Solving Fuzzy Nonlinear Equations by a General Iterative Method

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1 Journal of Uncertain Systems Vol.4, No.3, pp , 200 Online at: Solving Fuzzy Nonlinear Equations by a General Iterative Method Anjeli Garg, S.R. Singh * Department of Mathematics, Faulty of Science, Banaras Hindu University, Varanasi, Uttar Pradesh-22005, INDIA Received 7 March 2009; Revised 2 August 2009 Abstract The present paper aims to study solution of fuzzy nonlinear equations, whose some parameters are fuzzy numbers. The fuzzy numbers have been presented in parametric form and a general iterative method has been proposed for the numerical solution of a system of fuzzy nonlinear equations. The proposed method has also been illustrated by an example to show the efficiency of the developed algorithm. 200 World Academic Press, UK. All rights reserved. Keywords: fuzzy number, parametric form, membership function, fuzzy nonlinear system, general iteration method Introduction The concept of fuzzy numbers and arithmetic operations on it was introduced by Zadeh [2, 30] which was further enriched by Mizumoto and Tanaka [2]. Later on Dubois and Prade [3] made a significant contribution by introducing the concept of LR fuzzy numbers and presented a computational formula for operations on fuzzy numbers. With the development in the theory of fuzzy numbers, one of the major area emerged for application of these fuzzy numbers, is the solution of equations whose parameters are fuzzy numbers. Solution of the system of fuzzy equations is required in various areas such as Physics, Chemistry, Economics and many financial systems. Buckley and Qu [9] studied the solution of linear and quadratic equations when parameters are either real or complex fuzzy numbers. The study used the extension principle approach and envisaged that many simple fuzzy equations have no solution. Further, Buckley and Qu [0, ] addressed this problem and gave a new solution concept for such cases. Friedman, Ming and Kandel [6] proposed a general model for solving a nn nn fuzzy linear system whose coefficient matrix is crisp and right side column is an arbitrary fuzzy number vector. Wang, Zhong and Ha[27] also proposed an iterative algorithm for solving a system of fuzzy linear equations. Allahviranloo [3] presented algorithm for numerical solution of fuzzy system of linear equations based on iterative Jacobi and Gauss Siedel methods. Asady, Abbasbandy and Alavi [2] considered the solution of a mm nn fuzzy general linear system for a case when mm nn. Different approaches to solve fuzzy linear systems have also been given by several authors, such as Abbasbandy and Jafarian [4], Abbasbandy, Ezzati and Jafarian [5], Muzzioli and Reynaerts [22], Dehghan, Hashemi and Ghatee [5] and Wang and Zheng [28]. In real life problem fuzzy system of equations also occur in nonlinear forms and are not solved by methods commonly used for linear systems, Abbasbandy and Asady [] considered this problem and presented a Newton s method for solving fuzzy nonlinear equations, which was applied by Kajani, Asady and Venchen [9] for solving a dual fuzzy nonlinear systems. Selekwa and Collins [23] considered the numerical solution of systems of qualitative nonlinear algebraic equations by fuzzy logic. Ujevic [26] presented a method for solution of nonlinear equations based on derived quadrature rules. The solution of fuzzy nonlinear equations by steepest descent method was considered by Abbasbany and Jafarian [6], where as Jafari and Varsha [8] presented a revised adomian decomposition method. For the purpose, Abbasbandy [7] further proposed Newton s method for solving a system of fuzzy nonlinear equations, where as Saavedra, Mangueira, Cano and Flores [24] described solution of nonlinear fuzzy systems by decomposition of incremental fuzzy numbers. Recently, Shokri [25] proposed an approach of midpoint Newton s method for the solution of system of fuzzy nonlinear equations. Here, the motivation of the present work is * Corresponding author. singh_shivaraj@rediffmail.com (S.R. Singh).

