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（PC）矩阵推导优先向量（priority vector）的新方法。我们假设专家提供的PC矩阵中每个条目均附带对该特定判断的置信度评估。随后，通过给定PC矩阵对应的图表示，找出所有生成树（spanning tree）。对于每棵生成树，可获得唯一优先向量，其权重对应于构成该树条目的置信水平。最终优先向量通过对所有生成树所得优先向量的聚合得到。置信水平采用实数（清晰数）与三角模糊数建模。文中还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，我们提出了生成树数量上限的新公式，使决策者能够预先了解所提方法对给定PC矩阵的计算复杂度。