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Colorado School of Mines
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Machine Placement Level
401 402 403 404 405 406 407 408 409 410 411 j 412 Total
560 10- 13 12 i H DR ■ ■ ■ ■ m ... 1160
600 10- 13 12 ’ _____i i 40
mm mm
792 43- 46 45 2 — ■ ■ — — 1390
|n leu in
820 14- 15 16 i a e.m O É l n 'pu 1440
820 16- 17 16 2 □ CZ! LZJ □ 520
820 38- 41 40 1 ■ t i 1370
immfwm
820 42- 46 45 i jh i CJHLJHi 2140
849 14- 15 16 i 1 30
849 18-22 19 i 0 □ □ □ Cl □ D 0 a a I l Ï 3 980
849 23- 26 25 i 1___11___18 11 i8 11 1S 1 i-----! i-J LZUtD 1740
849 31- 34 33 i 2160
849 35- 38 37 1 E Z*(EZH (EZ^c^C 3<C 3<c3e|[ze<nei|[Bei 1870
849 39- 41 40 1 □ a ezi e u e i ■ O ■ n mu | Œ ecj peu 1630
849 42- 46 45 i F n me 320
878 23- 26 25 i ____ . 1 »a a _ _ □ □ |i:....: . 480
878 27- 30 28 i a mm n ■ m i a i mo...tt Tv- 2160
878 31- 34 33 i 1 1 mm 730
878 35-38 37 1 j| n e = am oe Ice |: e i:jê r:u F::e 1430
[MB |0UI |0O# l^Di INO |Wi |BPi !mW )KB |HH |BI
Development Ore jm | 1560
\üD
Development Waste JQ |Q □ p [□ |Q [□ |Z3 jü LJ p 996
Total Kton/Month 11943 2033 2043 2073 1983 2023 2073 1983 1893 2003 2053 20431 24146
F
Total Kton/Day J 62.7 70.1 65.9 69.1 64.0 67.4 66.9 64.0 63.1 64.6 68.4 65.9
B1 Kton/Day I 9"4 10.3 9.7 9.7 9.7 10.3 9.7 9.4 9.7 9.7 10.7 10.6
B2 Kton/Day 30.4 33.9 33.0 34.8 31.1 32.4 32.7 30.7 31.8 32.0 33.8 32.0
D3 Kton/Day j 22.9 25.9 23.2 24.7 23.2 24.7 24.5 23.9 21.7 22.9 24.0 23.2
Month 401 402 403 404 405 406 407 408 409 410 411 412 I
Figure 22. Illustration of a third year production schedule for the Kiruna Mine with the
original objective function
1 I
i
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Table 11. Calculation of the percent deviation by using du and dd values for original
objective function
Time Period du Ore Type du Time Period dd Ore Type dd
Month B1, B2, D3 Kton Month B1, B2, D3 Kton
1 B1 73 1 B2 54
1 D3 32 2 B2 94
2 B1 23 3 B2 134
2 03 52 4 B2 114
3 B1 43 4 03 18
3 03 62 5 B2 134
4 B1 43 5 03 8
5 B1 23 6 B2 74
6 B1 33 6 03 48
7 B1 63 7 B2 24
7 03 22 9 B2 89
8 B1 73 10 B2 214
8 B2 6 11 B2 244
8 03 62 12 82 134
9 B1 58 13 B1 17
9 03 12 13 B2 184
10 B1 53 14 B1 27
10 03 12 14 B2 164
11 B1 33 14 D3 78
11 03 32 15 B2 64
12 B1 13 15 03 18
12 D3 92 16 B2 59
13 D3 42 16 D3 58
15 B1 13 17 B2 44
16 B1 28 17 03 48
17 B1 23 18 B2 14
18 B1 23 18 03 58
19 B1 3 19 B2 4
19 03 52 20 B1 17
20 D3 102 20 B2 114
21 03 2 21 B1 7
22 D3 12 21 B2 114
23 03 2 22 B1 7
24 03 2 22 B2 84
26 B1 3 23 B1 17
26 B2 6 23 B2 4
26 D3 22 24 B1 27
27 B1 3 24 B2 54
27 B2 46 25 B1 7
28 B2 66 25 B2 34
28 03 12 25 D3 18
29 B1 3 27 03 8
30 B1 13 28 B1 7
30 03 12 29 B2 14
31 B1 3 29 D3 8
31 B2 36 30 B2 4
31 03 32 32 B1 7
32 D3 12 32 B2 24
34 B1 3 33 B1 7
34 B2 16 33 B2 24
35 B1 23 33 03 78
35 B2 36 34 D3 18
36 B1 33 35 D3 8
36 B2 16 36 03 8
sum 1,615 2,939
Total Production Target 72,072 Kton
Total Deviation 4,554 Kton
Deviation 6.32 %
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5.2.2. Production Scheduling by Minimizing Deviation from "Planned Production
Demands" with Weight on the Time Periods
As mentioned in Chapter 3, the deviation terms in the objective by time period
and /or ore type can be weighted to emphasize the importance of meeting demands as
closely as possible in a specific time period and/or for a specific ore type. A weighted
scheme was implemented where weight for first year is 100, weight on second year is 100
and weight on third year is 1.
