An Interval Type-2 Dual Fuzzy Polynomial Equations and Ranking Method of Fuzzy Numbers

Authors: Nurhakimah Ab. Rahman, Lazim Abdullah

Abstract:

According to fuzzy arithmetic, dual fuzzy polynomials cannot be replaced by fuzzy polynomials. Hence, the concept of ranking method is used to find real roots of dual fuzzy polynomial equations. Therefore, in this study we want to propose an interval type-2 dual fuzzy polynomial equation (IT2 DFPE). Then, the concept of ranking method also is used to find real roots of IT2 DFPE (if exists). We transform IT2 DFPE to system of crisp IT2 DFPE. This transformation performed with ranking method of fuzzy numbers based on three parameters namely value, ambiguity and fuzziness. At the end, we illustrate our approach by two numerical examples.

Keywords: Dual fuzzy polynomial equations, Interval type-2, Ranking method, Value.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1090506

References:

[1] S. Abbasbandy and M. Amirfakhrian, "Numerical approximation of fuzzy functions by fuzzy polynomials”, Applied Mathematics and Computation, vol. 174, pp. 669-675, 2006.
[2] S. Abbasbandy, M. Otadi and M. Mosleh, "Numerical solution of a system of fuzzy polynomials by fuzzy neural network,” Information Science 178, pp. 1948-1960, 2008.
[3] H. Rouhparvar, "Solving fuzzy polynomial equation by ranking method,” First Joint Congress on Fuzzy and Intelligent Systems, Ferdowsi University of Mashhad, Iran, pp. 1-6, 2007.
[4] A. Noor’ani, J. Kavikumar, M. Mustafa and S. Nor, "Solving dual fuzzy polynomial equation by ranking method,” Far East Journal Mathematics and Science 51, no. 2, pp. 151-163, 2011.
[5] H.M. Nehi, H.R. Maleki and M. Mashinchi, "A canonical representation for the solution of fuzzy linear system and fuzzy linear programming problem,” Journal Applied Mathematics and Computing, vol. 20, pp. 345-354, 2006.
[6] S. Abbasbandy and M. Otadi, "Numerical solution of fuzzy polynomials by fuzzy neural network,” Applied Mathematics and Computation, vol. 181, pp. 1084-1089, 2006a.
[7] S. Abbasbandy and M. Otadi, "Fuzzy neural network approach to fuzzy polynomials,” Mathware & Soft Computing, 13, pp. 127-134, 2006b.
[8] S. Abbasbandy and M. Alavi, "A method for solving fuzzy linear systems,” Iranian Journal of Fuzzy Systems, vol. 2, pp. 37-43, 2005.
[9] S. Abbasbandy and B. Asady, "Newton’s method for solving fuzzy nonlinear equations,” Applied Mathematics and Computation, vol. 159, pp. 349-356, 2004.
[10] J.J. Buckley and Y. Qu, "Solving linear and quadratic fuzzy equations,” Fuzzy Sets and Systems, vol. 35, pp. 43-59, 1990.
[11] M. Delgado, M.A. Vila and W. Voxman, "On a canonical representation of fuzzy numbers,” Fuzzy Sets and Systems 93, pp. 125-135, 1998a.
[12] M. Delgado, M.A. Vila and W. Voxman, "A fuzziness measure for fuzzy numbers: Application,” Fuzzy Sets and Systems 94, pp. 205-216, 1998b.
[13] J.M. Mendel, R.I. John and F.L. Liu, "Interval type-2 fuzzy logic systems made simple,” IEEE Transactions on Fuzzy Systems, vol. 14(6), pp. 808-821, 2006.
[14] M. Otadi and M. Mosleh, "Solution of fuzzy polynomial equations by modified Adomian decomposition method,” Soft Computing. 15, pp. 187-192, 2011.
[15] T. Allahviranloo and S. Asari, "Numerical solution of fuzzy polynomials by Newton-Raphson method,” Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol. 7, No. 4(27), pp. 17-24, 2011.
[16] G. Klir and T. Folger, "Fuzzy Sets,” Uncertainty and Information (Prentice-Hall, Englewood Cliffs, NJ), 1988.
[17] A. Noor’ani, J. Kavikumar, M. Mustafa and S. Nor, "Solving a system of fuzzy polynomials by ranking method,” Far East Journal of Applied Mathematics, 43(1), pp. 7-20, 2010.