Approximate Solution of Fuzzy Matrix Equations with LR Fuzzy Numbers

Affiliation(s)

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China.

Department of Public Courses, Gansu College of Chinese Medicine, Lanzhou, China.

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China.

Department of Public Courses, Gansu College of Chinese Medicine, Lanzhou, China.

ABSTRACT

In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.

In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.

Cite this paper

X. Guo and D. Shang, "Approximate Solution of Fuzzy Matrix Equations with LR Fuzzy Numbers,"*Advances in Pure Mathematics*, Vol. 2 No. 6, 2012, pp. 373-378. doi: 10.4236/apm.2012.26056.

X. Guo and D. Shang, "Approximate Solution of Fuzzy Matrix Equations with LR Fuzzy Numbers,"

References

[1] L A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X

[2] D. Dubois, and H. Prade, “Operations on Fuzzy Numbers,” International Journal of Systems Science, Vol. 9, No. 3, 1978, pp. 613-626. doi:10.1080/00207727808941724

[3] M. Friedman, M. Ma and A. Kandel, “Fuzzy Linear Systems,” Fuzzy Sets and Systems, Vol. 96, No. 2, 1998, pp. 201-209. doi:10.1016/S0165-0114(96)00270-9

[4] M. Friedman, M. Ma and A. Kandel, “Duality in Fuzzy Linear Systems,” Fuzzy Sets and Systems, Vol. 109, No. 1, 2000, pp. 55-58. doi:10.1016/S0165-0114(98)00102-X

[5] T. Allahviranloo, “Numerical Methods for Fuzzy System of Linear Equations,” Applied Mathematics and Computation, Vol. 155, No. 2, 2004, pp. 493-502. doi:10.1016/S0096-3003(03)00793-8

[6] B. Asady, S. Abbasbandy and M. Alavi, “Fuzzy General Linear Systems,” Applied Mathematics and Computation, Vol. 169, No. 1, 2005, pp. 34-40. doi:10.1016/j.amc.2004.10.042

[7] K. Wang and B. Zheng, “Inconsistent Fuzzy Linear Systems,” Applied Mathematics and Computation, Vol. 181, No. 2, 2006, pp. 973-981. doi:10.1016/j.amc.2006.02.019

[8] M. Dehghan, B. Hashemi and M. Ghatee, “Solution of the Full Fuzzy Linear Systems Using Iterative Techniques,” Chaos, Solitons & Fractals, Vol. 34, No. 2, 2007, pp. 316-336. doi:10.1016/j.chaos.2006.03.085

[9] S. Abbasbandy, M. Otadi and M. Mosleh, “Minimal Solution of General Dual Fuzzy Linear Systems,” Chaos, Solitons & Fractals, Vol. 37, No. 4, 2008, pp. 638-652. doi:10.1016/j.chaos.2006.10.045

[10] X. B. Guo and Z. T. Gong, “Undetermined Coefficient Method for Solving Semi-Fuzzy Matrix Equation,” Proceedings of 9th International Conference on Machine Learning and Cybernetics, Qingdao, 11-14 July 2010, pp. 596-600.

[11] Z. T. Gong and X. B. Guo, “Inconsistent Fuzzy Matrix Equations and Its Fuzzy Least Squares Solutions,” Applied Mathematical Modelling, Vol. 35, No. 1, 2011, pp. 1456-1469. doi:10.1016/j.apm.2010.09.022

[12] X. B. Guo, “Approximate Solution of Fuzzy Sylvester Matrix Equations,” The 7th International Conference on Computational Intelligence and Security, Sanya, 3-4 December 2011, pp. 52-57.

[13] X. B. Guo and D. Q. Shang, “Fuzzy Symmetric Solutions of Fuzzy Matrix Equations,” Advances in Fuzzy Systems, 2012, Article ID: 318069. doi:10.1155/2012/318069

[14] X. D. Zhang, “Matrix Analysis and Its Applications,” Tsinghua University and Springer Press, Beijing, 2004.

[15] A. Berman and R. J. Plemmons, “Nonnegative Matrices in the Mathematical Sciences,” Academic Press, New York, 1979.

[1] L A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X

[2] D. Dubois, and H. Prade, “Operations on Fuzzy Numbers,” International Journal of Systems Science, Vol. 9, No. 3, 1978, pp. 613-626. doi:10.1080/00207727808941724

[3] M. Friedman, M. Ma and A. Kandel, “Fuzzy Linear Systems,” Fuzzy Sets and Systems, Vol. 96, No. 2, 1998, pp. 201-209. doi:10.1016/S0165-0114(96)00270-9

[4] M. Friedman, M. Ma and A. Kandel, “Duality in Fuzzy Linear Systems,” Fuzzy Sets and Systems, Vol. 109, No. 1, 2000, pp. 55-58. doi:10.1016/S0165-0114(98)00102-X

[5] T. Allahviranloo, “Numerical Methods for Fuzzy System of Linear Equations,” Applied Mathematics and Computation, Vol. 155, No. 2, 2004, pp. 493-502. doi:10.1016/S0096-3003(03)00793-8

[6] B. Asady, S. Abbasbandy and M. Alavi, “Fuzzy General Linear Systems,” Applied Mathematics and Computation, Vol. 169, No. 1, 2005, pp. 34-40. doi:10.1016/j.amc.2004.10.042

[7] K. Wang and B. Zheng, “Inconsistent Fuzzy Linear Systems,” Applied Mathematics and Computation, Vol. 181, No. 2, 2006, pp. 973-981. doi:10.1016/j.amc.2006.02.019

[8] M. Dehghan, B. Hashemi and M. Ghatee, “Solution of the Full Fuzzy Linear Systems Using Iterative Techniques,” Chaos, Solitons & Fractals, Vol. 34, No. 2, 2007, pp. 316-336. doi:10.1016/j.chaos.2006.03.085

[9] S. Abbasbandy, M. Otadi and M. Mosleh, “Minimal Solution of General Dual Fuzzy Linear Systems,” Chaos, Solitons & Fractals, Vol. 37, No. 4, 2008, pp. 638-652. doi:10.1016/j.chaos.2006.10.045

[10] X. B. Guo and Z. T. Gong, “Undetermined Coefficient Method for Solving Semi-Fuzzy Matrix Equation,” Proceedings of 9th International Conference on Machine Learning and Cybernetics, Qingdao, 11-14 July 2010, pp. 596-600.

[11] Z. T. Gong and X. B. Guo, “Inconsistent Fuzzy Matrix Equations and Its Fuzzy Least Squares Solutions,” Applied Mathematical Modelling, Vol. 35, No. 1, 2011, pp. 1456-1469. doi:10.1016/j.apm.2010.09.022

[12] X. B. Guo, “Approximate Solution of Fuzzy Sylvester Matrix Equations,” The 7th International Conference on Computational Intelligence and Security, Sanya, 3-4 December 2011, pp. 52-57.

[13] X. B. Guo and D. Q. Shang, “Fuzzy Symmetric Solutions of Fuzzy Matrix Equations,” Advances in Fuzzy Systems, 2012, Article ID: 318069. doi:10.1155/2012/318069

[14] X. D. Zhang, “Matrix Analysis and Its Applications,” Tsinghua University and Springer Press, Beijing, 2004.

[15] A. Berman and R. J. Plemmons, “Nonnegative Matrices in the Mathematical Sciences,” Academic Press, New York, 1979.