Priorities in multi-criteria decision-making (MCDM) convey the relevance preference of one criterion over another, which is usually reflected by imposing the non-negativity and unit-sum constraints. The processing of such priorities is different than other unconstrained data, but this point is often neglected by researchers, which results in fallacious statistical analysis. This article studies three prevalent fallacies in group MCDM along with solutions based on compositional data analysis to avoid misusing statistical operations. First, we use a compositional approach to aggregate the priorities of a group of DMs and show that the outcome of the compositional analysis is identical to the normalized geometric mean, meaning that the arithmetic mean should be avoided. Furthermore, a new aggregation method is developed, which is a robust surrogate for the geometric mean. We also discuss the errors in computing measures of dispersion, including standard deviation and distance functions. Discussing the fallacies in computing the standard deviation, we provide a probabilistic criteria ranking by developing proper Bayesian tests, where we calculate the extent to which a criterion is more important than another. Finally, we explain the errors in computing the distance between priorities, and a clustering algorithm is specially tailored based on proper distance metrics.

翻译：多准则决策中的优先级传达了某一准则相对于另一准则的重要性偏好，通常通过施加非负性和单位求和约束来体现。此类优先级的处理方式不同于其他无约束数据，但这一要点常被研究者忽视，从而导致错误的统计分析。本文研究了群体多准则决策中三种普遍存在的谬误，并基于成分数据分析提出了避免误用统计操作的解决方案。首先，我们采用成分方法聚合一组决策者的优先级，结果表明成分分析的结果与归一化几何均值一致，这意味着应避免使用算术均值。此外，我们开发了一种新的聚合方法，作为几何均值的稳健替代方案。同时，我们讨论了计算离散度量（包括标准差和距离函数）时的错误。针对标准差计算中的谬误，我们通过开发适当的贝叶斯检验提供了概率性准则排序，从而计算某一准则相较于另一准则的重要程度。最后，我们解释了计算优先级间距离时的错误，并基于合适的距离度量专门设计了一种聚类算法。