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A review of underground stope boundary optimization algorithms
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A review of underground stope boundary optimization algorithms ⁎
AS Nhleko , T. Tholana, PN Neingo Lecturer, School of Mining Engineering, University of the Witwatersrand, Johannesburg, South Africa
A R T I C L E I N F O
A B S T R A C T
Keywords: Optimization Stope boundary Deterministic Stochastic Algorithm Underground mining
Optimization is a key aspect of the mine design and planning process. A number of algorithms and techniques have been developed to optimize mines. However, most of these techniques focus on open pit optimization to an extent that some authors argue that open pit limit optimization has reached saturation level. Optimization of underground mines only received attention in recent decades and has been focused on three main areas including stope boundary optimization. This paper reviews and analyzes literature on algorithms developed to date for stope boundary optimization. There has been an increase in the number of algorithms developed to optimize stope layout. Most of these algorithms are heuristic, consider stope dimension as one of the constraints and optimize layouts in three dimensional space. However, all these algorithms are based on deterministic orebody models, therefore, fail to consider uncertainty intrinsic in ore deposits. Also, none of these algorithms guarantee optimal stope layout solution in three dimension. Consequently, there is a need for further research in the field of stope boundary optimization.
1. Introduction
mines is computationally more complex than open pit mines and hence the less algorithms that give optimum solutions for underground mines. Unlike in open pit mines, the main challenge in developing a generic methodology for the optimization of underground mines is that there is a wide range of underground mining methods available and their application varies among mine sites, hence each deposit requires a specialized optimization solution (Alford et al., 2007; Sandanayake, 2014). Underground mines also require consideration of other constraints not applicable to open pit mines including ventilation and size of equipment to fit into stopes. Also, in open pit mines, for any given slope angle there is a single option available to remove a given block. Nevertheless, for underground mines there are several options available (Sandanayake, 2014). All these factors make developing optimization solutions for underground mines more computationally complex and hence the lesser amount of research work currently available compared to open pit mines. Alford et al. (2007, p. 574) stated that “the complexity of the underground mine design problem and the unique mine design solutions sought for each ore body suggest that there will never be an elegant solution method analogous to that which exists for open pit mining”. Musingwini (2016) stated that because of this computationally complex nature of optimizing underground mines most of the current work on optimization in underground mine planning is mostly academic. Topal and Sens (2010) mentioned that optimization techniques in underground mines have been mainly focused on three main strategic mine planning areas:
Optimization in general involves either maximizing or minimizing an objective function against a given set of constraints. The objective can be minimizing inputs to a process because they are scarce, minimizing undesired outputs such as waste or maximizing desired outputs such as Net Present Value (NPV). Irrespective of the objective, the basis of optimization is a mathematical model that represents the problem that is then solved using an algorithm. To date, optimization algorithms have played an important role in mine planning and decision-making though more faster and efficient techniques still require to be developed (Little, 2012). Optimization techniques for mining operations date back to the 1960s with initial research developments and application in surface mining. In open pit mining, considerable strides have been made in developing algorithms for open pit optimization. In contrast however, fewer algorithms have been developed for underground mine optimization resulting in most underground mines operating on sub-optimal mine plans particularly in the area of optimization of stope boundary and layout definition (Little, 2012). Optimization in underground mines only started in the 1970s as an extension of open pit optimization applications and to date only a few optimization techniques have been developed to solve underground optimization problems. However, Ataee-Pour (2005) indicated that most of those few optimization techniques for underground optimization have proved not to provide optimum solutions. This is because the optimization of underground ⁎
Corresponding author. E-mail address: [email protected] (A. Nhleko).
https://doi.org/10.1016/j.resourpol.2017.12.004 Received 31 July 2017; Received in revised form 1 December 2017; Accepted 6 December 2017 0301-4207/ © 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Nhleko, S., Resources Policy (2017), https://doi.org/10.1016/j.resourpol.2017.12.004
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1. Optimization of stope boundaries; 2. Optimization of location of development and infrastructure; and 3. Optimization of production schedules.
methods; equivalent to an ultimate pit limit in open pit mining. Defining an optimum stope limit is fundamental in optimizing value from the extraction of a mineral deposit. It is a critical element of strategic mine planning. Little (2012) mentioned that the aim of stope boundary optimization is to select the best combination of blocks to form a series of stopes based on value measures such as grade or profit while satisfying physical mining and geotechnical constraints. The process to define an optimum stope layout combines thousands of blocks into a set of stopes, such that, the undiscounted value is maximized whilst satisfying physical and geotechnical constraints (Sandanayake et al., 2015). Algorithms that have been developed for the optimization of ultimate stope limits are categorized as either rigorous or heuristic (Ataee-Pour, 2006). These exact/rigorous and heuristic algorithms are reviewed in the following sub-sections.
A number of optimization techniques and algorithms have been developed for optimization of each of those areas in underground mines. However, this paper focuses on stope boundary optimization because stope boundary optimization provides the first opportunity to mine planners to optimize and maximize the NPV of a mining project. It is therefore, the aim of this paper to review and analyze literature available on underground stope boundary optimization algorithms. In this paper the phrases ‘stope boundary’, ‘ultimate stope limits’, ‘stope envelope’ and ‘stope layout’ are used interchangeably. 2. Orebody modelling
3.1. Exact algorithms Irrespective of the mining method, either underground or surface, mining starts with prospecting and exploring for mineral resources of economic interest. Based on borehole data and geological information, a geological block model is created. The geological modelling process starts by dividing the orebody into regular blocks in three dimensions, with each block containing its characteristic data; most importantly grade, volume and density. From the geological block model subsequent evaluation processes are done including geostatistical techniques applied to estimate the quantity and quality of the mineral deposit and economic evaluation to convert the geological block model to an economic block model which is one of the key input for optimizing stope boundaries. However, there are other algorithms that generate a stope layout solution based on cut-off grade and head grade such as Floating Stope Algorithm. As mentioned earlier, stope boundary optimization is one of the first opportunity available to mine planners to optimize the long term value of a mining project. Erdogan et al. (2016) mentioned that it may be considered as the starting point in the full optimization process for underground mines when both development and production schedule are considered. The economic orebody limits must be defined first which then allows the optimal location of key development access routes such as shafts, declines, tunnels and raises to be identified. Therefore, incorrect definition of stope boundaries results in incorrect placement of underground infrastructure which may require that the mine design be modified later in the life of the mine (LOM). In other words, an optimum stope boundary determines the efficacy of the mine design and subsequently, the long term production schedule, which then informs the cash flow profile and ultimately, the NPV of a project. Therefore, it is important that its in-situ representation is accurately modelled and understood. Traditionally, to determine whether it will be economic to mine a particular block, the geological block model is converted to an economic block model by applying geological and economic parameters to each block to determine all economically mineable blocks to be included in the ultimate mining limits. The economic modelling process is based on calculating the revenue derived from each block and the cost of mining each block, comparing these two values on a block by block basis to get Block Economic Values (BEVs). For each set of cost and revenue parameters applied, the BEVs distinguish payable and unpayable ore blocks. A block will be economic to mine if the revenue from mining is greater than the cost of mining and processing, that is, if the block economic value is positive. From the economic block model, stopes are created and the following section review algorithms that have been developed to optimize stope boundaries.
Exact algorithms are those algorithms based on a mathematical model and hence they guarantee an optimum solution. These include the Dynamic Programing, Downstream Geostatistical and Branch and Bound algorithms. This section reviews the exact algorithms with the main focus being their shortfalls concerning generation of optimal solutions in 3 dimensional (3D) space. 3.1.1. Dynamic Programming algorithm Riddle (1977) developed an algorithm based on the dynamic programming technique in order optimize stope layouts for block caving mines. The algorithm is a modification of the Dynamic Programming algorithm for optimizing open pit limits by Johnson and Sharp (Shahriar et al., 2007). In the algorithm the relation of height mined is constrained by draw control. Riddle (1977) describes the optimization process by the algorithm mentioning that it starts by first assuming that there is no footwall region and then a minimum number of adjacent draw-points that should be mined in any discrete mining unit are established. The algorithm then assumes that one footwall region is added within the section and then it also establishes a minimum number of adjacent draw-points that should be left in a discrete, non-mined footwall region. Each combination of mined and non-mined draw-points is investigated and the profit of each combination is taken. If the profit with a footwall is greater or equal to the maximum profit obtained for the no-footwall case then a more optimum layout exists with a footwall case and the process is repeated to the two areas divided by a footwall. For example, if the north block is more optimal with a footwall, the process is continued by adding the southern region to the previous optimal condition analyzing the north end for further footwall (Riddle, 1977). This process is continued until no more optimum footwall case is found or it is no longer practical to add further footwalls. In his review of stope layout optimization techniques Ataee-Pour (2005) highlighted the limitations of the algorithm. DP generates optimum stope layouts in two dimension. The optimum 2D sections can be combined to generate a 3D stope layout. However, stope constraints can be violated during the process, thus, optimality in 3D is not guaranteed. The other limitation is that the algorithm is only applicable to block caving mines and is not applicable to other mining methods. 3.1.2. Downstream Geostatistical approach Developed by Deraisme et al. (1984), the approach defines an optimal economic stope design based on downstream geostatistics. Downstream geostatistics is defined by Deraisme p. 583) et al. (1984) as “the methodology for the study of the influence of mining constraints on mining recovery and ore quality”. This approach introduces the application of geostatistics to determine the boundary of mineable ore in an orebody. It can be applied when the mining method to be used is cut and fill or sublevel stoping. It uses downstream geostatistics to build 2D sectional numerical models of the orebody and delineate the ore to
3. Algorithms for stope boundary optimization A stope is a mining area in underground mining which consists of a number of mining blocks. An optimum stope boundary is the limit for material that can be mined economically through underground mining 2
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Fig. 1. Compared outlines of the cut and fill mining method on a section. (Source: Ataee-Pour, 2000).
The cumulative economic values are placed at the beginning and ending points of the blocks (Ataee-Pour, 2000). This approach can generate optimal solutions in one dimensional space along the mining direction, thus, it can only be applied to simple orebodies. It does not take into account the slope constraints, solving mixed integer programming problems is time consuming when working with a large number of blocks (Ataee-Pour, 2000; Bai, 2013; Nhleko and Musingwini, 2017).
be mined. According to Ataee-Pour (2000), when non-linear geostatistics technique failed to estimate mineable reserves, Downstream Geostatistical approach was introduced as an alternative. The approach was introduced when linear and non-linear geostatistics failed to estimate mineable reserves (Ataee-Pour, 2000). The determination of the mineable reserves is controlled by mining constraints of the specific mining method selected to be used to exploit the orebody. Fig. 1 shows the comparative outlines of a section using the Downstream Geostatistics approach and the manual approach. It can be seen that the results are almost identical. This approach can generate 3D solutions, however, it does not take into account the economic profit of the block as its morphological operations only regulate the geometry (Ataee-Pour, 2000; Bai, 2013). This approach cannot guarantee true optimality as it neglects the profitability of the stopes included in the stope boundary.
