Comparison of Classical Method, Extension Principle and α-Cuts and Interval Arithmetic Method in Solving System of Fuzzy Linear Equations ()
1. Introduction
There are many linear equation systems in many areas of science and engineering. According to Moore [1] , exact numerical data might be unrealistic, but there could be considered uncertain data as more aspects of a real word problem. Fuzzy data are being used as a natural way to describe uncertain data. So, we need to solve those linear systems in which all parameters or some of them are fuzzy numbers. Friedman et al. [2] [3] applied an embedding method for solving
, where A is a nonsingular crisp matrix. There are many other numerical methods for solving fuzzy linear system such as Jacobi, Gauss-Seidel, Adomiam decomposition method and SOR iterative method [4] [5] [6] [7] . Dehgan in [8] , [9] introduced the full fuzzy system in which b and A are fuzzy vector and fuzzy matrix, respectively. Then Kumar in [10] obtained an exact solution of fully fuzzy linear system by solving linear programming.
In 1965, Lotfi A. Zadeh [11] , professor of electrical engineering at the University of California (Berkley), published the first of his paper on his new theory of Fuzzy Sets and System. After the development of fuzzy set theory, researchers have successfully applied this in economics. Buckley [12] applied fuzzy mathematics in finance; in 1992 Buckley devised a technique to solve fuzzy equations in economics and finance. The above methods made us inspired to work on the solution techniques and finally, we got something. The objective of this paper is to present three different and effective methods to solve the system of fuzzy linear equation. Furthermore, we show the comparison among the methods with the help of numerical calculation as well as graphical representations. The paper organized as follows. In Section 2 we set some basic definitions and notation list. Section 3 deals with the methods. The applications of the models are presented in Section 4 and finally, Section 5 shows the results of the models.
2. Notations and Definitions
2.1. Notations List
2.2. Fuzzy Sets
A fuzzy set [3] is a class of objects with a continuum of the grade of membership. Let X be a space of points. A fuzzy set A in X is characterized by a membership function which associates with each points in X a real number
in the interval
with the value of
at x representing the grade of membership of x in A. Thus the nearer to the value of
to unity, the higher the grade of membership of x in A.
2.3. Fuzzy Linear Equation
Fuzzy linear equations are similar to ordinary linear equations in classical mathematics. A fuzzy linear equation is of the form
or
or
where
and
are given fuzzy numbers and
is an unknown fuzzy number by which the equation is satisfied.
2.4. The System of Fuzzy Linear Equations
A system of fuzzy linear equations is of the form
, where
is a
matrix of fuzzy numbers
,
is an unknown
vector of fuzzy numbers
by which the equation is satisfied and
is a
vector of fuzzy numbers
. Now we can write corresponding
system for all
as follows:
3. Methods
3.1. Classical Solution
We denote the classical solution of
as
, if it exists.
Substitute the α-cuts of
,
and
for
,
and
respectively in the system of linear equations
(1)
(2)
(3)
After substituting the α-cuts of
,
and
for
,
and
in the Equations (1)-(3), we get the following three interval equations
,
(4)
(5)
(6)
We now need to simplify these equations.
Assuming that all the
and
are triangular fuzzy numbers and put
in Equations (4)-(6). Then we obtain the crisp linear system of equations
The sign of the solutions
and
determines the sign of the unknown fuzzy numbers
and
. Let us assume for this discussion that all the
and all the
, so that we try for
. We get from Equations (4)-(6)
(7)
(8)
(9)
which yields a
crisp system of linear equations as below
(10)
(11)
(12)
(13)
(14)
(15)
We solve this system for
and
.
Using matrix notation this system can be written as
(16)
Using
,
and
, (17)
we get a crisp system of the form
.
For obtaining the fuzzy solution for the fully fuzzy linear system of equations, the necessary condition is that the coefficient matrix of the converted crisp system is invertible
.
The system (5.2.29) can be partitioned into two system as
and
That is,
and
. The solution of these crisp systems determines
and
, which are used to reconstruct the components of
fuzzy vector
.
After solving for the
and
we check to see if the intervals
,
define continuous fuzzy numbers for
. What is needed is:
1)
,
2)
, and
3)
for
(equality for triangular shaped fuzzy numbers).
3.2. Extension Principle Solution
We denote the extension principle solution of a system of fuzzy linear equations by
and it always exists but may, or may not satisfy the original system of fuzzy linear equations. That is,
may, or may not be true.
