A Hybrid Model for Risk-Based Strategic Planning in Open-Pit Mining: Integrating Deterministic, Stochastic, and ISO 31000 Approaches

Abstract

The strategic planning of open-pit mining projects is highly influenced by geological variability, economic fluctuations, and operational uncertainties. Traditional deterministic optimization models often fail to account for these uncertainties, leading to potentially misleading economic evaluations. This paper proposes a hybrid model that integrates deterministic and stochastic optimization methods, following the principles of the ISO 31000 risk management framework, to comprehensively quantify uncertainty through key parameters affecting strategic mine planning. Monte Carlo simulations were applied to generate probability distributions of input parameters, including metal prices, mining and processing costs, and ore grade variability, allowing for a more robust financial assessment. The results demonstrate that while the deterministic approach estimates an NPV of USD 130.8 million, the stochastic model yields an average NPV of USD 155.5 million with a standard deviation of USD 76.5 million, highlighting the significant variability in financial outcomes. Risk assessment using Value at Risk (VaR) and Conditional Value at Risk (CVaR) further quantifies potential financial losses, revealing a 3% probability of project unprofitability. The developed methodology provides a structured approach to integrating uncertainty into mine planning, enabling more reliable economic evaluations and improving decision-making in strategic mining operations.

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. Dimitrakopoulos [4,5] emphasizes the significance of stochastic frameworks for enhancing mine planning amid uncertainty, showcasing their ability to reduce risks. Jelvez et al. [6,7] enhance this by utilizing conditional simulations to better represent grade variability and improve production scheduling. Gilani and Sattarvand [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, Martinez [9] underscores the advantages of employing probabilistic modeling rather than deterministic estimates to achieve more accurate investment evaluations. Furthermore, Marcotte et al. [10] and Acorn et al. [11] concentrate on heuristic and stochastic methodologies to enhance the determination of mine boundaries, thus improving resource management utilization. In the meantime, Koushavand et al. [12] and Sattarvand 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. Baek et al. [14] investigated the effect of price changes on open-pit limits, while Del Castillo and Dimitrakopoulos [15] studied both market and geological factors affecting long-term profitability. The models proposed by Tholana and Musingwini [16] allow for a more comprehensive analysis of the economic value of blocks under uncertainty. Gligoric [17] and Rahmanpour 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 (Environmental, Social, and Governance) considerations [26,27], net present value (NPV) is widely regarded as a key indicator of project success in the mining industry. However, NPV estimation often overlooks natural variations in key parameters. Erdem et al. [28] proposed a methodology for quantifying risk in NPV estimates using Monte Carlo simulations, enhancing the understanding of how uncertainty impacts profitability. Monte Carlo simulations generate distributions of key parameters, allowing for the analysis of various scenarios and the identification of the most unfavorable outcomes [29]. Similarly, many other authors [30,31,32,33,34,35,36] stress the importance of risk assessment in estimating mineral reserves to minimize potential losses and improve the accuracy of long-term planning.

Most studies address only one or two sources of uncertainty in the strategic planning process, which limits the impact of all relevant factors when estimating the economic value of a block in a geological model (the fundamental unit for optimizing mine boundaries and estimating mineral reserves). This approach can result in imprecise estimates and suboptimal decisions in complex mining projects. Therefore, this work aims to develop and validate a hybrid risk assessment model that integrates deterministic methods with stochastic techniques, in accordance with the principles of the ISO 31000-2018 risk management standard [37]. The objective is to thoroughly quantify the uncertainty of key parameters that affect the economic value of a block and enhance risk management in open-pit mining. To achieve this, the study focuses on implementing Monte Carlo simulations for financial risk assessment, integrating Value at Risk (VaR) and Conditional Value at Risk (CVaR) for precise risk quantification, and establishing a structured methodology applicable across different mining conditions.

To implement the proposed hybrid model, Monte Carlo simulations are employed to generate a diverse set of potential scenarios, capturing uncertainties in key parameters such as metal prices, mining and processing costs, and geological variability. These factors directly influence the economic value of individual blocks and, consequently, the financial viability of the entire project. To enhance risk assessment, the model integrates Value at Risk (VaR) and Conditional Value at Risk (CVaR) as key indicators, quantifying potential financial losses and facilitating decision-making based on predefined risk tolerance levels. By systematically addressing uncertainty, this framework enables more reliable and informed strategic planning, ultimately contributing to improved long-term project profitability. In order to emphasize the practical benefits and validity of the presented model, a case study was conducted.

The structure of this paper consists of five chapters, as illustrated in Figure 1. Section 2 offers a comprehensive overview of the methodology, focusing on developing a hybrid model that integrates deterministic and stochastic optimization approaches. Furthermore, it incorporates fundamental principles and key steps of risk assessment in accordance with the ISO 31000-2018 standard, ensuring a structured framework for identifying, analyzing, and evaluating risks in strategic mine planning. In Section 3, we identify sources of uncertainty for a specific case study involving a polymetallic copper, gold, and silver ore deposit and define the values and distributions for the basic input parameters necessary for implementing the model. Section 4 presents the results of our analysis related to the basic economic and technical optimization parameters and risk indicators. Finally, Section 5 discusses the overall results of the analysis, provides the conclusions of the paper, and offers recommendations for future research.

