## 5 Testing the Proposed Algorithm
### Boxplots of execution times for each input instance
From Table 6, it was observed that the algorithm we proposed managed to achieve the optimal values proven by the solvers in all cases (instances 3, 5, 8, and 10) except for one (instance 11). In the rest of the cases, the local-based PSO exhibited a favorable performance compared to both solvers, achieving new lower bounds. It is worth mentioning that the proposed algorithm significantly improved the result in the case of the difficult instance 7. Additionally, in Table 7, the execution times are reported. Please bear in mind that, as was reported by [PERSON] et al. [12], the software tools used in that work were CPLEX and Gurobi, probably two of the most powerful commercial solvers in the market. Nevertheless, the execution times concerning those tools refer to the time when the best score was achieved, either with proven optimality or not. In the cases where the solvers failed to prove the optimality of the solution, they were allowed an extra running time of several days with no success. Despite this fact, the corresponding reported time was the time of achieving their best result. The current subsection would be incomplete if we did not report the effect of the refining schemas on the outcome of our algorithm. Next, we present the effect of both refining schemas on the score produced by the main part of our algorithm in Table 8.  
It was observed that the influence of the refining schemas on the outcome was of great importance. Specifically, the cost reduction percentage was between 38.9% and 100%. The average reduction percentage was 71.2% and the median was 76.2%. Another observation was that at the hard-to-solve cases, which were the instances 7, 1, 2, and 4, the average reduction percentage was 83.5% and the median was 87%. This fact probably indicates that the refining schemas' effect increased as the instances became harder. This is reasonable, considering that the execution of the second refining procedure was triggered only in case \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Instance** & **Gurobi** & **CPLEX** & **Local-Based PSO** &
\begin{tabular}{c} **Optimality** \\ **Proven?** \\ \end{tabular} \\ \hline
1 & 8 & 8 & **7** & No \\
2 & 16 & 14 & **13** & No \\
3 & 5 * & 5 * & 5 * & Yes \\
4 & 5 & 6 & **3** & No \\
5 & 0 * & 0 * & 0 * & Yes \\
7 & 12 & 11 & **6** & No \\
8 & 11 * & 11 * & 11 * & Yes \\
9 & 2 & 2 & 2 & No \\
10 & 10 * & 10 * & 10 * & Yes \\
11 & 17 * & 18 & 19 & Yes \\ \hline \hline \end{tabular} Bold: indicates new lower bound. *: indicates optimal value.  
\end{table} 
Table 6: Commercial solvers [12] versus local-based PSO. where the first process returned a solution featuring idle hours, which was more likely in case of harder instances. Following the presentation of the proposed algorithm's results and the analysis of its performance, it could be useful for the interested reader to refer to the respective website [47], which we maintained in relation to the current work. There, she/he can find more information about the presented algorithm and the used input files. Additionally, she/he can access the executable file of the programmed algorithm and instructions about how to reproduce and verify the results. Moreover, all the timetables produced by the algorithm with detailed analysis of their final cost are available at this site.