## 4 Numerical Experiments
### Experimental Results
number of classrooms required, the overall classroom utilization rate, and the standard deviation in the utilization rates. In the comparison of the classroom utilization rates, the GA and POGA-DP exhibit overall similar occupancy rates. However, POGA-DP achieves a higher occupancy in certain instances (e.g., Instances 5, 8, 10, and 13). For example, in Instance 5, the occupancy rate of the GA is 27.461%, whereas POGA-DP increases it to 35.268%, indicating that POGA-DP may prioritize improving the classroom utilization during the optimization process. Nevertheless, in some cases (e.g., Instances 8 and 10), POGA-DP instead reduces the occupancy rate. For instance, in Instance 8, the GA achieves an occupancy rate of 45.739%, while POGA-DP only reaches 40.733%, suggesting that POGA-DP's optimization strategy does not solely aim to maximize the utilization but also takes balanced resource allocation into account. In terms of standard deviation, the GA exhibits greater fluctuations in certain instances, such as Instance 10 (GA: 1.290%,; POGA-DP: 0.952%) and Instance 15 (GA: 1.525%; POGA-DP: 0.885%). This indicates that the GA has larger variations in the utilization rates in these cases, whereas POGA-DP demonstrates relatively greater stability. However, in some instances (e.g., Instance 4), POGA-DP shows a higher fluctuation (1.604%) compared to that with the GA (0.628%), suggesting that POGA-DP may lead to more dynamic resource adjustments in specific cases, potentially affecting the overall scheduling stability. Despite the slightly higher variability in isolated cases, POGA-DP maintains a generally superior level of stability and significantly enhances the resource utilization efficiency across multiple instances. In terms of the number of classrooms required, the GA method generally demands more classrooms. For example, in Instance 9, the GA requires 58 classrooms, whereas POGA-DP only needs 37, indicating that POGA-DP can reduce the classroom demand while maintaining the same or even higher occupancy rates, thereby improving the resource utilization efficiency. This trend is evident in multiple instances (e.g., Instances 5, 8, 10, 14, and 15). Notably, in Instance 15, the GA requires 82 classrooms, whereas POGA-DP only requires 58, achieving nearly a 30% reduction in the resource usage. However, in certain cases, the number of classrooms required by the GA and POGA-DP remains the same \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**Occupancy**} & \multicolumn{2}{c}{**Occupancy Std. Dev.**} & \multicolumn{2}{c}{**Classroom Usage**} \\ \cline{2-7}  & **GA** & **POGA-DP** & **GA** & **POGA-DP** & **GA** & **POGA-DP** \\ \hline Instance 1 & 66.000\% & 66.000\% & 0.000\% & 0.000\% & 3 & 3 \\ Instance 2 & 49.611\% & 48.542\% & 0.371\% & 0.329\% & 6 & 5 \\ Instance 3 & 37.422\% & 38.374\% & 0.642\% & 0.770\% & 12 & 10 \\ Instance 4 & 34.790\% & 36.586\% & 0.628\% & 1.604\% & 13 & 13 \\ Instance 5 & 27.461\% & 35.268\% & 1.184\% & 0.886\% & 25 & 17 \\ Instance 6 & 45.720\% & 47.382\% & 0.255\% & 0.270\% & 9 & 6 \\ Instance 7 & 51.909\% & 51.871\% & 0.176\% & 0.276\% & 14 & 11 \\ Instance 8 & 45.739\% & 40.733\% & 0.590\% & 0.629\% & 34 & 23 \\ Instance 9 & 43.357\% & 45.067\% & 0.870\% & 0.608\% & 58 & 37 \\ Instance 10 & 46.111\% & 42.073\% & 1.290\% & 0.952\% & 72 & 49 \\ Instance 11 & 51.435\% & 55.250\% & 0.157\% & 0.000\% & 6 & 6 \\ Instance 12 & 50.357\% & 51.972\% & 0.247\% & 0.095\% & 8 & 8 \\ Instance 13 & 45.708\% & 50.612\% & 0.406\% & 0.368\% & 20 & 15 \\ Instance 14 & 40.199\% & 39.042\% & 1.045\% & 0.641\% & 48 & 38 \\ Instance 15 & 42.111\% & 43.995\% & 1.525\% & 0.885\% & 82 & 58 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Classroom utilization in the scheduling scheme. (e.g., Instances 1, 11, and 12), suggesting that both methods exhibit a similar optimization performance under specific conditions.