Optimization of ore production scheduling strategy using NSGA-II-GRA in open-pit mining

Abstract

To address the challenges posed by ore grade fluctuations, extraction cost variations, and market price instability in open-pit ore production strategies, an optimization model integrating a solution algorithm integrating dynamic simulation is proposed. First, Monte Carlo simulation is used to generate random variables, simulating ore grade and extraction cost fluctuations at different extraction points. Then, the Non-dominated Sorting Genetic Algorithm II with Grey Relational Analysis (NSGA-II-GRA) is employed for an initial solution to obtain the preliminary Pareto-optimal solution set. Furthermore, a market price fluctuation constraint equation is introduced during the optimization process to conduct a second iterative optimization of the initial solution set, generating optimized plans under different price fluctuation intervals. Finally, by integrating subjective and objective weights, the weighted grey relational analysis method is applied to select the optimal Ore Production Scheduling Strategy. The optimization results indicate that, while achieving the optimization objectives, calcium oxide grade and production target achievement rate increased by 7% and 7.17%, respectively, while ore output increased by 16.7% and 20.5% under different price fluctuation intervals.

Introduction

In the mining production process, formulating an ore extraction plan that aligns with corporate development strategies is crucial, as it not only affects economic benefits but also determines the long-term sustainability of the mine. The extraction strategy must be flexible enough to dynamically adjust production plans under varying conditions to maximize operational efficiency. Mining enterprises typically rely on manual compilation methods to formulate extraction strategies. However, this process is complex and influenced by various uncertainties, such as ore grade at each mining site, extraction costs, market prices, and the operational environment of the mine. Manual compilation cannot fully account for these uncertainties; thus, relying solely on manually designed extraction strategies in open-pit mines has become outdated^1,2,3,4,5. Such strategies, based on deterministic data indicators, are reactive in responding to real-world production conditions⁶. The research capabilities of a country are closely related to its national strength and economic background⁷. Consequently, countries such as the United States and Australia began studying uncertainty issues earlier than China.

The impact of geological uncertainty on mine planning has received extensive attention, leading to the development of various optimization methods, including geological modeling optimization, uncertainty quantification, and dynamic scheduling⁸. In 1998, Dimitrakopoulos et al. proposed a solution for addressing geological uncertainties in mine planning by integrating geological simulation with stochastic optimization algorithms to determine the optimal mining strategy⁹. That same year, Sevim and Lei combined heuristic algorithms with dynamic programming to identify key variables in open-pit mining strategies for hard rock deposits¹⁰. Benndorf J introduced a mine production scheduling approach based on stochastic integer programming (SIP), which considers deviations in ore quantity and quality due to geological uncertainty. The SIP model integrates equally probable orebody scenarios to control the risks associated with production target deviations over long-term operations. Results indicated that compared to conventional planning, stochastically generated mine plans effectively mitigate project risks and enhance project value¹¹. Nelis incorporated uncertainty from drill hole data into a mixed-integer programming model, aiding planners in determining optimal mining cut-offs and improving short-term operational planning and profitability¹². Groeneveld et al. developed a hybrid model that combines mixed-integer programming and Monte Carlo simulation to determine optimal strategies for mine design under stochastic conditions. This model encompasses multiple design decisions (mine layout, storage, equipment) and various uncertainties (operating costs, prices, and recovery rates)¹³. Tolouei and Moosavi proposed two hybrid algorithms: one integrating the Lagrangian relaxation method with particle swarm optimization, and another combining the Lagrangian relaxation method with the Bat Algorithm, both aimed at optimizing long-term open-pit production scheduling under ore grade uncertainty¹⁴. Danish et al. incorporated equally probable geological simulations into the optimization process, effectively reducing production deviation risks¹⁵. Deressa et al. developed an uncertainty management and economic optimization approach by integrating geological modeling and pit optimization techniques¹⁶. Seyyed-Omid proposed a stochastic integer programming (SIP) model that integrates equally probable scenarios, risk block models, and E-Type block models to address the challenge of achieving production targets under geological uncertainty, demonstrating its effectiveness¹⁷. Additionally, numerous researchers have employed intelligent optimization algorithms, dynamic programming, and simulation-based approaches to solve mine design and production scheduling problems^{18,19,20,21,22,23,24,25}.

Although the impact of geological uncertainty on mine planning has been extensively studied, considering only geological factors is insufficient to ensure the economic feasibility of production plans. Consequently, research on economic uncertainty has gained increasing attention in recent years. Baek et al. proposed a novel method to quantitatively assess the uncertainty of open-pit mine optimization results caused by fluctuations in mineral prices. By generating multiple mineral price scenarios using Monte Carlo simulation and historical price data, a probabilistic model was developed to represent uncertainty in open-pit mining optimization²⁶. Dehghani et al. demonstrated using binary tree techniques that when price and cost uncertainties are considered, they significantly impact the net present value (NPV) of open-pit mining projects²⁷. In mine planning projects, Rahmanpour and Osanloo developed an algorithm that incorporates multiple risk values associated with ore price uncertainty to determine the least-risk mining sequence²⁸. Kasa et al. employed dynamic (variable) metallurgical recovery rates parameters for NPV estimation, effectively improving optimization success rates and significantly reducing economic uncertainty in mine optimization²⁹.