3.2. Heuristic algorithms Heuristic algorithms are not supported by a mathematical formulation and hence provide a solution which is not necessarily an optimum one but, one that is close to an optimum solution. Silver (2004, p. 936) defined heuristic as “a method which, on the basis of experience or judgement, seems likely to yield a reasonable solution to a problem but which cannot be guaranteed to produce the mathematically optimal solution”. Examples of heuristic stope optimization algorithms include the Octree Division algorithm, the Floating Stope algorithm, the Multiple Pass Floating Stope algorithm, the Maximum Value Neighbourhood algorithm, Topal and Sens (2010)’s algorithm, Sandanayake (2014)’s algorithm and Simulated Annealing algorithm.
3.1.3. Branch and Bound algorithm for integer programming Mining problems that can be solved using linear programming, integer programming and mixed integer programming are generally too large and difficult to solve as a whole. Therefore, by applying the Branch and Bound approach, it is possible to divide the problem into sub-problems which are relatively easier to solve. As cited in AtaeePour (2005) Ovanic and Young in 1995 developed an algorithm to optimize stope boundary based on branch and bound approach. The algorithm uses mixed integer approach called Type-Two Special Ordered Sets (SOS2). The objective of this approach is to identify the start and end locations for mining given stope parameters. The objective is realized by the application of the piecewise linear functions. The first function sums all block values along the row that will be included in the stope layout, while the second function sums the block values to be excluded in the stope layout. The difference between these two functions is the sum of all block values lying between start and finish positions (Ataee-Pour, 2000; Little, 2012). A hypothetical situation where a row of blocks with their respective block positions and net economic block values is shown in Fig. 2. The objective function is to determine the optimum stope layout that maximizes the block economic values. The block positions along the panel are considered as reference values in the piecewise linear function. Whereas the economic values are considered to be attribute values, however, the cumulative economic values of the blocks are used.
3.2.1. Octree Division algorithm Cheimanoff et al. (1989) developed the Octree Division algorithm, a rule-based algorithm that is contained in the GEOCAD package that converts resources to mining reserves by identifying mineable volumes in 3D. The algorithm gathers borehole data, geostatistical analysis data, shapes and any other geological objects to build a geological block model. The algorithm then converts the geological block model to mineable reserves taking to account mining and economic constraints. The mineable reserves are divided into sub-volumes, i.e. minimum allowable stope size, for further economic evaluations. These sub-volumes are assessed based on the following rules (Ataee-Pour, 2000):
• A sub-volume containing no mineralised vein and satisfies the constraints is removed; • A sub-volume that has the minimum allowable stope size or has all the mineralised vein is stored in the mineable block; • A sub-volume containing a proportion of the mineralised vein, i.e. •
Fig. 2. A row of economic value blocks. (Source: Sandanayake, 2014).
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the sub-volume is located partly inside the ore and partly outside the ore, and does not have the minimum allowable stope dimension is divided into eight equal sub-volumes; The algorithm is terminated when there is no sub-volume in the list that has not been evaluated.
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main disadvantage is that it produces overlapping stopes with shared mining blocks. In mining, a block cannot physically belong to more than one stope. Therefore, the generated results require manual adjustments in order to exclude the shared blocks from the final stope layout. The manual adjustment is dependent on the input of the mine planning engineer (Ataee-Pour, 2000; Little, 2012; Sandanayake, 2014). After the manual adjustment, the derived economic value of the contributing stopes will be different from the actual economic value generated by the algorithm. Thus, this algorithm does not guarantee a true optimal solution. 3.2.3. Multiple Pass Floating Stope algorithm The Multiple Pass Floating Stope (MPFS) algorithm was developed by Cawrse in 2001 as an extension of the Floating Stope algorithm to try and improve the functionality of Datamine software (Sens, 2011). It allows more envelopes to be generated by applying a multiple-optimization process. This process is implemented in three steps (Sandanayake, 2014):
Fig. 3. Illustration of the Octree division approach. (Source: Sandanayake, 2014).
• Definition of input parameters such as head grade, cut-off grade and maximum waste; • File generation: creation of economic stope envelopes for each set of parameters; • File management: conversion of data files or the statistical files into
Fig. 3 shows an example of the Octree Division algorithm. The removal of sub-volumes is performed by using the syntactic pattern recognition technique (Ataee-Pour, 2000). Fig. 3(a) presents the division of a block into eight equal sub-volumes while Fig. 3(b) shows sub-volumes being removed from the mineable block by the algorithm. The Octree Division algorithm generates 3D stope layouts. However, blocks with minimum allowable sizes are included in the final stope layout irrespective of the amount of ore contained. Thus, the net value of the stope is reduced if blocks containing waste are included in the final design. Therefore, the algorithm does not guarantee that the final stope layout is the optimal solution.
a Microsoft Excel compatibles (CSV) format.
This process provides the mine planning engineer with extra information when selecting the best stope layouts. However, it does not eliminate the drawbacks of the Floating Stope algorithm. Therefore, this process cannot guarantee true optimal stope layouts. According to Little (2012), the Australian Minerals Industry Research Association P884 research project, Planning & Rapid Integrated Mine Optimization (PRIMO) developed the Vulcan stope optimizer which appears to address some of the shortcomings of the Floating Stope algorithm. For example, it is able to specify mineable shapes instead of general areas. The specified mineable shapes can be for a range of cut-off grades, mining dimensions & orientations, and mining methods.
3.2.2. Floating Stope algorithm The Floating Stope algorithm was developed by Alford (1995) and is incorporated into Datamine's Mineable Reserve Optimizer (MRO) software. The approach of this algorithm is analogous to the Moving Cone algorithm for open pit limit optimization. It is typically applied for preliminary resource appraisal, selection of the stoping method and detailed mine design. The first step in this algorithm is to specify the cut-off grade in order to separate ore and waste blocks. Subsequently, a rectangular block representing a stope with minimum stope dimensions is specified. The main constraint is the geometry of the stope, which is translated into the minimum stope dimension. The specified stope is floated in a fixed block model relative to its origin, with predefined float increments in three orthogonal directions (three dimensions). Two envelopes are used during this floating process as shown in Fig. 4 that is outer and inner envelopes. The inner envelope is made of the union of all the best grade stope shapes which may be defined from all of the blocks above the cutoff grade that can be exploited. Whereas the outer envelope is the union of all possible stope positions for each block, defined from all of the blocks above the cut-off grade that can be mined (Ataee-Pour, 2000; Little, 2012). The main advantage of this algorithm is it simplicity. However, its
3.2.4. Maximum Value Neighbourhood algorithm Ataee-Pour (2000) developed the Maximum Value Neighbourhood algorithm (MVN) based on the principle of the Floating Stope algorithm. This algorithm defines the mining blocks called neighbourhoods (NB), which should be mined together to satisfy the minimum stope size constraint. These blocks are constrained by the geotechnical and mining requirements. Each block in the block model is assigned its block BEV. The minimum stope size is achieved by identifying the set of feasible neighbourhoods for each block of the model, which provides the maximum net value. During the optimization process, checks are done to exclude unnecessary blocks such as those with negative economic values, already flagged or those where the MVN of the block is negative (Little, 2012; Sandanayake, 2014). Fig. 5 shows an example of the MVN technique for a one dimensional (1D) block model using a row of six mining blocks. These mining blocks labelled a-f for ease of reference and the value on each block represents the particular block's BEV. The minimum stope size is defined by the neighbourhood of three blocks. The algorithm provides two MVN solutions, i.e. b, c, d and e. The challenge is that selecting either of the neighbourhoods may result in violation of the minimum stope width constraint. This can be illustrated in these options:
• Option 1: if neighbourhood (i) is selected from step 2, it may be • Fig. 4. Inner and outer envelopes for a single block. (Source: Little, 2012).
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impossible to mine block b because it is a single stope thus violating the minimum stope size constraint; Option 2: if neighbourhood (ii) in step 3 is selected, it may be impossible to mine block e, as it is a single block and does not satisfies the minimum stope size requirements.
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Fig. 5. Illustration of the problem with the MVN algorithm. (Source: Sandanayake, 2014).
The final stope includes a negative block, as seen in Fig. 5, in order to satisfy the minimum stope size constraint. It is worth noting that should the MVNs be re-examined, these blocks might not form part of the ultimate stope. Therefore, this algorithm generates different solutions depending on the starting point, thus, it fails to guarantee a true optimal solution. The main shortcomings of this algorithm are (Little, 2012; Bai, 2013; Sandanayake, 2014):
stopes. The results from Matlab are visualized, in MINESIGHT software, based on the pre-set defined parameters (Erdogan et al., 2016; Sandanayake, 2014). Matrix (2013). This algorithm allows stope layouts to be generated according to different stope design strategies, such as to (Topal and Sens, 2010):
• Generate optimum stope boundaries using one fixed stope size; • Use a range of given stope sizes in order to generate optimum stope boundaries; • Find an optimum stope size and generate the stope layout based on the stope size; • Generate stope designs starting with stopes with the highest profit
• The ultimate stope layout is affected by the starting location of evaluation; • Blocks examined earlier are given preferential treatment in the optimization process; • It does not consider the stope wall slope parameter as a constraint; • The cost of mining size is not considered but only individual blocks.
per cubic meter.
The main advantage of this algorithm is that it can generate stope layout solutions in 3D. Conversely, the stopes included in the final layout are chosen in a descending order while rejecting overlapping stopes. Therefore, it disregards the likelihood of multiple stope combinations, which may be derived from a given stope set. Some of these multiple stope combinations may yield a stope layout with higher total economic value (Topal and Sens, 2010). Fig. 6 presents a hypothetical example illustrating the drawback of this algorithm. The stope size is set at five blocks long the x axis and four blocks along the y axis. The algorithm selected Stope 1 & Stope 2 as the ultimate stope layouts. However, the most profitable combination is that made of Stope 3 & Stope 4. The ultimate stopes selected by the algorithm have a total economic value of 15 while the most profitable combination have an economic value of 17. Thus, the algorithm fails to generate the most profitable stope layout.
However, big stopes can provide less cost per tonne compared to small stopes, consequently, affecting the neighbourhoods selected for the ultimate stope layout.
This algorithm is based on a simple premise and yields ease in computational implementation. However, its drawbacks indicate that it does not ensure true optimality.
