Multi-Criteria Decision Analysis (MCDA) is extensively used across diverse industries to assess and rank alternatives. Among numerous MCDA methods developed to solve real-world ranking problems, TOPSIS remains one of the most popular choices in many application areas. TOPSIS calculates distances between the considered alternatives and two predefined ones, namely the ideal and the anti-ideal, and creates a ranking of the alternatives according to a chosen aggregation of these distances. However, the interpretation of the inner workings of TOPSIS is difficult, especially when the number of criteria is large. To this end, recent research has shown that TOPSIS aggregations can be expressed using the means (M) and standard deviations (SD) of alternatives, creating MSD-space, a tool for visualizing and explaining aggregations. Even though MSD-space is highly useful, it assumes equally important criteria, making it less applicable to real-world ranking problems. In this paper, we generalize the concept of MSD-space to weighted criteria by introducing the concept of WMSD-space defined by what is referred to as weight-scaled means and standard deviations. We demonstrate that TOPSIS and similar distance-based aggregation methods can be successfully illustrated in a plane and interpreted even when the criteria are weighted, regardless of their number. The proposed WMSD-space offers a practical method for explaining TOPSIS rankings in real-world decision problems.

翻译：摘要：多准则决策分析广泛应用于各行业，用于评估和排序备选方案。在众多为解决现实排序问题而开发的MCDA方法中，TOPSIS仍是许多应用领域中最受欢迎的选择之一。TOPSIS计算备选方案与两个预定义方案（即理想解与反理想解）之间的距离，并根据所选的距离聚合方式对备选方案进行排序。然而，TOPSIS内部运作机制的解释较为困难，尤其是在准则数量较多时。为此，近期研究表明TOPSIS聚合可用备选方案的均值（M）和标准差（SD）表示，从而构建了MSD空间——一种用于可视化和解释聚合的工具。尽管MSD空间极具实用性，但它假设各准则同等重要，这使其难以应用于现实排序问题。本文通过引入权重缩放均值与标准差的概念，将MSD空间推广至加权准则情形，提出了WMSD空间。我们证明，即使准则被赋予权重且数量任意，TOPSIS及类似基于距离的聚合方法仍可在二维平面中有效展示并解释。所提出的WMSD空间为解释现实决策问题中的TOPSIS排序提供了一种实用方法。