J. Cent. South Univ. (2019) 26: 3126-3139

DOI: https://doi.org/10.1007/s11771-019-4241-1

A value adding approach to hard-rock underground mining operations: Balancing orebody orientation and mining direction through meta-heuristic optimization

Martha E. VILLALBA MATAMOROS, Mustafa KUMRAL

Department of Mining and Materials Engineering, McGill University,3450 University Street, Montreal, QC H3A 0E8, Canada

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract:

Underground mines require complex construction activities including the shaft, levels, raises, winzes and ore passes. In an underground mine based on stoping method, orebody part(s) maximizing profit should be determined. This process is called stope layout optimization (SLO) and implemented under site-specific geotechnical, operational and economic constraints. For practical purpose, the design obtained by SLO shows consecutive stopes in one path, which assists in defining the mining direction of these stopes. However, this direction may not accommodate the spatial distribution of the ore grade: if the orebody orientation and mining direction differ, the value of the mining operation may decrease. This paper proposes an approach whereby paths in the SLO are defined as decision variables to avoid the cost of mining in the wrong direction. Furthermore, in the genetic-based formulation, which accounts for orebody uncertainty, a robust cluster average design process is proposed to improve SLO’s performance regarding metal content. A case study in narrow gold vein deposit shows that the profit of an underground mining operation could be underestimated by 25%-48% if the algorithm ignores stope layout orientation.

Key words:

underground mine planning; orebody uncertainty; orebody orientation; mining direction; stope layout optimization；

Cite this article as:

Martha E. VILLALBA MATAMOROS, Mustafa KUMRAL. A value adding approach to hard-rock underground mining operations: Balancing orebody orientation and mining direction through meta-heuristic optimization [J]. Journal of Central South University, 2019, 26(11): 3126-3139.

1 Introduction

Planning of stope-based underground mining operations involves a series of activities such as orebody modeling, stope layout design, main access layout development (e.g., shaft, major raises, level spacing and primary drifts, and roadway network systems), equipment selection, and stope sequencing. Given that the problem size of the underground mine planning is large, existing computational resources cannot solve it optimally. Therefore, the problem is broken into sub-problems and sequentially solved. Some of these sub-problems are 1) development (i.e., selection of shaft location, shafts dimensioning, level or drift selection and dimensioning and decline design), 2) stope layout optimization (SLO), and 3) stope sequencing. To maximize operation performance, these activities should be synchronized [1]. The critical issue is whether the development or the stope layout sub-problem should be solved first. If the development is designed first, the mining direction will have to accommodate the selected shaft location and drifts. This design may not comply with vein direction and could lead to undervaluing project profit. If the stope layout is designed first, the drift orientation that follows the stope layout orientation may not maximize recovery of the orebody because SLO does not consider the mining direction as a decision criterion. SLO should include stope layout orientations as decision variables to ensure the maximum orebody mining recovery to minimize conflict between stope orientation and mining direction. Divide and conquer type of approaches does not guarantee to find optimal solutions. These two sub-problems should be solved simultaneously.

SLO research shows that flexible stope dimensions lead to better mining ore recovery in proportion to higher profits [2-5]. Respecting geotechnical and mining considerations, a sector in the deposit has consecutive stopes with equal excavation cross-section which may change in their lengthwise direction. In practice, these stopes are placed to try to follow the main orebody orientation to ensure high mining recovery of the orebody, and rotated geological block model—following main orebody orientation—may assist on this purpose. However, the rotated geological model option is challenged in complex deposits because the orebody orientations are not explicit in some deposits and may change by sectors.

The performance of a problem-solving exercise can be measured as a function of completeness (finding a solution), optimality (find an optimal solution), time complexity (computing time to find a solution) and space complexity (memory to perform the search) [6]. Orebodies are discretized into blocks, and some attributes of the block (e.g., grade and geology) are estimated/ simulated using geostatistical techniques. SLO requires the block model, as the main input parameter, to seek the stope configuration which generates the highest profit created through block aggregation under site-specific economic, operations and geotechnical considerations. Block size and shape can challenge SLO. Large and non-cubic blocks can limit the stope design in the block model’s parallel and orthogonal orientation. Although smaller blocks provide a better definition of the stope layout in any orientation, as well as efficient internal dilution management, they reduce SLO performance concerning computational time. A balance between block size and SLO performance should be assessed.

