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

Authors: Ahmed Elmoasry; Mudassir Shams; Naveed Yaqoob; Nasreen Kausar; Yaé Ulrich Gaba; Naila Rafiq
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

In this research article, we propose efficient numerical iterative methods for estimating roots of interval-valued trapezoidal fuzzy nonlinear equations. Convergence analysis proves that the order of convergence of numerical schemes is 3. Some real-life applications are considered from optimization as numerical test problems which contain interval-valued trapezoidal fuzzy quantities in parametric form. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in literature.


---

## Body

## 1. 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 [1–8]. Hence, it is necessary to approximate the roots of fuzzy nonlinear equation.
(1)Fr=c.The standard analytical technique like the Buckley and Qu method [9–12] is not suitable for solving the equations like
(2)ar6+br4−cr3+dr−e=f,r+cosr=g,rlnr+er−11+r2+tanr=h,where a, b, c, d, e, f, g,and h are fuzzy numbers and r is a fuzzy variable.We therefore look towards numerical iterative schemes which approximate the roots of fuzzy nonlinear equations. To approximate roots of fuzzy nonlinear equations, Abbsbandy and Asady [13] used Newton’s method, Allahviranloo and Asari [14] used the Newton-Raphson method, Mosleh [15] used the Adomian decomposition method, and Ibrahim et al. give the Levenberg-Marquest method (see also [16–23]).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 Section2, 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.


## 2. Preliminaries

Definition 1.
A fuzzy number is a fuzzy set liker:R⟶I=0,1 which satisfies [24–27].
(1)
r is upper semicontinuous(2)
ra=0 outside some interval a1,a2(3)
There are real numbersb1,b2 such that a1≤b1≤b2≤a2 and
(i)
ra is monotonic increasing on a1,b1(ii)
ra is monotonic decreasing on b2,a2(iii)
ra=1, for b1≤a≤b2
We denote byE, the set of all fuzzy numbers. An equivalent parametric form is also given in [19] as follows.Definition 2 [28].
A fuzzy numberr in parametric form is a pair rL,rU of function rLτ,rUτ,0≤τ≤1, which satisfies the following requirements:
(1)
rLτ is a bounded monotonic increasing left continuous function(2)
rUτ is a bounded monotonic decreasing left continuous function(3)
rLτ≤rUτ,0≤τ≤1
A popular fuzzy number is the generalized interval-valued trapezoidal fuzzy numberA, denoted by A=a1,a2,a3,a4;w^,0<w^<1, a fuzzy number with membership function as follows:
