Content uploaded by Senthilkumar Sethurathinam
Author content
All content in this area was uploaded by Senthilkumar Sethurathinam on Jul 01, 2020
Content may be subject to copyright.
BISECTION METHOD FOR FUZZY
NONLINEAR EQUATIONS
L.S. Senthilkumar1 and K. Ganesan2
1,2Department of Mathematics, SRM University, Kattankulathur, Chennai-603203.
Email: senthilkumar.l@ktr.srmuniv.ac.in ,ganesan.k@ktr.srmuniv.ac.in
ABSTRACT. In this paper a new algorithm based
on bisection method is proposed to solve fuzzy
nonlinear equations. In this method fuzzy
nonlinear equation has been converted into
parametric form and obtained a fuzzy solution.
Fuzzy polynomial is expressed in parametric
form, it has been evaluated at a fuzzy number in
its parametric form itself and then it converted
into triangular fuzzy number for comparison
purpose. A numerical example is provided to
illustrate the proposed method in this paper.
Keywords: Bisection method, F(R)- the set of
triangular fuzzy numbers, Ranking of fuzzy
numbers, Fuzzy nonlinear equation.
I. INTRODUCTION
Zadeh[18] proposed fuzzy logic in 1965.
This logic is applicable in many fields of science.
Sometimes outcome of a system depends on the roots
of fuzzy algebraic equations. Some fuzzy nonlinear
equations were solved in [Rogers [11], Gautama [4],
Subash [16]]. A fuzzy modified Newton-Raphson
method is a Newton-Raphson method for solving non
linear equations. Subash [16] proposed a new method
using linear interpolation method to solve a non
linear equation () = 0 . It is a modification of
fuzzy Newton-Raphson method that converges
rapidly to exact solution. One can use Numerical
Methods to compute the roots of a non-linear
equation to any desired degree of accuracy. Here it is
noted that a non-linear equation over linear fuzzy
real numbers is called a fuzzy non-linear equation.
Gautam [4] proposed a new method to solve
a fuzzy non-linear equation with the help of
Bisection Algorithm. Javad [5] solved fuzzy
nonlinear equations using numerical methods.
Mahmoud, et. al [7] discussed a fuzzy nonlinear
equations. Noor’ani, et.al [10] used ranking method
to solve a system of fuzzy polynomials.
Waziri [17] solved dual fuzzy nonlinear
equation using Broyden's and Newton's methods.
Kajani [6] solved dual fuzzy nonlinear equations.
Senthilkumar and Ganesan[9] proposed Harmonic
mean method to solve Fuzzy Nonlinear Equation.
Now a modified bisection method to solve FNLE is
proposed in this paper. For evaluating fuzzy
polynomial at a fuzzy number, computation of higher
powers of fuzzy number is essential. This is
discussed by Giachelti and young[19]. They used six
parameters to define parameterized fuzzy numbers.
Squeres and square root of fuzzy numbers was
discussed by Bashar M.A and shirin[3]. They are so
many methods for triangular approximation of fuzzy
numbers.
Saneifard[13] suggests a new approach to the
problem of defuzzification using weighted metric
between two fuzzy numbers. Saneifard[14] solved
the problem of defuzzification in computing with
regular weighted function. Abbasbandy and
Asady[20] introduced a fuzzy trapezoidal
approximation using a metric between two fuzzy
numbers. Abbasbandy and Amirfakhiran[22] have
used value and ambiguity of fuzzy number and have
solved an optimization problem to obtain the nearest
triangular fuzzy number.
In this paper we present a new approach for
solving non-linear fuzzy equations with fuzzy
coefficients. We use bisection method to solve it .For
this fuzzy function is evaluated at a given fuzzy
number. For this the function is converted into the
Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 12,Number 1 (2016)
© Research India Publications : http://www.ripublication.com
271
parametric form
f (x )
=
f (r ),f (r )
, it is evaluated at a
given fuzzy number and then converted into
triangular fuzzy number for comparison purpose
(Adabitabarfirozja and Eslampia[1]). Ranking of
fuzzy numbers (Abbasbandy [2]) is defined based on
this new representation of fuzzy numbers.
The present paper has been arranged as
follows: Section II states some of the preliminary
results pertaining the fuzzy arithmetic operations.
The fuzzy nonlinear equation under study is stated in
Section III. Section IV deals with the bisection
method algorithm to solve fuzzy nonlinear equation.
