A Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Partitioned Heronian Mean Operator
Abstract
:1. Introduction
2. Preliminaries
2.1. Shapley Fuzzy Measure
- (1)
- and;
- (2)
- and, then.
2.2. PHM
- (1)
- Idempotency: If, then.
- (2)
- Permutability: Ifis a permutation of, then.
- (3)
- Boundedness: Ifand, then.
2.3. NSs and SVNSs
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- .
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- .
- (1)
- If, thenis preferable to, which is represented as;
- (2)
- Ifand, thenis preferable to, which is denoted by;
- (3)
- If,and, thenis preferable to, which is denoted by;
- (4)
- If,and, thenis indifferent to, which is represented by.
3. Single-Valued Neutrosophic PHM Operators
3.1. SVNPHM Operator
- (1)
- As , then Equation (7) reduces to:
- (2)
- When and , Equation (7) reduces to:
- (3)
- When , Equation (7) becomes:
3.2. WSVNSPHM Operator
- (1)
- As , Equation (12) reduces to:
- (2)
- When and , Equation (12) reduces to
- (3)
- When , Equation (12) becomes
4. Single-Valued Neutrosophic MCDM Method with Incomplete Weight Information
5. Example
5.1. Decision-Making Process
- Step 2. Compute closeness coefficients
5.2. Sensitivity Analysis
5.3. Comparison Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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<0.2,0.9,0.6> | <0.5,0.5,0.4> | <0.5,0.3,0.4> | <0.5,0.3,0.3> | <0.6,0.6,0.5> | |
<0.2,0.7,0.5> | <0.6,0.6,0.3> | <0.4,0.2,0.6> | <0.6,0.1,0.2> | <0.5,0.4,0.4> | |
<0.2,0.8,0.5> | <0.4,0.6,0.5> | <0.5,0.2,0.4> | <0.4,0.1,0.3> | <0.6,0.7,0.5> | |
<0.2,0.9,0.6> | <0.4,0.5,0.4> | <0.5,0.4,0.3> | <0.5,0.2,0.2> | <0.3,0.8,0.6> | |
<0.1,0.9,0.6> | <0.3,0.7,0.6> | <0.4,0.6,0.5> | <0.5,0.1,0.2> | <0.5,0.4,0.4> |
<0.6,0.1,0.2> | <0.4,0.5,0.5> | <0.5,0.3,0.4> | <0.5,0.3,0.3> | <0.5,0.4,0.6> | |
<0.5,0.3,0.2> | <0.3,0.4,0.6> | <0.4,0.2,0.6> | <0.6,0.1,0.2> | <0.4,0.6,0.5> | |
<0.5,0.2,0.2> | <0.5,0.4,0.4> | <0.5,0.2,0.4> | <0.4,0.1,0.3> | <0.5,0.3,0.6> | |
<0.6,0.1,0.2> | <0.4,0.5,0.4> | <0.5,0.4,0.3> | <0.5,0.2,0.2> | <0.6,0.2,0.3> | |
<0.6,0.1,0.1> | <0.6,0.3,0.3> | <0.4,0.6,0.5> | <0.5,0.1,0.2> | <0.4,0.6,0.5> |
0.6259 | 0.7101 | 0.3090 | 0.2743 | 0.7101 | |
0.8305 | 0.8334 | 1 | 0.4415 | 0 | |
0.5119 | 0.3660 | 0.6340 | 0.1791 | 0.5279 | |
0 | 0.5729 | 0.3090 | 0.3483 | 0.4495 | |
0.8305 | 0 | 0 | 0.8209 | 0.2899 |
Parameter | Score Value | Final Rank Order | ||||
---|---|---|---|---|---|---|
0.2210 | 0.1841 | 0.2218 | 0.2354 | 0.2802 | ||
0.2741 | 0.2288 | 0.2654 | 0.2823 | 0.3371 | ||
0.3333 | 0.2759 | 0.3120 | 0.3388 | 0.3939 | ||
0.3627 | 0.2986 | 0.3357 | 0.3679 | 0.4232 | ||
0.3796 | 0.3116 | 0.3497 | 0.3848 | 0.4410 | ||
0.3905 | 0.3200 | 0.3588 | 0.3957 | 0.4528 |
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Tian, C.; Peng, J.J.; Zhang, Z.Q.; Goh, M.; Wang, J.Q. A Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Partitioned Heronian Mean Operator. Mathematics 2020, 8, 1189. https://doi.org/10.3390/math8071189
Tian C, Peng JJ, Zhang ZQ, Goh M, Wang JQ. A Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Partitioned Heronian Mean Operator. Mathematics. 2020; 8(7):1189. https://doi.org/10.3390/math8071189
Chicago/Turabian StyleTian, Chao, Juan Juan Peng, Zhi Qiang Zhang, Mark Goh, and Jian Qiang Wang. 2020. "A Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Partitioned Heronian Mean Operator" Mathematics 8, no. 7: 1189. https://doi.org/10.3390/math8071189
APA StyleTian, C., Peng, J. J., Zhang, Z. Q., Goh, M., & Wang, J. Q. (2020). A Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Partitioned Heronian Mean Operator. Mathematics, 8(7), 1189. https://doi.org/10.3390/math8071189