The solution time is less than the original model. The optimal solution is obtained
in 8 seconds for three-year monthly time horizon. Figures 23, 24 and 25 present the
complete three-year schedule.
The ratio of the total tons of deviation to the total tons demanded is 7.09 %.
74.20 % of the 7.09 % results from the first 2 years compared to 85.29 % for the original
schedule. These numbers are produced by the same procedure that is explained in the
original case. Table 12 presents the output for dukt and ddkt for this scenario. As a result,
the weighting on the time period has an effect on not only the running time but also on
the quality of the objective.
Different weights can be used with each ore type and/or time period. The mine
planner can carry these weights and produce different strategic plans according to the
company’s planning objectives and market conditions.
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Colorado School of Mines
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Machine Placement Level
201 202 203 204 205 206 207 208 209 210 211 212 Total
765 42- 43 45 i ■ 690
765 44- 46 45 2 H i □BE QflB HE 1680
792 14- 15 16 i F I EU EU E U |EU |EU EU l a CB tB tB |H 1630
792 19-21 19 2 □ 130
792 33- 34 33 2 ru eu EU CU I ! 540
792 35- 37 37 1 HE i l l EB 1 420
792 39- 42 40 i m3 |n I n i n | n m in |E U |e u b u r u t u 1760
792 43- 46 45 2 i ■ m m m m 750
818 16- 17 16 2 0 D D □ 650
820 16- 17 16 2 P a 150
820 18- 19 19 i Em EmEmemEmEmEmiEmlc 1160
820 23- 26 25 i c Em MU Em Itm izi] cm i 1120
820 27- 28 28 i i i I I EU ■u cu imi im jim tm d 1250
820 29- 30 28 2 r i HI I 260
820 31- 32 33 j tu IZJ i i |n |n |n |n |n n t u E U lu 1650
820 33- 34 33 2 [EE m ■ ■ LH n i l a | a tu t a imi fcmi 1600
820 35- 37 37 1 m mm » Hi Bi UB f Bi FEB FEE 1EBI 1690
820 38- 41 40 i * a a n 360
849 27- 28 28 1 I m ie CB tB t » l a 930
849 29- 30 28 2 ■ GB CH o s im a n a 1210
849 31- 34 33 i a m IB cb pa p i a )a 820
849 35- 38 37 i X ...JÜ BE 240
Development Ore BEE ifln EH BE iflH 1560
Development Waste
a !n □ m 996
Total Kton/Month 2053 1983|1973 1913 1883 1873 1973 2043 1893 1853 1843 1963 23246
Total Kton/Day 66.2 70.8 63.6 63.8 60.7 62.4 63.6 65.9 63.1 59.8 61.4 63.3
B1 Kton/Day 11.9 11.4 11.0 11.3 10.3 11.0 11.3 11.6 10.8 10.6 10.0 9.7
B2 Kton/Day 29.8 31.5 27.2 28.8 27.2 29.8 29.5 30.1 27.9 24.0 24.4 27.2
D3 Kton/Day
24.5 27.9 25.5 23.7 23.2 21.7 22.9 24.2 24.3 25.2 27.0 26.5
Month
201 202 203 204 205 206 207 208 209 210 211 212
Figure 23. Illustration of a first year production schedule for the Kiruna Mine with the
weighted objective function
□ : n
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Machine Placement Level 301 302 303 304 305 306 307 308 309 310 311 312 Total
765 44- 46 45 2 140
792 14- 15 ■ m m 340
792 39- 42 ■ ■ a l a j u leu j g \tzm 970
792 43- 46 45 2 1920
820 14- 15 16 i !■ :■ ! ■ «■ i™ |3 i m m c e I 980
820 16- 17 16 2 J D E=j jCZZl IEZ3 jE=j iEZ3 i_ j le u tz i |l:j |_ j |_ j I 1560
820 33- 34 33 2 j P |U \M ~J ] □ |C 3 C 3 O 950
820 35- 37 1200
in
820 38- 41 40 i 1690
820 42-46 45 i 680
849 23- 26
849 27- 28
849 29- 30
849 31- 34 i n j n j n n |izi i n ezi j
849 35- 38 37 1
849 39- 41 40 1 3 Z3 210
878 27- 30 28 j im ne ™ L.ecBKT m IB ■ 1480
