ABSTRACT Complete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the “spanning tree” and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197–208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results.

中文翻译：

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从所有生成树计算的权重向量的算术（几何）平均值的（对数）最小二乘最优性，用于不完全加法（乘法）成对比较矩阵

摘要 完全和不完全加法/乘法成对比较矩阵应用于偏好建模、多属性决策和排名。本文证明了两种众所周知的方法的等价性。从所有生成树计算的权重向量的算术（几何）平均值被证明是（对数）最小二乘问题的最佳选择，不仅是完整的，正如最近在 Lundy, M., Siraj, S. , Greco, S. (2017)：“生成树”和行几何平均偏好向量的数学等价及其对偏好分析的影响，欧洲运筹学杂志 257(1) 197-208，但也适用于不完整矩阵. 与完整的情况不同，其中一个明确的公式，即矩阵元素的行算术/几何平均值，对于（对数）最小二乘问题存在，不完整的情况需要完全不同的新证明。最后，用于计算电路电位的基尔霍夫定律与我们的结果有关。