The consistency test is a vital component of pairwise comparison matrices if meaningful results are to be guaranteed, and it has been studied extensively since the analytic hierarchy process was developed by Saaty. However, when using the existing methods, it is imperative to carry out matrix operations, which are usually not intuitional. In this paper, a direct method, from the perspective of 3 tuples, is proposed to test and improve the consistency of a pairwise comparison matrix, which is far more intuitional and easier to understand. In the proposed 3 tuples iterative method, only simple mathematical calculations are needed, without the need for matrix operations. Furthermore, the calculation of Saaty’s consistency ratio is only conducted to verify the proposed method. Some related theorems and propositions are proved mathematically or verified by random simulations. The proposed method is applied to some published examples to verify its effectiveness and practicability. Finally, we provide an improved iterative method, which is based on the 3 tuples iterative method. The improved iterative method and the 3 tuples iterative method are closely linked and each has its own different perspective. Moreover, they are applied on different occasions and reinforce one another.

中文翻译：

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层次分析法的直接一致性测试和改进方法

如果要确保有意义的结果，一致性测试是成对比较矩阵的重要组成部分，并且自Saaty开发分析层次过程以来，对此进行了广泛的研究。但是，当使用现有方法时，必须进行矩阵运算，这通常不是直觉的。本文从3个元组的角度出发，提出了一种直接的方法来测试和提高成对比较矩阵的一致性，这更加直观，更易于理解。在提出的三元组迭代方法中，仅需要简单的数学计算，而无需矩阵运算。此外，Saaty一致性比率的计算仅是为了验证所提出的方法。一些相关的定理和命题可以通过数学证明或通过随机模拟进行验证。将该方法应用于一些公开的实例，以验证其有效性和实用性。最后，我们提供了一种改进的迭代方法，该方法基于3个元组的迭代方法。改进的迭代方法和3个元组的迭代方法紧密相连，并且各自有不同的观点。而且，它们被应用在不同的场合并且彼此增强。改进的迭代方法和3个元组的迭代方法紧密相连，并且各自有不同的观点。而且，它们被应用在不同的场合并且彼此增强。改进的迭代方法和3个元组的迭代方法紧密相连，并且各自有不同的观点。而且，它们被应用在不同的场合并且彼此增强。