(3)Ar=w^r−a1a2−a1ifa1<r<a2,w^ifa2≤r≤a3,w^a4−ra4−a3a3<r<a4,0otherwise.
AssumeFTNw^ be the family of all w^-trapezoidal fuzzy number, i.e.,
(4)FTNw^=A=a1,a2,a3,a4;w^,a1≤a2≤a3≤a4;0<w^<1.Definition 3 [29].
LetAL∈FTNw∧L and AU∈FTNw∧UA level w∧L,w∧U− interval-valued trapezoidal fuzzy number A, denoted by
(5)A=AL,AU=a1L,a2L,a3L,a4L;w∧L,a1U,a2U,a3U,a4U;w∧U,is an interval-valued fuzzy number on set R with
(6)ALr=w∧Lr−a1La2L−a1Lifa1L<r<a2L,w∧Lifa2L≤r≤a3L,w∧La4L−ra4L−a3La3L<r<a4L,0otherwise,lowertrapezoidalfuzzynumberAUr=w∧Ur−a1Ua2U−a1Uifa1U<r<a2U,w∧Uifa2U≤r≤a3U,w∧Ua4U−ra4U−a3Ua3U<r<a4U,0otherwise.uppertrapezoidalfuzzynumberwhere a1L≤a2L≤a3L≤a4L, a1U≤a2U≤a3U≤a4U, 0≤w∧L≤w∧U≤1, a1U≤a1L, and a1L≤a4U. This interval-valued trapezoidal fuzzy number is shown in Figure 1. Moreover, ALr≤AUr, which means the grade of membership r∈A=ALr,AUr, and the latest and greatest grade of membership at r are ALr and AUr, respectively. We therefore denote the family of all interval-valued trapezoidal fuzzy number by Fw∧L,w∧U=A=ALr,AUr, i.e.,
(7)Fw∧L,w∧U=A=ALr,AUr=a1L,a2L,a3L,a4L;w∧L,a1U,a2U,a3U,a4U;w∧U:a1U≤a1L;a4L≤a4UALr∈FTNw∧L,AUr∈FTNw∧U,0≤w∧L≤w∧U≤1.Definition 4 [29].
Aw∧L,w∧U is said to be nonnegative Fw∧L,w∧U iff a1U≥0 and denoted by F+w∧L,w∧U.Definition 5 [30].
Twow∧L,w∧U− interval-valued trapezoidal fuzzy numbers.
A=a1L,a2L,a3L,a4L;w∧L,a1U,a2U,a3U,a4U;w∧U and B=b1L,b2L,b3L,b4L;w∧L,b1U,b2U,b3U,b4U;w∧U are said to be equal iff A=B, i.e., aiL=biL and aiU=biU for all i=1,2,3,4.Definition 6. [30].
Extend addition, scalar multiplication, and extend multiplication inw∧L,w∧U interval-valued trapezoidal fuzzy number are defined as if A=a1L,a2L,a3L,a4L;w∧L,a1U,a2U,a3U,a4U;w∧U and B=b1L,b2L,b3L,b4L;w∧L,b1U,b2U,b3U,b4U;w∧U∈Fw∧L,w∧U and k∈R; then,
(8)A⊕B=a1L+b1L,a2L+b2L,a3L+b3L,a4L+b4L;w∧La1U+b1U,a2U+b2U,a3U+b3U,a4U+b4U;w∧U,kA=ka1L,ka2L,ka1L,ka1L;w∧L,ka1U,ka2U,ka1U,ka1U;w∧U,k>0,ka4L,ka3L,ka2L,ka1L;w∧L,ka4U,ka3U,ka2U,ka1U;w∧U,k<0,0,0,0,0;w∧L,0,0,0,0;w∧U,k=0,A⊗B=a1L∗b1L,a2L∗b2L,a3L∗b3L,a4L∗b4L;w∧La1U∗b1U,a2U∗b2U,a3U∗b3U,a4U∗b4U;w∧U,a1U,b1U≥0.Definition 7 [28].
Letσ,0∈R. The signed distance between σ and 0 is dσ,0=σ.Definition 8 [31].
LetA∈Fw∧L,w∧U; then, alpha-cut set of A denotes and is defined by
(9)Aα=ALα,AUα=AlUα,AlLα∪AtLα,AtUα;0≤α≤w∧LAlUα,AtUα;w∧L≤α≤w∧U,where
(10)AlLα=a1L+a2L−a1Lαw∧L,AtLα=a4L+a4L−a3Lαw∧L,AlUα=a1U+a2U−a1Uαw∧U,AtUα=a4U+a4U−a3Uαw∧U,Figure 1
Alpha-cut level of interval-valued trapezoidal fuzzy numberA.


