## 3 Methodology
### Algorithms for Determining and Evaluating the Optimal Amount of Ore Hailing Equipment  
To incorporate the dual objectives of the model of minimizing both the completion time and cost, an iterative approach was utilized in which models are created using only one objective function. As shown in Algorithms 1 and 2, the iterative process allows for the exploration of the trade-off between completion time and costs, providing a set of solutions that balance both objectives. Considering the tactical-level analysis, the following procedure was proposed to determine the optimal amount of hauling equipment, as detailed in Algorithm 1. The algorithm starts by defining the input data, which include \(\mathcal{M}\), \(\mathcal{V}\), \(\mathcal{T}\), \(p_{i}^{h}\), \(p_{j}^{l}\), \(h\), and \(P\). Then, defining the output data, which include the minimum number of hauling equipment (\(v_{opt}\)), and the hauling equipment' assignment to loading equipment (\(Schedule\)) including \(C_{max}\). Then, assign \(v\) to the magnitude of \(\mathcal{V}\), the total number of hauling equipment available (line 1). The variable \(Continue\) is set to true to initiate the loop (line 2). As long as it remains true, a model is created (considering Equations (2)-(13) of PMSPOHO model, input data and \(v\)), and a \(Result\) that minimizes the \(C_{max}\) is obtained (lines 3-5). A feasibility evaluation is conducted, where if the \(Result\) is feasible, \(v\) is reduced by one unit (attempting to solve the problem with fewer hauling equipment) (lines 6-8). If it is not feasible, \(v_{opt}\) is increased by one unit, and the loop terminates (lines 9-13). This algorithm returns \(v_{opt}\) and \(Schedule\) (line 14). ```
Input: \(\mathcal{M}\), \(\mathcal{V}\), \(\mathcal{T}\), \(p_{i}^{h}\), \(p_{j}^{l}\), \(h\), \(P\);

Output: \(v_{opt}\): minimum hauling equipment, _Schedule_: hauling equipment' assignment to loading equipment including the \(C_{max}\). ```
1:\(v\leftarrow|\mathcal{V}|\)
2:\(Continue\leftarrow\)true
3:while\(Continue\)do
4:\(model\leftarrow\)Createmodel(\(Equations(2)\)-(13), Input, \(v\))
5:\(Result\leftarrow\)Solve(\(model\), Minimize)
6:ifIsFeasible(\(Result\))then
7:\(v\gets v-1\)
8:\(Result\), \(Schedule\)
9:else
10:\(v_{opt}\gets v+1\)
11:\(Continue\leftarrow\)false
12:endif
13:endwhile
14:return\(v_{opt}\), Schedule ```
**Algorithm 1** Optimal hauling equipment determinator. The following procedure was proposed for an analysis of the relationship between \(k\) and \(C_{max}\), detailed in Algorithm 2. This pseudocode allows for observing the behavior of \(k\) as \(C_{max}\) is incremented. The algorithm starts by defining the input data, which includes the same factors as in Algorithm 1: \(k_{1}^{h}\), \(k_{j}^{l}\), and \(v\). Then, defining the output data includes a list of costs and a list of termination times (\(C_{max}\)). Define \(v\) (line 1). The \(model_{lb}\) is created (considering Equations (2)-(13) of PMSPOHO model and input data) (line 2), and a \(Result_{lb}\) from the minimization of \(C_{max_{lb}}\) is obtained (line 3), extracting this result (line 4). A \(model_{ub}\) is created (considering Equations (1), (3)-(13) of PMSPOHO model and input data) (line 5), and a \(Result_{ub}\) from the minimization of \(k\) is obtained (line 6), extracting \(C_{max_{ub}}\) for this result (line 7). ```
1:Input: \(\mathcal{M}\), \(\mathcal{V}\), \(\mathcal{T}\), \(p_{i}^{h}\), \(p_{j}^{l}\), \(k_{i}^{l}\), \(k_{j}^{l}\), \(h\), \(P\), and \(v\);
2:Output: ListPair(\(C_{max},k\)): List of cost respect to \(C_{\max}\).