## Fuzzy linear systems of the form $$A_{1}x+b_{1}=A_{2}x+b_{2}$$.(English)Zbl 1095.15004

The linear system $$Ax=b$$, where the elements $$a_{ij}$$ of the matrix $$A$$ and the elements $$b_i$$ of the vector $$b$$ are represented with interval values, is called an interval linear system. Similarly, the linear system $$Ax=b$$, where the elements $$a_{ij}$$ of the matrix $$A$$ and the elements $$b_i$$ of the vector $$b$$ are fuzzy numbers, is called a fuzzy linear system. Interval linear systems can be considered as a special case of fuzzy linear systems.
In this paper, the link between interval linear systems and fuzzy linear systems is illustrated. Also, a generalization of the vector solution obtained by J. J. Buckley and Y. Qu [Fuzzy Sets Syst. 43, 33–43 (1991; Zbl 0741.65023)] to the most general fuzzy system $$A_1 x + b_1=A_2 x + b_2$$, with $$A_1$$ and $$A_2$$ square matrices of fuzzy coefficients and $$b_1$$ and $$b_2$$ fuzzy number vectors, is proposed. The conditions under which the system has a vector solution are given and it is shown that the linear systems $$Ax = b$$ and $$A_1 x + b_1=A_2 x + b_2$$, with $$A = A_1 - A_2$$ and $$b = b_2 - b_1$$, have the same vector solutions. Finally, a simple algorithm, which is reduced to an interval analysis technique, to solve the system $$Ax = b$$, with $$A$$ and $$b$$ matrices with fuzzy elements, is introduced.

### MSC:

 15A06 Linear equations (linear algebraic aspects) 08A72 Fuzzy algebraic structures 65F05 Direct numerical methods for linear systems and matrix inversion 65G30 Interval and finite arithmetic

### Keywords:

interval linear systems; fuzzy number; fuzzy vector; algorithm

Zbl 0741.65023
Full Text:

### References:

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