) = 0.7747, E(a3) = 0.7531, E(a4) = 0.8035 | |
p = 1, q = 0.5 | E(a1) = 0.7342, E(a2) = 0.7545, E(a3) = 0.7453, E(a4) = 0.7891 | |
p = 1, q = 2 | E(a1) = 0.7343, E(a2) = 0.7548, E(a3) = 0.7457, E(a4) = 0.7889 | |
p = 0, q = 1 | E(a1) = 0.7437, E(a2) = 0.7730, E(a3) = 0.7570, E(a4) = 0.8019 | |
p = 0.5, q = 1 | E(a1) = 0.7356, E(a2) = 0.7541, E(a3) = 0.7467, E(a4) = 0.7885 | |
p = 2, q = 1 | E(a1) = 0.7330, E(a2) = 0.7553, E(a3) = 0.7445, E(a4) = 0.7895 | |
p = 2, q = 2 | E(a1) = 0.7334, E(a2) = 0.7530, E(a3) = 0.7441, E(a4) = 0.7877 |
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Fan, C.; Ye, J.; Hu, K.; Fan, E. Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods. Information 2017, 8, 107. https://doi.org/10.3390/info8030107
Fan C, Ye J, Hu K, Fan E. Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods. Information. 2017; 8(3):107. https://doi.org/10.3390/info8030107
Chicago/Turabian StyleFan, Changxing, Jun Ye, Keli Hu, and En Fan. 2017. "Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods" Information 8, no. 3: 107. https://doi.org/10.3390/info8030107
APA StyleFan, C., Ye, J., Hu, K., & Fan, E. (2017). Bonferroni Mean Operators of Linguistic Neutrosophic Numbers and Their Multiple Attribute Group Decision-Making Methods. Information, 8(3), 107. https://doi.org/10.3390/info8030107