Cycle products and efficient vectors in reciprocal matrices
arxiv(2023)
摘要
Given an $n$-by-$n$ reciprocal (pair-wise comparison) matrix $A=[a_{ij}]$ and
a positive vector $w=\left[ \begin{array} [c]{ccc}% w_{1} & \cdots & w_{n}%
\end{array} \right] ^{T},$ the directed graph $G(A,w)$ has an edge from $i$ to
$j$ if and only if $w_{i}\geq a_{ij}w_{j}.$ Vector $w$ is efficient for $A$ if
and only if $G(A,w)$ is strongly connected if and only if (new to this
literature) $G(A,w)$ contains a Hamiltonian cycle. We show an exact
correspondence between such cycles and Hamiltonian cycle products in $A$ that
are $\leq1.$ This facilitates a way to describe all efficient vectors for $A$
via cycle products, and applications, such as that the set of vectors efficient
for $A$ determines $A,$ and that, if $G(A,u)$ and $G(A,v)$ present a common
cycle, all vectors on the line segment joining $u$ and $v$ are efficient. A
sufficient condition for convexity of the efficient set of vectors follows.
Order reversals in an efficient vector are also analyzed. The description of
the set of efficient vectors for an $n$-by-$n$ column perturbed consistent
matrix is given. It is the union of at most $(n-1)(n-2)/2$ convex subsets.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要