Document Type : Research Paper

**Authors**

^{1}
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

^{2}
Public Health College, Gansu University of Traditional Chinese Medicine, Lanzhou 730000, China

**Abstract**

In this paper, the fuzzy matrix equation $A\widetilde{X}B=\widetilde{C}$ in which $A,B$ are $n \times n$

crisp matrices respectively and $\widetilde{C}$ is an $n \times n$ arbitrary LR fuzzy numbers matrix, is investigated. A new numerical procedure for calculating the fuzzy solution is designed and a sufficient condition for the existence of strong fuzzy solution is derived. Some examples are given to illustrate the proposed method.

crisp matrices respectively and $\widetilde{C}$ is an $n \times n$ arbitrary LR fuzzy numbers matrix, is investigated. A new numerical procedure for calculating the fuzzy solution is designed and a sufficient condition for the existence of strong fuzzy solution is derived. Some examples are given to illustrate the proposed method.

**Keywords**

[1] S. Abbasbandy, A. Jafarian, R. Fzzati, Conjugate gradient method for fuzzy symmetric positive definite system of

linear equations, Applied Mathematics and Computation, 171 (2005), 1184{1191.

[2] S. Abbasbandy, R. Ezzati, A. Jafarian, LU decomposition method for solving fuzzy system of linear equations,

Applied Mathematics and Computation, 172 (2006), 633{643.

[3] S. Abbasbandy, M. Otadi, M. Mosleh, Minimal solution of general dual fuzzy linear systems, Chaos, Solitions and

Fractals, 29 (2008), 638{652.

[4] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation,

153 (2004), 493{502.

[5] T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathe-

matics and Computation, 162 (2005), 189{196.

[6] T. Allahviranloo, The adomian decomposition method for fuzzy system of linear equations, Applied Mathematics

and Computation, 163 (2005), 553{563

[7] T. Allahviranloo, F. H. Lot, M. K. Kiasari, M. Khezerloo, On the fuzzy solution of LR fuzzy linear systems, Applied

Mathematical Modelling, 37 (2013), 1170{1176.

[8] T. Allahviranloo, M. Ghanbari, R. Nuraei, A note on fuzzy linear systems, Fuzzy Sets and Systems, 177(1) (2011),

87-92.

[9] T. Allahviranloo, N. Mikaeilvand, M. Barkhordary, Fuzzy linear matrix equations, Fuzzy Optimization and Decision

Making, 8 (2009), 165{177.

[10] M. Amirfakhrian, M. Fallah, R. Rodriguez-Lopez, A method for solving fuzzy matrix equations, Soft Computing,

22 (2018), 2095{2103.

linear equations, Applied Mathematics and Computation, 171 (2005), 1184{1191.

[2] S. Abbasbandy, R. Ezzati, A. Jafarian, LU decomposition method for solving fuzzy system of linear equations,

Applied Mathematics and Computation, 172 (2006), 633{643.

[3] S. Abbasbandy, M. Otadi, M. Mosleh, Minimal solution of general dual fuzzy linear systems, Chaos, Solitions and

Fractals, 29 (2008), 638{652.

[4] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation,

153 (2004), 493{502.

[5] T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathe-

matics and Computation, 162 (2005), 189{196.

[6] T. Allahviranloo, The adomian decomposition method for fuzzy system of linear equations, Applied Mathematics

and Computation, 163 (2005), 553{563

[7] T. Allahviranloo, F. H. Lot, M. K. Kiasari, M. Khezerloo, On the fuzzy solution of LR fuzzy linear systems, Applied

Mathematical Modelling, 37 (2013), 1170{1176.

[8] T. Allahviranloo, M. Ghanbari, R. Nuraei, A note on fuzzy linear systems, Fuzzy Sets and Systems, 177(1) (2011),

87-92.

[9] T. Allahviranloo, N. Mikaeilvand, M. Barkhordary, Fuzzy linear matrix equations, Fuzzy Optimization and Decision

Making, 8 (2009), 165{177.

[10] M. Amirfakhrian, M. Fallah, R. Rodriguez-Lopez, A method for solving fuzzy matrix equations, Soft Computing,

22 (2018), 2095{2103.

[11] B. Asady, S. Abbasbandy, M. Alavi, Fuzzy general linear systems, Applied Mathematics and Computation, 169

(2005), 34{40.

[12] N. Babbar, A. Kumar, Abhinav Bansal, Solving fully fuzzy linear system with arbitrary triangular fuzzy numbers,

Soft Computing, 17 (2013) 691-702.

