A Novel Approach to Multi-Attribute Group Decision-Making based on Interval-Valued Intuitionistic Fuzzy Power Muirhead Mean
Abstract
:1. Introduction
2. Preliminaries
2.1. The Power Average and Muirhead Mean Operators
2.2. Interval-Valued Intuitionistic Fuzzy Sets
- If, then;
- If, thenif , then ;if , then .
3. Power Muirhead Mean Operators for Interval-Valued Intuitionistic Fuzzy Sets
3.1. The Interval-Valued Intuitionistic Fuzzy Power Muirhead Mean (IVIFPMM) Operator
- (1)
- ;
- (2)
- ;
- (3)
- , if.
3.2. The Interval-Valued Intuitionistic Fuzzy Weighted Power Muirhead Mean (IVIFWPMM) Operator
4. A Method to MAGDM in the Interval-Valued Intuitionistic Fuzzy Context
5. Case Analysis
5.1. The Decision-Making Process
5.2. Sensitivity Analysis
5.3. Comparison Analysis
6. Conclusion Remarks
Author Contributions
Funding
Conflicts of Interest
References
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G1 | G2 | G3 | G4 | |
---|---|---|---|---|
x1 | ([0.5, 0.6],[0.3, 0.4]) | ([0.4, 0.6],[0.2, 0.3]) | ([0.6, 0.8],[0.1, 0.2]) | ([0.6, 0.7],[0.1, 0.3]) |
x2 | ([0.7, 0.8],[0.1, 0.2]) | ([0.5, 0.6],[0.1, 0.2]) | ([0.3, 0.5],[0.3, 0.4]) | ([0.7, 0.8],[0.1, 0.2]) |
x3 | ([0.4, 0.6],[0.2, 0.3]) | ([0.5, 0.7],[0.1, 0.2]) | ([0.4, 0.6],[0.3, 0.4]) | ([0.5, 0.6],[0.2, 0.3]) |
x4 | ([0.6, 0.7],[0.1, 0.3]) | ([0.5, 0.6],[0.2, 0.3]) | ([0.4, 0.5],[0.2, 0.4]) | ([0.4, 0.7],[0.2, 0.3]) |
G1 | G2 | G3 | G4 | |
---|---|---|---|---|
x1 | ([0.7, 0.8],[0.1, 0.2]) | ([0.5, 0.6],[0.1, 0.3]) | ([0.4, 0.5],[0.2, 0.4]) | ([0.5, 0.8],[0.1, 0.2]) |
x2 | ([0.5, 0.6],[0.2, 0.3]) | ([0.6, 0.7],[0.2, 0.3]) | ([0.5, 0.5],[0.2, 0.3]) | ([0.6, 0.7],[0.1, 0.2]) |
x3 | ([0.4, 0.5],[0.1, 0.2]) | ([0.6, 0.8],[0.1, 0.2]) | ([0.5, 0.7],[0.2, 0.3]) | ([0.6, 0.7],[0.1, 0.3]) |
x4 | ([0.5, 0.6],[0.2, 0.3]) | ([0.4, 0.5],[0.3, 0.4]) | ([0.6, 0.8],[0.1, 0.2]) | ([0.5, 0.8],[0.1, 0.2]) |
G1 | G2 | G3 | G4 | |
---|---|---|---|---|
x1 | ([0.6, 0.6],[0.2, 0.3]) | ([0.5, 0.8],[0.1, 0.2]) | ([0.5, 0.7],[0.1, 0.2]) | ([0.6, 0.7],[0.2, 0.3]) |
x2 | ([0.7, 0.8],[0.1, 0.2]) | ([0.4, 0.5],[0.3,0.4]) | ([0.6, 0.7],[0.1, 0.2]) | ([0.5, 0.6],[0.2, 0.3]) |
x3 | ([0.6, 0.6],[0.2, 0.3]) | ([0.5, 0.7],[0.2, 0.3]) | ([0.6, 0.8],[0.1, 0.2]) | ([0.5, 0.6],[0.3, 0.4]) |
x4 | ([0.4, 0.5],[0.3, 0.4]) | ([0.6, 0.8],[0.1, 0.2]) | ([0.5, 0.6],[0.2, 0.3]) | ([0.7, 0.8],[0.1, 0.2]) |
G1 | G2 | |
x1 | ([0.6217, 0.7057],[0.1676, 0.2749]) | ([0.4701, 0.6581],[0.1247, 0.2737]) |
x2 | ([0.6250, 0.7293],[0.1354, 0.2387]) | ([0.5289, 0.6305],[0.1755, 0.2811]) |