## 3. Construction of Iterative Scheme (MM)

In order to approximate the roots of interval-valued trapezoidal fuzzy nonlinear equationFr=c, we propose the following two-step iterative scheme as follows:
(11)FlLrlL,rtL,rlU,rtU,τ=clLτ,FtLrlL,rtL,rlU,rtU,τ=ctLτ,FlUrlL,rtL,rlU,rtU,τ=clUτ,FtUrlL,rtL,rlU,rtU,τ=ctUτ.∀τ∈0,1.Suppose thatr=αlL,αtL,αlU,αtU is the solution of above system and r0=rl0L,rt0L,rl0U,rt0U is approximate solutions of the system, t denote the alpha-cut parameter; then,
(12)αlLτ=rl0Lτ+h1τ,αlLτ=rt0Lτ+k1τ,αlUτ=rl0Uτ+h2τ,αtUτ=rt0Uτ+k2τ.By using Taylor’s series ofFlL,FtL,FlU,FtU about rl0Lτ,rt0Lτ,rl0Uτ,rt0Uτ, then we have the following:
(13)FlLαlL,αtL,αlU,αtU,τ=FlLrl0L,rt0L,rl0U,rl0U,τ+h1FlLrl0L,rt0L,rl0U,rt0U,τrl0L+k1FlLrl0L,rt0L,rl0U,rt0U,τrt0L+h2FlLrl0L,rt0L,rl0U,rt0U,τrl0U+k2FlLrl0L,rt0L,rl0U,rt0U,τrt0U+Oh1,k1,h2,k22,FtLαlL,αtL,αlU,αtU,τ=FtLrl0L,rt0L,rl0U,rl0U,τ+h1FtLrl0L,rt0L,rl0U,rt0U,τrl0L+k1FtLrl0L,rt0L,rl0U,rt0U,τrt0L+h2FtLrl0L,rt0L,rl0U,rt0U,τrl0U+k2FtLrl0L,rt0L,rl0U,rt0U,τrt0U+Oh1,k1,h2,k22,FlUαlL,αtL,αlU,αtU,τ=FlUrl0L,rt0L,rl0U,rl0U,τ+h1FlUrl0L,rt0L,rl0U,rt0U,τrl0L+k1FlUrl0L,rt0L,rl0U,rt0U,τrt0L+h2FlUrl0L,rt0L,rl0U,rt0U,τrl0U+k2FlUrl0L,rt0L,rl0U,rt0U,τrt0U+Oh1,k1,h2,k22,FtUαlL,αtL,αlU,αtU,τ=FtUrl0L,rt0L,rl0U,rl0U,τ+h1FtUrl0L,rt0L,rl0U,rt0U,τrl0L+k1FtUrl0L,rt0L,rl0U,rt0U,τrt0L+h2FtUrl0L,rt0L,rl0U,rt0U,τrl0U+k2FtUrl0L,rt0L,rl0U,rt0U,τrt0U+Oh1,k1,h2,k22.Ifrl0Lτ,rt0Lτ,rl0Uτ,rt0Uτ are close to αlLτ,αlLτ,αlUτ,αtUτ, then h1τ,k1τ,h2τ,k2τ are small enough. Assume all partial derivatives of h1τ,k1τ,h2τ,k2τ exist and bounded; then, we have the following:
(14)FlLrl0L,rt0L,rl0U,rl0U,τ+h1FlLrl0L,rt0L,rl0U,rt0U,τrl0L+k1FlLrl0L,rt0L,rl0U,rt0U,τrt0L+h2FlLrl0L,rt0L,rl0U,rt0U,τrl0U+k2FlLrl0L,rt0L,rl0U,rt0U,τrt0U=clLτ,FtLrl0L,rt0L,rl0U,rl0U,τ+h1FtLrl0L,rt0L,rl0U,rt0U,τrl0L+k1FtLrl0L,rt0L,rl0U,rt0U,τrt0L+h2FtLrl0L,rt0L,rl0U,rt0U,τrl0U+k2FtLrl0L,rt0L,rl0U,rt0U,τrt0U=ctLτ,FlUrl0L,rt0L,rl0U,rl0U,τ+h1FlUrl0L,rt0L,rl0U,rt0U,τrl0L+k1FlUrl0L,rt0L,rl0U,rt0U,τrt0L+h2FlUrl0L,rt0L,rl0U,rt0U,τrl0U+k2FlUrl0L,rt0L,rl0U,rt0U,τrt0U=clLτ,FtUrl0L,rt0L,rl0U,rl0U,τ+h1FtUrl0L,rt0L,rl0U,rt0U,τrl0L+k1FtUrl0L,rt0L,rl0U,rt0U,τrt0L+h2FtUrl0L,rt0L,rl0U,rt0U,τrl0U+k2FtUrl0L,rt0L,rl0U,rt0U,τrt0U=ctLτ.Sinceh1τ,k1τ,h2τ,k2τ are unknown quantities, they are obtained by solving the following equations:
(15)J∗rl0L,rt0L,rl0U,rt0U,τh1τk1τh2τk2τ=clLτ−FlLrl0L,rt0L,rl0U,rt0U,τctLτ−FtLrl0L,rt0L,rl0U,rt0U,τclUτ−FlUrl0L,rt0L,rl0U,rt0U,τctUτ−FtUrl0L,rt0L,rl0U,rt0U,τ,where
