## 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.

### MSC:

 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65F05 Direct numerical methods for linear systems and matrix inversion 08A72 Fuzzy algebraic structures

Zbl 0444.94049
Full Text:

### References:

  Bazarra, M.S.; Jarvis, J.J.; Sherali, H.D., Linear programming and network flows, (1990), Wiley New York  Buckley, J.J.; Qu, Y., Solving systems of linear fuzzy equations, Fuzzy sets and systems, 43, 33-43, (1991) · Zbl 0741.65023  M. Dehghan, M. Ghatee, B. Hashemi, Some computations on fuzzy matrices, submitted for publication. · Zbl 1161.65031  M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems, Applied Mathematics and Computation, in press, doi:10.1016/j.amc.2005.07.033. · Zbl 1137.65336  DeMarr, R., Nonnegative matrices with nonnegative inverses, Proceedings of the American mathematical society, 307-308, (1972) · Zbl 0257.15002  Dubois, D.; Prade, H., Operations on fuzzy numbers, International journal of systems science, 613-626, (1978) · Zbl 0383.94045  Dubois, D.; Prade, H., Systems of linear fuzzy constraints, Fuzzy sets and systems, 3, 37-48, (1980) · Zbl 0425.94029  Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049  Friedman, M.; Ming, M.; Kandel, A., Fuzzy linear systems, Fuzzy sets and systems, 96, 201-209, (1998) · Zbl 0929.15004  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  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  A.A. Gvozdik, Solution of fuzzy equations, UDC, 518.9 (1985) 60-67. · Zbl 0593.65031  Hansen, E., Interval arithmetic in matrix computations, part I, SIAM journal on numerical analysis, 2, 308-320, (1965) · Zbl 0135.37303  Hansen, E., Interval arithmetic in matrix computations, part II, SIAM journal on numerical analysis, 4, 1-9, (1967) · Zbl 0209.46601  M.F. Kawaguchi, T. Da-Te, A calculation method for solving fuzzy arithmetic equations with triangular norms, in: 2nd IEEE International Conference on Fuzzy Systems, San Francisco, CA, USA, 1993, March 28-April 1.  Kreinovich, V.; Lakeyev, A.V.; Rohn, J.; Kahl, P.T., Computational complexity and feasibility of data processing and interval computations, Applied optimization, vol. 10, (1998), Springer · Zbl 0945.68077  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  Sakawa, M., Fuzzy sets and interactive multiobjective optimization, (1973), Plenum press New York and London  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  Watkins, D.S., Fundamentals of matrix computations, (2002), Wiley-Interscience Pub. New York · Zbl 1005.65027  Ye, Y., Interior point algorithms, theory and analysis, (1997), Wiely New York · Zbl 0943.90070  Zadeh, L.A., Fuzzy sets, Information and control, 8, 338-353, (1965) · Zbl 0139.24606  Zhao, R.; Govind, R., Solutions of algebraic equations involving generalized fuzzy numbers, Information science, 56, 199-243, (1991) · Zbl 0726.65048  M. Dehghan, B. Hashemi, M. Ghatee, Solution of the fully fuzzy linear systems using iterative techniques, submitted for publication. · Zbl 1144.65021  M. Dehghan, M. Ghatee, B. Hashemi, Inverse of a fuzzy matrix of fuzzy numbers, submitted for publication. · Zbl 1181.65046  S. Muzzioli, H. Reynaerts, Fuzzy linear systems of the form A1x+b1=A2x+b2, Fuzzy Sets and Systems, in press. · Zbl 1095.15004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.