Significant research has been devoted to the SLO problem in the last two decades. ATAEE-POUR [2] suggested the maximum value neighborhood, where the stopes are selected from the evaluation of variable neighborhoods by each block. TOPAL et al [7] and SANDNAYAKE et al [4, 8] proposed heuristic stope layout approach constrained to geotechnical and physical considerations and used one predefined stope size at the time. VILLALBA et al [5] presented a heuristic approach to control stope layout inherent dilution and use variable stopes to provide flexibility during the profit maximization since orebody has intricate patterns. The maximum value neighborhood, floating stope approach, minable shape optimizer and two applications were compared by ERDOGAN et al [9]. Their solution times fluctuate between 5 h and 12 s and the most profitable solution compared with the real mine case is the minable shape optimizer approach. OVANIC et al [10] found the optimal solution for one-dimensional space only. Therefore, the heuristics defined above were significant progress to solve stope layout 3D problems in a reasonable time. The complexity of stope layout 3D problem is associated with geometrical, geotechnical, and operational constraints as well as the size of the problem. To improve the heuristic solution,stope layout research considers the application of intelligent search methods [11], such as annealing by MANCHUK et al [12] to find a stope layout solution that maximizes profit using an initial stope design as an initial solution. ALFORD et al [13] proposed a formulation based on a floating stope and adapted annealing for finding non-overlapping stopes. NIKBIN et al [14] proposed a polynomial-time dynamic programming algorithm that guarantees the optimal solution for a given row or column. This 1D algorithm is combined with a greedy algorithm as a hybrid approach to make it capable of solving 3D stope layout problems.

Moving from the deterministic to the stochastic solution, GRIECO et al [15] formulated a probabilistic stope layout mixed integer model. The geotechnical and operational constraints are satisfied by the use of rings; however, the definition of minable rings is before optimization, and the use of a level of risk neglects the joint local uncertainty. VILLAIBA et al [16] developed a stochastic optimization model to account for orebody uncertainty: 1) the quantification of stope layout uncertainty, 2) the assessment of an average design whereby feasibility evaluation breeds an initial population, and 3) the improvement of this initial population over generations using genetic algorithms (GA) operators. The first stage generates multiple stope layout scenarios based on equally probable orebody realizations, whereas mining operations implement a single stope layout scenario. Thus, an average design is calculated, which solution is improved through GA regarding feasibility and profit; however, this average design process is sensitive to the order in which the stope layout scenarios are evaluated.

Primary main access, such as shafts, ramps, levels, and drifts [17] should ensure lower haulage costs for underground mining networks. The shaft interacts with the system of ramps, drifts, and levels that access ore zones. GLIGORIC et al [18] presented shaft location selection as a decision criterion in network optimization. This selection considers access points (input parameter) nodes, transportation costs on the Steiner minimal tree, shaft development, operational costs and availability of the transportation system. Access points and primary main access layouts are fixed locations, which consider orebody location a relevant parameter in the decision-making process [19]; however, orebody patterns are uncertain because they are generated using sparse data. For instance, sectors with low mineral content may be not good placed for fixed locations because some may become profitable over time with newly acquired data. Otherwise, mine deposits contain locations or potential place for fixed locations that are not part of the stope layout in any scenario since these locations have low or no presence of mineral and are away from profitable ore sectors.

This paper explores a new approach in underground hard-rock mine planning where the average design process from VILLALBA et al [16] is improved by iterating stope layout scenarios orders until converging to the design with the highest metal contain. The originality of this paper also rests on finding a balance between orebody orientation and mining direction through SLO. That is, mining direction is treated as a decision criterion. Orebody uncertainty is incorporated into SLO with mining direction knowledge to define a configuration of stopes with knowledge about possible grade fluctuations. However, it could also identify “no profitable” sectors that have minimum probabilities to become profitable for the given current data, geological model, and economic parameters. Thus, stochastic stope layout optimization with optimistic projected economic values may assist in identifying “no profitable” sectors. These sectors could be considered potential places to locate primary main access with low risk of expensive relocation costs and access points in shaft location.

2 Model formulation

Stope layout design and mining direction are interrelated. If stope orientation and vein orientation differ, mining may advance in a direction orthogonal to veins, reducing metal production. The stope layout design provides information about the mining direction of levels or drifts because the direction of consecutive stopes imposes the mining direction. If the two directions differ, a balancing direction will be needed. This balancing direction is a decision-making problem and should be treated as a decision variable in SLO.

In conventional mining applications, using sparse data, the blocks are estimated with models. Model outputs are used as input parameters to mine design optimizations; however, the estimated models or single average models have imperfect knowledge of the mine deposit which is usually known as a significant source of uncertainty and risk. Geostatistical simulation assists to quantify orebody uncertainty and to reproduce grade variability to understand the extension of metal veins. Unlike conventional estimation techniques, simulation techniques provide multiple images of the deposit, which assist in quantifying the uncertainty of the actual description of the deposit [20-24]. Geostatistical simulation provides an opportunity to apply stochastic optimization approaches to mine planning [25-28].

Each image of the orebody generates a stope layout scenario, and N images produce N scenarios, which profit distribution corresponds to their space of uncertainty. The average value of these scenarios can be higher than the deterministic solution (Figure 1). Thus, optimization under uncertainty may deliver a higher profit because orebody realizations (or images) help to explore more search space in the SLO than an orebody single average does. Beyond the potential stochastic solution with higher profit showed in this stope layout problem, the main strength of this stochastic approach relies on providing well-informed solutions by accounting for orebody uncertainty.

