A two-step mathematical programming framework for undercut horizon optimization in block caving mines

Resources Policy 65 (2020) 101586

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A two-step mathematical programming framework for undercut horizon optimization in block caving mines Roberto Noriega a, Yashar Pourrahimian a, *, Eugene Ben-Awuah b a

Mining and Rock Science Development and Innovation Lab (MRDIL), Department of Civil & Environmental Engineering, School of Mining and Petroleum Engineering, University of Alberta, 6-243 Donadeo Innovation Centre for Engineering, 9211 116th st, Edmonton, Alberta, T6G 1H9, Canada b Bharti School of Engineering, Laurentian University, F220 935 Ramsey Lake Road, Sudbury, Ontario, P3E 2C6, Canada

A R T I C L E I N F O

A B S T R A C T

Keywords: Block caving Mixed-integer programming Undercut horizon Economic envelope Production scheduling

The definition of caving economic limits is one of the initial steps in the planning and design of caving projects. This paper proposes a binary optimization framework to integrate the caving envelope and production schedule that maximizes the net present value of the project under technical constraints that model the caving operation mechanics. The constraints considered in the framework are mining capacities, draw rates, maximum and minimum column heights, horizontal and vertical precedences, undercut development rates and the maximum relative adjacent height of draw between columns. An early-start algorithm is used to reduce the number of decision variables and a sliding-time heuristic is applied to significantly reduce the computing time. The framework is implemented in a MATLAB environment with CPLEX as the optimization engine. A case study is presented for the section of a copper deposit, where different horizons were evaluated to select the optimal undercut level and define the caving envelope and initial production schedule. Results were obtained in under 20 min, which allows the method to be efficiently used to evaluate multiple scenarios.

1. Introduction The use of underground mining methods to extract mineral resources is becoming more widespread as near-surface deposits are depleted and operating pits reach their final economic limits. Amongst underground mining techniques, block caving methods are favored due to their low operating costs and high production capacities. However, caving methods incur high capital expenditure and have long development times. Therefore, the decision-making process at the early stages of the mine life cycle is critical for the success of a caving project (Flores, 2014). The operating principle in block caving mines involves the excavation of a void, commonly referred to as the undercut level, to weaken the rock mass above and allow it to cave under gravity forces and internal rock stresses (Brown, 2007; Laubscher, 1994). The caved material then flows through drawpoints connected from the undercut level to a production level directly below, where it is extracted most commonly by LHDs towards a materials re-handling level and then brought to the surface (Hustrulid and Bullock, 2001; Shelswell et al., 2018; Usmani et al., 2014). The traditional planning process in block caving mines starts with the computation of a resource model or block model, in which every

block has an estimated value of an attribute of interest in the deposit, such as mineral grades and metallurgical parameters. The mining units are then defined as draw columns. The draw column is the most appropriate shape to represent the mining extraction method in block caving, as the material is continuously extracted through the draw­ points. Due to the large heights of extraction, the draw-zone cross-sec­ tion is similar to a cylinder or column (Castro et al., 2007; Rustan, 2000; Sahupala et al., 2010). Within the deposit model, an elevation is selected to serve as the undercut level, where a mining footprint or outline is defined with the objective of maximizing the economic value and ore tonnage of the projected cave. A drawpoint layout is designed, speci­ fying the spacing and orientation of tunnels, based on the geotechnical characteristics of the rock mass as well as the operational requirements of the project. The extraction zone associated with each drawpoint is represented as a draw cone. The draw cone is divided into slices along its vertical axis, with each slice intersecting with the block model to compute tonnages and grades, amongst other attributes. An economic envelope or boundary is calculated from the slice model by determining the desired height of draw from each draw column. This envelope is then scheduled to obtain a long-term plan (Rubio, 2002). The selection of the elevation for the placement of the undercut level

* Corresponding author. E-mail addresses: [email protected] (R. Noriega), [email protected] (Y. Pourrahimian), [email protected] (E. Ben-Awuah). https://doi.org/10.1016/j.resourpol.2020.101586 Received 21 April 2019; Received in revised form 8 July 2019; Accepted 7 January 2020 Available online 16 January 2020 0301-4207/© 2020 Elsevier Ltd. All rights reserved.

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is one of the first decisions to make when planning a block caving mine, and it effectively constrains the potential economic value and mineable reserves of the project. Further to this, once an undercut is being developed it is neither possible nor very expensive to adjust to another level. The selection of the undercut level is closely tied to the economic envelope obtained from the caving of the overlying rock mass, taking into consideriation geotechnical and operational constraints such as maximum column heights, cave back angles and the required hydraulic radius of the footprint (Laubscher et al., 2017). Furthermore, the esti­ mated profit of the caving outline is discounted due to the vertical precedence of extraction within each draw column as the material is drawn. Profit is also controlled by the desired draw rate and the hori­ zontal mining advancement direction within the undercut as the col­ umns are opened at different periods. The definition of a caving envelope to evaluate the economic pro­ jection of an undercut elevation can be compared to the problem of defining the final pit limits and pushbacks in open-pit mining, where multiple optimization procedures have been proposed (Elkington and Durham, 2011; Lerchs and Grossman, 1965; Underwood and Tolwinski, 1998; Whittle and Rozman, 1991). Several optimization algorithms for the long-term scheduling of block caving mines have been developed and applied in the past few years addressing different operational challenges (Khodayari and Pourrahimian, 2015a, 2019; Malaki et al., 2017; Pourrahimian et al., 2012, 2013; Rubio and Diering, 2004; Sepúlveda et al., 2018; Smoljanovic et al., 2011). However, these al­ gorithms target the drawpoint scheduling problem. They assume that an undercut elevation has been selected, a drawpoint layout designed, and that an economic envelope— commonly in the form of the best height of draw for each column— is already known. The next section of this paper reviews relevant literature related to undercut level evaluation in block caving. Section 3 highlights the development of the theoretical frameworks for undercut level optimi­ zation and Section 4 discusses the application of the proposed binary integer programming (BIP) model with a case study. The paper con­ cludes in Section 5.