In summary, current research on uncertain open-pit production planning has primarily focused on geological uncertainty, while studies on economic uncertainty remain relatively limited. Existing studies mainly address market changes by optimizing ore grade or adopting static pricing strategies but lack optimization methods that account for dynamic market price fluctuations. This study constructs a multi-objective optimization model integrating Monte Carlo simulation and NSGA-II for a large-scale limestone open-pit mine. Monte Carlo simulation is used not only to generate uncertain scenarios for ore grade and market prices but also to formulate optimization constraints, ensuring solution feasibility under various uncertainty conditions. During optimization, NSGA-II is first used to obtain an initial solution set, followed by the introduction of a market price fluctuation constraint for a secondary iterative optimization, enabling the extraction strategy to dynamically adjust based on market changes. Compared to traditional static scheduling methods, this optimization framework considers the impact of market uncertainty on mining operations and employs weighted grey relational analysis to select the optimal solution. This approach ensures that the optimization results balance cost control and economic profitability, ultimately enhancing the operational robustness of open-pit mines under market fluctuations and providing scientific decision-making support for complex market environments.

Construction of a ore production scheduling strategy planning model considering uncertainty

In open-pit mining projects, decision-making complexity often arises from various uncertainties, with project risks mainly stemming from these uncertainties, as they directly impact goal setting and achievement. Through studies on long-term Ore Production Scheduling Strategy models, Dimitrakopoulos³⁰ further categorized the uncertainties in mining projects, which include ore grade and quantity uncertainty, technical uncertainty (e.g., slope restrictions and drilling capacity during mining), economic uncertainty (e.g., capital costs, operating costs, and product prices), and time uncertainty (e.g., truck and loading times). Among these, the uncertainties in ore grade, operating costs, and product prices have a particularly significant impact on the project.

Objective functions

The expressions for the objective functions of extraction cost and ore market value are shown in Eqs. (1) and (2).

(1) Objective Function for Minimizing extraction costs: Due to varying distances from each extraction point to the dumping point, transportation costs differ. Additionally, geological factors such as geological structure complexity, orebody depth, and rock hardness lead to variations in extraction costs at different extraction points. Therefore, minimizing extraction costs is a reasonable objective. Since extraction costs at each extraction point are random variables that fluctuate under different conditions, effectively managing these uncertainties is crucial during planning.

$$\begin{aligned} \min f_1=E\sum _{i=1}^{n}C_ix_i \end{aligned}$$

(1)

where E is the expectation operator, $x_i$ represents the quarterly Ore Production Scheduling Strategy for each platform, and $C_i$ is the random variable representing the extraction cost of the ore.

(2) Objective Function for Maximizing Ore Market Value: To enhance the economic value and profitability of the mine, ore extraction strategies for each extraction point must be carefully formulated, considering market demand and price fluctuations. This ensures that production decisions are financially optimal, dynamically adjusting mining strategies to respond flexibly to market changes.

$$\begin{aligned} \max f_2=E\gamma P_0\sum _{i=1}^{n}G_ix_i \end{aligned}$$

(2)

where $P_0$ represents the initial market price of the ore, $G_i$ is the random variable representing the ore grade of calcium oxide, and $\gamma$ is the rate of market price fluctuation.

Constraints

The open pit mine exit strategy is influenced by factors such as the production principles of the quarry, the capacity of the equipment and the market price of the ore. According to the actual situation of the mine constraints are shown in formula (3) to formula (9).

(1) Ore Grade Constraint Equation: Due to differences in ore grades at each extraction point, ore grade must be constrained to meet beneficiation requirements and support sustainable development. This aims to control fluctuations within a defined range while maintaining magnesium oxide content within a reasonable range, as excess magnesium oxide in limestone negatively impacts its performance and economic value.

$$\begin{aligned} G_{\min }\le \frac{\sum _{i=1}^{n}G_ix_i}{\sum _{i=1}^{n}x_i}\le G_{\max } \end{aligned}$$

(3)

$$\begin{aligned} V_{\min }\le \frac{\sum _{i=1}^{n}V_ix_i}{\sum _{i=1}^{n}x_i}\le V_{\max } \end{aligned}$$

(4)

where $G_{\min }$ and $G_{\max }$ represent the lower and upper bounds of calcium oxide ore grade, respectively, and $V_{\min }$ and $V_{\max }$ represent the lower and upper bounds of magnesium oxide content at each platform.

(2) Ore Output Constraint Equation: The ore production volume during the planning period must meet the requirements of the long-term Ore Production Scheduling Strategy, meaning that the total output must satisfy the production volume target for the planning period.

$$\begin{aligned} \sum _{i=1}^{n}x_i\eta _i\le D_{T,\max } \end{aligned}$$

(5)

where $D_{T,\max }$ represents the upper limit of the total output of all platforms and $\eta _i$ represents the completion rate of the ore extraction plan for each platform.

(3) Production Target Achievement Rate Constraint: This constraint regulates the Production Target Achievement Rate (PTAR), an operational indicator that measures the degree of execution of the planned ore extraction volume, keeping it within a reasonable range to assess execution efficiency. It ensures flexibility in the production process, enabling adaptation to uncertainties while maintaining acceptable target fulfillment levels, ultimately optimizing resource allocation and overall efficiency.

$$\begin{aligned} C_{\min }\le \frac{\sum _{i=1}^{n}x_i\eta _i}{\sum _{i=1}^{n}x_i}\le C_{\max } \end{aligned}$$

(6)

where $C_{\min }$ represents the upper limit of the production target achievement rate and $C_{\max }$ represents the lower limit of the completion rate of the production target achievement rate.