2 Journal of Uncertain Systems, Vol.4, No.3, pp , to develop a general computational algorithm, which is based on the theory of fixed point iteration, for solutions of fuzzy nonlinear equations. All the previous works of solving fuzzy system of nonlinear equations have used the Newton s approach. Here, in this paper, we have extended the theory of general iterative method for solving fuzzy nonlinear systems, where right hand side of equation was taken to be a fuzzy number. The algorithm for the method has been developed and illustrated by numerical example for finding the positive fuzzy real roots of a fuzzy nonlinear system of equations. The present work has been organized in the coming section; Section 2 describes and presents fuzzy numbers in parametric form and the arithmetic operations on it. Section 3 of the paper describes the method and algorithm to the general iteration method for solution fuzzy system of nonlinear equations followed by numerical illustration of the method in Section 4 and the conclusion is placed in Section 5. 2 Fuzzy Number and its Parametric Form In this section, some basics of fuzzy numbers and its parametric representation as presented in [,2,3,2,30 ] are being viewed and some of the needed are being reproduced to make the study self contained. Definition A fuzzy number is a fuzzy set, uu : R II = [0,], which satisfies ) uu is upper semi-continuous; 2) There are real numbers aa, bb, cc, dd such that cc aa bb dd and uu (xx) is monotonically increasing on [cc, aa], uu (xx) is monotonically decreasing on [bb, dd], uu (xx) =, aa xx bb; 3) uu (xx) = 0, if xx lies outside the interval [cc, dd]. Definition 2 A fuzzy number uu in parametric form is a pair uu, uu of function uu(rr), uu(rr) 0 rr, which satisfies the following conditions ) uu(rr) is a bounded monotonic increasing left continuous function over [0, ]; 2) uu(rr) is a bounded monotonic decreasing left continuous function over [0, ]; 3) uu(rr) uu(rr), 0 rr. Trapezoidal Fuzzy Number A fuzzy number uu is the trapezoidal fuzzy number defined as uu = (xx 0, yy 0, σσ, ββ) with interval defuzzifier [xx 0, yy 0 ] and left fuzziness σσ and right fuzziness ββ, where the membership function is and its parametric form is σσ (xx xx 0 + σσ) xx xx 0 xx xx 0 xx [xx uu (xx) = 0, yy 0 ] ββ (yy 0 xx + ββ) yy 0 xx yy 0 + ββ 0 ooooheeeeeeeeeeee uu(rr) = xx 0 σσ + σσσσ and uu(rr) = yy 0 + ββ ββββ. Further, if xx 0 = yy 0, then uu = (xx 0, σσ, ββ) is called the triangular fuzzy number. Arithmetic Operations The addition and scalar multiplication of fuzzy numbers in parametric form are defined using the interval arithmetic and are presented as following. For any two fuzzy numbers uu = uu, uu and vv = vv, vv, we define addition uu + vv and multiplication by scalar kk as Addition uu + vv (rr) = uu(rr) + vv(rr), uu + vv (rr) = uu(rr) + vv(rr). Scalar Multiplication kkkk (rr) = kkuu(rr), kkkk (rr) = kkuu(rr) for kk > 0 kkkk (rr) = kkuu(rr), kkkk (rr) = kkuu(rr) for kk < 0.

