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**Computational methods for solving fully fuzzy linear systems.**
*(English)*
Zbl 1101.65040

Summary: Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modelling, which can formulate uncertainty in actual environment. In addition, this is an important sub-process in determining inverse, eigenvalue and some other useful matrix computations, too. One of the most practicable subjects in recent studies is based on LR fuzzy numbers, which are defined and used by D. Dubois and H. Prade [Fuzzy sets and systems. Theory and applications. (1980; Zbl 0444.94049)] with some useful and easy approximation arithmetic operators on them. Recently M. Dehghan, M. Ghatee, and B. Hashemi [Some computations on fuzzy matrices (to appear)] extended some matrix computations on fuzzy matrices, where a fuzzy matrix appears as a rectangular array of fuzzy numbers.

In continuation to our previous work, we focus on fuzzy systems in this paper. It is proved that finding all of the real solutions which satisfy in a system with interval coefficients is NP-hard. The same result can similarly be derived for fuzzy systems. So we employ some heuristics based methods on Dubois and Prade’s approach, finding some positive fuzzy vector \(\widetilde x\) which satisfies \(\widetilde A\widetilde x\), where \(\widetilde A\) and \(\widetilde b\) are a fuzzy matrix and a fuzzy vector, respectively. We propose some new methods to solve this system that are comparable to the well known methods such as the Cramer’s rule, Gaussian elimination, LU decomposition method (Doolittle algorithm) and its simplification. Finally we extend a new method employing linear programming for solving square and non-square (over-determined) fuzzy systems. Some numerical examples clarify the ability of our heuristics.

In continuation to our previous work, we focus on fuzzy systems in this paper. It is proved that finding all of the real solutions which satisfy in a system with interval coefficients is NP-hard. The same result can similarly be derived for fuzzy systems. So we employ some heuristics based methods on Dubois and Prade’s approach, finding some positive fuzzy vector \(\widetilde x\) which satisfies \(\widetilde A\widetilde x\), where \(\widetilde A\) and \(\widetilde b\) are a fuzzy matrix and a fuzzy vector, respectively. We propose some new methods to solve this system that are comparable to the well known methods such as the Cramer’s rule, Gaussian elimination, LU decomposition method (Doolittle algorithm) and its simplification. Finally we extend a new method employing linear programming for solving square and non-square (over-determined) fuzzy systems. Some numerical examples clarify the ability of our heuristics.

### MSC:

65F20 | Numerical solutions to overdetermined systems, pseudoinverses |

65F05 | Direct numerical methods for linear systems and matrix inversion |

08A72 | Fuzzy algebraic structures |

### Keywords:

fuzzy number; fuzzy approximate arithmetic; fully fuzzy linear system; over-determined fuzzy linear system of equations; Cramer’s rule; Gaussian elimination; fuzzy LU decomposition; Doolittle algorithm; linear programming; numerical examples### Citations:

Zbl 0444.94049
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\textit{M. Dehghan} et al., Appl. Math. Comput. 179, No. 1, 328--343 (2006; Zbl 1101.65040)

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### References:

[1] | Bazarra, M. S.; Jarvis, J. J.; Sherali, H. D., Linear Programming and Network Flows (1990), Wiley: Wiley New York |

[2] | Buckley, J. J.; Qu, Y., Solving systems of linear fuzzy equations, Fuzzy Sets and Systems, 43, 33-43 (1991) · Zbl 0741.65023 |

[5] | DeMarr, R., Nonnegative matrices with nonnegative inverses, Proceedings of the American Mathematical Society, 307-308 (1972) · Zbl 0257.15002 |

[6] | Dubois, D.; Prade, H., Operations on fuzzy numbers, International Journal of Systems Science, 613-626 (1978) · Zbl 0383.94045 |

[7] | Dubois, D.; Prade, H., Systems of linear fuzzy constraints, Fuzzy Sets and Systems, 3, 37-48 (1980) · Zbl 0425.94029 |

[8] | Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press: Academic Press New York · Zbl 0444.94049 |

[9] | Friedman, M.; Ming, M.; Kandel, A., Fuzzy linear systems, Fuzzy Sets and Systems, 96, 201-209 (1998) · Zbl 0929.15004 |

[10] | Giachetti, R. E.; Young, R. E., Analysis of the error in the standard approximation used for multiplication of triangular and trapezoidal fuzzy numbers and the development of a new approximation, Fuzzy Sets and Systems, 91, 1-13 (1997) · Zbl 0915.04003 |

[11] | Giachetti, R. E.; Young, R. E., A parametric representation of fuzzy numbers and their arithmetic operators, Fuzzy Sets and Systems, 91, 185-202 (1997) · Zbl 0920.04008 |

[13] | Hansen, E., Interval arithmetic in matrix computations, Part I, SIAM Journal on Numerical Analysis, 2, 308-320 (1965) · Zbl 0135.37303 |

[14] | Hansen, E., Interval arithmetic in matrix computations, Part II, SIAM Journal on Numerical Analysis, 4, 1-9 (1967) · Zbl 0209.46601 |

[16] | Kreinovich, V.; Lakeyev, A. V.; Rohn, J.; Kahl, P. T., Computational Complexity and Feasibility of Data Processing and Interval Computations. Computational Complexity and Feasibility of Data Processing and Interval Computations, Applied Optimization, vol. 10 (1998), Springer · Zbl 0945.68077 |

[17] | Rao, S. S.; Chen, L., Numerical solution of fuzzy linear equations in engineering analysis, International Journal for Numerical Methods in Engineering, 42, 829-846 (1998) · Zbl 0911.73077 |

[18] | Sakawa, M., Fuzzy Sets and Interactive Multiobjective Optimization (1973), Plenum press: Plenum press New York and London |

[19] | Wagenknecht, M.; Hampel, R.; Schneider, V., Computational aspects of fuzzy arithmetics based on Archimedean t-norms, Fuzzy Sets and Systems, 123, 49-62 (2001) · Zbl 0997.65071 |

[20] | Watkins, D. S., Fundamentals of Matrix Computations (2002), Wiley-Interscience Pub.: Wiley-Interscience Pub. New York · Zbl 1005.65027 |

[21] | Ye, Y., Interior Point Algorithms, Theory and Analysis (1997), Wiely: Wiely New York · Zbl 0943.90070 |

[22] | Zadeh, L. A., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606 |

[23] | Zhao, R.; Govind, R., Solutions of algebraic equations involving generalized fuzzy numbers, Information Science, 56, 199-243 (1991) · Zbl 0726.65048 |

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