elevation and an initial footprint; however, it does not consider the interaction between each column, as a certain cave back slope and relative extraction rates are required to maintain a desirable cave shape. Moreover, it does not explicitly account for the horizontal mining advancement direction within the undercut and the potential effect on the economic value from the discounting associated with the opening of columns at different periods. Recent versions have been developed to include a scheduling option within the FF module; however, the po­ tential workflow becomes time-consuming due to the requirement to iteratively formulate multiple scenarios for each considered level, even though an exact optimal solution is not guaranteed (Villa, 2014). A methodology proposed by Elkington et al. (2012) uses an integer-programming formulation to determine a 3D cave outline over multiple cut-off grades for a deposit block model. To represent the geometric characteristics of a caving operation, the objective is to maximize the metal content above a certain cut-off constrained to a minimum mining footprint area, a minimum and maximum column height, a maximum adjacent height of draw and a minimum horizontal pillar distance between caves. The method makes it possible to define alternative caving outlines or grade shells at different cut-offs to aid the planner in identifying high-grade areas, guiding the cave development and selecting the best extraction level placement. Some of the disad­ vantages of this method are that it does not maximize economic value explicitly, as it maximizes metal content within the different caving outlines. Furthermore, no time factor is considered in the model. In other words, the model does not take into account the economic dis­ counting effect due to the extraction sequences of a block caving operation. An approach presented by Vargas et al. (2014) introduced geological uncertainty into the process of selecting the elevation and defining the caving outline by building multiple realizations of the block model. These conditional simulations are evaluated in the PCBC FF module to determine the best undercut elevation. Afterwards, a version of the final pit limit procedure is adapted to mimic the geometry of the caving outline and define an economic envelope. The blocks below the selected elevation are eliminated, and the deposit block model is turned upside down. Next, the final pit limit algorithm is used to calculate a mining envelope with the added restriction of allowing for vertical walls up to a certain percentage of the height of the column, then controlling the cave back slopes through the precedence constraints for the blocks on top of the considered percentage. The final step is to filter the result to remove individual columns with no neighbors included in the footprint and then smooth the outline. The algorithm is repeated for multiple simulations, obtaining the undiscounted economic value and ore tonnage of the en­ velope. A value-at-risk evaluation is performed to quantify and sum­ marize the variability of the economic values and tonnages associated with different risk levels to aid the decision maker. The methodology does not account for the discounting effect of the vertical (within col­ umns) and horizontal (within the undercut), since the method used to calculate the envelope is a variation of the final pit limit optimization algorithm, which maximizes undiscounted profit. This could lead to a significant difference between the envelope value and the actual ex­ pected NPV of the project, as well as the shape of the estimated caving outline. Another commercial software application, the CaveLogic™ module developed by Maptek™, was introduced by Arancibia and Soto (2018). This module allows the user to calculate an economic envelope and schedule it from a deposit block model following geotechnical and operational parameters. The mining footprint definition at each poten­ tial undercut level starts with the calculation of the optimum height of draw for each column individually, then smooths and contours the outline by using a genetic algorithm (GA) and image processing tech­ niques. No further information on the details of the development and application of the GA procedure is provided as CaveLogic is a private software application. The process is then repeated for different undercut elevations to provide key indicators such as undiscounted economic

2. Literature review Current industry practice regarding the undercut elevation and mining footprint definition is based on the methodology introduced by Diering (2000, 2010) in GEOVIA PCBC™ software, which is used by virtually every operating block caving mine or potential project. GEO­ VIA PCBC™ includes the Footprint Finder (FF) module to determine the optimal level of extraction as the first step in the designing and planning of a block caving mine. FF takes as an input the deposit block model including the economic valuation of each block, commonly determined based on the estimated mineral grades, mining costs and revenue fac­ tors, accounting for dilution with a vertical mixing model integrated based on the Laubscher model (1994). Each elevation is evaluated independently, generating vertical columns based on the block loca­ tions. The best height of draw (BHOD) for each column is calculated as the height that yields the maximum economic benefit, limited by a defined maximum height of draw. A discounted economic value is ob­ tained for each column independently by adding the economic values of each block starting from the selected undercut, discounted by a vertical mining rate usually expressed as meters per year, up to the determined BHOD. Additionally, development costs can be included to account for the opening of each column. The columns with a positive economic value are considered to be included in the mining footprint at the current elevation. The value and ore tonnage of each potential undercut is calculated as the summation of the economic values and ore tonnages of all the columns included within the footprint respectively. These values (economic and ore tonnage) are used as the main criteria to select the optimal level to start the extraction. The practical workflow makes it easy to examine multiple levels quickly and select the undercut 2

