Entropy Weight-TOPSIS Model for MCDA

• Entropy Weight-TOPSIS Model is a multi-criteria decision analysis method that integrates entropy-based criterion weighting with TOPSIS to rank alternatives by their closeness to an ideal solution.
• The method reduces subjective bias by using data dispersion metrics to assign weights, demonstrating reliable application in research evaluation, supply chain performance, education ranking, and network analysis.
• Recent enhancements include hybrid modifications, non-extensive entropy functions, and advanced aggregation techniques, which improve robustness in complex and uncertain decision-making environments.

The Entropy Weight-TOPSIS Model is a multi-criteria decision analysis (MCDA) technique that integrates objective criterion weighting via entropy with the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method. This hybrid approach enables decision makers to rank alternatives by their relative proximity to an ideal solution, reducing subjective bias in the assignment of criterion weights through the incorporation of entropy-based dispersion metrics. Its recent applications range from research evaluation and supply chain performance to education system ranking and network analysis.

1. Theoretical Foundations

The Entropy Weight-TOPSIS Model synthesizes information-theoretic concepts and spatial aggregation. In the entropy weighting phase, the degree of information contained in evaluation criteria is quantified, with higher dispersion leading to higher objective weights. Let the normalized performance of alternative $i$ on criterion $j$ be $Z_{ij}$, and proportion $P_{ij}=\frac{Z_{ij}}{\sum_{i=1}^n Z_{ij}}$. Shannon entropy for criterion $j$ is calculated as:

$$E_j = -\frac{1}{\ln n}\sum_{i=1}^n P_{ij}\ln P_{ij}$$

The entropy reduction coefficient, $G_j = 1-E_j$, is then normalized to produce the objective criterion weights:

$W_j = \frac{G_j}{\sum_{j=1}^m G_j}$

In the TOPSIS stage, the weighted normalized matrix is established, positive and negative ideal solutions are identified, and Euclidean distances $S_i^+$ and $S_i^-$ from these ideals are computed. The final relative closeness index $C_i$ is:

$C_i = \frac{S_i^-}{S_i^+ + S_i^-}$

Here, $C_i$ quantifies the degree to which alternative $i$ approaches the ideal solution.

2. Entropy Weighting: Objective Criterion Importance

Entropy weighting is data-driven, reducing reliance on subjective judgment. Indicators with greater variation contain more information and thus are assigned higher weights. This method has been employed in scientific research evaluation to robustly combine multiple performance indicators (Shi et al., 26 Mar 2025), in supply chain management to improve capacity discrimination among suppliers (Liao et al., 2023), and in higher education assessment to differentiate provincial performance (Yang et al., 11 Aug 2025).

Advanced versions use non-extensive entropy, such as Tsallis entropy with a parameter $q$ to further generalize weighting under incomplete or noisy data:

$$\tilde{e}_j = \frac{\sum_{i=1}^m p_{ij}^q - 1}{1-q}$$

Solving $\tilde{e}_j = W_r(j)$ (where $W_r$ is a grey relational correction weight) yields individual $q$ values for refined weight calibration (Liao et al., 2023).

3. TOPSIS Aggregation and Ranking

TOPSIS ranks alternatives by their Euclidean distances to ideal and anti-ideal points in the weighted normalized criterion space. The use of entropy-derived weights ensures that each criterion's influence reflects its objective information content. This reduces bias and enhances result credibility compared to methods employing subjective weights (Shi et al., 26 Mar 2025).

Recent methodological advances generalize TOPSIS by decomposing ranking into weight-scaled mean (WM) and standard deviation (WSD) of utilities. Proposed parameterizations allow decision makers to explicitly modulate sensitivity to central tendency versus dispersion (Susmaga et al., 10 Apr 2025, Susmaga et al., 2023). Visualization in WMSD-space enables transparent assessment of how weighted mean and variability influence rankings. For entropy-weighted TOPSIS, weights derived by entropy methods are directly incorporated into these weighted spatial aggregations (Susmaga et al., 2023).

4. Model Extensions and Hybridizations

The entropy weight-TOPSIS framework is versatile and has been extended in several directions. Hybrid models integrate additional weighting schemes (e.g., CRITIC), employ random weight intervals for sensitivity analysis, and aggregate multiple rankings using statistical measures such as the mode (Basilio et al., 5 Apr 2025).