2. Methodology

The process of strategic planning in mining project development occurs under conditions of complex geological and economic uncertainties. These uncertainties make it much more difficult to make key decisions, such as defining open-pit boundaries and estimating mineral reserves. To enhance the management of these challenges, this study proposes a hybrid methodology that integrates deterministic and stochastic approaches for strategic mine planning. The model considers key factors influencing the economic value of individual blocks and the overall project viability. Additionally, it aligns with the ISO 31000-2018 standard for risk management [37], ensuring a structured and systematic approach. Unlike traditional approaches that rely solely on either deterministic or stochastic risk assessments, this methodology integrates both, leveraging their complementary strengths. Deterministic models provide precise optimization results but fail to capture uncertainty, while stochastic models incorporate risk variability but often lack direct decision-making applicability. By combining these techniques, our hybrid model offers a more robust and adaptable framework for evaluating financial and operational risks in strategic mine planning. Furthermore, incorporating the ISO 31000-2018 framework ensures a structured and systematic approach to risk management, improving decision support for mining projects under complex geological and economic conditions.

The methodology consists of the following stages:

1. Establishing the context, where the objectives and scope of analysis are defined;

2. Risk identification, where key parameters are defined, and the uncertainty associated with them is analyzed;

3. Risk analysis, including risk quantification through stochastic methods and pit limits optimization;

4. Risk evaluation, where distributions of economic indicators are analyzed and conclusions are made about the acceptability of risks;

5. Risk treatment, where strategies are developed to manage the identified risks.

This approach enables the identification and quantification of risks within the strategic planning process of open-pit mines while supporting management in making decisions based on data. The phases of this methodology are outlined below, emphasizing practical application in the mining industry.

2.1. Establishing the Context

In the initial phase of risk analysis (Establishing the Context), the primary objective is to define the overall framework for risk assessment and decision-making in open-pit mine planning. Strategic planning in mining projects operates under complex geological, economic, and operational uncertainties, which must be systematically managed to ensure financial and technical feasibility.

This study follows the ISO 31000-2018 risk management framework [37], which emphasizes a structured approach to identifying, analyzing, and treating risks associated with mining operations. In this phase, the broader decision-making criteria and risk acceptance thresholds are defined, ensuring that subsequent risk identification and analysis steps are aligned with industry best practices. To establish a structured risk assessment process, the following key elements are defined:

• Economic indicators as measures of value—most companies have multiple and often conflicting corporate objectives [38,39], requiring a comprehensive evaluation of financial performance. While net present value (NPV) is widely used as the primary metric for project evaluation and strategic decision-making, additional financial indicators, such as the internal rate of return (IRR) and payback period (PBP), are also considered to address varying investment criteria and risk tolerance. These metrics provide a broader perspective on project feasibility, balancing long-term profitability with short-term financial stability. Net present value (NPV) is calculated according to the following formula:

  (1)$NPV=C_0+{\displaystyle \frac{C_1}{{\left(1+r\right)}^1}}+{\displaystyle \frac{C_2}{{\left(1+r\right)}^2}}+\dots +{\displaystyle \frac{C_t}{{\left(1+r\right)}^{\mathrm{t}}}}$

  where

  • C₀—initial cash flow (often negative);

  • C₁—cash flow at time 1;

  • C₂—cash flow at time 2;

  • C_t—cash flow at time t;

  • t—the time period of the investment;

  • r—the discount rate.

• Risk acceptance criteria—different levels of risk will be acceptable to different stakeholders (e.g., management, investors, etc.), depending on their risk tolerance and strategic priorities. The level of risk tolerance can vary significantly between organizations, influenced by factors such as corporate policies, financial stability, industry standards, and management preferences. However, in general, a probability of financial loss below 10% is considered favorable for investment and project development. For the purposes of this case study, a probabilistic approach is adopted, where the acceptable risk level is defined such that the probability of obtaining a negative NPV (P(NPV < 0)) must be less than 10%, ensuring that the likelihood of financial failure remains below a critical threshold.

• Data sources and scope—the data for this analysis were collected from multiple sources to ensure a comprehensive assessment of economic and technical uncertainties. Key inputs include geological resource models, historical and empirical data from comparable open-pit mining operations, industry benchmarks, and technical documentation related to mining and processing technologies. Market data, including metal price trends, were sourced from relevant financial and commodity exchanges. Additionally, techno-economic studies and investment reports provided insights into capital and operational cost structures, while regulatory and industry standards guided the selection of economic and risk assessment parameters.

This phase sets the foundation for Step 2: Risk Identification, where the specific technical and economic parameters influencing mine planning and financial projections are identified.

2.2. Risk Identification

In this phase, the focus is on identifying key economic and technical parameters that influence the financial viability of the mining project. The first step in strategic open-pit mine planning is the optimization of the pit boundaries and the estimation of geological reserves, using various algorithms [40]. This process involves transforming the geological model into an economic block model, where the fundamental unit is the economic value of the block (EVB). Incorporating both economic and technological parameters into the block model, each block is assigned a value based on its potential revenue from metal recovery, adjusted for extraction and processing costs, as well as technical constraints such as mining recovery and processing efficiency. The economic value of the block (EVB) can be expressed as shown in Equation (2):

(2)$EVB=EVB \left(ore\right)-EVB \left(waste\right)$

where

• EVB—the economic value of the block;

• EVB (ore)—the value of the ore in the block;

• EVB (waste)—the cost of production.

