State-of-the-art optimization methods for short-term mine planning

1.1 Relation between Long-term and short-term Mine planning

Good mining planning is the basis for the best financial results for a mining company. Short-term mine planning is set up to align with the company strategy. One of the main differences between short-term mine planning and long- and medium-term mine planning is time discretization. Short-term planning should be performed with a granularity of less than one year. This granularity can be extended for up to two years (Blom et al. 2019). Short-term mine planning is required to achieve the goals set by a long-term plan by designing push-backs. Generally, the objective is to generate the highest cash flow value and Net Present Value (NPV) over the LOM plan (Juarez et al. 2014; King 2014; Otto and Musingwini 2020) while minimizing costs and respecting production targets within the LOM plan (Jewbali and Dimitrakopoulos 2018; Noriega and Pourrahimian 2022). This involves incorporating more sophisticated mathematical analysis and advanced geostatistics techniques, performed by a specific and adequate softwares (for example: Datamine RM (Resources Modelling), Datamine UG (Underground), Snowden and NPV). However, different models and algorithms have been are applied to enhance performance and optimize short-term operations, ultimately contributing to improved production targets, reducing cost, and increasing efficiency. Short-term mine planning deals with the equipment and resources allocated on a monthly, weekly, daily, or shift-by-shift basis under the control of the LOM (Noriega and Pourrahimian 2022). Thus, the success of truck fleet and shovel allocation to mining face areas depends strongly on the best dispatching strategy for fleet management. Models and algorithms for the fleet management systems have been reviewed by Moradi Afrapoli and Askari-Nasab (2019). Short-term solutions often deals to determine destination for different qualities of materials: High grade (HG) sockipiles and Medium grade (MG) stockpiles for milling, Low grade (LG) stockpiles for leaching, waste to dump (Deutsch 2023).

Few papers have described short-term planning with granularity in days, weeks, and months. Table 2 specifies the main difference between short- and long-term mine planning, taking into account the discretization time, block model characteristics, design, mining precedence, cost, and equipment. These plans take into account the objective functions and constraints to be considered and the level of details for mine operations to be modelled. This includes constraints specific to each mine, their capacities, block sequencing, blending, plant requirements, and pit slopes (Caccetta and Hill 2003; Askari-Nasab et al. 2010; Marques et al. 2013; Badiozamani and Askari-Nasab 2014; Blom et al. 2019).

To ensure the link between long-term and short-term planning is maintained according to the company strategy, mining organizations apply Medium-term planning as step between these two planning. Tactical plan up to 3 years can be considered as Medium-term planning. For this review article, the short-term planning granularity refers up to 3 months, for mining operations.

Figure 1 illustrates the relationship between long-term and short-term mine plans based on the scope area limit of this review article.

Figure 1. Relation between Long-term and Short-term Mine planning and area of the study.

1.2 Research gaps for short-term planning

In the last decade, some studies have discussed short-term mine planning problems and technical solutions on a monthly basis (Askari-Nasab et al. 2010; Tabesh and Askari-Nasab 2013; Espinoza et al. 2013; Blom et al. 2019; Letelier et al. 2020; Salman et al. 2021; Menezes and dos Santos Corrêa 2022; Deutsch 2023).

However, short-term mine planning optimization for open-pit mines from weekly to monthly horizon granularity remains timidly explored. Most of papers listed in Figure 1, are discussing short-term planning with a time discretization up to one year.

In addition, short-term planning models must be solved quickly because of the dynamic nature of mining operations. Mining polygon optimization reduces the computational expenses of mine planning models by reducing the number of variables involved. Therefore, more research is required to develop more efficient mathematical models. The primary goal of this study is to review existing publications on short-term mine planning and mining polygon optimization methods to galvanize researchers to work on the quicker solvability of short-term mine planning models.

The first step focuses on the techniques used to create and optimize mining polygons, which is an important step in scheduling. In the second step, we will focus deeply to review different approaches for short-term mine planning optimization. Before concluding and stating the next orientation of this work, we will discuss all approaches reviewed on the recent improvements for overcoming uncertainties in short-term mine planning.

3.1 Dig-limits optimizations

The dig-limit optimization approach consists of defining the best possible limits between ore waste on a bench basis to ensure the profitability of mining operations based on the definition (Sari and Kumral 2018). The definition of polygons depends on the level of block model preparation (grade control program), which affects the grade uncertainty (P et al. 2002; Neufeld et al. 2003; Richmond and Beasley 2004; Vasylchuk and Deutsch 2019).

