# Adaptive Cluster-Based Normalization for Robust TOPSIS in Multicriteria Decision-Making
###### Abstract  
In multicriteria decision-making (MCDM), methods such as TOPSIS are essential for evaluating and comparing alternatives across multiple criteria. However, traditional normalization techniques often struggle with datasets containing outliers, large variances, or heterogeneous measurement units, which can lead to skewed or biased rankings. To address these challenges, this paper proposes an adaptive, cluster-based normalization approach, demonstrated through a real-world logistics case study involving the selection of a host city for an international event. The method groups alternatives into clusters based on similarities in criterion values and applies logarithmic normalization within each cluster. This localized strategy reduces the influence of outliers and ensures that scaling adjustments reflect the specific characteristics of each group. In the case study--where cities were evaluated based on cost, infrastructure, safety, and accessibility--the cluster-based normalization method yielded more stable and balanced rankings, even in the presence of significant data variability. By reducing the influence of outliers through logarithmic normalization and allowing predefined cluster profiles to reflect expert judgment, the method improves fairness and adaptability. These features strengthen TOPSIS's ability to deliver accurate, balanced, and context-aware decisions in complex, real-world scenarios. type=multi-objective, clustering,  
However, despite its advantages, certain challenges remain--particularly related to clustering and normalization. Traditional clustering methods tend to assign alternatives to fixed categories, even when those alternatives share features with multiple groups. This rigid classification can lead to misinterpretation and the loss of valuable information, especially when the data involves uncertainty or overlapping characteristics. Likewise, widely used normalization techniques--such as Min-Max and Z-score--often struggle to manage large data variations, skewed distributions, and extreme values, which can distort rankings and introduce bias into decision-making. To overcome these limitations, this study proposes two innovative methods designed to make the TOPSIS framework more flexible, reliable, and user-friendly. The first is Clustering Using Fuzzy Numbers and Centroid-Based Distance Allocation, a novel clustering approach that incorporates fuzzy numbers to represent uncertainty in the evaluation of alternatives. Unlike traditional clustering methods such as K-Means--where cluster centroids are determined based on the data of existing elements--our approach defines the centroid of each cluster a priori, based on expert judgment and ideal conditions for each criterion. Alternatives are then evaluated using fuzzy numbers to account for uncertainty, and their distances to the predefined cluster centroids are computed using crisp values derived from these fuzzy assessments. This process avoids arbitrary assignments and provides a more structured, interpretable classification framework that reflects both expert intent and the inherent imprecision of real-world data. The second innovation is logarithmic normalization in TOPSIS, a transformation technique that smooths extreme variations, preserves proportional differences, and prevents any single criterion from dominating the final rankings. A key benefit of this technique is that, while it enhances stability and accuracy, it remains as straightforward to apply as traditional methods like Min-Max or Z-score normalization, making it a practical enhancement for decision-makers. By integrating these two methodological advancements, this study improves both the clustering and normalization components of the TOPSIS framework, addressing key limitations in traditional approaches while maintaining simplicity and efficiency. Unlike traditional data-driven clustering algorithms, the proposed method allows decision-makers to define ideal cluster profiles independently of the dataset.