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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.

15A06Linear equations (linear algebra)
08A72Fuzzy algebraic structures
65F05Direct methods for linear systems and matrix inversion (numerical linear algebra)
65G30Interval and finite arithmetic