Generally, in mining industry practice, the dig-limit is delimited manually by a geologist after obtaining the grade control results before blasting ore. After delimitation of the ore according to the cut-off grade, the geologist and mining teams can decide the destination of the material to be blasted: ore to the plant for high grade (HG), ore to the stockpile for the low-grade (LG), and waste to the waste dump.

Several techniques have been used to perform dig-limits and ore polygons ( Table 3). Sari and Kumral (2018) used Mixed Integer Linear Programming Richmond (2007), used local search algorithms Richmond and Beasley (2004), used a heuristic approach with a floating cone algorithm to find an optimal solution for the grade control problem. To determine the optimal final destination for materials (ore or waste) (Vasylchuk and Deutsch 2019), also used a simple heuristic approach for mineable dig-limit optimization considering the excavation constraint while maximizing profit. Heuristic algorithms are known to solve problems faster and more efficiently, while sacrificing optimality and accuracy.

The genetic algorithms have been applied by Ruiseco et al. (2016) to automatically generate a dig-limit considering the selectivity of the fleet for the Selective Mining Unit (SMU) while maximizing profitability. In this approach, the dig-limit is considered as a chromosome and each SMU is considered as a gene. The first population is generated by a random set of feasible dig-limits that are combined with others to reach the optimized dig-limit. The genetic algorithm for dig-limits is better than manual methods; this algorithm is a near-optimal solution Sari and Kumral (2018).

Simulated annealing was applied by Isaaks et al. (2014) using the minimum expected loss method to find the optimal dig-limits constrained by a minimum width. Neufeld et al. (2003) used a semi-automatic algorithm for the dig-limit with the initial polygons by the user. Norrena and Deutsch (2001) and P et al. (2002) proposed an algorithm that optimizes dig-limit boundaries with maximum profit, taking into consideration the equipment’s limitations (digability).

The Heuristic approach was used by Vasylchuk and Deutsch (2019), who proposed a 2-D optimization dig-limit with multiple locations and destinations by maximizing profits. Contrary to Norrena and Deutsch (2001) and P et al. (2002), this algorithm uses a rectangular form of and for the dig limit, and there are no smoothing operations after their generation.

3.2 Clustering algorithms

Clustering algorithms attempt to identify groups or clusters of elements with the same properties in datasets. In short-term planning, clustering is used to aggregate blocks with the same properties to generate mining cuts. A mining cut is a SMU that generates a practical mining schedule (Eivazy and Askari-Nasab 2012).

Clustering has been described by Tabesh and Askari-Nasab (2013) as the process of combining related things in a manner that maximizes intra-cluster similarity and inter-cluster dissimilarity ( Figure 2).

Figure 2. Before and after clustering data.

Clustering algorithms for block aggregation have been developed in the literature, principally by Askari-Nasab et al. (2010); Tabesh and Askari-Nasab (2011); Tabesh and Askari-Nasab (2013) and Badiozamani and Askari-Nasab (2014) to deal with uncertainties in short-term planning and to create mining polygons that can be used as planning units in the next planning stages (Tabesh and Askari-Nasab 2019). Similar block units are grouped into a similarity index with the control of the shape and size of the polygons.

Two clustering models can be used: hierarchical clustering, partitional clustering, and k-means clustering. K-Means consists of choosing the number K of clusters before selecting at random K points and the centroids from the datasets. Each data point was then assigned to the nearest point to create K clusters around the centroid. This process consists of computing and placing the new centroid of each cluster. Each data point was assigned to the new nearest centroid if the outcome was unsatisfactory. If any reassignment occurred, we had to compute again to obtain a good result. Compared to hierarchical clustering, the K-means clustering method is reported to be faster but less accurate (Feng et al. 2010).

Hierarchical clustering is categorized in two types: agglomerative and divisive.

Agglomerative is a bottom-up approach useful for mine planning that uses the similarity of an object to merge to form one object. First, each data point is made into a single-point cluster to form N clusters. The two closest data points are merged to form one cluster using Euclidean distance. The number of similar clusters will be reduced from N to N1 clusters. The algorithm will continue to have at the end one huge cluster grouping similar objects.

Divisive algorithms are the reverse of the agglomerative algorithms. This divides the units of a cluster into two smaller groups. Consequently, the divisive algorithms stop when the number of clusters is equal to the number of objects. The common concepts between the two groups are the definition of similarity measures and ways to update the similarity values when new clusters are defined (Tabesh and Askari-Nasab 2013).

