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The impacts of slope angle approximations on open pit mining production scheduling

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Open pit mining production scheduling requires slope angles to be defined as input to the engineering/optimization processing steps. The slope angles are approximated through appropriate parameters and methods, in an attempt to reproduce the safety requirements of an open pit mining operation. Different methods can result in considerably dissimilar results for the pit configuration. This paper presents the volatility of reported mineral reserves and cashflows to different slope angle approximation methods, when applied to open pit mining production scheduling. Two methods were revisited and parameters were changed to create a set of scenarios. 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 comparison between both methods shows the importance of the selection of the appropriate parameters and methods for each deposit. The ultimate pits produced by the blocks precedence method have shown a variation of up to 10.4% in reserves, with errors up to 7.8 degrees in slope angle approximations; against no variation and 0% error for the surface based method. For the mining schedule, using similar parameters for both methods, the discounted cashflow and production indicators differences are also analysed and reported herein.
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The impacts of slope angle approximations on open pit
mining production scheduling
Filipe S. Beretta*
Mining Engineer Consultant, SRK Consulting, Cardiff, UK fberetta@ymail.com
Alexandre Marinho
MiningMath Associates, Belo Horizonte, Brazil, CEO, alexandre.marinho@miningmath.com
ABSTRACT
Open pit mining production scheduling requires slope angles to be defined as input to the
engineering/optimization processing steps. The slope angles are approximated through appropriate
parameters and methods, in an attempt to reproduce the safety requirements of an open pit mining
operation. Different methods can result in considerably dissimilar results for the pit configuration. This
paper presents the volatility of reported mineral reserves and cashflows to different slope angle
approximation methods, when applied to open pit mining production scheduling. Two methods were
revisited and parameters were changed to create a set of scenarios. 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 comparison
between both methods shows the importance of the selection of the appropriate parameters and methods
for each deposit. The ultimate pits produced by the blocks precedence method have shown a variation of
up to 10.4% in reserves, with errors up to 7.8 degrees in slope angle approximations; against no variation
and 0% error for the surface based method. For the mining schedule, using similar parameters for both
methods, the discounted cashflow and production indicators differences are also analysed and reported
herein.
INTRODUCTION
Conventional open pit mine production scheduling involves a group of activities to be performed and
decisions to be taken by mine planners, such as pit optimization with nested pits, pushback design,
blending, cut-off grade optimization, etc. These steps are widely used in the industry as an alternative to
the Direct Block Scheduling technique (Johnson 1968) which has not been able to be performed in the past
decades, due to computational limitations and lack of efficient algorithms.
The current technology adopts different methods to model the approximation of slope angles so as to
represent the geotechnical assessment of the deposit. This paper presents the impacts on the resulting
mine sequencing and the discounted cashflows for a copper project for two different methods. The
commercial softwares GEOVIA Whittle and MiningMath SimSched, which are based on blocks
precedence (Whittle) and mining surfaces (SimSched) are revisited. Results and comparisons follow.
SLOPE ANGLE APPROXIMATION METHODS
The method based on blocks precedence assumes that a given block cannot be mined if a given set of
blocks from levels above are not extracted. To approximate the slope angle configuration, the method
uses the blocks centroids and a maximum number of levels parameter set by the user.
Figure 1 shows, in a section view, that a given overall slope angle is not always achievable with the
desired degree of accuracy, causing imprecisions in the resultant slope angles.
Source: Whittle, 1998.
Figure 1: Imprecisions of slope angle approximations resulting from the precedence method
The second method considered for this study is based on the direct representation of surfaces. In this
case, the slope angles are represented by surfaces, not by dependencies among blocks. The representation
is based on the cells of a grid, that considers the same dimensions of the blocks in X and Y, but with the
3rd dimension free to assume any elevation value within its column. The center of each cell could be seen
as a point in space, which could be triangulated to generate a tri-dimensional surface. Slope angles are
controlled by comparing elevations of adjacent cells.
Based on this representation, different mine scheduling methods have been proposed (Goodwin et al.
2005; Marinho 2013). Figure 2 shows the elevations assumed for each cell in vertical sections,
guaranteeing surfaces with accurate slope angles.
Figure 2: Examples of surfaces respecting a 45 degrees slope angle setup
CASE STUDY AND METHODOLOGY
This study considers a copper porphyry deposit discretized into mining blocks of 25x25x10 m3. Figure 3
shows section and plan views of the three geological domains considered.