A crucial aspect of risk identification is determining how the economic value of each block is assigned in the optimization process. Since the parameters that define EVB are variable (changing throughout the mine’s lifespan and depending on the applied methodology), expressing EVB as a single, fixed value does not fully capture the economic complexity of mining projects. Instead, it is essential to integrate stochastic modeling approaches that account for fluctuations in market conditions, geological properties, and operational costs.

2.2.1. Key Sources of Uncertainty in EVB

Risk identification involves assessing the primary sources of uncertainty that influence strategic decision-making in open-pit mining. These uncertainties are categorized into four main groups [41], as shown in Table 1. Each of these parameters affects the economic value of the block:

1. Geological uncertainty—variability in orebody geometry, ore grade, and specific gravity, affecting recoverable metal content.

2. Market uncertainty—fluctuations in metal prices and exchange rates, directly impacting the revenue calculation.

3. Cost uncertainty—variability in mining, processing, and selling costs, affecting net profitability.

4. Recovery uncertainty—the efficiency of mining and processing operations, determining the actual amount of recovered metal.

These uncertainties play a crucial role in shaping the financial and operational risks associated with mine development. Their proper identification allows for a more structured risk analysis in the following stages of the model.

Each of these uncertainty categories is incorporated into the model through probability distributions, which serve as input parameters for Monte Carlo simulations. This ensures that the full range of possible economic and geological variations is considered when optimizing strategic mine planning.

2.2.2. Stochastic Uncertainty Assessment Using Monte Carlo Simulations

Metropolis and Ulam first introduced the Monte Carlo method [42]. Monte Carlo methods are numerical methods for solving mathematical problems by modeling random variables and statistically evaluating their characteristics.

In general, the Monte Carlo method involves the following 4 steps [43]:

1. The generation of a static model (process functions);

2. Defining input parameters through probability distribution functions;

3. The generation of random variables from the set of distribution of input parameters;

4. Analysis of the results obtained.

Establishing the input data for a Monte Carlo simulation is influenced by numerous criteria and can be approached in various ways. Consequently, extensive market analyses and future demand forecasts are utilized to project metal prices. To assess production costs, industry standards and evaluations from comparable mining operations (of the same deposit type, employing similar technology and equipment, etc.) are referenced. Geological uncertainty is assessed by examining the deterministically developed geological model, the density of exploration activities, permissible assessment errors (according to various standards), and additional factors.

This aligns with the introduction, where numerous authors have proposed different approaches to address different types of uncertainties, including geology, metal prices, production costs, and recovery factors.

After establishing the input parameters, a number of simulations are conducted to generate various geological models and combinations of parameters, resulting in a comprehensive collection of input scenarios. The number of simulations remains unlimited and is primarily determined by the analyst’s resources. An increase in the number of simulations clearly correlates with enhanced model reliability.

A student license for the @Risk software (version 8.2.2) [44] was used to conduct Monte Carlo simulations.

2.3. Risk Analysis

The main goal of the third phase of the model’s development is to quantify the financial and technical risk through scenario-based analysis of the project’s economic indicators and the technical characteristics of the open pit mine.

Building on the previously conducted Monte Carlo simulations for the uncertain parameters, this phase involves performing a set of open-pit mine optimizations, with each optimization corresponding to a Monte Carlo iteration, ensuring a structured risk assessment. The Lerchs–Grossmann (LG) algorithm [45], an industry standard method for pit optimization, was applied using Whittle software, version 4.7.3 [46]. This process generates distributions of key financial indicators, including NPV, cash flow, IRR, and payback period, allowing for a structured comparison between deterministic and stochastic results, and providing insight into the variability of project outcomes under uncertain conditions.

Additionally, in this phase, the obtained distributions of results are analyzed, identifying average values (expected values), determining variability through measures like standard deviation and confidence intervals, and identifying extreme cases (minimum and maximum values).

The results from this phase establish a foundation for the next step, where risk evaluation techniques, including Value at Risk (VaR) and Conditional Value at Risk (CVaR), are applied to further assess the probability and severity of unfavorable financial outcomes. These evaluations guide decision-making regarding project feasibility and risk mitigation strategies.

2.4. Risk Evaluation

The fourth phase of the model focuses on quantifying financial risk acceptability and determining the probability of adverse economic scenarios that could impact the project’s feasibility. Utilizing the results obtained from the risk analysis phase, which produced probability distributions for key economic indicators, this stage evaluates risk thresholds and financial stability through structured quantitative measures.

As previously defined in Section 2.1 (Establishing the Context), an acceptable risk level is set at P(NPV < 0) < 10%, meaning that the probability of obtaining a negative NPV should not exceed 10% to ensure financial feasibility. In this phase, this threshold is quantitatively assessed using Value at Risk (VaR) and Conditional Value at Risk (CVaR), providing a structured basis for risk-informed decision-making.

2.4.1. Financial Risk Evaluation Using VaR and CVaR

To assess financial risk, the model incorporates two key indicators: Value at Risk (VaR) and Conditional Value at Risk (CVaR). These risk metrics quantify the probability and severity of unfavorable economic conditions, particularly in scenarios where Net Present Value (NPV) could become negative.

Value at Risk (VaR) serves as an industry standard tool for assessing risk. It provides a statistical evaluation of value with a specified probability. This concept is grounded in Harry Markowitz’s optimal portfolio theory [47], which posits that variance is a key measure of risk.