[13] A. Ben-Israel, T. N. E. Greville, Generalized inverses: Theory and applications, second edition, Springer-Verlag,

New York, 2003.

[14] A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic press, New York, 1979.

[15] W. S. W. Daud, N. Ahmad, G. Malkawi, A preliminary study on the solution of fully fuzzy Sylvester matrix

equation, In AIP Conference Proceedings, 1775 (2016), 30{36.

[16] M. Dehghan, B. Hashemi, M. Ghatee, Solution of the full fuzzy linear systems using iterative techniques, Chaos,

Solitons and Fractals, 34 (2007) 316{336.

[17] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems, Applied Mathematics and Computation, 175

(2006), 645{674.

[18] K. Dookhitram, R. Lollchund, R. K. Tripathi, M. Bhuruth, Fully fuzzy Sylvester matrix equation, Journal of

Intelligent and Fuzzy Systems, 28 (2015), 2199{2211.

[19] D. Dubois, H. Prade, Operations on fuzzy numbers, Journal of Systems Science, 9 (1978), 613{626.

[20] S. A. Edalatpanah, Modied iterative methods for solving fully fuzzy linear systems, In I, Management Association

(Ed.) Fuzzy Systems: Concepts, Methodologies, Tools, and Applications, (2017), 55{73.

[21] M. Friedman, M. Ma, A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96 (1998), 201{209.

[22] R. Ghanbari, Solutions of fuzzy LR algebraic linear systems using linear programs, Applied Mathematical Mod-

elling, 39 (2015), 5164-5173.

[23] R. Goetschel, W. Voxman, Elementary calculus, Fuzzy Sets and Systems, 18 (1986), 31{43.

[24] Z. T. Gong, X. B. Guo, Inconsistent fuzzy matrix equations and its fuzzy least squares solutions, Applied Mathe-

matical Modelling, 35 (2011), 1456{1469.

[25] Z. T. Gong, X. B. Guo, K. Liu, Approximate solution of dual fuzzy matrix equations, Information Sciences, 266

(2014), 112-133.

[26] X. B. Guo, D. Q. Shang, Approximate solutions of LR fuzzy Sylvester matrix equations, Journal of Applied Math-

ematics, 2013 (2013), 1{10.

[27] X. B. Guo, K. Zhang, Solving fuzzy matrix equation of the form eXA = eB, Journal of Intelligent and Fuzzy Systems,

32 (2017), 2771{2778.

[28] X. B. Guo, Y. L. Han, Further investigation to dual fuzzy matrix equation, Journal of Intelligent and Fuzzy Systems,

33 (2017), 2617{2629.

[29] J. J. Kaur, A. Kumar, A note on approximate solution of dual fuzzy matrix equations, Information Sciences, 418

(2017), 184{185.

[30] M. Ma, M. Friedman, A. Kandel, Duality in fuzzy linear systems, Fuzzy Sets and Systems, 109 (2000), 55{58.

[31] S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems, 1(2) (1978) 97{111.

[32] H. S. Najafi, S. A. Edalatpanah, An improved model for iterative algorithms in fuzzy linear systems, Computational

Mathematics and Modeling, 24 (2013), 443{451.

[33] H. S. Najafi, S. A. Edalatpanah, H-Matrices in fuzzy linear systems, International Journal of Computational

Mathematics, 6 (2013), 52{56.

(2005), 34{40.

[12] N. Babbar, A. Kumar, Abhinav Bansal, Solving fully fuzzy linear system with arbitrary triangular fuzzy numbers,

Soft Computing, 17 (2013) 691-702.

[13] A. Ben-Israel, T. N. E. Greville, Generalized inverses: Theory and applications, second edition, Springer-Verlag,

New York, 2003.

[14] A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic press, New York, 1979.

[15] W. S. W. Daud, N. Ahmad, G. Malkawi, A preliminary study on the solution of fully fuzzy Sylvester matrix

equation, In AIP Conference Proceedings, 1775 (2016), 30{36.

[16] M. Dehghan, B. Hashemi, M. Ghatee, Solution of the full fuzzy linear systems using iterative techniques, Chaos,

Solitons and Fractals, 34 (2007) 316{336.

[17] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems, Applied Mathematics and Computation, 175

(2006), 645{674.

[18] K. Dookhitram, R. Lollchund, R. K. Tripathi, M. Bhuruth, Fully fuzzy Sylvester matrix equation, Journal of

Intelligent and Fuzzy Systems, 28 (2015), 2199{2211.

