Document Type : Research Paper

**Authors**

Department of Mathematics, Firoozkooh Branch, Islamic Azad Univer- sity, Firoozkooh, Iran

**Abstract**

In this paper, we use parametric form of fuzzy number and we convert

a fuzzy linear system to two linear system in crisp case. Conditions for the existence of a minimal solution to $mtimes n$ fuzzy linear equation systems are derived and a numerical procedure for calculating the minimal solution is designed. Numerical examples are presented to illustrate the proposed method.

a fuzzy linear system to two linear system in crisp case. Conditions for the existence of a minimal solution to $mtimes n$ fuzzy linear equation systems are derived and a numerical procedure for calculating the minimal solution is designed. Numerical examples are presented to illustrate the proposed method.

**Keywords**

[1] S. Abbasbandy and M. Alavi, A new method for solving symmetric fuzzy linear systems,

Mathematics Scientic Journal, Islamic Azad University of Arak, 1 (2005), 55-62.

[2] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm

fuzzy integral equations of the second kind, Chaos, Solitons & Fractals, 31(1) (2007), 138-

146.

[3] S. Abbasbandy, A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric

positive denite system of linear equations, Appl. Math. Comput., 171(2) (2005), 1184-1191.

[4] S. Abbasbandy, R. Ezzati and A. Jafarian, LU decomposition method for solving fuzzy system

of linear equations, Appl. Math. Comput., 172(1) (2006), 633-643.

[5] S. Abbasbandy, J. J. Nieto and M. Alavi, Tuning of reachable set in one dimensional fuzzy

dierential inclusions, Chaos, Solitons & Fractals, 26(5) (2005), 1337-1341.

[6] S. Abbasbandy, M. Otadi and M. Mosleh, Minimal solution of general dual fuzzy linear

systems, Chaos, Solitons & Fractals, 37(4) (2008), 1113-1124.

[7] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Appl. Math. Com-

put., 155(2) (2004), 493-502.

[8] T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equa-

tions, Appl. Math. Comput., 162(1) (2005), 189-196.

[9] T. Allahviranloo, The Adomian decomposition method for fuzzy system of linear equations,

Appl. Math. Comput., 163(2) (2005), 553-563.

Mathematics Scientic Journal, Islamic Azad University of Arak, 1 (2005), 55-62.

[2] S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear Fredholm

fuzzy integral equations of the second kind, Chaos, Solitons & Fractals, 31(1) (2007), 138-

146.

[3] S. Abbasbandy, A. Jafarian and R. Ezzati, Conjugate gradient method for fuzzy symmetric

positive denite system of linear equations, Appl. Math. Comput., 171(2) (2005), 1184-1191.

[4] S. Abbasbandy, R. Ezzati and A. Jafarian, LU decomposition method for solving fuzzy system

of linear equations, Appl. Math. Comput., 172(1) (2006), 633-643.

[5] S. Abbasbandy, J. J. Nieto and M. Alavi, Tuning of reachable set in one dimensional fuzzy

dierential inclusions, Chaos, Solitons & Fractals, 26(5) (2005), 1337-1341.

[6] S. Abbasbandy, M. Otadi and M. Mosleh, Minimal solution of general dual fuzzy linear

systems, Chaos, Solitons & Fractals, 37(4) (2008), 1113-1124.

[7] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Appl. Math. Com-

put., 155(2) (2004), 493-502.

[8] T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equa-

tions, Appl. Math. Comput., 162(1) (2005), 189-196.

[9] T. Allahviranloo, The Adomian decomposition method for fuzzy system of linear equations,

Appl. Math. Comput., 163(2) (2005), 553-563.

[10] S. E. Amrahov and I. N. Askerzade, Strong solutions of the fuzzy linear systems, Computer

Modeling in Engineering & Sciences, 76 (2011), 207-216.

[11] B. Asady, S. Abbasbandy and M. Alavi, Fuzzy general linear systems, Appl. Math. Comput.,

169(1) (2005), 34-40.

[12] S. Barnet, Matrix methods and applications, Clarendon Press, Oxford, 1990.

[13] W. Cong-Xin and M. Ming, Embedding problem of fuzzy number space, Fuzzy Sets and

Systems, 44(1) (1991), 33-38.

[14] D. Dubois and H. Prade, Operations on fuzzy numbers, J. Systems Sci., 9(6) (1978), 613-626.

[15] R. Ezzati, Solving fuzzy linear systems, Soft Computing, 15(1) (2011), 193-197.

[16] M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96(2)

(1998), 201-209.

[17] M. Friedman, M. Ming and A. Kandel, Duality in fuzzy linear systems, Fuzzy Sets and

Systems, 109(1) (2000), 55-58.

[18] N. Gasilov, A. G. Fatullayev and S. E. Amrahov, Solution of non-square fuzzy linear systems,

Journal of Multiple-Valued Logic and Soft Computing, 20(1) (2013), 221-237.

