# Adaptive Cluster-Based Normalization for Robust TOPSIS in Multicriteria Decision-Making
## 3 Materials and Methods
### Clustering Using Fuzzy Numbers and Centroid-Based Distance Allocation  
To group alternatives into meaningful clusters, we implement a fuzzy clustering approach. This method involves the following steps:  
Step 1: Representation of alternatives with fuzzy numbers--each alternative's criteria's values are converted into fuzzy numbers (\(a\), \(b\), \(c\)) representing the lower bound, central value, and upper bound, respectively. This allows for a more flexible representation of uncertainty in the decision-making process. Step 2: In this step, the cluster centroids are determined qualitatively by the user, identifying the optimal ranking for each criterion within the cluster. This process employs fuzzy numbers, represented as values \((a,b,c)\) ranging between 0 and 1. These results will allow us to measure the distance between the cluster centroids, which represent the ideal position for each cluster based on the selected criteria--and the scores of each alternative. Beyond this primary purpose, the resulting centroids are also used to determine the criterion weights, which will later be applied in the TOPSIS method. Formula (1) calculates the centroid representing the optimal position within a given cluster. \[C_{cwj}=\left(\frac{a_{w}+b_{w}+c_{w}}{3}\right)_{j}, \tag{1}\]  
where \(a_{w},b_{w},c_{w}\) are the fuzzy number components representing the optimal position within a given cluster \(j\). Step 3: Determination of alternative centroids--using a fuzzy clustering approach, the centroids of each alternative are determined. The centroid \(C_{Aij}\) for each alternative \(i\) is computed as follows:  
\[C_{Aij}=\left(\frac{a_{i}+b_{i}+c_{i}}{3}\right)_{j}, \tag{2}\]  
where \(a_{i},b_{i},c_{i}\) are the fuzzy number components of alternative \(i\) in the criterion \(j\). This formula is applied to all alternatives across all considered criteria to establish the ranking of the alternatives. Equations (1) and (2) define two distinct types of centroids within the proposed method: cluster centroids and alternative centroids. The cluster centroid represents the ideal position of a given cluster across all criteria and is computed based on predefined fuzzy values that characterize the cluster's optimal conditions. This centroid remains fixed for all alternatives within the cluster, serving as a reference point for comparison. In contrast, the alternative centroid is calculated based on the fuzzy scores assigned to a specific alternative, meaning it varies from one alternative to another. While the cluster centroid reflects the overall profile of a group, the alternative centroid captures the individual positioning of an alternative within the decision space. This distinction is crucial for the clustering process, as it enables a more flexible classification of alternatives while maintaining a structured evaluation framework. Step 4: The new decision matrix, obtained from Step 3, is normalized using the cost (lower is better) and benefit (higher is better) formulas (Formulas (2) and (3)). \[N_{i}=\frac{\max(X)-X_{i}}{\max(X)-\min(X)}, \tag{3}\]  
\[N_{i}=\frac{X_{i}-\max(X)}{\max(X)-\min(X)}, \tag{4}\]  
where \(N_{i}\) is the normalized value of the alternative \(i\), \(X_{i}\) is the original value of the alternative \(i\), \(\max(X)\) is the maximum value in the criterion, and \(\min(X)\) is the minimum value in the criterion. These formulas scale the values between 0 and 1, ensuring a fair comparison between alternatives while maintaining the meaning of cost and benefit criteria.