Figure 3: Plan view and a vertical section representing the blocks by domain
A pit optimization (Beretta and Marinho 2014) and a mine production scheduling have been performed
using both methodologies described in the previous section. All the parameters were kept fixed and the
number of levels was varied for the Whittle results, to find a better approximation of the required slope
angles. The surface based method was executed in a single run, using SimSched, which has a hybrid
algorithm implemented composed by mixed-integer programming with proprietary heuristics that does
not require a number of levels to be set. The parameters considered are described in Table 1.
Table 1: Input parameters
Parameters
Units
Value
Geotechnical
Waste
(Deg)
40
Ore 1
(Deg)
60
Ore 2
(Deg)
30
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
($/t)
6,050
RESULTS AND DISCUSSION
Pit optimization
The pit optimization results and comparisons were published in Beretta and Marinho (2014). For a matter
of completion, the main results and conclusions of this previous work are highlighted here. The errors on
the slope angles approximations achieved 7.8% for the blocks precedence method, varying up to 10.4% in
reserves and 2.8% in cashflow, when changing the parameter maximum number of levels. No errors were
found for the surface methods results, as the angles are guaranteed to be 100% accurate, as described
previously and shown in Figure 2, given that there is no approximation when comparing elevations of
two adjacent cells. The results are shown in Table 2 and Figure 4. Beretta and Marinho (2014) also shows
that, if the errors are corrected, the method based on surfaces returns the highest undiscounted cashflow
for this case.
Table 2: Results of the pit optimizations performed
Method
Maximum error
Degrees
Ore
Mt
Strip Ratio
Aver. Grade
% Cu
Cashflow
M$
t/t
Blocks
Precedence
7.8
128.8
1.06
0.87
1,745.10
1.4
115.9
1.09
0.89
1,705.75
0.6
116.7
1.12
0.89
1,696.87
Surface
0.0
133.0
1.13
0.82
1,707.20
Figure 4: Vertical section 1 showing the pit optimizations resulting surfaces
Nested pits
Whittle uses nested pitshells as basis to calculate the mine production scheduling, which are defined by a
set of runs of the Lerchs-Grossmann (LG) algorithm with different selling prices (Revenue Factors). Table
3 shows the nested pits obtained by varying the revenue factor from 0.01 up to 1.00.
Table 3: Whittle nested pits
Pit
Revenue
Factor
Max. of 10 Benches Search
Max. of 20 Benches Search
Max. of 35 Benches Search
Ore (Mt)
Waste (Mt)
Ore (Mt)
Waste (Mt)
Ore (Mt)
Waste (Mt)
1
0.46
5.4
19.6
5.3
19.9
5.3
20.2
2
0.47
6.1
20.1
6.9
23.2
6.7
22.9
3
0.48
7.2
21.9
7.2
22.9
7.9
24.2
4
0.49
8.6
24.3
8.6
25.3
8.5
24.6
5
0.5
11.6
30.8
9.5
26.8
9.2
26.0
6
0.51
12.9
31.5
12.2
30.9
12.4
31.6
7
0.52
13.1
31.4
12.3
30.8
12.6
31.5
8
0.53
14.4
32.8
13.7
33.3
13.1
31.5
9
0.54
15.1
34.1
16.2
37.7
15.1
36.0
10
0.55
17.2
36.8
16.9
37.6
16.9
38.1
11
0.56
18.2
37.5
18.0
38.5
17.7
38.2
12
0.57
18.6
37.3
18.3
38.3
18.6
39.2
13
0.58
19.5
38.1
19.3
39.1
19.6
40.3
14
0.59
22.1
42.3
20.3
39.9
20.9
41.7
15
0.6
23.1
42.2
22.9
43.5
22.4
42.7
16
0.61
26
46.3
25.1
46.5
24.9
46.4
17
0.62
35.7
68.7
34.9
69.2
26.2
47.6
18
0.63
37
68.2
36.1
68.6
36.1
69.4
19
0.64
41.9
85.1
37.6
69.9
37.7
70.9
20
0.65
44.2
86.4
38.5
69.3
42.8
87.6
21
0.66
45.8
89
44.6
90.5
43.9
87.8
22
0.67
48
88.9
46.5
89.9
45.7
88.1
23
0.68
50.9
91.9
48.0
90.6
47.8
91.7
24
0.69
51.8
91.2
50.4
92.8
50.5
93.8
25
0.7
54.1
94.9
52.0