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Fig. 1. Aggregation of blocks at the undercut layout into production units based on the desired drawpoints’ spacing (left), and definition of MUs based on the equivalent minimum draw rate from the undercut elevation up to the maximum desired column height (right).

value and ore tonnage for the planner to select an optimum undercut placement. Once the undercut elevation has been decided, the footprint is scheduled. A sequence is required as an input to define the mining directions and sub-areas to guide the exploitation; the scheduling pro­ cess is performed by maximizing the tonnage that can be removed in a specific period for each column. A mining production plan and a scheduled economic envelope are obtained as the final result for the caving project. The main advantage of this methodology is that it pro­ vides a near optimal solution in a matter of seconds, which makes it possible to explore different scenarios efficiently. However, the defini­ tion of the envelope is not integrated within the scheduling process to define the optimal boundary. Moreover, the sequencing is required as an

input to define the production schedule, which could lead to sub-optimal solutions. This paper provides an optimization framework for the definition of an optimal caving envelope and its use in the selection of an undercut elevation for the decision-making process in block-caving planning. A BIP framework is formulated in two steps. The first is to build mining units within a particular undercut elevation. Those units are represen­ tative of the block caving extraction process as the potential solution space. The second is to schedule the units to estimate an optimal enve­ lope that accounts for geotechnical and operational parameters, as well as the economic discounting from the vertical and horizontal mining extraction. This method integrates the boundary definition and

Fig. 2. Iterative optimization workflow for the selection of the best undercut elevation in block caving mines. 3

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scheduling process within an exact optimization model.

Table 1 Overview of indices, parameters and sets used in the layout aggregation optimization.

3. Methodology

Indices

In block-caving operations, given a particular undercut elevation, the footprint at the extraction level below is divided into multiple con­ nected drawpoints from which the broken ore is drawn. The main as­ sumptions followed by the optimization framework presented here are that production is carried out identically by adjacent drawpoints within the same drawbell. At the undercut section, blocks are aggregated to represent production units (PU) based on the desired drawpoints’ spacing dimensions between drifts (A) and across the minor pillars (B) (Fig. 1). Furthermore, over the vertical extension of the defined PU, blocks are grouped based on the required minimum draw rate to sustain caving, from the base of the PU up to the maximum desired column height, into mining units (MU). The height of the MU is determined as that which will make the total tonnage within the unit equivalent to the minimum draw rate required per period. The resulting MUs are used as the base for scheduling (Fig. 1). These assumptions help reduce the number of variables associated with the resource model at the block level support, while the resolution loss associated with the block ag­ gregation procedure is still representative of the block caving extraction scheme. Two BIP models were developed to define the optimal caving limits and schedule from the resource model, with relevant technical and economic parameters. The initial model determines the optimal aggre­ gation layout at the specified undercut elevation to maximize the metal content of the PUs, upon which the MUs are constructed. The proposed model performs the aggregation by optimizing the arrangement of production units in order to maximize the metal content in the columns above them. Metal content was chosen rather than eco­ nomic value to consider the whole orebody instead of only high-grade areas that could lead to impractical mining footprints. Block caving re­ quires a defined extraction layout, or footprint, to guarantee appropriate caving conditions. This means that there is limited selectivity, and gaps within it could lead to major geotechnical problems and dilution. Moreover, by using metal content as the criteria, the whole column up to the set maximum column height for each particular production unit is considered for the posterior production scheduling algorithm to decide the boundaries along with the mine plan, considering the whole layout and set of conditions. The use of economic value would require the definition of the BHOD to calculate the profit of each production unit. The economic value of a draw column depends on its height as it ac­ cesses more or less material. Current practice defines the economic value using the cumulative value along the vertical profile of the column, setting the height at the maximum. However, this does not consider the interaction of adjacent columns, in which extracting additional ore with a lower value could allow higher extraction heights on adjacent col­ umns. This could potentially generate a higher NPV based on the cave back slope constraints and mining advancement direction. This aggre­ gation step provides the scheduling algorithm with an arrangement of mining units that contains the maximum metal in the deposit. Providing the mining units in this way can help with a decision on the sequence of extraction and boundaries, giving the BHOD profile of the cave as an output of the optimization model. The second BIP model schedules the MUs based on block caving operational constraints in order to maximize the NPV of the project, defining the footprint and caving economic limits as well as an initial schedule for reserve evaluation. The optimization framework has a direct application as a method to select the best undercut elevation in block caving mines. The proposed methodology is summarized in Fig. 2. A set of potential undercut elevations is initially defined. This can be defined over all vertical levels of the orebody or selected based on the minimum ore area at each potential layout that has to be opened in order to sustain caving within the rock mass. The framework can be iteratively repeated over each potential candidate undercut elevation,

c 2 f1; :::; Cg

Index for all possible production units

b 2 f1; :::; Bg

Index for all possible ore blocks in a given undercut level

Sets Set containing all PUs that overlap with block b, with number of elements NðOb Þ

