求解模糊线性系统的几何方法
A Geometric Approach to Solve Fuzzy Linear Systems
AI总结：
提出基于线性变换几何表示的方法，将解视为实向量模糊集，并证明当系数矩阵为广义置换矩阵时解为模糊数向量。
摘要：
本文研究了系数矩阵为清晰实数、右端项为三角模糊数向量的线性系统。提出了一种基于线性变换几何表示的新方法来求解。该方法利用了三角模糊数向量在n维空间中构成一个矩形棱柱，且平行六面体在线性变换下的像仍为平行六面体的事实。所提方法阐明了为什么一般情况下不同方法不能生成模糊数解。几何上证明了若系数矩阵为广义置换矩阵，则模糊线性系统（FLS）的解为模糊数向量，与右端项无关。本文与以往FLS论文最重要的区别在于，解被寻求为向量（实分量）的模糊集，而非模糊数向量。解集中的每个向量以一定可能性求解给定的FLS。所提方法也可应用于右端项为参数形式模糊数向量的情况。然而，此时解的α-截集不能通过几何相似性确定，需要额外计算。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