(4) Mining Equipment Capacity Constraint Equation: Each extraction point is constrained by the performance limits of the equipment used. When formulating the ore extraction strategy, these limitations must be considered to ensure that production tasks are reasonably assigned without exceeding the equipment’s capacity.

$$\begin{aligned} N_sW_s\alpha \le \sum _{i=1}^{n}x_i\le N_sW_s \end{aligned}$$

(7)

$$\begin{aligned} N_tW_t\beta \le \sum _{i=1}^{n}x_i\le N_tW_t \end{aligned}$$

(8)

$N_s$ represents the number of excavation equipment, and $W_s$ represents the rated excavation capacity of the equipment within the planned period; $N_t$ represents the number of transportation equipment, and $W_t$ represents the rated transportation capacity of the equipment within the planned period, and $\alpha$ and $\beta$ are the actual working coefficients of the equipment.

(5) Ore Market Price Fluctuation Constraint Equation: An exponential decay function is adopted to model the effect of market price fluctuations on production adjustments, allowing the adjustment magnitude to be responsive to price volatility while maintaining operational stability. When the market price of ore decreases, to maintain mine profitability, production can be increased to spread out fixed costs. Conversely, when prices rise, production should be expanded within reasonable limits to maximize economic benefits during periods of high prices. This strategy allows for capturing additional profit from higher prices by adjusting production volume accordingly. As the magnitude of price changes increases, production volume becomes more sensitive to market prices.

$$\begin{aligned} x_i(t+1)=x_i(t)+\alpha _i\left( 1-\exp ^{-\frac{|\vartriangle P|}{\Gamma }}\right) \end{aligned}$$

(9)

where $x_i(t+1)$ represents the planned output of each platform after being affected by market price fluctuations, $x_i(t)$ represents the planned output of each platform unaffected by market price fluctuations, $\alpha _i$ represents the sensitivity coefficient of $x_i(t)$ to changes in market price, $\vartriangle P$ represents the rate of change of the current market price relative to the initial price P0, and $\Gamma$ represents the lag effect coefficient of market price changes.

Handling of random variables

Monte Carlo simulation estimates uncertainty and risk by generating a large number of random samples and performing statistical analysis. In the uncertainty-based multi-objective planning model, ore grade and extraction cost are two key uncertain factors. To dynamically simulate the fluctuations of these indicators, this study employs the linear congruential method within Monte Carlo simulation to generate uniformly distributed random numbers³¹. The recursive formula is presented as Eq. (10):

$$\begin{aligned} x_{t+1}=(aX_t+c)\quad \textrm{mod} \quad m \end{aligned}$$

(10)

where the initial value $X_0$ is called the seed; a, c, and m (all positive integers) are referred to as the multiplier, increment, and modulus, respectively; and $X_n$ represents the n-th pseudo-random number. Equation (10) employs the linear congruential method to generate a pseudo-random number sequence, where parameter settings influence the periodicity and uniformity of the distribution. This ensures a sufficiently long random number cycle, thereby simulating the periodic uncertainty in open-pit mining production, such as fluctuations in ore grade persisting over multiple mining cycles. By performing normalization, a uniformly distributed random number sequence can be obtained Eq. (11):

$$\begin{aligned} U_t=\frac{X_n}{m} \end{aligned}$$

(11)

The generated uniform random numbers approximate a true sequence of identically distributed random variables. The sequence $X_0, X_1, X_2, \ldots$ will repeat after a maximum of m steps, making the sequence periodic, with a period not exceeding m. To generate random samples of ore grade and other uncertain factors that conform to actual distributions, Eq. (11) normalizes the pseudo-random number $X_n$ into a uniform random number $U_t$ in the (0,1) interval.

Handling of the expectation in objective functions

The expectation operator in the objective function is calculated using the Monte Carlo integration method³², which estimates the expected value of a function by employing random sampling. This approach approximates the value of the integral by generating a large number of random sample points. For each pseudo-random number sample generated, the value of the objective function is computed. The function values at these sample points are then used to estimate the average or expected value of the function over a given range. The main integration formula is represented in formula (12).

$$\begin{aligned} I\approx \frac{b-a}{N}\sum _{i=1}^{N}f(x_i) \end{aligned}$$

(12)

where I represents the integral value and N denotes the number of samples (i.e., the number of random sample points generated). The larger N is, the higher the estimation accuracy typically becomes. Furthermore, $f(x_i)$ represents the value of the function be-ing integrated at the i-th sample point $x_i$, and $x_i$ refers to the i-th random sample point, which is usually generated uniformly or according to a specific distribution within the integration interval. The Monte Carlo integration in Eq. (12) is used to estimate the expected value of the objective function. For the ore market value objective $f_2$, the integral value I represents the expected revenue of the mine for a future cycle, considering fluctuations in ore grade and market price. By increasing the sample size N, the estimation error can be significantly reduced, with a deviation from the theoretical value of less than 1%.

Based on the Law of Large Numbers and the Central Limit Theorem it is ensured that, as the number of samples N increases, the estimated value I converges to the true value. In mining projects, Monte Carlo integration is used to simulate uncertainties in ore grade and market prices, optimizing mining strategies to maximize profit and minimize costs.

Solution algorithm

The production planning model is optimized using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) combined with the weighted grey relational analysis method. Figure 1 illustrates the complete process of this algorithm. Based on the initial NSGA-II solution, an ore market price fluctuation constraint equation is added for a second solution iteration on the solution set. The grey relational analysis method then utilizes the combined weights derived from the entropy weight method and analytic hierarchy process (AHP) to filter the final solution set obtained from the second iteration, selecting the solution with the highest relational degree as the optimal solution.

Fig. 1

NSGA-II weighted grey relational analysis algorithm flowchart.

