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# A Geometric Approach to Solve Fuzzy Linear Systems

Nizami Gasilov1, Şahin Emrah Amrahov2, Afet Golayoğlu Fatullayev1, Halil İbrahim Karakaş1, Ömer Akın3
Ankara University, Computer Engineering Department, Ankara, Turkey.
TOBB ETU, Ankara, Turkey.

Computer Modeling in Engineering & Sciences 2011, 75(3&4), 189-204. https://doi.org/10.3970/cmes.2011.075.189

### Abstract

In this paper, linear systems with a crisp real coefficient matrix and with a vector of fuzzy triangular numbers on the right-hand side are studied. A new method, which is based on the geometric representations of linear transformations, is proposed to find solutions. The method uses the fact that a vector of fuzzy triangular numbers forms a rectangular prism in n-dimensional space and that the image of a parallelepiped is also a parallelepiped under a linear transformation. The suggested method clarifies why in general case different approaches do not generate solutions as fuzzy numbers. It is geometrically proved that if the coefficient matrix is a generalized permutation matrix, then the solution of a fuzzy linear system (FLS) is a vector of fuzzy numbers irrespective of the vector on the right-hand side. The most important difference between this and previous papers on FLS is that the solution is sought as a fuzzy set of vectors (with real components) rather than a vector of fuzzy numbers. Each vector in the solution set solves the given FLS with a certain possibility.
The suggested method can also be applied in the case when the right-hand side is a vector of fuzzy numbers in parametric form. However, in this case, alpha-cuts of the solution cannot be determined by geometric similarity and additional computations are needed.

### Keywords

Fuzzy linear system, triangular fuzzy number, generalized permutation matrix.

Gasilov, N., Amrahov, �. E., Fatullayev, A. G., Karakaş, H. �., Akın, �. (2011). A Geometric Approach to Solve Fuzzy Linear Systems. CMES-Computer Modeling in Engineering & Sciences, 75(3&4), 189–204. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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