VaR is calculated through 4 steps [48]:

1. Determine the time period during which the organization wants to assess the potential loss. This can be for one day, a month, a year, or for the entire life of the mine in our case.

2. Choose the confidence level used in VaR estimation, which typically ranges from 90% to 99%, and can occasionally exceed that. The 99% confidence level is used primarily due to regulatory requirements, as stakeholders are concerned with potential losses from catastrophic scenarios.

3. Establish the probability distributions of outcomes for the portfolio or assets we are considering. Several methods can be used for this, and the distribution will be represented as a curve that often resembles a normal distribution curve, illustrating the likelihood of a loss occurring.

4. Calculate VaR by examining the amount of losses associated with the area below the critical confidence level on the distribution, as illustrated in Figure 2.

However, this risk assessment measure has its drawbacks, as noted in the study by Artzner et al. [49], which points out that VaR does not specify the maximum size of potential losses, nor does it account for the size of less likely losses. Due to these shortcomings, Rockafellar and Uryasev [50] proposed an improved risk assessment measure known as Conditional Value at Risk (CVaR). CVaR roughly (or exactly, under certain conditions) equals the average of the worst-case loss scenarios for a certain percentage [51].

The risk evaluation framework in this study follows three structured steps:

1. Generating probability distributions of net present value (NPV) and other key economic indicators through 100 stochastic pit optimizations.

2. Calculating VaR and CVaR at the 90%, 95%, and 97% confidence levels to establish financial risk thresholds.

3. Interpreting risk thresholds to determine financial stability and assess the necessity of implementing risk mitigation strategies.

The selection of confidence levels follows both industry best practices and project-specific analysis:

• The 90% confidence level (VaR 90%) represents a benchmark threshold, providing a probabilistic estimate of financial loss in 10% of scenarios.

• The 95% confidence level (VaR 95%) identifies more extreme financial risks, indicating potential losses that occur in only 5% of cases.

• The 97% confidence level (VaR 97%) was determined based on 100 pit optimizations, showing that at this threshold, the probability of financial loss is only 3%, defining the project’s financial stability limit.

2.4.2. Application of VaR and CVaR in Strategic Mine Planning

The outcomes of the risk evaluation process serve as a decision-support tool for strategic mine planning. By integrating VaR and CVaR assessments, the model provides quantitative risk thresholds that guide project feasibility analysis and financial decision-making:

• VaR 97% (low-risk scenario): The project is financially stable, with less than a 3% probability of a negative NPV. No immediate intervention is needed.

• VaR 95% (moderate-risk scenario): Some risk exposure exists, and mitigation strategies such as adjusting production schedules or reviewing operating costs may be required.

• VaR 90% (high-risk scenario): The project has a higher likelihood of financial instability, requiring actions such as revising the mining plan or modifying extraction sequences to mitigate risk.

By integrating these risk thresholds into the planning framework, the model ensures that investment decisions are based on quantitative risk assessments, reducing exposure to unexpected financial instability.

2.5. Risk Treatment

Based on the risk evaluation phase, appropriate risk treatment options are defined to ensure project feasibility while maintaining alignment with strategic and operational objectives. The available risk treatment strategies include:

• Risk avoidance: Eliminating high-risk pit limits or excluding blocks with a high probability of financial loss, reducing exposure to extreme economic uncertainty.

• Risk reduction: Adjusting mine boundaries to lessen the impact of high-risk factors, such as enhancing geological research and improving the reliability of geological models or conducting a more thorough analysis of market trends and future demand for raw materials to better predict prices and costs.

• Acceptance of risks: If the probability of obtaining a negative NPV remains below the predefined risk tolerance threshold (P(NPV < 0) < 10%), the project is considered financially viable, justifying investment and project implementation. The final pit limits are determined based on the optimization scenarios that meet this criterion, ensuring that the selected design maintains economic feasibility under uncertainty.

To ensure practical applicability, risk treatment strategies are prioritized based on three key factors:

1. Financial impact: Risks that could lead to significant financial losses or affect project viability are considered high-priority and require immediate mitigation strategies.

2. Probability of occurrence: Risks with a high likelihood of occurring are addressed with proactive measures, while lower-probability risks may be monitored over time.

3. Operational constraints: Some risks may be difficult to mitigate due to technological limitations, environmental regulations, or economic feasibility, influencing their prioritization.

Based on these factors, risk treatment strategies are ranked as follows:

• High-priority risks (e.g., critical financial exposure, high regulatory uncertainty)—require immediate corrective action, such as mine plan adjustments or operational cost reductions.

• Medium-priority risks (e.g., moderate financial impact, manageable technical risks)—addressed through contingency planning and flexible operational strategies.

• Low-priority risks (e.g., minor fluctuations in market conditions)—can be monitored and accepted within predefined risk tolerance limits.

The proposed methodology aligns with the ISO 31000-2018 risk management framework, ensuring a structured and systematic approach to risk assessment and mitigation.

Figure 3 graphically represents the proposed methodology, showing the integration of risk assessment phases, from initial context establishment to final risk treatment decisions.

3. Case Study

To implement the development model, the proposed methodology was applied to a real case study, specifically on a porphyry copper deposit with associated elements of gold and silver. The porphyry copper mineralization is situated in the Western Balkan region, within an area primarily composed of hydrothermally altered andesitic rocks, extending approximately 1.5 km in length and reaching a maximum width of about 600 m (with an average width of 400 to 500 m), oriented toward the east–northeast. The deposit has an oval shape in horizontal projection, elongated in the NNW-SE direction. In vertical projection, it possesses an irregular shape and is located beneath an elevation of −100 m.

