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General fuzzy linear systems. (English) Zbl 1122.15005

Linear systems \(Ax =b\) with fuzzy right-hand side (\(b\) and \(x\) are vectors of fuzzy numbers) were introduced by M. Friedman, M. Ma and A. Kandel [Fuzzy Sets Syst. 96, No. 2, 201–209 (1998); comment and reply ibid. 140, 559–561 (2003; Zbl 0929.15004)], where the method of solutions based on associated \(2n \times 2n\) nonnegative matrix \(S\) for an \(n\times n\) square matrix \(A\) and the notions of weak and strong fuzzy solutions was presented. These results are now generalized to the case of an \(m \times n\) matrix \(A\) by using the generalized inverses of the associated matrix \(S\) (in particular the Moore-Penrose inverse). Moreover, in the case of an inconsistent system, the solution of the associated \(2m \times 2n\) linear system is replaced by the approximate solution from the least squares method.

MSC:

15A06 Linear equations (linear algebraic aspects)
15B48 Positive matrices and their generalizations; cones of matrices
15A09 Theory of matrix inversion and generalized inverses
08A72 Fuzzy algebraic structures
15B33 Matrices over special rings (quaternions, finite fields, etc.)

Citations:

Zbl 0929.15004
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Full Text: DOI

References:

[1] Badard, R., The law of large numbers for fuzzy processes and the estimation problem, Inform. Sci., 28, 161-178 (1982) · Zbl 0588.60004
[2] Tanaka, H.; Uejima, S.; Asai, K., Linear regression analysis with fuzzy model, IEEE Trans. Syst. Man Cybernet., 12, 903-907 (1982) · Zbl 0501.90060
[3] Buckley, J. J., Fuzzy eigenvalues and input-output analysis, Fuzzy Sets Syst., 34, 187-195 (1990) · Zbl 0687.90021
[4] Friedman, M.; Ming, Ma; Kandel, A., Fuzzy linear systems, Fuzzy Sets Syst., 96, 201-209 (1998) · Zbl 0929.15004
[5] Allahviranloo, T., Numerical methods for fuzzy system of linear equations, Appl. Math. Comput., 155, 493-502 (2004) · Zbl 1067.65040
[6] Allahviranloo, T., Successive over relaxation iterative method for fuzzy system of linear equations, Appl. Math. Comput., 162, 189-196 (2005) · Zbl 1062.65037
[7] Babolian, E.; Sadeghi Goghary, H.; Abbasbandy, S., Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Appl. Math. Comput., 161, 733-744 (2005) · Zbl 1062.65143
[8] Allahviranloo, T., The Adomian decomposition method for fuzzy system of linear equations, Appl. Math. Comput., 163, 553-563 (2005) · Zbl 1069.65025
[9] Allahviranloo, T., A comment on fuzzy linear systems, Fuzzy Sets Syst., 140, 559 (2003) · Zbl 1050.15003
[10] Asady, B.; Abbasbandy, S.; Alavi, M., Fuzzy general linear systems, Appl. Math. Comput., 169, 34-40 (2005) · Zbl 1119.65325
[11] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1026.15004
[12] Plemmons, R. J., Regular nonnegative matrices, Proc. Amer. Math. Soc., 39, 26-32 (1973) · Zbl 0273.20051
[13] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016
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