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 yi, 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 = (aij), 1 < i < n; 1 < j < n is an n x n crisp matrix and yi, 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 yir 1 < i < n are TFN, in Equation 3.63 we have an FLS with TFN, whereas if the coefficient matrix A = (aij)and yi, 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 а-cut form.

Let [aL, aN, aR] 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 a-Cuts.

Algorithm 3.1

Matinfar et al. (2008)

x

Left

Right

Left

Right

x1

  • 45 23
  • ---1--a
  • 11 11

10 12 - - a 11 11

-4 + 2a

-1 - a

x2

  • 2 4
  • ---1--a
  • 11 11

68 46 - a 11 11

1 + a

5 - 3a

x3

  • 71 6
  • --1--a
  • 11 11

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

 
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