-I
878 31- 34 33 1 200
Development Ore I— I— pa I— I— ca —
B L I 1560
Development Waste !□ !□ [□ 0 [□ !□ [□ p 996
Total Kton/Month 1853 1813 1943 1943 2013 2053 2123 1993 1883 1963 1993 2013 i 23586
Total Kton/Day 59.8 64.8 62.7 64.8 64.9 68.4 68.5 64.3 62.8 63.3 66.4 64.9 |
Bl Kton/Day 8.7 9.6 9.7 10.2 10.0 10.3 9.7 9.0 9.7 9.4 9.7 9.4
B2 Kton/Day 25.9 29.8 29.8 30.9 30.1 32.4 31.7 28.5 29.8 30.7 32.4 29.8
D3 Kton/Day 25.2 25.4 23.2 23.7 24.8 25.7 27.1 26.8 23.3 23.2 24.3 25.8
Month 301 302 303 304 305 306 307 308 309 310 311 312
Figure 24. Illustration of a second year production schedule for the Kiruna Mine with the
weighted objective function
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Colorado School of Mines
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Machine Placement Level
401 402 403 404 405 406 407 408 409 410 411 412 Total
■ ■ ■ ■
560 10- 13 12 i 1050
640 9- 10 9 2 440
792 43- 46 45 2 1390
ri si n j 500
820 14- 15 16 i
820 16- 17 16 2 1 1 [CZ] |LZI |LZ3 520
820 38- 41 40 È 1880
820 42- 46 45 2140
849 18- 22 19 □ p p p p p p p p p p a 980
849 23- 26 25 c lnn Id leu l a l a l a | c L n P 1740
849 31- 34 2160
849 35- 38 1690
849 39- 41 la [□ lE3 lu |B3
849 42- 46
878 23- 26 25 p EZ] p[
878 27- 30 28
878 31- 34 33
878 35- 38 37 340
■■ ■■ 'mtm mm :h * 1 :■« !■■ »■ iu üeô"
Development Ore
Development Waste n !□ n n n p n n m n n p i 996
Total Kton/Month 1983 2093 2123 2133 2013 2043 2083 2103 2153 2163 2173 20631 25126
Total Kton/Day 64.0 72.2 68.5 71.1 64.9 68.1 67.2 67.8 71.8 69.8 72.4 66.5
Bl Kton/Day I 10.0 11.4 11.0 11.0 10.3 11.7 11.0 11.3 11.3 11.0 11.3 11.0
B2 Kton/Day ; 3i.i 35.3 33.3 34.4 30.7 31.4 31.7 32.0 36.8 35.3 36.8 32.0
D3 Kton/Day 22.9 25.5 24.2 25.7 23.9 25.0 24.5 24.5 23.7 23.5 24.3 23.5
Month 401 402 403 404 405 406 407 408 409 410 411 412
Figure 25. Illustration of a third year production schedule for the Kiruna Mine with the
weighted objective function
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Colorado School of Mines
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Table 12. Calculation of the percent deviation by using du and dd values for weighted
objective function
Time Period du Ore Type du Time Period dd Ore Type dd
Month Bl, B2, D3 Kton Month Bl, B2, D3 Kton
i Bl 73 i B2 54
i D3 32 2 B2 94
2 Bl 23 3 B2 134
2 D3 52 4 B2 114
3 B1 43 4 03 18
3 D3 62 5 82 134
4 B1 43 5 03 8
5 Bl 23 6 B2 84
6 Bl 33 6 03 78
7 B1 53 7 B2 64
8 B1 63 7 03 18
8 03 22 8 B2 44
9 Bl 28 9 B2 139
9 03 2 10 62 234
10 Bl 33 11 B2 244
10 03 52 12 B2 134
11 Bl 3 13 B1 27
11 03 82 13 B2 174
12 Bl 3 14 Bl 27
12 03 92 14 B2 144
13 03 52 14 03 18
15 Bl 3 15 B2 54
16 Bl 8 15 03 8
17 Bl 13 16 B2 49
17 03 42 16 03 18
18 Bl 13 17 B2 44
18 03 42 18 B2 4
19 Bl 3 20 Bl 17
19 B2 6 20 B2 94
19 03 112 21 B1 7
20 03 102 21 B2 84
23 03 2 21 03 28
24 03 72 22 Bl 7
25 Bl 13 22 B2 24
26 B1 33 22 03 8
26 B2 46 23 B1 7
26 03 12 23 B2 4
27 B1 43 24 Bl 7
27 B2 56 24 B2 54
27 03 22 25 B2 14
28 Bl 33 25 03 18
28 B2 56 29 B2 24
28 03 42 30 B2 34
29 B1 23 33 03 18
29 03 12
30 Bl 53
30 03 22
31 B1 43
31 B2 6
31 03 32
32 B1 53
32 B2 16
32 03 32
33 Bl 43
33 B2 126
34 Bl 43
34 B2 116
34 03 2
35 Bl 43
35 B2 126
35 03 2
36 Bl 43
36 B2 16
36 03 2
sum 2,497 2,611
