Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran


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


Main Subjects

[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 di erential 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.