[19] D. Dubois, H. Prade, Operations on fuzzy numbers, Journal of Systems Science, 9 (1978), 613{626.

[20] S. A. Edalatpanah, Modied iterative methods for solving fully fuzzy linear systems, In I, Management Association

(Ed.) Fuzzy Systems: Concepts, Methodologies, Tools, and Applications, (2017), 55{73.

[21] M. Friedman, M. Ma, A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96 (1998), 201{209.

[22] R. Ghanbari, Solutions of fuzzy LR algebraic linear systems using linear programs, Applied Mathematical Mod-

elling, 39 (2015), 5164-5173.

[23] R. Goetschel, W. Voxman, Elementary calculus, Fuzzy Sets and Systems, 18 (1986), 31{43.

[24] Z. T. Gong, X. B. Guo, Inconsistent fuzzy matrix equations and its fuzzy least squares solutions, Applied Mathe-

matical Modelling, 35 (2011), 1456{1469.

[25] Z. T. Gong, X. B. Guo, K. Liu, Approximate solution of dual fuzzy matrix equations, Information Sciences, 266

(2014), 112-133.

[26] X. B. Guo, D. Q. Shang, Approximate solutions of LR fuzzy Sylvester matrix equations, Journal of Applied Math-

ematics, 2013 (2013), 1{10.

[27] X. B. Guo, K. Zhang, Solving fuzzy matrix equation of the form eXA = eB, Journal of Intelligent and Fuzzy Systems,

32 (2017), 2771{2778.

[28] X. B. Guo, Y. L. Han, Further investigation to dual fuzzy matrix equation, Journal of Intelligent and Fuzzy Systems,

33 (2017), 2617{2629.

[29] J. J. Kaur, A. Kumar, A note on approximate solution of dual fuzzy matrix equations, Information Sciences, 418

(2017), 184{185.

[30] M. Ma, M. Friedman, A. Kandel, Duality in fuzzy linear systems, Fuzzy Sets and Systems, 109 (2000), 55{58.

[31] S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems, 1(2) (1978) 97{111.

[32] H. S. Najafi, S. A. Edalatpanah, An improved model for iterative algorithms in fuzzy linear systems, Computational

Mathematics and Modeling, 24 (2013), 443{451.

[33] H. S. Najafi, S. A. Edalatpanah, H-Matrices in fuzzy linear systems, International Journal of Computational

Mathematics, 6 (2013), 52{56.

[34] R. Nuraei, T. Allahviranloo, M. Ghanbari, Finding an inner estimation of the solution set of a fuzzy linear system,

Applied Mathematical Modelling, 37 (2013), 5148{5161.

[35] M. Otadi, M. Mosleh, Solving fully fuzzy matrix equations, Applied Mathematical Modelling, 36 (2012), 6114{6121.

[36] R. J. Plemmons, Regular nonnegative matrices, Proceedings of the American Mathematical Society, 39 (1973),

26-32.

[37] M. L. Puri, D. A. Ralescu, Differentials for fuzzy functions, Journal of Mathematics Analyisis and Application, 91

(1983), 552{558.

[38] C. X. Wu, M. Ma, Embedding problem of fuzzy number space: Part I, Fuzzy Sets and Systems, 44 (1991), 33{38.

[39] C. X.Wu, M. Ma, Embedding problem of fuzzy number space: Part III, Fuzzy Sets and Systems, 46 (1992), 281{286.

[40] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.

[41] B. Zheng, K. Wang, General fuzzy linear systems, Applied Mathematics and Computation, 181 (2006), 1276{1286.

Applied Mathematical Modelling, 37 (2013), 5148{5161.

[35] M. Otadi, M. Mosleh, Solving fully fuzzy matrix equations, Applied Mathematical Modelling, 36 (2012), 6114{6121.

[36] R. J. Plemmons, Regular nonnegative matrices, Proceedings of the American Mathematical Society, 39 (1973),

26-32.

[37] M. L. Puri, D. A. Ralescu, Differentials for fuzzy functions, Journal of Mathematics Analyisis and Application, 91

(1983), 552{558.

[38] C. X. Wu, M. Ma, Embedding problem of fuzzy number space: Part I, Fuzzy Sets and Systems, 44 (1991), 33{38.

[39] C. X.Wu, M. Ma, Embedding problem of fuzzy number space: Part III, Fuzzy Sets and Systems, 46 (1992), 281{286.

[40] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338{353.

[41] B. Zheng, K. Wang, General fuzzy linear systems, Applied Mathematics and Computation, 181 (2006), 1276{1286.

September and October 2019

Pages 33-44