3.2.5. Topal and Sens heuristics algorithm Topal and Sens (2010) developed an algorithm for stope boundary optimization, which is based on heuristics. The algorithm converts the mining block model to a block model with consistent dimensions of the mining blocks called a regularized block model. Consequently, stopes constrained by the height, length and width are generated from the regularized block model. Thereafter, Matlab software is used to implement the algorithm based on the economic values of the generated 5
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terms of number of blocks in the x, y and z direction and the defined stope size is floated along the x, y and z direction in the economic block model to identify all possible stopes (Sandanayake, 2014) as shown in Fig. 8. Fig. 8 shows that for the given blocks, after floating the stope dimension, four possible stopes are created within the economic block model. The algorithm then calculates stope economic values and only stopes with positive stope values are selected. Fig. 9 gives a 2D demonstration of the stope creation process. The BEVs for each block are written on the blocks. After floating the stope dimensions of 3 × 3 blocks, nine possible stope scenarios are created but, only stope i, iv, v and ix have positive stope values of 17, 6, 15 and 16 respectively. From the set of stopes with positive stope values, the algorithm discards overlapping stopes resulting with a set of non-overlapping stope sets and then the set with the highest economic value is selected as the optimum solution (Sandanayake, 2014). The limitation of this algorithm is the use of fixed values in calculating BEVs which are used to create the economic block model. This means that there is a chance that discarded stopes, that is, those with negative economic values can in reality have positive economic values if all possible geological and economic parameters used in the BEV calculation are considered. Therefore, the algorithm also fails to generate reliable optimum stope boundaries.
Fig. 6. The problem with Sens and Topal's algorithm. (Source: Sandanayake, 2014).
3.2.6. Network Flow algorithm Bai (2013) developed a heuristic algorithm to optimize stope boundary when using sublevel stoping mining method. The algorithm optimize the stope design by following this process:
• Define location and centreline of a vertical raise; • Define cylindrical coordinate system (r, θ, z) based on the position of the centreline of the raise, see Fig. 7(a); • Creates linkage between blocks based on the hangingwall and footwall stopes as geotechnical constraints; • Construct a graph based on the vertical arc for the constraints, see Fig. 7(b); • Select a block for inclusion into the stope subject to the maximum
3.3. Comparison of stope boundary optimization algorithms
distance from the raise and the horizontal width.
Table 1 shows a comparison of the most common algorithms developed to date for stope boundary optimization. Analysis of the literature was done based on category of algorithms used, dimensional space of optimization, constraints considered, applicable mining method, and etcetera.
After the raise position has been defined, any block in the coordinate system can be expressed as change in r, θ and z. Where r is the distance of a block from the centre line of the raise, θ represents the length of the block and z represents the height of the block (Bai, 2013). The algorithm stope value is maximized by identifying the best position of the raise and its vertical extent. However, Sandanayake (2014) stated that the disadvantage of this method is that it is limited to sublevel stoping mining method and small mineralised orebodies.
3.3.1. Time series for algorithm development The time series for algorithm development for the common algorithms on stope layout optimization is shown in Fig. 10. Fig. 10 shows that initial research work in stope boundary optimization started after 1970, specifically in 1977 by Riddle who developed the Dynamic Programming algorithm as already discussed. All optimization work in mining is founded on Operations Research (OR) techniques which date back to the 1930s during the World War II. Despite this long history of OR techniques evolution, their adoption in underground mining optimization only started almost five decades later.
3.2.7. Sandanayake's heuristic algorithm Sandanayake (2014) developed a heuristic algorithm for optimizing stope boundaries. The algorithm starts by regularizing a resource model, that is, converting an irregular geological block model into a regularly sized geological block model which is then converted into an economic block model. The algorithm then defines the stope sizes in
Fig. 7. (a) Block model in cylindrical coordinate system. (b) Typical arcs based on constraints.
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Fig. 8. Stope generation Sandanayake, 2014).
process.
(Source:
3.3.4. Dimensional space of optimization Fig. 12 shows that only 10% (Branch and Bound) of the algorithms reviewed generate optimal solutions but, in one dimensional space. However, in 2D it fails to guarantee an optimal solution. Only 20% of the algorithms generate stope layout solutions in 2D. These are the Downstream Geostatistical and Dynamic Programming algorithms. These algorithms can be extended to 3D problems, however, they do not guarantee an optimal solution as discussed in Sections 3.1.1 and 3.1.3. The majority (70%) of the developed algorithms provide solutions in three dimensional space. However, none of these algorithms can guarantee generation of an optimal solution.
Nevertheless, there has been an increase in the development of stope boundary optimization algorithms since 1977 with most of the algorithms developed in the period between 1990 and 2010. The period 2010–2016 is made of only six years not 10 years as compared to other periods. Therefore, it can be forecasted that this increasing trend will continue since to date there is no algorithm that has yet been developed that guarantee true optimality. 3.3.2. Category of algorithm Fig. 11 shows the proportion of heuristic and exact algorithms. The figure shows that most algorithms developed to date are heuristic. Heuristic algorithms do not generate optimal layouts but generates a solution which might not be the best solution. This is likely one of the reasons why there is not yet an algorithm that provide optimum stope boundaries.
3.3.5. Constraints considered during optimization Fig. 13 shows the algorithms and the constraints considered during the optimization process. The Network Flow algorithm considers four constraints, that is, the geotechnical constraints on the footwall and the hangingwall slopes, the maximum distance of a block from the raise and the horizontal width required to bring the most distant mining block to the raise (Bai, 2013). Whereas, Sandanayake's algorithm and Geostatistical approach consider three constraints. Sandanayake's algorithm optimizes stope boundaries considering mining constraint, such as the minimum stope dimensions, stope size variations, pillar separation and level optimization (Sandanayake, 2014). Whilst, the Downstream Geostatistical algorithm considers the stope dimension, hangingwall and footwall slope angles. Dynamic programming, MVN and Octree Division algorithm consider two constraints. The Dynamic Programming algorithm considers stope dimension and footwall slope angle during optimization while
3.3.3. Mining methods Table 1 shows that all of the heuristic algorithms are applicable to all mining methods. This can be alluded to the fact that heuristic algorithms generate a solution which may or may not be optimal. Whereas exact algorithms have been applied to specific mining methods with the exception of the Branch and Bound algorithm that has been applied to all mining methods. The Branch and Bound algorithm is applicable to all mining methods because it only considers one dimension which is in a linear form for instance, stope length. Even though these optimization techniques can be applied to most mining methods, most mining methods on which they have been applied are sub-level open stoping, cut and fill and block caving. 7
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Fig. 9. Generation of possible stopes in 2D. (Source: Sandanayake, 2014).
instance, when optimizing a large economic mining block model, the search time for optimal blocks increases.
Octree Division algorithm considers geometric constraints and economic constraints. The MVN algorithm considers stope dimension and level location. Geometric constraints being those constraints that define practical working space dimensions based on the geotechnical characteristics of the orebody and the economic constraint is the cut-off grade and the mining cost (Cheimanoff et al., 1989). The remainder of the algorithms consider only stope dimension as a constraint during the optimization process. The information depicted in Table 2 is the result of a case study conducted by Erdogan et al. for a gold deposit using a sublevel open stoping mining method. Table 2 shows the number of constraints and solution times for the different algorithms. The Floating Stope algorithm does not generate a series of stopes but it produces inner and outer envelopes as discussed in Section 3.2.1. The number of stopes generated by the algorithms differ based on the stopes each algorithm considers feasible to be included in the final stope layout. The solution time is affected by the search area and number of constraints. For
3.3.6. Algorithms developed into software There are a few stope boundary optimization algorithms that have been developed into software commercially available (Table 3). The Octree Division algorithm has been developed into a production scheduling tool called BONANZA, a part of the GEOCAD package that converts geological resources to geological reserves. The Floating Stope algorithm has been developed into Datamine and Maptek software packages. In Datamine software, the Mineral Reserve Optimizer (MRO) is a tool developed based on the principles of the Floating Stope algorithm. The MRO generates envelopes within which stopes can be designed and it can also be used for mineral reserve estimation. In order to generate optimal mineable shapes, the mine planner can use an optional add-on from the Datamine software called Mineable Shape Optimizer (MSO). 8
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Table 1 Comparison of the common stope layout optimization algorithms. (Adopted from Nhleko and Musingwini, 2017). Algorithm
Classification
Mining method
Dimensional space
Optimization criteria
Constraints
True optimality
Dynamic Programming Downstream Geostatistical
Exact Exact
2D 2D
Economic value Geometry
Exact Heuristic
1D 3D
Economic value Grade/vein
Floating Stope Multiple Pass Floating stope Maximum Value Neighbourhood (MVN) Topal and Sens heuristic Network Flow
Heuristic Heuristic Heuristic
All All All
3D 3D 3D
Economic value Economic value Economic value
Stope dimension; draw control Stope dimension; hangingwall and footwall slope angle Stope dimension (stope length) Stope dimension, cut-off grade and mining cost Stope dimension Stope dimension Stope dimension
No No
Branch and Bound Octree Division
Block caving Cut-and-fill; sublevel stoping All All
Heuristic Heuristic
Sublevel stoping
3D 3D
Heuristic
Sublevel stoping
3D
Stope dimension Stope dimension, footwall angle and hanging wall angle Stope dimension, pillar separation and level location
No No
Sandanayake
Economic value Grade/Economic value Economic value
Yes No No No No
No
Fig. 12. Dimensional space of optimization. Fig. 10. Algorithm development time series.
Fig. 11. Categories of algorithms developed for stope boundary optimization.
Fig. 13. Constraints applied by the algorithms.
Maptek software package has developed Vulcan Stope Optimizer (VSO) based on the Floating Stope algorithm. The VSO generates stope shapes to specification in an optimal way (Maptek, 2017). MineSight stope optimizer found in the MineSight 3D software package utilises the MVN algorithm to generate stope layouts that yield maximum value. The Multiple Pass Floating stope algorithm is also embedded in Datamine software package.