3 208 A. Garg and S.R. Singh: Solving Fuzzy Nonlinear Equations Multiplication of Two Fuzzy Numbers Multiplication of two fuzzy numbers uu = uu, uu and vv = vv, vv can be defined in parametric form using interval arithmetic as uu vv = uu(rr), uu(rr) vv(rr), vv(rr) = min uuvv, uuvv, uuvv, uuvv, max uuvv, uuvv, uuvv, uuvv. ) If uu > 0 and vv > 0, then 2) If uu < 0 and vv > 0, then and 3) If uu < 0 and vv < 0, then and 4) If uu > 0 and vv < 0, then and uu(rr), uu(rr) vv(rr), vv(rr) = uu(rr) vv(rr), uu(rr) vv(rr) ; uu = uu(rr), uu(rr) uu vv = uu(rr), uu(rr) vv(rr), vv(rr) = uu(rr)vv(rr), uu(rr)vv(rr) ; uu = uu(rr), uu(rr) and vv = vv(rr), vv(rr) uu vv = uu(rr)vv(rr), uu(rr)vv(rr) ; vv = vv(rr), vv(rr) uu vv = uu(rr)vv(rr), uu(rr)vv(rr). Inverse operation Inverse of a fuzzy number uu is defined considering different cases. Here it is to be noted that origin does not belong to the 0-cut of uu, i.e. uu is either strictly positive or strictly negative. And if uu < 0 uu(rr) < 0 and uu(rr) < 0, then uu > 0 uu(rr) > 0 and uu(rr) > 0, uu = uu(rr), uu(rr) = uu(rr), uu(rr). uu = uu(rr), uu(rr), uu = uu(rr), uu(rr) = uu(rr), = uu(rr) uu(rr), uu(rr). Division Division of a fuzzy number uu by a fuzzy number vv can be defined by multiplication of uu with inverse of vv as discussed above. Here again, origin does not lie in the 0-cut of vv. uu vv = uu vv and then considering different possible cases, value of division of uu by vv, i.e., uu can be calculated. vv 3 A General Iteration Method In this section we consider the solution of fuzzy nonlinear system ff(xx, yy ) = cc, gg(xx, yy ) = cc 2. First, we rewrite system of equation () using parametric representation of fuzzy numbers xx, yy, cc and cc 2 as ff xx, xx, yy, yy, rr = cc, ff 2 xx, xx, yy, yy, rr = cc, gg xx, xx, yy, yy, rr = cc 2, gg 2 xx, xx, yy, yy, rr = cc 2. () (2)

4 Journal of Uncertain Systems, Vol.4, No.3, pp , Then the system (2) can be equivalently written as where xx = FF xx, xx, yy, yy, rr, yy = GG xx, xx, yy, yy, rr, xx = FF 2 xx, xx, yy, yy, rr, yy = GG 2 xx, xx, yy, yy, rr FF xx, xx, yy, yy, rr = xx + αα ff xx, xx, yy, yy cc, GG xx, xx, yy, yy, rr = yy + ββ gg xx, xx, yy, yy cc 2, FF 2 xx, xx, yy, yy, rr = xx + γγ ff 2 xx, xx, yy, yy cc, GG 2 xx, xx, yy, yy, rr = yy + δδ gg 2 xx, xx, yy, yy cc 2. (3) (4) Let ζζ, ηη be the exact solution of the above system, where ζζ = ζζ, ζζ and ηη = ηη, ηη, then it will satisfy the given equations rr [0,], ζζ = FF ζζ, ζζ, ηη, ηη, rr, ηη = GG ζζ, ζζ, ηη, ηη, rr, ζζ = FF 2 ζζ, ζζ, ηη, ηη, rr, ηη = GG 2 ζζ, ζζ, ηη, ηη, rr, (5) ζζ ηη and its solution vector can be represented as. ζζ ηη Now, if (xx 0, yy ) 0 be a suitable initial approximation to ζζ, ηη, then we study the fixed point iteration