1

THE IMPACTS OF SLOPE ANGLE APPROXIMATIONS ON PIT OPTIMIZATION

Filipe Beretta - SRK Consulting (UK) - fberetta@srk.co.uk

Alexandre Marinho - MiningMath Associates - alexandre.marinho@miningmath.com

RESUMO

Durante o processo de otimização de cavas, é necessário definir método e parâmetros para

aproximações dos ângulos de talude. Este artigo apresenta a sensibilidade dos resultados

da otimização ao método e parâmetros considerados. O método baseado na precedência de

blocos, adotado pelo programa GEOVIA Whittle, é examinado, comparando os resultados

para variações do parâmetro ‘número máximo de bancadas’ para um depósito de cobre. O

método baseado em superfícies, implementado pelo software MiningMath SimSched, é

explorado e comparado com os resultados anteriores. As cavas finais produzidas pelo

método de precedência de blocos mostraram uma variação de até 10.4% nas reservas e

2.8% nos fluxos de caixa, com erros de até 7.8 graus nas aproximações dos ângulos; contra

nenhuma variação e 0% de erro para o método baseado em superfícies.

Ângulo de talude; otimização de cava; Whittle; SimSched.

ABSTRACT

During the pit optimisation process, methods and parameters for slope angle approximations

must be defined. This paper presents the sensitivity of the optimisation results to the method

and parameters considered. The method based on blocks precedence, adopted in GEOVIA

Whittle software, is examined, comparing the results for the variation of the ‘maximum

number of levels’ parameter. The method based on mining surfaces, implemented in

MiningMath SimSched software, is explored and compared with the previous results. The

ultimate pits produced by the blocks precedence method have shown a variation of up to

10.4% in reserves and 2.8% in cashflow, with errors up to 7.8 degrees in slope angle

approximations; against no variation and 0% error for the surface based method.

Slope angle; pit optimization; Whittle; SimSched.

2

INTRODUCTION

Every mining project must present economic forecasts, so that investors are able to make

decisions related to the future mining operations. The mineral deposit is traditionally

discretized into mining blocks, and mine planners must decide which blocks should be

exploited, when and whether they should be processed or not. This decision process is

defined as the mine scheduling problem (Johnson, 1968 [1]).

Considering the computational complexity of giving an optimal schedule directly from the

resource orebody model, called Direct Block Scheduling (Jelvez, 2012 [2]), the historical

developments have guided mine planners to solve these problems into steps: ultimate pit

limits with nested pits; mining phases; operational schedule; blending; cutoff optimization;

stockpiles; etc. Given that each step is obtained separately, the final mine schedule is not

guaranteed to be optimal, even if each individual decision could be taken optimally.

The ultimate pit problem is the first step of this simplified process. The objective is to

determine the limits of the economic extraction of the deposit in order to maximize the project

cashflow. It should be noted that only slope angles are taken into account; the costs of time,

production and operational constraints are ignored at this stage. The ultimate pit with its

nested pits is a well-accepted approximation of the reality, which should be taken only as a

guide for the following mine planning steps. The ultimate pit does not have to be, necessarily,

the actual final state of the mine after the whole scheduling process.

The available software technology has allowed pit optimisation to be performed from a

simple scoping study up to complex on-site problems, as discussed by Dagdalen (2001) [3].

For decades, different pit optimization algorithms have been developed (Zhao and Kim, 1991

[4]; Hochbaum, 2008 [5]). One of the most popular methods (Lerchs and Grossmann, 1965

[6]) maximizes the project undiscounted cashflow using graph theory and discretizes the pit

space, usually by changing the metal price with revenue factors, which generates nested pits

(Whittle, 1999 [7]).

Used in many technical publications, such as Amankwah (2011) [8] and do Carmo et al.

(2006) [9], the Lerchs and Grossmann (LG) algorithm efficiency has been attested for

decades. LG is based on a fixed economic value for each block, considering that block

destinations are pre-determined. The input parameters are the economic aspects related to

the deposit, as well as the geotechnical slope angles.

3

In order to obtain the ultimate pit solution, the input parameters must be determined with the

appropriate level of accuracy. Economic parameters are normally well-analysed prior to the

optimization process. However, other inputs that affect the calculations are, in several

opportunities, neglected by the software user. Some parameters, despite being considered of

low impact, can result in noticeable economic differences that deserve the user’s attention,

depending on the characteristics of the deposit being considered.

Some of these parameters are the slope angles and the maximum number of levels

considered for the slopes approximation, as presented by Gallagher and Kear (2001) [10].

