处理判断矩阵次序一致性和基数一致性的优化方法

An optimization approach for managing ordinal and cardinal consistencies for pairwise comparison matrices

基数一致性和次序一致性均能对决策信息的理性程度进行刻画.当AHP判断矩阵(又称为互反判断矩阵、乘性偏好关系)具有基数和/或次序不一致时,很难对方案进行合理排序.然而,以往的针对判断矩阵一致性的研究很少同时考虑个体次序一致性和基数一致性.本文提出了用于解决AHP中判断矩阵个体一致性问题的优化模型.首先,对判断矩阵的次序一致性的满足条件进行了分析,导出了次序一致性条件对应的不等式和等式约束表达形式,从而能在优化模型中显式表示次序一致性.基于对次序一致性的显式表示,本文提出了三个优化模型,第一个模型使得修正后的判断矩阵满足次序一致性,第二个模型使得修正后的判断矩阵满足基数一致性,第三个模型则同时控制了次序一致性和基数一致性.与已有的个体不一致性调整方法相比,本文的模型解决了次序一致性以及同时满足次序一致性和基数一致性的优化建模问题,能在预定目标下直接得到最优结果。从而为决策者提供更加精准的交互反馈意见.最后,通过算例的比较分析验证了本文模型的优越性和有效性.

Both ordinal consistency and cardinal consistency are able to characterize the rationality degree of individual preferences.In the analytic hierarchy process(AHP),it is difficult to reasonably rank the alternatives when pairwise comparison matrices(PCMs,also called multiplicative preference relations)are significantly cardinal and/or ordinal inconsistent.However,few previous researches have simultaneously considered individual ordinal consistency and cardinal consistency.In this paper,an optimization approach is proposed to solve the individual consistency problems in AHP.Firstly,the conditions of satisfying the ordinal consistency of PCM are analyzed,and the expressions of inequality constraints corresponding to the ordinal consistency conditions are derived,so that the ordinal consistency can be controlled explicitly in the optimization model.Three optimization models are proposed:The first one is used to get a revised preference relation meeting ordinal consistency;the second one is used to obtain a revised preference relation with acceptable cardinal consistency level;and the third one is used to control both kinds of consistencies.Compared with the existing individual inconsistency adjustment methods,the proposed models can directly get the optimal results under the predetermined goals so as to provide a more accurate interactive feedback for decision makers.Finally,the superiority and validity of the proposed models are verified by using a classical example.