模糊关系不等式 A o X o B ≤ C 的解
Solutions to fuzzy relation inequality A o X o B ≤ C

对于普通的矩阵乘积, 当一个方程或者不等式有解时, 有可能存在无数多个解, 而直接求解它们又是很困 难的. 同样, 对于有限论域上采用最大–最小合成算子的模糊关系方程或者不等式也存在着类似的问题. 不幸的是, 研究此类问题的文献相对较少. 本文致力于研究模糊关系不等式A o X o B ≤ C的一种新求解方法. 首先, 利用两 个重要的公式, 将所考虑的模糊关系不等式转化成较简单的形式. 对于模糊关系不等式的可解性给出一个充分必 要条件. 它表明模糊关系不等式A o X o B ≤ C的解可以由有限个节点解来刻画. 然后, 利用矩阵的半张量积, 给出 具体的求解算法. 最后, 介绍了具有模糊关系不等式限制的格化线性规划, 来说明本文所提出方法的有效性.
For traditional matrix product, there may exist infinite solutions, in a sense that some matrix equations or inequalities can be solved. Furthermore, it is very difficult to solve them directly. Similarly, for max-min composition in finite course, fuzzy relational equations or inequalities may also have the same trouble. Unfortunately, there are few papers referring to the problem. This paper devotes to deriving a new method of solving fuzzy relation inequality (FRI) in terms of A?X ?B 6 C. First of all, two important formulas are proved. Then, the considered FRIs are converted into simplified ones taking use of the two transformations. While for solvability of FRIs, a necessary and sufficient condition is obtained. It illustrates that the solutions of considered FRIs can be depicted by finite ones. Via semi-tensor product (STP) of matrices, a concrete algorithm is derived. Finally, with FRIs constraints latticized linear programming is presented to demonstrate effectiveness of the proposed methods.