The sensitivity of the pit optimization to these parameters is going to be explored within this

paper. The current technology and the commercial softwares available on the market adopt

different methods to achieve or improve the approximation of overall slope angles so as to

represent the geotechnical assessment of the deposit.

This paper illustrates the impact of the slope angles approximations on ultimate pit reserves

and cashflows, and the accuracy of those approximations for a copper project. The

commercial softwares GEOVIA Whittle and MiningMath SimSched, which are based on

different methods for slope angles approximations, are used herein. The respective methods

of blocks precedence (Whittle) and mining surfaces (SimSched) are revisited. Results and

comparisons follow.

SLOPE ANGLE APPROXIMATION METHODS

The reality of open pit mines is that, at the start of the operations, at the end of each mining

period, at the end of the life-of-mine, or at any point in time, there is a surface in the field that

represents the state of the mine at that time. Humans need to approximate this reality into

computational models so that computers are able to recognize, present and take decisions

that represent the reality reasonably well. Herein, the two most common methods to

approximate slope angles have been chosen to assess their impacts on the reserves and the

cashflow of a real project.

For decades, mining professionals have represented mines into tri-dimensional blocks with

values. The method based on blocks precedence, adopted by Whittle, attempts to create a

list of block predecessors for each block, which means that, for removing block B, it is

necessary to remove the set of blocks PB (predecessors of block B), in order to give access

to block B. Figure 1 shows, in a section view, that a given overall slope angle is not always

4

achievable with the desired degree of accuracy, causing imprecisions in the resultant

ultimate pit slope angles. The black line is an example of one wall of the mine respecting

exactly the desired slope angle; the blocks are represented in light grey in the background.

Each green line represents one possible approximation for the black line, which would be the

desirable overall slope angle of the mining surface. The slope angles approximations are

based on block centroids. In order to limit the vertical distance of the search, the blocks

precedence method uses a parameter referred as the ‘maximum number of levels’. Set by

the user, this parameter determines the maximum number of blocks vertically searched for

the slope angles approximations (red lines in Figure 1), and, for simplicity, it will be called

herein by ‘number of levels’. The measure of the differences between the input desired and

the resulting slope angles is referred as the error of approximation.

Figure 1: Imprecision of blocks resulting from the precedence method (Whittle, 1998 [11])

In Figure 1, the bottom left block can only be mined if all other blocks above the chosen

green limit approximation are mined. In this example, if the number of levels is set to 1, only

the blocks from the first level above each block are considered in its list of predecessors;

therefore, by propagation, only the blocks crossed by the red lines would have to be mined,

which generates a steeper slope angle than requested. The best approximation was given by

the highest number of levels setup. If this parameter is increased, the list of predecessors is

extended to more levels, which also increases accuracy and processing times.

Figure 2 presents a feasible solution, looking only at blocks and considering a 45 degrees

overall slope angle. Note that, for each 50 m horizontally (two blocks), the vertical advance

also has 50 m (five blocks), which is a fair approximation of reality.

5

Figure 2: A feasible solution for the blocks precedence method

However, if the mining surface is represented by the basis (or centroid) of each block, there

will be situations like the one presented in Figure 3. Note that the reasonable approximation

of Figure 2, in reality, has regions with slope angles steeper than the requested 45 degrees.

Figure 3: Slope angle approximation errors

In this two-dimensional example, if one wants to guarantee that the situation in Figure 3

never happens, a conservative setup would be necessary, like the one illustrated in Figure 4.

Note that this configuration forces the overall slope angle to 38.7 degrees, which is over-

conservative and could result in economic losses.

Figure 4: A conservative slope angle approximation using blocks precedence

Most of the academic results and commercial softwares are based on the blocks precedence

method. Whittle has a report that calculates the errors associated with each setup of the

number of levels. An example is presented in Figure 5.

6

Figure 5: Example of the slope profile Whittle report

The second method evaluated herein is based directly on a representation of the mining

surfaces. Slope angles are controlled on the surface, not by dependencies between blocks.

Each horizontal red line represented in Figure 3 and Figure 4 could be seen as a part of the

mining surface being considered. These red lines are called grid cells, or simply cells. The

name ‘grid’ is associated with the fact that there is one cell for each column of blocks in the

orebody model. Each cell can assume any position within its column. This representation is

well-known by users of softwares such as GEOVIA GEMS, where it is called a Surface

Elevation Grid (SEG), or MineSight 3D, where it is called a Grid Surface File (GSF). The

center of each cell could be seen as a point in space, which could be triangulated so as to

generate a tri-dimensional surface.

