# Adaptive Cluster-Based Normalization for Robust TOPSIS in Multicriteria Decision-Making
## 5 Results and Discussion  
In this section, we apply the proposed models, including the new clustering approach and the logarithmic normalization method--within the TOPSIS framework for the presented case study. The results are detailed step by step, then analyzed and compared with those obtained using traditional methods. Table 4 presents the processing of the data from Table 3. Using the fuzzy number of each alternative for each criterion, the corresponding centroid is calculated (columns 2 to 5), using Equation (2). These centroids are then normalized using the Min-Max method (columns 6 to 9), using Equations (3) and (4). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **(C1)** & **(C2)** & **(C3)** & **(C4)** & **(C1)** & **(C1)** & **(C1)** & **(C1)** \\  & **Centroid** & **Centroid** & **Centroid** & **Centroid** & **Centroid** & **Centroid** & **Centroid** & **Centroid** \\ \hline City A & 14 & 5 & 7 & 6 & 0.92 & 0.33 & 0.6 & 0.4 \\ City B & 11 & 3 & 4 & 4 & 1.00 & 0.00 & 0 & 0 \\ City C & 39 & 9 & 9 & 9 & 0.00 & 1.00 & 1 & 1 \\ City D & 37 & 9 & 9 & 9 & 0.08 & 1.00 & 1 & 1 \\ City E & 20 & 7 & 7 & 8 & 0.68 & 0.67 & 0.6 & 0.8 \\ City F & 18 & 6 & 6 & 8 & 0.76 & 0.50 & 0.4 & 0.8 \\ City G & 16 & 5 & 5 & 6 & 0.84 & 0.33 & 0.2 & 0.4 \\ City H & 23 & 4 & 4 & 7 & 0.60 & 0.17 & 0 & 0.6 \\ City I & 35 & 8 & 8 & 8 & 0.16 & 0.83 & 0.8 & 0.8 \\ City J & 37 & 9 & 8 & 9 & 0.08 & 1.00 & 0.8 & 1 \\ City K & 16 & 6 & 6 & 7 & 0.84 & 0.50 & 0.4 & 0.6 \\ City L & 20 & 5 & 5 & 7 & 0.68 & 0.33 & 0.2 & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Normalized centroids for the four considered criteria across selected cities. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Cost (C1) (SK)**} & \multicolumn{3}{c}{**Infrastructure (C2)**} & \multicolumn{3}{c}{**Safety (C3)**} & \multicolumn{3}{c}{**Accessibility (C4)**} \\ \hline  & a & b & c & a & b & c & a & b & c & a & b & c \\ City A & 11 & 14 & 16 & 4 & 5 & 6 & 6 & 7 & 8 & 5 & 6 & 7 \\ City B & 9 & 11 & 14 & 2 & 3 & 4 & 3 & 4 & 5 & 3 & 4 & 5 \\ City C & 34 & 39 & 45 & 8 & 9 & 10 & 8 & 9 & 10 & 8 & 9 & 10 \\ City D & 32 & 37 & 43 & 8 & 9 & 10 & 8 & 9 & 10 & 8 & 9 & 10 \\ City E & 18 & 20 & 23 & 6 & 7 & 8 & 6 & 7 & 8 & 7 & 8 & 9 \\ City F & 16 & 18 & 20 & 5 & 6 & 7 & 5 & 6 & 7 & 7 & 8 & 9 \\ City G & 14 & 16 & 18 & 4 & 5 & 6 & 4 & 5 & 6 & 5 & 6 & 7 \\ City H & 20 & 23 & 25 & 3 & 4 & 5 & 3 & 4 & 5 & 6 & 7 & 8 \\ City I & 29 & 35 & 41 & 7 & 8 & 9 & 7 & 8 & 9 & 7 & 8 & 9 \\ City J & 32 & 37 & 43 & 8 & 9 & 10 & 7 & 8 & 9 & 8 & 9 & 10 \\ City K & 14 & 16 & 18 & 5 & 6 & 7 & 5 & 6 & 7 & 6 & 7 & 8 \\ City L & 18 & 20 & 23 & 4 & 5 & 6 & 4 & 5 & 6 & 6 & 7 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Clustered data of cities for event hosting analysis. The centroids calculated for each alternative are then used to compute their distances to the optimal scores defined for each cluster, as outlined in Table 2.  
Table 5 presents these distances, calculated using the Euclidean norm, as specified in Equation (5). As the table shows, the distance between each alternative and the ideal cluster values varies. To assign each alternative to a cluster, we select the one with the minimum distance, as described in Equation (6). The final column in Table 5 displays the shortest distance for each alternative, with the assigned cluster highlighted in bold. Table 6 presents the results aggregated by cluster, revealing a distribution that aligns well with the intended definitions of each group. Cluster 1--Cost-Effective Cities with Moderate Infrastructure--includes alternatives with the lowest costs, while the other criteria generally exhibit moderate values, confirming the coherence of the classification. Cluster 2--High-Investment Cities with World-Class Infrastructure--comprises alternatives that match the profile of high-cost cities offering top-tier scores in infrastructure, safety, and accessibility.