93.4
52.1
94.5
26
0.71
55.4
95.4
55.9
99.9
54.1
98.3
27
0.72
59.1
99.3
56.9
99.4
56.8
100.6
28
0.73
59.9
98.6
57.6
98.8
57.9
100.3
29
0.74
62.3
99.6
60.4
100.7
60.4
101.5
30
0.75
63.5
98.8
62.8
102.7
61.5
100.5
31
0.76
64.4
100.5
63.9
103.8
63.4
104.2
32
0.77
66.3
100.2
65.9
104.0
65.7
104.6
33
0.78
67.9
99.8
67.8
103.6
68.0
104.9
34
0.79
72.2
104.9
70.3
104.2
69.1
103.9
35
0.8
73.2
104
71.7
104.4
71.7
105.4
36
0.81
74.1
105
72.0
104.3
72.0
105.1
37
0.82
77
106.5
75.5
109.5
75.1
109.0
38
0.83
78.8
106
78.2
109.4
77.7
109.0
39
0.84
81.2
105
81.4
112.4
80.7
112.0
40
0.85
82
105.7
81.8
112.5
81.7
113.7
41
0.86
84.8
106
83.8
111.8
84.3
114.0
42
0.87
87.8
105.6
87.4
113.4
86.3
112.4
43
0.88
92.5
112.8
88.3
113.3
89.0
116.0
44
0.89
94
111.9
95.8
120.8
91.9
116.8
45
0.9
99.5
117.3
97.9
120.3
96.5
120.1
46
0.91
100.2
116.9
98.6
120.0
98.2
120.8
47
0.92
103.9
121.1
101.1
120.8
101.0
122.4
48
0.93
105.2
121.2
102.7
122.6
103.4
126.1
49
0.94
108.1
122.4
105.6
124.3
105.0
125.6
50
0.95
109.3
123.2
106.1
123.9
105.6
125.1
51
0.96
112.6
123.3
108.7
124.8
108.0
125.3
52
0.97
114.6
124.6
111.5
126.7
110.6
126.7
53
0.98
116.6
125.5
113.9
128.4
111.2
126.1
54
0.99
120.6
127.5
115.2
127.5
113.1
125.7
55
1
128.8
137.1
115.9
126.9
116.8
130.3
Mining Schedule
Whittle performs the steps of pit optimization with nested pits, pushback design, mining schedule, cut-
off optimization, etc. sequentially. SimSched performs the whole scheduling in one single run through a
direct block scheduling algorithm based on mixed integer programming (MIP) plus proprietary heuristics
(Guimarães and Marinho 2014). Algorithms developed in this framework are able to include additional
restrictions within the same optimization and decide also the optimal destination for each block.
For this comparison, operational parameters were ignored, to avoid the discussion on which result is
more or less operational. For the scheduling, the discount rate assumed was 10% per period; the
processing capacity was set to 5Mt and the total mining capacity to 15Mt. A cut-off grade optimization
(Rendu 2008) has been manually performed in Whittle through the setup of the minimum grade to be
sent to the plan, period by period, as an interactive process that requires time and previous
understanding of the deposit, as shown in Table 4.
Table 4: Cut-off grade optimization parameters
Period
(from)
Cut-off
(% Cu)
1
0.00
2
0.70
6
0.50
7
0.00
8
0.25
11
0.00
Stockpiles were not included in this comparison. Figure 5 shows the profile of the plant throughput for
the Whittle and SimSched results. Note that Whittle allows for blocks to be mined partially, while
SimSched does not.
Figure 5: Ore production profile
The comparison of other production indicators is shown in Table 5. Note that the indicators and the
number of periods are directly affected by the method and the number of levels considered. For the
Whittle results, the less conservative setup (10 levels) allowed for higher ore production and NPV, in
exchange for higher errors in slope angles approximations. The direct block scheduling with surfaces
produced the highest NPV for this deposit, as shown in Figure 6. There is no optimality GAP assessment,
as the software used has heuristics built in.
Table 5: Comparison of production indicators
Period
Mine Total Production (Mt)
Plant Input Grade (% Cu)
Plant Output Cu Content (kt)
SimSched
Whittle - Max.
number of Benches:
SimSched
Whittle - Max.
number of Benches:
SimSched
Whittle - Max.
