Einstein Aggregation Operators under Bipolar Neutrosophic Environment with Applications in Multi-Criteria Decision-Making
Abstract
:1. Introduction
- To suggest different bipolar neutrosophic Einstein AOs as well as desired properties to study;
- Based on BNN, establish a multi-criteria DM approach in the direction of real life problem-solving;
- Give a numerical description of amulti-criteria DM example.
2. Preliminaries
- i.
- In conditionthenis greater than, denoted by;
- ii.
- In conditionand aa, thenis greater than, denoted by;
- iii.
- In condition, aaandin that case,is superior to, denoted by;
- iv.
- In condition, aaandin that case,is equal to, denoted by;
- i.
- Then, if and only if
- ii.
- if and only if
- iii.
- The union is defined as below:
- iv.
- The intersection is defined as:
- v.
- Let be BNSs.
- i.
- In conditionthenis greater than, denoted by
- ii.
- In conditionandthenis greater than, denoted by
- iii.
- In conditionandin that case,is superior to, denote by
- iv.
- In conditionandin that case,is equal to, denoted by
3. Bipolar Neutrosophic Einstein Average AOs
3.1. Bipolar NeutrosophicEinstein Weighted-Average Aggregation Operators
3.2. BN Einstein OrderedWeighted Average Aggregation Operators
3.3. BN Einstein Hybrid Average Aggregation Operators
4. Bipolar Neutrosophic Einstein Geometric AOs
4.1. Bipolar Neutrosophic EinsteinWeighted Geometric AO
4.2. BN Einstein OrderedWeighted Geometric Aggregation Operators
4.3. BN Einstein HybridGeometric Aggregation Operators
5. Multi-Criteria Group DM Problem Based on BN Einstein Aggregation Operators
5.1. Algorithm
5.2. Illustrative Example
6. Comparison
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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L1 | L2 | L3 | L4 | |
---|---|---|---|---|
G1 | (0.2,0.6,0.5,−0.3,−0.9,−0.6) | (0.4,0.6,0.7,−0.5,−0.4,−0.3) | (0.8,0.3,0.5,−0.6,−0.1,−0.9) | (0.5,0.4,0.6,−0.8,−0.5,−0.4) |
G2 | (0.5,0.7,0.3,−0.8,−0.5,−0.7) | (0.3,0.4,0.6,−0.8,−0.7,−0.6) | (0.4,0.6,0.9,−0.5,−0.4,−0.5) | (0.3,0.8,0.9,−0.1,−0.4−0.3) |
G3 | (0.5,0.7,0.8,−0.4,−0.7,−0.4) | (0.3,0.7,0.7,−0.5,−0.3,−0.2) | (0.5,0.3,0.4,−0.6,−0.7,−0.9) | (0.4,0.6,0.5,−0.3,−0.5,−0.8) |
G4 | (0.8,0.5,0.4,−0.7,−0.6,−0.5) | (0.2,0.4,0.5,−0.8,−0.6,−0.3) | (0.3,0.7,0.4,−0.5,−0.7,−0.5) | (0.9,0.4,0.6,−0.5,−0.4,−0.7) |
L1 | L2 | L3 | L4 | |
---|---|---|---|---|
G1 | (0.4,0.6,0.5,−0.7,−0.4,−0.8) | (0.5,0.4,0.6,−0.8,−0.5,−0.7) | (0.4,0.6,0.5,−0.4,−0.8,−0.5) | (0.5,0.6,0.3,−0.4,−0.6,−0.8) |
G2 | (0.4,0.7,0.5,−0.6,−0.3,−0.9) | (0.4,0.7,0.8,−0.3,−0.5,−0.4) | (0.2,0.5,0.7,−0.6,−0.5,−0.4) | (0.8,0.4,0.2,−0.8,−0.1,−0.4) |
G3 | (0.6,0.3,0.6,−0.3,−0.7,−0.8) | (0.6,0.4,0.6,−0.7,−0.5,−0.8) | (0.6,0.3,0.2,−0.1,−0.4,−0.7) | (0.5,0.6,0.7,−0.3,−0.5,−0.6) |