(16)J∗=FlLrl0Lrl0L,rt0L,rl0U,rt0U,τFlLrt0Lrl0L,rt0L,rl0U,rt0U,τFlLrl0Url0L,rt0L,rl0U,rt0U,τFlLrt0Url0L,rt0L,rl0U,rt0U,τFtLrl0Lrl0L,rt0L,rl0U,rt0U,τFtLrt0Lrl0L,rt0L,rl0U,rt0U,τFtLrl0Url0L,rt0L,rl0U,rt0U,τFtLrt0Url0L,rt0L,rl0U,rt0U,τFlUrl0Lrl0L,rt0L,rl0U,rt0U,τFlUrt0Lrl0L,rt0L,rl0U,rt0U,τFlUrl0Url0L,rt0L,rl0U,rt0U,τFlUrt0Url0L,rt0L,rl0U,rt0U,τFtUrl0Lrl0L,rt0L,rl0U,rt0U,τFtUrt0Lrl0L,rt0L,rl0U,rt0U,τFtUrl0Url0L,rt0L,rl0U,rt0U,τFtUrt0Url0L,rt0L,rl0U,rt0U,τ.yl0Lτyt0Lτyl0Uτyt0Uτ=rl0Lτrt0Lτrl0Uτrt0Uτ+h1τk1τh2τk2τ,rl1Lτrt1Lτrl1Uτrt1Uτ=yl0Lτyt0Lτyl0Uτyt0Uτ+Z∗h1τk1τh2τk2τ,Z=4J∗∗yl0L,yt0L,yl0U,yt0U,τ−2J∗rl0L,rt0L,rl0U,rt0U,τ−1∗J∗yl0L,yt0L,yl0U,yt0U,τ−J∗∗rl0L,rt0L,rl0U,rt0U,τ,J∗∗=FlLyl0Lyl0L,yt0L,yl0U,yt0U,τFlLyt0Lyl0L,yt0L,yl0U,yt0U,τFlLyl0Uyl0L,yt0L,yl0U,yt0U,τFlLyt0Uyl0L,yt0L,yl0U,yt0U,τFtLyl0Lyl0L,yt0L,yl0U,yt0U,τFtLyt0Lyl0L,yt0L,yl0U,yt0U,τFtLyl0Uyl0L,yt0L,yl0U,yt0U,τFtLyt0Uyl0L,yt0L,yl0U,yt0U,τFlUyl0Lyl0L,yt0L,yl0U,yt0U,τFlUyt0Lyl0L,yt0L,yl0U,yt0U,τFlUyl0Uyl0L,yt0L,yl0U,yt0U,τFlUyt0Uyl0L,yt0L,yl0U,yt0U,τFtUyl0Lyl0L,yt0L,yl0U,yt0U,τFtUyt0Lyl0L,yt0L,yl0U,yt0U,τFtUyl0Uyl0L,yt0L,yl0U,yt0U,τFtUyt0Uyl0L,yt0L,yl0U,yt0U,τ.J∗=J∗rlnL,rtnL,rlnU,rtnU,τ, J∗∗=J∗∗ylnL,ytnL,ylnU,ytnU,τ, and the next approximation for rlLτ,rtLτ,rlUτ,rtUτ is found by using recursive scheme as follows:
(17)rlLn+1τrtLn+1τrlUn+1τrtUn+1τ=ylLnτytLnτylUnτytUnτ+Z∗h1nτk1nτh2nτk2nτ,ylLnτytLnτylUnτytUnτ=rlLnτrtLnτrlUnτrtUnτ+h1nτk1nτh2nτk2nτ,Z=4J∗∗ylLn,ytLn,ylUn,ytUn,τ−2J∗rlLn,rtLn,rlUn,rtUn,τ−1∗J∗ylLn,ytLn,ylUn,ytUn,τ−J∗∗rlLn,rtLn,rlUn,rtUn,τ,J∗rlnL,rtnL,rlnU,rtnU,τh1nτk1nτh2nτk2nτ=clLτ−FlLrlnL,rtnL,rlnU,rtnU,τctLτ−FtLrlnL,rtnL,rlnU,rtnU,τclUτ−FlUrlnL,rtnL,rlnU,rtnU,τctUτ−FtUrlnL,rtnL,rlnU,rtnU,τ,J∗=FlLrlnLrlnL,rtnL,rlnU,rtnU,τFlLrtnLrlnL,rtnL,rlnU,rtnU,τFlLrlnUrlnL,rtnL,rlnU,rtnU,τFlLrtnUrlnL,rtnL,rlnU,rtnU,τFtLrlnLrlnL,rtnL,rlnU,rtnU,τFtLrtnLrlnL,rtnL,rlnU,rtnU,τFtLrlnUrlnL,rtnL,rlnU,rtnU,τFtLrtnUrlnL,rtnL,rlnU,rtnU,τFlUrlnLrlnL,rtnL,rlnU,rtnU,τFlUrtnLrlnL,rtnL,rlnU,rtnU,τFlUrlnUrlnL,rtnL,rlnU,rtnU,τFlUrtnUrlnL,rtnL,rlnU,rtnU,τFtUrlnLrlnL,rtnL,rlnU,rtnU,τFtUrtnLrlnL,rtnL,rlnU,rtnU,τFtUrlnUrlnL,rtnL,rlnU,rtnU,τFtUrtnUrlnL,rtnL,rlnU,rtnU,τ,J∗∗=FlLylnLylnL,ytnL,ylnU,ytnU,τFlLytnLylnL,ytnL,ylnU,ytnU,τFlLylnUylnL,ytnL,ylnU,ytnU,τFlLytnUylnL,ytnL,ylnU,ytnU,τFtLylnLylnL,ytnL,ylnU,ytnU,τFtLytnLylnL,ytnL,ylnU,ytnU,τFtLylnUylnL,ytnL,ylnU,ytnU,τFtLytnUylnL,ytnL,ylnU,ytnU,τFlUylnLylnL,ytnL,ylnU,ytnU,τFlUytnLylnL,ytnL,ylnU,ytnU,τFlUylnUylnL,ytnL,ylnU,ytnU,τFlUytnUylnL,ytnL,ylnU,ytnU,τFtUylnLylnL,ytnL,ylnU,ytnU,τFtUytnLylnL,ytnL,ylnU,ytnU,τFtUylnUylnL,ytnL,ylnU,ytnU,τFtUytnUylnL,ytnL,ylnU,ytnU,τ,∀τ∈0,1.For initial guess, one can use the fuzzy number(18)r0=rl0U0,rl0L0,rl0L1,rt0L1,rl0U1,rt0U1,rt0L0rt0U0,Remark 9.