Ob Parameters Mc

Metal content of production unit c

Table 2 Overview of the decision variables used in the layout aggregation optimization. Decision Variables xc 2 f0; 1g yb 2 f0; 1g

Binary variable. Takes the value of 1 if unit c is included in the layout and 0 if not. Binary variable to ore blocks to avoid gaps in the undercut layout

summarizing the results based on different indicators such as the caving project NPV and ore tonnage after scheduling. This allows the planner to evaluate different candidate undercut elevations, in order to make an informed decision based on an optimal production schedule that satisfies the block caving operational constraints. 3.1. Undercut layout aggregation BIP model Table 1 and Table 2 describe the indices, sets, parameters and de­ cision variables used in the optimization model to obtain the production units at a specific undercut level. The optimization model starts by building all possible PUs in the layout, by aggregating the blocks based on the desired drawpoint spacing between drifts and across the minor pillars. The metal content for each PU is calculated by summing the metal content of the blocks contained within the PU over the vertical extension of the orebody up until the waste contact is found or the desired maximum column height is reached. A binary variable is associated with each PU to determine whether it is included in the layout footprint. The objective function in Equation (1) maximizes the metal content of the final undercut layout footprint to be scheduled in the next step. Equations (2) and (3) show the constraints used in this optimization model, to select non-intersecting but adjacent PU in the footprint. P X

Max

(1)

Mc ⋅xc 1

N ðOb Þ

X

xc � yb

8b 2 f1; :::; Bg

(2)

8b 2 f1; :::; Bg

(3)

c¼1

yb ¼ 1

The solution to this step provides the combination of PUs, which represents the drawpoints from which the actual production is carried out, at a particular undercut elevation that maximizes the metal content. The results from this step will be used to schedule and maximize the NPV of the project in the next step. This effectively reduces the number of variables for the next step while still providing a solution space that is representative of the block caving extraction scheme. After the PUs have been defined on the undercut layout, the MUs are built from each PU 4

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with a height corresponding to the equivalent minimum draw rate required up to the maximum column height or waste contact of the orebody.

is controlled by limiting the extraction of a specific MU only when the MU directly below has been extracted. The lowest MU from each PU does not have vertical precedence. Constraint (6) additionally enforces continuous mining in the PU, which is required in caving operations in order to avoid compaction of the columns, with each period extracting at least one MU equivalent to the minimum draw rate. The horizontal precedence is defined over a convex V-shape mining advancement front, which is common industry practice as it has been found that it helps to control the stresses induced as the cave front progresses. This horizontal precedence is determined based on a selected starting point and the mining front angle, by finding the first V-shape that contains the center of the PU and selecting the directly adjacent PUs as precedent (Fig. 3).

3.2. Scheduling BIP model Table 3 and Table 4 show the different indices, sets, parameters and decision variables used in the scheduling BIP optimization model. This model determines an optimal production schedule from the defined MUs to obtain the economic caving limits, undercut sequence schedule, initial production schedule and reserves. The objective function presented in Equation (4) is composed of two terms. The first term maximizes the NPV of the project based on the discounted profit from the extraction of the MUs. This discount is influenced by the horizontal mining direction as well as the vertical physical precedence of the MUs. The second term considers the devel­ opment cost incurred to open each PU. This would be equivalent to the cost of preparing the two adjacent drawpoints within a drawbell to start production. In this particular formulation, the development cost is associated with the lowest MU in a PU.

Max

T X C X U �� X MetalPricet ⋅Rec⋅gu;c ⋅Tonu;c t¼1

c¼1 u¼1

ztu;c � ztV u;c þ ztV u;c1

8u 2 f2; :::; Ug8tf2; :::; Tg8c 2 f1; :::; Cg

(6)

u;c

NðH u;c Þ � ztu1c ;c �

T NðH XÞ X t¼1

c¼1

ztu1c ;c

8c 2 f1; :::; Cg8t 2 f1; :::; Tg (7)

Constraint (8) limits the extraction of MU from each PU to the maximum draw rate. This helps maintain safe conditions along the

� � ðMineCostt þ ProcessCostt Þ⋅Tonu;c t ⋅z u;c ð1 þ iÞt

(4)

" # T X C X DevCostu1c ⋅ztu1 ;c c

t¼1

ð1 þ iÞt

c¼1

caved material within the columns and avoid geotechnical risks.

Constraint (5) refers to the maximum mining capacity from all the PUs. This mining capacity would be limited by the projected equipment and mining infrastructure capacities and can accommodate different values per period in order to model ramp-up and ramp-down scenarios. U X

Tonu ⋅ ztu;c � M

t

8t 2 f1; :::; Tg8c 2 f1; :::; Cg

Cc X

Tonu;c � ztu;c � DR

8u 2 f1; :::; Ug8t 2 f1; :::; Tg8c 2 f1; :::; Cg

c¼1

(8)

The maximum number of PUs that can be opened per period is controlled by Constraint (9). The undercut development rate is a com­ mon parameter used in the planning of block caving mines. The rate

(5)

u¼1

Constraints (6) and (7) control the vertical and horizontal prece­ dence of the different MUs and PUs respectively. The vertical precedence

Table 4 Overview of the economic and technical parameters used in the scheduling optimization.

