## 4 Experimental Results and Evaluation
### Experiment A  
As noted above, this experiment aimed to investigate and identify the optimal parameters for the ABC algorithm with the Pareto-based approach (including the number of bees or the bee colony size (\(C_{s}\)), the loop quantity (\(I_{o}\)), mutation rate (\(M_{r}\)), and crossover rate (\(C_{r}\)). The \(\overset{\sim}{C}\) metric was used for efficiency comparison. An experimental design utilizing a three-level factorial design was conducted to determine the optimal setting of the parameter values, as outlined in Table 6.  
The computational runs were replicated 20 times using a different case study. There are two case studies available to assess the performance of the parameters: one with a small knapsack problem in \"Problem1\" and the other with a large knapsack problem in \"Problem2\". In practice, the available computation time is limited. Therefore, the iterations were fixed at 50,000, as test runs indicated that this number was sufficient to achieve convergent results. In this research, the \(\overset{\sim}{C}\) metric was used to compare the non-dominant answer groups of the methods individually. For example, to measure the effectiveness of levels 1 and 2, the denotation \(\overset{\sim}{C}\)(1,2) measures the number of members of 2 dominated by 1. Thus, \(|2|\) is the number of solutions in this method. The smaller the value of \(\overset{\sim}{C}\)(1,2), the more strongly the indication is that level 2 is better than level 1. The \(\overset{\sim}{C}\) metric results are presented in Tables 7-9 for the function evaluation, crossover rate, and mutation rate, respectively.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{**Levels**} & \multicolumn{3}{c}{**Function Evaluation**} & \multirow{2}{*}{**Crossover Rate (\(C_{r}\))**} & \multirow{2}{*}{**Mutation Rate (\(M_{r}\))**} \\ \cline{3-3} \cline{5-6}  & & **Bee Colony Size (\(C_{s}\))** & & & \\ \hline
1 & 1 & 10 & 5000 & 0.1 & 0.1 \\
2 & 2 & 50 & 1000 & 0.5 & 0.5 \\
3 & 3 & 100 & 500 & 1.0 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Key parameters of ABC algorithm with Pareto-based approach.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Problem** & **C(1,2)** & **C(2,1)** & **C(1,3)** & **C(3,1)** & **C(2,3)** & **C(3,2)** & **Best** \\ \hline Problem 1 & 0.286 & 1.000 & 1.000 & 0.200 & 1.000 & 0.286 & **2** \\ Problem 2 & 0.200 & 0.750 & 0.667 & 0.500 & 1.000 & 0.600 & **Size \(\times\) Iter** \\ \hline Average & 0.243 & 0.875 & 0.833 & 0.350 & 1.000 & 0.443 & **50 \(\times\) 1000** \\ \hline \hline \end{tabular}
\end{table}
Table 7: Function evaluation results given by \(\overset{\sim}{C}\) metric.  
In Tables 7-9, the \(\overset{\sim}{\text{C}}\) metric values that are less than or close to 0 are chosen. We evaluated the function evaluation values presented in Table 7 and found the optimal parameter for function evaluation is \"levels2\" with a bee colony size of 50 and a loop quantity of 1000. As in Table 8, we evaluated the crossover rate values and found that the optimal parameter for crossover rate values is \"levels2,\" which is equal to 0.5. Finally, we evaluated the mutation rate in Table 9, and it was found that the optimal parameter for the mutation rate was \"levels1\", which was 0.1. Therefore, we summarize the results of the function evaluation, crossover rate, and mutation rate using values of 50 \(\times\) 1000, 0.5, and 0.1, respectively, in Table 10.