Based on the exploration work, a block model of the deposit was created using the Geovia Surpac software package, version 7.7 [52]. The inverse square distance and kriging methods were employed to evaluate the content in the calculation blocks. The model was verified through trend analysis. The model consists of 935,150 ore blocks, each measuring 15 × 15 × 15 m, as shown in Figure 4. The average standard density in the model is 2.5 tonnes/m³, and the average grade is 0.276% Cu. The cut-off value used is 0.15% Cu in the ore.

The low metal grade and potentially high investment costs make this case study an excellent candidate for further comprehensive analyses.

The geological block model provided the foundation for applying the proposed methodology. For confidentiality reasons, other specific aspects of the block model, such as the owner’s identity and precise location, among others, have not been disclosed.

In the applied hybrid model for risk management, the economic parameters in the development of mining projects—such as NPV, cash flow, IRR, and payback period—will serve as the primary measures of value in the decision-making process for this specific case study. The main goal will be to estimate the risk that the mining project may be unprofitable, as well as to assess the potential losses that could occur.

Alongside the deterministic geological block model, a set of essential techno-economic parameters has been established for pit optimization and mineral reserve assessment, utilizing a deterministic (conventional) approach. These parameters are detailed in Table 2.

The ore production capacity essentially represents the maximum capacity of the ore flotation plant, while the waste capacity is equal to the minimal quantities needed to reach the ore capacity (this was determined through several iterations). The slope angles are 37 degrees in all directions.

Mining recovery relies on numerous factors, including geological, technical, economic, and human elements [53,54,55]. For these purposes, a value of 95% was adopted in line with the applied technology. Processing recovery for various types of metals is determined based on the technical characteristics of flotation and metallurgical processing.

Mining and processing costs are determined using historical and empirical data from open-pit mines of comparable size and technology, where copper ore is mined.

Metal prices were adopted as average monthly prices for the period from January 2015 to November 2024 (the last 10 years), with the data taken from the London Metal Exchange (LME) [56].

The discount rate for mining projects typically ranges between 8% and 12% [57]. For this analysis, a value of 10% was chosen.

Finally, the last item used for calculating NPV with a deterministic approach is the initial investment in the project, which amounts to 400 million USD. This figure is defined in line with the available techno-economic documentation, market prices, and historical data. The rationale behind such a substantial investment lies in the requirement for high-capacity equipment, a processing plant, and significant infrastructure work, primarily the relocation of the railway that runs through the center of the deposit, while other supporting facilities and equipment have a lesser impact. The spatial relationship of the railway line to the ore body is illustrated in Figure 5.

For the needs of the stochastic approach, the previously described geological model and the listed parameters provided the foundation for defining probability distributions for each parameter individually. The simulation was conducted for 100 optimizations, requiring the generation of 100 different geological models and 100 values for all parameters. As mentioned earlier, @Risk software (version 8.2.2) and Monte Carlo simulation were utilized to generate probability distributions.

It is crucial to recognize that the impact of geological uncertainty was evaluated by simulating the variability of copper, gold, and silver grades, as well as standard density. The distribution established for each parameter remains consistent across every EVB calculation, with the exception of the distribution for grade and standard density. For these two parameters (considering their impact on EBV uncertainty), the distribution is defined around the estimated value for each block individually.

The definition of probability distributions and their basic parameters for the stochastic approach was established based on the following:

• Geological parameters (grade and standard density)—although international standards such as JORC [58] and NI 43-101 [59] do not define the exact limits of permissible error in estimating mineral resources and reserves, various studies [60,61] suggest that the accuracy of the estimation depends on the degree of exploration and the available data. Following this approach, error intervals of ±15% and ±30% were adopted in this research, which align with the precision levels characteristic of the Measured and Indicated Resources categories (pert distribution). This ensures transparency in presenting geological uncertainty, which is essential for economic evaluations of mining projects.

• Metal prices—the price distributions for copper, gold, and silver were modeled using a beta distribution based on the average prices over the past ten years [56]. However, considering the significant price increase in the last five years and the growing global demand for these metals [62,63], the chosen distributions suggest a higher probability of future prices exceeding the long-term average values. Specifically, the mean historical prices used as reference points are USD 7083/t for copper, USD 51,831/kg for gold, and USD 648/kg for silver. The distributions reflect an asymmetry toward higher price values, indicating that while there is still some probability of prices falling below these averages, the likelihood of sustained higher prices in the future is significantly greater. This approach provides a more realistic projection for economic evaluations in mining projects, accounting for recent market trends and anticipated demand growth.

• Production costs—the base values for mining costs (USD 2.50/t) and processing costs (USD 5.50/t) were determined based on historical production data from similar open-pit mines in the same region and taking into account the type of equipment planned for use. A beta distribution was utilized, reflecting the expectation that future costs will likely surpass the historical average, akin to the approach used for metal prices. This assumption stems from the observed rise in production costs in recent years, influenced by increasing equipment and material costs [64], along with persistent inflation [65]. The distribution applied considers these economic trends, ensuring a more accurate risk assessment in cost projections for the mining project.

• The distributions for mining recovery, processing recovery, selling costs, and investment costs were established based on historical data, anticipated operational efficiency, and the reliability of available documentation.

