TY - EJOU
AU - Gasilov, Nizami
AU - Amrahov, Şahin Emrah
AU - Fatullayev, Afet Golayoğlu
AU - Karakaş, Halil İbrahim
AU - Akın, Ömer
TI - A Geometric Approach to Solve Fuzzy Linear Systems
T2 - Computer Modeling in Engineering \& Sciences
PY - 2011
VL - 75
IS - 3&4
SN - 1526-1506
AB - 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.
KW - Fuzzy linear system
KW - triangular fuzzy number
KW - generalized permutation matrix
DO - 10.3970/cmes.2011.075.189