1. Introduction
In the real world, many of our scientific problems turn into problems related to solving linear system of equations. Parameters involved in such equations are generally determined through estimation, experiments and modeling. Thus, the parameters often involve some uncertainty or impreciseness. Therefore, our preferred choice is to choose fuzzy parameters rather than crisp parameters. Intuitionistic fuzzy parameters are more flexible in describing uncertainty with membership and non-membership functions with hesitancy function than fuzzy parameters. To handle this uncertainty or impreciseness, Zadeh [
1] introduced the concept of fuzzy set theory. Since then, there have been several generalizations of fuzzy set theory made by researchers. One of them is intuitionistic fuzzy set theory, which was introduced by Atanassov [
2,
3]. Friedman et al. [
4] proposed a general model to solve
FSLE, in which the coefficient matrix is crisp and the right-hand side is an arbitrary fuzzy vector. Iterative methods for solving FSLE are given by Allahviranloo [
5]. The SOR method to solve FSLE is presented by Allahviranloo [
6]. To solve IFLS, several authors provided different approaches. Atti et al. [
7] developed an approach to solve IFLS, in which they converted
IFLS into four
crisp linear systems of equations. Saw et al. [
8] proposed the Jacobi iterative method to solve IFLS. They converted
IFLS into one
crisp linear system of equations. In the present work, we extended the well-known Gauss–Seidel and SOR methods to solve IFLS.
2. Materials and Methods
The
intuitionistic fuzzy system of linear equations may be written as
In matrix-vector form, the above system may be written as , where the coefficient matrix is a crisp real matrix, is a column vector of fuzzy numbers and is the vector of fuzzy unknowns.
Definition 1. An intuitionistic fuzzy number vector given by , , , is called solution of (1) if: Hence, from (
1), we have four crisp
linear systems for all
i which can be extended to a
crisp linear system, as follows:
,
where
are determined as follows:
, ,
and which are not determined are zero.
Additionally, and
.
From the structure of S, it is clear that contains the positive entries of the matrix A, while contains the negative entries of the matrix A and . We represent S as where and .
Theorem 1. Let the matrix S be strictly diagonally dominant. The Gauss–Seidel iterate converges to for any (see [9], p. 120). Theorem 2. The matrix S is non-singular iff and are both non-singular. (see [4]) Proof. The matrix S is non-singular iff is non-singular. Now, is non-singular iff and is non-singular. □
Theorem 3. Let S be non-singular. Then, the unique solution X of Equation (1) is always a intuitionistic fuzzy vector for arbitrary vector Y, if is non-negative. (see [5]) Theorem 4. Matrix A in Equation (1) is strictly diagonally dominant if the matrix S is strictly diagonally dominant. (see [8]) 2.1. Gauss–Seidel Iterative Scheme
Without loss of generality, suppose that for all .
Let , where
, ,
and suppose .
From , we have
+ =
So, the Gauss–Seidel iterative technique reads as:
The results in the matrix-vector form of the Gauss–Seidel iterative technique are , where
,
, .
From Theorem (1) and (4), the Gauss–Seidel iterates converge to the unique solution , for any . The stopping criterion with tolerance is
, , , .
2.2. SOR Iterative Scheme
If we decomposed the matrix as , with diagonal component , and strictly lowered triangular component and upper triangular component , then the decomposed matrix S became similar to , where
, ,
From
, we rewrite the system as
Using relaxation parameter
, we rewrite the above system in the new form as
=
−
So, the SOR iterative technique read as:
This can be written in matrix-vector form as where
,
, .
3. A Practical Application
The authors of [
10] considered the electrical circuit shown in
Figure 1, where
and
are the input voltages, and
and
are the output voltages. The circuit is a kind of summing amplifier with two inputs and two outputs. The relationship between input and output voltages is as follows:
.
They considered the output voltages as type-2 fuzzy numbers. In this paper, we treat the same example, but we consider the output voltages as intuitionistic fuzzy numbers, as considered by the authors in [
7]:
and
.
Here, we are looking at how to calculate the input voltages when the output voltages are known but uncertain. That is, is “about 16 volts” and is “about –16 volts”. Different experts may have different viewpoints on the output voltage’s uncertainty.
Now, if we choose to focus on one expert’s interpretation, then the linear system shown in Equation (
1) will be type-1 FLSE, as mentioned in [
10].
In addition, we consider the hesitation of the expert, which is quite natural when making a decision, because of characteristics of human beings applying knowledge and skills. Then, the linear system shown in Equation (
1) will be an intuitionistic fuzzy linear system of equations. This is more realistic than type-1 FLSE.
If we want to consider more than one expert’s opinion, then we obtain the system of equations as type-2 FLSE, originally considered in [
10].
Now, if we consider different experts’ opinions individually, together with their hesitations, then we can take, for example, the arithmetic average of different IFNs to determine the output voltages, and the system can be better represented as IFLS.
In this case, the above system reduces to
The exact and approximated solutions are plotted and compared for .
The exact and approximated solutions are plotted and compared for .