Numerical examples are given in Section V to
illustrate the proposed method.
II. PRELIMINARIES
The some basic concepts and related results of
triangular fuzzy numbers are recalled.
Definition 1. A fuzzy set
a
%
defined on the set of
real numbers R is said to be a fuzzy number if its
membership function
a : R [0,1]
→
%has the following
characteristics:
(i)
a
%
is convex
(
)
{
}
[ ]
1 2 1 2
1 2
i.e
for all x , x and
a(
λx 1- λx ) min a(x ), a(x ) ,
0,1∈ λ ∈
+ ≥
% % %
.
R .
(ii)
a
%
is normal i.e., there exists an
x R
∈
such that
(
)
a x 1
=
%
(iii)
a
%
is Piecewise continuous.
Definition 2. A fuzzy number
a
%
on
R
is said to be
a triangular fuzzy number F(R) or Linear Real Fuzzy
Number (LRFN) if its membership function
: R [0,1]
µ →
has the following characteristics:
0,if x a
x a
(x) ,if a x b
b a
c x ,if b x c
c b
≤
−
µ = ≤ ≤
−
−
≤ ≤
−
(1)
The triangular fuzzy number is denoted by
a = (a, b, c).
%
F(R)
is used to denote the set of all
triangular fuzzy numbers. Also if
m b
=
represents the
modal value or midpoint,
( b a )
α = −
represents the
left spread and
(c b)
β = −
represents the right spread
of the triangular fuzzy number
a = (a, b, c)
% then the
triangular fuzzy number
a
%
can be represented by the
triplet
(
)
a ,m, .
= α β
% i.e.,
(
)
a = (a, b, c) = , m, .
α β
%
Here it is noted that any real number b can be written
as a linear fuzzy real number (b,b,b) and so R
⊆
F(R). As a linear fuzzy real number, consider (b,b,b)
to represent the real number b itself.
Definition 3. A triangular fuzzy number
a F(R)
∈
% can also be represented as a pair
(
)
a a, a
=
%
of functions
(
)
(
)
a r and a r
for
0 r 1
≤ ≤
which
satisfies the following requirements:
(i)
(
)
a r
is a bounded monotonic increasing left
continuous function.
(ii)
(
)
a r
is a bounded monotonic decreasing left
continuous function.
(iii)
(
)
(
)
a r a r
≤,
0 r 1
≤ ≤
A. Ranking of triangular fuzzy numbers
Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 12,Number 1 (2016)
© Research India Publications : http://www.ripublication.com
272
Many different approaches for the ranking of
fuzzy numbers have been proposed in the literature
[2]. For an arbitrary triangular fuzzy number
1 2 3
a = (a , a , a )
% with parametric form
(
)
a a(r), a(r)
=
%,
0
a(1) a(1)
a2
+
=
the magnitude of the triangular
fuzzy number
a
%
is defined by:
1
0
0
1
Mag(a) (a a a )f (r)dr
2
−
−
= + +
∫
%
(2)
where the function
f (r)
is a non-negative and
increasing function on
[0,1]
with
f (0) 0,
=
f (1) 1
=
and 1
0
1
f (r ) dr
2
=
∫. The function
f (r)
can be
considered as a weighting function. In real life
applications,
f (r)
can be chosen by the decision
maker according to the situation. In this paper, for
convenience,
f (r) r
=
.
Hence the equation (2) becomes
1 2 3
a 9a a
Mag(a)
12
+ +
=
%
The magnitude of a triangular fuzzy number
a
%
synthetically reflects the information on every
membership degree, and meaning of this magnitude
is visual and natural.
Mag(a)
%
is used to rank fuzzy
numbers. The larger
Mag(a)
%
, the larger fuzzy
number. For any two triangular fuzzy numbers
a
%
and
b
%
in F(R), define the ranking of
a
%
and
b
%
by
comparing the
Mag(a)
%
and
Mag(b)
%
on R as follows:
(i)
a b
≥
%
% if and only if
Mag(a)
%
≥
Mag(b)
%
(ii)
a b
≤
%
% if and only if
Mag(a)
%
≤
Mag(b)
%
(iii)
a b
≈
%
% if and only if
Mag(a)
%
=
Mag(b)
%
B. Arithmetic operation on triangular fuzzy
numbers
Arithmetic operations:
From [16], for given two fuzzy real numbers
x1 =(a1,b1,c1) , x2 =(a2,b2,c2) we define addition,
subtraction, multiplication and division by
(i) x1+ x2 = (a1+a2,b1+b2,c1+c2)
(ii) x1- x2 = (a1-c2,b1-b2,c1-a2)
(iii) x1 x2= (min(a1a2,a1c2,a2c1,c1c2), b1b2,
max(a1a2,a1c2,a2c1,c1c2))
(iv)
= ( min(
,
,
,
),
,
max(
,
,
,
) ).
