基于偏序集的模糊互补矩阵次序一致性检验与调整

针对模糊互补矩阵次序一致性检验和调整存在争议和繁杂问题,提出了模糊互补矩阵次序一致性检验与调整的偏序集表示方法。在定义了偏序集、模糊互补矩阵、截集矩阵等相关概念基础上,求出模糊互补矩阵B的0.5水平截集矩阵,证明了模糊互补判断矩阵次序一致性和0.5水平截集矩阵为偏序关系矩阵的等价性;模糊互补判断矩阵完全次序一致性的充要条件是任意截集均满足传递性;任意截集满足传递性和偏序关系矩阵的等价性。结果表明,利用0.5水平截集矩阵转换为矩阵来检验模糊互补矩阵的次序一致性;通过调整每个截集矩阵满足传递性并赋值,能够达到模糊互补矩阵完全次序一致性。最后,通过两个算例验证该检验和调整方法的合理性和可行性。

Order Consistency Test and Adjustment of Fuzzy Complementary Matrices Based on Partial Ordered Set

A new representation method based on the partial ordered set is proposed to solve the controversial and complicated problem of ordinal consistency test and adjustment of fuzzy complementary matrices. Based on the definitions of partial ordered set, fuzzy complementary matrices, truncated set matrix and other related concepts, the truncated set matrix with the level of 0.5 is obtained. The ordinal consistency of fuzzy complementary judgment matrix and the equivalence of the truncated set matrix with the level of 0.5 as partial order relation matrix are proved. The sufficient and necessary condition for the complete ordinal consistency of fuzzy complementary judgment matrix is that any truncated set satisfies transitivity, and any truncated set satisfies transitivity and the equivalence of partial order relation matrix. The results show that the ordinal consistency of the fuzzy complementary matrix can be checked by transforming the truncated set matrix with the level of 0.5 into the Hasse matrix. The complete ordinal consistency of fuzzy complementary matrix can be achieved by adjusting each truncated set matrix to satisfy transitivity and assignment. Finally, two examples are given to verify the rationality and feasibility of the test and adjustment method.