Non-dominated sorting genetic algorithm

The NSGA algorithm is a multi-objective optimization algorithm based on the Pareto optimal solution, which was proposed by Srinivas and Deb in 1995. The advantage of NSGA-II over other algorithms is that it can search for solutions in multiple directions³³. In addition, it does not depend on the continuity and concavity of the objective function³⁴. The optimization process consists of two stages: the initial iteration and the second iteration. The initial iteration primarily obtains the basic optimized solution set, while the second iteration further incorporates market price fluctuation constraints to enhance the model’s responsiveness to economic uncertainties.

During the initial iteration stage, Monte Carlo simulation is used to quantify the uncertainty of ore grade and mining costs, simulating the quarterly ore output fluctuations across three mining sites. An initial population of 300 individuals is generated, each corresponding to a different quarterly ore extraction plan. The objective function evaluation is performed through 1,000 random samples to estimate the average mining costs and market value under uncertain conditions. A penalty function is applied to enforce 14 constraints, including mining capacity, ore grade, and equipment limitations.In the non-dominated sorting phase, NSGA-II classifies the solution set by identifying individuals that are not dominated by any other solutions as the first Pareto front. The remaining solutions are then iteratively sorted into subsequent Pareto fronts until all individuals are categorized. To ensure uniform solution distribution, crowding distance is computed using Eq. (13) to quantify the isolation degree of each solution in the objective space.

$$\begin{aligned} d_i=\sum _{t=1}^{n}\left| \frac{f_t(i+1)-f_t(i-1)}{f_{t}^{\max }-f_t^{\min }}\right| \end{aligned}$$

(13)

where $f_t$ is the optimization objective, in order of the size of the objective value; d is the crowding distance; $f_t(i+1)$, $f_t(i-1)$ is the value of the objective function of the neighboring individuals of individual i; and $f_t^{\max }$ and $f_t^{ \min }$ are the maximum and minimum values of the objective function, respectively. The crowding distance $d_i$ measures the degree of isolation of solution i in the objective space. A larger divalue indicates that the neighboring solutions ($i+1$ and $i-1$) are farther apart, meaning the solution is located in a sparse region and contributes more to maintaining the diversity of the Pareto front. When two solutions have similar costs but significantly different market values, the crowding distance helps identify and prioritize the retention of such solutions, preventing premature convergence to a local optimum. In the genetic operations phase, tournament selection is used to select high-fitness individuals from the parent population, followed by crossover and mutation operations to generate the offspring population. The parent and offspring populations are then merged, and the 300 best individuals are retained. The algorithm iterates for 500 generations to obtain the preliminary Pareto optimal solution set.

Once non-dominated sorting and crowding distance computation are completed, crowding distance comparison is further employed to filter optimized solutions and maintain Pareto front diversity. Each individual is ranked based on non-dominated sorting and crowding distance to ensure a well-distributed solution set. First, individuals from lower-ranking fronts are prioritized. If two individuals belong to the same front, the one with a larger crowding distance is selected. The selection rule is defined as: Define $i \prec n_j$ if $i_{\textrm{rank}}<i_{\textrm{distance}}$, or if $i_{\textrm{rank}}=i_{\textrm{distance}}$ and $i_{\textrm{rank}}>i_{\textrm{distance}}$. That is, solutions with larger crowding distances are prioritized to ensure uniform distribution in the objective space and avoid excessive solution concentration in local regions. If two solutions have equal crowding distances, one is randomly selected to further maintain diversity. Finally, NSGA-II completes the selection of optimal solutions by integrating crowding distance sorting and tournament selection, ensuring stability and adaptability in different market fluctuation scenarios.

During the second iteration, Eq. (9) is introduced to incorporate market price fluctuation constraints, ensuring that the optimized solutions remain adaptable to dynamic market conditions. The quarterly ore output is dynamically adjusted based on the market price variation rate to align with evolving market conditions. NSGA-II is then employed to re-optimize the solutions obtained from the first iteration.In the solution selection phase, a similarity merging strategy is used to eliminate redundant solutions where the quarterly ore output difference is less than 5000 tons. Ultimately, 21 sets of Pareto optimal solutions are obtained, each corresponding to different market price scenarios.

Weighted grey relational analysis

Analytic hierarchy process (AHP)

The analytic hierarchy process (AHP), proposed by Professor Saaty in the early 1970s, is a hierarchical weight decision analysis method³⁵. It represents a complex problem as an ordered hierarchical structure, and decision options are ranked based on human judgment. The specific steps are as follows: First, the complex decision problem is broken down into different levels, including the goal level, criterion level, and option level. Next, within each level, pairwise comparisons are made between elements of the same level to assess their relative importance concerning an element of the upper level, using a scale of 1 to 9 to represent the degree of relative importance. Then, the weight of each factor is calculated by determining the eigenvector of the judgment matrix and normalizing it. A consistency check is performed to ensure that the matrix’s consistency is reasonable. If the consistency ratio (CR) is less than 0.1, the matrix is considered consistent. Finally, the weights of each level are combined to derive the overall weight of each option. A consistency check is also necessary at this stage to ensure that the overall consistency is reasonable. Its consistency indicator CI and consistency ratio CR are calculated as shown in Eqs. (14) and (15).

$$\begin{aligned} CI&=\frac{\lambda _{\max }-n}{n-1} \end{aligned}$$

(14)

$$\begin{aligned} CR&=\frac{CI}{RI} \end{aligned}$$

(15)

where n is the order of the matrix and RI represents the random index.

Entropy weight method

The entropy weight method is a technique based on the principle of information entropy to determine weights³⁶. Its core idea is to evaluate the importance of each criterion by measuring the amount of information it carries and thus determining the weight. The specific steps are as follows: First, the information entropy of each criterion is calculated using the formula (16).

$$\begin{aligned} E_i=-\frac{1}{\ln (n)}\sum _{j=1}^{n}p_{ij}\cdot \ln (p_{ij}) \end{aligned}$$

(16)

where $p_{ij}$ represents the proportion of the i-th criterion in the j-th sample. The larger the information entropy $E_i$ smaller the variation in this criterion across the samples, indicating lower importance.