Table 3 Overview of the subscripts, sets and decision variables used in the scheduling optimization.

Parameters

Indices

MetalPricet

Metal price per ton of metal in period t ($/t)

Rec

Metallurgical recovery of the processing operations (%)

gu;c

Grade of mining unit u of production unit c (g/t)

u 2 f1;:::;Ug

Index for all mining units

t 2 f1; :::; Tg

Index for all periods

c 2 f1; :::; Cg

Index for all production units

u1c

Index for the first or lowest mining unit within production unit c

Su1c Heightu

Planar surface area of production unit c (m2), based on the first (lowest) MU of the PU Height of mining unit u (m)

Cc

Set containing the mining units that are within production unit c. Each set has a total number of elements of NðCc Þ

MineCostt

Mining cost per ton of ore in period t ($/t) Processing cost per ton of ore in period t ($/t)

Vu;c

Single element set containing the mining unit directly below unit u of production unit c. Used for vertical precedence constraints Set containing the production units that have to be opened before the extraction of the mining unit u of production unit c, based on the mining direction. Each set has a total number of elements of NðHu;c Þ

ProcessCostt DevCostu1c

Development cost of production unit c, accounted for when extracting the first (lowest) MU, u1c , of production unit c ($)

Tonu;c

Sets

Hu;c

Ac

i M

MaxOpRatet

Decision Variables ztu;c

2 f0; 1g

Discount rate (%)

t

DR ; DR

Set containing all adjacent production units to the unit c for cave back slope constraint. Each set has a total number of elements of NðAc Þ

MaxAbsDiffc’ c

Binary variable controlling the decision to extract the mining unit u of production unit c in period t

5

Tonnage of mining unit u of production unit c (t)

Maximum mining capacity in period t (t) Minimum and maximum draw rate of production units per period (t/ period) Maximum undercutting rate (m2/period) Maximum allowable absolute difference in height between production unit c and each member c’ of its adjacent set Ac to maintain cave back-slope constraints

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Fig. 3. Determination of the different precedence constraints. On the left, the horizontal precedent PUs for the unit highlighted in red are those shaded in gray, determined by the V-shape advancement front. On the right, each MU (e.g. 2, 3, and 4) within each PU is constrained by the extraction of the MU directly itself. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

controls the number of PUs that can be advanced and opened for pro­ duction. The undercut development rate is restricted by the develop­ ment equipment capacity and geotechnical considerations to maintain a safe caving environment. The constraint is generated for every period and works by considering the sum of the areas at the undercut layout of the first MU from each PU on the period it is extracted, constraining it to a maximum development rate expressed in m2 of excavation per period. C X c¼1

Su1c ⋅ ztu1c ;c � MaxOpRatet

8t 2 f1; :::; Tg

(9)

Constraint (10) limits the maximum relative height of draw between adjacent PUs at any period in order to control the cave back slope, to ensure a smooth caving profile, and minimize dilution and geotechnical risks. The MU’s height is used to control the slope profile, rather than tonnages drawn, to make it more intuitive to mine planners. The adja­ cent units for each PU are defined in a similar way as the horizontal precedency was established; however, an additional circular search neighborhood is defined as proposed by Nezhadshahmohammad et al. (2017) to guarantee a tighter control on the adjacent draw profile be­ tween PUs over the production schedule. The radius of this search neighborhood can be increased to enforce the draw profile constraint over larger areas around each PU, while the minimum suggested value would be equal to the largest dimension of the PU in order to include the directly adjacent units. U h X

Heightu ⋅ztu;c

u¼1

U h i X

Fig. 4. The sliding time window heuristic (STWH) as applied to solve the scheduling BIP optimization model.

complex problem due to the different precedence and adjacent relative height constraints included to model the caving operation, as well as a large number of variables that could arise in massive orebodies. It is desired that the method can provide a quality solution in a reasonable amount of time, in order to allow for the possibility of conducting sensitivity analysis and evaluating multiple scenarios. The precedence constraints of the mining units, both vertical and horizontal, together with the mining capacities, maximum draw rate and relative adjacent height of draw constraints, can be used in order to establish the earliest possible period that a MU can be extracted. Therefore, all variables related to the extraction of the MU at an earlier period can be eliminated from the problem. This earliest start algorithm, introduced by Topal (2008), is useful to reduce the number of variables and improve the computing time required to solve the model. The al­ gorithm was implemented such that it will initially find the earliest possible period that a PU can be opened, based on the precedence constraints between PU, the maximum relative adjacent height of draw and maximum mining capacity. Afterwards, for each MU within the PU, the earliest possible period of extraction can be determined based on the maximum draw rate defined. A sliding time window heuristic (STWH) is implemented in order to reduce the computing times further and make the optimization frame­ work more useful to mine planners. The STWH was first successfully introduced by Cullenbine et al. (2011) to produce quick solutions for the block scheduling problem in open-pit mines, and has been implemented in further research efforts on mine sequencing optimization

i c Heightu ⋅ztu;c’ �MaxAbsDiff c’ c 8c2f1;:::;Cg 8c’2A 8t

u¼1

2f1;:::;Tg (10) Finally, Constraint (11) ensures that each MU is only extracted once. It also gives freedom to the model to decide whether or not a MU is extracted, providing the BHOD for each PU and the undercut footprint limits as part of the solution. T X

ztu;c � 1

8u 2 f1; :::; Ug8c 2 f1; :::; Cg

(11)

t¼1

3.3. Solving the optimization framework The proposed optimization framework has been developed and implemented in a MATLAB environment using CPLEX 12.7 (IBM, 2016) as the optimization engine. The initial BIP model to define the layout of the PUs at the undercut footprint can be solved within seconds due to its simplicity. However, the BIP scheduling algorithm becomes a very 6

