## 6 Conclusions  
Container liner shipping companies operate within a complex environment where they must balance profitability and service reliability, all while navigating evolving regulatory policies such as surcharges imposed on China-built ships calling at U.S. ports. These challenges complicate strategic planning related to route deployment, ship assignment, and demand acceptance, requiring integrated decision-making frameworks. To address these challenges, this study develops a bi-objective MIP model for the HSR-DA problem. We establish a series of theorems to analyze the mathematical structure of the model. Specifically, we prove that the route operation decision variables and ship assignment variables must remain integers due to the practical infeasibility of fractional route operation and ship assignment, while the chartering constraints form a TU matrix, which allows the corresponding variables to be relaxed to continuous values without loss of integrality. Additionally, we demonstrate that the Pareto frontier consists of a finite union of line segments. Within each segment, it is continuous and piecewise linear, while generally being non-convex and exhibiting discontinuities. Based on these theorems, we reformulate the original MIP into a PRMIP model. We demonstrate that the PRMIP model can generate the same optimal objective value as the one obtained by the MIP model when solving the same instance, while significantly improving computational efficiency. Furthermore, we can always find an optimal solution to the PRMIP model such that it is equivalent to an optimal solution to the MIP model. These theoretical insights provide a solid foundation for enhancing solution efficiency and understanding the structural characteristics of the Pareto frontier. Computational experiments conducted with real-world-inspired scenarios highlight the practical effectiveness of the proposed model. The results demonstrate a clear trade-off between maximizing profitability and maximizing transport volumes, represented through an explicitly characterized Pareto frontier. Notably, the model consistently avoids costly surcharges for China-built ships through strategic fleet assignment and chartering decisions, underscoring its ability to inform operational strategies under complex regulatory constraints. In scenarios prioritizing profit maximization, the accepted transport volume decreases while the average freight rate increases, confirming the necessity of targeted and strategic route selection and demand acceptance. Sensitivity analysis further reveals that variations in freight rates significantly affect the structure of optimal decisions. Higher freight rates provide greater flexibility, allowing operators to selectively expand transport volumes while enhancing profitability simultaneously. Conversely, lower freight rates necessitate stringent selectivity in demand acceptance, reinforcing the importance of adaptive decision making in response to market fluctuations. Future work can extend our model by addressing several key areas to enhance its robustness and practical applicability:  
1. Incorporating transshipment and empty container repositioning: The current model excludes transshipment and empty container repositioning, which are critical for operational efficiency. Future work can optimize cargo routing and empty container flows, potentially using foldable containers or leasing strategies to reduce costs and improve container utilization [77; 78; 79]. * Incorporating variable sailing speeds: Allowing speed to be a decision variable can build a more comprehensive framework. This extension can capture the critical trade-off between fuel costs and service efficiency by directly linking sailing speed to operational fuel expenses and strategic ship allocation. Higher speeds increase fuel consumption, while lower speeds extend voyage times, which may require additional ships to maintain fixed service frequencies. Recent studies provide valuable insights in this field.