Interval and Fuzzy Systems of Equations
The linear system
where
the coefficient matrix A = (a), 1 < i < n; 1 < j < n is an n x n crisp matrix y„ 1 < i < n are interval numbers, is called an interval linear system
If the coefficient matrix A = (aj and y_{i}, 1 < i < n both are interval then the system is called a fully interval linear system, whereas in Equation 3.63, if the coefficient matrix A = (a_{ij}), 1 < i < n; 1 < j < n is an n x n crisp matrix and y_{i}, 1 < i < n are fuzzy numbers, it is called a fuzzy linear system (FLS). If the coefficient matrix A = (a_{{]}) and y,, 1 < i < n both are fuzzy then the system is called fully fuzzy linear system (FFLS).
Linear System of Equations with Triangular Fuzzy Numbers
If the coefficient matrix A = (aj), 1 < i < n and 1 < j < n is an n x n crisp matrix and y_{ir} 1 < i < n are TFN, in Equation 3.63 we have an FLS with TFN, whereas if the coefficient matrix A = (a_{ij})and y_{i}, 1 < i < n both are TFN, then the system is called FFLS with TFN. Now, an algorithm is proposed to solve linear system of equations with TFN.
Algorithm 3.1
Step 1. TFN is written in аcut form.
Let [a^{L}, a^{N}, a^{R}] be a triangular fuzzy number, then it may be represented as
Step 2. Now, the intervals are transferred into the crisp form using the transformation given in Equations 3.44 through 3.47.
Step 3. We get a system of linear equations with crisp values. This system may be solved by any standard method used for crisp values.
Step 4. Finally, the solution vector is
Example 3.2
Let us consider the following TFN system of linear equations:
Equation 3.64 may be transformed into the following interval form:
The preceding equations may be written as
TABLE 3.1
Comparison between the Fuzzy Solutions in Terms of aCuts.
Algorithm 3.1 
Matinfar et al. (2008) 

x 
Left 
Right 
Left 
Right 
^{x}1 

10 12   a 11 11 
4 + 2a 
1  a 
^{x}2 

68 46  a 11 11 
1 + a 
5  3a 
^{x}3 

83 6  a 11 11 
6 + a 
8  a 
Now, solving Equation 3.66, we may get the solution vector as
The obtained results are compared with the results (Matinfar et al. 2008) for different membership functions. The results are presented in Table 3.1.
Example 3.3
Next, let us consider the fully fuzzy system of linear equations Ax = b, where
The matrix A and vector b may be transformed into the interval form using Algorithm 3.1 and accordingly, we get
Finally, the solution vector is obtained as