Figure 1 Uncertainty space of N=100 stope layout scenarios (distribution), deterministic solution (orange line), and distribution mean (black line)

The optimization aims to maximize project profit under site-specific geotechnical, operational and economic constraints. During optimization, the ore/waste classification, processing recovery, and variable dilution for each stope make the optimization problem non-linear, similar to other mine planning optimization procedures. Thus, the stope layout value using the estimated model (single scenario) will not represent the average value of the stope layouts (Figure 1). The scenarios are built based on N equally probable orebody images, and their space of uncertainty provides the potential mean value; however, this value is not linked to any design of N scenarios. Since mining implementation may require one design, N stope layout scenarios are used to calculate their average design. This single design is a robust scenario because expected variations of the orebody grade are taking into account, given current data.

2.1 Stope average design improvement

For surface mining, CUBA et al [29] identified a set of representatives mining sequences using a modified hierarchical clustering technique. The mining sequences are illustrated as decision network branches, and the representative set is the branches that are more likely to occur. These selected mining sequences account for the variability of the mining deposits. An adaptation of this concept was implemented by VILLALBA et al [5] for underground mining where a single common design or average stope layout is computed from N stope layout solutions (or N mining regions) given orebody realizations. These mining regions are dissimilar because their configuration process depends on the spatial grade variability, orebody uncertainty, and SLO parameters.

The process follows mainly these steps: 1) compute the minimum Euclidian distance from blocks to mining regions and 2) if minimum distance is less than a translation ratio (a_j), blocks of mining regions are clustered. The translation ratio could be defined from the maximum stope size to deal with the dissimilarities among mining regions. After all mining regions are assessed in order 1 to N, each block stored the frequency of occurrence. The blocks which frequency represents more than half (N-1) regions define the common design. This design has a value which can change when the scenarios are evaluated in a different order than 1 to N region. From N region, the method starts evaluating the selected first region which blocks are linked with the remaining N-1 regions.Considering that a block is linked with the best fitting from other regions only one time, the following regions have every time fewer blocks to evaluate. For instance, following order 1 and having the minimum distance ≤a_j,the block 1 of stope 1, scenario 1 (Figure 2) and block 1 of scenario 2 and 3 are linked, while following order 3, block 1 of stope 3, scenario 3 is linked with only block 1 of scenario 1. Thus, choosing the first region to evaluate and order of regions in the evaluation may mainly influence the value of the average design. In Eq.(1), n may represent the region to start the clustering evaluation or order evaluation of scenarios; N regions imply at least N potential average designs or iterations.

In each iteration n, a binary decision variable z_naijk may take a value of one if a block k of stope i, at sector j is part of the average design using translation ratio and following orientation a. After N iterations, the metal-contain value, given tonnage (w_jk) and grade (g_jk), converges to the maximum value which is selected as the average design solution.

                    (1)

The average design is a well-informed design because it accounts for all possible variations of the grade, and optimal mining direction that ensures higher mining recovery. This process also identifies locations where feasible stopes configuration can be possible, that is, Kr(j) domain contains the blocks that have been chosen at least once to be part of any stope layout scenario given N equally probably images of the orebody. In this paper, the domain Kr(j) also may assist the decision-making during the haulage mining network design since Kr(j) identifies orebody parts that are unprofitable in all possible scenarios of the orebody, which is called here as “waste sectors” because these have almost zero chance to become profitable. These “waste sectors” may be potential locations for fixed access points, shaft, and other installations to avoid the high cost of relocation because of being built in a profitable sector. However, optimistic projected economic parameters should also be considered to identify better the deposits parts that do not have any prospects for future profitable extraction, and the Australasian Joint Ore Reserves Committee, JORC [30] suggested excluding these deposits parts even from the mineral resource. Thus, the location of fixed installations may consider a better informed “waste sector” provided by reduced domain, which accounts for orebody uncertainty and optimistic projected economic parameters.

Figure 2 Order evaluation influence on average design process

The clustering process considers mining directions but an additional step using GAs ensures that an average design is feasible matching mainly geotechnical and operational restrictions and improves their feasible solutions.

2.2 Applying GA to find near-optimal stope layout accounting for mining directions

GA improves the fitness value and allows us to escape from the local optimal by using operators in generating new solutions and considering a set of solutions instead of a single initial solution [31, 32]. For each generation, the new solutions inherit the best parts of the previous generation solutions to obtain better solutions in terms of fitness. The fitness value is calculated each solution h=1, …, H(e) of generation e=1, …, E using a modification of the stope layout algorithm proposed by VILLALBA et al [5], where additional mining directions are represented by index a=1, …, A (Eq. (2)). The fitness is the maximum value of the mining stopes configuration where internal dilution is minimized given ore grade spatial variability and geotechnical and operational considerations.

1, …, A, 1, …, H(e)           (2)

This fitness evaluation accounts for a constraint (3) that ensures that stopes have minimum percentage r of the average design. A binary variable (c_jk) to distinguish average design blocks.

1, …, I 1, …, J,

1, …, A, 1, …, H(e)           (3)

Constraint (4) ensures that a block can be selected as part of stope layout no more than once, where decision variable x_haijk decides if block k is part (or not) of stope i, in sector j, in the direction a, for solution h of generation e.

1, …, Kr(j), 1, …, A,

1, …, H(e)                       (4)

The mining directions are connected to the slice predecessor constraint (5) where blocks (η_aij) in height and width directions must be mined along block k. Ω_k' defines the set of slice predecessors, for each k in the direction a.

