## 6 Discussion  
This study contributes to the field by integrating bounded rationality into the analysis of freeway toll rate optimization, addressing a gap in traditional models that rely solely on utility maximization. By incorporating user preferences and perceptual demand, the proposed agent-based simulation model provides a more realistic representation of route selection behavior, offering a novel approach to understanding the dynamic relationships among toll rates, travel demand, and revenue. Compared to the existing literature, which often assumes perfect rationality or relies on static models, this study's dynamic and behaviorally grounded framework enhances the accuracy and applicability of toll rate optimization strategies. The findings have significant theoretical and practical implications. Theoretically, this study advances the understanding of how bounded rationality influences travel behavior and toll revenue, providing a foundation for future research on complex decision-making processes in transportation systems. Practically, the model offers actionable insights for policymakers and freeway operators to design toll rate schemes that maximize revenue while maintaining travel demand. For instance, the identification of a theoretical toll rate range and the analysis of speed differentials between freeways and parallel roads can inform more effective traffic management strategies. However, this study has several limitations. First, the agent-based simulation model is relatively simplistic. In the actual scenario, route selection behavior is not only affected by the travel cost and preference, such as traffic conditions, but weather, travel purpose, and other factors will also affect the results of route selection. Moreover, the traffic condition also affects the travel cost, and the complex relationship of agents' decision making is not well reflected, which leads to the discrepancy between the results of the model analysis and the actual situation. Due to the sensitivity of travel demand and toll collection data of the freeway system, this study lacks side-by-side comparison data and historical cumulative data, therefore, the model results are not verified by empirical data, which makes the model not that convincing. Therefore, a natural next step is to further improve the agents' decision-making model to make it more consistent with the actual decision-making process. At the same time, the simulation model should be verified and improved by carrying out cooperative programs with freeway enterprises to accumulate the changes in traffic demand and toll collection before and after the adjustment of rate schemes in different regions, making the analysis results more reliable. In conclusion, this study provides valuable insights into freeway toll rate optimization by incorporating bounded rationality and dynamic relationships among toll rates, travel demand, and revenue. While the findings are promising, further refinement and empirical validation are necessary to fully realize the model's potential in guiding transportation policy and improving system efficiency. Conceptualization, J.S. ; methodology, J.R.; software, [PERSON]; validation, [PERSON]; formal analysis, [PERSON]; investigation, [PERSON]; resources, [PERSON]; data curation, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, [PERSON] and Z.W. ; visualization, [PERSON]; supervision, [PERSON]; project administration, [PERSON]; funding acquisition, [PERSON] All authors have read and agreed to the published version of the manuscript. This article was supported by the Guangxi Computing Center Co., Ltd. and Guangxi Communications Investment Group Corporation Ltd.  
Not applicable. The datasets used and analyzed during the current study are available from the corresponding author upon reasonable request. The authors declare that this study received funding from Guangxi Computing Center Co., Ltd.