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Fig. 5. Section of the copper deposit used for the case study to evaluate block caving extraction.

Fig. 6. Plan views of the deposit at different elevations.

(Dimitrakopoulos and Ramazan, 2008; Lamghari and Dimitrakopoulos, 2016; Rimele et al., 2018), including a block caving long-term planning application by Dirkx et al. (2018). The STWH works by repeatedly solving a relaxed version of the problem, outside of a defined time window for each period, while keeping the solutions obtained fixed into the next iteration until the last time period, T, is solved. A time window of size τ, with τ
iteration. The time window is now moved to the next period (t þ 1) with the model formulation modified accordingly. All variables within the time window are again added in their regular form, with the variables outside the time window relaxed to be continuous. The process is repeated, moving the time window one period at a time until a solution is obtained for all periods (Fig. 4). The length of the time window has a direct influence on the quality of the solution, indicated by the NPV, as well as the computing time required to solve it. By running different computational experiments with the case study presented below, it was found that a time window of τ ¼ 1 provides a quality solution in a few minutes, with the small im­ provements in quality obtained for larger windows completely out­ weighed by the larger increase in computation time required to solve the 7

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Table 5 Technical parameters for the implementation of the optimization framework for the case study. Technical Parameters PU Dimensions Maximum Column Height Minimum Column Height No. of Periods

40m � 20m 320m 60m 12

T

Minimum Draw Rate

DR

Maximum Draw Rate

DR

Maximum Mining Capacity

Mt

Undercut Development Rate

MaxOpRatet

20,000 (m2/period)

Maximum Relative Height of Draw Point of Entry Dilution Mining Starting Point Mining Direction Azimuth Convex Front Angle

MaxAbsDiffc’ c

80 (kton/period)

80 (kton/period) (~40 m/ period) 180 (kton/period) (~80 m/ period) 3000–12,000 (kton/period)

65% 280 mE, 113 mN 15� 160�

Fig. 7. Plan view at elevation 1980m showing the PBEVs, the convex front angle and the decision for mining direction.

model. 4. Case study

Table 6 Economic parameters used for the case study evaluation.

The optimization framework was tested on the section of a copper deposit in its prefeasibility stage to evaluate extraction using block caving methods. The deposit is defined over a block size of 20m � 20m and contains a total of 242 Mt of ore with an average copper grade of 0.854%, extending vertically from the horizon of 1800–2720 m above sea level. A 3D view of the deposit is shown in Fig. 5, alongside with the grade-tonnage curve. The high-grade ore is located on the southeast corner and extends vertically through the different levels of the deposit. Fig. 6 shows two plan views of the deposit at different elevations where the high-grade area can be seen to extend on the uppermost levels. The optimization framework takes as input the resource block model along with the specified technical and economical parameters to define the best elevation to place the undercut level, as well as the final economical caving limits and its optimal extraction sequence. The technical parameters used for the case study are shown in Table 5 and are defined to model a typical caving project. Table 5 resource model initially diluted using the Laubscher mixing method with a point-ofentry dilution of 65% (Laubscher, 1994). The PU dimesions to model the caving operation were defined at 40m by 20m. These dimensions represent spacing between drifts of 40m and 20m spacing across the minor pillars; this is representative of current design practice where typical spacing is around 30m by 15m with some variations. The se­ lection of the PU dimensions is constrained by the block size dimensions of the resource model, which could be reblocked in order to provide a better representation of the caving project. The minimum column height refers to the minimum extraction height needed after breaking the rock mass, to sustain continuous caving conditions along the production life of the draw column and its adjacent area. This parameter depends on the geotechnical conditions of the orebody, as well as the desired size of the mine, as larger caves would require large columns for adequate caving. Current industry practice reports minimum column heights of around 50m–100m (for the largest mines). For the case study, the minimum column height was set at 60m to match industry practice and the block height of 20m of the input resource block model. The maximum column height sets the extraction limit and vertical boundary for each draw column. Current practice determines the best economic height of draw of each column individu­ ally before any scheduling efforts. The economic height of draw is also bounded by a maximum column height as larger columns present more geothechnical risks and need to be controlled to sustain caving over their lifespan. Current operating mines report extraction height of 200m–250m, with some of the largest projects operating or expected to

Selling Price ($/t)

MetalPricet

6000

Mining Cost ($/t)

MineCost

9.3

Processing Cost ($/t)

ProcessCostt

18.4

Recovery (%)