G4 | (0.2,0.3,0.4,−0.7,−0.6,−0.8) | (0.8,0.5,0.4,−0.7,−0.4,−0.6) | (0.8,0.4,0.5,−0.7,−0.5,−0.1) | (0.6,0.5,0.8,−0.7,−0.6,−0.4) |
L1 | L2 | L3 | L4 | |
---|---|---|---|---|
G1 | (0.5,0.6,0.4,−0.7,−0.4,−0.3) | (0.2,0.5,0.6,−0.4,−0.7,−05) | (0.5,0.7,0.2,−0.9,−0.5,−0.3) | (0.5,0.6,0.3,−0.7,−0.5,−0.2) |
G2 | (0.9,0.2,0.4,−0.5,−0.4,−0.8) | (0.5,0.1,0.2,−0.9,−0.6,−0.4) | (0.1,0.4,0.8,−0.6,−0.5,−0.3) | (0.6,0.5,0.3,−0.9,−0.3,−0.5) |
G3 | (0.4,0.5,0.6,−0.1,−0.6,−0.5) | (0.3,0.4,0.8,−0.5,−0.4,−0.3) | (0.4,0.6,0.3,−0.4,−0.5,−0.3) | (0.5,0.4,0.9,−0.5,−0.4,−0.7) |
G4 | (0.1,0.4,0.5,−0.4,−0.8,−0.7) | (0.4,0.3,0.6,−0.2,−0.7,−0.5) | (0.7,0.5,0.6,−0.4,−0.3−0.9) | (0.1,0.5,0.7,−0.5,−0.8,−0.3) |
L1 | L2 | |
---|---|---|
G1 | (0.4234,0.6000,0.4581,−0.6501,−0.4842,−0.6306) | (0.3783,0.4568,0.6096,−0.5917,−0.5810,−0.5943) |
G2 | (0.6940,0.4453,0.4360,−0.5766,−0.3620,−0.8517) | (0.4219,0.3288,0.4750,−0.5426,−0.5640,−0.4224) |
G3 | (0.5161,0.4057,0.6187,−0.2024,−0.6627,−0.6704) | (0.4632,0.4251,0.6870,−0.5952,−0.4423,−0.6001) |
G4 | (0.2462,0.3555,0.4381,−0.5675,−0.6938,−0.7403) | (0.6286,0.4014,0.4839,−0.4527,−0.5567,−0.5351) |
L3 | L4 | |
G1 | (0.4940,0.6011,0.3536,−0.5954,−0.6522,−0.4973) | (0.5000,0.5776,0.3233,−0.5453,−0.5520,−0.5868) |
G2 | (0.1818,0.4670,0.7589,−0.5895,−0.4905,−0.3718) | (0.6950,0.4726,0.2807,−0.7190,−0.2130,−0.4321) |
G3 | (0.5161,0.4021,0.2533,−0.2160,−0.4764,−0.6073) | (0.4905,0.5135,0.7552,−0.3708,−0.4614,−0.6659) |
G4 | (0.7293,0.4649,0.5274,−0.5482,−0.4504,−0.6005) | (0.4884,0.4893,0.7387,−0.5952,−0.6796,−0.3989) |
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Jamil, M.; Afzal, F.; Akgül, A.; Abdullah, S.; Maqbool, A.; Razzaque, A.; Riaz, M.B.; Awrejcewicz, J. Einstein Aggregation Operators under Bipolar Neutrosophic Environment with Applications in Multi-Criteria Decision-Making. Appl. Sci. 2022, 12, 10045. https://doi.org/10.3390/app121910045
Jamil M, Afzal F, Akgül A, Abdullah S, Maqbool A, Razzaque A, Riaz MB, Awrejcewicz J. Einstein Aggregation Operators under Bipolar Neutrosophic Environment with Applications in Multi-Criteria Decision-Making. Applied Sciences. 2022; 12(19):10045. https://doi.org/10.3390/app121910045
Chicago/Turabian StyleJamil, Muhammad, Farkhanda Afzal, Ali Akgül, Saleem Abdullah, Ayesha Maqbool, Abdul Razzaque, Muhammad Bilal Riaz, and Jan Awrejcewicz. 2022. "Einstein Aggregation Operators under Bipolar Neutrosophic Environment with Applications in Multi-Criteria Decision-Making" Applied Sciences 12, no. 19: 10045. https://doi.org/10.3390/app121910045
APA StyleJamil, M., Afzal, F., Akgül, A., Abdullah, S., Maqbool, A., Razzaque, A., Riaz, M. B., & Awrejcewicz, J. (2022). Einstein Aggregation Operators under Bipolar Neutrosophic Environment with Applications in Multi-Criteria Decision-Making. Applied Sciences, 12(19), 10045. https://doi.org/10.3390/app121910045