Downstream Geostatistical approach which simulates grade variability, is that they are based on fixed, deterministic economic block models of the orebody that fail to consider uncertainty in geological and economic parameters used in block economic value (BEV) calculation. Traditional models of calculating BEVs are deterministic in nature; single values of geological (grade) and economic (price and costs) parameters are used in the BEV calculation resulting with a deterministic economic model of the orebody. Fig. 14 shows a simplified example of a deterministic BEV calculation model. From the deterministic economic block model, practical and optimum stope designs are done around the economic block model using
3.3.7. Risk and uncertainty In addition to the limitations discussed above for each algorithm the common limitation of all these algorithms and techniques, except the 9
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scheduling). This also makes mines fail to obtain optimum value from their finite mineral resources. Risk and uncertainty is intrinsic to mining, something which mine planners must account for. McGill (2007) citing Croll (1999) showed that in a study that reviewed 11 major projects, grade estimation of the mineral reserve, metal price forecast, valuation methodology are the three factors that pose the greatest risk in mineral projects evaluation. It is impossible to predict these parameters over the life of mine and therefore static and deterministic methods of orebody evaluation fail to consider the uncertainty of these variables. They may result in misleading forecasts of the orebody in terms of size and quality. Therefore, there is a need for more accurate stochastic orebody modelling methods that consider uncertainty and hence minimizing the need for subjective judgements. Despite uncertainty being all about mining, it is often ignored. Sabour, p. 54) et al. (2008) highlighted the consequences of ignoring uncertainty and stated that “In monetary terms, this may cause a loss of multimillion dollars due to either the foregone extra profits that could be gained during the periods of favourable market conditions or the net realized capital losses as a result of the unexpected weak market”. Therefore, the importance of incorporating uncertainty in economic block modelling and stope boundary optimization cannot be overemphasized if robust mine designs and plans are to be developed and optimum value is to be created from the exploitation of mineral resources. Deterministic approaches to stope boundary optimization are also likely to result in changes being made to mine designs later in the project implementation phase. With the importance of accessing the mineral deposit early for early cash flows and subsequently optimum value, coupled with the current constrained business environment in terms of access to and availability of capital, value is lost when designs are modified after the project has started. Some modifications to the mine design later in the life of mine (LOM) may require substantial capital which may not be available or may compromise the NPV of the project. Musingwini (2016) stated that unlike in open pit mining, once an underground mine layout has been designed and development infrastructure put in place the mine will be ‘locked-in’ to that layout and any changes may be capital intensive. The shortcomings of deterministic planning methods are also discussed by Castillo and Dimitrakopoulos (2014) and Magagula et al. (2015) who stated that most mine planning methods offer a static solution to a dynamic problem which causes a gap between plans and actual results. To minimize all these shortcomings and ensure mine plans incorporate uncertainty in geological, economic and technical parameters there is a need for a paradigm shift from deterministic to stochastic BEV calculation, where these key parameters are modelled appropriately to incorporate their variability in the BEV calculation process. The application of probabilistic methods in BEV stope boundary optimization will improve confidence levels in mine designs and subsequently in production schedules. Since an economic block model is one of the main input for the mine planning process in general and for the stope optimization process in particular with everything else subordinating it, its real representation (in terms of quantity/size and quality/metal content) should be as accurate as possible. Incorporating uncertainty in the economic block modelling process ensures that uncertainty is addressed early in the mine design and planning processes. The importance of addressing uncertainty early in the mine planning process is to set a sound foundation for creating robust mine plans and designs later in the planning process and this is critical for project success. Without a realistic economic block model, efforts to optimize later stages of the planning process may fail to maximize the economic potential of an orebody. Stochastic economic block modelling approaches result in an economic block model that mirrors the true behavior of the key inputs for the economic modelling process resulting in a realistic representation of the orebody upon which reliable stope boundary optimization, mine designs can be done. Stochastic modelling enables the development of
Table 2 Solution times for certain optimization algorithms. Source:Adopted from Erdogan et al. (2016). Technique
Constraint(s)
Number of stopes generated
Solution times (h: mm:ss)
Mine Shape Optimizer Floating Stope algorithm MVN algorithm Sens and Topal algorithm Sandanayake algorithm
Stope dimension, pillar dimension Stope dimension
41
00:00:50
N/A (envelopes)
00:02:13
Stope dimension Stope dimension
18 66
00:01:56 00:00:12
Stope dimension, pillar dimension, mining level
64
05:17:00
Table 3 Algorithms developed into software. Algorithm
Extension
Software
Octree Division algorithm
None
Floating Stope algorithm
Mineable Reserve Optimiser Vulcan stope optimizer None
GEOCAD (BONANZA) Datamine Maptek
MSSTOPE
MineSight
Floating Stope algorithm Multiple Pass Floating Stope process Maximum Value Neighbourhood algorithm
Datamine
Fig. 14. A deterministic BEV calculation model.
the algorithms reviewed above. The assumption with this deterministic approach of economic block modelling is that geological and economic parameters and the resulting block economic values are known with certainty. This is likely to result in economic models that overestimate or underestimate the quantity and quality of mineable resources. Hence, algorithms founded on deterministic economic block models are therefore, likely to result in suboptimum stope boundaries because they do not consider uncertainty associated with quantifying ore deposits. This results in unreliable mine designs and plans. Therefore, there is a need for a paradigm shift from deterministic to stochastic BEV calculation to result in stochastic economic orebody models. 3.3.7.1. The need for stochastic economic orebody evaluation. While, the deterministic method assumes certainty in the geological and economic parameters, this assumption however, ignores the known fact that considerable uncertainties are associated with geological and economic parameters used to calculate BEVs. Realistically, these input parameters are variable hence are uncertain but, the traditional deterministic modelling techniques do not incorporate that uncertainty which compromises the success of optimization efforts of subsequent processes (development infrastructure location and production 10
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comprehensive, robust and integral mine plans that accurately account for risks and uncertainties throughout the life of mine. This will significantly increase confidence during mine design and improve reliability in the location and sizing of development infrastructure.
Operations Research in Natural Resources [online]. Springer, pp. 561–577. (Available from). 〈http://link.springer.com/book/10.1007/978-0-387-71815-6〉 (Accessed 19 August 2016). Ataee-Pour, M., 2005. A critical survey of the existing stope layout optimization techniques. J. Min. Sci. 41 (5), 447–466. Ataee-Pour, M., 2006. The MVN multiple pass algorithm for optimization of stope boundaries. Iran. J. Min. Eng. 1 (2), 7–20. Bai, X., 2013. Optimization of underground stope with network flow methods. PhD Thesis. Universite de Montreal. Castillo, F.D., Dimitrakopoulos, R., 2014. Joint effect of commodity price and geological uncertainty over the life of mine and ultimate pit limit. Min. Technol. 123 (4), 207–219. Cheimanoff, N., Deliac, E., Mallet, J., 1989. GEOCAD: an alternative CAD and artificial intelligence tool that helps moving from geological resources to mineable reserves. In: Proceedings of the 21st International Symposium on the Application of Computers and Operations Research in the Mineral Industry, pp. 471–478. Deraisme, J., De Fouquet, C., and Fraisse, H., 1984. Geostatistical orebody model for computer optimization of profits from different underground mining methods. 18th APCOM IMM. Erdogan, G., Cigla, M., Topal, E., Yavuz, M., 2016. Implementation and comparison of four stope boundary optimisation algorithms in an existing underground mine, 6th Internation Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey, 5–7 October 2016. Little, J., 2012. Simultaneous optimization of stope layouts and production schedules for long-term underground mine planning. PhD Thesis. School of Mechanical and Mining Engineering. The University of Queensland. Magagula, N.S., Musingwini, C., Ali, M.M., 2015. Multinomial logistic regression analysis of a stochastic mine production system. In: Proceedings of the 23rd International Symposium on Mine Planning and Equipment Selection (MPES2015): Smart Innovation in Mining. Johannesburg, South Africa: The Southern African Institute of Mining and Metallurgy, pp. 331–339. Maptek, 2017. Stope optimiser. INTERNET. 〈http://www.maptek.com/pdf/vulcan/ modules/Maptek_Vulcan_Stope_Optimiser.pdf〉, (Accessed 26 July 2017). McGill, J.E., 2007. Technical Risk Assessment Techniques in Mineral Resource Management with Special Reference To The Junior and Small-scale Mining Sectors. University of Pretoria, Pretoria (Available from). 〈http://www.repository.up.ac.za/ bitstream/handle/2263/27609/00dissertation.pdf?sequence=1&isAllowed=y〉 (Accessed 14 September 2016). Musingwini, C., 2016. Optimization in underground mine planning-developments and opportunities. J. S. Afr. Inst. Min. Metall. 116 (9), 809–820. Nhleko, A.S., Musingwini, C., 2017. Development of an algorithm for stope boundary optimization for underground mines. In: Proceedings of the 3rd Young Professionals Conference, 9-10 March 2017, Innovation Hub, Pretoria, pp. 241–252. Riddle, J.M., 1977. A dynamic programming solution of a block-caving mine layout. In: Presented at the Proceeding Proceedings of the 14th International Symposium on Application of Computers and Operations Research in the Minerals Industries. Society for Mining, Metallurgy and Exploration, pp. 767–780. Sabour, S.A., Dimitrakopoulos, R., Kumral, M., 2008. Mine design selection under uncertainty. Min. Technol. 117 (2), 53–63. Sandanayake, D.S.S., 2014. Stope boundary optimization in underground mining based on a heuristic approach. PhD Thesis. Curtin University, Western Australian School of Mines (Available from). 〈http://espace.library.curtin.edu.au/webclient/ StreamGate?folder_id=0&dvs=1471863949334~699&usePid1=true&usePid2= true〉 (Accessed 19 February 2016). Sandanayake, D.S.S., Topal, E., Asad, M.W.A., 2015. Designing an optimal stope layout for underground mining based on a heuristic algorithm. Int. J. Min. Sci. Technol. 25 (5), 767–772. Sens, J., 2011. Stope Mine Design Optimisation using Various Algorithms for the Randgold Kibali Project. Shahriar, A.P.K., Oraee, K., Bakhtavar, P.S.E., 2007. A Study on the optimization Algorithms for Determining Open-Pit and Underground Mining Limits. In: Presented at the VII-th International Scientific Conference, SGEM 2007, SGEM. Silver, E.A., 2004. An overview of heuristic solution methods. J. Oper. Res. Soc. 55 (9), 936–956. Topal, E., Sens, J., 2010. A new algorithm for stope boundary optimization. J. Coal Sci. Eng. 16 (2), 113–119.
4. Conclusion Optimization plays a pivotal role in mine design and planning. There are several algorithms that have been developed to optimize underground mines. However, this paper focuses on stope boundary optimization because it is the first opportunity available to mine planners to optimize value of mining projects. Subsequent mine design and planning processes are subordinate to the orebody, therefore, the optimum definition of mineable areas is crucial in ensuring that realistic and optimum designs and schedules are generated. The existing literature was reviewed and analyzed, and several findings were realized. The time series analysis showed that the development of stope boundary optimization techniques started in the late 1970s and since then, there has been an increase in the number of algorithms developed. All the algorithms reviewed considered stope dimension as one of the constraints. Also, about 70% of the algorithms define stope boundary in 3D space. Since mining is a 3D problem, it is important that algorithms should be able to optimize in 3D. It was also found that most of the algorithms are applicable to all mining methods except the Dynamic Programming and Downstream Geostatistical algorithm, which are mining method specific. Most (70%) of the algorithms in the literature reviewed are heuristic, thus, true optimality cannot be guaranteed by these algorithms. The other common limitation of all the algorithms is that they are based on static orebody models, therefore, true optimality is also not guaranteed. Mining is a dynamic process which requires dynamic optimization techniques but these algorithms offer static solutions to a dynamic problem. There is a need for further research in developing algorithms that will provide true optimal and robust stope boundary solutions. Disclaimer The opinions and interpretations expressed in this paper are those of the authors and not of the organizations to which they are affiliated. References Alford, C., 1995. Optimisation in underground mine design. Presented at the Application of Computers and Operations Research in the Minerals Industries (APCOM xxv 1995), Brisbane, Australia, 9-14 July 1995. : Australasian Institute of Mining and Metallurgy, Carlton, Victoria, pp. 213–218. Ataee-Pour, M., 2000. A heuristic algorithm to optimise stope boundaries, PhD Thesis, Faculty of Engineering. University of Wollongong (INTENRET). 〈http://ro.uow.edu. au/cgi/viewcontent.cgi?article=3923&context=theses〉 (Accessed 15 February 2016). Alford, C., Brazil, M., Lee, D.H., 2007. Optimization in Underground Mining. Handbook of
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Contents lists available at ScienceDirect
Resources Policy journal homepage: www.elsevier.com/locate/resourpol
A review of underground stope boundary optimization algorithms ⁎
AS Nhleko , T. Tholana, PN Neingo Lecturer, School of Mining Engineering, University of the Witwatersrand, Johannesburg, South Africa
A R T I C L E I N F O
A B S T R A C T
Keywords: Optimization Stope boundary Deterministic Stochastic Algorithm Underground mining
Optimization is a key aspect of the mine design and planning process. A number of algorithms and techniques have been developed to optimize mines. However, most of these techniques focus on open pit optimization to an extent that some authors argue that open pit limit optimization has reached saturation level. Optimization of underground mines only received attention in recent decades and has been focused on three main areas including stope boundary optimization. This paper reviews and analyzes literature on algorithms developed to date for stope boundary optimization. There has been an increase in the number of algorithms developed to optimize stope layout. Most of these algorithms are heuristic, consider stope dimension as one of the constraints and optimize layouts in three dimensional space. However, all these algorithms are based on deterministic orebody models, therefore, fail to consider uncertainty intrinsic in ore deposits. Also, none of these algorithms guarantee optimal stope layout solution in three dimension. Consequently, there is a need for further research in the field of stope boundary optimization.
