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VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9737
FUZZY FALSE POSITION METHOD FOR SOLVING FUZZY NONLINEAR
EQUATIONS
Muhammad Zaini Ahmad, Nor Aifa Jamaluddin, Elyana Sakib, Wan Suhana Wan Daud and Norazrizal Aswad
Abdul Rahman
Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, Arau, Perlis, Malaysia
E-Mail: mzaini@unimap.edu.my
ABSTRACT
In this paper, we focus on extended numerical methods for solving fuzzy nonlinear equations. An extension of
false position method into fuzzy setting is proposed for solving such equations and it will be referred to as fuzzy false
position method. An algorithm for this process of solving will be provided. For the purpose of optimization, genetic
algorithm will also be incorporated in order to find the best solution for the problem under consideration. Two numerical
examples with graphical representations are provided to illustrate the efficiency of the proposed method. The results
showed that the proposed method is able to find the best solution for fuzzy nonlinear equations.
Keywords: false position method, nonlinear equation, fuzzy nonlinear equation, numerical methods, fuzzy false position method.
INTRODUCTION
In recent years, system of simultaneous nonlinear
equations plays a major function in various fields such as
applied mathematics, engineering, statistics and social
sciences. Therefore, much attention has been given into
developing numerical methods to work out with these
systems.
In such cases, to model the real world
engineering systems, normally there will be problems of
fuzziness in some of the equation’s parameters, hence
fuzzy numbers are employed rather than crisp number. At
first, the concept of fuzzy numbers which include its
arithmetic operation were introduced by Zadeh [1]. Then,
it was followed by Dubois and Prade [2]. One of the
major applications of fuzzy arithmetic is nonlinear
systems whose parameters are totally or partially
represented by fuzzy numbers [3, 4].
However, in solving fuzzy nonlinear equations,
the classical numerical methods cannot be applied
directly. Indeed, standard analytical techniques proposed
by Buckley and Qu [5], are not suitable for solving the
fuzzy equations. Hence, it is necessary to examine and
develop new numerical methods to find the roots of fuzzy
nonlinear equations. Thus, in [6, 7, 8, 9, 10], Newton’s
method, Broyden’s method and Fixed Point’s method
were used to solve this type of equations. Another method
used is the false position method. The advantage of this
method is that it takes account of the observation of the
approximate solution. This is done by drawing a secant
line from two function values where one is negative and
the other one is positive. Then, the root is estimated as the
position where it crosses the x-axis or when the function
value is equal to zero. Once the false position method
comes close to the root, it converges quickly.
Genetic algorithm is a global optimization
method used for solving both constrained and
unconstrained problem inspired by natural evolution. The
algorithm repeatedly modifies a population of individual
solutions. The idea is by guessing solutions, then
combining the fittest solution to create a new generation
of solutions which is better. It is a popular strategy to
optimize non-linear systems with a large number of
variables. Genetic algorithm might not ensure an optimal
solution, but it is able to give a good approximation in a
reasonable amount of time.
In this paper, False Position method is proposed
to solve these fuzzy nonlinear equations. In the next
section, we will recall some basic definitions of fuzzy
numbers based on Zadeh’s extension principle. Later, we
will study the classical False Position method and its
algorithm. This include the genetic algorithm for
optimization process. Then, in subsequent section, we will
extend the False Position method into fuzzy setting for
solving fuzzy nonlinear equations. Finally, two numerical
examples with graphical representations will be
demonstrated to illustrate the proposed algorithm and
conclusion will be made.
BASIC CONCEPTS
In this section, we review some important
definitions of fuzzy numbers.
Definition 1. [2, 1, 11] A fuzzy number is a fuzzy set like
which satisfies the following
properties:
1. is upper semi continuous,
2. is outside some interval ,
3. there are real numbers such that ,
a. is monotonic increasing on ,
b. is monotonic decreasing on ,
c. , .
