# Adaptive Cluster-Based Normalization for Robust TOPSIS in Multicriteria Decision-Making
## 3 Materials and Methods
### Logarithmic Normalization: An Adaptive Approach for TOPSIS  
Normalization is a fundamental step in multi-criteria decision-making (MCDM) methods such as TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution). It ensures that criteria with different units and scales can be meaningfully compared. Traditional normalization methods, such as Min-Max and Z-score, effectively rescale data but may not be suitable for datasets with high variance, extreme outliers, or non-linear distributions. In such cases, logarithmic normalization emerges as an alternative technique that dynamically adjusts to data distributions, making it particularly effective for handling data spanning multiple orders of magnitude. By compressing large numerical variationswhile amplifying smaller differences, this approach ensures a more balanced contribution of criteria to the final decision, preserving the relative ranking among alternatives. Logarithmic normalization is mathematically expressed as shown in Equation (7),  
\[X=\frac{\log(X)-\log(X_{\text{min}})}{\log(X_{\text{max}})-\log(X_{\text{min}})}, \tag{7}\]  
where \(X\) represents the normalized value, \(X\) is the original value, and \(X_{\text{min}}\) and \(X_{\text{max}}\) denote the minimum and maximum values within a given criterion. This transformation is particularly beneficial in decision-making scenarios where some criteria exhibit exponential growth patterns, such as financial metrics, environmental indicators, and energy consumption data. By using a logarithmic scale, the influence of extreme values is harmonized, ensuring that all criteria contribute meaningfully to the decision-making process in TOPSIS, for instance. A key strength of logarithmic normalization is its adaptive nature. Unlike fixed range methods, it automatically adjusts to varying data magnitudes, dynamically scaling values to ensure fair comparisons across criteria. This makes it particularly effective for datasets with highly skewed distributions or large numerical differences. Moreover, it enhances decision stability, reducing the dominance of criteria with disproportionately large values while ensuring that smaller values remain distinguishable. Another advantage of logarithmic normalization is its ability to enhance differentiation among alternatives. By redistributing values in a way that emphasizes proportional differences, it ensures that the ranking process in TOPSIS remains representative and reliable, even when dealing with highly dispersed datasets. This is particularly beneficial in cases where criteria exhibit non-linear relationships, allowing for a more accurate reflection of each alternative's performance. The implications of logarithmic normalization in TOPSIS are significant. By integrating this approach, rankings become more stable and reflective of real-world conditions, ensuring that decision-making processes remain robust and interpretable. Given its ability to balance differences across criteria without distorting rankings, logarithmic normalization offers an advanced scaling technique that aligns well with diverse decision-making scenarios. Despite its many advantages, logarithmic normalization has yet to be widely integrated into the TOPSIS framework, presenting an exciting opportunity for innovation. By introducing this approach, we can create a more adaptive way of handling criteria with high variance, non-linear distributions, and sensitivity to outliers. This integration helps improve ranking stability, ensures fairer comparisons between alternatives, and strengthens the overall decision-making process. What makes logarithmic normalization even more appealing is its ease of implementation. While it effectively balances data distribution and minimizes the impact of extreme values, it remains just as simple to apply as Min-Max or Z-score normalization. This means that decision-makers can benefit from its advantages without facing additional computational complexity or implementation challenges.