关于含缺失项的小型成对比较矩阵最优补全的一致性
On the coincidence of optimal completions for small pairwise comparison matrices with missing entries
AI总结：
该研究探讨含缺失项的成对比较矩阵的最优补全问题，证明Saaty与几何非一致性指标在四阶及以下矩阵中产生相同填充，五阶及以上则不然。
摘要：
不完整的成对比较矩阵包含一些缺失的判断。估计这些值的一种自然方法是在未知条目被替换为变量后，最小化一个合理的非一致性度量。为此目的广泛使用的两个非一致性指标是Saaty非一致性指标和几何非一致性指标，它们分别与特征向量和最小二乘对数优先级推导方法密切相关。已证明，对于阶数不超过四的不完整成对比较矩阵，这两种度量意味着相同的最优填充，但对于阶数至少为五的矩阵则不一定如此。
英文摘要：
Incomplete pairwise comparison matrices contain some missing judgements. A natural approach to estimate these values is provided by minimising a reasonable measure of inconsistency after unknown entries are replaced by variables. Two widely used inconsistency indices for this purpose are Saaty's inconsistency index and the geometric inconsistency index, which are closely related to the eigenvector and the logarithmic least squares priority deriving methods, respectively. The two measures are proven to imply the same optimal filling for incomplete pairwise comparison matrices up to order four but not necessarily for order at least five.
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