可能满意度与判断矩阵的一致性检验及改进

Possibility-Satisfiability Degree and the Test and Improvement of the Comparison Matrix' Consistency

对判断矩阵一致性的检验与改进是层次分析法中的重要问题.针对现有检验与调整方法存在的不足,将"可能满意度"的概念引入该领域,提出一种新的优化调整算法.该方法利用判断矩阵的特征值及其Frobenius范数,给出判断矩阵的可能度和满意度的定义与计算公式,分别考察一致性改进程度和相对原始判断矩阵的偏离程度,并将二者合并为一个衡量一致性改善效果的综合指标:判断矩阵的可能满意度.利用该指标,并结合Saaty的一致性比率(CR)标准,可以有效地控制不一致判断矩阵的改进方向和调整力度,在尽可能保留决策者原始判断信息条件下,逐步达到可接受的一致性.最后通过算例说明算法的有效性.

It is an important problem to test and improve the consistency of the comparison matrix in AHP. Owing to the shortcomings of existing methods, we applied the concept of 'possibility-satisfiability degree' into this realm, and put forward a new kind of optimal adjustment algorithm. It made use of the maximum eigenvalue and Frobenius-norm of the comparison matrix to define the possibility degree and satisfiability degree of that matrix, which could examine respectively the improvement extent and the degree of deviation between original matrix and the adjusted one. Then by combining the two measurements into a synthetical index: the possibility-satisfiability degree of the comparison matrix, the general effect of the adjustment could be embodied. Through this index accompanied with Saaty's CR standard, the direction and extent of the improvement would be controlled, and the optimal matrix with acceptable consistency could be available, which had preserved furthest the decision maker's primitive judgment information. The final example illustrated the flow and efficiency of the new method.