# Adaptive Cluster-Based Normalization for Robust TOPSIS in Multicriteria Decision-Making
## 2 Literature Review  
Multi-criteria decision-making (MCDM) encompasses a set of methodologies used to evaluate and prioritize multiple--often conflicting--factors in the decision-making process. These approaches are critical in fields such as environmental management, engineering, and economics, where complex decisions are frequently encountered [1]. One of the most widely applied MCDM techniques is the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). Its core principle is straightforward: the optimal alternative is the one closest to the ideal solution and farthest from the worst-case scenario. TOPSIS is particularly valued for its simplicity and its ability to effectively handle both qualitative and quantitative data [2]. Like any method, however, TOPSIS has its limitations. It can struggle with datasets characterized by uncertainty, outliers, or high variance, which may affect the consistency and reliability of its rankings. In response, researchers have explored methods to link input uncertainty with output uncertainty within the TOPSIS framework, highlighting the challenges of interpreting uncertain data in real-world decision-making contexts [3]. Refining these techniques can increase decision-makers' confidence in the results, thereby enhancing the overall value and applicability of MCDM methods across various industries [4]. Comparative studies have examined TOPSIS alongside other MCDM methods such as VIKOR, PROMETHEE, and AHP. For instance, one study evaluated four different techniques--AHP, TOPSIS, ELECTRE III, and PROMETHEE II--in the context of group decision-making for sewer network projects, offering valuable insights into their applicability and effectiveness [5; 6; 7]. Traditional clustering methods--such as K-Means and Hierarchical Clustering--have long served as fundamental tools for grouping similar data points in decision-making models. Their efficiency and ease of implementation contribute to their widespread use. However, these methods have notable limitations, especially when dealing with uncertainty, complex data distributions, or overlapping classifications. Because they rely on crisp boundaries, each data point is strictly assigned to a single cluster, which can result in inaccurate or overly simplistic groupings in real-world scenarios where data are often ambiguous and multidimensional [8]. To overcome these limitations, fuzzy clustering techniques--particularly Fuzzy C-Means (FCM)--offer a more flexible alternative. Unlike traditional clustering methods, FCM allows data points to belong to multiple clusters with varying degrees of membership, enabling more nuanced and adaptable classifications. This approach is especially valuable in domains such as medical diagnosis, image segmentation, and customer profiling, where real-world data rarely conforms to clearly defined categories [9]. Although FCM enhances clustering accuracy and adaptability, it also introduces considerable computational complexity. In contrast to K-Means, which follows a relatively simple iterative process, FCM requires more intensive calculations due to the continuous updating of membership probabilities and the optimization of an objective function. This iterative minimization process can become computationally expensive, particularly when working with large, high-dimensional datasets. As a result, FCM increases processing time and demands greater computational resources [10]. Another significant barrier to the adoption of fuzzy clustering methods is the need for programming and algorithmic expertise. Implementing these methods--particularly in large-scale applications--requires familiarity with programming languages such as Python v3, R v4, or MATLAB R2024a, as well as with specialized libraries like scikit-fuzzy or the Fuzzy Logic Toolbox.