Total Production Target 72,072 Kton
Total Deviation 5,108 Kton
Deviation 7.09 %
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Colorado School of Mines
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5.2.3. Production Scheduling by Minimizing Production Deviation Between
Production Time Periods
The production deviation between time periods can be minimized as well as the
deviations from the monthly production targets in order to get a smooth schedule. The
optimal solution is obtained in 40 seconds for a three-year time horizon. Figures 26, 27
and 28 show the complete three-year schedule obtained when using the objective function
which minimizes the production deviation between time periods.
The ratio of the total tons of deviation to the total tons demanded is 9.25 %. The
same procedure is followed to produce this number as explained in Section 5.2.1 and
5.2.2. Table 13 presents the output for du^ and dd^ for this scenario. Although the
production deviation increases from the planned production compared with the previous
strategies, this strategy produces a smoother schedule than the other strategies. In Table
14, the first column gives the time periods (month), the second column gives production
targets (Kton), the third, fourth and fifth columns give the total monthly scheduled
tonnages for three scenarios. The summation of the absolute differences between
production in each time periods for each scenario is 2,510 Kton for original case, 2,600
Kton for weighted case and 2,100 in this case.
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Machine Placement Level
201 202 203 204 205 206 207 208 209 210 211 212 Total
nin
765 42-43 45 i 690
765 44- 46 45
2 |M |1 M I I j— j— j— f f |M Im
1680
\m
79214- is 16 1 | p j n i n | n i n Ikii n r a m f ! ■ ïêâô"
792 19-21 19 2 t a ll I I I | | | | | I I I 130
792 33- 34 33 2 1 1 □ □ a 1 540
792 35-37 37 , : ■ « B BB 420
792 39- 42 40 ^ ho ID I n I n in in I n I n I n m in t i 1760
792 43- 46 45 2 750
□
818 16-17 16 2 □ □ 650
820 16- 17 16 2 □ □ 150
EZI EZI EZ] EZ3 |EZ] BZD EZZl |BZ )□
820 18- 19 19 1 1160
M IBZ] EZ] BZD EZ] HZ EZ] EZZ
820 23- 26 25 1 1120
820 27- 28 28 1 11 8 1 11 n Ie h ezi ezi ezi ïz i n 1250
IE EB
820 29- 30 28 2 260
820 31- 32 33 1 m ezi ezi n n n n E 1 *1 HI E l I I 1650
m H I EZI PZJ
820 33- 34 33 2 E IZJ IZJ 1600
820 35- 37 37
820 38- 41 40
849 27- 28 28 '■ 1= \rm jra pa toi ça I 930
849 29- 30 28 n re n i e t m ■ ■ ■ ■ vm 1510
849 31- 34 33 1; 820
849 35- 38 37 S 40
Development Ore |Bfl BE B E EM BE BE {BE BE B E BE BE MM i 1560
Development Waste □ □ □ [□ [□ [□ [□ [□ I ! : I sse
Total Kton/Month J2093 2083 2043 1933 1893 1893 1993 2043 1903 1823 1743 1903| 23346
Total Kton/Day I 67.5 74.4 65.9 64.4 61.1 63.1 64.3 65.9 63.4 58.8 58.1 61.4
Bl Kton/Day F 11.9 11.4 11.0 11.7 10.6 11.3 11.6 11.9 11.2 10.6 10.0 9.7
B2 Kton/Day 30.1 32.3 27.8 29.1 27.2 30.1 30.4 30.4 28.3 24.6 24.4 27.2
D3 Kton/Day J 25.5 30.7 27.1 23.7 23.2 21.7 22.3 23.5 24.0 23.5 23.7 24.5
Month
201 202 203 204 205 206 207 208 209 210 211 212
Figure 26. Illustration of a first year production schedule for the Kiruna Mine with
minimizing the production deviation between time periods
1 1 1 B S
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Machine Placement Level
