## 3 Methods
### Generation of Truck Travel Times  
In this study, we compared the simulation results of instances wherein different probability density functions of the measured truck travel time were used as the input variables. The simulation results were compared by creating five temporal elements of the TCT with the Monte Carlo simulation, using the uniform, triangular, normal, and observed probability distributions derived from actual truck travel time data. The uniform, triangular, normal, and observed probability distributions were used in this study because they are widely used as probability density functions and can be defined using the big data of an ICT system. It should be noted that the uniform, triangular and normal distributions are theoretical probability density functions defined mathematically; however, the observed probability distribution is an empirical one. The uniform and triangular distributions were generated by considering the maximum and minimum values. The normal distribution was generated by considering the mean and standard deviation (STD), and the observed probability distribution was obtained using actual accumulated data (Figure 6). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Type** & **Data** & **Unit** \\ \hline \multirow{4}{*}{Input} & \multirow{2}{*}{Time parameters} & Travel time of the empty truck & TEu & Minutes \\ \cline{3-4}  & & Travel time of the loaded truck & TEu & Minutes \\ \cline{3-4}  & & Working time & LT & Minutes \\ \cline{3-4}  & & Daily working time & & Minutes \\ \cline{3-4}  & & Number of trucks & & Numbers \\ \cline{3-4}  & & Capacity of a truck & & Tons \\ \cline{3-4}  & & Number of simulations & & Numbers \\ \hline Output & \multicolumn{2}{c}{Total amount of the loaded ore} & Tons \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of the input and output parameters of the truck haulage system simulation. Figure 6: Probability density functions of the truck travel time for loading point 203. (**a**) Uniform distribution, (**b**) triangular distribution, (**c**) normal distribution, (**d**) observed probability distribution. Considering the measured truck travel time, a random variable for truck travel time was created based on a Monte Carlo simulation. A Monte Carlo simulation is a numerical, experimental method to obtain the output variable value of the system calculation model by considering the statistical value of the input variable. The values of the input random variable were sampled based on the statistical distribution, and the output variable was calculated using a computational model [47]. To use the Monte Carlo simulation, a cumulative relative frequency graph of the TCT was created. The class width and relative frequency are required to generate the cumulative relative frequency graph of the TCT. Using the statistical analysis results, the class width required to create a cumulative relative frequency graph was calculated (Equation (2)). The frequency of each class was calculated according to the data, and the relative frequency of each class was calculated by dividing the total data recorded (Equation (3)). The cumulative relative frequency was calculated using the relative power of each class, as shown in Equation (4):  
\[\text{Class width}=\frac{\text{Maximum value}-\text{Minimum value}}{\text{ Number of classes}} \tag{2}\]  
\[F_{i}^{r}=\frac{f_{i}}{N} \tag{3}\]  
\[F_{n}^{c}=\sum_{i=1}^{n}F_{i}^{r} \tag{4}\]  
Using the cumulative relative frequency graph based on the probability function distribution, a random variable for truck travel time was created by Monte Carlo simulation, as follows: the cumulative relative frequency graph \(F(x)\) represents the probability \(P\) that the truck travel time \(X\) is less than or equal to \(x\) (Equation (5)). Because \(F(x)\) is a probability, it possesses a value from 0 to 1. Thus, we can determine the inverse function of \(F(x)\), where \(G(F(x))\) is the inverse function of \(F(x)\), and \(r\) is any real number between 0 and 1.