Rec

88.7

Discount Rate (%)

i

12

Development Cost ($/PU)

DevCostu1c

300,000

t

operate columns over 400m height. To establish the maximum column height for the case study, the individual BHOD for each production unit was calculated by evaluating the cumulative economic value based on the technical and economic assumptions listed. Each production unit has a different BHOD; however, the maximum BHOD of all the production units was 320m. Therefore, 320m was used as the maximum column height for the building of the mining units. This means that none of the production units are forced to reach the maximum column height or BHOD, but that the scheduling algorithm has the option to set the ver­ tical extraction limit somewhere in between the defined minimum and maximum heights in order to maximize the NPV using the operational constraints and considering the deposit as a whole. The draw rates are defined on a ton/period base for each PU, considering that each PU represents the two adjacent drawpoints within a drawbell. The minimum draw rate, DR , is used to define the MU by grouping the blocks within the projected PU vertically. For the case study, each MU has an equivalent height of around 40m. The definition of the draw rates is also constrained by the height of the blocks in the resource model. The maximum relative height of draw between PUs is used to approximate the cave back slope as the extraction progresses to allow for

safe extraction and efficient draw control. The MaxAbsDiff c’ c definition should consider the dimensions of the PU and the approximate height of the MU based on the spacing and draw rates evaluated. For the case study, a draw difference of 80 kton/period was considered, which rep­ resents one MU and a maximum cave back angle of 60� . The mining capacity is defined considering a ramp-up period of three years from 3000 to 12,000 kton/period. The mining direction and starting point were defined based on the methodology proposed by Khodayari and Pourrahimian (2015b). On a particular undercut elevation, a production block economic value (PBEV) was calculated as the economic value obtained after the

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Resources Policy 65 (2020) 101586

Fig. 8. NPV, ore tonnage and average grade of the caving envelope at different undercut elevations.

Table 6. These parameters were selected based on the current market and block caving operations. The models were tested on a Dell Precision T3500 computer with Intel(R) quad-core i7 CPU at 3.40 GHz, with 16 GB of RAM. The models were solved using CPLEX 12.7, which uses a branch-and-cut algorithm, with a MIP gap set at 1%; MIP gap is the relative tolerance on the gap between the best integer objective and the objective of the best node remaining.

Table 7 KPI’s obtained for the envelopes at the highest valued undercut elevations. Undercut Elevations NPV (M$) Ore tonnage (Mt) Average grade (%Cu)

2000m

2020m

2040m

1424 110.5 0.941

1430 110.7 0.940

1418 109.5 0.938

5. Results and discussion

extraction of each PU up to its BHOD, defined up to its maximum eco­ nomic value, along with its neighbor columns based on a search radius. This search radius was set up at 40m based on the PU dimensions. The PU with the highest PBEV was defined as the starting position, and the direction was defined to move from higher to lower value areas. The direction was set at an azimuth of 15� from which a convex front with an angle of 160� is defined. Fig. 7 shows the PBEV at elevation 1980 m, and the decision for the starting point at the highest value, moving towards the lower economic values of the deposit, defining the convex mining advancement front to build the precedences. The PBEV distribution followed a similar trend over different levels of the deposit. The economic parameters used for the case study are presented in

The undercut layout aggregation BIP optimization framework was used to evaluate multiple elevations in order to define the optimal un­ dercut level and caving limits. Based on the caving maximum and minimum heights defined, as well as an inspection of the grade distri­ bution at different elevations, the levels between 1800m and 2140m were considered for evaluation. Fig. 8 shows the KPI’s (NPV, ore tonnage and average grade) of the caving economic envelope obtained from placing the undercut at the different elevations. The elevations were selected to be placed at the bottom of the blocks of each corre­ sponding level of the resource model. The highest NPV was 1430M$, which was obtained by placing the

Fig. 9. Computing time required to run the optimization framework over different levels. 9

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Fig. 10. Optimal economic limits for the case study. On the left, the mining reserves (green) within the orebody resource. On the right, a plan view of the undercut level (Elevation 2020 m) showing the individual columns from the resource block model that are included within the mining footprint. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 11. Mining sequence that maximizes the NPV at the optimal elevation (left) and on the right, a more detailed view of the BHOD of the footprint obtained.

undercut at the elevation of 2020m, with total ore tonnage reserves of 111 Mt and an average grade of 0.94 %Cu. This value considers the discounted value from the mining sequence and the caving operational parameters. The economic value at the uppermost levels is reduced due to the proximity to the limits of the deposit, which restricts the avail­ ability of ore material to be included within the caving envelope. On the other hand, the envelopes at the lowermost levels have a lower average

grade. The envelopes obtained between the 2000m and 2040m levels have similar economic and reserve values that stand out from the rest (Table 7). Fig. 8 allows the planners to significantly narrow the possibilities for the undercut level placement within an economically optimal range. The different levels within this range can be further explored by integrating geotechnical criteria to select the one best suited for the caving of the

Fig. 12. Production schedule for the caving envelope at the optimal undercut elevation. 10

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Resources Policy 65 (2020) 101586

Fig. 13. Discounted cash flow and cumulative NPV obtained from the optimal caving envelope and sequence.