1, …, I, 1, …, J,

1,…, Kr(j), 1, …, A,

1, …, H(e)                       (5)

The Ω_k' for each block k at direction a must have an available number of blocks (Eq. (6)) equal to the blocks in height γ_j and width δ_j stope dimensions. The variable dw_a is related to the stope width dimension calculation and depends on the direction that stopes are being evaluated. For example, it could be equal to dy block dimensions if consecutive stopes face east or it could correspond to dx block dimensions if consecutive stopes face north.

1, …, I, 1, …, J,

1, …, A, 1, …, H(e)          (6)

Each stope must have a minimum length and precedence constraint (Eq. (7)) ensures this condition. φ_aij represents the number of blocks in stope direction where variable dl_a depends on the direction that stopes are being evaluated; for instance, if consecutive stopes are facing north dl_a=dy, each block k must be evaluated along a set of block successors (Ψ_k').

1, …, I, 1, …, J,

1, …, Kr(j), 1, …, A,

1, …, H(e)                       (7)

where

Due to changes in stresses, underground openings are subjected to failure after a continuous dynamic loading and enough time [33, 34]. Preserving stope stability ensures safety for the workforce and equipment [35]. Thus, the optimization considers hard constraints, the maximum stope sizes derived from geomechanical evaluations. The maximum lengthof each stope is related to decision variables (d_haij) and the second set of successor blocks γ_k' in constraint (8). The parameter θ_aij defines the number of consecutive blocks in the maximum length stope direction where the set of block successors γ_k' identified per block k being assessed.

1, …, I,

1, …, J, 1, …, Kr(j),

1, …, A, 1, …, H(e)           (8)

where

In orientation a, solution h and generation e, the decision variable d_haij in Eq. (9) controls the stope dimensions per sector. Equation (10) helps to compute the optimal length dimension of each stope through decision variable l_haij.

1, …, I, 1, …, J,

1, …, A, 1, …, H(e)           (9)

1, …, I, 1, …, J,

1, …, A, 1, …, H(e)          (10)

Following stope length direction, first slices of stopes are evaluated if succeeding blocks x_haijk’ are available (Eq. (11)). Binary parameter m_hajk' marks the availability of blocks for each direction a, solution h and generation e.

1, …, I, 1, …, J,

1, …, Kr(j), 1, …, A,

1, …, H(e)                      (11)

The internal slices, considered dilution, are gathered until reaching their allowed number of slices or decision variable (b_aj), whose value should not be larger than the maximum stope slices minus two slices (Eq. (12)), and the decision variable s_haij in Eq. (13) assists in defining the internal dilution slices of stope i at sector j,in direction a and solution h that maximize profit. Equation (14) links the optimal stope length and number of internal dilution slices. The constant 2 in Eqs. (12) and (14) correspond to the first and last slice of stopes, whose average grade must be ≥ cut-off grade. The option of SLO by sectors in this formulation may handle the dynamic nature of cut-off concerning time and location [36] since grades and mining cost can change through deposit space.

1, …, J, 1, …, A   (12)

1, …, J, 1, …, A,

1, …, H(e)                      (13)

1, …, I, 1, …, J,

1, …, A                         (14)

The stope slices are aggregated until matching the stope size constraints, and their average grade must be greater than or equal to the cut-off grade (Eq. (15)) unless they correspond to internal dilution slices that have an average grade lower than the cut-off grade but higher than a second cut-off grade. This second cut-off (cutoff_dilution) is lower than the cut-off that allows for considering slices that have some mineral content as internal dilution. These cut-offs depend on commodity price and metal recoveries as main variables; the impact of commodity price must be evaluated because of effect of stope layout design [37]. Equation (16) ensures that the average grade of blocks within each stope is greater than or equal to the cut-off grade. Furthermore, this stope layout formulation has two operational considerations; the stopes in both length and vertical directions should not overlap.

1, …, I,

1, …, J, 1, …, A, 1, …, H(e) (15)

1, …, I, 1, …, J, 1, …, A,

1, …, H(e)                      (16)

For each generation, the chromosomes or solutions are randomly chosen from a population. The selection operator may choose solutions that will be able to survive over time, leading to a population that tends toward an optimal solution over generations. To build the population each generation, the roulette-wheel method is applied, where the selection probability of each solution is related to its fitness value, that is, the new population is sampled from a probability distribution of fitness values [32, 38].

A locus is a location of a gene in the chromosome that corresponds to a block k in the solution h of size Kr(j). The allele of each locus is related to binary decision variable x_haijk, which takes the value of 1 if a block k is selected as part of stope layout design. Since a stope is a set of adjacent blocks that fulfill operational, economic, and geotechnical considerations, the crossover and mutation operators may decay the solution at any given generation because the stopes are destroyed, and good individuals are lost. To prevent the loss of good individuals, crossover and mutation operators do not consider the block locations of feasible stopes and infeasible stopes locations are the only candidates for modifications. Thus, some unfeasible stopes could be added to the solution because their blocks may have the option to become profitable using genetic operators since new arrangements of stopes are tested in each generation. Similar to other metaheuristic algorithms, the GA implemented in stope layout problem will need to be solved in multiple generations until a solution of the population converges. The details of a single generation are similar to VILLALBA et al [5]; however, the fitness evaluation was improved by adding mining directions criteria and using an improved cluster average design. The GA adaptation must ensure that offspring tends the best fitness and the population is diverse. Thus, the criteria for selecting block locations of unfeasible stope layout and locking locations of feasible solutions for crossover and mutation may help to improve fitness in each generation because the searching space of new stope configurations is expanded.

