## 4 Results
### Results of the Stochastic Approach  
In contrast to the deterministic approach, the stochastic optimization incorporated uncertainty through Monte Carlo simulations. A total of 100 optimizations were conducted, each with varying input parameters drawn from probability distributions defined in Section 3 (shown in Appendix A). This process generated a range of potential outcomes for key indicators, allowing for a probabilistic assessment of project viability. The resulting distributions for NPV, IRR, and payback period are shown in Figure 7, providing a more comprehensive view of financial risk in the project. Figure 6: Optimal pit shell with ore block representation—deterministic approach. Figure 7: Distributions for key economic indicators of the mining project: (**a**) NPV; (**b**) IRR; (**c**) payback period. Table 4 presents the numerical results of stochastic optimization, including economic and technical indicators, along with basic statistics indicators. In addition to the distributions and numerical results, cash flow was also generated for all scenarios and is shown in Figure 8.  
A stochastic approach to open-pit optimization quantifies variability in key economic and technical indicators, offering a broader view of potential project outcomes in uncertain conditions. The results indicate that the average net present value (NPV) is USD 155.5 million, exceeding the deterministic result of USD 130.8 million, but with a significant standard deviation of USD 76.5 million, suggesting a high degree of uncertainty. The lowest recorded NPV is USD \(-\)17 million, while in the most favorable scenarios, it reaches USD 327.9 million, confirming a wide range of possible financial outcomes. The internal rate of return (IRR) averages 17.93%, but in extreme cases, it can drop below 10%. The investment payback period varies from 2.82 to 5.66 years, with an average of 3.85 years. Additionally, the cash flow analysis reveals significant fluctuations throughout the project's lifespan, with some scenarios indicating prolonged periods of financial instability. These findings clearly demonstrate that the deterministic approach fails to capture the full spectrum of potential economic outcomes, while the stochastic analysis offers a more realistic insight into the potential variations and risks of the project. In addition to the probabilistic distributions of key economic indicators, the spatial variability of block extraction probabilities offers further insight into the effects of geological and economic uncertainties on mine planning. The probability-based pit design, derived from 100 stochastic optimizations, emphasizes the stability of core mining areas while pinpointing marginal zones where extraction is highly sensitive to changes in input parameters. These findings serve as a crucial supplement to the financial risk assessment, allowing for a more comprehensive evaluation of project viability by integrating economic \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Key Metric** & **Average** & **Standard** & **Minimum** & **Maximum** \\  & **Value** & **Deviation** & & \\ \hline Ore tonnage & 130,214,597 & 16,847,183 & 89,685,220 & 162,793,719 \\ Waste tonnage & 280,104,403 & 68,480,510 & 139,139,780 & 435,053,781 \\ NPV @10\% DC (USD) & 155,537,909 & 76,537,723 & \(-\)17,043,403 & 327,950,767 \\ IRR (\%) & 17.93 & 3.43 & 8.74 & 25.39 \\ Payback period & & & & \\ (years) & 3.85 & 0.58 & 2.82 & 5.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of stochastic optimization. Figure 8: Cash flow for different scenarios. variability with geological constraints. The visualized probability distributions aid strategic decision-making by marking areas of high certainty for extraction and zones that require further evaluation to reduce risk in long-term planning, as illustrated in Figures 9 and 10.