Abstract and Applied Analysis

General Solutions of Fully Fuzzy Linear Systems

T. Allahviranloo, S. Salahshour, M. Homayoun-nejad, and D. Baleanu

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Abstract

We propose a method to approximate the solutions of fully fuzzy linear system (FFLS), the so-called general solutions. So, we firstly solve the 1-cut position of a system, then some unknown spreads are allocated to each row of an FFLS. Using this methodology, we obtain some general solutions which are placed in the well-known solution sets like Tolerable solution set (TSS) and Controllable solution set (CSS). Finally, we solved two examples in order to demonstrate the ability of the proposed method.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2012), Article ID 593274, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449940

Digital Object Identifier
doi:10.1155/2013/593274

Mathematical Reviews number (MathSciNet)
MR3035363

Zentralblatt MATH identifier
1272.93061

Citation

Allahviranloo, T.; Salahshour, S.; Homayoun-nejad, M.; Baleanu, D. General Solutions of Fully Fuzzy Linear Systems. Abstr. Appl. Anal. 2013, Special Issue (2012), Article ID 593274, 9 pages. doi:10.1155/2013/593274. https://projecteuclid.org/euclid.aaa/1393449940


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References

  • M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy Sets and Systems, vol. 96, no. 2, pp. 201–209, 1998.
  • M. Ma, M. Friedman, and A. Kandel, “Duality in fuzzy linear systems,” Fuzzy Sets and Systems, vol. 109, no. 1, pp. 55–58, 2000.
  • T. Allahviranloo, “Numerical methods for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 155, no. 2, pp. 493–502, 2004.
  • T. Allahviranloo, “Successive over relaxation iterative method for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 162, no. 1, pp. 189–196, 2005.
  • T. Allahviranloo, “The Adomian decomposition method for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 163, no. 2, pp. 553–563, 2005.
  • T. Allahviranloo and M. Afshar Kermani, “Solution of a fuzzy system of linear equation,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 519–531, 2006.
  • T. Allahviranloo, E. Ahmady, N. Ahmady, and Kh. S. Alketaby, “Block Jacobi two-stage method with Gauss-Seidel inner iterations for fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 175, no. 2, pp. 1217–1228, 2006.
  • T. Allahviranloo, “A comment on fuzzy linear systems,” Fuzzy Sets and Systems, vol. 140, no. 3, p. 559, 2003.
  • S. Abbasbandy, R. Ezzati, and A. Jafarian, “LU decomposition method for solving fuzzy system of linear equations,” Applied Mathematics and Computation, vol. 172, no. 1, pp. 633–643, 2006.
  • S. Abbasbandy and A. Jafarian, “Steepest descent method for system of fuzzy linear equations,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 823–833, 2006.
  • T. Allahviranloo and M. Ghanbari, “On the algebraic solution of fuzzy linear systems based on interval theory,” Applied Mathematical Modelling, vol. 36, no. 11, pp. 5360–5379, 2012.
  • E. Amrahov and I. N. Askerzade, “Strong solutions of the fuzzy linear systems,” Computer Modeling in Engineering & Sciences, vol. 76, no. 3-4, pp. 207–216, 2011.
  • B. Asady, S. Abbasbandy, and M. Alavi, “Fuzzy general linear systems,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 34–40, 2005.
  • R. B. Kearfott, Rigorous Global Search: Continuous Problems, vol. 13 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
  • R. Ezzati, “Solving fuzzy linear systems,” Soft Comput, vol. 15, pp. 193–197, 2011.
  • N. Gasilov, E. Amrahov, A. G. Fatullayev, H. Karakaş, and O. Ak\in, “A geometric approach to solve fuzzy linear systems,” Computer Modeling in Engineering & Sciences, vol. 75, no. 3-4, pp. 189–203, 2011.