If one wants to control slope angles over surfaces, each cell elevation must be compared to

its adjacent cells. The slope angle is evaluated as illustrated by the diagonal red lines in

Figure 3 and Figure 4. Figure 6 shows two possible representations of surfaces and their

associated mined blocks in grey. For any mining surface given, the blocks that have

centroids above their cell elevations are considered within the ultimate pit, or mining period.

7

Figure 6: Examples of surfaces respecting a 45 degrees slope angle setup

This approximation is 100% accurate, with no need for any parameter adjustment. Surfaces

resulting from algorithms based on this approach will have no triangles with a slope angle

higher than requested. For complex deposits with highly variable geotechnical zones, there

are no extra difficulties in slope angle management. Based on this representation, different

mine scheduling methods have been proposed (Goodwin et al., 2005 [12]; Marinho, 2013

[13]; Guimarães and Marinho, 2014 [14]).

CASE STUDY AND METODOLOGY

The deposit considered in this study is a copper porphyry occurrence in a highly

metamorphosed setting, associated with significant sulphide mineralization. The mineralized

domains outcrop and are hosted within gneisses and amphibolites, bound by schists with

lower degree of metamorphism. The deposit was discretized into regular blocks of

25x25x10 m³.

This deposit was selected due to its average complexity in terms of geotechnical zones.

Figure 7 shows section views of the three geological domains considered. Overall slope

angles have been defined for each domain. The input slope angle for the waste is 40

degrees. For the base case, the mineralized domains 1 and 2 assume slope angles of 45

and 50 degrees, respectively.

8

Figure 7: Plan view and vertical section representing the blocks by domain

In order to measure the variability of reserves and cashflow, the number of levels was varied

for different pit optimisation runs, using Whittle. All the remaining parameters were kept fixed.

These parameters are described in Table 1.

The surface based method was executed in a single run, using SimSched, which does not

require a parameter setup.

9

Table 1: Input parameters for the optimization calculation.

Parameters Units Value

Geotechnical

Waste (Deg) 40

Ore 1 (Deg) 45

Ore 2 (Deg) 50

Mining Factors

Dilution (%) 10.0

Recovery (%) 90.0

Processing Recoveries

Ore 1 (%) 80.0

Ore 2 (%) 87.0

Operating Costs

Mining Costs ($/tmoved) 3.50

Incremental Mining Cost ($/bench) 0.05

Reference Level (m)

3,130

Processing Costs ($/tore) 18.00

General and

Administration Costs ($m/year) 40.0

($/tore) 4.00

Selling Costs (%) 3.1

($/tmetal) 187.55

Metal Price

Copper ($/t) 6,050

RESULTS AND DISCUSSION

Base case

The resulting reserves and cashflows, with respective physical ultimate pit limits, for the

number of levels set to 10, 20 and 40, are presented in Table 2 and Figure 8. Results for the

surface method are also included in the comparison. With the increase of the parameter

being tested, the maximum error decreases from 3.0 to 0.3 degrees, the reserves decrease

by 5.4%, the average grades have negligible changes and the cashflow decreases by 2.4%.

The maximum error is the largest difference found between the resulting slope angles and

the desired input value, considering all the faces created for the optimised pitshell.

10

The method of surfaces, implemented in SimSched, has no approximation errors and

presents higher cashflow, even when compared to the Whittle result with 1.8 degrees of

maximum error. As illustrated in Figure 4 and observed in Figure 8, in order to reduce the

errors associated with slope angle approximations for the blocks precedence method, the

assumptions must be more conservative, which results in a lower cashflows for the project.

Figure 8 shows the oscillations of the ultimate pit shells for the blocks precedence method,

while the ultimate pit for the surface method has clear straight lines, changing only in the

transition zones of the different domains. The solution for the parameter set to 10 levels has

steeper slope angles than the others. If the remaining solutions are compared, it can be

noted that the solution given by SimSched has more ore and less waste, but the ore has a

lower average grade than the Whittle results.

Table 2: Results of the different pit optimizations performed.

Method Number of

Levels

Maximum error Ore Strip Ratio Aver. Grade Cashflow

Degrees Mt t/t % Cu M$

Blocks

Precedence

10 3.0 119.3 1.16 0.89 1,706.04

20 1.8 115.8 1.21 0.89 1,670.25

40 0.3 112.9 1.20 0.90 1,665.33

Surface - 0.0 116.5 1.15 0.85 1,692.80

Figure 8: Vertical section 1 showing the results and the domains

11

Constant slope angle case

Another scenario was run with both methods using a constant input value for the slope

angles. The base case was changed assuming a slope value of 40 degrees for all domains.