Probability distributions for parameters affecting EVB are provided in Appendix A.

4. Results

The methodology was systematically applied to the porphyry copper deposit case study by first transforming the geological block model into an economic block model. This transformation involved assigning a financial value to each block using predefined economic parameters, including metal prices, processing costs, and recovery rates. The Monte Carlo simulation was applied by defining probability distributions based on historical data, industry benchmarks, and expert forecasts, ensuring a comprehensive representation of uncertainty. Furthermore, extreme scenarios were defined by analyzing outlier values within the distributions, which allowed for a structured evaluation of potential worst-case financial outcomes.

This structured approach ensures that each step of the methodology is directly reflected in the case study, enhancing the transparency and applicability of the proposed model.

In addition to economic factors, technical indicators are also analyzed, including the total quantities of ore and waste, the likelihood of optimized boundaries occurring in space and sections through the block model with generated mine boundaries. These results provide a more detailed understanding of the variability of the ore body and contribute to decision-making related to determining optimal pit limits.

A special focus is placed on quantifying risk using the indicators Value at Risk (VaR) and Conditional Value at Risk (CVaR), which allows for informed decision-making regarding the project’s viability under uncertain conditions. The results are presented through tables, histogram distributions, cumulative distribution functions, and visualizations of open-pit limits across various scenarios.

4.1. Results of the Deterministic Approach

The deterministic optimization was performed based on a fixed set of techno-economic parameters (Table 2 in Section 3). This approach provides a baseline scenario without considering uncertainty in the key input parameters. The results in Table 3 represent the optimal pit design under these deterministic conditions, showing the ore and waste tonnage, financial indicators, and overall feasibility of the project. Figure 6 shows the optimal pit shell obtained by the deterministic approach.

In this case, the open-pit optimization produces 142.3 million tons of ore and 281.6 million tons of waste material, yielding a net present value (NPV) of USD 130.8 million at a 10% discount rate. The internal rate of return (IRR) is estimated to be 17%, and the investment payback period is 3.84 years.

While this approach offers a foundation for cost-effectiveness analysis, it does not take into account the variability of key factors.

4.2. 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.

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 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 Figure 9 and Figure 10.

The results illustrated demonstrate the probability of block extraction in an open pit based on 100 stochastic optimizations and consider the variability in mine design due to uncertainties in economic and geological parameters. The first figure (plan view) shows the spatial distribution of extraction probability, where the orange and yellow areas indicate high extraction confidence (≥80%), and blue and green areas represent lower probabilities (<50%), reflecting uncertainty in the final boundaries of the pit. The second and third figures (cross-sections) provide a detailed vertical view of the deposit, showcasing the variation in extraction probability with depth. The gradual transition from a zone of high probability in the core to an area of lower probability at the periphery suggests that peripheral blocks are more sensitive to changes in economic and geological conditions. This visualization helps assess the robustness of pit designs, providing guidance for risk-informed mine planning decisions.

4.3. Risk Assessment—VaR and CVaR Analyses

To quantify financial risk, we applied Value at Risk (VaR) and Conditional Value at Risk (CVaR) at multiple confidence levels (90%, 95%, 97%). These indicators provide insights into the probability and magnitude of potential losses under adverse economic conditions. The results shown in Table 5 and in Figure 11 illustrate the cumulative distribution function of NPV outcomes, highlighting the thresholds where risk exposure becomes critical. This analysis directly supports the decision-making framework outlined in Section 2.5 on risk treatment.

The analysis of these data indicates that, at a 90% confidence interval, there is a 10% probability that the NPV will fall below USD 11.9 million, while the average loss in these adverse scenarios is USD 3.1 million. At a 95% confidence level, the probability of obtaining an NPV lower than USD 8.4 million is 5%, with an average loss of USD 14.1 million. The 90% confidence level serves as a moderate risk threshold, while the 95% level is a widely recognized standard for financial risk assessment in the industry.

Additionally, an important threshold was identified at 97% confidence, representing a 3% probability that the project could become unprofitable (NPV = 0). In this scenario, the average loss in the worst 3% of cases is USD 13.2 million, highlighting the need for a structured risk management approach. Given that the acceptable risk threshold is defined as P(NPV < 0) < 10%, these findings confirm that the project remains within an acceptable risk range, but additional mitigation strategies may be considered for scenarios approaching the 97% confidence threshold.

5. Discussion

The analyses conducted indicate the importance of incorporating uncertainty into the strategic planning process for open pit mines. The traditional deterministic approach, while simple and widely used in the mining industry, fails to adequately account for the financial risks associated with a project. The assumption that all key parameters are fixed does not reflect realistic fluctuations in market conditions, operating costs, and geological characteristics. The results of the analysis confirm that the deterministic approach presents positive financial indicators, with an NPV of 130.8 million dollars, an internal rate of return of 17%, and a payback period of 3.84 years. However, this assessment overlooks the potential for adverse outcomes that can significantly impact the project’s profitability.

The stochastic approach allows for a more detailed analysis of various possible scenarios and a more realistic presentation of economic profitability. By considering the uncertainty of the key input parameters through Monte Carlo simulations, the results indicate that the average NPV is USD 155.5 million, which is higher than the deterministic result, though it comes with a standard deviation of USD 76.5 million, reflecting a high degree of variability in financial outcomes. The maximum NPV reaches USD 327.9 million, while the most unfavorable scenarios show losses of USD −17 million, confirming a certain probability that the project may be unprofitable. Additionally, the internal rate of return fluctuates between 8.74% and 25.39%, while the investment return period spans from 2.82 to 5.66 years, indicating significant oscillations in the dynamics of capital return.

