## 3 Methodology
### Algorithms for Determining and Evaluating the Optimal Amount of Ore Hailing Equipment
3:Define \(v\)
4:\(model_{lb}\leftarrow\)Createmodel(\(Equations(2)\)-(13), Input)
5:\(Result_{lb}\leftarrow\)Solve(\(model_{lb}\), Minimize)
6:\(C_{max_{lb}}\leftarrow\)ExtractCmax(\(Result_{lb}\))
7:\(model_{ub}\leftarrow\)Createmodel(\(Equations\) (1), (3)-(13), Input)
8:\(Result_{ub}\leftarrow\)Solve(\(model_{ub}\), Minimize)
9:\(C_{max_{ub}}\leftarrow\)ExtractCmax(\(Result_{ub}\))
10:for\(C_{max_{uf}}\) in \(C_{max_{lb}}\) to \(C_{max_{ub}}\)do
11:\(model_{ref}\leftarrow\)Createmodel(\(Equations\) (1), (3)-(8), (10)-(13), Input)
12:\(Result_{ref}\leftarrow\)Solve(\(model_{ref}\), Minimize)
13:\(Cost\leftarrow\)ExtractCost(\(Result_{ref}\))
14: ListPair(\(C_{max},k\)) \(\leftarrow\)AddToList(\(C_{max_{uf}}\), Cost)
15:endfor
16:return\(C_{max},k\) ```

**Algorithm 2** Progressive cost-time evaluator

A progressive evaluation is carried out that iterates \(C_{max_{uf}}\) by 1 from \(C_{max_{lb}}\) to \(C_{max_{ub}}\) (line 8). For each value, a \(model_{ref}\) is created (considering Equations (1), (3)-(8), (10)-(13) of PMSPOHO model and input data) (line 9). \(Result_{ref}\) is obtained by minimizing \(k\) for each iteration (line 10); \(k\) is extracted and added to a list (\(Cost\)) for each corresponding \(C_{max_{uf}}\) (lines 11-12). This algorithm returns the list of pairs (\(C_{max_{uf}}\),\(k\)) (lines 14),providing a detailed evaluation of how \(k\) varies about changes in \(C_{max_{ref}}\). In the algorithm, \(C_{max_{ref}}\) serves as an intermediate reference value to iteratively evaluate the trade-off between minimizing \(C_{max}\) and \(k\). This iterative evaluation provides insights into the cost-time trade-offs in the model. ## 4 Computational Experiments and Results Analysis

This section presents the computational experiments to evaluate the performance of the proposed PMSPOHO model and algorithms. All experiments were conducted on a computer with a Windows 11 operating system, 12 th Gen Intel(R) Core(TM) i5-12450H, 2.50 GHz processor, and 8 GB RAM. The Python 3.11 programming language was used for the development. The Gurobi 10.0.1 commercial optimizer and CBC free optimizer were used with a search time limit of 3600 s and a GAP of 0.01. Based on actual data provided in the studies by [PERSON] [2], [PERSON] [3], and [PERSON] and [PERSON] [4], which include the numbers of loading and hauling equipment, loading times, hauling times, time horizons, and production targets, 34 experimental instances were generated. These instances are designed to simulate real-world scenarios, enabling a robust evaluation of our proposed methodologies under various conditions. This approach ensures that our experimental setup is based on realistic operational parameters, enhancing the applicability and relevance of our findings. It is important to note that a direct comparison with the authors' studies above was not conducted, as their research methodologies differ significantly from our proposed approach. ### Tuning the Values for the Optimizer Parameters

Gurobi Optimizer version 10.0.1 features an extensive range of parameters designed to fine-tune the optimizer performance. Given the vast array of options among possible configurations, identifying the optimal combination of parameters that enhance the overall model performance poses a significant challenge. The TuningAPI from Gurobi offers a valuable tool for exploring various parameter settings that can improve model performance in specific cases [56]. Considering the procedure indicated by [PERSON] et al. [57], a sample of seven instances was created to identify the appropriate parameter settings for our model using TuningAPI. This sample includes those instances where the computation time obtained by the PMSPOHO model using the default parameters (PMSPOHO-D) exceeds the meantime of its respective class by two standard deviations. Table 3 shows the results of Gurobi parameter tuning across seven different problem instances. Each table row represents an instance, with columns indicating the specific Gurobi parameters adjusted for that instance.