Let the components of
are
and
. In this method, we need to fuzzify the crisp solutions
(18)
(19)
(20)
using the extension principle.
Let
,
,
and
.
To obtain the first component
in
, we substitute
for
in
and evaluate using the extension principle.
It should be noted that,
,
that is, the determinant of the coefficients matrix must be nonsingular and invertible.
Let α-cut of
is
.
Then the α-cut of
can be written as
Or,
(21)
(22)
for
and
.
Similarly, α-cut of
is
. Then the α-cut of
is
(23)
(24)
for
and
.
And the α-cut of
is
. Then the α-cut of
is
(25)
(26)
for
and
.
After solving for the
and
we check to see if the intervals
,
define continuous fuzzy numbers for
. What is needed is:
1)
,
2)
, and
3)
for
(equality for triangular shaped fuzzy numbers).
Now by setting
we can check whether
is true or false.
If we set
, we get the crisp solution
from Equation (21) and Equation (22); crisp solution
from Equation (23) and Equation (24) and crisp solution
from Equation (25) and Equation (26) by assuming all the
and
are triangular shaped fuzzy numbers and
.
3.3. α-Cuts and Interval Arithmetic
We denote the α-cut and interval arithmetic solution of a system of fuzzy linear equations by
and it always exists but may, or may not satisfy the original system of fuzzy linear equations. That is,
may, or may not be true.
Let the components of
are
and
. In this method, we need to fuzzify the crisp solutions
(27)
(28)
(29)
using α-cut and interval arithmetic.
We substitute the α-cuts of
for
in Equations (27)-(29) and the simplifying using interval arithmetic we obtain the α-cuts of
and
.
It should be noted that
,
that is, the determinant of the coefficients matrix must be nonsingular and invertible.
Let α-cut of
is
.
To find the α-cut of
we substitute
for
in Equation (27). Then we get,
Let us assume that all
and all the
. Then by simplifying and using the interval arithmetic we get,
(30)
And
(31)
Again let α-cut of
is
. To find the α-cut of
we substtute
for
in Equations (28). Then we get,
Let us assume that all
and all the
. Then by simplifying and using the interval arithmetic we get,
(32)
And
(33)
And finally, let α-cut of
is
. To find the α-cut of
we substitute
for
in Equations (29). Then we get,
Let us assume that all
and all the
. Then by simplifying and using the interval arithmetic we get,
(34)
And
(35)
After solving for the
and
we check to see if the intervals
,
define continuous fuzzy numbers for
. What is needed is:
1)
,
2)
, and
3)
for
(equality for triangular shaped fuzzy numbers).
Now by setting
we can check whether
is true or false.
If we set
, we get the crisp solution
from Equation (30) and Equation (31); crisp solution
from Equation (32) and Equation (33) and crisp solution
from Equation (34) and Equation (35) by assuming all the
and
are triangular shaped fuzzy numbers and
.
4. Applications
4.1. Classical Method
Consider the system of fuzzy linear equation in matrix form
We solve this fuzzy matrix equation using the classical method.
The above system can be written as
(36)
Here
,
, and
.
Also, we have,
,
and
.
Then the α-cuts are:
,
,
,
,
,
.
Now substituting the α-cuts of
,
,
,
,
and
for
,
,
,
,
and
in the system (36) and we get
(37)
If we put
, we get the crisp solutions
,
and
. So we assume we can get a solution with
,
and
.
From Equation (37) we get,
(38)
We find that,
;
;
;
;
;
.
That is
,
and
are increasing functions of
and
,
and
are decreasing functions of
. Also
,
and
.
Hence,
,
and
defines the α-cuts of three fuzzy numbers respectively.
Now the support of
is
and modal of
is
;
The support of
is
and modal of
is
; and
The support of
is
and modal of
is
.
Therefore we can say that, the classical solution
exists and its components are continuous triangular shaped fuzzy numbers
,
and
.
The membership function of the triangularly shaped number
is
, for
,
and,
, for
.
Thus the membership function of
is
(39)
and its graph is shown in Figure 1.
The membership function of the triangularly shaped number
is
, for
,
and
, for
.
Thus the membership function of
is
(40)
and its graph is shown in Figure 2.
And the membership function of the triangular shaped number
is
Figure 1. Graph of the membership function of
.
Figure 2. Graph of the membership function of
.
, for
,
and
, for
.
Thus the membership function of
is
(41)
and its graph is shown in Figure 3.
Finally the graph of the classical solution
is shown in Figure 4.
