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

Muzzioli, S.; Reynaerts, H.

Fuzzy Sets Syst. 157, No. 7, 939-951 (2006).

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.

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

Keywords:

interval linear systems; fuzzy number; fuzzy vector; algorithm

Citations:

Zbl 0741.65023

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