Fig. 14. Caving heights at different periods of the mine life based on the economically optimal sequence.

rock mass. Also, each level can be evaluated individually under different technical or economic conditions to make a robust and well-informed decision. This is possible due to the low computing times required to run the optimization framework on each individual level. Fig. 9 shows those computing times. The time is counted from the importing of the resource block model up to the transferring of the PU and MU level solution back to the resource block model for visualization and analysis. The mean average time is 15.68 min, with lower times required at the uppermost levels as there would be less MU due to the limits of the deposit. For the purpose of the case study, level 2020m was selected as the best undercut placement level to visualize as it achieves the highest

NPV. Fig. 10 shows the caving economic envelope (the optimal mineable reserves) considered for the parameters described, as well as the foot­ print limits at the selected elevation. The individual blocks excluded from the footprint are also removed from the PU dimensions selected to model the caving extraction operation. The optimal mining sequence generated with the scheduling BIP model is presented in Fig. 11 along with the caving BHOD at the indi­ vidual block level. The undercut sequence is defined by the optimization model based on the precedences built from the selected starting point, mining direction and convex advancement front angle, and is con­ strained by the defined maximum undercut development rate. This sequence provides the optimal economic value and it can be evaluated 11

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Appendix A. Supplementary data

based on geotechnical criteria to ensure a proper cave development and that adjustments can be made if necessary. However, the sequence provides an upper bound for the potential discounted economic value of the project. The BHOD is also the output of the optimization model and it is optimized based on the global mining sequence. The production profile of the caving project is shown in Fig. 12 and the cash flow projections for the life of mine in Fig. 13. The production capacity satisfies the defined constraint limits with a ramp-up period in the first three years. The production in the first year is especially con­ strained based on the maximum number of PUs that can be developed and the maximum draw rate established; however, the highest grade is mined during this period due to the determined optimal undercut level. The discounted cash flow in the initial periods is lower as the mine is starting to develop, with the highest cash flow obtained in Period 5. After Period 5, the cash flow decreases progressively as the cave moves into lower-grade areas. The optimal draw sequence of the mine is presented in Fig. 14. The draw height of the block columns at different periods during the life of mine is shown with respect to the BHOD. The draw profile satisfies the different constraints added to model the caving operation, especially the relative adjacent height of draw. It can be observed, by the last periods, some of the columns are not drawn due to economic conditions.

Supplementary data to this article can be found online at https://doi. org/10.1016/j.resourpol.2020.101586. References Arancibia, M., Soto, F., 2018. Developing projects using CaveLogic TM. In: Potvin, Y., Jakubec, J. (Eds.), Proceedings of the Fourth International Symposium on Block and Sublevel Caving, Caving 2018. Australian Centre for Geomechanics, Vancouver, pp. 407–418. Brown, E.T., 2007. Block Caving Geomechanics. Julius Kruttschnitt Mineral Research Centre, The University of Queensland. Castro, R., Trueman, R., Halim, A., 2007. A study of isolated draw zones in block caving mines by means of a large 3D physical model. Int. J. Rock Mech. Min. Sci. 44, 860–870. https://doi.org/10.1016/j.ijrmms.2007.01.001. Cullenbine, C., Kevin Wood, R., Newman, A., 2011. A sliding time window heuristic for open pit mine block sequencing. Opt. Lett. 5, 365–377. Diering, T., 2000. PC-BC: a block cave design and draw control system. In: Proceedings MassMin 2000. 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6. Conclusions This paper presented a two-step BIP-based optimization framework for the definition of an economic envelope and production schedule that maximizes the NPV of a caving project. The optimization framework started by defining a PU over a given plan layout, based on the desired drawpoint spacing. The PU represents the two adjacent drawpoints within a drawbell in an operating cave. Based on the minimum draw rate, the blocks within the projected PU were grouped vertically along the column up to a defined maximum column height or the ore-waste limit. This grouping defines the MUs that are the basis for the sched­ uling BIP optimization model. The scheduling of the MUs maximizes the NPV with consideration of the development cost required to put each MU into production, con­ strained by technical constraints. These constraints are the physical precedences of the MU over the horizontal and vertical dimensions based on a convex advancement front, mining capacities, maximum draw rates, undercut development rates and a maximum relative adja­ cent height of draw between PUs over the time periods. An early start pre-processing algorithm and a sliding time window heuristic were applied to significantly reduce the computing time required to obtain a solution. The optimization framework was used on a section of a copper de­ posit. Technical and economic parameters representing current market conditions and caving operations were used to evaluate multiple un­ dercut elevations to define the optimal placement for the undercut level as well as the economic limits of the cave project and a production schedule. The solution obtained provides an initial estimate of the po­ tential economic value of the caving project as well as providing an initial sequence that can serve for further detailed engineering studies. Future research will focus on integrating geological and grade un­ certainty in the optimization framework, in order to generate a caving envelope that considers these uncertainties in the data. Another exten­ sion of the framework will focus on adapting the model to handle multiple production lift scenarios, which are common in large blockcaving mines. Acknowledgements Continued financial support from the Natural Sciences and Engi­ neering Research Council (NSERC) of Canada (RGPIN 201606163) is gratefully acknowledged.

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