SequencerlnL,rtnL,rlnU,rtnUn=0∞ converges to αlL,αtL,αlU,αtU iff ∀τ∈0,1, limn⟶∞rlnLτ=αlLτ, limn⟶∞rtnLτ=αtLτ, limn⟶∞rlnUτ=αlUτ, and limn⟶∞rtnUτ=αtUτ.Lemma 10.
LetFαlL,αtL,αlU,αtU=clL,ctL,clU,ctU, and if the sequence of rlnL,rtnL,rlnU,rtnUn=0∞ converges to αlL,αtL,αlU,αtU according to Newton’s method, then
(19)limn⟶∞Pn=0,where
(20)Pn=sup0≤τ≤1maxh1nτ,k1nτ,h2nτ,k2nτ.Proof.
It is obvious, because for all∀τ∈0,1 in convergent case,
(21)limn⟶∞h1nτ=limn⟶∞k1nτ=limn⟶∞h2nτ=limn⟶∞k2nτ=0.
Hence, it is proved.☐Finally, it is shown that under certain condition, the MM method for fuzzy equationFr=0 is cubic convergent. In compact form for Fr=0, the MM method can be written as follows:
(22)ynτ=rnτ−F′rnτ−1Frnτ,rn+1τ=ynτ−Z∗F′rnτ−1Frnτ,∀τ∈0,1,where
(23)Z=4J∗∗−2J∗−1∗J∗−J∗∗.Theorem 11.
LetF:H⊆Rn⟶Rn, be u-times Frĕchet differential function on a convex set H containing the root α of Fr=0; then, the MM method has cubic convergence and satisfies the following error equation.
(24)en+1=2∗A22−12A3−4A22en3+Oen4,where An=1/2!∗Frn,τ/F′rn,τ,n=2,3,⋯.Proof.
Leten=rn−α and en+1=rn+1−α, then by Taylor series of Frn,τ in the neighborhood of α if J∗ and J∗∗ exist. Then,
(25)Fr,τ=Frn,τ+F′rn,τr−rn+12!F′′rn,τr−rn2+.⋯
andFr,α=0.(26)Frn,τ=F′r,ξen+A2en2+A3en3+Oen4.
This gives(27)F′rn,τ−1Frn,τ=en+A2en2+2A2+2A3en3+⋯,yn−α=A2en2+−2A2+2A3en3+.⋯
ExpandingF′yn,τ about α, we have the following:
(28)F′yn,τ=1+2A22en2+2−2A22+2A3en3+⋯,Z∗F′rn,τ−1Frn,τ=−A2en2+4A22−32A3−4A22en3+⋯,rn+1−α=yn−α−A2en2+4A22−32A3−4A22en3+⋯,en+1=2∗A22−12A3−4A22en3+Oen4.
Hence, the theorem is proved.☐A well-known existing method in literature for solving triangular fuzzy nonlinear equation is classical Newton Raphson’s method. Interval-valued trapezoidal fuzzy version of well-known Newton method [13] (abbreviated as NN) for finding roots of interval-valued trapezoidal fuzzy nonlinear equation is as follows:
(29)rlLn+1τrtLn+1τrlUn+1τrtUn+1τ=rlLnτrtLnτrlUnτrtUnτ+h1nτk1nτh2nτk2nτ,where(30)J∗rlnL,rtnL,rlnU,rtnU,τh1nτk1nτh2nτk2nτ=clLτ−FlLrlnL,rtnL,rlnU,rtnU,τctLτ−FtLrlnL,rtnL,rlnU,rtnU,τclUτ−FlUrlnL,rtnL,rlnU,rtnU,τctUτ−FtUrlnL,rtnL,rlnU,rtnU,τ.