The K-means clustering algorithm with a heuristic approach have been used by Salman et al. (2021) to obtain mineable shapes using the Newman and Marvin datasets related to the copper deposit (Newman et al. 2010). This algorithm is applicable on a bench-by-bench basis and for metallic deposits.

Material destination homogeneity and mining direction constraints were applied to obtain minable cluster shapes. The objective was to reduce the costs of optimizing short-term mine planning. In contrast to the hierarchical clustering of Tabesh and Askari-Nasab (2019), this algorithm does not consider multiple elements with grade uncertainty.

Table 3 presents the results of different techniques for mining polygons using and completing the table of Sari and Kumral (2018).

4.1 Mixed Integer programming (MIP) methods

Mixed Integer programming (MIP) is a mathematical optimization in which the objective function and constraints are linear or non-linear, and variables are integer, discrete, and non-discrete in some cases.

The MIP model for mine planning was developed by Gershon (1982) and Gershon (1983) to maximize NPV with blending and processing constraints. This optimization model can also minimize cost, maximize production, and maximize NPV. In contrary to the stochastic optimization, Eivazy and Askari-Nasab (2012) mentioned this method doesn’t capture the cost uncertainty related to the change of input data (Geological Block model, cost, prices, recoveries and mining constraints).

The advantage of MIP in short-term planning is the limitation of the size problem to be solved, which requires close attention to a number of algorithmic parameters and solutions (Smith 1998), because fewer blocks are considered in problems than in long-term planning. Only the blocks scheduled to be extracted within the specified time horizon may be considered in the MIP model of the short-term planning problem (L’Heureux et al. 2013).

Depending on the mining operations to be modelled and the required level of detail for the planning, MIP programming has been widely used to perform short-term planning in the last decade (L’Heureux et al. 2013; Upadhyay and Nasab 2017; Upadhyay and Askari-Nasab 2018; Blom et al. 2019) MIP methods need the necessary decision variables and indices at time t to represent each mining operation (drilling and blasting, loading, hauling, stockpiling, etc.) to be modelled and included in the schedule for each block or block set of the geological model.

A MIP model has been developed by Eivazy and Askari-Nasab (2012) to minimize mining, hauling, stockpiling, rehandling, processing, and waste rehabilitation costs for short-term mine planning of an iron mine on a monthly basis to track long-term mine planning. They used a branch-and-cut algorithm using the TOMLAB/CPLEX optimizer to solve this problem by considering some constraints (precedence, mining and processing capacities, grade processing requirements) and multiple destinations and mining directions.

MIP in short-term mine planning has been also used by L’Heureux et al. (2013) to address the issue of shovel movement and production capacity of equipment on a daily to monthly basis. They used the CPLEX to solve the formulation. The objective is to minimize the cost of mining operations such as drilling, blasting, and extraction, respecting the precedence of these activities to constrain shovel movements. Shovels were assigned to a face (block or set of blocks) for the time horizon. Variable decisions were used for each period for shovel-to-face assignments, face-to-face movement of shovels, and drilling, blasting, and extraction decisions.

A multi-stage mine production scheduling (MPS) has been proposed by Kozan and Liu (2016) and Kozan and Liu (2018) to optimize multi-resource multi-stage timetables at the operational level and to determine how and when the equipment will be allocated to the selected blocks (or set of blocks) to perform mining operations (drilling, blasting, and extraction) over a short period of time under equipment capacity constraints for iron mines on a weekly basis. Their MIP model was solved by CPLEX optimization to minimize the mining operation cost (minimizing all of the equipment’s idle times at each stage), and maximize mining productivity and utilization of mining equipment through multiple processing stages.

To ensure shovel allocation decisions based on available faces to generate short-term production schedules capturing operational uncertainty, Upadhyay and Askari-Nasab (2018) proposed a Mixed Integer Linear Goal Programming (MILGP) model solved by CPLEX. The integer variable represents the number of truck trips between the faces and destination. The objective is to maximize production, optimize ore tonnage received at processing plants, minimize deviations in grade blending requirements from various ore destinations, and minimize shovel movement. Later, Upadhyay et al. (2021) improved their model to be more practical by allocating shovels continuously on mining cuts.

To solve the dig-limits between ore and waste problems for short-term open-pit mine operations, Sari and Kumral (2018) used mixed-integer linear programming (MILP) that are compatible with the maneuvering capability of the excavator and minimize dilution/loss, as opposed to free selection of ore and waste SMUs based on the cut-off grade. The objective is to maximize profit with block aggregation of ore or waste.

