## 3 Methodology
### Algorithms for Determining and Evaluating the Optimal Amount of Ore Hailing Equipment
The parameters include Symmetry, DegenMove, Heuristics, GomoryPasses, PreDepRow, MIPFocus, Method, CutPasses, and BranchDir. The third instance stands out significantly because it shows the highest variation (20.92%) between the default computational time (Tc-D) and the tuned computational time (Tc-T), indicating a significant improvement in performance with the adjusted parameters. In this instance, the parameters Heuristics, GomoryPasses, and PreDepRow were adjusted, resulting in a substantial reduction in computation time from 69.59 to 55.03 time units. This adjustment reduced the overall calculation time and optimized the use of computational resources, making the process more efficient and effective. The con \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Summary**} & \multicolumn{6}{c}{**Value Parameters**} \\ \hline
**Inst** & **p** & **m** & **P** & **Symmetry** & **Dependence** & **Huristic** & **Category-Passes** & **PreDepRow** & **MIPFocus** & **Method** & **CutPasses** & **BenchDir** & **Tc-D** & **Tc-T** & **Variation** \\ \hline
1 & 3 & 2 & 20 & 2 & & & & & & 0.8 & 0.79 & 1.29\% \\
2 & 6 & 2 & 20 & & 2 & & & & & 1.22 & 1.15 & 5.74\% \\
3 & 8 & 4 & 81 & & & 0 & 1 & 1 & & & 0.89 & 55.03 & 3.92\% \\
4 & 12 & 4 & 81 & & & & & & & & 46.54 & 44.4 & 4.67\% \\
5 & 15 & 4 & 81 & & & & & 0 & & & 15.33 & 15.64 & 1.27\% \\
7 & 20 & 4 & 100 & & & & & & & 1 & & 81.62 & 75.9 & 2.11\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters configurations found for the mathematical model. siderable difference between Tc-D and Tc-T underscores the importance of customization and fine-tuning model parameters to achieve maximum efficiency in solving complex mathematical problems. In this instance, the adjusted parameter Heuristics emerges as a key player in accelerating the solving process. By increasing the use of heuristic methods, which are solving strategies that employ simple and fast rules to generate satisfactory solutions quickly, the model explored and evaluated potential solutions more efficiently. This approach significantly reduced the time required to reach a feasible solution, demonstrating its practicality and effectiveness. Applying heuristics is a valuable tool, particularly in scenarios where speed is critical and some precision can be sacrificed for improved resolution speed. GomoryPasses, another adjusted parameter, controls the amount of Gomory cut passes. These passes are optimization techniques that strengthen the model linear relaxation, leading to faster convergence. Adjusting the amount of these passes can significantly impact the solution efficiency, balancing the need to improve the solution with the additional time required for each cut. PreDepRow, which handles the preprocessing of dependent rows, also fundamentally improved efficiency by reducing problem size and enhancing numerical stability. Effective preprocessing can simplify the model by eliminating redundancies and reducing overall complexity. In the third instance, adjusting PreDepRow significantly reduced computational time, noting the importance of proper model management. These adjustments underscore the importance of tuning parameters to enhance the efficiency solver, enabling faster and more accurate results. ### Benchmarking Results for Optimizers

Table 4 presents the benchmarking results of optimizers across 34 instances. The instances were organized considering the production target, as an increase in computational time (Tc) was noted as this target increased. The results were compared between the default model (PMSPOHO-D), the tuned model (PMSPOHO-T), and the model with the free optimizer (PMSPOHO-CBC). The critical parameters for comparison were Tc and GAP, which measure the complexity model and the efficiency solver. In this study, the GAP value is provided directly by the Gurobi optimizer as part of its solution output. The tuned model (PMSPOHO-T) consistently handles lower computational times than the other models.