## 4. Numerical Applications

Here, we present examples to illustrate the performance and efficiency of MM and NN methods for approximating roots of interval-valued trapezoidal fuzzy nonlinear equations. Examples1–3 are considered from Buckley and Qu [9]. All the computations are performed using CAS Maple 18 with 64 digits floating point arithmetic with stopping criteria as follows. Analytical, numerical approximate solutions, computational order of convergence [32], computational time in second, and residual error graph of interval-valued trapezoidal fuzzy nonlinear equation used in Examples 1–3 are shown in Figures 2–8(a) and 8(c), respectively. Algorithm 1 shows the implementation of MM iterative method on CAS Maple18.
(31)ien=Fr,τ<∈iien=rn+1τ−rnτ<∈,where en represents the absolute error. We take ∈=10−15.Figure 2
Analytical solution of Examples 1–3.Figure 3
Computational order of convergence.Figure 4
Computational time in seconds.Figure 5
Initial guessed values, analytical, and numerical approximate solution.Figure 6
Initial guessed values, analytical, and numerical approximate solution.Figure 7
Initial guessed values, analytical, and numerical approximate solution.Figure 8
Error graph of iterative methods MM and NN.
(a)(b)(c)In Figure2, left shows analytical solution of interval-valued trapezoidal fuzzy nonlinear equation used in Example 1, center shows for Example 2, and right shows for Example 3, respectively.Algorithm 1: (MM method).
Step 1. Transform Fr,τ=c into FlLrlL,rtL,rlU,rtU,τ=clLτ,FtLrlL,rtL,rlU,rtU,τ=ctLτ,FlUrlL,rtL,rlU,rtU,τ=clUτ,FtUrlL,rtL,rlU,rtU,τ=ctUτ.∀τ∈0,1.Step 2. Solve FlLrlL,rtL,rlU,rtU,τ=clLτ,FtLrlL,rtL,rlU,rtU,τ=ctLτ,FlUrlL,rtL,rlU,rtU,τ=clUτ,FtUrlL,rtL,rlU,rtU,τ=ctUτ. for τ=0 and τ=1 to obtain initial guess value.Step 3. Evaluate Fr,τ=c at initial guess point and compute Jacobian matrix J∗,J∗∗.Step 4. Use MM to compute next iterationynτ=rnτ−F′rnτ−1Frnτ,rn+1τ=ynτ−Z∗F′rnτ−1Frnτ,∀τ∈0,1,whereZ=4J∗∗−2J∗−1∗J∗−J∗∗.Step 5. For given ∈>0, if (i) en=Fr,τ<∈ and (ii) en=rn+1τ−rnτ<∈, then stop.Step 6. Set k=k+1 and go to step 1.Figure3 shows computational order of convergence of iterative methods MM and NN for finding roots of interval-valued trapezoidal fuzzy nonlinear equations used in Examples 1–3, respectively.In Figure3, 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.Figure4 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 Figure4, 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 dollarA1=10,20,30,40;2/3,5,15,35,43;1 at fuzzy interest rate r to earn maximum profit, so that after a year, they may withdraw 25000$ S1=45,55,75,95;2/3,80,90,110,120;1 approximately and after two years 900000$ S2=10,15,20,25;2/3,5,10,25,47;1 left. Find r so that A1 will be sufficient to cover S1 and S2. where r is an interval-valued trapezoidal fuzzy number whose support lies between 0,1 After a year, the amount in the account will be
(32)A1−S1+A1rτ.
At the end of second year, total amount left is(33)A1−S1+A1∗rτ+A1−S1−A1∗rτrτ,or
(34)A1rτ2+B∗rτ+D,where B=2A1−S1 and D=A1−S1. Therefore, we have to solve
(35)A1∗rτ2+B∗rτ+D=S2,or
(36)A1∗rτ2+B∗rτ=C,where C=S2−D. For fuzzy interest rate substituting values of A1, B, and C in above equation, we have the following:
(37)10,20,30,40;23,5,15,35,45;1rτ2+50,60,70,80;23,45,55,75,95;1rτ=80,90,110,120;23,75,85,115,125;1.
Without any loss of generality, assume thatr is positive; then, the parametric form of this equation is as follows:
(38)10+15τ,40−15τ,5+10τ,45−10τrτ2+50+15τ,80−15τ,45+10τ,95−20τrτ=80+15τ,120−15τ,75+10τ,125−10τ,10+15τrlLτ2+50+15τrlLτ=80+15τ,40−15τrtLτ2+80−15τrtLτ=120−15τ,5+10τrlUτ2+45+10τrlUτ=75+10τ,45−10τrtUτ2+95−20τrtUτ=125−10τ.Table1 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 1τMMNNrn+1−rnFrnrn+1−rnFrn0.03.1e-471.2e-364.1e-101.4e-80.11.1e-433.1e-323.1e-101.1e-80.21.4e-451.1e-320.1e-208.2e-70.35.6e-463.4e-341.7e-107.2e-90.41.3e-411.4e-321.5e-165.5e-90.54.4e-411.7e-327.2e-114.1e-90.66.1e-497.1e-373.2e-113.1e-90.78.1e-479.1e-334.1e-106.2e-90.81.3e-454.1e-326.3e-117.5e-90.91.6e-466.4e-337.8e-118.1e-917.7e-476.8e-335.1e-123.4e-9Table2 shows analytical solutions for Example 1.Table 2
Analytical solution for Example1.
τ0.00.10.20.30.40.50.60.70.80.91.0rlU0.91720.92680.93680.94730.95810.96940.98120.99361.00651.02001.0342rlL1.00001.00961.01971.03041.04181.05381.06661.08021.09471.11021.1269rtL1.27491.23941.20801.18001.15471.13181.11091.09181.07431.05811.0430rtU1.43711.39741.36201.33011.30121.27491.25071.22851.20801.18901.1712Figure5 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.To obtain initial guess, we use above system forτ=0 and τ=1; therefore,
(39)10rlL20+50rlL0=80,40rtL20+80rtL0=120,5rlU20+45rlU0=75,45rtU20+95rtU0=125,25rlL21+65rlL1=95,25rtL21+65rtL1=105,15rlU21+55rlU1=85,35rtU21+75rtU1=115.Consequently,rlU0=0.5, rlL0=0.5, rtL0=0.5, rtU0=0.5, and rlU0=rlL0=rtL0=rtU0=1/2. After 4 iterations, we obtain the solution with the maximum error less than 10−30. Now suppose r is negative, we have the following:
(40)10+15τrlLτ2+50+15τrtLτ=80+15τ,40−15τrtLτ2+80−15τrlLτ=120−15τ,5+10τrlUτ2+45+10τrtUτ=75+10τ,45−10τrtUτ2+95−20τrlUτ=125−10τ.Forτ=0, we have rlL0>rtL0; therefore negative root does not exist.Example 2.
Consider the interval-valued trapezoidal fuzzy nonlinear equation(41)0.65,0.73,0.87,0.95;23,0.6,0.7,0.9,1;1rτ2+0.25,0.33,0.47,0.55;23,0.2,0.3,0.5,0.6;1rτ=0.45,0.53,0.67,0.75;23,0.4,0.5,0.7,0.8;1.
Without any loss of generality, assume thatr is positive; then, the parametric form of this equation is as follows:
(42)0.65+0.12τ,0.95−0.12τ,0.6+0.1τ,1−0.1τrτ2+0.25+0.12τ,0.55−0.12τ,0.2+0.1τ,0.6−0.1τrτ=0.45+0.12τ,0.75−0.12τ,0.4+0.1τ,0.8−0.1τ,or
(43)0.65+0.12τrlLτ2+0.25+0.12τrlLτ=0.45+0.12τ,0.95−0.12τrtLτ2+0.55−0.12τrtLτ=0.75−0.12τ.0.6+0.1τrlUτ2+0.2+0.1τrlUτ=0.4+0.1τ,1−0.1τrtUτ2+0.6−0.1τrtUτ=0.8−0.1τ.Table3 clearly shows the dominance behavior of MM over NN in terms of absolute error on the same number of iterations n=4 for Example 2.Table 3τMMNNrn+1−rnFrnrn+1−rnFrn0.04.0e-494.0e-331.7e-131.7e-110.15.6e-475.6e-323.4e-131.2e-110.23.1e-438.8e-336.1e-143.6e-110.35.4e-486.1e-365.1e-137.1e-110.46.1e-497.9e-354.4e-145.6e-120.53.7e-484.0e-334.3e-133.4e-110.61.6e-488.2e-331.7e-134.6e-110.73.6e-494.3e-333.7e-136.7e-110.81.2e-493.6e-347.1e-145.6e-110.97.5e-468.9e-348.7e-141.2e-1118.1e-496.1e-359.1e-145.1e-11Table4 shows analytical solutions for Example 2.Table 4
Analytical solution for Example2.
τ0.00.10.20.30.40.50.60.70.80.91.0rlU0.64330.64370.64400.64430.64460.64500.64530.64570.64600.64640.6468rlL0.64500.64540.64580.64630.64670.64720.64770.64820.64870.64930.6498rtL0.66160.66060.65960.65860.65770.65680.65600.65520.65440.65370.6530rtU0.66660.66550.66450.66350.66250.66160.66070.65990.65910.65830.6576Figure6 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 2.To obtain initial guess, we use above system forτ=0 and τ=1; therefore,
(44)0.65rlL20+0.2rlL0=0.45,0.95rtL20+0.55rtL0=0.75,0.6rlU20+0.2rlU0=0.4,1.0rtU20+0.6rtU0=0.8,0.77rlL21+0.37rlL1=0.57,0.83rtL21+0.43rtL1=0.63,0.7rlU21+0.3rlU1=0.5,0.9rtU21+0.5rtU1=0.7.Consequently,rlU0=0.6, rlL0=0.6, rtL0=0.6, rtU0=0.6, and rlU0=rlL0=rtL0=rtU0=1/2. After 4 iterations, we obtain the solution with the maximum error less than 10−30. Now suppose r is negative, we have