1. Introduction
mines is computationally more complex than open pit mines and hence the less algorithms that give optimum solutions for underground mines. Unlike in open pit mines, the main challenge in developing a generic methodology for the optimization of underground mines is that there is a wide range of underground mining methods available and their application varies among mine sites, hence each deposit requires a specialized optimization solution (Alford et al., 2007; Sandanayake, 2014). Underground mines also require consideration of other constraints not applicable to open pit mines including ventilation and size of equipment to fit into stopes. Also, in open pit mines, for any given slope angle there is a single option available to remove a given block. Nevertheless, for underground mines there are several options available (Sandanayake, 2014). All these factors make developing optimization solutions for underground mines more computationally complex and hence the lesser amount of research work currently available compared to open pit mines. Alford et al. (2007, p. 574) stated that “the complexity of the underground mine design problem and the unique mine design solutions sought for each ore body suggest that there will never be an elegant solution method analogous to that which exists for open pit mining”. Musingwini (2016) stated that because of this computationally complex nature of optimizing underground mines most of the current work on optimization in underground mine planning is mostly academic. Topal and Sens (2010) mentioned that optimization techniques in underground mines have been mainly focused on three main strategic mine planning areas:
Optimization in general involves either maximizing or minimizing an objective function against a given set of constraints. The objective can be minimizing inputs to a process because they are scarce, minimizing undesired outputs such as waste or maximizing desired outputs such as Net Present Value (NPV). Irrespective of the objective, the basis of optimization is a mathematical model that represents the problem that is then solved using an algorithm. To date, optimization algorithms have played an important role in mine planning and decision-making though more faster and efficient techniques still require to be developed (Little, 2012). Optimization techniques for mining operations date back to the 1960s with initial research developments and application in surface mining. In open pit mining, considerable strides have been made in developing algorithms for open pit optimization. In contrast however, fewer algorithms have been developed for underground mine optimization resulting in most underground mines operating on sub-optimal mine plans particularly in the area of optimization of stope boundary and layout definition (Little, 2012). Optimization in underground mines only started in the 1970s as an extension of open pit optimization applications and to date only a few optimization techniques have been developed to solve underground optimization problems. However, Ataee-Pour (2005) indicated that most of those few optimization techniques for underground optimization have proved not to provide optimum solutions. This is because the optimization of underground ⁎
Corresponding author. E-mail address: [email protected] (A. Nhleko).
https://doi.org/10.1016/j.resourpol.2017.12.004 Received 31 July 2017; Received in revised form 1 December 2017; Accepted 6 December 2017 0301-4207/ © 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Nhleko, S., Resources Policy (2017), https://doi.org/10.1016/j.resourpol.2017.12.004
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1. Optimization of stope boundaries; 2. Optimization of location of development and infrastructure; and 3. Optimization of production schedules.
methods; equivalent to an ultimate pit limit in open pit mining. Defining an optimum stope limit is fundamental in optimizing value from the extraction of a mineral deposit. It is a critical element of strategic mine planning. Little (2012) mentioned that the aim of stope boundary optimization is to select the best combination of blocks to form a series of stopes based on value measures such as grade or profit while satisfying physical mining and geotechnical constraints. The process to define an optimum stope layout combines thousands of blocks into a set of stopes, such that, the undiscounted value is maximized whilst satisfying physical and geotechnical constraints (Sandanayake et al., 2015). Algorithms that have been developed for the optimization of ultimate stope limits are categorized as either rigorous or heuristic (Ataee-Pour, 2006). These exact/rigorous and heuristic algorithms are reviewed in the following sub-sections.
A number of optimization techniques and algorithms have been developed for optimization of each of those areas in underground mines. However, this paper focuses on stope boundary optimization because stope boundary optimization provides the first opportunity to mine planners to optimize and maximize the NPV of a mining project. It is therefore, the aim of this paper to review and analyze literature available on underground stope boundary optimization algorithms. In this paper the phrases ‘stope boundary’, ‘ultimate stope limits’, ‘stope envelope’ and ‘stope layout’ are used interchangeably. 2. Orebody modelling
3.1. Exact algorithms Irrespective of the mining method, either underground or surface, mining starts with prospecting and exploring for mineral resources of economic interest. Based on borehole data and geological information, a geological block model is created. The geological modelling process starts by dividing the orebody into regular blocks in three dimensions, with each block containing its characteristic data; most importantly grade, volume and density. From the geological block model subsequent evaluation processes are done including geostatistical techniques applied to estimate the quantity and quality of the mineral deposit and economic evaluation to convert the geological block model to an economic block model which is one of the key input for optimizing stope boundaries. However, there are other algorithms that generate a stope layout solution based on cut-off grade and head grade such as Floating Stope Algorithm. As mentioned earlier, stope boundary optimization is one of the first opportunity available to mine planners to optimize the long term value of a mining project. Erdogan et al. (2016) mentioned that it may be considered as the starting point in the full optimization process for underground mines when both development and production schedule are considered. The economic orebody limits must be defined first which then allows the optimal location of key development access routes such as shafts, declines, tunnels and raises to be identified. Therefore, incorrect definition of stope boundaries results in incorrect placement of underground infrastructure which may require that the mine design be modified later in the life of the mine (LOM). In other words, an optimum stope boundary determines the efficacy of the mine design and subsequently, the long term production schedule, which then informs the cash flow profile and ultimately, the NPV of a project. Therefore, it is important that its in-situ representation is accurately modelled and understood. Traditionally, to determine whether it will be economic to mine a particular block, the geological block model is converted to an economic block model by applying geological and economic parameters to each block to determine all economically mineable blocks to be included in the ultimate mining limits. The economic modelling process is based on calculating the revenue derived from each block and the cost of mining each block, comparing these two values on a block by block basis to get Block Economic Values (BEVs). For each set of cost and revenue parameters applied, the BEVs distinguish payable and unpayable ore blocks. A block will be economic to mine if the revenue from mining is greater than the cost of mining and processing, that is, if the block economic value is positive. From the economic block model, stopes are created and the following section review algorithms that have been developed to optimize stope boundaries.
Exact algorithms are those algorithms based on a mathematical model and hence they guarantee an optimum solution. These include the Dynamic Programing, Downstream Geostatistical and Branch and Bound algorithms. This section reviews the exact algorithms with the main focus being their shortfalls concerning generation of optimal solutions in 3 dimensional (3D) space. 3.1.1. Dynamic Programming algorithm Riddle (1977) developed an algorithm based on the dynamic programming technique in order optimize stope layouts for block caving mines. The algorithm is a modification of the Dynamic Programming algorithm for optimizing open pit limits by Johnson and Sharp (Shahriar et al., 2007). In the algorithm the relation of height mined is constrained by draw control. Riddle (1977) describes the optimization process by the algorithm mentioning that it starts by first assuming that there is no footwall region and then a minimum number of adjacent draw-points that should be mined in any discrete mining unit are established. The algorithm then assumes that one footwall region is added within the section and then it also establishes a minimum number of adjacent draw-points that should be left in a discrete, non-mined footwall region. Each combination of mined and non-mined draw-points is investigated and the profit of each combination is taken. If the profit with a footwall is greater or equal to the maximum profit obtained for the no-footwall case then a more optimum layout exists with a footwall case and the process is repeated to the two areas divided by a footwall. For example, if the north block is more optimal with a footwall, the process is continued by adding the southern region to the previous optimal condition analyzing the north end for further footwall (Riddle, 1977). This process is continued until no more optimum footwall case is found or it is no longer practical to add further footwalls. In his review of stope layout optimization techniques Ataee-Pour (2005) highlighted the limitations of the algorithm. DP generates optimum stope layouts in two dimension. The optimum 2D sections can be combined to generate a 3D stope layout. However, stope constraints can be violated during the process, thus, optimality in 3D is not guaranteed. The other limitation is that the algorithm is only applicable to block caving mines and is not applicable to other mining methods. 3.1.2. Downstream Geostatistical approach Developed by Deraisme et al. (1984), the approach defines an optimal economic stope design based on downstream geostatistics. Downstream geostatistics is defined by Deraisme p. 583) et al. (1984) as “the methodology for the study of the influence of mining constraints on mining recovery and ore quality”. This approach introduces the application of geostatistics to determine the boundary of mineable ore in an orebody. It can be applied when the mining method to be used is cut and fill or sublevel stoping. It uses downstream geostatistics to build 2D sectional numerical models of the orebody and delineate the ore to
3. Algorithms for stope boundary optimization A stope is a mining area in underground mining which consists of a number of mining blocks. An optimum stope boundary is the limit for material that can be mined economically through underground mining 2
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Fig. 1. Compared outlines of the cut and fill mining method on a section. (Source: Ataee-Pour, 2000).
The cumulative economic values are placed at the beginning and ending points of the blocks (Ataee-Pour, 2000). This approach can generate optimal solutions in one dimensional space along the mining direction, thus, it can only be applied to simple orebodies. It does not take into account the slope constraints, solving mixed integer programming problems is time consuming when working with a large number of blocks (Ataee-Pour, 2000; Bai, 2013; Nhleko and Musingwini, 2017).
be mined. According to Ataee-Pour (2000), when non-linear geostatistics technique failed to estimate mineable reserves, Downstream Geostatistical approach was introduced as an alternative. The approach was introduced when linear and non-linear geostatistics failed to estimate mineable reserves (Ataee-Pour, 2000). The determination of the mineable reserves is controlled by mining constraints of the specific mining method selected to be used to exploit the orebody. Fig. 1 shows the comparative outlines of a section using the Downstream Geostatistics approach and the manual approach. It can be seen that the results are almost identical. This approach can generate 3D solutions, however, it does not take into account the economic profit of the block as its morphological operations only regulate the geometry (Ataee-Pour, 2000; Bai, 2013). This approach cannot guarantee true optimality as it neglects the profitability of the stopes included in the stope boundary.