III. FUZZY NON-LINEAR ALGEBRAIC
EQUATION
A non-linear equation over linear fuzzy real
numbers is called a fuzzy non-linear equation. Here,
it is converted into a parametric form. For this
conversion, the first concern is the integral power of
triangular fuzzy number. It is discussed in Bashar
[3].
Consider a fuzzy non-linear equation with fuzzy
coefficients
n n 1
0 1 n
i i i i
f ( x ) a x a x . . . . a
w h e r e , x ( x , y , z ) a n d a ( a , b , c )
−
= + + +
= =
%% % %
%
(3)
Then the above polynomial can be represented in
parametric form
Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 12,Number 1 (2016)
© Research India Publications : http://www.ripublication.com
273
n
n
0
0
n 1
n 1
1 n
1 n
n n 1
0 1 n
n n 1
0 1 n
Now (1) can b e re p resent ed in p a ram e tric form
f( x ) (f ( r ), f ( r) ) [a , a ][ x , x ]
[a , a ] [x , x ] ... [a , a ]
[a x a x ... a ,
a x a x ... a ]
−
−
−
−
= = +
+ +
= + + +
+ + +
%%
A. Triangular Approximate of fuzzy Numbers
Delgado, Ma and Voxman [21] define two
parameters of fuzzy number u, the value and
ambiguity with respect to a reducing function s(r) by
Vs(u) and As(u) as follows:
1 1
s s
0 0
V (u) s(r )(u(r) u(r))dr , A (u) s(r)(u(r) u(r))d
r
= + = −
∫ ∫
Any fuzzy number given in parametric form can
be approximated to a triangular fuzzy number.
This was discussed by Adabitabarfirozja and
Eslampia[1]. They proposed a method for
finding the nearest triangular approximations of
fuzzy number using value and ambiguity
functions.
Theorem 1. If T(u)=(t1(u),t2(u),t3(u)) is nearest
triangular approximation of = ((∝), (∝)) based
on the nearest
(. ) and
(. ) , then
1
2
2
0
11
2 2
0
2
1
2
2
0
31
2 2
0
s ( r ) (1 r ) ( u ( r ) r t ( u ) ) d r
t ( u ) ,
s ( r )(1 r ) d r
t ( u ) u (1 ) u (1 ) ,
s ( r ) (1 r ) ( u ( r ) r t ( u ) ) d r
t ( u )
s ( r ) (1 r ) d r
− −
=
−
= =
− −
=
−
∫
∫
∫
∫
IV. BISECTION METHOD TO SOLVE
NONLINEAR FUZZY EQUATIONS
Consider a fuzzy nonlinear algebraic equation
1 1 1 1 2 2 2 2
1 1 1 2 2 2
f(x) 0
Let x (a , b , c ) and x (a , b , c )
be two initial approximate roots.