x3 | ([0.4533, 0.5582],[0.1469, 0.2504]) | ([0.5475, 0.7498],[0.1172, 0.2194]) |
x4 | ([0.5146, 0.6160],[0.1760, 0.3201]) | ([0.4843, 0.6221],[0.2049, 0.3112]) |
G1 | G2 | |
x1 | ([0.4961, 0.6708],[0.1356, 0.2711]) | ([0.5583, 0.7495],[0.1174, 0.2505]) |
x2 | ([0.4712, 0.5542],[0.1945, 0.2999]) | ([0.6163, 0.7187],[0.1169, 0.2191]) |
x3 | ([0.4965, 0.7004],[0.1942, 0.2996]) | ([0.5473, 0.6481],[0.1612, 0.3205]) |
x4 | ([0.5198, 0.6838],[0.1472, 0.2749]) | ([0.5291, 0.7729],[0.1243, 0.2271]) |
S | Ranking Orders | |
---|---|---|
S = (1, 0, 0, 0) | ||
S = (1, 1, 0, 0) | ||
S = (1, 1, 1, 0) | ||
S = (1, 1, 1, 1) |
G1 | G2 | G3 | G4 | |
---|---|---|---|---|
x1 | ([0.4, 0.5],[0.3, 0.4]) | ([0.4, 0.6],[0.2, 0.4]) | ([0.1, 0.3],[0.5, 0.6]) | ([0.3, 0.4],[0.3, 0.5]) |
x2 | ([0.6, 0.7],[0.2, 0.3]) | ([0.6, 0.7],[0.2, 0.3]) | ([0.4, 0.7],[0.1, 0.2]) | ([0.5, 0.6],[0.1, 0.3]) |
x3 | ([0.3, 0.6],[0.3, 0.4]) | ([0.5, 0.6],[0.3, 0.4]) | ([0.5, 0.6],[0.1, 0.3]) | ([0.4, 0.5],[0.2, 0.4]) |
x4 | ([0.7, 0.8],[0.1, 0.2]) | ([0.6, 0.7],[0.1, 0.3]) | ([0.3, 0.4],[0.1, 0.2]) | ([0.3, 0.7],[0.1, 0.2]) |
x5 | ([0.3, 0.4],[0.2, 0.3]) | ([0.3, 0.5],[0.1, 0.3]) | ([0.2, 0.5],[0.4, 0.5]) | ([0.3, 0.4],[0.5, 0.6]) |
Method | Ranking Orders | |
---|---|---|
Method introduced by Xu [22] | ||
Method given by He et al. [29] () | ||
Method presented by Xu and Chen [23] (s = t = 1) | ||
Method put forward by Yu and Wu [24] (p = q =1) | ||
Method proposed by Sun and Xia [25] (k = 2) | ||
Method developed by Liu and Li [26] (s = t = 1) | ||
Method raised by Liu [27] (s = t =1) | ||
Method proposed by Liu et al. [28] (k = 2) | ||
The proposed method based on the IVIFWPMM operator S = (0.5,0.5,0.5,0.5) |
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Xu, W.; Shang, X.; Wang, J.; Li, W. A Novel Approach to Multi-Attribute Group Decision-Making based on Interval-Valued Intuitionistic Fuzzy Power Muirhead Mean. Symmetry 2019, 11, 441. https://doi.org/10.3390/sym11030441
Xu W, Shang X, Wang J, Li W. A Novel Approach to Multi-Attribute Group Decision-Making based on Interval-Valued Intuitionistic Fuzzy Power Muirhead Mean. Symmetry. 2019; 11(3):441. https://doi.org/10.3390/sym11030441
Chicago/Turabian StyleXu, Wuhuan, Xiaopu Shang, Jun Wang, and Weizi Li. 2019. "A Novel Approach to Multi-Attribute Group Decision-Making based on Interval-Valued Intuitionistic Fuzzy Power Muirhead Mean" Symmetry 11, no. 3: 441. https://doi.org/10.3390/sym11030441