The results for this case are shown in Table 3 and the comparison between the resulting

surfaces is shown in Figure 9. This scenario shows the smallest differences among the

results, as there is no slope angle transition at the contacts between domains. When the

number of levels was set to 10, the maximum error was 2.0 degrees.

Table 3: Results for the scenarios with constant slope value.

Method Number of

Levels

Maximum error Ore Strip Ratio Aver. Grade Cashflow

Degrees Mt t/t % Cu M$

Blocks

Precedence

10 2.0 105.6 1.43 0.91 1,518.88

20 0.8 104.1 1.48 0.91 1,489.76

40 0.3 103.0 1.48 0.91 1,486.93

Surface - 0.0 101.6 1.39 0.87 1,508.00

Figure 9: Vertical section 1 showing the impact of the constant slope value

Abrupt transitions case

In order to show the sensitivity of the methods to abrupt transitions between slope profiles,

an exaggerated scenario was created considering the input values for the slope angle as 60

degrees for mineralization 1 and 30 degrees for mineralization 2. For the waste material, the

input slope angle remained 40 degrees. The same values for the number of levels were

12

considered for the blocks precedence method in comparison to a single run of the surface

method.

Due to its complexity, this is the most sensitive case. The scenario using 10 as the number of

levels results in a maximum error of 7.8 degrees, being the highest error found within this

study.

Among the blocks precedence results, the difference achieves more than 10% on the ore

reserves, and the cashflow varies 2.8% for the same group of tests. Table 4 summarizes the

results for the cases performed with abrupt transitions and Figure 10 shows the differences

of the resulting surfaces. As the Whittle software has limitations for the number of columns in

a slope cone, and it could not compute the precedence arcs for the 40 levels setup, the

number of levels was restricted to 35 for the last blocks precedence run.

Table 4: Results for the case with abrupt transition between slope profiles.

Method Number of

Levels

Maximum error

Degrees

Ore

Mt

Strip Ratio Aver. Grade

% Cu

Cashflow

M$ t/t

Blocks

Precedence

10 7.8 128.8 1.06 0.87 1,745.10

20 1.4 115.9 1.09 0.89 1,705.75

35 0.6 116.7 1.12 0.89 1,696.87

Surface - 0.0 133.0 1.13 0.82 1,707.20

Figure 10: Vertical section 1 showing the impact of the abrupt transitions case

13

Figure 11 shows details of the slope transition zone at the right wall of the pit in the section

view. It should be noted that in transition zones from a 30 degrees region to a 60 degrees

region, which happens from blue blocks to red blocks, the Whittle’s blocks precedence

method solution is optimistic, frequently assuming the highest value of 60 degrees. The

SimSched implementation of the surface method assumes the average slope angle at

transition zones, creating a more conservative surface.

Figure 11: Vertical section 1 showing details on transitions zones

Slope correction for the blocks precedence method

In order to perform a fair comparison, the surfaces generated by the Whittle implementation

of the blocks precedence method were corrected by the surface method algorithm

implemented in SimSched, so that the maximum error was set to zero and the same criteria

at transition zones was adopted. There are no parameters in Whittle for slopes

approximations capable of returning zero error and/or less optimistic criteria for transition

zones. Figure 12 shows the given surface when the number of levels was set to 10 and the

14

same surface after slope corrections. The corrected surface includes more ore and waste,

and the strip ratio increases by 6.6%.

Figure 12: Vertical section 1 showing one surface before and after slope corrections

Figure 13 shows the surfaces for the different number of levels after corrections with results

in Table 5.

Table 5: Results for the abrupt case after slope corrections.

Method Number of

Levels

Maximum error

Degrees

Ore

Mt

Strip Ratio

t/t

Aver. Grade

% Cu

Cashflow

M$

Blocks

Precedence

10 0.0 132.2 1.13 0.82 1,690.70

20 0.0 118.2 1.11 0.84 1,681.60

35 0.0 118.7 1.12 0.84 1,677.50

Surface - 0.0 133.0 1.13 0.82 1,707.20

15

Figure 13: Vertical section 1 showing the surfaces after slope corrections

Notice that for the same parameters of Table 1 plus the geotechnical setup of this

subsection, a given user could have run Whittle and taken the ultimate pit with 128.8Mt of ore

and 1,745.10M$ of cashflow from Table 4, while another user, who decided for a more

conservative setup, using a maximum of 35 levels and followed by slope corrections, could

have found 118.7Mt of ore and 1,677.50M$ of cashflow from Table 5. This represents an

8.5% difference in the ore reserves and a 4.0% difference in cashflow given only by

differences in slope approximations criteria.