Next, the weight of each criterion is calculated based on the information entropy using the formula (17).

$$\begin{aligned} \omega _i=\frac{1-E_i}{\sum _{i=1}^{m}(1-E_i)} \end{aligned}$$

(17)

This yields the relative weight of each criterion, with higher information content leading to a higher weight. Therefore, the entropy weight method dynamically adjusts the weight distribution by calculating the information content of each criterion, ensuring that criteria with higher discriminating power receive more attention in multi-objective optimization problems. This method is only used for weight calculation after optimization, while the Monte Carlo method is used for generating random data during the NSGA-II optimization process. The two do not logically conflict. The entropy weight method assigns weights based on the dispersion of optimization results rather than directly influencing the optimization process.

Combined weighting method

After calculating the weights of each objective function in the uncertain multi-objective optimization model using both the entropy weight method and the analytic hierarchy process, the following formula is used to combine them into an overall weight. The current methods for determining the combined weight include multiplicative and additive approaches. In this article, the multiplicative method is adopted to assign the combined weight³⁷, with the combination formula is shown in (18).

$$\begin{aligned} W_j=\frac{a_jb_j}{\sum _{j=1}^{n}a_jb_j} \end{aligned}$$

(18)

where $W_j$ represents the final weight of objective function j, $a_j$ is the weight of objective function j calculated using the AHP method, and $b_j$ is the weight of objective function j calculated using the entropy weight method.

Weighted grey correlation analysis

Grey correlation analysis is a method suitable for analyzing systems with uncertainty and incomplete information, and is usually used to assess the degree of association between different indicators³⁸. The weighted grey correlation analysis method requires³⁹, first, normalizing the raw data and, subsequently, quantifying the similarity between the reference series and each comparison series by calculating the degree of correlation between them. The formula for calculating the degree of association is shown in Eq. (19).

$$\begin{aligned} r_{ij}=\frac{\min _j(\vartriangle _{ij})+\rho \max _j(\vartriangle _{ij})}{\vartriangle _{ij}+\rho \max _j(\vartriangle _{ij})} \end{aligned}$$

(19)

where $\vartriangle _{ij}$ denotes the absolute difference between the j-th element of the reference sequence and the i-th element of the comparison sequence, and $\rho$ is the resolution factor, which is generally taken as 0.5.

Finally, the weighted sum of the correlation coefficients of each indicator and its comprehensive weight is calculated. Its calculation formula is as shown in Eq. (20).

$$\begin{aligned} \xi _i=\sum _{j=1}^{m}W_j\cdot r_{ij} \end{aligned}$$

(20)

where $\xi$ represents the weighted sum of the correlation coefficients of the different indicators and their combined weights. The allocation of weights reflects the relative importance of each criterion in the decision-making problem.

Multi-objective optimization evaluation metrics

The performance evaluation of the NSGA-II primarily considers three aspects: solution set quality, computational efficiency, and robustness. The quality of the solution set can be assessed using the following metrics: Hypervolume (HV) evaluates the coverage and quality of the solution set by calculating the hypervolume between the solution set and a reference point⁴⁰. A higher HV value indicates that the solution set occupies a larger area or volume in the objective space, representing higher solution quality. Spread measures the distribution of solutions by assessing the deviation of the Euclidean distances between adjacent solutions from the average distance⁴¹. A larger Spread value indicates a less uniform distribution of the solution set. Spacing (SP) is another critical metric for evaluating the uniformity of the solution set distribution⁴². It calculates the deviation of distances between adjacent solutions, reflecting the uniformity of the distribution⁴³. A smaller Spacing value implies a more uniform distribution of the solution set, enabling the optimization algorithm to explore the solution space more effectively. Additionally, Robustness (R) evaluates the stability of solutions under uncertainty or disturbances. This metric assesses robustness by introducing perturbations to the solutions and calculating the standard deviation of the deviations between the perturbed and original solutions⁴⁴. A higher Robustness value indicates stronger stability of the solution under parameter changes or random disturbances. These metrics collectively provide a comprehensive evaluation of the algorithm’s performance, offering a scientific basis for assessing its overall effectiveness and parameter sensitivity.

Engineering application and results analysis

Mine overview

A large limestone mine was selected as the research object. The mining area is located in the basement layer of the Weilong-Zhangwu Basin, within the northwest-west trending structural belt, at the core of the Yihuqiao syncline. The region has experienced relatively strong tectonic movements, with well-developed faults but weakly developed folds. The terrain belongs to the mid-to-low mountain geomorphic unit, with the highest elevation at 2,479.5 m and the lowest at 2,154 m, resulting in a relative height difference of 325.5 m. The CaO content ranges from 40.10% to 53.89%, with an average of 50.80%. The MgO content varies between 0.20% and 2.07%, averaging 0.83% with a coefficient of variation of 38.08%, indicating significant fluctuations and uneven distribution. The primary water recharge factor in the area is atmospheric precipitation, with an annual average precipitation of 465.8 mm and an average annual evaporation of 1,500.4 mm–3.2 times the precipitation, indicating that evaporation exceeds rainfall in this region.The mine adopts an open-pit mining method, employing a top-down, horizontal bench mechanized extraction approach with a bench height of 14 m. Drilling is carried out using down-the-hole drills, excavation is performed by hydraulic excavators, and transportation is handled by large mining dump trucks. The mine covers an area of 0.4944 square kilometers, with geological reserves of 72 million tons and an annual extraction capacity of 4.5 million tons. The maximum mining elevation is 1389 meters, while the minimum is 1160 meters. When formulating the Ore Production Scheduling Strategy, the mine used digital mining software to generate mining plans; however, uncertainties in grade and cost variations, as well as market price fluctuations, were not considered. To address this issue, uncertainty planning techniques were employed, combined with weighted grey relational analysis and intelligent optimization algorithms, to comprehensively evaluate various mining indicators and develop a quarterly Ore Production Scheduling Strategy. Figures 2 and 3 show the surface model and grade model of the mining area, respectively. The geological structure of the mine is complex, with significant fluctuations in CaO content.

