## Maximal- and minimal symmetric solutions of fully fuzzy linear systems.(English)Zbl 1220.65033

Summary: We propose a new method to obtain symmetric solutions of a fully fuzzy linear system (FFLS) based on a 1-cut expansion. To this end, we solve the 1-cut of a FFLS (in the present paper, we assumed that the 1-cut of a FFLS is a crisp linear system or equivalently, the matrix coefficient and right hand side have triangular shapes), then some unknown symmetric spreads are allocated to each row of a 1-cut of a FFLS. So, after some manipulations, the original FFLS is transformed to solving $$2n$$ linear equations to find the symmetric spreads.
However, our method always give us a fuzzy number vector solution. Moreover, using the proposed method leads to determining the maximal- and minimal symmetric solutions of the FFLS which are placed in a tolerable solution set and a controllable solution set, respectively. However, the obtained solutions could be interpreted as bounded symmetric solutions of the FFLS which are useful for a large number of multiplications existing between two fuzzy numbers. Finally, some numerical examples are given to illustrate the ability of the proposed method.

### MSC:

 65F05 Direct numerical methods for linear systems and matrix inversion 15B15 Fuzzy matrices 65C30 Numerical solutions to stochastic differential and integral equations

INTOPT_90
Full Text:

### References:

 [1] Friedman, M.; Ming, Ma.; Kandel, A., Fuzzy linear system, Fuzzy Sets and Systems, 96, 209-261 (1998) [2] Allahviranloo, T., Successive over relaxation iterative method for fuzzy system of linear equations, Appl. Math. Comput., 162, 189-196 (2005) · Zbl 1062.65037 [3] Allahviranloo, T., A comment on fuzzy linear systems, Fuzzy Sets and Systems, 140, 559 (2003) · Zbl 1050.15003 [4] Allahviranloo, T.; Afshar Kermani, M., Solution of a fuzzy system of linear equation, Appl. Math. Comput., 175, 519-531 (2006) · Zbl 1095.65036 [5] Allahviranloo, T.; Ahmady, E.; Ahmady, N.; Shams Alketaby, Kh., Block Jacobi two stage method with Gauss-Siedel ineer iterations for fuzzy system of linear equations, Appl. Math. Comput., 175, 1217-1228 (2006) · Zbl 1093.65032 [6] Abbasbandy, S.; Ezzati, R.; Jafarian, A., LU decomposition method for solving fuzzy system of equations, Appl. Math. Comput., 172, 633-643 (2006) · Zbl 1088.65023 [7] Abbasbandy, S.; Jafarian, A., Steepest descent method for system of fuzzy linear equations, Appl. Math. Comput., 175, 823-833 (2006) · Zbl 1088.65026 [8] T. Allahviranloo, S. Salahshour, Fuzzy symmetric solutions of fuzzy linear systems, J. Comput. Appl. Math., in press (doi:10.1016/j.cam.2010.02.042; T. Allahviranloo, S. Salahshour, Fuzzy symmetric solutions of fuzzy linear systems, J. Comput. Appl. Math., in press (doi:10.1016/j.cam.2010.02.042 · Zbl 1220.65032 [9] Allahviranloo, T., Numerical methods for fuzzy system of linear equationa, Appl. Math. Comput., 155, 493-502 (2004) · Zbl 1067.65040 [10] Allahviranloo, T., The Adomian decomposition method for fuzzy system of linear equations, Appl. Math. Comput., 163, 553-563 (2005) · Zbl 1069.65025 [11] Ming, Ma.; Friedman, M.; Kandel, A., Duality in fuzzy linear systems, Fuzzy Sets and Systems, 109, 55-58 (2000) · Zbl 0945.15002 [12] Wang, K.; Zhend, B., Inconsistent fuzzy linear systems, Appl. Math. Comput., 181, 973-981 (2006) · Zbl 1122.15004 [13] Wanga, X.; Zhong, Z.; Ha, M., Iteration algorithms for solving a system of fuzzy linear equations, Fuzzy Sets and Systems, 119, 121-128 (2001) · Zbl 0974.65035 [14] Zheng, B.; Wang, K., General fuzzy systems, Appl. Math. Comput., 181, 1276-1286 (2006) · Zbl 1122.15005 [15] Buckley, J. J.; Qu, Y., Solving system of linear fuzzy equations, Fuzzy Sets and Systems, 43, 33-43 (1991) · Zbl 0741.65023 [16] Buckley, J. J.; Qu, Y., Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems, 38, 43-59 (1990) · Zbl 0713.04004 [17] Buckley, J. J.; Qu, Y., Solving fuzzy equations: a new concept, Fuzzy Sets and Systems, 39, 291-301 (1991) · Zbl 0723.04005 [18] Muzzilio, S.; Reynaerts, H., Fuzzy linear system of the form $$A_1 x + b_1 = A_2 x + b_2$$, Fuzzy Sets and Systems, 157, 939-951 (2006) · Zbl 1095.15004 [19] Dehghan, M.; Hashemi, B.; Ghatee, M., Computational methods for solving fully fuzzy linear systems, Appl. Math. Comput., 179, 328-343 (2006) · Zbl 1101.65040 [20] Dehghan, M.; Hashemi, B.; Ghatee, M., Solution of the fully fuzzy linear systems using iterative techniques, Chaos Solitons Fractals, 34, 316-336 (2007) · Zbl 1144.65021 [21] Vroman, A.; Deschrijver, G.; Kerre, E. E., Solving systems of linear fuzzy equations by parametric functions—an improved algorithm, Fuzzy Sets and Systems, 14, 1515-1534 (2007) · Zbl 1121.65026 [22] T. Allahviranloo, N. Mikaelvand, Non-zero solutions of fully fuzzy linear systems, Int. J. Appl. Comput. Math. (in press).; T. Allahviranloo, N. Mikaelvand, Non-zero solutions of fully fuzzy linear systems, Int. J. Appl. Comput. Math. (in press). [23] Dubois, D.; Prade, H., Fuzzy Sets and Systems: Theory and Applications (1980), Academic Press · Zbl 0444.94049 [24] Goetschel, R.; Voxman, W., Elementary calculus, Fuzzy Sets and Systems, 18, 31-43 (1986) · Zbl 0626.26014 [25] Zimmermann, H. J., Fuzzy Sets Theory and Applications (1985), Kluwer: Kluwer Dorrecht [26] Baker Kearfott, R., Rigorous Global Search: Continuous Problems (1996), Kluwer Academic Publishers: Kluwer Academic Publishers The Netherlands · Zbl 0876.90082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.