number of Benches:
10
20
35
10
20
35
10
20
35
1
15.2
15.0
15.0
15.0
1.4
0.8
0.8
0.8
63
31
30
29
2
14.9
8.3
8.6
8.8
1.7
1.3
1.3
1.3
73
57
57
56
3
14.5
10.7
10.1
9.9
1.4
1.6
1.6
1.6
62
69
68
69
4
15.1
11.7
12.8
14.5
1.2
1.2
1.2
1.2
51
53
52
52
5
15.2
11.5
11.6
10.7
1.3
1.1
1.2
1.2
57
49
51
50
6
14.6
10.8
10.9
10.1
1.2
0.9
0.9
0.9
51
40
40
38
7
14.9
15.0
15.0
15.0
1.2
0.9
0.8
0.9
51
33
27
34
8
15.4
8.3
7.6
8.8
1.3
0.7
0.7
0.7
58
29
30
29
9
14.6
7.0
6.9
7.0
1.2
0.9
0.9
0.9
52
40
40
40
10
13.4
5.5
5.3
5.4
1.1
1.1
1.1
1.1
48
49
50
50
11
14.1
15.0
15.0
15.0
1.0
0.7
0.8
0.7
42
26
23
24
12
13.7
6.7
9.8
7.3
0.9
0.8
0.6
0.7
40
37
28
32
13
10.1
9.2
6.0
10.3
0.9
1.1
1.1
1.2
38
49
49
51
14
10.8
8.3
9.1
8.3
0.9
0.9
0.9
0.9
37
37
40
37
15
13.3
13.2
11.2
10.9
0.8
0.8
0.8
0.8
37
36
35
36
16
8.9
8.1
9.8
10.2
0.8
0.8
0.8
0.8
35
34
35
34
17
10.1
11.7
9.2
9.3
0.8
0.8
0.8
0.8
33
35
35
33
18
10.1
9.2
12.5
14.7
0.7
0.7
0.8
0.8
30
32
33
32
19
12.3
12.1
12.4
11.7
0.8
0.7
0.7
0.8
33
29
32
32
20
6.3
8.2
13.6
11.0
0.7
0.7
0.7
0.7
17
31
29
30
21
11.5
5.9
8.4
0.8
0.7
0.7
0.7
30
31
29
22
12.4
11.2
13.5
0.7
0.7
0.7
29
29
30
23
11.5
12.9
11.0
0.7
0.7
0.7
29
29
28
24
15.0
0.3
0.6
0.7
22
1
25
7.7
0.6
27
26
2.3
0.7
14
Total
98.2
124.9
111.8
112.6
1.1
0.9
0.9
0.9
907
946
872
877
Figure 6: Cumulative NPV profile
CONCLUSIONS
This study shows the impact of different methods adopted for pit optimization and mining production
scheduling. The direct block scheduling with slope approximations through surfaces, adopted by
SimSched, shows no errors in geotechnical approximations, differently from the LG-based techniques,
adopted by Whittle, where positive errors are encountered.
For the pit optimization, Whittle has shown a variation of up to 10.4% in reserves, with errors up to 7.8
degrees in slope angle approximations; against a single run with 0% error for the SimSched solution.
For the mine scheduling, Whittle required a set of steps and many runs to setup the parameters properly.
It was necessary nearly 10 hours to find the best parameters and execute the steps presented herein for
one case using Whittle. These steps and successive tests were not required by SimSched as it performs
direct block scheduling, which took around 1 minute for scheduling this deposit. Higher NPV and better
production control were found in SimSched results, while Whittle presented variations depending on the
maximum number of levels considered.
REFERENCES
Johnson, T. B. (1968) Optimum open pit mine production scheduling. PhD thesis, Operations Research
Department, University of California, Berkeley, May 1968.
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.
Marinho, A. (2013) Surface Constrained Stochastic Life-of-Mine Production Scheduling. MSc. Thesis,
McGill University, Montreal, Qc, 119 p.
Beretta, F., Marinho, A. (2014). Impacts of Slope Angle Approximations on Pit Optimization. 8th Brazilian
Congress of Surface Mining, Belo Horizonte, Brazil.
Guimarães, O., Marinho, A. (2014) Sequenciamento direto de blocos. 8th Brazilian Congress of Surface
Mining, Belo Horizonte, Brazil.
Rendu, J. M. (2008). An introduction to cut-off grade estimation, Society for Mining, Metallurgy, and
Exploration, Inc.(SME), Colorado, USA.
*Corresponding author: Filipe Beretta MSc., SRK Consulting, Mining Engineer Consultant, 5th Floor,
Churchill House 17 Churchill Way Cardiff - CF10 2HH Wales UK. Phone: +44 (0) 2920 348 150.
Email: fberetta@ymail.com
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Surface Constrained Stochastic Life-of-Mine Production Scheduling
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Impacts of Slope Angle Approximations on Pit Optimization. 8th Brazilian Congress of Surface Mining
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Guimarães, O., Marinho, A. (2014) Sequenciamento direto de blocos. 8th Brazilian Congress of Surface Mining, Belo Horizonte, Brazil.
An introduction to cut-off grade estimation
  • J M Rendu
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