301 302 303 304 305 306 307 308 309 310 311 312 Total
560 1 0- 13 12 i 240
765 44- 46 45 2 140
792 14- 15 16 j m m 340
a jca lea [a lea [a la
792 39- 42 40 1 970
792 43- 46 45 2 LJi* 1920
f (■ jM 1C |C1 J i fJ U pi
820 14- 15 16 i 980
820 16- 17 16 2 |c lo |c5 l a |o l o |o l a l a 1560
820 33- 34 33 2 a l a a a a a a i ! I 950
820 35- 37 37 j 1200
820 38- 41 40 i WJÊ I n I n [d |n h n ! n I n ^ 3 |gzi lu 1690
820 42- 46 45 1 — ■ 680
849 23- 26 1 c a 'a a c |c j Ç ] l e 1290
849 27- 28 3 1640
28 1 c= m i a i a c a a a a |m = i # = i K O !
849 29- 30 28 2 m m l u l a i u l a U l a jjm |m J1460
849 31- 34 33 1 la la a a e i 1 1 1930
849 35- 38 37 1 1930
849 39- 41 40 1 30
878 27- 30 760
Development Ore eri ■■ — ■ ■ ■■ 1560
□ □ □
Development Waste 1—1 1—1 rn r 996
Total Kton/Month 1843 1733 1833 1823 1893 1933 1983 1933 1823 1853 j 1823 1793 22266
Total Kton/Day 59.5 61.9 59.1 60.8 ei.i 64.4 64.0 62.4 60.8 59.8 eo.a 57.8
Bl Kton/Day 8.7 9.6 9.7 9.5 9.0 9.7 9.0 8.7 9.0 8.7 9.3 8.4 I
B2 Kton/Day 25.6 28.3 28.2 29.9 29.5 31.1 30.1 26.9 27.1 26.2 27.1 25.6
D3 Kton/Day 25.2 23.9 21.3 21.3 22.6 23.7 24.8 26.8 24.7 24.8 24.3 23.9
Month 301 302 303 304 305 306 307 308 309 310 311 312 I
Figure 27. Illustration of a second year production schedule for the Kiruna Mine with
minimizing the production deviation between time periods
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Machine Placement Leve 401 402 403 404 405 406 407 408 409 410 411 412 Total
560 10- 13 12 1280
■ I i i p » i
792 43- 46 45 1390
820 14- 15 16 1 1 e i [O jD jin in in w n po pzi azi pa 1500
□ r n |m im
820 16- 17 16 520
820 38- 41 40 ■ .jp " # i[ 1880
820 42- 46
849 18- 22
849 23- 26
849 31- 34
i ■! w- ■! pi papcjpgK iL —f ■
849 35- 38
en en en pen pen pen
849 39-41 40 i Q m m m EZI pen 1510
849 42- 46 45 i 180
878 23- 26 25 30
878 27- 30 28 2160
878 31- 34 33 40
Development Ore «■ !■■ « mm 'mm eni lee mm mm ma »■ 1560
Development Waste j___996
Total Kton/Month 1863 1943 1943 1953[1843 1853 1883 1903 [l753 1763 1763 1653 22116
Total Kton/Day 60.1 67.0 62.7 65.1 59.5 61.8 60.7 61.4 58.4 56.9 58.8 53.3
Bl Kton/Day 9.4 10.7 9.7 9.7 9.4 10.0 9.7 10.3 10.3 9.7 10.0 9.0
B2 Kton/Day 27.8 32.9 31.4 32.8 29.5 29.8 29.8 29.8 30.4 29.5 29.4 24.3
D3 Kton/Day 22.9 23.4 21.6 22.7 20.6 22.0 21.3 21.3 17.7 17.7 19.3 20.0 I
Month 401 402 403 404 405 406 407 408 409 410 411 412 j
Figure 28. Illustration of a third year production schedule for the Kiruna Mine with
minimizing the production deviation between time periods
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Table 13. Calculation of the percent deviation by using du and dd values for
minimization of the production deviation between time periods
Time Period du Ore Type du Time Period dd Ore Type dd
Month Bl, B2, D3 Kton Month B1, B2, D3 Kton
1 B1 73 1 B2 44
1 03 62 2 B2 74
2 B1 23 3 B2 114
2 03 132 4 B2 104
3 B1 43 4 03 18
3 03 112 5 B2 134
4 B1 53 5 03 8
5 B1 33 6 B2 74
6 Bl 43 6 03 78
7 Bl 63 7 82 34
8 B1 73 7 03 38
8 03 2 8 B2 34
9 B1 38 9 B2 129
10 B1 33 9 03 8
10 03 2 10 B2 214
11 B1 3 11 B2 244
12 B1 3 11 03 18
12 03 32 12 B2 134
13 03 52 13 B1 27
15 B1 3 13 B2 184
19 03 42 14 Bl 27
20 03 102 14 B2 184
21 03 12 14 03 58
