# Operation Optimization of the Sea Container Fleet Based on the Double-Level Planning Model
## 2 The Establishment of the Model
### The Establishment of Comprehensive Optimization Model  
#### Variable Relationships  
1. The relationship between air cargo volume and port container demand  
On route \(r\), if the ships can only call at each port once, its route system is that shown in Figure 2.  
Figure 1: Schematic diagram of route configuration. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Definition** \\ \hline \(r\) & Route. \(r\in\{1,2,\ldots,R\}\) \\ \(v\) & Type of vessel. \(v\in\{1,2,\ldots,V\}\) \\ \(m_{r}\) & Total number of ports of call on route \\ \(k\) & Segment. \(k=1,2,\ldots,m_{r}\) \\ \(Q_{r}^{k}\) & Cargo volume on sector \(k\) of route \\ \(d_{r}^{i}\) & Port on route \(i\) to port \(j\) weekly container demand (TEU) \\ \(L_{r}\) & Distance of route (miles) \\ \(C_{ap,v}\) & Rated capacity of \(v\) type of container (TEU) \\ \(S_{\min}\) & Minimum speed of type ship (kn) \\ \(S_{\max}\) & Maximum speed of type ship (kn) \\ \(t_{r}\) & Time (in days) required for a ship to complete a voyage on route \(r\) \\ \(T_{port}\) & Ship’s port call and berthing time on route \(r\) (days) \\ \(C_{cr}\) & Total weekly cost of ship on route \(r\) (USD) \\ \(C_{v}\) & Daily operating costs for type of vessel \\ \(K_{v}\) & Fuel consumption constants for ship types \\ \(P_{f}\) & Heavy oil price (USD/t) \\ \(P_{l}\) & Diesel price (USD/t) \\ \(F_{1,\rho}\) & Weekly by-engine fuel consumption per vessel \(v\) \\ \(F_{f,\rho}\) & Main engine fuel consumption per day per vessel \(v\) \\ \(F_{f}\) & Average daily total oil consumption of individual main engines \\ \(Z\) & Total weekly operating costs for ships on each route \\ \(x_{v,r}\) & When the \(r\) route is equipped with the \(V\)-type ship, it is defined as 1; otherwise, \\ \(x_{v,r}\) & it is set to 0. \\ \(n_{v,r}\) & Number of \(V\)-type ships on route \(r\) \\ \(S_{v}\) & The speed of a \(V\)-shaped ship \\ \(t_{s}\) & Carbon tax costs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Various parameters involved in ocean shipping. Figure 2: Schematic structure of the route system. Here, the seaport inside the dashed circle is situated on the opposite side of the intercontinental route. The ports linked by the dashed line do not engage in goods exchange. On the route, assuming there are \(n_{1r}\) ports at the starting point A and \(n_{2r}\) ports at the ending point B, there are \((n_{1r}+n_{2r})\) segments in total on the route \(r\). The relationship between freight volume and container transportation demand on each section of the route \(r\) can be represented as follows:  
On the route, assuming that there is \(1\) port at the starting point A and \(2\) ports at the ending point B. \[Q_{r}^{k}=\left\{ \begin{array}{ll}\sum\limits_{i=1}^{k}\sum\limits_{j=n_{ 1r}+1}^{n_{1r}+n_{2r}}d_{r}^{ij}+\sum\limits_{i=n_{1r}+1}^{n_{1r}+n_{2r}}\sum \limits_{j=k+1}^{n_{1r}}d_{r}^{ij}\\ \hskip 113.811024 ptk=1,2,\ldots,n_{1r}-1\\ \sum\limits_{i=n_{1r}+1}^{n_{1r}+n_{1r}+n_{2r}}d_{r}^{ij},k=n_{1r}\\ \sum\limits_{i=n_{1r}+1}^{n_{1r}+n_{2r}}d_{r}^{ij}+\sum\limits_{i=n_{1r}+1}^{k }\sum\limits_{j=1}^{n_{1r}}d_{r}^{ij}\\ \hskip 113.811024 ptk=n_{1r}+1,\ldots,n_{1r}+n_{2r}-1\\ \hskip 113.811024 pt\sum\limits_{i=n_{1r}+1}^{n_{1r}+n_{2r}}\sum\limits_{j=1}^{n _{1r}}d_{r}^{ij},k=n_{1r}+n_{2r}\end{array} \right. \tag{1}\]
2. The relationship between speed, roundtrip time, and the number of ships  
The roundtrip time of each voyage of the ship on each route is mainly determined by the water travel time and the berthing time in the port. The specific expression is as follows:  
\[t_{r}=T_{port}+L_{r}/24\sum\limits_{v=1}^{V}x_{vr}s_{v} \tag{2}\]  
In the ocean liner route, the frequency of the service is usually arranged on a weekly basis. Therefore, the number of \(v\)-shaped ships that are equipped for route \(r\) can be represented as follows:  
\[n_{vr}=x_{vr}[t_{r}/7] \tag{3}\]
3. The relationship between the speed and fuel consumption of ships  
The fuel cost encompasses the diesel consumption of the auxiliary engine and the heavy oil consumption of the main engine.