## 2 Materials and Methods
### Application of Method in an Open-Pit Mine  
In order to demonstrate its usefulness in a real-world situation, this approach is used on an active open-pit mining project. There are 25 trucks in total in the operation, each with a handling capability of 40 tons; five excavators and seven dump sites total. Dividing the overall output objective by the truck handling capacity helps one to obtain the necessary assignment numbers for every excavator, as shown in Table 2. This method guarantees that the production needs determine the assignment numbers in proportion. Key input values for the optimization model, these figures direct the truck distribution to excavators so effectively reaching the target production levels. When multiple trucks need to be assigned within a single timestamp, the travel time data from Table 3 is used to update the excavator ready times, providing a more accurate representation of excavator availability by accounting for delays arising from sequential truck assignments.  
Examples of truck locations and excavator ready times at specific timestamps are presented in Tables 4 and 5, respectively. Full tables are omitted for brevity due to their limited relevance to the discussion. Table 4 presents a thorough picture of the operational situation throughout the shift under review. Every column shows a particular second within the shift, enabling time-resolved study. Rows line up with the trucks working during the shift. If the vehicle needs an assignment at that specific second, the relevant cell for each truck contains its ID number. On the other hand, should a vehicle not need an assignment, its cell is empty with a value of 0. Table 4's last five rows provide the projected ready times for the excavators, therefore indicating the time left before any excavator may be used to service another truck. This structure enables exact decision-making inside the binary integer programming model by clearly synchronizing the depiction of truck requests and excavator availability.  
\begin{table}
\begin{tabular}{c c} \hline
**Excavator IDs** & **\# of Required Assignment** \\ \hline
0 & 280 \\
1 & 74 \\
2 & 50 \\
3 & 249 \\
4 & 246 \\ \hline \end{tabular}
\end{table}
Table 2: Required assignment number of excavators.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Excavator IDs** & **0** & **1** & **2** & **3** & **4** \\
**Dump Site IDs** & **0** & **1** & **2** & **3** & **4** \\ \hline
0 & 230 & 250 & 300 & 450 & 500 \\
1 & 250 & 350 & 500 & 400 & 200 \\
2 & 350 & 500 & 500 & 400 & 200 \\
3 & 500 & 370 & 700 & 650 & 550 \\
4 & 1000 & 500 & 200 & 500 & 750 \\
5 & 500 & 1000 & 500 & 210 & 1000 \\
6 & 210 & 500 & 1000 & 210 & 500 \\ \hline \end{tabular}
\end{table}
Table 3: Dump site to excavator travel times (in seconds).  
Table 5 details the location of trucks at specific seconds during the shift under consideration. Each cell corresponding to a truck is populated with the ID of the dump site where the truck is located at that moment. Trucks that are on the road are marked with the number 99 for coding purposes. This distinction ensures clarity in identifying trucks in transit, as the number 0 is reserved to represent the index of the first dump site. This coding scheme enables a clear and unambiguous representation of truck locations, whether they are stationary at a dump site or actively traveling, which is critical for accurately updating excavator-ready times when multiple trucks require assignment within a single timestamp. By incorporating the spatial distribution of trucks, this methodology ensures that excavator availability is accurately represented, reflecting the dynamic nature of truck movements and assignments in the open-pit mining operation.