## 5 Discussion  
We increased the number of iterations from 20 to 1000 and measured the fitness values of the different algorithms for Instance 15, with the results shown in Figure 5. The experimental results indicate that in the early stages of iteration, POGA-DP performed similarly to the GA, slightly outperforming PSO and ACO. However, after 40 iterations, the fitness value of POGA-DP rapidly increased to 961, nearly reaching the level of the GA, the PSM, and ACO after 500 iterations and significantly surpassing PSO's 733. This demonstrates that as the number of iterations increases, POGA-DP can continuously optimize the solution quality, gradually finding better solutions and showcasing its strong optimization capability in handling complex course scheduling problems.  
In previous studies, GAs and PSO have typically shown various limitations when addressing large-scale complex optimization problems. For instance, GAs often suffer from a slow convergence speed, while PSO tends to become trapped in local optima. In this experiment, the GA and PSO maintained these characteristics: the GA exhibited a slow improvement in the fitness values with fewer iterations, and PSO stagnated after reaching a certain number of iterations. The PSM's performance also aligned with the literature, as it quickly converged in terms of its fitness with fewer iterations but showed a diminished improvement afterward. POGA-DP's outstanding performance in this experiment demonstrated its advantage in improving the fitness values, particularly after 100 iterations, where its growth rate significantly surpassed that of the other algorithms. This improvement is closely related to the algorithm's design, which integrates dynamic programming with a genetic algorithm.  
One limitation of this study lies in the lack of diversity in the experimental datasets. While the experiments demonstrated the  
To address the aforementioned limitations, future research could introduce more diverse datasets to further validate the performance of POGA-DP, including optimization problems of varying scales and constraint conditions to assess its generality. Additionally, optimizing the parameter configurations of the GA and PSO, such as through adaptive parameter adjustments or hybrid algorithms, may further enhance their performance. Moreover, increasing the number of iterations in the experiments or testing other algorithm combinations (e.g., hybrid algorithms) could also provide more insights for evaluating the algorithms.  
Future research could further explore the performance of POGA-DP by increasing the complexity of the experimental conditions, such as varying the number of joint and independent courses or considering more dynamic constraints, like classroom availability and teachers' time limitations. Additionally, comparing POGA-DP with other cutting-edge intelligent optimization algorithms (e.g., combining deep learning with genetic algorithms) could help validate its performance in more complex environments. This would provide the academic community with a more comprehensive evaluation of the algorithm and offer more valuable solutions for complex course scheduling problems in practical applications.  
Conceptualization: [PERSON] and D.W. Methodology: X.H. Software: X.H. Validation: D.W. Formal analysis: X.H. Investigation: [PERSON]. Resources: D.W. Data curation: [PERSON]. Writing--original draft preparation: [PERSON]. Writing--review and editing: [PERSON]. Visualization: [PERSON]. Supervision: [PERSON] and [PERSON]. Project administration: [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.  
The authors are grateful to the anonymous reviewers for their insightful comments, which certainly improved this paper.  
Figure 5: Performance analysis of fitness across different iterations.