AI 中文总结
研究互反矩阵和向量相关有向图凝聚结构，在\(w\)低效时给出结构特征与构造性结果，如通过修改互反项实现效率、构造增强矩阵，还提出将\(w\)转换为有效向量的程序。
AI 中文摘要
互反矩阵是层次分析法（AHP）中的基本工具，优先级向量通常从成对比较中得出。正向量的效率可通过与互反矩阵\(A\)和向量\(w\)相关的有向图\(G_{A,w}\)的强连通性来表征。本文研究\(w\)低效时\(G_{A,w}\)的凝聚有向图结构，特别关注Perron向量。给出了该结构的特征并得出一些构造性结果。展示了如何通过修改一对互反项实现效率，构造了一个其有效向量扩展\(w\)的增强互反矩阵。最后提出将\(w\)转换为\(A\)的有效向量的程序。
英文摘要
Reciprocal matrices are a fundamental tool in the Analytic Hierarchy Process (AHP), where priority vectors are typically derived from pairwise comparisons. The efficiency of a positive vector, in the sense of Pareto optimality, can be characterized through the strong connectivity of a directed graph $G_{A,w}$ associated with a reciprocal matrix $A$ and a vector $w$. In this paper, we investigate the structure of the condensation digraph of $G_{A,w}$ in the case where $w$ is inefficient, with particular emphasis on the Perron vector. We provide a characterization of this structure and derive several constructive results. In particular, we show how efficiency can be achieved by modifying a single pair of reciprocal entries, and we construct an augmented reciprocal matrix whose efficient vector extends $w$. Finally, we propose a procedure to transform $w$ into an efficient vector for $A$.
