Ghanbari, R., Mahdavi-Amiri, N., Yousefpour, R. (2010). Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach. Iranian Journal of Fuzzy Systems, 7(2), 1-18. doi: 10.22111/ijfs.2010.167

Reza Ghanbari; Nezam Mahdavi-Amiri; Rohollah Yousefpour. "Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach". Iranian Journal of Fuzzy Systems, 7, 2, 2010, 1-18. doi: 10.22111/ijfs.2010.167

Ghanbari, R., Mahdavi-Amiri, N., Yousefpour, R. (2010). 'Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach', Iranian Journal of Fuzzy Systems, 7(2), pp. 1-18. doi: 10.22111/ijfs.2010.167

Ghanbari, R., Mahdavi-Amiri, N., Yousefpour, R. Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach. Iranian Journal of Fuzzy Systems, 2010; 7(2): 1-18. doi: 10.22111/ijfs.2010.167

Exact and approximate solutions of fuzzy LR linear systems: New algorithms using a least squares model and the ABS approach

^{1}Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

^{2}Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

^{3}Department of Mathematics, Mazandaran University, Babolsar, Iran

Abstract

We present a methodology for characterization and an approach for computing the solutions of fuzzy linear systems with LR fuzzy variables. As solutions, notions of exact and approximate solutions are considered. We transform the fuzzy linear system into a corresponding linear crisp system and a constrained least squares problem. If the corresponding crisp system is incompatible, then the fuzzy LR system lacks exact solutions. We show that the fuzzy LR system has an exact solution if and only if the corresponding crisp system is compatible (has a solution) and the solution of the corresponding least squares problem is equal to zero. In this case, the exact solution is determined by the solutions of the two corresponding problems. On the other hand, if the corresponding crisp system is compatible and the optimal value of the corresponding constrained least squares problem is nonzero, then we characterize approximate solutions of the fuzzy system by solution of the least squares problem. Also, we characterize solutions by defining an appropriate membership function so that an exact solution is a fuzzy LR vector having the membership function value equal to one and, when an exact solution does not exist, an approximate solution is a fuzzy LR vector with a maximal membership function value. We propose a class of algorithms based on ABS algorithm for solving the LR fuzzy systems. The proposed algorithms can also be used to solve the extended dual fuzzy linear systems. Finally, we show that, when the system has more than one solution, the proposed algorithms are flexible enough to compute special solutions of interest. Several examples are worked out to demonstrate the various possible scenarios for the solutions of fuzzy LR linear systems.

bibitem{ABS} J. Abaffy, C. Broyden and E. Spedicato, {it A class of direct methods for linear systems}, Numerische Mathematik, {bf45} (1984), 361-376.

bibitem{ABSBOOK} J. Abaffy and E. Spedicato, {it ABS projection algorithms: mathematical techniques for linear and nonlinear equations}, Ellis Horwood, Chichester, 1989.

bibitem{Abbasbandy}S. Abbasbandy and M. Alavi, {it A method for solving fuzzy linear system}, Iranian Journal of Fuzzy Systems, {bf2} (2005), 37-43.

bibitem{abramovich}F. Abramovich, M. Wagenknecht and Y. I. Khurgin, {it Solution of LR-type fuzzy systems of linear algebraic equations}, Busefal, {bf35} (1988), 86-99.

bibitem{Store} R. Bulrisch and J. Stoer, {it Introduction to numerical analysis (texts in applied mathematics)}, 2nd edition, Springer, 1992.

bibitem{Buckley} J. J. Buckley and Y. Qu, {it Solving systems of linear fuzzy equations}, Fuzzy Sets and Systems, {bf43} (1991), 33-43.

bibitem{dub} D. Dubois and H. Prade, {it Fuzzy sets and systems theory and applications}, Academic Press, New York, 1980.

bibitem{Egrvary} E. Egervary, {it On rank-diminishing operations and their applications to the solution of linear equations}, ZAMP, {bf9} (1960), 376-386.

bibitem{iabs} H. Esmaeili, N. Mahdavi-Amiri and E. Spedicato, {it A class of ABS algorithms for Diophantine linear systems}, Numerische Mathematik, {bf91} (2001), 101ï؟½-115.

bibitem{Frie1} M. Friedman, M. Ma and A. Kandel, {it Fuzzy linear systems}, Fuzzy Sets and Systems, {bf96} (1998), 201-209.

bibitem{Fuller}R. Fuller, {it On stability in possibilistic linear equality systems with Lipschitzian fuzzy numbers}, Fuzzy Sets and Systems, {bf34} (1990), 347ï؟½-353.

bibitem{Khoram} M. Khorramizadeh and N. Mahdavi-Amiri, {it Integer extended ABS algorithms and possible control of intermediate results for linear Diophantine systems}, 4OR, {bf7(2)} (2009), 145- 167.

bibitem{FrieD} M. Ma, M. Friedman and A. Kandel, {it Duality in fuzzy linear systems}, Fuzzy Sets and Systems, {bf109} (2000), 55-58.

bibitem{Golab} G. H. Golub and C. F. Van Loan, {it Matrix computations}, Baltimore, MD, 3rd edition, Johns Hopkins University Press, 1996.

bibitem{mah2} N. Mahdavi-Amiri and S. H. Nasseri, {it Duality results and a dual simplex method for linear programming problems with trapezoidal fuzzy variables}, Fuzzy Sets and System, {bf158} (2007), 1961-1978.

bibitem{mal1} H. R. Maleki, M. Tata and M. Mashinchi, {it Linear programming with fuzzy variables}, Fuzzy Sets and Systems, {bf109} (2000), 21-33.

bibitem{Minc} H. Minc, {it Nonnegative matrices}, Wiley, New York, 1988.

bibitem{Muz1} S. Muzzioli and H. Reynaerts, {it Fuzzy linear systems of form $A_{1}x+b_{1}=A_2x+b_2$}, Fuzzy Sets and Systems, {bf157} ( 2006), 939-951.

bibitem{fuzzy logic} H. T. Nguyen and E. A. Walker, {it A first course in fuzzy logic}, Chapman & Hall, 2000.

bibitem{Nicolai}S. Nicolai and E. Spedicato, {it A bibliography of the ABS methods}, Optimization Methods and Software, {bf8} (1997), 171-183.

bibitem{spidicato1} E. Spedicato, E. Bodon, A. D. Popolo and N. Mahdavi-Amiri, {it ABS methods and ABSPACK for linear systems and optimization: a review}, 4OR, {bf1} (2003), 51-66.

bibitem{spedicato2} E. Spedicato, Z. Xia and L. Zhang, {it ABS algorithms for linear equations and optimization}, Journal of Computational and Applied Mathematics, {bf124} (2000), 155-170.

bibitem{vorman} A. Vroman, G. Deschrijver and E. Kerre, {it Solving systems of linear fuzzy equations by parametric functions-an improved algorithm}, Fuzzy Sets and Systems, {bf158} (2007), 1515 ï؟½-1534.

bibitem{Wang} X. Wang, Z. Zhong and M. Ha, {it Iteration algorithms for solving a system of fuzzy linear equations}, Fuzzy Sets and Systems, {bf119} (2001), 121-128.

bibitem{zim2} H. J. Zimmermann, {it Fuzzy set theory and its applications}, Third ed., Kluwer Academic, Norwell, 1996.