A Geometric Approach to Solve Fuzzy Linear Systems
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, -cuts of the solution can not be determined by geometric similarity and additional computations are needed.
Keywords
Cite
@article{arxiv.0910.4049,
title = {A Geometric Approach to Solve Fuzzy Linear Systems},
author = {N. Gasilov and Şahin Emrah Amrahov and A. Golayoglu Fatullayev and H. I. Karakas and O. Akin},
journal= {arXiv preprint arXiv:0910.4049},
year = {2011}
}