# Adaptive Cluster-Based Normalization for Robust TOPSIS in Multicriteria Decision-Making
## 5 Results and Discussion

Table 9: Min–Max normalization results. Table 10. These weights are derived from the optimal values within each cluster and are essentially obtained by normalizing these values using the Min-Max method. The TOPSIS method was then applied using the weights derived for each cluster (as shown in Table 10) and the normalized data from both approaches. Table 11 presents the results obtained using logarithmic normalization, ranking the alternatives within their respective clusters. Table 12 shows the results using Min-Max normalization, allowing for a direct comparison between the two normalization techniques. The application of the TOPSIS method using both Min-Max and logarithmic normalization identified the top-ranked cities within each cluster. The results show that City K (Cluster 1) and City C (Cluster 2) consistently achieved the highest scores across both normalization methods, while City E (Cluster 3) exhibited some variation depending on the technique used. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Cluster** & \begin{tabular}{c} **(C1) Cluster** \\ **Centroid** \\ \end{tabular} & \begin{tabular}{c} **(C2) Cluster** \\ **Centroid** \\ \end{tabular} & \begin{tabular}{c} **(C3) Cluster** \\ **Centroid** \\ \end{tabular} & \begin{tabular}{c} **(C4) Cluster** \\ **Centroid** \\ \end{tabular} & \
\begin{tabular}{c} **Weight C1** \\ **Weight C2** \\ \end{tabular} &
\begin{tabular}{c} **Weight C3** \\ **Weight C4** \\ \end{tabular} \\ \hline
1 & 1.0 & 0.6 & 0.5 & 0.5 & 0.38 & 0.23 & 0.19 & 0.19 \\
2 & 0.1 & 0.9 & 1.0 & 0.9 & 0.03 & 0.32 & 0.34 & 0.31 \\
3 & 0.6 & 0.6 & 0.6 & 0.6 & 0.25 & 0.25 & 0.25 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Criterion weights for each cluster. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**City** & **Cluster** & **C1** & **C2** & **C3** & **C4** & **D+** & **D-** & **TOPSIS Score** \\ \hline City A & 1 & 0.36 & 0.21 & 0.19 & 0.18 & 0.03 & 0.08 & 0.74 \\ City B & 1 & 0.33 & 0.16 & 0.15 & 0.15 & 0.10 & 0.00 & 0.00 \\ City G & 1 & 0.38 & 0.21 & 0.16 & 0.18 & 0.03 & 0.08 & 0.69 \\ City K & 1 & 0.38 & 0.23 & 0.18 & 0.19 & 0.01 & 0.10 & **0.89** \\ \hline City C & 2 & 0.03 & 0.32 & 0.34 & 0.31 & 0.00 & 0.03 & **1.00** \\ City D & 2 & 0.03 & 0.32 & 0.34 & 0.31 & 0.00 & 0.03 & 0.98 \\ City I & 2 & 0.03 & 0.31 & 0.32 & 0.30 & 0.03 & 0.00 & 0.00 \\ City J & 2 & 0.03 & 0.32 & 0.32 & 0.31 & 0.02 & 0.02 & 0.57 \\ \hline City E & 3 & 0.24 & 0.25 & 0.25 & 0.25 & 0.01 & 0.08 & **0.89** \\ City F & 3 & 0.23 & 0.23 & 0.23 & 0.25 & 0.03 & 0.06 & 0.67 \\ City H & 3 & 0.25 & 0.19 & 0.19 & 0.24 & 0.08 & 0.02 & 0.18 \\ City L & 3 & 0.24 & 0.22 & 0.22 & 0.24 & 0.05 & 0.03 & 0.38 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: TOPSIS results using logarithmic normalization, bold numbers indicate the best alternative within each cluster identified by the method. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**City** & **Cluster** & **C1** & **C2** & **C3** & **C4** & **D+** & **D-** & **TOPSIS Score** \\ \hline City A & 1 & 0.23 & 0.15 & 0.19 & 0.13 & 0.18 & 0.36 & 0.66 \\ City B & 1 & 0.00 & 0.00 & 0.00 & 0.00 & 0.52 & 0.00 & 0.00 \\ City G & 1 & 0.38 & 0.15 & 0.06 & 0.13 & 0.16 & 0.43 & 0.73 \\ City K & 1 & 0.38 & 0.23 & 0.13 & 0.19 & 0.06 & 0.50 & **0.89** \\ \hline City C & 2 & 0.03 & 0.32 & 0.34 & 0.31 & 0.00 & 0.56 & **1.00** \\ City D & 2 & 0.02 & 0.32 & 0.34 & 0.31 & 0.02 & 0.56 & 0.97 \\ City I & 2 & 0.00 & 0.00 & 0.00 & 0.00 & 0.56 & 0.00 & 0.00 \\ City J & 2 & 0.02 & 0.32 & 0.00 & 0.31 & 0.34 & 0.45 & 0.57 \\ \hline City E & 3 & 0.10 & 0.25 & 0.25 & 0.25 & 0.15 & 0.44 & **0.75** \\ City F & 3 & 0.00 & 0.17 & 0.17 & 0.25 & 0.28 & 0.34 & 0.55 \\ City H & 3 & 0.25 & 0.00 & 0.00 & 0.00 & 0.43 & 0.25 & 0.37 \\ City L & 3 & 0.10 & 0.08 & 0.08 & 0.00 & 0.37 & 0.15 & 0.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: TOPSIS results using Min–Max normalization, bold numbers indicate the best alternative within each cluster identified by the method.