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**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.

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.

Reviewer: Václav Burjan (Praha)

### 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 |

### Citations:

Zbl 0741.65023
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\textit{S. Muzzioli} and \textit{H. Reynaerts}, Fuzzy Sets Syst. 157, No. 7, 939--951 (2006; Zbl 1095.15004)

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### References:

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