as if kk tth iterate is known, then (kk + ) tth iterate can be found as Using vector notation XX kk+ = gg(xx kk ) where xx kk+ = FF xx kk, xx kk, yy kk, yy kk, rr, yy kk+ = GG xx kk, xx kk, yy kk, yy kk, rr, xx kk+ = FF 2 xx kk, xx kk, yy kk, yy kk, rr, yy kk+ = GG 2 xx kk, xx kk, yy kk, yy kk, rr. xx FF xx, xx, yy, yy, rr kk yy kk XX kk = GG xx, xx, yy, yy, rr and gg(xx) = xx kk FF 2 xx, xx, yy, yy, rr yy kk. GG 2 xx, xx, yy, yy, rr If above iterative method converges, then lim kk xx kk = ζζ, lim kk yy kk = ηη. Now subtracting system of equations (6) from system of equations (5) respectively, we get (6)

5 20 A. Garg and S.R. Singh: Solving Fuzzy Nonlinear Equations ζζ xx kk+ = FF ζζ, ζζ, ηη, ηη, rr FF xx kk, xx kk, yy kk, yy kk, rr, ηη yy kk+ = GG ζζ, ζζ, ηη, ηη, rr GG xx kk, xx kk, yy kk, yy kk, rr, ζζ xx kk+ = FF 2 ζζ, ζζ, ηη, ηη, rr FF 2 xx kk, xx kk, yy kk, yy kk, rr, ηη yy kk+ = GG 2 ζζ, ζζ, ηη, ηη, rr GG 2 xx kk, xx kk, yy kk, yy kk, rr. Now, let εε kk = ζζ xx kk, εε kk = ζζ xx kk, δδ kk = ηη yy kk and δδ kk = ηη yy kk be errors in the k th iterate. Then, εε kk+ = FF xx kk + εε kk, xx kk + εε kk, yy kk + δδ kk, yy kk + δδ kk, rr FF xx kk, xx kk, yy kk, yy kk, rr, δδ kk+ = GG xx kk + εε kk, xx kk + εε kk, yy kk + δδ kk, yy kk + δδ kk, rr GG xx kk, xx kk, yy kk, yy kk, rr, εε kk+ = FF 2 xx kk + εε kk, xx kk + εε kk, yy kk + δδ kk, yy kk + δδ kk, rr FF 2 xx kk, xx kk, yy kk, yy kk, rr, δδ kk+ = GG 2 xx kk + εε kk, xx kk + εε kk, yy kk + δδ kk, yy kk + δδ kk, rr GG 2 xx kk, xx kk, yy kk, yy kk, rr. (7) (8) Further, using Taylor series expansion of FF, FF 2, GG and GG 2 about xx kk, xx kk, yy kk, yy kk, rr and neglecting the second and higher powers of εε kk, εε kk, δδ kk and δδ kk, we get, rr [0,], εε kk+ = εε kk FF xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk FF yy xx kk, xx kk, yy kk, yy kk, rr + εε kk FF xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk FF yy xx kk, xx kk, yy kk, yy kk, rr, δδ kk+ = εε kk GG xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk GG yy xx kk, xx kk, yy kk, yy kk, rr + εε kk GG xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk GG yy xx kk, xx kk, yy kk, yy kk, rr, εε kk+ = εε kk FF 2xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk FF 2yy xx kk, xx kk, yy kk, yy kk, rr + εε kk FF 2xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk FF 2yy xx kk, xx kk, yy kk, yy kk, rr, δδ kk+ = εε kk GG 2xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk GG 2yy xx kk, xx kk, yy kk, yy kk, rr + εε kk GG 2xx xx kk, xx kk, yy kk, yy kk, rr + δδ kk GG 2yy xx kk, xx kk, yy kk, yy kk, rr, which can be written in matrix form as εε kk+ FF xx FF yy FF xx FF yy δδ kk+ GG xx GG yy GG xx GG εε kk yy = δδ kk εε kk+ δδ kk+ FF 2xx FF 2yy FF 2xx FF 2yy εε GG 2xx GG 2yy GG 2xx GG 2yy kk δδ kk where the partial derivatives FF xx, FF yy, FF xx, FF yy,. are evaluated at kk tth iterate. It can also be written as (9) (0) where ee (kk) = εε kk δδ kk εε kk δδ kk and the Jacobian matrix is given by ee (kk+) = AA kk ee (kk) FF xx FF yy FF xx FF yy GG xx GG yy GG xx GG yy AA(xx) =. () FF 2xx FF 2yy FF 2xx FF 2yy GG 2xx GG 2yy GG 2xx GG 2yy AA kk is the Jacobian matrix of the iteration functions FF and GG evaluated at (xx kk, yy ). kk Now a sufficient condition for convergence of the solution for kk and rr [0,] is AA kk < where. is some suitable norm. If we use the maximum absolute row sum norm, we get the conditions

6 Journal of Uncertain Systems, Vol.4, No.3, pp , FF xx + FF yy + FF xx + FF yy <, GG xx + GG yy + GG xx + GG yy <, FF 2xx + FF 2yy + FF 2xx + FF 2yy <, GG 2xx + GG 2yy + GG 2xx + GG 2yy < and the necessary and sufficient condition for convergence is that for each k, ρρ(aa kk ) <, where ρρ(aa kk ) is the spectral radius of the matrix AA kk. Further, if (xx, 0 yy ) 0 is the close approximation of ζζ, ηη, then we usually test the conditions at the initial approximation (xx, 0 yy ) 0 only. Remark : Sequence xx nn, xx nn and yynn, yy nn converge to ζζ, ζζ and ηη, ηη respectively if and only if nn=0 nn=0 rr [0,], Lemma Let ff ζζ, ηη = 0 gg ζζ, ηη = 0. ηη, ηη respectively, then lim nn xx nn (rr) = ζζ(rr), lim nn xx nn (rr) = ζζ(rr), lim nn yy nn (rr) = ηη(rr), lim nn yy nn (rr) = ηη(rr). If the sequence of xx nn, xx nn nn=0 where PP nn = sup max 0 rr εε nn (rr), εε nn (rr), δδ nn (rr), δδ nn (rr). lim nn PP nn = 0 Proof: Proof is obvious, because for all r [ 0,] in the convergent case and yy nn, yy nn nn=0 lim nn εε nn (rr) = lim nn εε nn (rr) = lim nn δδ nn (rr) = lim nn δδ nn (rr) = 0. (2) converge to ζζ, ζζ and Theorem Let D be a closed, bounded and convex set in the plane. Assume that the components of F and G, i.e. FF, FF 2, GG aaaaaa GG 2, are continuously differentiable with respect to xx, xx, yy and yy and further assume that ) gg(dd) DD; 2) λλ max AA(xx) <. Then a) XX = gg(xx) has a unique solution αα DD; b) for any initial point XX 0, the iteration will converge in D to αα; c) αα XX nn+ ( GG(αα) + εε nn ) αα XX nn with εε nn 0 aaaa nn. Proof: See [8]. 4 Numerical Illustration In this section, we implement the developed algorithm for solution of fuzzy nonlinear equations for positive real roots. Consider a fuzzy nonlinear system as xx2 + 3xx + yy = 5, + 3yy (3) 2 = 4. xx 2 Now assuming the fuzzy numbers xx and yy be positive fuzzy numbers and are being represented as xx = xx, xx and yy = yy, yy and let fuzzy numbers 4 and 5 be written in parametric form as 4 = ( rr, rr), 5 = ( rr, rr).

