## 1 Introduction
The HSR-DA problem jointly optimizes route operation, heterogeneous ship assignments, transport demand acceptance, and chartering decisions with the dual objectives of maximizing weekly profit and total transport volume. A bi-objective mixed-integer programming (MIP) model is developed to support integrated decision making, incorporating practical constraints such as ship requirements, fleet availability considering chartering, accepted transport volume, and transport capacity limits. To enhance computational efficiency, this study further introduces a partially relaxed mixed-integer programming (PRMIP) model by leveraging the TU properties of the chartering constraint matrices, which significantly reduces computational complexity without compromising optimality. Overall, the proposed bi-objective MIP model provides a practical and implementable optimization framework for shipping companies seeking to adapt their operations optimally in response to challenging regulatory landscapes. In particular, the multifaceted contributions of this paper are outlined as follows:  
1. We establish a bi-objective MIP model for jointly optimizing the heterogeneous ship routing and transport demand acceptance, subject to ship requirement constraints, fleet availability and demand acceptance limits. This formulation enables shipping companies to effectively evaluate and strategically balance the often competing objectives of maximizing weekly profit and maximizing total transport volume, thereby facilitating more informed decision making in their operational planning and resource allocation under the new regulatory pressures. 2. We prove the necessity of maintaining integrality for route operation variables and ship assignment variables and develop a computationally efficient PRMIP model by leveraging the totally unimodular (TU) property of the chartering constraint matrices, allowing for continuous relaxation of the ship chartering variables without compromising optimality. The model is validated through a real-world case study, which shows that the PRMIP model achieves identical optimal solutions to the MIP model with significantly reduced computational time. 3. We analyze the structural properties of the Pareto frontier for the bi-objective MIP model and find that it consists of a finite union of continuous, piecewise linear segments. We also show that the global frontier is generally non-convex and can exhibit discontinuities due to the discrete nature of ship assignment and route operation decisions. This structural insight provides a precise characterization of the optimal trade-offs between profit maximization and transport volume, illuminating the set of efficient solutions. 4. Sensitivity analyses demonstrate that as freight rates increase, the Pareto frontier shifts to offer simultaneously improved profitability and transport volume potential. This shift shows diminished marginal trade-offs between the two objectives, enhancing operational flexibility. 5. A case study based on a global liner network, considering the impact of U.S. surcharges, yields three major managerial insights: (i) optimal fleet deployment under the surcharges consistently involves avoiding the assignment of China-built ships to routes serving U.S. ports to minimize regulatory cost impacts; (ii) strategic selection of an operating point on the profit-volume Pareto frontier is crucial, as an extreme focus on one objective leads to disproportionate sacrifices in the other, underscoring the practical value of balanced operational strategies; and (iii) prioritizing profitability naturally drives route network rationalization and selective acceptance of higher-value cargo, which in turn reduces overall operational costs and optimizes fleet utilization. The remainder of this paper is structured as follows. Section 2 provides an overview of the relevant literature on liner shipping optimization, fleet deployment, demand acceptance, and the impact of regulatory policies.