then x [ x , x ] and x [x , x ]
=
= =
= =
%%
% %
% %
Using the expressions
1 1 1 2 2 2
x [ x , x ] a n d x [ x , x ]
= =
% %
in Eq (3.1), we get the following
1
1
1
n n 1
1 1
0 1 n
n n 1
1 1
0 1 n
2
2
2
n n 1
2 2
0 1 n
n n 1
2 2
0 1 n
f ( x ) ( f ( r ), f ( r ))
[ a x a x . .. a ,
a x a x . .. a ] a n d
f ( x ) ( f ( r ), f ( r ))
[ a x a x . .. a ,
a x a x . . . a ]
N o w b o th th e s e t w o f u z zy n u m b e r s c a n b e
c o n v e r t e d i n to t w o t r ia n g u l a r f u z z y n u m b
−
−
−
−
= =
+ + +
+ + +
= =
+ + +
+ + +
%%
%%
1 1 1 2 1 3 1
2 1 2 2 2 3 2
e r s
f ( x ) ( t ( f ) , t ( f ) , t ( f ) ) a n d
f( x ) ( t ( f ) , t ( f ) , t ( f ) )
=
=
%%
%%
1
2
2 1
1
0
1 1 1
2 2
0
1
2 1 1
1
212 1
0
3 1 1
2 2
0
s ( r ) (1 r ) ( f ( r ) r t ( f ) ) d r
t ( f ) ,
s ( r ) (1 r ) d r
t ( f ) f (1 ) f (1 ) ,
s ( r ) (1 r ) ( f ( r ) r t ( f ) ) d r
t ( f ) ,
s ( r ) (1 r ) d r
− −
=
−
= =
− −
=
−
∫
∫
∫
∫
Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 12,Number 1 (2016)
© Research India Publications : http://www.ripublication.com
274
1
2
2 2
2
0
1 2 1
2 2
0
2
2 2 2
1
222 2
0
3 2 1
2 2
0
s (r ) (1 r )( f ( r ) rt (f )) d r
t (f ) ,
s (r)(1 r ) dr
t (f ) f (1) f (1), , ( 4 .1)
s (r ) (1 r )( f ( r ) rt (f )) d r
t (f )
s (r)(1 r ) dr
− −
=
−
= =
− −
=
−
∫
∫
∫
∫
If mag
1
(f(x ))
%
%
and mag 2
(f(x ))
%
%
are of opposite
sign, then there exists a root between
1
x
%
and
2
x
%
Let it be
3
x
%
and
1 2
3
x x
x
2
+
=
% %
%If mag 3
(f(x ))
%
%
and
mag
1
(f(x ))
%
%
are of opposite sign, then root lies
between
3
µ
and
1
µ
Let it be
4
x
%
and
1 3
4
x x
x
2
+
=
% %
%Proceeding in this way, a sequence
of roots
{
}
x
can be obtained and it converges to
the exact root.
V. NUMERICAL EXAMPLE
Consider a fuzzy nonlinear equation
3 2
x x 1 0
+ − =
% % discussed in Gautam[4]
Let 1
x (0,0.5,1.0)
=
% 2
x (0.5,1.0,1.5)
=
% be the
two initial roots of the equation
Then
1
1
2
2
3 2
1
3 2
1
[x , x ] [0.5r,1 0.5r ] and
[x , x ] [0.5 0.5r,1.5 0.5r]
Then f (x ) (0.5r) (0.5r) 1and
f (x) (1 0.5r) (1 0.5r) 1
= −
= + −
= + −
= − + − −
3 2
2
3 2
2
f (x) (0.5(1 r)) (0.5(1 r)) 1and
f (x) (1.5 0.5r) (1.5 0.5r) 1
= + + + −
= − + − −
Then from Eq.(4.1)
1
2 1
2
1 2
1 2
3 3
3
3
f (x ) ( 0.66, 0.625,0.27)
f (x ) ( 0.725 ,1,1.23) and mag(f (x )) 0 and
mag(f (x )) 0
Therefore root lies be tween x and x
x x
Let it be x .Then x (0.25, 0.75,1.25) and
2
f (x ) ( 1.11, 0.02, 4.25)
Mag(f (x )) 0.Therefo
= − −
= − <
>
+
= =
= − −
>
%
% %
%
%
%
3 1
4
re root lies between x and x
x (0.13,0.63,1.13)=
Proceeding in this way, a sequence of roots
{
}
x
%
can be obtained.
TABLE 1 ROOTS OF THE POLYNOMIAL
No. of iterations
Roots
1 (0.25,0.75,1.25)
2 (0.13,0.63,1.13)
3 (0.19,0.69,1.19)
4 (0.16,0.66,1.16)
VI. CONCLUSION
Fuzzy polynomial is converted into a new
parametric form using the bisection algorithm. The
roots of the polynomial has been obtained using the
bisection algorithm. The numerical results are
provided to illustrate the theoretical results (see
Table 1).
REFERENCES
[1] Adabitabarfirozja. M and Eslampia. Z,
“Triangular approximations of fuzzy number with
value and ambiguity functions”, Int. J. Industrial
Mathematics, Vol. 5, pp 41-45, 2013.
[2] Abbasbandy.S and Hajjari. T , “A new approach
for ranking of trapezoidal fuzzy numbers”,
Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 12,Number 1 (2016)
© Research India Publications : http://www.ripublication.com
275
Computers and Mathematics with Applications,
57 ,pp 413–419, 2009.