Other approaches integrate fuzzy optimization for both expert and attribute weighting, further blending subjective and objective information in group decision-making (Hu et al., 2023). The integration with algorithmic modifications (e.g., grey relational analysis, non-extensive entropy correction) increases stability and applicability, especially under small sample sizes or data uncertainty (Liao et al., 2023). ICA-TOPSIS variants utilize independent component analysis to “unmix” inter-dependent criteria prior to TOPSIS aggregation, producing robust rankings even when criteria are not statistically independent (Pelegrina et al., 2020).

5. Empirical Applications and Performance

The entropy weight-TOPSIS model is widely applied for evaluating complex systems:

• In research evaluation, entropy-weighted criteria capture citation diversity, interdisciplinarity, and performance, supporting the ranking of journals and research teams (Shi et al., 26 Mar 2025).
• For higher education measurement, entropy weight-TOPSIS models have objectively ranked provinces and analyzed dynamic causal relationships using panel VAR models, informing resource allocation and policy (Yang et al., 11 Aug 2025).
• In supply capacity evaluation, non-extensive entropy-weighted TOPSIS introduces increased robustness, accurate discrimination, and stability across dynamic and incomplete datasets (Liao et al., 2023).
• Hybrid and iterative random weight EC-TOPSIS models help reduce uncertainty in rankings driven by social media metrics, producing robust, balanced classifications (Basilio et al., 5 Apr 2025).
• In network science, entropy is used to assess structural importance and predict links by combining path entropy and weights—a conceptual parallel to criterion informativeness and strength in TOPSIS (Xu et al., 2016).

Comparative analyses consistently demonstrate that entropy weight-TOPSIS achieves greater reliability, robustness against data noise, and reduced bias compared to traditional approaches based strictly on subjective weights or unweighted aggregation.

6. Geometric, Information-Theoretic, and Dynamical Interpretations

The geometric view of entropy-driven weighting maps decision criteria to statistical hypersurfaces, with entropy quantifying spread and uncertainty (Angelelli et al., 2019). Changes in evaluation functions induce entropy variations described by differential relations:

$$\delta S = \mathbb{E}(\delta f) - \delta (\mathbb{E}(f))$$

Weight updates follow replicator dynamics akin to evolutionary game theory, leading to self-consistent entropy optimization. Integral characteristics link local weighting to global performance indices, supporting aggregated decision analysis.

Generalizations of TOPSIS modulate sensitivity to mean and dispersion, and entropy weighting provides an objective, data-driven baseline for these control parameters (Susmaga et al., 10 Apr 2025). Visualization in WMSD-space clarifies the impact of entropy-derived weights on rankings and aggregations (Susmaga et al., 2023).

7. Limitations, Controversies, and Ongoing Directions

While entropy-based weighting reduces subjective bias and enhances credibility, its effectiveness depends on sufficient data variability and appropriate normalization. In situations with weak data dispersion or homogeneous samples, entropy may not discriminate well among criteria. The choice of entropy function (Shannon vs. Tsallis), incorporation of correction mechanisms (e.g., grey relational analysis), and hybridization with other statistical or algorithmic weighting approaches are areas of active research aimed at improving accuracy and flexibility in dynamic, uncertain multi-criteria environments.

A plausible implication is that ongoing development of entropy weight-TOPSIS and associated hybrid models will continue to shape best practices in MCDA, particularly for applications that require transparent, robust, and adaptable criterion weighting under evolving data conditions.

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Table: Summary of Core Computations in the Entropy Weight-TOPSIS Model

Step	Formula	Purpose
Data normalization	$Z_{ij}$ via min-max or z-score	Standardize scales of indicators
Entropy calculation	$$E_j = -\frac{1}{\ln n} \sum_{i=1}^n P_{ij} \ln P_{ij}$$	Quantify criterion dispersion
Weight assignment	$W_j = \frac{1 - E_j}{\sum_{j=1}^m(1 - E_j)}$	Objective criterion weighting
Weighted aggregation (TOPSIS)	$C_i = \frac{S_i^-}{S_i^+ + S_i^-}$	Rank alternatives by proximity to ideal

The Entropy Weight-TOPSIS Model integrates objective entropy-based weights with distance aggregation, yielding robust, bias-minimized rankings in complex decision environments.