22 03 42 15 B2 104
23 03 2 15 03 68
24 03 12 16 B1 12
26 B1 13 16 B2 79
27 B1 3 16 03 88
28 B2 6 17 B1 17
30 B1 3 17 B2 64
31 B1 3 17 03 28
32 B1 23 18 B1 7
33 B1 13 18 B2 44
34 B1 3 18 03 18
35 B1 3 19 B1 17
19 B2 44
20 B1 27
20 B2 144
21 B1 27
21 B2 164
22 B1 27
22 B2 164
23 Bl 17
23 B2 164
24 B1 37
24 B2 184
25 B1 7
25 B2 114
25 03 18
26 B2 24
26 03 48
27 B2 4
27 03 58
28 B1 7
28 03 48
29 B1 7
29 B2 64
29 03 88
30 B2 84
30 03 68
31 82 54
31 03 68
32 82 54
32 03 68
33 82 64
33 03 198
34 82 64
34 03 178
35 B2 94
35 03 148
36 Bl 17
36 82 224
36 03 108
sum 1,160 5,504
Total production target 72,072 Kton
Total Deviation 6,664 Kton
Deviation 9.25 %
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5.3. COMPARISON BETWEEN THE PROPOSED MODEL AND THE
CURRENT SCHEDULING METHOD
At the Kiruna Mine, long term mine planning is done manually with the help of
the computer. The manual mine scheduling process is described in detail in Section 2.4.
A great deal of experience and time is needed to produce production schedules manually.
Furthermore, a final schedule may easily contain some infeasibilities, especially in the
“out-years”.
The ratio of the tons of iron ore mined constituting a deviation from planned
production to the total tons of iron ore mined is 10-20% for the manually generated
schedules. However, it is difficult to accurately compare solution quality of the manual
and automatically generated schedules because despite the fact that deviations are higher
for the manually-generated schedules, mine sequencing constraints for these schedules
are also sometimes violated in order to generate a usable schedule. Therefore, the
deviations in the manually-generated schedules serve only as a lower bound on the actual
deviations, rendering the manually-generated schedules even less desirable that they
would appear.
Using the mixed integer programming model developed for Kiruna Mine,
optimal schedules are produced in less than 100 seconds and without violating any
operational constraints. The ratio of the tons of iron ore mined constituting a deviation
from planned production to the total tons of iron ore mined for the scenario evaluated in
this thesis is about 6.32%. Table 15 presents a comparison results for the manual and
automatically-generated schedules.
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CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1. CONCLUSIONS
The objective of this thesis is to develop a multi time period long-term production
scheduling model for underground sublevel caving operations. The developed
optimization model is examined for LKAB’s Kiruna Mine. Because of the large number
of integer variables in the model, it is impossible for the original MIP model to be solved
using current computer technology. Three different strategies are developed in order to
reduce the number of integer variables of the original MIP model. These are pre
processing the data by determining the monthly production amounts for three ore types
for each production area and development of early start and late start algorithms. By
utilizing all these strategies, the solution time for a three-year production schedule with
monthly fidelity is reduced to less than 100 seconds.