Fig. 2

Surface model of the mining area.

Fig. 3

Grade model of the mining area.

The mine adopts a top-down, horizontal layered bench mining method, with a bench height of 15 m and a slope angle of $75^{\circ }$. It uses medium and deep hole blasting technology with electronic detonators for sequential initiation. The mine’s development system includes a shaft and adit, with a shaft depth of 165 m, a diameter of 6 m, and an adit length of 800 m. The average daily ore output is 15,000 tons, with a monthly production capacity of 400,000 tons and a maximum quarterly output of 1.20374 million tons. Currently, three platforms are under development, with geological reserves as follows: the 1265 platform with 203,000 tons, the 1250 platform with 3.396 million tons, and the 1235 platform with 8.857 million tons. As of the end of 2023, the mine has development reserves for more than 24 months, mining reserves for more than 12 months, and minable reserves for more than 6 months. The mine operates on a three-shift system (8 h per shift), with each platform having one ore extraction point. The mining area is equipped with 22 mining trucks and 5 electric shovels.

The mine utilizes computer-aided scheduling for annual planning, while monthly and quarterly plans are primarily determined based on manual experience. This involves estimating a general range through experience and then implementing it. However, neither annual nor monthly plans take into account uncertainties in the production process, such as variations in ore grade, mining and transportation costs, or fluctuations in market prices. To address this issue, an optimization approach combining the NSGA-II with weighted grey relational analysis is applied. Table 1 presents the parameters of the optimization algorithm, while Table 2 provides the basic parameters of each ore extraction site.

Table 1 NSGA-II algorithm parameters.

Table 2 Basic parameters of each extraction point.

In the constraints on production volume due to market price fluctuations, the sensitivity parameter $\alpha$ should not be assigned a fixed value, as this would limit the effectiveness of the constraint equation. Instead, a range can be defined for $\alpha$. If market price fluctuations are significant, the value of $\alpha$ should be larger to reflect the high sensitivity of production volume to price changes. Conversely, if market price fluctuations are minor, the value of $\alpha$ can be appropriately reduced, as the sensitivity of production volume to price changes is relatively low. This approach aligns with economic principles. Therefore, as shown in Table 3, the sensitivity parameter is divided into six intervals for both positive and negative price fluctuation ranges. The Monte Carlo algorithm is used to allow the sensitivity parameter to vary randomly within these intervals. When market price fluctuations approach zero, this constraint does not participate in the iterative process.

Table 3 Sensitivity parameter values.

The Pareto front from the initial iteration is shown in Fig. 4, revealing a large number of solutions, which highlights the global search capability of the NSGA-II algorithm. Subsequently, based on the initial solutions, a market price fluctuation constraint was introduced for a second iteration. Due to the addition of this new constraint, the original solutions were divided into six sets according to the market price fluctuation range. Since each solution has a different sensitivity parameter $\alpha$, their spatial distribution varies. Therefore, a three-dimensional plot of the solution sets was generated with $\alpha$ as the vertical axis, as shown in Fig. 5. The Pareto front projection on the xy-plane demonstrates a relatively uniform distribution. To optimize the solutions, the data were sorted according to ore output, and the relationships between unit extraction cost and ore market value with respect to calcium oxide grade, magnesium oxide grade, and production target achievement rate (An operational indicator that measures the actual execution of the planned ore extraction volume.) were analyzed. From the fluctuation curves shown in Fig. 6, it is evident that unit mining cost is positively correlated with magnesium oxide grade, both reaching their peak values at an output of 103,000 tons, and dropping to their lowest at 113,000 tons. Conversely, unit extraction cost is generally negatively correlated with calcium oxide grade, reaching its lowest cost and highest calcium oxide grade at 113,000 tons. Overall, both unit extraction cost and production target achievement rate decrease. Meanwhile, market value and calcium oxide grade show an upward trend with increasing ore output, peaking at 112,000 tons. Magnesium oxide grade and production target achievement rate, however, are negatively correlated with market value. Based on this, the six sets of solutions generated after the second iteration were analyzed using weighted grey relational analysis to select the optimal solution for each market price fluctuation direction.

Fig. 4

Initial iteration Pareto front.

Fig. 5

Pseudo-3D Pareto front and objective function projection incorporating market price fluctuation parameters. (a) Objective function values when market price fluctuation is between −10% and −8%; (b) Objective function values for market price fluctuation between −7% and −5%; (c) Objective function values for market price fluctuation between −4% and −2%; (d) Objective function values for market price fluctuation between 2% and 4%; (e) Objective function values for market price fluctuation between 5% and 7%; (f) Objective function values for market price fluctuation between 8% and 10%.

Fig. 6

Statistics of production indicator fluctuations. (a) Changes in unit extraction cost and calcium oxide grade by output; (b) Changes in unit extraction cost and magnesium oxide grade by output; (c) Changes in unit extraction cost and plan completion rate by output; (d) Changes in ore market value and calcium oxide grade by output; (e) Changes in ore market value and magnesium oxide grade by output; (f) Changes in ore market value and production target achievement rate by output.