3 Case study

To illustrate the relevance of orientation along SLO and the robustness of improved average design, the proposed formulation was tested using data from a narrow vein of a gold deposit sector located in a volume of 140 m in the east, 188 m in the north and 150 m in the vertical direction. 109 composites of gold data and three-dimensional geological solid served as input data to assist in calculating the vein grades, with gold values ranging from 0.017 to 28.26 g/t. These composites were transformed into normal score units; then experimental semi- variograms were calculated to measure spatial correlation in gold data. Each experimental semi-variogram was modeled by nested spherical models with the maximum, medium, and minimum ranges of 200, 190 and 90 m, respectively, a principal orientation of anisotropy of 310° azimuth and minor orientation plunge 50°.

This variogram model was the primary input parameter into the sequential Gaussian simulation which evaluates deposits at point scale in Gaussian space 100 times (or realizations). These realizations must then be back-transformed into their original units. The average grade of these equally probable realizations is the estimated orebody at point scale. The orebody realizations, upscaled into blocks (Figure 3, left), were used to evaluate the stope layout scenarios.

The block model had 50400 units, each unit with a dimension of 4 m (E-W), 6 m (N-S) and 5 m (height). Among these units, 25697 flagged as the main geological domain (narrow vein) for grade evaluation. This paper considers N=100 realizations to avoid any bias in the subset selection and capture better the orebody uncertainty in SLO since the number of realizations is case-dependent.

The estimated single orebody model (also at block scale) was used to evaluate the deterministic stope layout for comparison purposes. Figure 4 illustrates the spatial distribution of the estimated gold value inside the narrow vein, where the average gold grade was 3.91 g/t.

In addition to the orebody model as an input parameter, the SLO requires satisfying the following considerations:

(a) Each stope must have an average grade ≥3.0 g/t, cutoff.

(b) The internal dilution slices must have average grades <3.0 and ≥0.5 g/t, cutoff_dilution, and the grades of barren blocks < 0.5 g/t.

(c) Block k, at sector j, with the grade (g_jk)<3.0 g/t is penalized by adjusting its recovery with a linear equation f(g_jk)=0.132g_jk +0.584 where the slope and intersect, respectively, are given by 0.132=(R-R_dilution)/(cutoff-cutoff_dilution) and 0.584= -cutoff×0.132+R, where metal recovery, R=0.98, metal recovery for internal dilution minimum grade, R_dilution=0.65.

Figure 3 N orebody realizations (left) and respective stope layout scenarios (right)

(d) Stopes can range from 12 m× 12 m× 15 m to 24 m× 12 m× 15 m (length×width×height) in any orientation that provides the best mining ore recovery, which ensures a higher profit.

Figure 4 Estimated gold grade spatial distribution of narrow vein

Thus, the stope layout algorithm was modified to consider eight orientations clustered in four groups that evaluate consecutive stopes in: 1) north or south orientation, 2) 45°NE or 235°azimuth,3) east or west orientation, and 4) 135° or 315° azimuth. The stope layout was assessed for these four groups to measure the cost of choosing the wrong orientation and finding the optimal one. After solving the modified SLO, north-south was determined to be the optimal direction to mine consecutive stopes (Figure 5, left). The design delivers US$ 18711481 of profits (Table 1). By comparison, Group 4 yielded a 48% less profit, Group 2 yielded 40% less profit, and Group 3 (Figure 5, right) yielded 25% less profit. Stopes length fluctuates between 12 m and 24 m with smaller height and width dimensions where 10% more stopes were found along the optimal north-south orientation than the east-west orientation (Figure 5, left).

Thus, consecutive stopes in the north-south directions maximize the ore recover. Neglecting stope orientation during SLO may generate less profit after mine production schedule. 100 orebody realizations generate 100 stope layout scenarios (blue wireframe in Figure 6) that can assist in identifying potential places for stope layout. Shafts, ramps, major raises, level spacing, and primary drifts could be located elsewhere to avoid the high cost of relocation: they were identified 100 times as non-profitable areas. The potential places for stope layout represent an area of uncertainty for stope layout.

Figure 5 Stope layouts following north-south orientation (left) and east-west orientation (right)

Table 1 Stope layout results based on orientation

Figure 6 Wireframe of potential places for stope layout and GA solution

The C++ based optimization of 100 scenarios for given orebody realizations took 121 h and 58 min in a dual-core processor computer (3.70 GHz, 16 GB RAM). Mine production requires one scenario; therefore, the average design of these 100 scenarios was computed using a translation ratio of 31 m to find similarities among stope layouts which locations differed among scenarios. The order evaluation of the scenarios will influence the average design. Thus, 100 iterations with different scenario orders were also evaluated to find the design with the maximum gold quantity. For the GA-based approach, the average design with the best gold quantity reached at 79 iterations with 39260 oz (Figure 7), which is 3% higher than the deterministic solution.