As mentioned in Chapter 2.2, the mined ore is sent to four different mills
according to its quality. Bl ore is transported to BF plant, a fines production plant located
in Kiruna. B2 ore is transported to the SVP pelletizing plant located in Svappavaara or to
the KK2 pelletizing plant located in Kiruna. D3 ore is transported to the KK3 pelletizing
plant located in Kiruna. The mine has to produce 297 Kton of ore as Bl, 977 kton of ore
as B2 and 728 Kton of ore as D3 per month in order to run these mills at full capacity. By
minimizing the over and under production tonnage from the target for each ore type and
each time period, the model generates schedules that maximize profit for the company.
For example, if the mine produces lower than production targets, the mill will not be able
to operate at frill capacity. As a result of not being able to meet the current demand, the
company will loose money. Also, if the mine produces higher than current production
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104
target, the mill will not be able to process all the ore. Because the mine can not stockpile
no more than 6000 tons, the mine has to reduce the current production. As a result of this
decision, the machine utilization will go down dramatically and the production cost for
per ton of ore will go up.
As discussed in Chapter 5.3, the proposed schedule produces 6 % deviation from
the production target compared with 10-20 % current computer assisted manual
scheduling procedures. Also, while the proposed schedule satisfies all the operational
constraints, the current mine schedule violates several constraints depending on the "how
lucky is the scheduler". Furthermore, in order to produce 3-year production schedule with
monthly fidelity requires 3 to 5 days with the current mine schedule compared to less
than a 100 seconds with the scheduling procedure developed in this thesis. The proposed
scheduling method allows the management to examine different production scheduling
strategies a very short time.
This research has made the following major contributions:
1. This thesis presents a way to successfully to solve multi-period production
scheduling problems for a large-scale underground mine.
2. MIP model running time depends exponentially on the number of integer
variables in the model. The number of integer variables is reduced
dramatically by preprocessing the production data and carefully formulating
the model. For the thirty six time periods, corresponding production blocks for
each machine placement for the Kiruna Mine data set would have required 36
(time periods) * 1,173 (blocks) = 42,228 integer variables in the model. The
proposed model has 36 (time periods) * 56 (machine placements) = 2,016
integers.
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105
3. It is possible to further reduce the number of integer variables by assigning an
earliest and a latest possible start date for each machine placement. Unique
algorithms have been developed to determine the earliest and latest possible
start time periods of the machine placements without violating optimality.
4. The MIP model formulation includes constraints that limit the number of
active production areas within each shaft group at a given time without
introducing any new binary integer variables.
5. The Kiruna Mine has adopted the optimization model developed in this thesis
as part of its production scheduling system.
6.2. RECOMENDATIONS FOR FUTURE RESEARCH
In this research, MIP is applied successfully to solve multi-period production
scheduling for a large scale underground mine. Based on the conclusion of the research in
the dissertation, the following recommendations can be made for further research:
Mixed integer programming is applied to the sublevel caving mining method in
this research. Although, every mining method has its own structure, production
scheduling of the mine requires similar concepts. The possibility of applying the
techniques and ideas that are developed in this thesis to different mines and mining
methods should be investigated.
The resulting model creates a long-term monthly production schedule for the
Kiruna Mine. Integration of this long-term strategic scheduling model with a short-term
(daily-weekly) scheduling model can be developed.
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109
Martin, R. K., 1999, Large Scale Linear and Integer Optimization: Massachusetts:
Kluwer Academic Publishers Corporation.
Mutmansky, J. M., 1980, “Computing Operations Research Techniques for Production
Scheduling”, Computer for 80's, pp. 615-625.
Newman, A., 2002, “Advanced Computational Optimization Notes”, Colorado School of
Mines, Division of Economics and Business, Golden, Colorado.
Newman, A., Topai, E , Kuchta, M , 2002, “An Efficient Optimization Model for Long-
Term Production Planning at LKAB's Kiruna Mine”, working paper.
Smith, M. L , 1998, “Optimizing Short-term Production Schedules in Surface Mining:
Integrating Mine Modeling Software with AMPL/CPLEX”, International Journal of
Surface Mining, pp. 149-155.
Tang, X., Xiong, G, and Li, X., 1993, “An Integrated Approach to Underground Gold
Mine Planning and Scheduling Optimization”, 24th International APCOM Symposium
Proceedings, Montreal, Quebec, Canada, pp. 148-154.
Thys, B. J., 1969, “Optimum Open Pit Mine Production Scheduling”, International
Symposium on Computer Application and Operations Research in The Mineral Industry,
Salt Lake City, Utah.
Topai, E , 1998, “Long and Short Term Production Scheduling of the Kiruna Iron Ore
Mine, Kiruna, Sweden”, Master of Science Thesis, Colorado School of Mines, Golden,
Colorado.
