## 2 Problem Description and Modelling
### Container Transport Vehicle Scheduling Optimisation Model Construction  
The primary objective of scheduling container transport vehicles is to streamline the execution of the transportation plan. This involves ensuring seamless coordination between the container transport vehicles and outbound vehicles to minimise wait times for loading. By optimising these operations, the transport vehicles can operate more consistently, thus reducing the overall time required to complete the transportation plan. Additionally, it aims to complete the transportation tasks while minimising total costs and reducing the time taken for container transit. Therefore, the objective function is formulated to achieve the lowest possible total costs and the shortest container transportation time while ensuring the completion of the transportation tasks. Therefore, the objective function is established as follows. \[ \begin{split}\text{min }f1&=\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M}\sum\limits_{t=1}^{T}X_{ij}^{t}\times A_{ij}^{t}\times D_{ i}\times P_{j}+\sum\limits_{j=1}^{M}\sum\limits_{t=1}^{T}\text{waiting}\text{T}_{j}^{t} \times P_{2}\\ \text{min}f2&=\sum\limits_{j=1}^{M}\sum\limits_{t=1 }^{T}\text{working}\text{T}_{j}^{t}\\ \text{min}F&=\alpha\times f1+\beta\times f2\end{split}  \tag{1}\]  
where the first item is the total cost of container transportation and is equal to the price of container transport vehicle transportation plus the cost of waiting time for container transport vehicle, and the second item is the container turnaround time. The third item is the total objective, requiring the lowest total cost and the shortest container turnaround time, and each of them is given a weight: \(\alpha\) is the first and \(\beta\) is the second. Constraints:  
\[\text{T}2_{j}^{t}=\text{T}1_{j}^{t}+\frac{\text{D}_{j}^{t}}{\text{V}_{j}} \tag{2}\]  
\[\text{T}3_{j}^{t}=\text{T}2_{j}^{t}+\text{waiting}\text{T}_{j}^{t} \tag{3}\]  
\[\text{T}4_{j}^{t}=\text{T}3_{j}^{t}+\frac{A_{j}^{t}}{s} \tag{4}\]  
\[\text{T}5_{j}^{t}=\text{T}4_{j}^{t}+\frac{\text{D}_{j}^{t}}{\text{V}_{j}} \tag{5}\]  
\[\text{T}6_{j}^{t}=\text{T}5_{j}^{t}+\frac{\text{D}_{j}^{t}}{b} \tag{6}\]  
\[\text{T}1_{j}^{t}= \begin{cases}0&t=1\\ \text{T}6_{j-1}^{t}&t>1\end{cases}  \tag{7}\]  
\[\text{waiting}\text{T}_{j}^{t}=\text{T}3_{j}^{t}-\text{T}2_{j}^{t} \tag{8}\]  
\[\text{working}\text{T}_{j}^{t}=\text{T}5_{j}^{t}-\text{T}1_{j}^{t} \tag{9}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Instructions** & **Unit** \\ \hline \(\text{T}4_{j}^{t}\) & Container transport vehicles j at t missions, time of departure from coal plant & time slot \\ \(\text{T}5_{j}^{t}\) & Container transport vehicles j in t missions, return time to yard & time slot \\ \(\text{T}6_{j}^{t}\) & Container transport vehicles j in t missions, time to unload cargo & time slot \\ \(\text{waiting}\text{T}_{j}^{t}\) & Container transport vehicles j waiting time at task t before coal loading begins & time slot \\ \(\text{working}\text{T}_{j}^{t}\) & Length of time container transport vehicles j are out of and back to the yard in t missions & time slot \\ \(totalQ_{i}\) & Total coal loading at coal preparation plant i & t \\ \(\text{X}_{j}^{t}\) & Decision variable, container transport vehicles j at t tasks to go to the target coal processing plant id & \\ \(\text{A}_{j}^{t}\) & Decision variable, the amount of coal pulled by container transport vehicles j at t missions to the & t \\ \(\text{target coal preparation plant}\) & & t \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)\[A_{j}^{t}\leq L_{j} \tag{10}\]  
\[\text{T3}_{j}^{t}\
otin[\text{T3}_{i}^{t},\text{T4}_{i}^{t}] \tag{11}\]  
\[S_{i}\leq totalQ_{i} \tag{12}\]  
\[A_{j}^{t}=2W \tag{13}\]  
(2) indicates that the time of arrival of the container transport vehicle at the coal preparation plant is equal to the time of departure from the yard plus the time of travel of the container transport.