In five-week time horizons, Nelis and Morales (2022) used MIP optimization to define mineable mining cuts and mining sequences for a copper mine. The objective is to perform mining cuts and dig-limits by considering the cutoff grade, mining precedence (representative of SMU), and defining the destination of materials in each bench to produce a better evaluation of real NPV in the long term. The mining polygons complied with operational constraints such as the size of the loading equipment.

4.2 Stochastics optimization methods

Stochastic optimization has recently been developed for short-term planning (Matamoros and Dimitrakopoulos 2016; Jewbali and Dimitrakopoulos 2018). These solutions are suitable for strategic mine planning. It has been relatively well developed for long-term planning over the last decade and is one of the best methods for mine planning optimization. This integrates the parameter uncertainties used in the plan. In contrast to deterministic programming, which applies a constant parameter, stochastic programming uses a probabilistic approach that uses a probability distribution of each parameter to be applied to each set of possible realizations.

Mining parameters (grades, operating costs, commodity prices, recoveries) and the operational constraints estimated on the data at the time of planning are considered risky assets because of their fluctuation during the execution of this plan (Osanloo and Rahmanpour 2017).

The goal of stochastic programming for short-term mine planning is to minimize this risk or maximize the predicted objective value of the decision variables. Two decision variable stages can be defined according to Blom et al. (2019), the first-stage decision variables define the plan to be executed, and the variables for the second stage are determined by each potential scenario and indicate the changes that would need to be made to the plan if the scenario came to pass.

The application of a real option pricing theory from financial uncertainty and risk as stochastic optimization for medium- and short-term planning has been advised by Li and Knights (2009). This model takes into account fuel price fluctuations. To minimize operating costs and overcome this variation, the authors proved a significant economic benefit by incorporating stochastic analysis in truck dispatching. The objective is to ensure that the trucks dump waste to the closest waste dump during the period of high fuel price to reduce the number of trucks used. In periods of low fuel prices, waste is hauled directly to the under-filled dumps with the target to balance the storage volume across the set of available dumps.

As uncertainties cause deviation from the long-term plan, Matamoros and Dimitrakopoulos (2016) presented stochastic integer programming for a short-term mining plan for the iron open-pit mine case. The first stage is to minimize mining costs (loading, hauling, and availability of faces and equipment) and maximize fleet utilization. The second stage is to minimize the cost over the range of recourse costs associated with deviations from the target plan in terms of geology (ore tons, grade, and deleterious elements) and fleet (mechanical availability and hauling time) uncertainties. The authors demonstrated that the solution of this stochastic formulation is better than a deterministic plan with a lower cost because it takes into account the fluctuations of the ore quality and fleet parameters. This model was improved by Quigley and Dimitrakopoulos (2020) by generating a short-term plan that minimizes the shovel movement and production deviation (ton and grade sent to the plant allocated). This model has been applied to copper mines.

Contrary to the deterministic approach of mine planning, which consists of scheduling production first and then allocating fleets, Both and Dimitrakopoulos (2020), have developed a new model for stochastic optimization by using a metaheuristic solution to solve the problem that simultaneously optimizes the short-term extraction sequence and fleet management. The model considers geological, equipment performance, and truck cycle–time uncertainties.

Combining Fuzzy linear programming (non-linear programming) and optimization of risky assets Osanloo and Rahmanpour (2017) has applied portfolio optimization to short-term planning to minimize mining costs and risky assets, and to maximize the expected return of the portfolio by investing a particular amount of money. For authors, the uncertainties come from the capacity of mining faces to comply with the plant requirements (tonnage and grade) after material blending (portfolio) of different faces, taking into account their geological characteristics (return). In relation to the limestone mine, the extraction capacity of each location was defined, and each mining face was considered as the extent of investment for each asset. All parameters are bounded numbers and are fuzzified (equipment capacities, grades, recoveries, etc.).

To help mine complexes respond to new information (grade control results) for adapting short-term mining plans for material-feeding plants, Paduraru and Dimitrakopoulos (2018) proposed state-dependent policies. The authors applied this model to gold and copper mines with six possible destinations. The objective is to assist and make the best decision to reassign the update blocks mined estimate to the new destination, to improve revenue, and to minimize the processing cost via stockpiling management.

The last work with stochastic planning was carried out by Jewbali and Dimitrakopoulos (2018) by combining long- and short-term planning through simulated grade control data to obtain a new estimate of ore body unavailable at the time of schedule setup. To reduce the deviation from long-term planning and satisfy the production plan, the authors proposed a multi-stage approach to the planning process by generating possible data for future grade control plans. Stochastic integer programming was applied to simulate the new ore body estimate via the grade control data generated. The objective was to maximize the NPV in long-term planning and minimize the cost of the production target within the long-term plan.