(45)0.65+0.12τrlLτ2+0.25+0.12τrlLτ=0.45+0.12τ,0.95−0.12τrtLτ2+0.55−0.12τrtLτ=0.75−0.12τ,0.6+0.1τrlUτ2+0.2+0.1τrlUτ=0.4+0.1τ,1−0.1τrtUτ2+0.6−0.1τrtUτ=0.8−0.1τ.Forτ=0, we have rlL0>rtL0, therefore, negative root does not exist.Example 3.
Consider the interval-valued trapezoidal fuzzy nonlinear equation(46)0.45,0.53,0.67,0.75;23,0.4,0.5,0.7,0.8;1rτ3+0.65,0.73,0.87,0.95;23,0.6,0.7,0.9,1;1sinrτ=0.25,0.33,0.47,0.55;23,0.2,0.3,0.5,0.6;1.
Without any loss of generality, assume thatr is positive; then, the parametric form of this equation is as follows:
(47)0.45+0.12τ,0.75−0.12τ,0.4+0.1τ,0.8−0.1τrτ3+0.65+0.12τ,0.95−0.12τ,0.6+0.1τ,1.0−0.1τrτ=0.25+0.12τ,0.55−0.12τ,0.2+0.1τ,0.6−0.1τ,or
(48)0.45+0.12τrlLτ3+0.65+0.12τsinrlLτ=0.25+0.12τ,0.75−0.12τrtLτ3+0.95−0.12τsinrtLτ=0.55−0.12τ,0.4+0.1τrlUτ3+0.6+0.1τsinrlUτ=0.2+0.1τ,0.8−0.1τrtUτ3+1.0−0.1τsinrtUτ=0.6−0.1τ.Table5 clearly shows the dominance behavior of MM over NN in terms of absolute error on the same number of iterations n=4 for Example 3.Table 5τMMNNrn+1−rnFrnrn+1−rnFrn0.02.8e-435.4e-331.1e-91.6e-70.13.8e-443.0e-337.6e-103.4e-70.24.7e-435.8e-333.5e-104.2e-70.39.1e-432.7e-336.1e-116.1e-70.46.2e-505.6e-395.1e-125.8e-70.53.6e-516.1e-384.6e-116.5e-70.64.8e-507.8e-393.1e-118.9e-70.79.1e-513.6e-394.2e-111.3e-70.88.1e-488.4e-407.7e-114.1e-70.99.8e-493.5e-395.1e-115.6e-716.1e-524.0e-383.3e-111.2e-7Table6 shows analytical solutions for Example 3.Table 6
Analytical solution for Example3.
τ0.00.10.20.30.40.50.60.70.80.91.0rlU0.31730.32660.33550.34400.35220.36000.36750.37470.38170.38830.3947rlL0.36000.36900.37750.38570.39350.40090.40800.41480.42130.42760.4336rtL0.50050.49710.49350.48980.48600.48210.47800.47380.46940.46480.4601rtU0.51370.51120.50870.50600.50330.50050.49770.49470.49170.48860.4854Figure7 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 3.To obtain initial guess, we use above system forτ=0 and τ=1; therefore,
(49)0.45rlL30+0.65sinrlL0=0.25,0.75rtL30+0.95sinrtL0=0.55,0.4rlU30+0.6sinrlU0=0.2,0.8rtU30+1sinrtU0=0.6,0.57rlL31+0.77sinrlL1=0.37,0.63rtL31+0.83sinrtL1=0.43,0.5rlU31+0.7sinrlU1=0.3,0.7rtU31+0.9sinrtU1=0.5.Consequently,rlU0=0.5, rlL0=0.3, rtL0=0.5, rtU0=0.3, and rlU0=rlL0=rtL0=rtU0=1/2. After 4 iterations, we obtain the solution with the maximum error less than 10−30. Now suppose r is negative, we have
(50)0.45+0.12τrlLτ3+0.65+0.12τsinrtLτ=0.25+0.12τ,0.75−0.12τrtLτ3+0.95−0.12τsinrlLτ=0.55−0.12τ,0.4+0.1τrlUτ3+0.6+0.1τsinrtUτ=0.2+0.1τ,0.8−0.1τrtUτ3+1.0−0.1τsinrlUτ=0.6−0.1τ.Forτ=0, we have hence rlL0>rtL0; therefore, negative root does not exist.Figures8(a)–8(c) show residual falls for iterative methods MM and NN for interval-valued trapezoidal fuzzy nonlinear equation used in Examples 1–3, respectively.


## 5. Conclusion

In this research paper, we constructed highly efficient two-step numerical iterative method to approximate roots of interval-valued trapezoidal fuzzy nonlinear equations. A set of real-life applications from optimization are considered as a numerical test examples showing the practical performance and dominance efficiency of MM over NN method on the same number of iterations. From Tables1–6 and Figures 1–8, we observe that numerical results of MM methods are better in terms of absolute error and CPU time as compared to NN method. Considering the same ways as in this article, we can establish higher order and efficient numerical iterative methods for solving system of fuzzy nonlinear equations.



---

*Source: 6369129-2021-09-06.xml*