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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.

##### MSC:
 15A06 Linear equations (linear algebra) 08A72 Fuzzy algebraic structures 65F05 Direct methods for linear systems and matrix inversion (numerical linear algebra) 65G30 Interval and finite arithmetic
##### Keywords:
interval linear systems; fuzzy number; fuzzy vector; algorithm