## 3 Research Findings
### Parameter Tuning for Metaheuristic Algorithms  
To tune the parameters, a response variable has been utilized. This response variable is a combination of five provided criteria, and its value is calculated using the following formula. Since the criteria do not have equal importance, weighting coefficients are assigned to them. \[R_{i}=\frac{\overline{NPF_{1}}+\overline{MSI_{2}}+\overline{SM_{3}}+\overline{ CPU-time}}{w_{1}+w_{2}+w_{3}+w_{4}} \tag{29}\]  
Factors and levels for the NSGA-II algorithm:  
The factors and their corresponding levels used for the NSGA-II algorithm are defined according to Table 1.  
Table 1 defines the levels of key factors employed in the NSGA-II algorithm. These factors--population size, crossover probability, and mutation probability--play a crucial role in determining the algorithm's performance in solving multi-objective optimization problems. The levels listed in the table were selected based on prior studies and experimental validation to explore a broad range of possible configurations. Population Size: The number of solutions considered in each generation. Larger population sizes allow for a more diverse search but require more computational resources. Crossover Probability: Determines the likelihood of combining parent solutions to create offspring, which is essential for exploring new areas of the solution space. Mutation Probability: Specifies how often mutations occur, promoting diversity and preventing premature convergence to local optima. This table serves as the foundation for setting up the experimental design, ensuring a systematic and efficient exploration of parameter configurations to identify the optimal settings for the NSGA-II algorithm. By referring to the standard array tables in the Taguchi method and using Minitab software (version 20), L9(34) orthogonal arrays are selected as the most suitable design for models with three to six factors. The orthogonal arrays for this design are shown in Table 2.  
Table 2 presents the orthogonal arrays used for parameter tuning in the NSGA-II algorithm. The configurations listed here ensure systematic exploration of variable interactions, contributing to enhanced algorithm efficiency. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Levels of Factors** & & & \\ \hline Parameters & 1 & 2 & 3 \\ Population Size & 50 & 70 & 100 \\ Crossover Probability & 0.2 & 0.5 & 0.8 \\ Mutation Probability & 0.2 & 0.3 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Levels of factors used for the NSGA-II algorithm [22]. Since the value of \(R_{i}\) varies in each problem and is not directly usable, the Relative Percent Deviation (RPD) is used for each problem. \[\text{RPD}=\frac{\text{Alg}_{\text{sol}}-\text{Min}_{\text{sol}}}{\text{Min}_{ \text{sol}}}\times 100 \tag{30}\]  
In the above equation, \(\text{Alg}_{\text{sol}}\) and \(\text{Min}_{\text{sol}}\) represent the values of \(R_{i}\) for each iteration of the experiment and the best solution obtained, respectively. After converting the value of \(\text{Ri}_{i}\) to RPD, according to the Taguchi parameter design structure, the S/N ratio is calculated based on RPD. Then, the average S/N ratio of the experiments is calculated for each parameter level. The best value for each parameter is the one with the lowest average among these averages; in other words, the optimal levels of factors are those that yield the minimum average ratio. After conducting the Taguchi experiment, the results, the average of averages, and the average S/N ratio for each level of factors in the NSGA-II algorithm for the model presented in Figures 2 and 3 are shown. Figure 2 displays the \"Average of Averages\" plot for the NSGA-II algorithm, summarizing the performance of the algorithm across different levels of experimental factors. This plot aggregates the results from multiple experimental runs, providing a visual representation of the algorithm's overall performance.