Figure 7 Gold mining recovery over 100 iterations with different scenario orders

The stope layout blocks of 100 scenarios were aggregated into a single scenario, where each unique location is associated with a relative frequency (blocks with greater than or equal 0.5 relative frequency represent the average design). This single scenario defined the Kr(j) domain of 2424 blocks, which is 90% smaller than K(jthe) domain of 25697 blocks and shows a reduction in the computational time of GA.

Since the average design process did not follow the stope layout constraints, the average design feasibility and fitness value are assessed 12120 iterations, where each stope should have >10% of the average design. The initial population of forty solutions (Figure 8 as generation 0) is sampled from the cumulative distribution of 229 feasible solutions, which minimum, and maximum fitness values are US$12745600 and US$18870100, respectively. The chromosome or solution has 2142 blocks where feasible stope layout blocks [482-711] are flagged with 1. The infeasible stope locations were included, increasing percentage of blocks [35%-65%] with a value of 1, to improve the performance of genetic operators. Since new stope configurations are evaluated for each generation, some infeasible blocks may become feasible.

Figure 8 Convergence of stope layout solution after 250 generations

Several combinations were tested to choose the main parameters of GA, such as the initial population size, crossover and mutation rate. Also, the calibration of these parameters is further improved by using experimental designs [39]. The sensitivity analysis of these parameters showed that profit might fluctuate up to 7%. In this case study, GA used the initial population size of 40, a crossover rate of 0.4, and a mutation rate of 0.10. GA-based approach took 25 h and 20 min of computing time of 250 generations to find the best feasible solution, which started to converge at approximately the 47th generation with no improvements in consecutive generations (Figure 8).

GA-based approach explores and exploits the solution space efficiently by using GA operators. Thus, the stochastic stope layout solution following the north-south orientation (Figure 6) has 7.2% more profit, 25% fewer barren blocks, and 4% higher average stope grade than the deterministic solution. The main contribution of this approach is to find the optimal direction to minimize losses, which can account for 25% and 48% of the profit when using the wrong orientation. The case study for narrow vein illustrates the benefit of the proposed approach.

4 Conclusions

There is a strong relationship between mining direction and stope layout design in underground mining operations. To date, stope layout approaches have not addressed the flexibility to evaluate multiple stope layout orientations. In practice, the stope layout design tends to follow the principal orientation of the orebody; however, complex deposits, with no evident orientation as a single narrow vein, may require SLO tools that also assist in finding the optimal stope layout orientation. Otherwise, the underground mine planning process may undervalue a project. This paper proposes an approach to evaluate stope layout orientations along the stope layout design. A case study in a portion of a narrow vein deposit demonstrated value-adding potential by considering different stope layout orientations. Consecutive stopes in the north-south orientation maximize ore mining recovery. Other orientations generated 25%-48% less profit, indicating that SLO that neglects orebody orientation may be less profitable.

The proposed approach can assist mining management in creating a development plan. To avoid the high cost of relocation, the access point and production shaft can be located close to the beginning of the stope mining location and perpendicular to consecutive stope layout orientation. Furthermore, the union of stope layouts scenarios can help to identify the locations where the primary mine accesses should not be placed. Also, non-profitable locations (with a low probability of being a stope) are identified multiple times through the set of stope layout scenarios.

The approach presented here explores the solution space efficiently by evaluating possible new stope layouts through GA. The use of the average design (calculated from 100 stope layout scenarios using orebody realizations) as part of the GA initial solution yielded a well-informed solution. The average design process and the stope layout solution following the north-south orientation delivered 7.2% more profit, 25% fewer barren blocks, and 4% more average stope grade than the deterministic solution. A case study with a single narrow vein illustrated the benefit of the proposed approach; however, the approach will be even more valuable in deposits without clear orebody orientations. Future research will focus on optimizing GA parameters (e.g., population size, and mutation and crossover probabilities) to further increase project profitability.

References

[1] STADLER H. A framework for collaborative planning and state-of-the-art [M]// Supply Chain Planning. Springer, 2009: 1-26.

[2] ATAEE-POUR M. Optimisation of stope limits using a heuristic approach [J]. Mining Technology, 2004, 113(2): 123-128. DOI: 10.1179/037178404225004959.

[3] ATAEE-POUR M. A critical survey of the existing stope layout optimization techniques [J]. Journal of Mining Science, 2005, 41(5): 447-466. DOI: 10.1007/s10913-006- 0008-9.

[4] SANDANAYAKE D S, TOPAL E, ASAD M W A. A heuristic approach to optimal design of an underground mine stope layout [J]. Applied Soft Computing, 2015, 30: 595-603. DOI: 10.1016/j.asoc.2015.01.060.