[19] O. Kaleva, Fuzzy dierential equations, Fuzzy Sets and Systems, 24(3) (1987), 301-317.

[20] A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold,

New York, 1985.

[21] G. J. Klir, U. S. Clair and B. Yuan, Fuzzy set theory: foundations and applications, Prentice-

Hall Inc., 1997.

[22] D. Kincaid and W. Cheney, Numerical analysis, Mathematics of scientic computing. Second

Edition. Brooks/Cole Publishing Co., Pacic Grove, CA, 1996.

[23] M. Ming, M. Friedman and A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems,

108(1) (1999), 83-90.

[24] M. Otadi, New solution of fuzzy linear matrix equations, Theory of Approximation and

Applications, 9(1) (2013), 55-66.

[25] M. Otadi and M. Mosleh, Simulation and evaluation of dual fully fuzzy linear systems by

fuzzy neural network, Applied Mathematical Modelling, 35(10) (2011), 5026-5039.

[26] M. Otadi and M. Mosleh, Solving fully fuzzy matrix equations, Applied Mathematical Mod-

elling, 36(12) (2012), 61146121.

[27] M. Otadi and M. Mosleh, Minimal solution of non-square fuzzy linear systems, Journal of

Fuzzy Set Valued Analysis, Article ID jfsva-00105, doi: 10.5899/2012/jfsva-00105, 2012.

[28] M. Otadi, M. Mosleh and S. Abbasbandy, Numerical solution of fully fuzzy linear systems

by fuzzy neural network, Soft Comput, 15 (2011), 15131522.

[29] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals, 22(5) (2004),

1039-1046.

[30] K. Wang, G. Chen and Y. Wei, Perturbation analysis for a class of fuzzy linear systems, J.

of Comput. Appl. Math., 224(1) (2009), 54-65.

[31] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,

Information Sciences, 8(3) (1975), 199-249.

Modeling in Engineering & Sciences, 76 (2011), 207-216.

[11] B. Asady, S. Abbasbandy and M. Alavi, Fuzzy general linear systems, Appl. Math. Comput.,

169(1) (2005), 34-40.

[12] S. Barnet, Matrix methods and applications, Clarendon Press, Oxford, 1990.

[13] W. Cong-Xin and M. Ming, Embedding problem of fuzzy number space, Fuzzy Sets and

Systems, 44(1) (1991), 33-38.

[14] D. Dubois and H. Prade, Operations on fuzzy numbers, J. Systems Sci., 9(6) (1978), 613-626.

[15] R. Ezzati, Solving fuzzy linear systems, Soft Computing, 15(1) (2011), 193-197.

[16] M. Friedman, M. Ming and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96(2)

(1998), 201-209.

[17] M. Friedman, M. Ming and A. Kandel, Duality in fuzzy linear systems, Fuzzy Sets and

Systems, 109(1) (2000), 55-58.

[18] N. Gasilov, A. G. Fatullayev and S. E. Amrahov, Solution of non-square fuzzy linear systems,

Journal of Multiple-Valued Logic and Soft Computing, 20(1) (2013), 221-237.

[19] O. Kaleva, Fuzzy dierential equations, Fuzzy Sets and Systems, 24(3) (1987), 301-317.

[20] A. Kaufmann and M. M. Gupta, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold,

New York, 1985.

[21] G. J. Klir, U. S. Clair and B. Yuan, Fuzzy set theory: foundations and applications, Prentice-

Hall Inc., 1997.

[22] D. Kincaid and W. Cheney, Numerical analysis, Mathematics of scientic computing. Second

Edition. Brooks/Cole Publishing Co., Pacic Grove, CA, 1996.

[23] M. Ming, M. Friedman and A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems,

108(1) (1999), 83-90.

[24] M. Otadi, New solution of fuzzy linear matrix equations, Theory of Approximation and

Applications, 9(1) (2013), 55-66.

[25] M. Otadi and M. Mosleh, Simulation and evaluation of dual fully fuzzy linear systems by

fuzzy neural network, Applied Mathematical Modelling, 35(10) (2011), 5026-5039.

[26] M. Otadi and M. Mosleh, Solving fully fuzzy matrix equations, Applied Mathematical Mod-

elling, 36(12) (2012), 61146121.

[27] M. Otadi and M. Mosleh, Minimal solution of non-square fuzzy linear systems, Journal of

Fuzzy Set Valued Analysis, Article ID jfsva-00105, doi: 10.5899/2012/jfsva-00105, 2012.

[28] M. Otadi, M. Mosleh and S. Abbasbandy, Numerical solution of fully fuzzy linear systems

by fuzzy neural network, Soft Comput, 15 (2011), 15131522.

[29] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals, 22(5) (2004),

1039-1046.

[30] K. Wang, G. Chen and Y. Wei, Perturbation analysis for a class of fuzzy linear systems, J.

of Comput. Appl. Math., 224(1) (2009), 54-65.

[31] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning,

Information Sciences, 8(3) (1975), 199-249.

January and February 2015

Pages 89-99