## 3 Research Findings
### Factors and Levels for the MOPSO Algorithm  
Level of factors are defined as presented in Table 4. Moreover, the orthogonal arrays corresponding to Table 4 are defined as Table 5.  
Figure 3: Average S/N ratio plot for NSGA II algorithm [22]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{3}{c}{**Levels of Factors**} & \multicolumn{1}{c}{**Optimal Level**} \\ \hline
**Parameters** & **1** & **2** & **3** & **of Factors** \\ \hline nPop & 50 & 70 & 100 & 70 \\ pc & 0.2 & 0.5 & 0.8 & 0.2 \\ Pm (Mutation Probability) & 0.2 & 0.3 & 0.4 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal levels of factors used for the NSGA-II algorithm [22]. Table 5 provides the orthogonal array L9(35) used in the MOPSO algorithm for experimental design. The orthogonal array is a powerful tool for conducting experiments efficiently, allowing the investigation of multiple factors simultaneously with a reduced number of experiments. This table lists all the possible factor combinations used to evaluate the performance of the MOPSO algorithm. By applying the orthogonal array, the study ensures that the experimental setup covers a broad range of factor levels, which helps in understanding how each parameter affects the algorithm's ability to solve the hub location problem under various conditions. After conducting the Taguchi experiment, the results of the average means and the average S/N ratio for each level of the factors in the MOPSO algorithm for the presented model are shown in Figures 4 and 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Levels of Factors**} \\ \hline
**Parameters** & **1** & **2** & **3** \\ \hline nPop & 50 & 75 & 100 \\ nRep & 70 & 100 & 150 \\ W & 0.5 & 0.6 & 0.7 \\ C1 & 1 & 1.25 & 1.5 \\ C2 & 1 & 1.25 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Levels of factors used for the MOPSO algorithm. \begin{table}
\begin{tabular}{c c c c c c} \hline
**C2** & **C1** & **W** & **nRep** & **nPop** & **Experiment Number** \\ \hline
2 & 1 & 1 & 1 & 1 & 2 \\
3 & 1 & 1 & 1 & 1 & 3 \\
1 & 2 & 2 & 2 & 1 & 4 \\
2 & 2 & 2 & 2 & 1 & 5 \\
3 & 2  
Figure 4 displays the \"Average Means\" plot for the MOPSO algorithm, which summarizes the mean performance across different experimental setups. This plot provides an overview of how the algorithm's performance changes with varying factor levels. By visualizing the average means, it helps identify the optimal parameter combinations for achieving the best performance. The plot is particularly useful for understanding the relative influence of different factors on the MOPSO algorithm's performance, allowing researchers to fine-tune the algorithm for better efficiency. Figure 5 illustrates the \"Average Signal-to-Noise (S/N) Ratio\" plot for the MOPSO algorithm. The S/N ratio is a key measure of robustness, showing how consistently the algorithm performs under varying conditions. A higher S/N ratio indicates better stability and reliability in the algorithm's output. This plot helps in identifying the most stable and effective parameter settings, ensuring that the MOPSO algorithm performs optimally under different scenarios and contributes to achieving the goals of the hub location problem. Figure 4: Average means plot for the MOPSO algorithm [22]. Figure 5: Average S/N ratio plot for the MOPSO algorithm [22]. Based on the above plots, the optimal levels of factors are obtained as described in Table 6.  
Table 6 displays the levels of factors for the MOPSO algorithm derived from empirical studies. The data here illustrate the optimal parameter settings identified for maximizing performance and efficiency. Table 6 outlines the optimal levels of factors used in the MOPSO algorithm based on empirical studies. The table presents the best-performing parameter settings for maximizing the algorithm's efficiency and performance in solving the hub location problem. It includes the values for the population size nPop, the number of generations nRep, the inertia weight W, and the cognitive and social coefficients C1 and C2.