[5] VILLALBA M E, KUMRAL M. Heuristic stope layout optimization accounting for variable stope dimensions and dilution management [J]. International Journal of Mining and Mineral Engineering, 2017, 8(1): 1-18. DOI: 10.1504/ IJMME.2017.10003210.

[6] RUSSELL S, NORVIG P. Artificial intelligence: A modern approach [M]. 3rd ed. Pearson Education, Inc., 1995.

[7] TOPAL E, SENS J. A new algorithm for stope boundary optimization [J]. Journal of Coal Science & Engineering (China), 2010, 16(2): 113-119. DOI: 10.1007/s12404- 010-0201-y.

[8] SANDANAYAKE D S, TOPAL E, ASAD M W A. Designing an optimal stope layout for underground mining based on a heuristic algorithm [J]. International Journal of Mining Science and Technology, 2015, 25(5): 767-772. DOI: 10.1016/j.ijmst.2015.07.011.

[9] ERDOGAN G, CIGLA M, TOPAL E, YAVUZ M. Implementation and comparison of four stope boundary optimization algorithms in an existing underground mine [J]. International Journal of Mining, Reclamation and Environment, 2017, 31(6): 389-403. DOI: 10.1080/ 17480930.2017. 1331083.

[10] OVANIC J, YOUNG D S. Economic optimization of stope geometry using separable programming with special branch and bound techniques [C]// MITRI H S. The 3rd Canadian Conference on Computer Applications in the Mineral Industry. Montreal, 1995: 129-135.

[11] VOβ S. Meta-heuristics: The state of the art [M]// NAREYEK A. Workshop on Local Search for Planning and Scheduling. Springer, 2000: 1-23.

[12] MANCHUK J, DEUTSCH C V. Optimizing stope designs and sequences in underground mines [J]. SME Transactions, 2008, 324: 67-75.

[13] ALFORD C, HALL B. Stope optimisation tools for selection of optimum cut-off grade in underground mine design [C]// Project Evaluation Conference. Melbourne, 2009: 137-144.

[14] NIKBIN V, ATAEE-POUR M, SHAHRIAR K, POURRAHIMIAN Y. A 3D approximate hybrid algorithm for stope boundary optimization [J]. Computers & Operations Research, 2018. DOI: 10.1016/j.cor.2018.05.012.

[15] GRIECO N, DIMITRAKOPOULOS R. Managing grade risk in stope design optimisation: probabilistic mathematical programming model and application in sublevel stoping [J]. Mining Technology, 2007, 116(2): 49-57. DOI: 10.1179/ 174328607X191038.

[16] VILLALBA M E, KUMRAL M. Underground mine planning: Stope layout optimization under uncertainty using genetic algorithms [J]. International Journal of Mining, Reclamation and Environment 2019, 33(5): 353-370. DOI: 10.1080/17480930.2018.1486692.

[17] MORIN M A. Underground hardrock mine design and planning: A system’s perspective [D]. Kingston, Ontario: Queen's University, 2001.

[18] GLIGORIC Z, BELJIC C, SIMEUNOVIC V. Shaft location selection at deep multiple orebody deposit by using fuzzy TOPSIS method and network optimization [J]. Expert Systems with Applications, 2010, 37(2): 1408-1418. DOI: 10.1016/j.eswa.2009.06.108.

[19] BRAZIL M, THOMAS D A, WENG J F, RUBINSTEIN J H, LEE D H. Cost optimisation for underground mining networks [J]. Optimization and Engineering, 2005, 6(2): 241-256. DOI: 10.1007/s11081-005-6797-x.

[20] JOURNEL A G, HUIJBREGTS C J. Mining geostatistics [M]. Academic Press, 1978.

[21] DEUTSCH C V, JOURNEL A G. Geostatistical software library and user’s guide [M]. New York: Oxford University Press, 1998.

[22] DIMITRAKOPOULOS R. Conditional simulation algorithms for modelling orebody uncertainty in open pit optimisation [J]. International Journal of Surface Mining, Reclamation and Environment, 1998, 12(4): 173-179. DOI: 10.1080/09208118908944041.

[23] DEUTSCH C V. Geostatistical reservoir modeling [M]. New York: Oxford University Press, 2002.

[24] JOURNEL A G, KYRIAKIDIS P C. Evaluation of mineral reserves: A simulation approach [M]. New York: Oxford University Press, Inc., 2004.

[25] DIMITRAKOPOULOS R, RAMAZAN S. Stochastic integer programming for optimising long term production schedules of open pit mines: methods, application and value of stochastic solutions [J]. Mining Technology, 2008, 117(4): 155-160. DOI: 10.1179/174328609X417279.

[26] DIMITRAKOPOULOS R. Stochastic optimization for strategic mine planning: A decade of developments [J]. Journal of Mining Science, 2011, 47(2): 138-150. DOI: 10.1134/S1062739147020018.

[27] BENNDORF J, DIMITRAKOPOULOS R. Stochastic long-term production scheduling of iron ore deposits: Integrating joint multi-element geological uncertainty [J]. Journal of Mining Science, 2013, 49(1): 68-81. DOI: 10.1007/978-3-319-69320-0_12.