To determine the weights, three mining engineers familiar with the mine’s operations were invited to conduct pairwise comparisons of six key indicators and construct a judgment matrix based on the Analytic Hierarchy Process (AHP). The criterion layer includes mining costs, market value, calcium oxide grade, magnesium oxide grade, ore output, and production target achievement rate. The objective layer aims to optimize production plans, while the alternative layer consists of multiple sets of ore extraction strategies. Since this step only requires determining the weights of the six indicators in the criterion layer without evaluating specific alternatives, only the criterion layer undergoes pairwise comparisons. Expert ratings adopt a 1–9 scale and their reciprocals, and the aggregated judgment matrix (21) is as follows:

$$\begin{aligned} \begin{bmatrix} 1 & 1 & 3 & 3 & 5 & 7 \\ 1 & 1 & 3 & 3 & 5 & 7 \\ 1/3 & 1/3 & 1 & 1 & 3 & 5 \\ 1/3 & 1/3 & 1 & 1 & 3 & 5 \\ 1/5 & 1/5 & 1/3 & 1/3 & 1 & 3 \\ 1/7 & 1/7 & 1/5 & 1/5 & 1/3 & 1 \end{bmatrix} \end{aligned}$$

(21)

The consistency test on the judgment matrix yielded the following results:

$$\begin{aligned} W&=[0.31670.31670.05310.13020.13020.0531]^T\lambda _{\max }=6.0774\\ CI&=0.0155~RI=1.24~CR=0.0125~(CR<0.10,~\mathrm {consistency~condition~met}) \end{aligned}$$

where W is the feature vector (normalized to vector), $\lambda _{\max }$ is the maximum feature root, CI is the metric judgment matrix deviation consistency index, RI is the average random consistency index of the judgment matrix, and CR is the random consistency ratio.

As shown in Table 4, $\alpha _i$ is the weight obtained with the hierarchical analysis method, $\beta _i$ is the weight obtained with the entropy value method, and the two weights are integrated and combined using formula (1) to obtain the final weight $W_i$. After the solution set obtained after the second iteration of NSGA-II is preprocessed using normalization, the grey correlation coefficients of each index in each scheme are calculated as shown in Table 5. Finally, the weighted sum of the integrated weight A and the correlation coefficients of each index is calculated, and according to the ranking, the weighted sum of the weighted sums of the combined weights A and the correlation coefficients of each index is calculated. Finally, the weighted sum of the obtained composite weight $W_i$ and the correlation coefficient of each index is weighted, and the optimal solution is output according to the sorting result.

Table 4 Weight calculation results.

Table 5 Relational coefficient matrix.

Optimization results

The optimization results achieved the goal of maximizing ore market value and minimizing unit extraction cost under both increased and decreased production scenarios, with quarterly ore production, average CaO grade, average MgO grade, ore market value, unit extraction cost, and production target achievement rate all meeting the requirements of the original plan. According to the optimization results, the production target achievement rate is superior to the original plan regardless of whether production is increased or decreased. As shown in Table 6, Solution 1 and Solution 2 represent the optimal solutions for price fluctuation intervals in the positive and negative ranges, respectively. Compared to the original plan, ore output increased by 16.7% and 20.5%, and unit extraction cost decreased by 0.29% and 0.38%, respectively. In both the positive and negative intervals, ore output varied across different extraction points. Due to differences in ore grades, Solution 1 mainly increased the output at extraction point III, while Solution 2 primarily increased the output at extraction point I. By adjusting the ore blending strategy, the indicators for ore grade, value, cost, and production target achievement rate were optimized. o further assess the economic benefits of different optimization methods, Quarterly Total Cash Flow (QTCF) was used as a comparative metric to evaluate the economic performance of different optimization approaches. QTCF is defined as:

$$\begin{aligned} \textrm{ATCF}=\mathrm {Ore~Market~Value}-\mathrm {Quarterly~Ore~Output}\times \mathrm {Unit~Mining~Cost} \end{aligned}$$

(22)

Ore Market Value is measured in ten thousand CNY and represents the market revenue from the mined ore during the quarter. Quarterly Ore Output is measured in tons, while Unit Mining Cost is measured in CNY per ton. The results indicate that NSGA-II-GRA outperforms Genetic Algorithm (GA) in terms of Quarterly Total Cash Flow (QTCF), with Solution 2 achieving a 32.8% increase compared to the original plan, demonstrating greater profitability. Additionally, Solution 1 shows a 5.03% improvement over the original plan, effectively mitigating economic losses during market downturns. In contrast, GA exhibits limited optimization performance, further validating the applicability and economic advantages of NSGA-II-GRA under various market fluctuation scenarios. When the market price of ore tends to decrease, production should be increased to spread fixed costs and thereby reduce unit production costs, ensuring the mine’s profitability and enhancing its competitiveness. Conversely, when the market price rises, production should be increased under reasonable conditions to maximize economic benefits during periods of high prices. This strategy takes advantage of the profit mar-gin created by high prices, increasing production to generate more revenue. From the ore grade modeling map, it is evident that the ore grade distribution in this mine is highly complex, with significant fluctuations and a high proportion of low-grade ore. Therefore, grade optimization and ore extraction optimization are crucial. By implementing an optimized strategy, the utilization efficiency of high-grade ore can be improved, while also enhancing the sustainability of mining operations, ensuring rational resource development, and maximizing economic benefits.

Table 6 Comparative analysis of optimization results.