4.3 Metaheuristic and Heuristic Methods

Metaheuristics is a mathematical optimization technique with the goal of efficiently exploring the search space and finding near-optimal solutions. However, there is no guarantee of feasibility and optimality of the resulting solutions, although it provides good near-optimal solutions within a reasonable time. Fathollahzadeh et al. (2021) classified heuristic and metaheuristic methods. There are a wide variety of metaheuristic approaches, and most solutions used in short-term mine planning are as follows:

The Tabu Search (TS) is a metaheuristic method for mathematical optimization that is used for local or neighborhood searches and it has been by Glover (1986).

This optimization method iteratively moves from one potential solution to a better solution in the same neighborhood space. As the search progresses, Tabu search carefully explores the neighborhood of each solution and saves the results in a list of potential solutions to consider. If a potential solution appears in the tabu list, it cannot be revisited until it expires. The size of the tabu list should be large enough to prevent cycling but not so large that it prohibits too many moves. Liu and Ong (2004) and TS ability to prevent visiting previously recorded solutions (Mousavi et al. 2016a).

Simulate annealing (SA) algorithm was inspired by the analogy of thermodynamics, in which metals are cooled and annealed (Kirkpatrick et al. 1983). It is a metaheuristic approach known for finding a better solution when one is obstructed by a non-improving solution, and it can be viewed as a special form of TS, where a move becomes a tabu with a specified probability.

Shaw (1998) applied a large neighborhood search (LNS) has been applied by Shaw (1998). Unlike common neighborhood search, LNS uses a destroy-and-repair mechanism to define a new solution. According to this mechanism, part of the solution is destroyed, and then a repair technique rebuilds the solution (Pisinger and Ropke 2010). For mine planning, this method involves moving blocks between periods and destinations.

Few studies have described metaheuristic problems in short-term mine planning. Shishvan and Sattarvand (2015) developed an Ant Colony Optimization (ACO) metaheuristic to solve an extended MBS-type problem applied to a copper–gold mine for long-term mine planning.

One of the last works for a metaheuristic approach for short-term planning was proposed by Mousavi et al. (2016a) comparing metaheuristic algorithms: Tabu Search (TS), simulated annealing (SA), and a hybrid solution SA and TS to solve the Open Pit Blocks Sequencing Problem (OPBS). These methods include mining constraints, stockpiling, ore blending, machine workspace to extract a given block from four possible sides, and dynamic destination assignment on a daily or weekly basis. The objective of the proposed OPBS model is to minimize stockpiling costs, including rehandling, holding costs, and satisfying processing requirements. To improve the TS and SA solutions, the authors proposed a hybrid TS-SA algorithm where SA embedded in the TS framework is used to accept some non-improving moves to allow a more diversified investigation of the solution space.

Mousavi et al. (2016a) proposed another hybrid metaheuristic short-term planning optimization method. To solve the Model Block Sequencing (MBS) problem, they hybridized the simulated annealing (SA), large neighborhood search (LNS), and branch-and-bound (B&B) approaches used by Eivazy and Askari-Nasab (2012). A partial neighborhood solution (PNS) is constructed in each iteration of the SA. Then, a full. neighbourhood solution (NS) is achieved by assigning destinations to blocks using the B&B algorithm. The objective is to minimize stockpiling costs, including rehandling, holding costs, and satisfying processing requirements.

Liu and Kozan (2012) proposed a hybrid shifting bottleneck procedure (HSBP) algorithm combined with the Tabu Search (TS) metaheuristic algorithm has been developed to deal with the parallel-machine job-shop scheduling (PMJSS) problem. The objective is to improve the existing solutions to a short-term planning problem with equipment assignment.

Kumral and Dowd (2005) proposed a simulated annealing metaheuristic combined with Lagrangian relaxation and a multi-objective simulated annealing model (MOSA) to determine the optimal short-term production to reach an optimal or near-optimal schedule.

Similar to metaheuristic methods, Menezes and dos Santos Corrêa (2022) proposed a heuristic method, a new integrated mathematical method that considers quality, metallurgical recovery, mass balance, and stockpiling for an iron mine. The objective is to minimize operating costs by using a set of algorithms to manage the variations in production capacity for mining supply chains and meeting customer demands. To find the optimal solution for short-term mine planning, the authors used heuristic algorithms (relax, fix, fix, and optimize) and mixed-integer programming (local branching) using CPLEX.