## Fuzzy linear systems.(English)Zbl 0929.15004

Fuzzy Sets Syst. 96, No. 2, 201-209 (1998); comment and reply ibid. 140, 559-561 (2003).
Fuzzy linear systems were considered by many authors [cf. e.g. R. Fullér, Fuzzy Sets Syst. 34, No. 3, 347-353 (1990; Zbl 0696.15003); R. Zhao and R. Govind, Inf. Sci. 56, No. 1-3, 199-243 (1991; Zbl 0726.65048); J. J. Buckley, T. Feuring and Y. Hayashi, Int. Ser. Intell. Technol. 11, 213-232 (1997; Zbl 0893.65016)]. Here we have a linear system with fuzzy right-hand sides, where fuzzy numbers are based on definition by R. Goetschel and W. Voxman [Fuzzy Sets Syst. 10, 87-99 (1983; Zbl 0521.54001)]. The method of solution depends on a construction of $$2n\times 2n$$ crisp linear system with a nonnegative matrix. The paper contains many examples of practical computations.

### MSC:

 15A06 Linear equations (linear algebraic aspects) 15B48 Positive matrices and their generalizations; cones of matrices 15B33 Matrices over special rings (quaternions, finite fields, etc.) 03E72 Theory of fuzzy sets, etc.

### Citations:

Zbl 0696.15003; Zbl 0726.65048; Zbl 0893.65016; Zbl 0521.54001
Full Text:

### 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] Buckley, J. J., Fuzzy eigenvalues and input-output analysis, Fuzzy Sets and Systems, 34, 187-195 (1990) · Zbl 0687.90021 [3] Chang, S. S.L.; Zadeh, L. A., On fuzzy mapping and control, IEEE Trans. Systems Man Cybernet., 2, 30-34 (1972) · Zbl 0305.94001 [4] Cong-Xin, W.; Ming, M., Embedding problem of fuzzy number space: Part I, Fuzzy Sets and Systems, 44, 33-38 (1991) · Zbl 0757.46066 [5] Cong-Xin, W.; Ming, M., Embedding problem of fuzzy number space: Part III, Fuzzy Sets and Systems, 46, 281-286 (1992) · Zbl 0774.54003 [6] Diamond, P., Fuzzy least squares, Inform. Sci., 46, 144-157 (1988) · Zbl 0663.65150 [7] Dubois, D.; Prade, H., Operations on fuzzy numbers, J. Systems Sci., 9, 613-626 (1978) · Zbl 0383.94045 [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.; Kandel, A., Fundamentals of Computer Numerical Analysis, ((1994), CRC Press: CRC Press Boca Raton, FL), 307-351 [10] Goetschell, R.; Voxman, W., Eigen fuzzy number sets, Fuzzy Sets and Systems, 16, 75-85 (1985) · Zbl 0581.04007 [11] Goetschell, R.; Voxman, W., Elementary Calculus, Fuzzy Sets and Systems, 18, 31-43 (1986) [12] Heshmaty, B.; Kandel, A., Fuzzy linear regression and its applications to forecasting in uncertain environments, Fuzzy Sets and Systems, 15, 159-191 (1985) · Zbl 0566.62099 [13] McAllister, M. N., Symmetric matrices of fuzzy numbers: The eigenvalue problem for the polynomial case, (Proc. Internat. Symp. on Fuzzy Systems and Knowledge Engineering (1987), Guangzhou-guiyang: Guangzhou-guiyang P.R. China), 640-645 [14] Minc, H., Nonnegative Matrices (1988), Wiley: Wiley New York · Zbl 0638.15008 [15] Mizumoto, M.; Tanaka, K., The four operations of arithmetic on fuzzy numbers, Systems Comput. Controls, 7, 5, 73-81 (1976) [16] Mizumoto, M.; Tanaka, K., Some properties of fuzzy numbers, (Gupta, M. M.; Ragade, R. K.; Yager, R. R., Advances in Fuzzy Set Theory and Applications (1979), North-Holland: North-Holland Amsterdam), 156-164 [17] Nahmias, S., Fuzzy variables, Fuzzy Sets and Systems, 1, 2, 97-111 (1978) · Zbl 0383.03038 [18] Negoita, C. V.; Ralescu, D. A., Applications of Fuzzy Sets and Systems Analysis (1975), Wiley: Wiley New York · Zbl 0326.94002 [19] Peeva, K., Fuzzy linear systems, Fuzzy Sets and Systems, 49, 339-355 (1992) · Zbl 0805.04005 [20] Prade, H., Operations research with fuzzy data, (Wang, P. P.; Chang, S. K., Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems (1980), Plenum Press: Plenum Press New York), 115-169 [21] Puri, M. L.; Ralescu, D. A., Differentials for fuzzy functions, J. Math. Anal. Appl., 91, 552-558 (1983) · Zbl 0528.54009 [22] Sanchez, E., Eigen fuzzy sets and fuzzy relation, J. Math. Anal. Appl., 81, 399-421 (1981) · Zbl 0466.04003 [23] Tanaka, H., Fuzzy data analysis by probabilistic linear models, Fuzzy Sets and Systems, 24, 363-375 (1987) · Zbl 0633.93060 [24] Tanaka, H., Linear regression analysis with fuzzy model, IEEE Trans. Systems Man Cybernet., SMC-14, 325-328 (1984) · Zbl 0551.90061 [25] Tanaka, H.; Ishibuchi, H.; Yoshikawa, S., Exponential possibility regression analysis, Fuzzy Sets and Systems, 69, 305-318 (1995) · Zbl 0842.62062 [26] Yager, R. R., On the lack of inverses on fuzzy arithmetic, Fuzzy Sets and Systems, 4, 73-82 (1980) · Zbl 0433.03033 [27] Yager, R. R., Fuzzy prediction based upon regression models, Inform. Sci., 26, 45-63 (1982) · Zbl 0509.62087 [28] Zadeh, L. A., The concept of a linguistic variable and its application to approximate reasoning, Inform. Sci., 8, 199-249 (1975) · Zbl 0397.68071 [29] Zimmerman, H.-J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems, 1, 45-55 (1978) · Zbl 0364.90065
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.