3.2. Heuristic algorithms Heuristic algorithms are not supported by a mathematical formulation and hence provide a solution which is not necessarily an optimum one but, one that is close to an optimum solution. Silver (2004, p. 936) defined heuristic as “a method which, on the basis of experience or judgement, seems likely to yield a reasonable solution to a problem but which cannot be guaranteed to produce the mathematically optimal solution”. Examples of heuristic stope optimization algorithms include the Octree Division algorithm, the Floating Stope algorithm, the Multiple Pass Floating Stope algorithm, the Maximum Value Neighbourhood algorithm, Topal and Sens (2010)’s algorithm, Sandanayake (2014)’s algorithm and Simulated Annealing algorithm.
3.1.3. Branch and Bound algorithm for integer programming Mining problems that can be solved using linear programming, integer programming and mixed integer programming are generally too large and difficult to solve as a whole. Therefore, by applying the Branch and Bound approach, it is possible to divide the problem into sub-problems which are relatively easier to solve. As cited in AtaeePour (2005) Ovanic and Young in 1995 developed an algorithm to optimize stope boundary based on branch and bound approach. The algorithm uses mixed integer approach called Type-Two Special Ordered Sets (SOS2). The objective of this approach is to identify the start and end locations for mining given stope parameters. The objective is realized by the application of the piecewise linear functions. The first function sums all block values along the row that will be included in the stope layout, while the second function sums the block values to be excluded in the stope layout. The difference between these two functions is the sum of all block values lying between start and finish positions (Ataee-Pour, 2000; Little, 2012). A hypothetical situation where a row of blocks with their respective block positions and net economic block values is shown in Fig. 2. The objective function is to determine the optimum stope layout that maximizes the block economic values. The block positions along the panel are considered as reference values in the piecewise linear function. Whereas the economic values are considered to be attribute values, however, the cumulative economic values of the blocks are used.
3.2.1. Octree Division algorithm Cheimanoff et al. (1989) developed the Octree Division algorithm, a rule-based algorithm that is contained in the GEOCAD package that converts resources to mining reserves by identifying mineable volumes in 3D. The algorithm gathers borehole data, geostatistical analysis data, shapes and any other geological objects to build a geological block model. The algorithm then converts the geological block model to mineable reserves taking to account mining and economic constraints. The mineable reserves are divided into sub-volumes, i.e. minimum allowable stope size, for further economic evaluations. These sub-volumes are assessed based on the following rules (Ataee-Pour, 2000):
• A sub-volume containing no mineralised vein and satisfies the constraints is removed; • A sub-volume that has the minimum allowable stope size or has all the mineralised vein is stored in the mineable block; • A sub-volume containing a proportion of the mineralised vein, i.e. •
Fig. 2. A row of economic value blocks. (Source: Sandanayake, 2014).
3
the sub-volume is located partly inside the ore and partly outside the ore, and does not have the minimum allowable stope dimension is divided into eight equal sub-volumes; The algorithm is terminated when there is no sub-volume in the list that has not been evaluated.
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main disadvantage is that it produces overlapping stopes with shared mining blocks. In mining, a block cannot physically belong to more than one stope. Therefore, the generated results require manual adjustments in order to exclude the shared blocks from the final stope layout. The manual adjustment is dependent on the input of the mine planning engineer (Ataee-Pour, 2000; Little, 2012; Sandanayake, 2014). After the manual adjustment, the derived economic value of the contributing stopes will be different from the actual economic value generated by the algorithm. Thus, this algorithm does not guarantee a true optimal solution. 3.2.3. Multiple Pass Floating Stope algorithm The Multiple Pass Floating Stope (MPFS) algorithm was developed by Cawrse in 2001 as an extension of the Floating Stope algorithm to try and improve the functionality of Datamine software (Sens, 2011). It allows more envelopes to be generated by applying a multiple-optimization process. This process is implemented in three steps (Sandanayake, 2014):
Fig. 3. Illustration of the Octree division approach. (Source: Sandanayake, 2014).
• Definition of input parameters such as head grade, cut-off grade and maximum waste; • File generation: creation of economic stope envelopes for each set of parameters; • File management: conversion of data files or the statistical files into
Fig. 3 shows an example of the Octree Division algorithm. The removal of sub-volumes is performed by using the syntactic pattern recognition technique (Ataee-Pour, 2000). Fig. 3(a) presents the division of a block into eight equal sub-volumes while Fig. 3(b) shows sub-volumes being removed from the mineable block by the algorithm. The Octree Division algorithm generates 3D stope layouts. However, blocks with minimum allowable sizes are included in the final stope layout irrespective of the amount of ore contained. Thus, the net value of the stope is reduced if blocks containing waste are included in the final design. Therefore, the algorithm does not guarantee that the final stope layout is the optimal solution.
a Microsoft Excel compatibles (CSV) format.
This process provides the mine planning engineer with extra information when selecting the best stope layouts. However, it does not eliminate the drawbacks of the Floating Stope algorithm. Therefore, this process cannot guarantee true optimal stope layouts. According to Little (2012), the Australian Minerals Industry Research Association P884 research project, Planning & Rapid Integrated Mine Optimization (PRIMO) developed the Vulcan stope optimizer which appears to address some of the shortcomings of the Floating Stope algorithm. For example, it is able to specify mineable shapes instead of general areas. The specified mineable shapes can be for a range of cut-off grades, mining dimensions & orientations, and mining methods.
3.2.2. Floating Stope algorithm The Floating Stope algorithm was developed by Alford (1995) and is incorporated into Datamine's Mineable Reserve Optimizer (MRO) software. The approach of this algorithm is analogous to the Moving Cone algorithm for open pit limit optimization. It is typically applied for preliminary resource appraisal, selection of the stoping method and detailed mine design. The first step in this algorithm is to specify the cut-off grade in order to separate ore and waste blocks. Subsequently, a rectangular block representing a stope with minimum stope dimensions is specified. The main constraint is the geometry of the stope, which is translated into the minimum stope dimension. The specified stope is floated in a fixed block model relative to its origin, with predefined float increments in three orthogonal directions (three dimensions). Two envelopes are used during this floating process as shown in Fig. 4 that is outer and inner envelopes. The inner envelope is made of the union of all the best grade stope shapes which may be defined from all of the blocks above the cutoff grade that can be exploited. Whereas the outer envelope is the union of all possible stope positions for each block, defined from all of the blocks above the cut-off grade that can be mined (Ataee-Pour, 2000; Little, 2012). The main advantage of this algorithm is it simplicity. However, its
3.2.4. Maximum Value Neighbourhood algorithm Ataee-Pour (2000) developed the Maximum Value Neighbourhood algorithm (MVN) based on the principle of the Floating Stope algorithm. This algorithm defines the mining blocks called neighbourhoods (NB), which should be mined together to satisfy the minimum stope size constraint. These blocks are constrained by the geotechnical and mining requirements. Each block in the block model is assigned its block BEV. The minimum stope size is achieved by identifying the set of feasible neighbourhoods for each block of the model, which provides the maximum net value. During the optimization process, checks are done to exclude unnecessary blocks such as those with negative economic values, already flagged or those where the MVN of the block is negative (Little, 2012; Sandanayake, 2014). Fig. 5 shows an example of the MVN technique for a one dimensional (1D) block model using a row of six mining blocks. These mining blocks labelled a-f for ease of reference and the value on each block represents the particular block's BEV. The minimum stope size is defined by the neighbourhood of three blocks. The algorithm provides two MVN solutions, i.e. b, c, d and e. The challenge is that selecting either of the neighbourhoods may result in violation of the minimum stope width constraint. This can be illustrated in these options:
• Option 1: if neighbourhood (i) is selected from step 2, it may be • Fig. 4. Inner and outer envelopes for a single block. (Source: Little, 2012).
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impossible to mine block b because it is a single stope thus violating the minimum stope size constraint; Option 2: if neighbourhood (ii) in step 3 is selected, it may be impossible to mine block e, as it is a single block and does not satisfies the minimum stope size requirements.
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Fig. 5. Illustration of the problem with the MVN algorithm. (Source: Sandanayake, 2014).
The final stope includes a negative block, as seen in Fig. 5, in order to satisfy the minimum stope size constraint. It is worth noting that should the MVNs be re-examined, these blocks might not form part of the ultimate stope. Therefore, this algorithm generates different solutions depending on the starting point, thus, it fails to guarantee a true optimal solution. The main shortcomings of this algorithm are (Little, 2012; Bai, 2013; Sandanayake, 2014):
stopes. The results from Matlab are visualized, in MINESIGHT software, based on the pre-set defined parameters (Erdogan et al., 2016; Sandanayake, 2014). Matrix (2013). This algorithm allows stope layouts to be generated according to different stope design strategies, such as to (Topal and Sens, 2010):
• Generate optimum stope boundaries using one fixed stope size; • Use a range of given stope sizes in order to generate optimum stope boundaries; • Find an optimum stope size and generate the stope layout based on the stope size; • Generate stope designs starting with stopes with the highest profit
• The ultimate stope layout is affected by the starting location of evaluation; • Blocks examined earlier are given preferential treatment in the optimization process; • It does not consider the stope wall slope parameter as a constraint; • The cost of mining size is not considered but only individual blocks.
per cubic meter.
The main advantage of this algorithm is that it can generate stope layout solutions in 3D. Conversely, the stopes included in the final layout are chosen in a descending order while rejecting overlapping stopes. Therefore, it disregards the likelihood of multiple stope combinations, which may be derived from a given stope set. Some of these multiple stope combinations may yield a stope layout with higher total economic value (Topal and Sens, 2010). Fig. 6 presents a hypothetical example illustrating the drawback of this algorithm. The stope size is set at five blocks long the x axis and four blocks along the y axis. The algorithm selected Stope 1 & Stope 2 as the ultimate stope layouts. However, the most profitable combination is that made of Stope 3 & Stope 4. The ultimate stopes selected by the algorithm have a total economic value of 15 while the most profitable combination have an economic value of 17. Thus, the algorithm fails to generate the most profitable stope layout.
However, big stopes can provide less cost per tonne compared to small stopes, consequently, affecting the neighbourhoods selected for the ultimate stope layout.
This algorithm is based on a simple premise and yields ease in computational implementation. However, its drawbacks indicate that it does not ensure true optimality.
