## 3 Case Study
### Calculations
#### 3.1.1 Development of the Calibration Model  
Step 1. Gathering of existing information and generation of probabilistic distribution functions. From a MineStar report, all the data were reviewed to later generate the probabilistic distribution functions and density functions of the fixed times and loaded tonnages through Arena Input Analyzer. Those results are presented in Tables 2 and 3. The functions employed by Arena Software were the normal distribution (NORM) and the beta distribution (BETA). It should be noted that Arena Input Analyzer recommends employing the function with the lowest square error after performing two goodness-of-fit tests. Those functions included Normal, Weibull, Beta, Triangular, Erlang, Gamma, Lognormal, Uniform, and Exponential. Furthermore, the information gathered from the MineStar report includes the results obtained in the operation, so it also involves the operator factor as their work affects the operating times and payloads. In addition, the attitude factor is implicit in the velocities employed for this work. A demonstration of the Arena Input Analyzer to estimate the best distribution that fits the material loaded from the shovel 1 sample is shown next in Table 1. With 267 samples, the errors obtained from all the distributions available are as follows:  
Figure 2: Layout of the Operations Area. Furthermore, Table 2 sums up the payload distribution, number of samples and errors obtained from the Kolmogorov-Smirnov and Chi-Square tests from all the loading equipment, whereas Table 3 shows the information related to the fixed times for trucks. Next, the velocities employed along with the Dijkstra algorithms are presented in Table 4. \begin{table}
\begin{tabular}{c c c c c} \hline
**Location** & **Distribution** & **Expression** & **Sample Size** & \(p\)**-Value KS** & **Error** \\
**Equipment** & **Distribution** & **Expression** & **Sample Size** & \(p\)**-Value KS** & **Error** \\ \hline \hline \begin{tabular}{c} \end{tabular} & Normal & NORM(375, 18.9) & 267 samples & \textgreater{}0.15 & 0.002106 \\ \begin{tabular}{c} \end{tabular} & Normal & NORM(370, 16.1) & 240 samples & \textgreater{}0.15 & 0.006607 \\ \begin{tabular}{c} \end{tabular} & Normal & NORM(376, 15.7) & 256 samples & \textgreater{}0.15 & 0.003483 \\ \
\begin{tabular}{c} \end{tabular} & Beta & \(321+96\times\) BETA(1.82, 2.96) & 96 samples & \textgreater{}0.15 & 0.007573 \\
\begin{tabular}{c} \end{tabular} & Normal & NORM(357, 13.9) & 75 samples & \textgreater{}0.15 & 0.007145 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PDF of the loaded tonnages for rope shovels and FEL. \begin{table}
\begin{tabular}{c c c} \hline  
Step 2. Creation and calibration of the model. On the day to be simulated, there were five pieces of loading equipment, including three rope shovels and two FELs. Moreover, trucks operated under a mechanical availability of 83.43%, and at the end of the day, approximately 330,000 tons were moved. Table 5 shows the moved material and the material and destinations reached that day. From the gathered information, it was determined that trucks worked 10-h shifts a day with 2 h corresponding to operating delays as a result of shift change, equipment check, blasting, or lunch breaks. Step 3. Validation of results  
There were 100 simulations considered fulfilling a difference of less than 1% between the half-width indicators of the loaded tonnages using loading equipment and total production and those indicators' mean values. Thus, the following results (Table 6) from the calibration model were obtained: