## 3 Methodology
### Modeling  
**Model assumptions. ** The assumptions of the model are as follows, based on the work presented by [PERSON] [2]:  
* Only one ore unloading point was considered. * The refueling time is negligible. * The loading and hauling equipment is available at all times. * The ore unloading waiting time is insignificant. Figure 2: Schedule considering ore hauling operation. **Notation. ** Table 2 provides the notation for the model proposed, showing the type and definition, respectively. **Objective function. ** The first objective function (Equation (1)) seeks to minimize \(k\) associated to use equipment. The second objective function (Equation (2)) seeks to minimize \(C_{max}\) of all trips. \[\min\ k \tag{1}\] \[\min\ C_{max} \tag{2}\]  
**Constraints. ** With respect to constraints, the set of constraints in Equation (3) ensures the assignment of only one hauling vehicle \(i\) to one loading vehicle \(j\) at time \(t\). The set of constraints in Equation (4) ensures only one loading equipment \(j\) per hauling equipment \(i\) at time \(t\). \[\sum_{i\in V}x_{ijt}\leq 1\quad\forall j\in\mathcal{M},t\in \mathcal{T} \tag{3}\] \[\sum_{j\in M}x_{ijt}\leq 1\quad\forall i\in\mathcal{V},t\in \mathcal{T} \tag{4}\]  
The set of constraints in Equations (5) and (6) ensures that the loading and hauling equipment are used for the required duration when assigned. \[\sum_{\tau=t+1}^{t+p_{j}^{l}}y_{j\tau}\geq(p_{j}^{l}-1)\cdot x_{ijt}\quad \forall i\in\mathcal{V},j\in\mathcal{M},t\in\mathcal{T}\setminus(t\leq h-p_{j }^{l}) \tag{5}\] \[\sum_{\tau=t+1}^{t+p_{j}^{l}+p_{i}^{l}}z_{i\tau}\geq(p_{i}^{h}+p_ {j}^{l}-1)\cdot x_{ijt}\quad\forall i\in\mathcal{V},j\in\mathcal{M},t\in \mathcal{T}\setminus(t\leq h-p_{i}^{h}-p_{j}^{l}) \tag{6}\]  
The set of constraints in Equations (7) and (8) ensures that loading and hauling equipment are not double-booked and are used only when available. \[x_{ijt}\leq 1-y_{ji}\quad\forall i\in\mathcal{V},j\in\mathcal{M},t \in\mathcal{T} \tag{7}\] \[x_{ijt}\leq 1-z_{ii}\quad\forall i\in\mathcal{V},j\in\mathcal{M},t \in\mathcal{T} \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Var**} & \multicolumn{1}{c}{**Type**} & \multicolumn{1}{c}{**Definition**} \\ \hline \(\mathcal{M}\) & Set & Set of loading equipment \(\mathcal{M}=\{1,2,\ldots,m\}\). \\ \(\mathcal{V}\) & Set & Set of hauling equipment \(\mathcal{V}=\{1,2,\ldots,v\}\). \\ \(\mathcal{T}\) & Set & Set of units of time: \(\mathcal{T}=\{1,2,\ldots,h\}\). \\ \(p_{i}^{l}\) & Parameter & Loading time for loading equipment \(j\). \\ \(p_{i}^{h}\) & Parameter & Hauling time for hauling equipment \(i\). \\ \(p\) & Parameter & The production target is the total amount of trips to be made. \\ \(h\) & Parameter & Time horizon to complete the production target. \\ \(k_{i}^{l}\) & Parameter & Cost associated with use loading equipment. \\ \(k_{i}^{h}\) & Parameter & Cost associated with use hauling equipment. \\ \(x_{ijt}\) & Variable & Principal decision variable, takes the value of 1 if the hauling equipment \(i\) is using the loading equipment \(j\) at time \(t\), 0 otherwise. \\ \(y_{ji}\) & Variable & The auxiliary decision variable takes the value of 1 if loading equipment \(j\) is using at time \(t\), 0, otherwise. \\ \(z_{it}\) & Variable & The auxiliary decision variable takes the value of 1 if hauling equipment \(i\) is using at time \(t\), 0, otherwise. \\ \(C_{max}\) & Variable & Completion time of the production target \\ \(k\) & Variable & Costs associated with equipment usage \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation for the model proposed. The set of constraints in Equation (9) ensures \(C_{max}\). \[C_{max}\geq t\cdot x_{ijt}+p_{i}^{h}+p_{j}^{l}\quad\forall i\in\mathcal{V},j\in \mathcal{M},t\in\mathcal{T} \tag{9}\]  
The set of constraints in Equation (10) ensures that \(P\) is met. \[\sum_{i}^{V}\sum_{j}^{M}\sum_{t}^{T}x_{ijt}\geq P\quad\forall i\in\mathcal{V},j \in\mathcal{M},t\in\mathcal{T} \tag{10}\]  
Equation (11) calculates \(k\).