## Appendix A
This relaxed linear programming (LP) solution improves both objectives compared to strategy (1) with the integer solution \(x_{r_{1}}=1\), and achieves a higher value in the first objective while keeping the second objective unchanged compared to strategy (2). Critically, \(x_{r_{1}}=1/3\) is practically invalid as we cannot operate only one-third of the route. Hence, the LP relaxation of \(x_{r}\) inherently produces non-integer solutions conflicting with the real-world integer operation requirement, proving \(x_{r}\) cannot be freely relaxed. \(\square\)  
**Proof of Theorem 2. ** We demonstrate this via a synthetic counterexample. Assume that there is only one candidate route \(r_{1}\), which excludes U.S. ports and needs three ships for operation, i.e., \(N_{r_{1}}=3\). There is only one OD pair \((o,d)\), and its weekly transport demand is 3500 TEUs, i.e., \(D_{od}=3500\) TEUs. The revenue earned per TEU for this OD pair is set as \(p_{od}=800\) USD. Consider there are two ship types, \(s_{1}\) and \(s_{2}\). The company now owns one ship of type \(s_{1}\) and two ships of type \(s_{2}\), i.e., \(A_{s_{1}}=1\), \(A_{s_{2}}=2\). Ship type \(s_{1}\) has a capacity of \(Q_{s_{1}}=6000\) TEUs, a charter-in cost \(c_{s_{1}}^{\rm in}=0.5\) million USD, a charter-out revenue \(c_{s_{1}}^{\rm out}=0.4\) million USD, and a transport cost of \(F_{s_{1}r_{1}}+B_{s_{1}r_{1}}=2.7\) million USD. Ship type \(s_{2}\) has a capacity of \(Q_{s_{2}}=3000\) TEUs, a charter-in cost \(c_{s_{2}}^{\rm in}=0.25\) million USD, a charter-out revenue \(c_{s_{2}}^{\rm out}=0.2\) million USD, and a transport cost of \(F_{s_{2}r_{1}}+B_{s_{2}r_{1}}=0.9\) million USD. Since the route does not include any U.S. ports, the extra penalty costs are zero. Then, for the original model, there will be multiple possible route operation and ship assignment solutions. Through calculation, the two solutions that maximize the first objective and the second objective, respectively, are in the Pareto-optimal solutions as follows: (1) Operating route \(r_{1}\), i.e., \(x_{r_{1}}=1\). In this case, the company deploys three ships of type \(s_{2}\), i.e., \(n_{s_{1}r_{1}}=0\), \(n_{s_{2}r_{1}}=3\); then, one ship of type \(s_{1}\) is chartered out and one ship of type \(s_{2}\) is chartered in, i.e., \(n_{s_{1}}^{\rm in}=n_{s_{2}}^{\rm out}=0\), \(n_{s_{1}}^{\rm in}=1\). Since \(Q_{s_{1}}\cdot n_{s_{1}r_{1}}/N_{r_{1}}+Q_{s_{2}}\cdot n_{s_{2}r_{1}}/N_{r_{1} }=6000\times 0+3000\times 3/3=3000\) TEUs, the weekly accepted transport demand is \(z_{odr_{1}}=3000\) TEUs. Then, the weekly profit in this case can be calculated as follows:  
\[p_{od}z_{odr_{1}}-n_{s_{1}r_{1}}/N_{r_{1}}(F_{s_{1}r_{1}}+B_{s_{ 1}r_{1}})-n_{s_{2}r_{1}}/N_{r_{1}}(F_{s_{2}r_{1}}+B_{s_{2}r_{1}})\] \[-n_{s_{1}r_{1}}/N_{r_{1}}\cdot B_{s_{1}r_{1}}^{\rm extra}-n_{s_{2 }r_{1}}/N_{r_{1}}\cdot B_{s_{2}r_{1}}^{\rm extra}+c_{s_{1}}^{\rm out}n_{s_{1} }^{\rm out}-c_{s_{2}}^{\rm in}n_{s_{2}}^{\rm in}\] \[=800/10^{6}\times 3000-0-0.9-0-0.4-0.25=1.65\ {\rm million USD}.\]  
The total weekly transport volume in this case is 3000 TEUs. (2) Operating route \(r_{1}\), i.e., \(x_{r_{1}}=1\). In this case, the company deploys one ship of type \(s_{1}\) and two ships of type \(s_{2}\), i.e., \(n_{s_{1}r_{1}}=1\), \(n_{s_{2}r_{1}}=2\), and no ship is chartered in or chartered out, i.e., \(n_{s_{1}}^{\rm in}=n_{s_{2}}^{\rm in}=n_{s_{1}}^{\rm out}=n_{s_{2}}^{\rm out}=0\). Since \(Q_{s_{1}}\cdot n_{s_{1}r_{1}}/N_{r_{1}}+Q_{s_{2}}\cdot n_{s_{2}r_{1}}/N_{r_{1} }=6000\times 1/3+3000\times 2/3=400\) TEUs \(>D_{od}\), the demand is fully satisfied. Then, the weekly profit in this case can be calculated as follows:  
\[p_{od}z_{odr_{1}}-n_{s_{1}r_{1}}/N_{r_{1}}(F_{s_{1}r_{1}}+B_{s_{ 1}r_{1}})-n_{s_{2}r_{1}}/N_{r_{1}}(F_{s_{2}r_{1}}+B_{s_{2}r_{1}})\] \[-n_{s_{1}r_{1}}/N_{r_{1}}\cdot B_{s_{1}r_{1}}^{\rm extra}-n_{s_{ 2}r_{1}}/N_{r_{1}}\cdot B_{s_{2}r_{1}}^{\rm extra}\] \[=800/10^{6}\times 3500-\frac{1}{3}\times 2.7-\frac{2}{3}\times 0.9- 0-0=1.3\ {\rm million USD}.\]  
The total weekly transport volume in this case is 3500 TEUs.