Abstract: In order to solve the consistency problem of AHP, an adjustment method based on basic loops analysis is proposed, which can solve the logical inconsistency and numerical inconsistency at the same time while preserving most of the original comparison information. Both logical inconsistency and numerical inconsistency are caused by the decision-maker's(DM's)inaccurate judgment, in which the numerical inconsistency can be improved by reducing the consistent ratio (CR) of the pairwise comparison matrix (PCM), while logical inconsistency can only be solved by eliminating all the three way loops from the PCM. Therefore, by breaking down the n order PCM into three order basic matrices, among which the basic matrices with three-way cycle are called basic loops, so that the problem is equally transformed into consistency improvement problem of basic loops. By comparing the consistency of the basic loops, two methods, namely maximum CR sum method and optimization method, are proposed to determine the most inconsistent element (MIE), and an optimization model is designed to modify the MIE. Finally, the feasibility of our method is verified by several examples, and the comparison with existing methods shows that this method is more effective.

Key words: analytical hierarchy process(AHP), basic loop, logical consistency, numerical consistency, consistent ratio(CR)

摘要： 为解决AHP一致性问题,提出一种基于基本回路修正的调整方法,能够同时解决数值不一致和逻辑不一致问题,同时保证对原始信息的修改量最小。数值不一致和逻辑不一致均由决策者的不准确判断引起,其中数值不一致可以通过降低一致性比率(CR)值进行改善,而逻辑不一致只有将判断矩阵中所有三阶回路去除才能得到解决。因此,通过对n阶判断矩阵进行基本矩阵分解,得到C³_n个3阶的基本矩阵,其中存在三阶回路的称为基本回路,从而将判断矩阵的一致性修正问题转化为基本回路的一致性修正问题。通过对基本回路的一致性比较,提出了两种确定最不一致元素的方法,即CR和最大法和优化法,并设计了优化模型对最不一致元素进行修正。最后,通过算例分析验证了本文方法的可行性,与已有方法的对比结论证明了本文方法更为有效。

关键词: 层次分析法, 基本回路, 逻辑一致性, 数值一致性, 一致性比率