# Adaptive Cluster-Based Normalization for Robust TOPSIS in Multicriteria Decision-Making
## 5 Results and Discussion
Cluster 3--Balanced Cities with a Mix of Features--includes alternatives with intermediate cost levels and criteria ratings that fall between those of Clusters 1 and 2. This consistency reinforces the validity of the proposed clustering method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Cluster** & \begin{tabular}{c} **(C1)** \\ **Centroid** \\ \end{tabular} & \begin{tabular}{c} **(C2)** \\ **Centroid** \\ \end{tabular} & \
\begin{tabular}{c} **(C3)** \\ **Centroid** \\ \end{tabular} &
\begin{tabular}{c} **(C4)** \\ **Centroid** \\ \end{tabular} \\ \hline City A & 1 & 14 & 5 & 7 & 6 \\ City B & 1 & 11 & 3 & 4 & 4 \\ City G & 1 & 16 & 5 & 5 & 6 \\ City K & 1 & 16 & 6 & 6 & 7 \\ \hline City C & 2 & 39 & 9 & 9 & 9 \\ City D & 2 & 37 & 9 & 9 & 9 \\ City I & 2 & 35 & 8 & 8 & 8 \\ City J & 2 & 37 & 9 & 8 & 9 \\ \hline City E & 3 & 20 & 7 & 7 & 8 \\ City F & 3 & 18 & 6 & 6 & 8 \\ City H & 3 & 23 & 4 & 4 & 7 \\ City L & 3 & 20 & 5 & 5 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Result of the distribution of alternatives using the proposed clustering method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **(C1) Distance** & **(C2) Distance** & **(C3) Distance** & **min** \\ \hline City A & **0.31** & 1.18 & 0.46 & 0.31 \\ City B & **0.93** & 1.84 & 1.11 & 0.93 \\ City C & 1.26 & **0.17** & 0.92 & 0.17 \\ City D & 1.20 & **0.15** & 0.87 & 0.15 \\ City E & 0.43 & 0.74 & **0.23** & 0.23 \\ City F & 0.39 & 0.98 & **0.34** & 0.34 \\ City G & **0.43** & 1.31 & 0.57 & 0.43 \\ City H & 0.76 & 1.36 & **0.74** & 0.74 \\ City I & 0.94 & **0.22** & 0.57 & 0.22 \\ City J & 1.13 & **0.22** & 0.79 & 0.22 \\ City K & **0.21** & 1.07 & 0.33 & 0.21 \\ City L & 0.50 & 1.16 & **0.49** & 0.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Evaluation of distances to the centroids of each cluster. Based on these results, we can conclude that the proposed model produces outcomes consistent with expectations. This means that analyzing the distribution of alternatives across the different clusters confirms that the results are logical and align with the expected distribution of alternatives within each cluster. Table 7 compares the proposed clustering method with the Fuzzy K-Means approach, revealing that the results are nearly identical--with one notable exception: City K is assigned to Cluster 1 by the proposed method, whereas Fuzzy K-Means places it in Cluster 3. Although this difference may appear minor, it highlights an important distinction in how each method interprets distances and assigns alternatives to clusters. Overall, the strong alignment between the two methods supports the effectiveness and reliability of the proposed approach as a viable alternative to traditional fuzzy clustering techniques. More importantly, assigning City K to Cluster 1 appears to be a more appropriate classification. The city shares a low-cost profile, which is a defining characteristic of Cluster 1. In fact, City K has the same cost value as City G, which was placed in Cluster 1 by the Fuzzy K-Means method. The only differences between the two are minor, such as a one-point variation in other criteria--making them highly comparable. Therefore, grouping City K with City G in Cluster 1 is more consistent with the underlying logic of the clustering process. This supports the conclusion that the proposed method offers a more accurate and contextually sound classification. Another key advantage of the proposed method lies in its simplicity and computational efficiency when compared to Fuzzy K-Means, which depends on multiple iterative calculations and a more complex optimization process. In contrast, the proposed method uses a direct and intuitive approach by assigning each alternative to the nearest centroid, eliminating the need for repeated recalculations. Fuzzy K-Means, on the other hand, involves continuous re-evaluation of centroids, which increases computational demands--particularly for larger datasets.