## 2 Problem Formulation
### Port Cargo Demand  
By substituting Equations (2) and (3) into Equation (1), a more specific utility function is obtained as follows:  
\[U_{l}(x)=\left\{\begin{aligned} U_{A}(x)=v-P_{A}-\sum \limits_{m=1}^{M}w_{m}\bigg{(}\delta^{m}x+\epsilon\frac{x}{\overline{V}^{m}} \bigg{)}-C_{A}^{sea}-d_{A}\frac{Q_{A}}{h_{A}}\\ U_{B}(x)=v-P_{B}-\sum\limits_{m=1}^{M}w_{m}\bigg{(}\delta^{m}(b-x)+ \epsilon\frac{b-x}{\overline{V}^{m}}\bigg{)}-C_{bB}^{land}-C_{B}^{sea}-d_{B} \frac{Q_{B}}{h_{B}}\end{aligned}\right.,\ i=A,B \tag{4}\]  
In the interval [0,1], there must exist a critical point \(x\) where the utility of the shipper's choice of port A and port B for transshipment are equal, that is, there is a point \(x(x=\hat{x})\) that makes \(U_{A}(\hat{x})=U_{B}(\hat{x})\).  
\[v-P_{A}-\sum\limits_{m=1}^{M}w_{m}\bigg{(}\delta^{m}\hat{x}+\epsilon\frac{ \hat{x}}{\overline{V}^{m}}\bigg{)}-C_{A}^{sea}-d_{A}\frac{Q_{A}}{h_{A}}=v-P_{ B}-\sum\limits_{m=1}^{M}w_{m}\bigg{(}\delta^{m}(b-\hat{x})+\epsilon\frac{b-\hat{x}}{ \overline{V}^{m}}\bigg{)}-C_{bB}^{land}-C_{B}^{sea}-d\frac{Q_{B}}{h_{B}} \tag{5}\]  
To facilitate narration, we replace \(\frac{d}{h_{A}}\) and \(\frac{d}{h_{B}}\) by \(H_{A}\) and \(H_{B}\), respectively. According to Equation (5), it can be further obtained as follows:  
\[\hat{x}=\frac{P_{B}-P_{A}+C_{B}^{sea}-C_{A}^{sea}+C_{bB}^{land}+b\sum\limits_ {m=1}^{M}w_{m}\bigg{(}\delta^{m}+\epsilon\frac{1}{\overline{V}^{m}}\bigg{)}+ H_{B}Q_{B}-H_{A}Q_{A}}{2\sum\limits_{m=1}^{M}w_{m}\bigg{(}\delta^{m}+ \epsilon\frac{1}{\overline{V}^{m}}\bigg{)}} \tag{6}\]  
As shown in Figure 3, when \(x<\hat{x}\), the transportation cost for selecting port A (green line) is less than the transportation cost for selecting port B (blue line), indicating that the utility of selecting port A to transport goods to port K is greater than that of selecting port B. Consequently, the shipper will prioritize selecting port A for transporting to destination port K. Conversely, when \(x>\hat{x}\), the transportation cost for selecting port A (green line) is greater than the transportation cost for selecting port B (blue line), and the shipper will prioritize selecting port B for transporting goods to port K. In other words, the critical point \(\hat{x}\) determines the shippers' choice: those on the left side will select port A, while those on the right side will choose port B.  
Since the hinterland of the port is a unit linear space, with port A at position 0, according to the assumption that \(Q_{A}+Q_{B}=1\), the cargo demand functions of port A and port B under perfect competition mode are as follows:  
\[Q_{A}=\frac{P_{B}-P_{A}+C_{B}^{sea}-C_{A}^{sea}+C_{bB}^{land}+b\sum\limits_{m= 1}^{M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+H_{ B}}{2\sum\limits_{m=1}^{M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{ \overline{V}^{m}}\Big{)}+H_{A}+H_{B}} \tag{7}\]  
\[Q_{B}=\frac{P_{A}-P_{B}+C_{A}^{sea}-C_{B}^{sea}-C_{bB}^{land}+(2-b)\sum\limits _{m=1}^{M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+ H_{A}}{2\sum\limits_{m=1}^{M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{ \overline{V}^{m}}\Big{)}+H_{A}+H_{B}} \tag{8}\]  
It can be seen in Equations (7) and (8) that the cargo throughput of the two ports is related to each other's port charges, maritime transportation costs, cross-border land transportation costs, and port congestion costs, that is, port charges, maritime transportation costs, cross-border land transportation costs, and port congestion costs are the control variables affecting port cargo throughput. When any three of them are determined, the port cargo throughput is a monotonic function of the uncontrolled variables (\(Q_{i}(P_{i},C_{ik}^{sea},C_{bB}^{land},H_{i})\)).