Towards Explainable TOPSIS: Visual Insights into the Effects of Weights and Aggregations on Rankings

TL;DR

This work addresses the interpretability gap in TOPSIS when criteria carry weights. It introduces Weight-Scaled MSD-space (WMSD-space), built on a Weighted Utility Space (VS) and weight-scaled means/standard deviations, to visualize aggregations and rankings in a 2D plane independent of the number of criteria. The IA-WMSD property formalizes how distances to the ideal and anti-ideal relate to the weighted mean and dispersion, enabling direct visualization of aggregation functions I_w, A_w, and R_w. Case studies on student grades and the Index of Economic Freedom demonstrate weight-driven shifts in rankings and show how WMSD-space supports cross-expert comparison and sensitivity analysis without sacrificing interpretability. Overall, WMSD-space provides a practical, explainable visualization framework for weighted TOPSIS in real-world decision problems and paves the way for interactive tools and extensions to uncertainty and other distance-based MCDA methods.

Abstract

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.

Figures (11)

• Figure 1: The dataset that will serve as the running example for explaining different representations of objects analyzed in this paper. (A) The original dataset (decision matrix) describing $m = 4$ students (alternatives) using final grades from $n = 3$ subjects (criteria). (B) The same dataset depicted as a subset of the criteria space, i.e., of all possible alternatives described by the three criteria describing students. (C) The same alternatives presented as a subset of utility space, the re-scaled equivalent of criteria space. (D) The analyzed students represented in MSD-space, a space defined by the mean (M) and standard deviation (SD) of the utility space descriptions of the alternatives. (E) Alternatives represented in weighted utility space, with weights $\mathbf{w} = [0.5, 0.6, 1.0]$. (F) Alternatives represented in WMSD-space, a space defined by the weight-scaled mean (WM) and weight-scaled standard deviation (WSD) of the weighted utility space descriptions of the alternatives.
• Figure 2: A depiction of the IA-MSD property in $\mathit{US}$ and MSD-space for a three-dimensional problem. (A) Vector orthogonality depicted in $\mathit{US}$. (B) Illustration of the IA-MSD property in MSD-space. The re-scaled $\delta^{01}_2$ lengths of vectors $\mathbf{\overline{u}}$ and $\mathbf{u} - \mathbf{\overline{u}}$ from panel A correspond to the values of $mean(\mathbf{u})$ and $std(\mathbf{u})$ depicted in MSD-space. (C) Color encoding of the aggregation function $\mathsf{R}(\mathbf{u})$, with blue representing the least preferred and red the most preferred values.
• Figure 3: Vector orthogonality presented in (A) $\mathit{US}$ and (B) $\mathit{VS}$, for $n = n_p = 2$. The weight vector used to transform the presented $\mathit{US}$ into $\mathit{VS}$ is $\mathbf{w} = [1.0, 0.5]$. The diagonal $D_{\mathbf{0}}^{\mathbf{1}}$ is the blue line segment between vertices $\mathbf{0}$ and $\mathbf{1}$ in $\mathit{US}$. Analogously, $D_{\mathbf{0}}^{\mathbf{w}}$ is the blue line segment between vertices $\mathbf{0}$ and $\mathbf{w}$ in $\mathit{VS}$.
• Figure 4: An illustration of the IA-WMSD property in (A) $\mathit{VS}$ and (B) WMSD-space, for $\mathbf{w} = [1.0, 0.5]$ and a point $\mathbf{v} = [0.75, 0.25]$. The illustration shows how the re-scaled lengths $\delta^{01}_{\mathbf{w}}$ of vectors $\mathbf{\overline{v}}$ and $\mathbf{v} - \mathbf{\overline{v}}$ are equal to the weight-scaled mean (WM) and standard deviation (WSD), which define the WMSD-space. The diagonal $D_{\mathbf{0}}^{\mathbf{w}}$ is the blue line segment between vertices $\mathbf{0}$ and $\mathbf{w}$.
• Figure 5: Visualizations of WMSD-space for the number of criteria (A) $n=3$, (B) $n=4$, (C) $n=5$, each for three different sets of weights depicted by different line types. Notice that the dotted light gray line on each subplot corresponds to uniform weights and, therefore, the special case of WMSD-space, which is MSD-space. It is also worth noting how the arithmetic mean of the weight ($mean(\mathbf{w})$) corresponds to the maximal x-axis coordinate of WMSD-space.

Theorems & Definitions (4)

• Theorem 1: IA-MSD Property
• Definition 1: MSD-space
• Theorem 2: IA-WMSD Property
• Definition 2: WMSD-space