3.2.5. Topal and Sens heuristics algorithm Topal and Sens (2010) developed an algorithm for stope boundary optimization, which is based on heuristics. The algorithm converts the mining block model to a block model with consistent dimensions of the mining blocks called a regularized block model. Consequently, stopes constrained by the height, length and width are generated from the regularized block model. Thereafter, Matlab software is used to implement the algorithm based on the economic values of the generated 5
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terms of number of blocks in the x, y and z direction and the defined stope size is floated along the x, y and z direction in the economic block model to identify all possible stopes (Sandanayake, 2014) as shown in Fig. 8. Fig. 8 shows that for the given blocks, after floating the stope dimension, four possible stopes are created within the economic block model. The algorithm then calculates stope economic values and only stopes with positive stope values are selected. Fig. 9 gives a 2D demonstration of the stope creation process. The BEVs for each block are written on the blocks. After floating the stope dimensions of 3 × 3 blocks, nine possible stope scenarios are created but, only stope i, iv, v and ix have positive stope values of 17, 6, 15 and 16 respectively. From the set of stopes with positive stope values, the algorithm discards overlapping stopes resulting with a set of non-overlapping stope sets and then the set with the highest economic value is selected as the optimum solution (Sandanayake, 2014). The limitation of this algorithm is the use of fixed values in calculating BEVs which are used to create the economic block model. This means that there is a chance that discarded stopes, that is, those with negative economic values can in reality have positive economic values if all possible geological and economic parameters used in the BEV calculation are considered. Therefore, the algorithm also fails to generate reliable optimum stope boundaries.
Fig. 6. The problem with Sens and Topal's algorithm. (Source: Sandanayake, 2014).
3.2.6. Network Flow algorithm Bai (2013) developed a heuristic algorithm to optimize stope boundary when using sublevel stoping mining method. The algorithm optimize the stope design by following this process:
• Define location and centreline of a vertical raise; • Define cylindrical coordinate system (r, θ, z) based on the position of the centreline of the raise, see Fig. 7(a); • Creates linkage between blocks based on the hangingwall and footwall stopes as geotechnical constraints; • Construct a graph based on the vertical arc for the constraints, see Fig. 7(b); • Select a block for inclusion into the stope subject to the maximum
3.3. Comparison of stope boundary optimization algorithms
distance from the raise and the horizontal width.
Table 1 shows a comparison of the most common algorithms developed to date for stope boundary optimization. Analysis of the literature was done based on category of algorithms used, dimensional space of optimization, constraints considered, applicable mining method, and etcetera.
After the raise position has been defined, any block in the coordinate system can be expressed as change in r, θ and z. Where r is the distance of a block from the centre line of the raise, θ represents the length of the block and z represents the height of the block (Bai, 2013). The algorithm stope value is maximized by identifying the best position of the raise and its vertical extent. However, Sandanayake (2014) stated that the disadvantage of this method is that it is limited to sublevel stoping mining method and small mineralised orebodies.
3.3.1. Time series for algorithm development The time series for algorithm development for the common algorithms on stope layout optimization is shown in Fig. 10. Fig. 10 shows that initial research work in stope boundary optimization started after 1970, specifically in 1977 by Riddle who developed the Dynamic Programming algorithm as already discussed. All optimization work in mining is founded on Operations Research (OR) techniques which date back to the 1930s during the World War II. Despite this long history of OR techniques evolution, their adoption in underground mining optimization only started almost five decades later.
3.2.7. Sandanayake's heuristic algorithm Sandanayake (2014) developed a heuristic algorithm for optimizing stope boundaries. The algorithm starts by regularizing a resource model, that is, converting an irregular geological block model into a regularly sized geological block model which is then converted into an economic block model. The algorithm then defines the stope sizes in
Fig. 7. (a) Block model in cylindrical coordinate system. (b) Typical arcs based on constraints.
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Fig. 8. Stope generation Sandanayake, 2014).
process.
(Source:
3.3.4. Dimensional space of optimization Fig. 12 shows that only 10% (Branch and Bound) of the algorithms reviewed generate optimal solutions but, in one dimensional space. However, in 2D it fails to guarantee an optimal solution. Only 20% of the algorithms generate stope layout solutions in 2D. These are the Downstream Geostatistical and Dynamic Programming algorithms. These algorithms can be extended to 3D problems, however, they do not guarantee an optimal solution as discussed in Sections 3.1.1 and 3.1.3. The majority (70%) of the developed algorithms provide solutions in three dimensional space. However, none of these algorithms can guarantee generation of an optimal solution.
Nevertheless, there has been an increase in the development of stope boundary optimization algorithms since 1977 with most of the algorithms developed in the period between 1990 and 2010. The period 2010–2016 is made of only six years not 10 years as compared to other periods. Therefore, it can be forecasted that this increasing trend will continue since to date there is no algorithm that has yet been developed that guarantee true optimality. 3.3.2. Category of algorithm Fig. 11 shows the proportion of heuristic and exact algorithms. The figure shows that most algorithms developed to date are heuristic. Heuristic algorithms do not generate optimal layouts but generates a solution which might not be the best solution. This is likely one of the reasons why there is not yet an algorithm that provide optimum stope boundaries.
3.3.5. Constraints considered during optimization Fig. 13 shows the algorithms and the constraints considered during the optimization process. The Network Flow algorithm considers four constraints, that is, the geotechnical constraints on the footwall and the hangingwall slopes, the maximum distance of a block from the raise and the horizontal width required to bring the most distant mining block to the raise (Bai, 2013). Whereas, Sandanayake's algorithm and Geostatistical approach consider three constraints. Sandanayake's algorithm optimizes stope boundaries considering mining constraint, such as the minimum stope dimensions, stope size variations, pillar separation and level optimization (Sandanayake, 2014). Whilst, the Downstream Geostatistical algorithm considers the stope dimension, hangingwall and footwall slope angles. Dynamic programming, MVN and Octree Division algorithm consider two constraints. The Dynamic Programming algorithm considers stope dimension and footwall slope angle during optimization while
3.3.3. Mining methods Table 1 shows that all of the heuristic algorithms are applicable to all mining methods. This can be alluded to the fact that heuristic algorithms generate a solution which may or may not be optimal. Whereas exact algorithms have been applied to specific mining methods with the exception of the Branch and Bound algorithm that has been applied to all mining methods. The Branch and Bound algorithm is applicable to all mining methods because it only considers one dimension which is in a linear form for instance, stope length. Even though these optimization techniques can be applied to most mining methods, most mining methods on which they have been applied are sub-level open stoping, cut and fill and block caving. 7
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Fig. 9. Generation of possible stopes in 2D. (Source: Sandanayake, 2014).
instance, when optimizing a large economic mining block model, the search time for optimal blocks increases.
Octree Division algorithm considers geometric constraints and economic constraints. The MVN algorithm considers stope dimension and level location. Geometric constraints being those constraints that define practical working space dimensions based on the geotechnical characteristics of the orebody and the economic constraint is the cut-off grade and the mining cost (Cheimanoff et al., 1989). The remainder of the algorithms consider only stope dimension as a constraint during the optimization process. The information depicted in Table 2 is the result of a case study conducted by Erdogan et al. for a gold deposit using a sublevel open stoping mining method. Table 2 shows the number of constraints and solution times for the different algorithms. The Floating Stope algorithm does not generate a series of stopes but it produces inner and outer envelopes as discussed in Section 3.2.1. The number of stopes generated by the algorithms differ based on the stopes each algorithm considers feasible to be included in the final stope layout. The solution time is affected by the search area and number of constraints. For
3.3.6. Algorithms developed into software There are a few stope boundary optimization algorithms that have been developed into software commercially available (Table 3). The Octree Division algorithm has been developed into a production scheduling tool called BONANZA, a part of the GEOCAD package that converts geological resources to geological reserves. The Floating Stope algorithm has been developed into Datamine and Maptek software packages. In Datamine software, the Mineral Reserve Optimizer (MRO) is a tool developed based on the principles of the Floating Stope algorithm. The MRO generates envelopes within which stopes can be designed and it can also be used for mineral reserve estimation. In order to generate optimal mineable shapes, the mine planner can use an optional add-on from the Datamine software called Mineable Shape Optimizer (MSO). 8
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Table 1 Comparison of the common stope layout optimization algorithms. (Adopted from Nhleko and Musingwini, 2017). Algorithm
Classification
Mining method
Dimensional space
Optimization criteria
Constraints
True optimality
Dynamic Programming Downstream Geostatistical
Exact Exact
2D 2D
Economic value Geometry
Exact Heuristic
1D 3D
Economic value Grade/vein
Floating Stope Multiple Pass Floating stope Maximum Value Neighbourhood (MVN) Topal and Sens heuristic Network Flow
Heuristic Heuristic Heuristic
All All All
3D 3D 3D
Economic value Economic value Economic value
Stope dimension; draw control Stope dimension; hangingwall and footwall slope angle Stope dimension (stope length) Stope dimension, cut-off grade and mining cost Stope dimension Stope dimension Stope dimension
No No
Branch and Bound Octree Division
Block caving Cut-and-fill; sublevel stoping All All
Heuristic Heuristic
Sublevel stoping
3D 3D
Heuristic
Sublevel stoping
3D
Stope dimension Stope dimension, footwall angle and hanging wall angle Stope dimension, pillar separation and level location
No No
Sandanayake
Economic value Grade/Economic value Economic value
Yes No No No No
No
Fig. 12. Dimensional space of optimization. Fig. 10. Algorithm development time series.
Fig. 11. Categories of algorithms developed for stope boundary optimization.
Fig. 13. Constraints applied by the algorithms.
Maptek software package has developed Vulcan Stope Optimizer (VSO) based on the Floating Stope algorithm. The VSO generates stope shapes to specification in an optimal way (Maptek, 2017). MineSight stope optimizer found in the MineSight 3D software package utilises the MVN algorithm to generate stope layouts that yield maximum value. The Multiple Pass Floating stope algorithm is also embedded in Datamine software package.