[28] VILLALBA M E, DIMITRAKOPOULOS R. Stochastic short-term mine production schedule accounting for fleet allocation, operational considerations and blending restrictions [J]. European Journal of Operational Research, 2016, 255(3): 911-921. DOI: 10.1016/j.ejor.2016.05.050.

[29] CUBA M A, BOISVERT J B, DEUTSCH C V. Simulated learning model for mineable reserves evaluation in surface mining projects [J]. SME Transactions, 2013, 334: 527-534.

[30] JORC. Australaisian code for reporting of exploration results, mineral resources and ore reserves [M]. AusIMM and AIG. 2012: 44.

[31] MITCHELL M. An introduction to genetic algorithms [M]. Massachusetts Institute of Technology, 1999.

[32] GOLDBERG D E. Genetic algorithms in search, optimization, and machine learning [M]. New York: Addison Wesley Longman, 1989.

[33] BAWDEN J W, NANTEL J, SPROTT D. Practical rock engineering in the optimization of stope dimensions- application and cost-effectiveness [J]. CIM Bulletin, 1989, 82(926): 63-70.

[34] WU A X, HUANG M Q, HAN B, WANG Y M, YU S F, MIAO X X. Orthogonal design and numerical simulation of room and pillar configurations in fractured stopes [J]. Journal of Central South University, 2014, 21(8): 3338-3344. DOI: 10.1007/s11771-014-2307-7.

[35] HEIDARZADEH S, SAEIDI A, ROULEAU A. Evaluation of the effect of geometrical parameters on stope probability of failure in the open stoping method using numerical modeling [J]. International Journal of Mining Science and Technology, 2018, 29(3): 399-408. DOI: 10.1016/j.ijmst. 2018.05.011.

[36] GU X W, WANG Q, CHU D Z, ZHANG B. Dynamic optimization of cutoff grade in underground metal mining [J]. Journal of Central South University of Technology, 2010, 17(3): 492-497. DOI: 10.1007/s11771-010-0512-6.

[37] SALAMA A, NEHRING M, GREBERG J. Evaluation of the impact of commodity price change on mine plan of underground mining [J]. International Journal of Mining Science and Technology, 2015, 25(3): 375-382. DOI: 10.1016/j.ijmst.2015.03.008.

[38] REEVES C. Genetic algorithms [M]// Handbook of Metaheuristics. New York: Kluwer Academic, 2003: 55-82.

[39] VILLALBA M E, KUMRAL M. Calibration of genetic algorithm parameters for mining-related optimization problems [J]. Natural Resources Research, 2019, 28(2): 443-456. DOI: 10.1007/s11053-018-9395-2.

(Edited by YANG Hua)

中文导读

硬岩地下开采的一种增值方法：通过元启发式算法优化平衡矿体走向和开采方向

摘要：地下矿山需要复杂的基建工作，包括竖井、中段、提升、盲井和溜井。使用空场法开采地下矿山时，要确定能使利润最大化的矿体部分，该过程称为采场布局优化(SLO)，需要在特定场地的岩土工程、运营和经济约束下实施。在实际应用中，通过采场布局优化得到的设计图显示了一条路径上的连续采场，这有助于确定这些采场的开采方向。但是，该方向可能无法适应矿石品位的空间分布：如果矿体走向和开采方向不同，开采作业的价值可能会降低。本文提出了一种把采场布局优化中的路径定义为决策变量的方法，以避免在错误方向上开采造成损失。此外，在考虑矿体不确定性的遗传算法中，提出了一种鲁棒聚类平均设计方法，以提高采场布局优化在金属含量方面的性能。对薄金矿脉的实例研究表明，如果算法中忽略采场布局方向，地下采矿作业的利润可能会被低估25%～48%。

关键词：地下矿井规划；矿体不确定性；矿体走向；开采方向；采场布局优化

Foundation item: Project(488262-15) supported by the Natural Sciences and Engineering Research Council of Canada

Received date: 2018-08-17; Accepted date: 2019-07-20

Corresponding author: Mustafa KUMRAL, PhD, Associate Professor; Tel: +1-514-398-3224; E-mail: mustafa.kumral@mcgill.ca; ORCID: 0000-0003-1370-7446

Abstract: Underground mines require complex construction activities including the shaft, levels, raises, winzes and ore passes. In an underground mine based on stoping method, orebody part(s) maximizing profit should be determined. This process is called stope layout optimization (SLO) and implemented under site-specific geotechnical, operational and economic constraints. For practical purpose, the design obtained by SLO shows consecutive stopes in one path, which assists in defining the mining direction of these stopes. However, this direction may not accommodate the spatial distribution of the ore grade: if the orebody orientation and mining direction differ, the value of the mining operation may decrease. This paper proposes an approach whereby paths in the SLO are defined as decision variables to avoid the cost of mining in the wrong direction. Furthermore, in the genetic-based formulation, which accounts for orebody uncertainty, a robust cluster average design process is proposed to improve SLO’s performance regarding metal content. A case study in narrow gold vein deposit shows that the profit of an underground mining operation could be underestimated by 25%-48% if the algorithm ignores stope layout orientation.