7 22 A. Garg and S.R. Singh: Solving Fuzzy Nonlinear Equations Thus the fuzzy system (3) in fuzzy parametric form be written as It can be suitably rewritten as xx2 + 3xx + yy = rr, xx 2 + 3yy 2 = rr, xx 2 + 3xx + yy = rr, xx 2 + 3yy 2 = rr. (4) where and the Jacobian matrix is xx = FF xx, xx, yy, yy, rr, yy = GG xx, xx, yy, yy, rr, xx = FF 2 xx, xx, yy, yy, rr, yy = GG 2 xx, xx, yy, yy, rr FF xx, xx, yy, yy, rr = xx + αα xx 2 + 3xx + yy rr, GG xx, xx, yy, yy, rr = yy + ββ xx 2 + 3yy rr, FF 2 xx, xx, yy, yy, rr = xx + γγ xx 2 + 3xx + yy rr, GG 2 xx, xx, yy, yy, rr = yy + δδ xx 2 + 3yy rr, (5) (6) FF xx FF yy FF xx FF yy GG xx GG yy GG xx GG yy AA kk = FF 2xx FF 2yy FF 2xx FF 2yy GG 2xx GG 2yy GG 2xx GG 2yy aaaa kk tth iiiiiiiiiiiiii. By the necessary condition of convergence of solution, we have + αα 2xx kk + 3 αα 0 0 = ββ 2xx kk + ββ 6yy kk 0 0. (7) γγ(2xx kk + 3) γγ 0 0 δδ(2xx kk ) + δδ(6yy kk ) Now if we choose the initial guess as then at rr = 0, it reduces to + αα 2xx kk αα <, ββ 2xx kk + + ββ 6yy kk <, + γγ(2xx kk + 3) + γγ <, δδ(2xx kk ) + + δδ(6yy kk ) <. xx 0 = 0.5 = ( rr, rr), = 0.5 = ( rr, rr), yy 0 xx 0 = 0.25, xx 0 = 0.75, yy 0 = 0.25, yy 0 = 0.75, (8) (9) and the above conditions (8) reduces to + 3.5αα + αα <, 0.5ββ + +.5ββ <, + 4.5γγ + γγ <,.5δδ δδ <. Now, a choice of αα = γγ = /4 and ββ = δδ = /6, satisfy those conditions in (20). Hence the equations for finding successive iterates become (20)

8 Journal of Uncertain Systems, Vol.4, No.3, pp , xx kk+ = xx 4 kk 2 xx + yy kk rr, yy kk+ = xx 6 kk 2 + 3yy 2 kk 6yy kk rr, xx kk+ = xx 4 kk 2 xx kk + yy kk rr, yy kk+ = xx 6 kk 2 + 3yy 2 kk 6yy kk rr. Computing the solutions only for three iterations by the proposed general iterative method, we obtain the solution of xx and yy with the maximum error of the order 0 2 and are placed in Table. Computations have also been carried out by Newton s iterative method and are placed in Table for comparison with the proposed method. Table : The solution of xx and yy Solution by proposed method (3 rd iteration) Solution by Newton s method (3 rd iteration) rr xx yy xx yy The efficiency of the proposed algorithm in comparison with the existing Newton s algorithm [, 7] can be further visualized in Figure to Figure 4. (2).2 Solution by proposed method for x(3rd iteration) r Figure : Solution for xx by proposed method

9 24 A. Garg and S.R. Singh: Solving Fuzzy Nonlinear Equations Solution by proposed method for y (3rd iteration) r Figure 2: Solution for yy by proposed method Solution by Newton's method for x(3rd iteration) r Figure 3: Solution for xx by Newton s method Solution by Newton's method for y (3rd iteration) r 5 Conclusions Figure 4: Solution for y by Newton s method Thus this paper describes a procedure for solution of fuzzy nonlinear equations by a general iteration method based on fixed point theory. The method may be useful in modeling of nonlinear systems arising in Economics and Financial problems, where the parameters have vagueness and some degree of fuzziness. The developed algorithm

10 Journal of Uncertain Systems, Vol.4, No.3, pp , has been implemented for numerical solution of a fuzzy nonlinear system of equations as an example to illustrate the efficiency of the developed algorithm. Acknowledgements The authors are highly thankful to the University Grants Commission, New Delhi, INDIA, for the provided financial assistance. References [] Abbasbandy, S., and B. Asady, Newton s method for solving fuzzy nonlinear equations, Applied Mathematics and Computation, vol.59, pp , [2] Asady, B., S. Abbasbandy, and M. Alavi, Fuzzy general linear systems, Applied Mathematics