Analysis and discussion

To further understand the applicability of the algorithm parameters to the mine ore extraction strategizing process, tests were conducted, and the algorithm’s performance under different parameter settings was recorded. The quality and diversity of the Pareto front were observed using indicators such as run time, solution count, hypervolume, spread, spacing, and robustness. A comprehensive evaluation was performed to assess the impact of various parameters on the algorithm’s performance.

As shown in Table 7 and Fig. 7a, 7b, 7c, hypervolume, spacing, and robustness decreased as population size increased, and eventually stabilized. As seen in Fig. 7b, spread performed poorly when the population size was set too low. The figures show that when the population size was around 300, all indicators performed well. Additionally, given the low dimensionality of the model, increasing the population size beyond this point did not improve solution quality and instead extended search time. Thus, a population size of 300 is considered reasonable. After determining the population size, tests were conducted to observe the changes in multi-objective algorithm performance indicators as the maximum number of generations increased from 200 to 800. As shown in Table 8, the setting of the maximum number of generations directly affects the algorithm’s exploration and exploitation capabilities. At 500 generations, a good balance was achieved between solution diversity, hypervolume, and run time.

Fig. 7

Algorithm performance evaluation metric fluctuation analysis. (a) Changes in hypervolume as population size increases; (b) Changes in spread as population size increases; (c) Changes in spacing as population size increases; (d) robustness as population size increases.

Table 7 Effect of different population settings on evaluation results at various maximum generations.

Table 8 Effect of different maximum generations on evaluation results.

As shown in Fig. 7a and b, the response trends of extraction costs and ore market value were analyzed by adjusting the sensitivity parameter $\alpha$ within the range of −30 to 30. The mean values of the solution sets for different $\alpha$ values were calculated and are presented in Fig. 8. When $\alpha$ increases from −30 to 0, extraction costs show an upward trend, whereas they gradually decline as $\alpha$ increases from 0 to 30. In contrast, ore market value decreases when $\alpha$ decreases and increases as $\alpha$ grows. The figures indicate that the objective function values are particularly sensitive to changes in $\alpha$ within the range of −9 to 9, demonstrating a significant and reasonable adjustment effect on both costs and market value. Moreover, restricting the range of $\alpha$ can prevent adverse impacts of extreme parameter values on algorithm performance. Excessively large or small $\alpha$ values may result in reduced solution quality or increased costs. Under varying operational conditions, the optimal value of $\alpha$ may differ. Considering the diversity of operational environments, it is essential to adjust the $\alpha$ value according to specific circumstances in practical applications.

Fig. 8

Variation of parameters in the solution set: (a) Average mining cost variation in solution set; (b) Average ore market value variation in solution set.

The proposed model effectively balances economic efficiency and geological uncertainty by integrating Monte Carlo simulation with NSGA-II optimization. However, two key limitations remain. First, while calcium and magnesium oxide grades are constrained, the model does not explicitly incorporate ore blending strategies or use the CaO/MgO ratio as a decision variable, which may limit its alignment with beneficiation requirements. Second, under extreme market fluctuations ($>\pm 15$%), production adjustments may be constrained by predefined sensitivity parameters ($\alpha$) and real-world operational delays, such as equipment mobilization and regulatory approvals, which are not fully quantified in the current framework.

Future research could address these limitations by incorporating a multi-stage dynamic optimization framework, such as model predictive control (MPC), to enable rolling horizon planning with adaptive sensitivity parameters, improving responsiveness to both short-term shocks and long-term trends. Expanding the objective functions to include environmental metrics (e.g., carbon emissions per ton) and resource recovery rates would further enhance the model’s applicability to sustainable mining. These refinements would extend the model’s core contribution–robust production scheduling under dual uncertainties–while aligning it with broader industrial priorities.

Model validation

The solution quality of NSGA-II-GRA and Genetic Algorithm (GA) was compared based on multiple evaluation indicators. As shown in Table 9, NSGA-II-GRA outperforms GA in all four indicators, with solutions closer to the Pareto optimal front, more uniform distribution, and higher stability under market fluctuations. Although GA has a slightly shorter computation time, its lack of non-dominated sorting and crowding distance mechanisms results in solutions concentrated in local regions, reducing global search capability. Overall, NSGA-II-GRA demonstrates superior solution quality and adaptability in open-pit mine production optimization.

Table 9 Comparison of solution quality between NSGA-II-GRA and GA.

Conclusions

The fluctuations in ore grade and extraction costs during open-pit mine production were addressed using the Monte Carlo method to simulate these uncertainties dynamically and establish a quarterly planning model. The model incorporates market price fluctuation constraints to enable adaptive production adjustments, aiming to minimize extraction costs and maximize ore market value. A solution approach combining the NSGA-II algorithm and weighted grey relational analysis was developed, considering unit extraction cost, ore market value, calcium oxide grade, magnesium oxide grade, ore output, quarterly total cash flow, and production target achievement rate as evaluation criteria.

The integration of subjective and objective weighting methods refined the solution selection process, improving accuracy and reliability. The optimized solutions outperformed the original plan across all indicators, demonstrating improved economic efficiency and operational feasibility. A comparative analysis with a traditional genetic algorithm (GA) further validated the advantages of NSGA-II-GRA in solution quality, diversity, and robustness. Additionally, sensitivity analysis confirmed the model’s adaptability to market fluctuations, with the sensitivity parameter $\alpha$ optimized within the range of $-9$ to 9.

The proposed method provides a flexible and effective approach for optimizing ore production under uncertain market conditions, helping mining companies respond dynamically to external changes. By integrating NSGA-II with grey relational analysis, the decision-making process incorporates multiple complex factors, reducing subjective bias and enhancing the scientific rigor of production planning.