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Maximal- and minimal symmetric solutions of fully fuzzy linear systems. (English) Zbl 1220.65033

Summary: We propose a new method to obtain symmetric solutions of a fully fuzzy linear system (FFLS) based on a 1-cut expansion. To this end, we solve the 1-cut of a FFLS (in the present paper, we assumed that the 1-cut of a FFLS is a crisp linear system or equivalently, the matrix coefficient and right hand side have triangular shapes), then some unknown symmetric spreads are allocated to each row of a 1-cut of a FFLS. So, after some manipulations, the original FFLS is transformed to solving $2n$ linear equations to find the symmetric spreads.

However, our method always give us a fuzzy number vector solution. Moreover, using the proposed method leads to determining the maximal- and minimal symmetric solutions of the FFLS which are placed in a tolerable solution set and a controllable solution set, respectively. However, the obtained solutions could be interpreted as bounded symmetric solutions of the FFLS which are useful for a large number of multiplications existing between two fuzzy numbers. Finally, some numerical examples are given to illustrate the ability of the proposed method.

##### MSC:
 65F05 Direct methods for linear systems and matrix inversion (numerical linear algebra) 15B15 Fuzzy matrices 65C30 Stochastic differential and integral equations
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