CONCLUSIONS

This study shows the sensitivity of the ultimate pit reserves and cashflow to the method and

the selected input parameters for slope angle approximations. The blocks precedence

method has errors associated to slope angles approximations, and requires the setup of the

number of levels parameter. The surface method has zero error, and requires no parameter

setup; therefore, it is more reliable in terms of physical and numerical results reported.

The results for both methods are equivalent for the case with a constant slope angle. The

major differences were noticed for the case with abrupt changes between slope profiles,

which translates to larger errors for the blocks precedence method. The ultimate pits

produced by Whittle have shown a variation up to 10.4% in ore reserves and 2.8% in

cashflow, with errors up to 7.8 degrees in slope angle approximations. It was observed that,

if the deposit is more complex in terms of geotechnical zones, the impacts on the final results

will be higher when varying the number of levels.

16

Both Whittle and SimSched algorithms look for maximum cashflow. If slope angle

consequences are ignored and accepted as reasonable, all cases show similar results

between methods with differences in cashflow no higher than 2%. If the Whittle surfaces are

corrected for a fair comparison, SimSched produced an ultimate pit with similar ore and

waste production, but 1% higher cashflow than the best of the three results produced by

Whittle.

The Lerchs-Grossmann algorithm offers cashflow maximization for the given input

parameters. Although it is a deterministic algorithm with a guaranteed maximum cashflow,

the user must be careful when defining the number of levels. A more conservative parameter

(a higher number of levels) results in a lower cashflow. The current algorithms based on

surfaces do not mathematically guarantee the highest cashflow, as they have heuristics built

in, but they do not lose value with slope angle approximations and guarantee zero error on

resulting surfaces.

REFERENCES

[1] Johnson, T. B. (1968) Optimum open pit mine production scheduling. PhD thesis,

Operations Research Department, Universiry of California, Berkeley, May 1968.

[2] Jelvez, E., Morales, N., Peypouquet, J., Reyes, P. (2012) Algorithms based on

aggregation for the open pit block scheduling problem, in MININ 2012, Santiago, Chile.

[3] - Dagdalen, K. (2001) Open Pit Optimization – Strategies for Improving Economics of

Mining Projects Through Mine Planning, 17th International Mining Congress and Exhibition of

Turkey, pp. 117-121.

[4] Zhao, Y., Kim, Y. C. (1991) A New Graph Theory Algorithm for Optimal Pit Design. SME

Transactions, vol. 290, pp. 1832-1838.

[5] Hochbaum DS (2008) The pseudoflow algorithm: A new algorithm for the maximum-flow

problem. Operations Research 56(4), pp. 992-1009.

[6] Lerchs, H. and Grossmann, L.F. (1965) Optimum Design of Open Pit Mines, Canadian

Institute of Mining Bulletin. Vol. 58, no. 633, pp. 47-54.

17

[7] Whittle, J. (1999) A decade of open pit mine planning and optimization — The craft of

turning algorithms into packages. 28th APCOM Symposium, SMEAIME, Golden, Colorado,

pp. 15-24.

[8] Amankwah, H. (2011) Mathematical Optimization Models and Methods for Open-Pit

Mining, Dissertation, Linköping University – Insitute of Technology.

[9] do Carmo, F.A.R. et. Al. (2006) Otimização econômica de explotações a céu aberto,

Revista Escola de Minas, Vol. 59, pp. 317-321.

[10] Gallagher, M.S. and Kear, R.M. (2001) Split shell open pit design concept applied at De

Beers Venetia Mine South Africa using the Whittle and Gemcom software, The Journal of

The South African Institute of Mining and Metallurgy, 401-410.

[11] Whittle, J. (1998) Four-X User Manual, Whittle Programming Pty Ltd.

[12] Goodwin, G. C., Seron, M. M., Middleton, R. H., Zhang, M., Hennessy, B. F., Stone, M.

S., Menabde, M. (2005) Receding horizon control applied to optimal mine planning.

Automatica, Vol. 42 (8), pp. 1337-1342.

[13] Marinho, A. (2013) Surface Constrained Stochastic Life-of-Mine Production Scheduling.

MSc. Thesis, McGill University, Montreal, Qc, 119 p.

[14] Guimarães, O., Marinho, A. (2014) Sequenciamento direto de blocos. Submitted to the

8th Brazilian Congress of Surface Mining.