  • A. Ghomashi, S. Salahshour, and H. Hakimzadeh, “Approximating solutions of fully fuzzy linear systems: a financial case study,” Journal of Intelligent and Fuzzy Systems. In press.
  • A. Kumar, N. Babbar, and A. Bansal, “A new computational method to solve fully fuzzy linear systems for negative coefficient matrix,” International Journal of Manufacturing Technology and Management, vol. 25, pp. 19–32, 2012.
  • N. Kumar and A. Bansal, “A new approach for solving fully fuzzy linear systems,” Advances in Fuzzy Systems, vol. 2011, Article ID 943161, 8 pages, 2011.
  • K. Wang and B. Zheng, “Inconsistent fuzzy linear systems,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 973–981, 2006.
  • X. Wang, Z. Zhong, and M. Ha, “Iteration algorithms for solving a system of fuzzy linear equations,” Fuzzy Sets and Systems, vol. 119, no. 1, pp. 121–128, 2001.
  • B. Zheng and K. Wang, “General fuzzy linear systems,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1276–1286, 2006.
  • J. J. Buckley and Y. Qu, “Solving systems of linear fuzzy equations,” Fuzzy Sets and Systems, vol. 43, no. 1, pp. 33–43, 1991.
  • J. J. Buckley and Y. Qu, “Solving linear and quadratic fuzzy equations,” Fuzzy Sets and Systems, vol. 38, no. 1, pp. 43–59, 1990.
  • J. J. Buckley and Y. Qu, “Solving fuzzy equations: a new solution concept,” Fuzzy Sets and Systems, vol. 39, no. 3, pp. 291–301, 1991.
  • S. Muzzioli and H. Reynaerts, “Fuzzy linear systems of the form ${A}_{1}x+{b}_{1}={A}_{2}x+{b}_{2}$,” Fuzzy Sets and Systems, vol. 157, no. 7, pp. 939–951, 2006.
  • M. Dehghan and B. Hashemi, “Iterative solution of fuzzy linear systems,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 645–674, 2006.
  • M. Dehghan, B. Hashemi, and M. Ghatee, “Computational methods for solving fully fuzzy linear systems,” Applied Mathematics and Computation, vol. 179, no. 1, pp. 328–343, 2006.
  • M. Dehghan, B. Hashemi, and M. Ghatee, “Solution of the fully fuzzy linear systems using iterative techniques,” Chaos, Solitons & Fractals, vol. 34, no. 2, pp. 316–336, 2007.
  • T. Allahviranloo and N. Mikaeilvand, “Non zero solutions of the fully fuzzy linear systems,” Applied and Computational Mathematics, vol. 10, no. 2, pp. 271–282, 2011.
  • A. Vroman, G. Deschrijver, and E. E. Kerre, “A solution for systems of linear fuzzy equations in spite of the non-existence of a field of fuzzy numbers,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 13, no. 3, pp. 321–335, 2005.
  • A. Vroman, G. Deschrijver, and E. E. Kerre, “Solving systems of linear fuzzy equations by parametric functions–-an improved algorithm,” Fuzzy Sets and Systems, vol. 158, no. 14, pp. 1515–1534, 2007.
  • A. Vroman, G. Deschrijver, and E. E. Kerre, “Solving systems of linear fuzzy equations by parametric functions,” IEEE Transactions on Fuzzy Systems, vol. 15, pp. 370–384, 2007.
  • T. Allahviranloo, S. Salahshour, and M. Khezerloo, “Maximal- and minimal symmetric solutions of fully fuzzy linear systems,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4652–4662, 2011.
  • D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, vol. 144 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1980.
  • R. Goetschel, Jr. and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31–43, 1986.
  • M. L. Puri and D. A. Ralescu, “Fuzzy random variables,” Journal of Mathematical Analysis and Applications, vol. 114, no. 2, pp. 409–422, 1986.
  • H. J. Zimmermann, Fuzzy Sets Theory and Applications, Kluwer, Dorrecht, The Netherlands, 1985.
  • T. Allahviranloo and S. Salahshour, “Fuzzy symmetric solutions of fuzzy linear systems,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4545–4553, 2011.