Numerical Scheme for Finding Roots of Interval-Valued Fuzzy Nonlinear Equation with Application in Optimization

Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia Department of Mathematics and Statistics, Riphah International University I-14, Islamabad 44000, Pakistan Department of Mathematics, Yildiz Technical University, Faculty of Arts and Science, Esenler, 34210 İstanbul, Turkey Quantum Leap Africa (QLA), AIMS Rwanda Centre, Remera Sector KN 3, Kigali, Rwanda Institut de Mathématiques et de Sciences Physiques (IMSP/UAC), Laboratoire de Topologie Fondamentale, Computationnelle et Leurs Applications (Lab-ToFoCApp), BP 613, Porto-Novo, Benin African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon Department of Mathematics, NUML, Islamabad, Pakistan


Introduction
One of the ancient problems of science and engineering in general and in mathematics is to approximate roots of a nonlinear equation. The nonlinear equations play a major role in the field of engineering, mathematics, physics, chemistry, economics, medicines finance, and in optimization. Many times the particular realization of such type of nonlinear problems involves imprecise and nonprobabilistic uncertainties in the parameter, where the approximations are known due to expert knowledge or due to some experimental data. Due to these reasons, several real-world applications contain vagueness and uncertainties. Therefore, in most of real-world problems, the parameter involved in the system or variables of the nonlinear functions are presented by a fuzzy number or interval-valued trapezoidal fuzzy number. The concept of fuzzy numbers and arithmetic operation with fuzzy numbers were first introduced and investigated in [1][2][3][4][5][6][7][8]. Hence, it is necessary to approximate the roots of fuzzy nonlinear equation.
The standard analytical technique like the Buckley and Qu method [9][10][11][12] is not suitable for solving the equations like ar 6 + br 4 − cr 3 + dr − e = f , r + cos r ð Þ = g, r ln r ð Þ + e r − 1 1 + r 2 + tan r where a, b, c, d, e, f , g,and h are fuzzy numbers and r is a fuzzy variable.
This research article is aimed at proposing efficient higher order iterative method as compared to well-known classical method, such as the Newton-Raphson method. Numerical test results, CPU time, and log of residual show the dominance efficiency of our newly constructed method over the classical Newton's method.
This paper is organized in five sections. In Section 2, we recall some fundamental results of interval-valued trapezoidal fuzzy numbers. In Section 3, we propose numerical iterative scheme for approximating roots of interval-valued trapezoidal fuzzy nonlinear equations and its convergence analysis. In Section 4, we illustrate some real-world applications from optimization as numerical test examples to show the performance and efficiency of the constructed method and conclusions in the last section. Section 5 is a conclusion section.
(1) r is upper semicontinuous (2) rðaÞ = 0 outside some interval ½a 1 , a 2 (3) There are real numbers b 1 , b 2 such that a 1 ≤ b 1 ≤ b 2 ≤ a 2 and (i) rðaÞ is monotonic increasing on ½a 1 , b 1 (ii) rðaÞ is monotonic decreasing on ½b 2 , a 2 We denote by E, the set of all fuzzy numbers. An equivalent parametric form is also given in [19] as follows.
Definition 2 [28]. A fuzzy number r in parametric form is a pair ðr L , r U Þ of function r L ðτÞ,r U ðτÞ,0 ≤ τ ≤ 1, which satisfies the following requirements: (1) r L ðτÞ is a bounded monotonic increasing left continuous function (2) r U ðτÞ is a bounded monotonic decreasing left continuous function A popular fuzzy number is the generalized intervalvalued trapezoidal fuzzy number A, denoted by A = ða 1 , a 2 , a 3 , a 4 ;ŵÞ, 0 <ŵ < 1, a fuzzy number with membership function as follows: Assume F TN ðŵÞ be the family of allŵ-trapezoidal fuzzy number, i.e.,        Journal of Function Spaces Definition 3 [29].
is an interval-valued fuzzy number on set R with . This interval-valued trapezoidal fuzzy number is shown in Figure 1. Moreover, A L ðrÞ ≤ A U ð rÞ, which means the grade of membership r ∈ A = ½A L ðrÞ, A U ðrÞ, and the latest and greatest grade of membership at r are A L ðrÞ and A U ðrÞ, respectively. We therefore denote the family of all interval-valued trapezoidal fuzzy number Definition 4 [29]. A ðw∧ L , w∧ U Þ is said to be nonnegative F ðw∧ L , w∧ U Þ iff a U 1 ≥ 0 and denoted by F + ðw∧ L , w∧ U Þ.
Step 6. Set k = k + 1 and go to step 1.
Definition 8 [31]. Let A ∈ Fðw∧ L , w∧ U Þ; then, alpha-cut set of A denotes and is defined by where

