## 4 Experimental Results and Evaluation
### Experiment B  
The aim of this experiment was to benchmark the performance of the proposed method using the optimal parameter settings, which were identified in Experiment A. To test whether the non-dominant set provided by the meta-heuristic method could yield an answer close to that of an exact method, the LINGO optimization solver was implemented, and its optimal solutions were compared with those of the proposed method. That is, the closeness or consistency of the optimal solutions given by the proposed method to the best solutions from the LINGO optimization solver was determined. If closeness or a match was achieved, it was inferred that the non-dominant set obtained from the meta-heuristic method was suitable and of appropriate quality. The comparative results are presented in Table 11. In this work, a dataset from the OR-Library database, which is a collection of test datasets for various Operations Research (OR) problems, was used. \begin{table}
\begin{tabular}{l l} \hline Function Evaluation (\(\mathbf{C_{s}\times L_{o}}\)) & 50 \(\times\) 1000 \\ \hline Crossover rate (\(\mathbf{C_{r}}\)) & 0.5 \\ \hline Mutation rate (\(\mathbf{M_{r}}\)) & 0.1 \\ \hline \end{tabular}
\end{table} 
Table 10: Optimal parameters as indicated by \(\overset{\sim}{\text{C}}\) metric. \begin{table}
\begin{tabular}{l l l l l l l l} \hline
**Problem** & **C(1,2)** & **C(2,1)** & **C(1,3)** & **C(3,1)** & **C(2,3)** & **C(3,2)** & **Best** \\ \hline Problem 1 & 0.200 & 1.000 & 0.667 & 0.000 & 1.000 & 0.200 & \\ Problem 2 & 0.333 & 0.750 & 0.750 & 0.250 & 0.500 & 0.333 & **2** \\ \hline Average & 0.267 & 0.875 & 0.708 & 0.125 & 0.750 & 0.267 & **Csss\({}_{\text{r}}\) = 0.5** \\ \hline \end{tabular}
\end{table} 
Table 8: Crossover rate evaluation results given by \(\overset{\sim}{\text{C}}\) metric. \begin{table}
\begin{tabular}{l l l l l l l l} \hline
**Problem** & **C(1,2)** & **C(2,1)** & **C(1,3)** & **C(3,1)** & **C(2,3)** & **C(3,2)** & **Best** \\ \hline Problem 1 & 0.750 & 0.200 & 1.000 & 0.000 & 0.250 & 0.500 & **1** \\ Problem 2 & 1.000 & 0.000 & 0.600 & 0.167 & 0.200 & 0.500 & **1** \\ \hline Average & 0.875 & 0.100 & 0.800 & 0.083 & 0.225 & 0.500 & \\ \hline \end{tabular}
\end{table} 
Table 9: Mutation rate evaluation results given by \(\overset{\sim}{\text{C}}\) metric. An optimal solution given by the Pareto approach that was equal to the LINGO optimization solver solution was chosen for comparison. Table 11 shows that the proposed method provided equivalent answers to the exact method for data from the OR-Library. Hence, it can be concluded that the proposed ABC algorithm with the Pareto-based approach is effective for solving the problem. Table 12 presents the average runtimes of the ABC algorithm using the Pareto-based approach for solving problems from the OR-Library. These values are compared with the resolution times of the exact method using the LINGO optimization solver. While the proposed method is slower than the exact method in small-scale problems, it should be noted that for more complex problems, the exact method is not able to solve the problem within an acceptable computational runtime. In such cases, the ABC algorithm with the Pareto-based approach takes more time to optimize but still delivers a solution within a reasonable computational runtime. The performance of the proposed algorithm was also evaluated using the well-known Loh and Nee (LN) test cases, and Table 13 shows that the proposed algorithm was able to solve the SCLP. However, the performance of the proposed algorithm will need to be further measured with _RI_ values to facilitate a comparison of results.