Risk assessment using the Value at Risk (VaR) and Conditional Value at Risk (CVaR) methods is particularly important in the analysis. The assessed confidence thresholds allow for a precise evaluation of the likelihood of unfavorable outcomes. It was found that at 90% confidence, there is a 10% chance that the NPV will drop below USD 11.9 million, with an average loss of USD −3.1 million in those adverse scenarios. At 95% confidence, there is a 5% chance that the NPV will be less than USD −8.4 million, and the average loss in these instances is USD −14.1 million. The 97% confidence threshold suggests a 3% probability that the project will be unprofitable (NPV = 0), with an average loss of USD −13.2 million in those situations.

A key aspect of the financial risk evaluation is the definition of an acceptable risk threshold, which in this study is set as P(NPV < 0) < 10%. This means that the probability of obtaining a negative NPV should remain below 10% to justify investment in the project. The results confirm that the project remains within this acceptable risk range, as the worst-case scenario at 97% confidence suggests a 3% probability of financial failure. These findings reinforce the importance of applying structured risk metrics such as VaR and CVaR to determine whether financial risks fall within a manageable threshold before committing to large-scale investments.

Although ISO 31000 is a general risk management framework, its principles of systematic risk identification, analysis, and treatment provide a robust structure that can be adapted to mining projects. This study demonstrates how ISO 31000 can be effectively applied to strategic mine planning by aligning its core risk assessment steps with industry-specific methodologies such as stochastic mine optimization and financial risk quantification. The integration of Monte Carlo simulations with ISO 31000 ensures a structured probabilistic approach, while the use of VaR and CVaR strengthens financial risk assessment, making this framework highly relevant for large-scale capital-intensive mining projects.

6. Conclusions

The proposed hybrid model, which combines deterministic and stochastic methods with risk assessment, represents a significant advance in the analysis of mining projects. Unlike purely deterministic models, which fail to capture parameter variability, and stochastic models, which can provide probabilistic insights that are difficult to translate into concrete, actionable decisions, our hybrid approach balances the strengths of both methodologies. This provides a structured and flexible framework for risk assessment in mining operations, especially in cases with high geological uncertainty and market price fluctuations. Its key advantage lies in its ability to examine a wide range of potential economic outcomes, allowing for a realistic assessment of the profitability of a project under conditions of uncertainty. The incorporation of Monte Carlo simulations allows for the generation of various scenarios that mirror real fluctuations in market and operational conditions, while the application of VaR and CVaR analyses enables precise risk quantification and the definition of project sustainability thresholds. The model also affords the flexibility to adjust mine boundaries based on market trends and shifts in key input parameters, thereby enhancing strategic decision-making.

Comparing the deterministic and stochastic approaches leads to several key conclusions. Deterministic analysis underestimates risk and fails to account for the possibility of negative NPV, while stochastic analysis offers a more realistic view of the project’s economic performance. Although the average NPV in the stochastic model presents better figures, there is substantial variability, which highlights the need for a more thorough risk assessment before making final investment decisions. Mine boundaries also change with input parameters, confirming that flexible exploitation and mine planning strategies are essential to minimize the negative impacts of market and operational fluctuations. It is important to note that the present model does not aim to offer an optimal solution regarding the mine’s boundaries; rather, it serves as a tool that provides decision-makers with clearer insight into the uncertainties and risks during project development.

Although the analysis provides detailed insight into the uncertainty of the mining project, certain limitations should be considered. Monte Carlo simulations were conducted with a limited number of iterations, which is efficient in terms of computational complexity but may be insufficient to accurately capture rare, extreme scenarios. Given its complex nature, it is evident that the analysis requires significant time resources, i.e., more engineer hours, compared to the conventional approach. However, since it defines indicators of significant value for the strategic decision-making process and for projects whose lifespan is often measured in decades with values in the hundreds of millions of dollars, this additional time spent on analysis is certainly justified. Increasing the number of iterations could enhance the stability of the estimate and quantify financial risks more precisely.

While this study primarily focuses on polymetallic deposits (copper, gold, silver), the proposed methodology can also be adapted to other deposit types, such as coal and non-metallic mineral deposits, by adjusting key input parameters. These adaptations would involve modifying geological uncertainty factors, incorporating economic models that account for coal price volatility, and considering regulatory constraints related to environmental policies. Future research should focus on validating the model’s robustness under different mining conditions.

Beyond deposit-type considerations, open-pit mining projects are also exposed to a range of non-financial risks that can significantly impact operational stability and long-term sustainability. Technical risks, such as potential equipment failures and geotechnical stability issues, present challenges that must be accounted for in strategic mine planning. Additionally, social risks stemming from regulatory changes, labor market conditions, and community relations can influence project feasibility. Environmental constraints, including emissions regulations and water management policies, further complicate long-term sustainability. Future research should explore the integration of these risk categories into the model to develop a more comprehensive and adaptable risk assessment framework, ensuring that decision-makers are equipped with a holistic understanding of uncertainties beyond financial risks.

