Utility Inspired Generalizations of TOPSIS

TL;DR

The paper addresses the limited ability of classic TOPSIS to regulate the influence of WM and WSD by introducing parametrized elliptic aggregations $(\mathsf{I}^\varepsilon, \mathsf{A}^\varepsilon, \mathsf{R}^\varepsilon)$ that elongate WMSD isolines in the WM–WSD plane, enabling explicit trade-offs between WM and WSD via the parameter $\varepsilon$. It also introduces two-dimensional lexicographic aggregations $(\mathsf{I}^L, \mathsf{A}^L, \mathsf{R}^L)$ and their parameterized variants to preserve WSD influence when WM fails to differentiate alternatives. A comprehensive case study on a bus-maintenance dataset demonstrates how rankings shift under different aggregations, showing that $\mathbf{b}_{24}$ remains dominant while others vary with $\varepsilon$ and lexicographic settings. The framework is visualized through WMSD-space, providing a practical tool for decision makers to tailor TOPSIS behavior toward or away from utility-based methods while preserving non-zero WSD impact. Overall, the work offers a versatile, interpretable toolkit for designing TOPSIS-like rankings with controlled preference structures.

Abstract

TOPSIS, a popular method for ranking alternatives is based on aggregated distances to ideal and anti-ideal points. As such, it was considered to be essentially different from widely popular and acknowledged `utility-based methods', which build rankings from weight-averaged utility values. Nonetheless, TOPSIS has recently been shown to be a natural generalization of these `utility-based methods' on the grounds that the distances it uses can be decomposed into so called weight-scaled means (WM) and weight-scaled standard deviations (WSD) of utilities. However, the influence that these two components exert on the final ranking cannot be in any way influenced in the standard TOPSIS. This is why, building on our previous results, in this paper we put forward modifications that make TOPSIS aggregations responsive to WM and WSD, achieving some amount of well interpretable control over how the rankings are influenced by WM and WSD. The modifications constitute a natural generalization of the standard TOPSIS method because, thanks to them, the generalized TOPSIS may turn into the original TOPSIS or, otherwise, following the decision maker's preferences, may trade off WM for WSD or WSD for WM. In the latter case, TOPSIS gradually reduces to a regular `utility-based method'. All in all, we believe that the proposed generalizations constitute an interesting practical tool for influencing the ranking by controlled application of a new form of decision maker's preferences.

Figures (19)

• Figure 1: A schema explaining different representations of data analyzed in this paper. (A) The original dataset (decision matrix) describing $m = 4$ students (alternatives) using final grades from $n = 3$ subjects (criteria). The same dataset is then depicted as a subset of (B) the criterion space, the set of all possible alternatives described by the three criteria describing students; (C) the utility space, the re-scaled equivalent of criterion space; (D) the weighted utility space, the preference-modified version of the utility space, with weights $\mathbf{w} = [0.5, 0.6, 1.0]$; (E) the WMSD-space, the space defined by the values of WM and WSD, with its shape determined by $\mathbf{w}$.
• Figure 2: Four exemplary alternatives (students) depicted in WMSD-space defined for $\mathbf{w}=[0.5, 0.6, 1.0]$ for aggregation $\mathsf{R}$. Color encodes the aggregation value, with blue representing the least preferred and red the most preferred values.
• Figure 3: An exemplary point $\mathbf{v} = [0.75, 0.25]$ depicted in WMSD-space defined by $\mathbf{w} = [1.0, 0.5]$ for aggregations (A) $\mathsf{I}$, (B) $\mathsf{A}$ and (C) $\mathsf{R}$. Colour encodes the aggregation value, with blue representing the least preferred and red the most preferred values. The isolines of aggregation $\mathsf{I}$ and $\mathsf{A}$ are concentric circles with center in $[mean(\mathbf{w}), 0]$ and $[0, 0]$, respectively. The isolines of $\mathsf{R}$ form two arch-like curves 'centred' in $[mean(\mathbf{w}), 0]$ and $[0, 0]$.
• Figure 4: WMSD-space defined by $\mathbf{w} = [1.0, 0.6, 0.5]$ depicted against circular aggregations: (A) $\mathsf{I}$, (B) $\mathsf{A}$ (C) $\mathsf{R}$ (equivalent to the corresponding elliptic aggregations for $\epsilon = 1$).
• Figure 5: WMSD-space defined by $\mathbf{w} = [1.0, 0.6, 0.5]$ depicted against elliptic aggregations for $1 < \epsilon = 1.86 < +\infty$: (A) $\mathsf{I}^\epsilon$, (B) $\mathsf{A}^\epsilon$ (C) $\mathsf{R}^\epsilon$.