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9738
Definition 2. [12] A fuzzy number in parametric form
is a pair of functions , for which
satisfies the following requirements.
1. is a bounded monotonic increasing left continuous
function,
2. is a bounded monotonic decreasing left
continuous function,
3. for .
A crisp number is represented by for
.
Definition 3. A triangular fuzzy number is a fuzzy
number represented with three points, , and
its membership function can be represented as
where , and hence its parametric form is
Let be the set of all triangular fuzzy numbers.
Definition 4. [12] The addition and scalar multiplication
of fuzzy numbers are defined by the extension principle
and can be equivalently represented as follows.
For arbitrary , , and
, the addition and multiplication by real
number are defined as
1. addition, ,
2. subtraction, ,
3. scalar multiplication,
If , then .
THE CLASSICAL FALSE POSITION METHOD
FOR SOLVING NONLINEAR EQUATION
The method of false position is a root finding
algorithm that hybrid the features from bisection and
secant method. It involves the bracketing of bisection
method and the secant line in secant method. It was
introduced to improve the bisection method which
converges at a fairly slow speed. Like bisection method, it
starts with two proper
values of (lower bound value) and (upper bound
value) for the current bracket, such that
This is based on the following intermediate value
theorem.
Theorem 1. If is continuous on and is a value
between and , then there exists a value in
such that .
In bisection method, sometimes it may not be
efficient because it does not take into consideration that
is much closer to the zero of the function as
compared to . The idea for False Position method,
on advantage of this observation, is to connect function
value at to the function value at by drawing a
secant line, and then estimates the root, , as where it
crosses the -axis.
Firstly, we write down two versions of the slope
of the secant line from to and from
to .
and
So,
The above equation can be solved to obtain the
next predicted root as
Through simple algebraic manipulations, it can
be expressed as
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9739
False position algorithm
The steps to apply the false-position method to
find the root of the equation are as follows.
1. Choose and as two guesses for the root such
that , or in other words,
changes sign between and .
2. Estimate the root, of the equation as
3. Now check the following.
a. If , this implies that the root lies
between and , then replace .
b. If , this implies that the root lies
between and , then replace .
c. If or , implies
that the root is . Stop the algorithm .
4. Find the new estimate root
5. Find the new absolute relative approximate error,
where estimated root from current iteration
and estimated root from previous iteration.
6. Compare the absolute relative approximate error,
with the pre-specified relative error tolerance, . If
, go back to step 3, else, stop the algorithm.
FUZZY FALSE POSITION METHOD FOR
SOLVING FUZZY NONLINEAR EQUATIONS
The method of False Position to solve fuzzy
nonlinear equations, can be described as follows.
1. Choose two initial values of fuzzy numbers for which
their function value have opposite sign. Let us
assume two initial values are
and .
To fulfill that condition, check .
2. Transform the initial fuzzy values and
into their parametric form
,
where
,
and
,
where
.
For iteration, we discretise in the form
, where and .
The discretised α are equally spaced, that is
, for , and .
In this study, is called the discretization spacing.
After discretisation, we have a set of with
elements.
3. Then, find for with
,
,
,
where
In order to find minimum and maximum of ,
we use the genetic algorithm approach. A genetic
algorithm is a method for solving both constrained and
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9740
unconstrained optimization problems. Unlike classical
algorithm which only generates a single point, this genetic
algorithm generates a population of points from given
interval. In addition, the best point in that population
approaches an optimal solution. This method generates
optimum solution at each iteration for every
with .
4. When , then , check .
a. If , this implies that the fuzzy root lies
between and , then .
b. If , this implies that the fuzzy root lies
between and , then .
c. If or , this implies
that the fuzzy root is .
5. Continue the iteration for every until the stopping
criteria as follows is fulfilled.
,
where estimated root from current iteration and
estimated root from previous iteration when
.