[3] Bashar M.A and S.Shirin “Squares and Square
roots of continuous fuzzy numbers”, Dhakauniv
.J.Sci, 53(2), pp 131-140, , 2005.
[4] Goutam Kumar Saha and Shapla Shirin(2013),
“Solution of Fuzzy Non-linear Equation using
Bisection Algorithm,” Dhaka Univ. J. Sci. Vol
61(1), pp 53-58, 2013.
[5] Javad Shokri, “Numerical Method for Solving
Fuzzy Nonlinear Equations”, Applied
Mathematical Sciences, Vol. 2, no. 24, pp 1191 –
1203, 2008.
[6] Kajani.M.T, Asady.B and Vencheh .A.H, “An
iterative method for solving dual fuzzy nonlinear
equations”,Applied Mathematics and
computation, Vol 167, pp 316-323, 2005.
[7] Mahmoud Paripour, Eisa Zarei, Ahmad
Shahsavaran, “Numerical solution for a system of
fuzzy nonlinear equations”, Journal of Fuzzy Set
Valued Analysis , pp 1-10, 2014.
[8] Ming Ma, Menahem Friedman and Abraham
Kandel, New fuzzy arithmetic. Fuzzy sets and
systems. Vol 108, pp 83-90, 1999.
[9] L.S. Senthilkumar and K. Ganesan, “Solving
Fuzzy Nonlinear Equation using Harmonic
Mean”, International Journal of Scientific &
Engineering Research, Volume 6, Issue 3, pp
229-232, March-2015 .
[10] Noor’ani .A, Kavikumar.J, Mustafa.M, “Solving
a system of fuzzy polynomials by ranking
method”,Far East journal of applied
Mathematics, Vol 43(1), pp 7-20, 2010.
[11] Rogers, F., Jun, Y., “Fuzzy Nonlinear
Optimization for the Linear Fuzzy Real Number
System”, International Mathematical Forum,
4(12), pp 589 – 596, 2009.
[12] Salehnegad .M Abbasbandy.S, Mosleh.M,
Otadi.M,“Canonical representation for
approximating solution of fuzzy polynomial
equations”, Iranian journal of optimization, Vol
3, pp 447-454, 2010.
[13] Saneifard. R , “A method for defuzzification by
Weighted Distance”, Int. J. Industrial
Mathematics, Vol.3, pp 209-217,2009.
[14] Saneifard. R, “On the problem of defuzzification
in computing with regular weighted function”,
Int. J. Industrial Mathematics, Vol. 2, pp 125-
133, 2011.
[15] Seied Mahmoud Khorasani and M. H. Dogani
Aghcheghloo,“Solving Fuzzy Nonlinear Equation
with Secant Method”, International Journal of
Algebra, Vol. 5, no. 6, pp 295 – 299, 2011.
[16] Subash, R. and Sathya (2011), “Numerical
solution of fuzzy modified Newton-Raphson
method for solving nonlinear equations”,
International Journal of Current Research, vol
3(11), pp 390-392, 2011.
[17] Waziri.M.Y, and Majid.Z.A, “A new approach
for solving dual fuzzy nonlinear equations using
Broyden's and Newton's methods”, Advances in
Fuzzy Systems, Vol 2012, Article No. 19, 2012.
[18] Zadeh L,A, “Fuzzy sets”, Information and
Control , Vol 8, pp 338–353, 1965.
[19] Ronald E.Giachetti and Robert E. Young, “A
Parametric representation of fuzzy numbers and
their arithmetic operators”, Fuzzy Sets and
Systems, Vol 91, pp185-202, 1997.
[20] Abbasbandy , Asady, “The nearest trapezoidal
fuzzy number to a fuzzy quantity”, Appl. Math.
Comput. Vol 156, pp 381-386, 2004.
[21] William Voxman, “Canonical representation of
discrete fuzzy numbers” Fuzzy Sets and Systems,
Vol . 118, pp 457-466, 2001.
[22] Abbasbandy, Amirfakhrian,”The nearest
approximation of a fuzzy quantity in parametric
form”,Appl.Math.Comput.Vol.172,pp624-
632,2006.
Global Journal of Pure and Applied Mathematics (GJPAM) ISSN 0973-1768 Volume 12,Number 1 (2016)
© Research India Publications : http://www.ripublication.com
276
