An Approach to the Total Least Squares Method for Symmetric Triangular Fuzzy Numbers
Abstract
:1. Introduction
2. Preliminaries—Materials and Methods
3. On Distance from a Triangular Fuzzy Number to a Certain Set of Fuzzy Numbers
4. Total Least Squares for Symmetric Triangular Fuzzy Numbers
4.1. Preliminary Results
4.2. Theoretical Results on the Total Least Squares for Symmetric Triangular Fuzzy Numbers
4.3. Concluding Theoretical Discussion
4.4. The Final Form of the Algorithm
- ;
- 1.1
- . The solution is .
- 1.2
- . The solution is .
- 1.3
- ;
- 1.3.1.
- . The solution is .
- 1.3.2.
- . The solutions are: , .
- 1.3.3.
- . The solution is .
- 1.4
- . The solution is .
- 1.5
- . The solution is .
- ;
- 2.1
- . The solution is .
- 2.2.
- . The solution is .
- 2.3.
- ;
- 2.3.1.
- . The solution is .
- 2.3.2.
- . The solutions are: , .
- 2.3.3.
- . The solution is .
- 2.4.
- . The solution is .
- 2.5.
- . The solution is .
5. Numerical Examples
6. Discussion
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Proofs of Propositions 1–4
Appendix B. Intermediate Calculations for Examples 1–4
1 | 3 | 2 | 2 | 1 | 6 | 2 | 9 | 4 | 4 | 1 |
2 | 5 | 1 | 9 | 3 | 45 | 3 | 25 | 81 | 1 | 9 |
3 | 12 | 3 | 7 | 4 | 84 | 12 | 144 | 49 | 9 | 16 |
Sum | 20 | 6 | 18 | 8 | 135 | 17 | 178 | 134 | 14 | 26 |
1 | 5 | 3 | 3 | 1 | 15 | 3 | 25 | 9 | 9 | 1 |
2 | 10 | 25 | 12 | 8 | 120 | 200 | 100 | 144 | 625 | 64 |
3 | 14 | 8 | 13 | 30 | 182 | 240 | 196 | 169 | 64 | 900 |
4 | 19 | 4 | 21 | 7 | 399 | 28 | 361 | 441 | 16 | 49 |
Sum | 48 | 40 | 49 | 46 | 716 | 471 | 682 | 763 | 714 | 1014 |
1 | 2 | 4 | 3 | 2 | 6 | 8 | 4 | 9 | 16 | 4 |
2 | 4 | 3 | 6 | 1 | 24 | 3 | 16 | 36 | 9 | 1 |
3 | 5 | 5 | 2.25 | 11.25 | 25 | 5.0625 | 25 | 35.375 | ||
Sum | 11 | 12 | 11.25 | 3+ | 41.25 | 11+ | 45 | 50.0625 | 50 | 40.375 |
1 | 3 | 2 | 2 | 8 | 6 | 16 | 9 | 4 | 4 | 64 |
2 | 5 | 1 | 9 | 2 | 45 | 2 | 25 | 81 | 1 | 4 |
3 | 12 | 3 | 7 | 9 | 84 | 27 | 144 | 49 | 9 | 81 |
Sum | 20 | 6 | 18 | 19 | 135 | 45 | 178 | 134 | 14 | 149 |
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Giuclea, M.; Popescu, C.-C. An Approach to the Total Least Squares Method for Symmetric Triangular Fuzzy Numbers. Mathematics 2025, 13, 1224. https://doi.org/10.3390/math13081224
Giuclea M, Popescu C-C. An Approach to the Total Least Squares Method for Symmetric Triangular Fuzzy Numbers. Mathematics. 2025; 13(8):1224. https://doi.org/10.3390/math13081224
Chicago/Turabian StyleGiuclea, Marius, and Costin-Ciprian Popescu. 2025. "An Approach to the Total Least Squares Method for Symmetric Triangular Fuzzy Numbers" Mathematics 13, no. 8: 1224. https://doi.org/10.3390/math13081224
APA StyleGiuclea, M., & Popescu, C.-C. (2025). An Approach to the Total Least Squares Method for Symmetric Triangular Fuzzy Numbers. Mathematics, 13(8), 1224. https://doi.org/10.3390/math13081224