Some Picture Fuzzy Dombi Heronian Mean Operators with Their Application to Multi-Attribute Decision-Making
Abstract
:1. Introduction
2. Preliminaries
2.1. Picture Fuzzy Sets
- 1
- ,
- 2
- ,
- 3
- ,
- 4
- .
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- .
2.2. Heronian Mean
3. The Picture Fuzzy Dombi Heronian Mean Operators
3.1. The Picture Fuzzy Dombi Heronian Mean (PFDHM) Operator
3.2. The Picture Fuzzy Dombi Weighted Heronian Mean (PFDWHM) Operator
3.3. The Picture Fuzzy Dombi Geometric Heronian Mean (PFDGHM) Operator
3.4. The Picture Fuzzy Dombi Weighted Geometric Heronian Mean (PFDWGHM) Operator
4. A Novel Approach to Multi-Attribute Decision-Making (MADM) Based on the Proposed Operators
4.1. Description of Atypical MADM Problem with Picture Fuzzy Information
4.2. An Algorithm for the Picture Fuzzy MADM Problem
5. Application Instance
5.1. The Decision-Making Process
5.2. Sensitivity Analysis
5.3. Comparative Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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G1 | G2 | G3 | G4 | |
---|---|---|---|---|
A1 | (0.53,0.33,0.09) | (0.89,0.08,0.03) | (0.42,0.35,0.18) | (0.08,0.89,0.02) |
A2 | (0.73,0.12,0.08) | (0.13,0.64,0.21) | (0.03,0.82,0.13) | (0.73,0.15,0.08) |
A3 | (0.91,0.03,0.02) | (0.07,0.09,0.05) | (0.04,0.85,0.10) | (0.68,0.26,0.06) |
A4 | (0.85,0.09,0.05) | (0.74,0.16,0.10) | (0.02,0.89,0.05) | (0.08,0.84,0.06) |
A5 | (0.90,0.05,0.02) | (0.68,0.08,0.21) | (0.05,0.87,0.06) | (0.13,0.75,0.09) |
Method | Ranking Results | |
---|---|---|
Wei’s [45] method based on picture fuzzy weighted average (PFWA) operator | ||
Wei’s [49] method based on picture fuzzy Hamacher weighted average operator (PFHWA) operator () | ||
The proposed method based on PFDWHM operator ) |
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Zhang, H.; Zhang, R.; Huang, H.; Wang, J. Some Picture Fuzzy Dombi Heronian Mean Operators with Their Application to Multi-Attribute Decision-Making. Symmetry 2018, 10, 593. https://doi.org/10.3390/sym10110593
Zhang H, Zhang R, Huang H, Wang J. Some Picture Fuzzy Dombi Heronian Mean Operators with Their Application to Multi-Attribute Decision-Making. Symmetry. 2018; 10(11):593. https://doi.org/10.3390/sym10110593
Chicago/Turabian StyleZhang, Hongran, Runtong Zhang, Huiqun Huang, and Jun Wang. 2018. "Some Picture Fuzzy Dombi Heronian Mean Operators with Their Application to Multi-Attribute Decision-Making" Symmetry 10, no. 11: 593. https://doi.org/10.3390/sym10110593
APA StyleZhang, H., Zhang, R., Huang, H., & Wang, J. (2018). Some Picture Fuzzy Dombi Heronian Mean Operators with Their Application to Multi-Attribute Decision-Making. Symmetry, 10(11), 593. https://doi.org/10.3390/sym10110593