In this paper, we propose a new method for the derivation of a priority vector from an incomplete pairwise comparisons (PC) matrix. We assume that each entry of a PC matrix provided by an expert is also evaluated in terms of the expert's confidence in a particular judgment. Then, from corresponding graph representations of a given PC matrix, all spanning trees are found. For each spanning tree, a unique priority vector is obtained with the weight corresponding to the confidence levels of entries that constitute this tree. At the end, the final priority vector is obtained through an aggregation of priority vectors achieved from all spanning trees. Confidence levels are modeled by real (crisp) numbers and triangular fuzzy numbers. Numerical examples and comparisons with other methods are also provided. Last, but not least, we introduce a new formula for an upper bound of the number of spanning trees, so that a decision maker gains knowledge (in advance) on how computationally demanding the proposed method is for a given PC matrix.

翻译：本文提出了一种从不完全成对比较矩阵导出优先向量的新方法。我们假设专家提供的成对比较矩阵中的每个条目还包含专家对该特定判断的置信度评估。随后，基于给定成对比较矩阵的对应图表示，找出所有生成树。对于每棵生成树，通过构成该树的条目对应的置信水平权重，获得唯一的优先向量。最终，通过对所有生成树得到的优先向量进行聚合，得到最终的优先向量。置信水平采用实数（清晰数）和三角模糊数进行建模。此外，本文提供了数值算例及与其他方法的比较。最后但同样重要的是，我们引入了一个关于生成树数量上界的新公式，使决策者能够提前了解所提方法在给定成对比较矩阵上的计算复杂度。