and Computation, vol.69, pp.34 40, [3] Allahviranloo, T., Numerical methods for fuzzy system of linear equations, Applied Mathematics and Computation, vol.55, pp , [4] Abbasbandy, S., and A. Jafarian, Steepest descent method for system of fuzzy linear equations, Applied Mathematics and Computation, vol.75, pp , [5] Abbasbandy, S., R. Ezzati, and A. Jafarian, LU decomposition method for solving fuzzy system of linear equations, Applied Mathematics and Computation, vol.72, pp , [6] Abbasbandy, S., and A. Jafarian, Steepest descent method for solving fuzzy nonlinear equations, Applied Mathematics and Computation, vol.74, pp , [7] Abbasbandy, S., and R. Ezzati, Newton s method for solving a system of fuzzy nonlinear equations, Applied Mathematics and Computation, vol.75, pp.89 99, [8] Atkinson, K.E., An introduction to Numerical Analysis, John Wiley and Sons, New York, 989. [9] Buckley, J.J., and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems, vol.38, pp.43 59, 990. [0] Buckley, J. J., and Y. Qu, Solving fuzzy equations: a new solution concept, Fuzzy Sets and Systems, vol.39, pp.29 30, 99. [] Buckley, J. J., and Y. Qu, Solving system of fuzzy linear equations, Fuzzy Sets and Systems, vol.43, pp.33 43, 99. [2] Chang, S.S.L., and L.A. Zadeh, On fuzzy mapping and control, IEEE Trans. Systems, Man cybernet, vol.2, pp.30 34, 972. [3] Dubois, D., and H. Prade, Operations on fuzzy numbers, Int. J. systems Science, vol.9, pp , 978. [4] Dubois, D., and H. Prade, Fuzzy Sets and System: Theory and Application, Academic Press, New York, 980. [5] Dehghan, M., B. Hashemi, and M. Ghatee, Computational methods for solving fully fuzzy linear systems, Applied Mathematics and Computation, vol.79, pp , [6] Friedman, M., M. Ming, and A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, vol.96, pp , 998. [7] Goetschel, R., and W. Voxman, Elementary calculus, Fuzzy Sets Systems, vol.8, pp.3 43, 986. [8] Jafari, H., D.G. Varsha, Revised adomian decomposition method for solving a system of nonlinear equations, Applied Mathematics and Computation, vol.75, pp. 7, [9] Kajani, M.T., B. Asady, and A.H. Venchen, An iterative method for solving dual fuzzy nonlinear equations, Applied Mathematics and Computation, vol.67, pp , [20] Moore, R.E., Methods and applications of interval analysis, SIAM Philadelphia PA, 979. [2] Mizumoto, M., and K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems Comput. Controls, vol.7, pp.73 8, 976. [22] Muzzioli, S., H. Reynaerts, Fuzzy linear systems of the form A x + b = A 2 x + b 2, Fuzzy Sets and System, vol.57, pp , [23] Selekwa, M.F., E.G. Collins Jr., Numerical solution for systems of qualitative nonlinear algebraic equations by fuzzy logic, Fuzzy Sets and Systems, vol.50, , [24] Saavedra, O.R., H.H.D. Mangueira, Y.C. Cano, and H.R. Flores, Solution of nonlinear fuzzy systems by decomposition of incremental fuzzy numbers, Information Sciences, vol.77, pp , [25] Shokri, J., On systems fuzzy nonlinear equations, Applied Mathematical Sciences, vol.2, pp , [26] Ujevic, N., A method for solving nonlinear equations, Applied Mathematics and Computation, vol.74, pp , [27] Wang, X., Z. Zhong, and M. Ha, Iteration algorithms for solving a system of fuzzy linear equations, Fuzzy Sets and Systems, vol.9, pp.2 28, 200. [28] Wang, K., and B. Zheng, Inconsistent fuzzy linear systems, Applied Mathematics and Computation, vol.8, pp , [29] Zheng, B., and K. Wang, General fuzzy linear systems, Applied Mathematics and Computation, vol.8, pp , [30] Zadeh, L.A., The concept of linguistic variable and its application to approximate reasoning, Information Sciences, vol.8, pp , 975.