Positive solution of non-square fully Fuzzy linear system of equation in general form using least square method

محورهای موضوعی : History and biography

R. Ezzati ¹ (Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran)
A. Yousefzadeh ² (Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran)

تاریخ دریافت : 1393/03/09 تاریخ پذیرش : 1393/06/03 تاریخ انتشار : 1392/12/10

کلید واژه: Fuzzy linear system, fuzzy number, Ranking Function, Fuzzy number vector solution,

چکیده مقاله :

In this paper, we propose the least-squares method for computing the positivesolution of a $m\times n$ fully fuzzy linear system (FFLS) of equations, where $m > n$, based onKaffman's arithmetic operations on fuzzy numbers that introduced in [18]. First, we considerall elements of coefficient matrix are non-negative or non-positive. Also, we obtain 1-cut of thefuzzy number vector solution of the non-square FFLS of equations by using pseudoinverse.If 1-cuts vector is non-negative, we solve constrained least squares problem for computingleft and right spreads. Then, in the special case, we consider 0 is belong to the support ofsome elements of coefficient matrix and solve three overdetermined linear systems and if thesolutions of these systems held in non-negative fuzzy solutions then we compute the solutionof the non-square FFLS of equations. Else, we solve constrained least squares problem forobtaining an approximated non-negative fuzzy solution. Finally, we illustrate the efficiencyof the proposed method by solving some numerical examples.

منابع و مأخذ:

[1] T. Allahviranloo, Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation, 155 (2004) 493-502.

 

[2] T. Allahviranloo, Successive over relaxation iterative method for fuzzy system of linear equations, Applied Mathematics and Computation, 162 (2005) 189-196.

 

[3] T. Allahviranloo, The Adomian decomposition method for fuzzy system of linear equations, Applied Mathematics and Computation, 163 (2005) 553-563.

 

[4] T. Allahviranloo, E. Ahmady, N. Ahmady and Kh. Shams Alketaby, Block Jacobi two-stage method with Gauss-Sidel inner iterations for fuzzy system of linear equations, Applied Mathematics and Computation, 175 (2006) 1217-1228.

 

[5] T. Allahviranloo, N. Mikaeilvand, Non Zero Solutions Of The Fully Fuzzy Linear Systems, Appl. Comput. Math. 10 (2) 271-282.

 

[6] J.J. Buckley and Y. Qu, Solving system of linear fuzzy equations, Fuzzy Sets and Systems, 43 (1991) 33-43.

 

[7] C. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets Syst. 95 (1998) 307-317.

 

[8] M. Dehghan, B. Hashemi and M. Ghatee, Solution of the fully fuzzy linear systems using iterative techniques, Chaos Solutions and Fractals 34 (2007) 316-336.

 

[9] M. Dehghan and B. Hashemi, Solution of the fully fuzzy linear systems using the decomposition procedure, Applied Mathematics and Computation, 182 (2006) 1568-1580.

 

[10] M. Dehghan, B. Hashemi, M. Ghatee, Computational methods for solving fully fuzzy linear systems, Appl Math and Comput 179 (2006) 328-343.

 

[11] M. Dehghan, B. Hashemi, Iterative solution of fuzzy linear systems, Applied Mathematics and Computation, 175 (2006) 645-674.

 

[12] D. Dubois, H. Prade, Fuzzy Sets and Systems; Theory and Applications, Academic Press, New York, 1980.

 

[13] M. Friedman, M. Ma, A. Kandel, Fuzzy linear systems, Proc. IEEE Int. Conf. Syst., Man, Cybernet. 1 (1996) 14-17. 

 

[14] M. Friedman, M. Ma, A. Kandel, Fuzzy linear systems, Fuzzy Sets Syst. 96 (1998) 201-209.

 

[15] R. Ghanbari , N.Mahdavi-Amiri, New solution of linear systems using ranking functions and ABS algorithms, Appl. Math. Comput. 34 (2010) 3363-3375.

 

[16] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317.

 

[17] A. Kumar, J.Kaur, P.Singh, A new method for solving fully fuzzy linear programming problems, Appl. Math. Comput. 35 (2011) 817-823.

 

[18] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic Theory and Applications, Van Nostrand Reinhold, New York,1985.

 

[19] M. Ming, M. Friedman, A. Kandel, General fuzzy least squares, Fuzzy Sets and Systems 88 (1997) 107118.

 

[20] R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.

 

[21] H. T. Nguyen, E.A. Wallker, A First Course in Fuzzy Logic, Chapman , Hall, 2000.

 

[22] V. Sundarapandian, Numerical Linear Algebra,New Dehli, 2008.

 

[23] H. J. Zimmermann, Fuzzy Set Theory and its Applications, third ed., Kluwer Academic , Norwell, 1996.

 

[24] L. A. Zadeh, A fuzzy-set-theoretic interpretation of linguistic hedges, Journal of Cybernetics 2 (1972) 4-34.

 

[25] L. A. Zadeh, The concept of the linguistic variable and its application to approximate reasoning, Information Sciences. 8 (1975) 199-249.

 

[26] R. Ezzati, S. Khezerloo, A. Yousefzadeh, Solving fully fuzzy linear system of equations in general form, Journal of Fuzzy Set Valued Analysis, Volume 2012, Article ID jfsva-00117, (2012) 1-11.