Construction of Iterative Scheme (MM)
In order to approximate the roots of interval-valued trapezoidal fuzzy nonlinear equation FðrÞ = c, we propose the following two-step iterative scheme as follows: ∀τ ∈ 0, 1 ½ : Þ is approximate solutions of the system, t denote the alpha-cut parameter; then, By using Taylor's series of F L l , F L t , F U l , F U t about ðr L l0 ðτÞ, r L t0 ðτÞ, r U l0 ðτÞ, r U t0 ðτÞÞ, then we have the following:  Table 3 τ Since h 1 ðτÞ, k 1 ðτÞ, h 2 ðτÞ, k 2 ðτÞ are unknown quantities, they are obtained by solving the following equations: where ,   Þ   2   6  6  6  6  6  4   3   7  7  7  7  7  5 , For initial guess, one can use the fuzzy number Remark 9. Sequence fðr L ln , r L tn , r U ln , r U tn Þg ∞ n=0 converges to ðα L l , α L t , α U l , α U t Þ iff ∀τ ∈ ½0, 1, lim n⟶∞ r L ln ðτÞ = α L l ðτÞ, lim n⟶∞ r L tn ðτÞ = α L t ðτÞ, lim n⟶∞ r U ln ðτÞ = α U l ðτÞ, and lim n⟶∞ r U tn ðτÞ = α U t ðτÞ.
and if the sequence of fðr L ln , r L tn , r U ln , r U tn Þg ∞ n=0 converges to ðα L l , α L t , α U l , α U t Þ according to Newton's method, then where Proof. It is obvious, because for all ∀τ ∈ ½0, 1 in convergent case, Hence, it is proved. ☐ Finally, it is shown that under certain condition, the MM method for fuzzy equation FðrÞ = 0 is cubic convergent. In compact form for FðrÞ = 0, the MM method can be written as follows: where Theorem 11. Let F : H ⊆ R n ⟶ R n , be u-times Frĕchet differential function on a convex set H containing the root α of FðrÞ = 0; then, the MM method has cubic convergence and satisfies the following error equation.
Proof. Let e n = r n − α and e n+1 = r n+1 − α, then by Taylor series of Fðr n , τÞ in the neighborhood of α if J * and J * * exist. Then, and Fðr, αÞ = 0: Expanding F ′ ðy n , τÞ about α, we have the following: where e n represents the absolute error. We take ∈ = 10 −15 .
In Figure 2, left shows analytical solution of intervalvalued trapezoidal fuzzy nonlinear equation used in Example  Figure 3 shows computational order of convergence of iterative methods MM and NN for finding roots of intervalvalued trapezoidal fuzzy nonlinear equations used in Examples 1-3, respectively.
In Figure 3, MM1-MM4 and NN1-NN4 show computational order of convergence of iterative method MM and NN for approximating roots of interval-valued trapezoidal fuzzy nonlinear equations used in Examples 1-3, respectively. Figure 4 shows computational time in seconds of iterative methods MM and NN for finding roots of interval-valued trapezoidal fuzzy nonlinear equations used in Examples 1-3, respectively.
In Figure 4, MM1-MM4 and NN1-NN4 show computational time in seconds of iterative method MM and NN for finding roots of interval-valued trapezoidal fuzzy nonlinear equation used in Examples 1-3, respectively.
Example 1 Application in optimization (a profit maximization problem). A corporation company wishes to invest one million dollar A 1 = hð10, 20, 30, 40 ; 2/3Þ, ð5, 15, 35, 43 ; 1Þi at fuzzy interest rate r to earn maximum profit, so that after a year, they may withdraw 25000$ S 1 = hð45, 55, 75, 95 ; 2/3Þ , ð80, 90, 110, 120 ; 1Þi approximately and after two years 900000$ S 2 = hð10, 15, 20, 25 ; 2/3Þ, ð5, 10, 25, 47 ; 1Þi left. Find r so that A 1 will be sufficient to cover S 1 and S 2 : where r is an interval-valued trapezoidal fuzzy number whose support lies between ½0, 1 After a year, the amount in the account will be At the end of second year, total amount left is or where B = 2A 1 − S 1 and D = A 1 − S 1 . Therefore, we have to solve or where C = S 2 − D. ð38Þ Table 1 clearly shows the dominance behavior of MM over NN in terms of absolute error on the same number of iterations n = 4 for Example 1. Table 2 shows analytical solutions for Example 1. Figure 5 shows initial guessed values, analytical, and numerical approximate solution graph of iterative methods MM and NN for interval-valued trapezoidal fuzzy nonlinear equation used in Example 1.