6. Compare the absolute relative approximate error
with the pre-specified relative error tolerance . If
, then go to step 3, else, stop the algorithm.
NUMERICAL APPLICATIONS
Example 1. We try to solve the following fuzzy nonlinear
equation with fuzzy false position method.
,
,
where the tolerance, . Firstly, choose
two initial fuzzy values for which their function values
have opposite sign: and where
and . Hence, .
Then, the parametric form of and are as follows.
where
and
where
.
For the first iteration, find
and
where
By using MATLAB, for with ,
the first iteration is generated as shown in Table-1.
Table-1. First iteration for example-1.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace .
Then, we continue the iteration with the new
and existing . The second iteration is shown in
Table-2.
Table-2. Second iteration for example-1.
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9741
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , we proceed the iteration with the
new and existing . The third iteration is shown
in Table-3.
Table-3. Third iteration for example-1.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , proceed the iteration with the
new and existing . The fourth iteration is shown
in Table-4.
Table-4. Fourth iteration for example-1.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , we proceed the iteration with the
new and existing . The fifth iteration is shown in
Table-5.
Table-5. Fifth iteration for Example-1.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
As , we stop the iteration. Hence, the
fuzzy root for
. The
graphical representation of iterations done is illustrated in
Figure-1.
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9742
Figure-1. Graphical representation of optimum solution
of a fuzzy non-linear equation, .
Example 2. Consider the following fuzzy nonlinear
equation.
,
,
where the tolerance .
To find the solution using fuzzy false position
method, first, we assume two initial fuzzy values for
which their function values have opposite sign:
and where and . Hence
.
Then, the parametric form of and are as follows.
where
and
where
For the first iteration, find
and
where
By using MATLAB, for with ,
the first iteration is generated as shown in Table-6.
Table-6. First iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace .
Then, we continue the iteration with the new
and existing . The second iteration is shown in
Table-7.
Table-7. Second iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9743
Since , we proceed the iteration with the
new and existing . The third iteration is shown in
Table-8.
Table-8. Third iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , we proceed the iteration with the
new and existing . The fourth iteration is shown
in Table-9.
Table-9. Fourth iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , we proceed the iteration with the
new and existing . The fifth iteration is shown in
Table-10.
Table-10. Fifth iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , we proceed the iteration with the
new and existing . The sixth iteration is shown
in Table-11.
Table-11. Sixth iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9744
Since , we proceed the iteration with the
new and existing . The seventh iteration is
shown in Table-12.
Table-12. Seventh iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , we proceed the iteration with the
new and existing . The eighth iteration is shown
in Table-13.
Table-13. Eighth iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
Since , we proceed the iteration with the
new and existing . The ninth iteration is shown
in Table-14.
Table-14. Ninth iteration for example-2.
For , substitute into . By
checking where , this
implies that the fuzzy root lies between and . Then,
replace . Check
As , we stop the iteration. Hence, the
fuzzy root for
. The
graphical representation of iterations done is illustrated in
Figure-2.
Figure-2. Graphical representation of optimum solution
of a fuzzy non-linear equation, .
CONCLUSIONS
VOL. 11, NO. 16, AUGUST 2016 ISSN 1819-6608
ARPN Journal of Engineering and Applied Sciences
©2006-2016 Asian Research Publishing Network (ARPN). All rights reserved.
www.arpnjournals.com
9745
We have studied False Position’s method for
solving fuzzy nonlinear equations instead of standard
analytical techniques which are not suitable. Several
modifications have been made to guarantee the convexity
of fuzzy solutions. First, the fuzzy nonlinear equation is
written in parametric form and then is solved step by step
using fuzzy false position’s algorithm. Genetic algorithm
has been proposed in order to determine the optimum
solution at each iteration.
ACKNOWLEDGEMENTS
This research is based upon work supported by
Ministry of Higher Education of Malaysia (MOHE) and
partially supported by Universiti Malaysia Perlis
(UniMAP).
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