Author Contributions

Conceptualization, P.M. and D.S. (Dejan Stevanovic); methodology P.M., D.S. (Dejan Stevanovic), and B.K.; writing—review and editing, P.M., D.S. (Dejan Stevanovic), B.K., D.S. (Dragana Slavkovic), and D.K.; supervision, D.S. (Dejan Stevanovic) and B.K. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Acknowledgments

The authors express their gratitude to the Ministry of Science, Technological Development and Innovation of the Republic of Serbia for supporting scientific research, which is essential for the advancement of a knowledge-based society.

Conflicts of Interest

Author Dragana Slavkovic was employed by the Institute of Transportation CIP Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be constructed as a potential conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Paper structure.

Figure 2. The VaR and CVaR for a 95% confidence level.

Figure 3. The structure of the methodological approach.

Figure 4. Block model of the deposit with constrained ore block.

Figure 5. The spatial position of the railway line and ore body.

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.

Figure 8. Cash flow for different scenarios.

Figure 9. Probability of block extraction—plan view.

Figure 10. Probability of block extraction—cross-section view.

Figure 11. Cumulative distribution function of project NPV for various confidence intervals: (a) 90%; (b) 95%; (c) 97%.

Table 1

Sources of uncertainty affecting the economic value of the block.

Geological Uncertainty	Market Uncertainty	Costs Uncertainty	Recovery Uncertainty
Geometry of orebody	Price of product	Mining costs	Mining recovery
Units of product—grade	Exchange rate	Processing costs	Processing recovery
Specific gravity		Selling costs

Table 2

Techno-economic parameters for deterministic pit optimization.

Parameter	Value
Annual capacity for ore	10,000,000 t/year
Annual capacity for waste	min for 10,000,000 t of ore
Slope angles	37°
Mining recovery	95%
Processing recovery	81.5% for Cu, 46.5% for Au, 18.6% for Ag
Selling costs	630.4 USD/t—Cu, 160 USD/kg—Au, 15 USD/kg—Ag
Production costs:	2.5 USD/t—Mining, 5.5 USD/t—Processing
Metal price	7083 USD/t—Cu, 51,831 USD/kg—Au, 648 USD/kg—Ag
Discount rate	10%
Cut-off	0.15%
Initial investment costs	400,000,000 USD

Table 3

The numerical results of the deterministic optimization.

Key Metric	Value
Ore tonnage	142,338,255 t
Waste tonnage	281,561,745 t
NPV @10% discount rate	USD 130,751,000
IRR	17%
Payback period	3.84 years

Table 4

Results of stochastic optimization.

Key Metric	Average Value	Standard Deviation	Minimum	Maximum
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

Table 5

VaR and CvaR for different confidence intervals.

Confidence Interval	VaR (NPV Millions of USD)	CvaR (NPV Millions of USD)
90%	11.9	−3.1
95%	−8.4	−14.1
97%	0	−13.2

Appendix A

Distributions for a parameter affecting the economic value of a block (EVB) in the optimization of open-pit mines.

Table A1

Probability distributions for parameters affecting EVB.

Cu grade (%)—pert distribution	Au grade (%)—pert distribution
[Image omitted. Please see PDF.]Min = 0.2295, Most likely = 0.2700, Max = 0.3105	[Image omitted. Please see PDF.]Min = 0.01275, Most likely = 0.0150, Max = 0.01725
Ag grade (%)—pert distribution	Standard density (t/m3)—pert distribution
[Image omitted. Please see PDF.]Min = 0.07225, Most likely = 0.0850, Max = 0.9775	[Image omitted. Please see PDF.]Min = 2.40, Most likely = 2.5, Max = 2.6
Mining costs (USD/t)—beta distribution	Processing costs (USD/t)—beta distribution
[Image omitted. Please see PDF.]Min = 2.2, Max = 3.0, α1 = 2.2, α2 = 1.8	[Image omitted. Please see PDF.]Min = 5.0, Max = 7.0, α1 = 2.5, α2 = 2.0
Price Cu (USD/t)—beta distribution	Price Ag (USD/kg)—beta distribution
[Image omitted. Please see PDF.]Min = 6500, Max = 8300, α1 = 2.5, α2 = 2.0	[Image omitted. Please see PDF.]Min = 550, Max = 800, α1 = 2.5, α2 = 2.0
Price Au (USD/kg)—beta distribution	Sell cost Cu (USD/t)—pert distribution
[Image omitted. Please see PDF.]Min = 48,000, Max = 62,000, α1 = 2.5, α2 = 2.0	[Image omitted. Please see PDF.]Min = 600, Most likely = 630, Max = 660
Sell cost Ag (USD/kg)—pert distribution	Sell cost Au (USD/kg)—pert distribution
[Image omitted. Please see PDF.]Min = 10, Most likely = 15, Max = 20	[Image omitted. Please see PDF.]Min = 145, Most likely = 160, Max = 175
Mining recovery (%)—normal distribution	Processing recovery Cu (%)—normal distribution
[Image omitted. Please see PDF.]Mean = 0.95, Std dev = 0.02	[Image omitted. Please see PDF.]Mean = 0.8148, Std dev = 0.018
Processing recovery Ag (%)—normal distribution	Processing recovery Au (%)—normal distribution
[Image omitted. Please see PDF.]Mean = 0.186, Std dev = 0.015	[Image omitted. Please see PDF.]Mean = 0.465, Std dev = 0.018
Investment cost (millions of USD)—pert distribution
[Image omitted. Please see PDF.]Min = 340, Most likely = 400, Max = 440

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