Abstract: Pairwise comparison matrix (PCM) is the basic mathematical tool of the analytic hierarch process (AHP). Ordinal consistency and acceptable consistency are two important concepts that capture the degree of consistency of a PCM, and ordinal consistency is considered the minimum requirement of rational judgments. However, the PCM provided by the decision maker usually does not exhibit ordinal consistency or acceptable consistency in a real case. It is therefore of great practical and theoretical significance to investigate the methods for improving the consistency of the PCM.
In recent years, though many methods have been proposed to improve ordinal consistency and acceptable consistency, there are still some gaps, as follows. (1)Existing strategies of eliminating the entries without ordinal consistency seem too single. (2)There are a few studies that make the modified PCM have both ordinal and acceptable consistency. (3)The process of solving some established optimization models is complicated, and the obtained results may not be optimal due to the strict constraints. (4)The deviation degree between the modified PCM and the original one may be higher, and even some of the revised elements exceed the 1/9-9 scale.
The process of forming the PCM is considered as a whole in the traditional AHP. In fact, the formation of the PCM is a complicated process, and it is related to psychology, knowledge and so on, especially when the number of alternatives is large. Recently, the leading principal submatrices model has been proposed as a new fundamental tool for decision analysis, where the typical AHP model serves as a special case. The advantage of the leading principal matrices model is that it provides a wise way to discover the irrational behavior of the decision-maker when evaluating the importance of alternatives.
In this paper, the leading principal submatrices model of the PCM is used to identify the entries without ordinal consistency, where the least number of modification entries are chosen as revision strategies. It is found that there may be more than one strategy to adjust the entries without ordinal consistency. Then, a minimal adjustment strategy is proposed. Three optimization models are established and optimized by using Gaussian quantum-behaved particle swarm optimization algorithm (GQPSO). The first one only considers ordinal consistency; the second one simply considers acceptable consistency; the third one considers both ordinal and acceptable consistency. Finally, the feasibility and efficiency of the models are verified by comparing other existing methods.
Different from previous studies, we propose a simple method to identify entries without ordinal consistency and solve the problem of simultaneously modifying ordinal inconsistency and acceptable consistency, which achieves intelligent resolution of optimization models and decision information adjustment minimization. The proposed models can provide the decision maker with more accurate and easily acceptable modification recommendations.
In the future, the proposed approach can be extended to investigate the consistency of additive reciprocal matrices, interval-valued comparison matrices, etc. Meanwhile, though the GQPSO algorithm achieves good performance in solving the established models, whether it outperforms other intelligent algorithms deserves further research.

Key words: pairwise comparison matrix, ordinal consistency, acceptable consistency, optimization model, Gaussian quantum-behaved particle swarm optimization algorithm

摘要： 互反判断矩阵是层次分析法的基本数学工具,其一致性定义及修正方法研究为导出可靠权重提供理论依据。本文提出互反判断矩阵的次序一致性和满意一致性修正的新方法,首先基于顺序主子式模型识别次序不一致性元素,建立元素调整最少的修正策略;接着建立次序一致性、满意一致性及次序一致性和满意一致性同时修正的三个优化模型;然后根据高斯量子行为粒子群优化算法实现优化模型求解,通过理论证明和比较分析验证所提模型的可行性和优越性。与已有模型比较表明,本文提出了次序不一致性元素的简便识别方法,解决了同时修正次序一致性和满意一致性的科学问题,实现了优化模型的智能求解和决策信息调整的最少化。

关键词: 互反判断矩阵, 次序一致性, 满意一致性, 优化模型, 高斯量子行为粒子群优化算法