Figure 3. Graph of the membership function of
.
Figure 4. Graph of the classical solution
.
4.2. Extension Principle Method
Consider the system of fuzzy linear equation in matrix form
We solve this fuzzy matrix equation using the extension principle method.
Here,
,
,
,
,
,
,
,
and
. Also, we have,
,
and
.
Then the α-cuts are:
,
,
,
,
,
.
Now the crisp solutions are
(42)
(43)
(44)
Since
,
and
are increasing functions of
,
and
; and decreasing functions of
,
and
, then
and
Similarly,
Also,
Here,
We find that,
;
;
;
;
;
.
That is,
,
and
are increasing functions of
and
,
and
are decreasing functions of
. Also
,
and
.
Hence,
,
and
define the α-cuts of three fuzzy numbers respectively.
Now the support of
is
and modal of
is
;
The support of
is
and modal of
is
;
and the support of
is
and modal of
is
.
Therefore we can say that, the extension principle solution
exists and its components are continuous triangular shaped fuzzy numbers
,
and
.
The membership function of the triangularly shaped number
, is
, for
,
and
, for
.
Thus the membership function of
is
(45)
and its graph is shown in Figure 5.
The membership function of the triangularly shaped number
is
, for
,
and
, for
.
Thus the membership function of
is
(46)
and its graph is shown in Figure 6.
And the membership function of the triangularly shaped number
is
Figure 5. Graph of the membership function of
.
Figure 6. Graph of the membership function of
.
, for
and
, for
.
Thus the membership function of
is
(47)
and its graph is shown in Figure 7.
Finally the graph of the extension principle solution
is shown in Figure 8.
Figure 7. Graph of the membership function of
.
Figure 8. Graph of the extension principle solution
.
4.3. α-Cut and Interval Arithmetic
Consider the system of fuzzy linear equation in matrix form
We solve this fuzzy matrix equation using α-cut and interval arithmetic.
Here,
,
,
,
,
,
,
,
and
. Also we have,
,
and
.
Then the α-cuts are:
,
,
,
,
,
.
Now the crisp solutions are
(48)
(49)
(50)
We replace
by
respectively in Equations (48)-(50).
Then,
Similarly,
Also,
Here,
We find that,
;
;
;
;
;
.
That is,
,
and
are increasing functions of
and
,
and
are decreasing functions of
. Also
,
and
.
Hence,
,
and
define the α-cuts of three fuzzy numbers respectively.
Now the support of
is
and modal of
is
;
The support of
is
and modal of
is
; and
The support of
is
and modal of
is
.
Therefore we can say that, the α-cut and interval arithmetic solution
exists and its components are continuous triangular shaped fuzzy numbers
,
and
.
The membership function of the triangularly shaped number
is
, for
, and
, for
.
Thus the membership function of
is
(51)
and its graph is shown in Figure 9.
The membership function of the triangular shaped number
is
, for
, and
, for
.
Figure 9. Graph of the membership function of
.
Thus the membership function of
is
(52)
and its graph is shown in Figure 10.
And the membership function of the triangularly shaped number
is
, for
, and
, for
.
Thus the membership function of
is
(53)
and its graph is shown in Figure 11.
Finally, the graph of the α-cut and interval arithmetic solution
is shown in Figure 12.
Figure 10. Graph of the membership function of
.
Figure 11. Graph of the membership function of
.
Figure 12. Graph of the α-cut and interval arithmetic solution
.
5. Results
Now we compare the classical solution
, extension principle solution
, and α-cut and interval arithmetic solution
for the system
From the above discussion, we get the solutions as follows
Figure 13. Comparison of the classical solution
, extension principle solution
, and α-cut and interval arithmetic solution
,
and
.
Here we see that, extension principle solution
and α-cut and interval arithmetic solution
are equal. That is,
. Hence we can say,
. The comparison among the classical solution
, extension principle solution
, and α-cut and interval arithmetic solution
is shown in Figure 13.
6. Conclusion
The system of fuzzy linear equations undoubtedly plays a vital role in presently applied mathematics. Here our intention was to establish some models of solving that system and we presented three different methods of with their applications. We came to know by the above discussion that among the three models extension principle solution
, and α-cut and interval arithmetic solution
give the same results. In the graphical representation we, find that the extension principle solution
, and α-cut and interval arithmetic solution
meet at the same point but the classical solution
is deviated a bit from the other two. Actually the solving techniques and their comparison are the ultimate findings of our work.