Downstream Geostatistical approach which simulates grade variability, is that they are based on fixed, deterministic economic block models of the orebody that fail to consider uncertainty in geological and economic parameters used in block economic value (BEV) calculation. Traditional models of calculating BEVs are deterministic in nature; single values of geological (grade) and economic (price and costs) parameters are used in the BEV calculation resulting with a deterministic economic model of the orebody. Fig. 14 shows a simplified example of a deterministic BEV calculation model. From the deterministic economic block model, practical and optimum stope designs are done around the economic block model using
3.3.7. Risk and uncertainty In addition to the limitations discussed above for each algorithm the common limitation of all these algorithms and techniques, except the 9
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scheduling). This also makes mines fail to obtain optimum value from their finite mineral resources. Risk and uncertainty is intrinsic to mining, something which mine planners must account for. McGill (2007) citing Croll (1999) showed that in a study that reviewed 11 major projects, grade estimation of the mineral reserve, metal price forecast, valuation methodology are the three factors that pose the greatest risk in mineral projects evaluation. It is impossible to predict these parameters over the life of mine and therefore static and deterministic methods of orebody evaluation fail to consider the uncertainty of these variables. They may result in misleading forecasts of the orebody in terms of size and quality. Therefore, there is a need for more accurate stochastic orebody modelling methods that consider uncertainty and hence minimizing the need for subjective judgements. Despite uncertainty being all about mining, it is often ignored. Sabour, p. 54) et al. (2008) highlighted the consequences of ignoring uncertainty and stated that “In monetary terms, this may cause a loss of multimillion dollars due to either the foregone extra profits that could be gained during the periods of favourable market conditions or the net realized capital losses as a result of the unexpected weak market”. Therefore, the importance of incorporating uncertainty in economic block modelling and stope boundary optimization cannot be overemphasized if robust mine designs and plans are to be developed and optimum value is to be created from the exploitation of mineral resources. Deterministic approaches to stope boundary optimization are also likely to result in changes being made to mine designs later in the project implementation phase. With the importance of accessing the mineral deposit early for early cash flows and subsequently optimum value, coupled with the current constrained business environment in terms of access to and availability of capital, value is lost when designs are modified after the project has started. Some modifications to the mine design later in the life of mine (LOM) may require substantial capital which may not be available or may compromise the NPV of the project. Musingwini (2016) stated that unlike in open pit mining, once an underground mine layout has been designed and development infrastructure put in place the mine will be ‘locked-in’ to that layout and any changes may be capital intensive. The shortcomings of deterministic planning methods are also discussed by Castillo and Dimitrakopoulos (2014) and Magagula et al. (2015) who stated that most mine planning methods offer a static solution to a dynamic problem which causes a gap between plans and actual results. To minimize all these shortcomings and ensure mine plans incorporate uncertainty in geological, economic and technical parameters there is a need for a paradigm shift from deterministic to stochastic BEV calculation, where these key parameters are modelled appropriately to incorporate their variability in the BEV calculation process. The application of probabilistic methods in BEV stope boundary optimization will improve confidence levels in mine designs and subsequently in production schedules. Since an economic block model is one of the main input for the mine planning process in general and for the stope optimization process in particular with everything else subordinating it, its real representation (in terms of quantity/size and quality/metal content) should be as accurate as possible. Incorporating uncertainty in the economic block modelling process ensures that uncertainty is addressed early in the mine design and planning processes. The importance of addressing uncertainty early in the mine planning process is to set a sound foundation for creating robust mine plans and designs later in the planning process and this is critical for project success. Without a realistic economic block model, efforts to optimize later stages of the planning process may fail to maximize the economic potential of an orebody. Stochastic economic block modelling approaches result in an economic block model that mirrors the true behavior of the key inputs for the economic modelling process resulting in a realistic representation of the orebody upon which reliable stope boundary optimization, mine designs can be done. Stochastic modelling enables the development of
Table 2 Solution times for certain optimization algorithms. Source:Adopted from Erdogan et al. (2016). Technique
Constraint(s)
Number of stopes generated
Solution times (h: mm:ss)
Mine Shape Optimizer Floating Stope algorithm MVN algorithm Sens and Topal algorithm Sandanayake algorithm
Stope dimension, pillar dimension Stope dimension
41
00:00:50
N/A (envelopes)
00:02:13
Stope dimension Stope dimension
18 66
00:01:56 00:00:12
Stope dimension, pillar dimension, mining level
64
05:17:00
Table 3 Algorithms developed into software. Algorithm
Extension
Software
Octree Division algorithm
None
Floating Stope algorithm
Mineable Reserve Optimiser Vulcan stope optimizer None
GEOCAD (BONANZA) Datamine Maptek
MSSTOPE
MineSight
Floating Stope algorithm Multiple Pass Floating Stope process Maximum Value Neighbourhood algorithm
Datamine
Fig. 14. A deterministic BEV calculation model.
the algorithms reviewed above. The assumption with this deterministic approach of economic block modelling is that geological and economic parameters and the resulting block economic values are known with certainty. This is likely to result in economic models that overestimate or underestimate the quantity and quality of mineable resources. Hence, algorithms founded on deterministic economic block models are therefore, likely to result in suboptimum stope boundaries because they do not consider uncertainty associated with quantifying ore deposits. This results in unreliable mine designs and plans. Therefore, there is a need for a paradigm shift from deterministic to stochastic BEV calculation to result in stochastic economic orebody models. 3.3.7.1. The need for stochastic economic orebody evaluation. While, the deterministic method assumes certainty in the geological and economic parameters, this assumption however, ignores the known fact that considerable uncertainties are associated with geological and economic parameters used to calculate BEVs. Realistically, these input parameters are variable hence are uncertain but, the traditional deterministic modelling techniques do not incorporate that uncertainty which compromises the success of optimization efforts of subsequent processes (development infrastructure location and production 10
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comprehensive, robust and integral mine plans that accurately account for risks and uncertainties throughout the life of mine. This will significantly increase confidence during mine design and improve reliability in the location and sizing of development infrastructure.
Operations Research in Natural Resources [online]. Springer, pp. 561–577. (Available from). 〈http://link.springer.com/book/10.1007/978-0-387-71815-6〉 (Accessed 19 August 2016). Ataee-Pour, M., 2005. A critical survey of the existing stope layout optimization techniques. J. Min. Sci. 41 (5), 447–466. Ataee-Pour, M., 2006. The MVN multiple pass algorithm for optimization of stope boundaries. Iran. J. Min. Eng. 1 (2), 7–20. Bai, X., 2013. Optimization of underground stope with network flow methods. PhD Thesis. Universite de Montreal. Castillo, F.D., Dimitrakopoulos, R., 2014. Joint effect of commodity price and geological uncertainty over the life of mine and ultimate pit limit. Min. Technol. 123 (4), 207–219. Cheimanoff, N., Deliac, E., Mallet, J., 1989. GEOCAD: an alternative CAD and artificial intelligence tool that helps moving from geological resources to mineable reserves. In: Proceedings of the 21st International Symposium on the Application of Computers and Operations Research in the Mineral Industry, pp. 471–478. Deraisme, J., De Fouquet, C., and Fraisse, H., 1984. Geostatistical orebody model for computer optimization of profits from different underground mining methods. 18th APCOM IMM. Erdogan, G., Cigla, M., Topal, E., Yavuz, M., 2016. Implementation and comparison of four stope boundary optimisation algorithms in an existing underground mine, 6th Internation Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey, 5–7 October 2016. Little, J., 2012. Simultaneous optimization of stope layouts and production schedules for long-term underground mine planning. PhD Thesis. School of Mechanical and Mining Engineering. The University of Queensland. Magagula, N.S., Musingwini, C., Ali, M.M., 2015. Multinomial logistic regression analysis of a stochastic mine production system. In: Proceedings of the 23rd International Symposium on Mine Planning and Equipment Selection (MPES2015): Smart Innovation in Mining. Johannesburg, South Africa: The Southern African Institute of Mining and Metallurgy, pp. 331–339. Maptek, 2017. Stope optimiser. INTERNET. 〈http://www.maptek.com/pdf/vulcan/ modules/Maptek_Vulcan_Stope_Optimiser.pdf〉, (Accessed 26 July 2017). McGill, J.E., 2007. Technical Risk Assessment Techniques in Mineral Resource Management with Special Reference To The Junior and Small-scale Mining Sectors. University of Pretoria, Pretoria (Available from). 〈http://www.repository.up.ac.za/ bitstream/handle/2263/27609/00dissertation.pdf?sequence=1&isAllowed=y〉 (Accessed 14 September 2016). Musingwini, C., 2016. Optimization in underground mine planning-developments and opportunities. J. S. Afr. Inst. Min. Metall. 116 (9), 809–820. Nhleko, A.S., Musingwini, C., 2017. Development of an algorithm for stope boundary optimization for underground mines. In: Proceedings of the 3rd Young Professionals Conference, 9-10 March 2017, Innovation Hub, Pretoria, pp. 241–252. Riddle, J.M., 1977. A dynamic programming solution of a block-caving mine layout. In: Presented at the Proceeding Proceedings of the 14th International Symposium on Application of Computers and Operations Research in the Minerals Industries. Society for Mining, Metallurgy and Exploration, pp. 767–780. Sabour, S.A., Dimitrakopoulos, R., Kumral, M., 2008. Mine design selection under uncertainty. Min. Technol. 117 (2), 53–63. Sandanayake, D.S.S., 2014. Stope boundary optimization in underground mining based on a heuristic approach. PhD Thesis. Curtin University, Western Australian School of Mines (Available from). 〈http://espace.library.curtin.edu.au/webclient/ StreamGate?folder_id=0&dvs=1471863949334~699&usePid1=true&usePid2= true〉 (Accessed 19 February 2016). Sandanayake, D.S.S., Topal, E., Asad, M.W.A., 2015. Designing an optimal stope layout for underground mining based on a heuristic algorithm. Int. J. Min. Sci. Technol. 25 (5), 767–772. Sens, J., 2011. Stope Mine Design Optimisation using Various Algorithms for the Randgold Kibali Project. Shahriar, A.P.K., Oraee, K., Bakhtavar, P.S.E., 2007. A Study on the optimization Algorithms for Determining Open-Pit and Underground Mining Limits. In: Presented at the VII-th International Scientific Conference, SGEM 2007, SGEM. Silver, E.A., 2004. An overview of heuristic solution methods. J. Oper. Res. Soc. 55 (9), 936–956. Topal, E., Sens, J., 2010. A new algorithm for stope boundary optimization. J. Coal Sci. Eng. 16 (2), 113–119.
4. Conclusion Optimization plays a pivotal role in mine design and planning. There are several algorithms that have been developed to optimize underground mines. However, this paper focuses on stope boundary optimization because it is the first opportunity available to mine planners to optimize value of mining projects. Subsequent mine design and planning processes are subordinate to the orebody, therefore, the optimum definition of mineable areas is crucial in ensuring that realistic and optimum designs and schedules are generated. The existing literature was reviewed and analyzed, and several findings were realized. The time series analysis showed that the development of stope boundary optimization techniques started in the late 1970s and since then, there has been an increase in the number of algorithms developed. All the algorithms reviewed considered stope dimension as one of the constraints. Also, about 70% of the algorithms define stope boundary in 3D space. Since mining is a 3D problem, it is important that algorithms should be able to optimize in 3D. It was also found that most of the algorithms are applicable to all mining methods except the Dynamic Programming and Downstream Geostatistical algorithm, which are mining method specific. Most (70%) of the algorithms in the literature reviewed are heuristic, thus, true optimality cannot be guaranteed by these algorithms. The other common limitation of all the algorithms is that they are based on static orebody models, therefore, true optimality is also not guaranteed. Mining is a dynamic process which requires dynamic optimization techniques but these algorithms offer static solutions to a dynamic problem. There is a need for further research in developing algorithms that will provide true optimal and robust stope boundary solutions. Disclaimer The opinions and interpretations expressed in this paper are those of the authors and not of the organizations to which they are affiliated. References Alford, C., 1995. Optimisation in underground mine design. Presented at the Application of Computers and Operations Research in the Minerals Industries (APCOM xxv 1995), Brisbane, Australia, 9-14 July 1995. : Australasian Institute of Mining and Metallurgy, Carlton, Victoria, pp. 213–218. Ataee-Pour, M., 2000. A heuristic algorithm to optimise stope boundaries, PhD Thesis, Faculty of Engineering. University of Wollongong (INTENRET). 〈http://ro.uow.edu. au/cgi/viewcontent.cgi?article=3923&context=theses〉 (Accessed 15 February 2016). Alford, C., Brazil, M., Lee, D.H., 2007. Optimization in Underground Mining. Handbook of
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