## 1 Introduction  
The strategic planning of open-pit mining operations is essential for the mining industry, as it maximizes profitability and sustainability in the resource extraction process. However, this process is becoming increasingly complex due to various unpredictable factors, including geological uncertainty, market price instability, and rising operating costs. Although widely applied, traditional deterministic models rely on assumptions about the fixed values of key parameters. These models often ignore the variability and risks associated with geological characteristics, cost estimations, and market conditions, leading to inaccurate estimates and misleading decisions [1, 2, 3]. The complexity of ore body geometry, variability in metal content, and limited exploration data present significant challenges for accurate modeling and optimization in open-pit mining. Geological uncertainty greatly impacts the open-pit boundary and the assessment of deposit value. This requires the use of stochastic methods to improve evaluation accuracy decision-making. [PERSON] [4; 5] emphasizes the significance of stochastic frameworks for enhancing mine planning amid uncertainty, showcasing their ability to reduce risks. [PERSON] et al. [6; 7] enhance this by utilizing conditional simulations to better represent grade variability and improve production scheduling. [PERSON] and [PERSON] [8] present ant colony optimization as a method to minimize deviations from the planned output and enhance operational efficiency and stability. In addressing the issue of financial risk, [PERSON] [9] underscores the advantages of employing probabilistic modeling rather than deterministic estimates to achieve more accurate investment evaluations. Furthermore, [PERSON] et al. [10] and [PERSON] et al. [11] concentrate on heuristic and stochastic methodologies to enhance the determination of mine boundaries, thus improving resource management utilization. In the meantime, [PERSON] et al. [12] and [PERSON] et al. [13] illustrate how geological uncertainty pulls through all stages of mine planning, thereby influencing the final extraction strategies and the long-term economic feasibility. Collectively, these studies affirm that incorporating geological uncertainty into optimization models enhances risk-adjusted economic evaluations, improves operational resilience, and contributes significantly to the long-term sustainability of mining projects. Besides the geological challenges, the volatility of metal prices significantly impacts the economic indicators of mining projects. Fluctuations in global markets, driven by changes in supply and demand, geopolitical tensions, and macroeconomic conditions, necessitate models that account for market risks. [PERSON] et al. [14] investigated the effect of price changes on open-pit limits, while [PERSON] and [PERSON] [15] studied both market and geological factors affecting long-term profitability. The models proposed by [PERSON] and [PERSON] [16] allow for a more comprehensive analysis of the economic value of blocks under uncertainty. [PERSON] [17] and [PERSON] et al. [18] also examine the integration of market risks and stochastic methods for improved planning accuracy. These strategies emphasize the necessity of incorporating price volatility into planning to lessen the impact of market uncertainties on economic outcomes. In addition to the two previously mentioned factors--geological and price uncertainty--the authors also investigated the impact of various techno-economic factors on the optimization of open-pit boundaries and the estimation of mineral reserves. This includes parameters such as operating costs [19; 20], deposit utilization and processing recovery [21; 22; 23; 24], and the time value of money [25]. Although the successful implementation of mining projects is influenced by many factors, including economic, technical, and ESG (Environment, Social, and Governance) considerations [26; 27], net present value (NPV) is widely regarded as a key indicator of project success in the mining industry.