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ABSTRACT
Common challenges associated with grade uncertainty involve failing to meet decisive
operational targets, which include (among others) the following: ore tonnage sent to the mill, total
metal processed at the mill, blending requirements on ore feed, total waste tonnage mined,
maximum allowable proportion of potentially deleterious materials (e.g., toxic elements such as
arsenic). These challenges reflect, to an important extent, the uncertainty involved in defining
precisely the mineral grades in an ore deposit.
This has motivated a vast body of research directed at improving understanding stochastic
mine planning techniques, with an aim of incorporating its tools to mine production scheduling.
One popular paradigm for stochastic mine planning consists of formulating fully stochastic linear
programming (SLP) models which adopt sets of realizations of the orebody to represent
uncertainty regarding grades (Dimitrakopoulos et al., 2014). Since constraints must be met with
total certainty, solutions from these formulations provide a decision maker with an absolute
aversion to risk, i.e., one who (invariably) favors the most certain of two possible outcomes,
regardless of their corresponding payoffs. Such production schedules may be too conservative in
satisfying the production targets, while simultaneously producing sub-optimal results in those
circumstances in which some flexibility in meeting targets exists. In a second paradigm, mine
planners overcome the shortcomings of traditional production scheduling by incorporating
geologic and grade uncertainty through geostatistical conditional simulations. However, this
means that it is conceivable that one could also potentially benefit from any favorable development
regarding previously “uncertain” domains of the ore deposit.
The work undertaken in this dissertation focuses on generating production schedules that
take into account grade uncertainty, as described by geostatistically simulated realizations of the
ore deposit, and provide optimized production schedules that also consider the desired degree of
risk in meeting the production planning outcomes. To do this, the production scheduling problem
is formulated as a large-scale linear program (LP) that considers grade uncertainty as characterized
by a resource block model. The large-scale LP problem is solved using an iterative decomposition
algorithm whose subproblems are multi-time-period sequencing problems. At each iteration, one
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ACKNOWLEDGMENTS
I would like to express my deepest gratitude to the many people, inside and outside the
CSM campus who have helped make this dissertation possible. First my advisor, Dr. Kadri
Dagdelen. I’ve always marveled at his unique ability to creatively think outside the box and, within
few minutes, untie apparently “impossible Gordian knots.” But most of all I’m indebted to him for
the infinite personal support and encouragement throughout this (at times improbable) Journey.
This I will surely carry with me for the rest of my life. I wish to thank the members of my jury
who generously accepted to serve on my thesis committee. I thank Dr. Priscilla Nelson, for her
great leadership in the Mining Department (and also for her maternal instincts...!). I would
obviously like to thank Dr. Alexandra Newman for her immense generosity, her selflessness and
for - throughout the years we have shared at Mines - always believing, even when I myself had
doubts. I wish to thank, Dr. Thys Johnson, the smartest man alive, for the incredible help he has
given me, and most of all, for his friendship and the brotherhood we have formed in this research
group. Dr. Ozbay has been the best Rock Mechanics professor I’ve ever had. I thank him for that,
and mostly, for having always kept the door of his office open to me. Literally, but also
figuratively. Dr. Rennie Kaunda arrived heaven-sent at a complicated time when the jury had to
be recomposed. I wish to thank him for being so supportive and helping make this day possible. I
also thank Dr. Amanda Hering (who left Mines) for her help and for having accepted my invitation.
I wish to thank the enormous generosity of Sociedade Mineira de Catoca (SMC) in funding
the greater part of my stay at Mines (including the time I have spent pursuing my MSc. Degree). I
am deeply indebted to my colleagues at SMC and, in particular, to its former General Manager Dr.
José M. A. Ganga Jr., for his amazing stewardship of SMC and the personal investment placed in
me.
Dr. Marcelo Godoy, this thesis would not be possible without your contribution. I wish to
acknowledge the funding from Newmont Company which was instrumental in prosecuting the
work described herein, but at least as important, were the wonderful suggestions and technical
advice you gave along this journey. Thanks a lot Marcelo. I certainly owe a great debt of gratitude
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to professors Marcos Goycoolea, Daniel Espinoza and Eduardo Moreno for the many exchanges
and advice throughout the latter year of the research.
I have been fortunate to be surrounded by very smart people at Mines. At the risk of unfairly
leaving many colleagues out, I wish to mention by name Jung Mi Lee and Ruyxiang Gu, from an
earlier stage, and Ismail Traore, Matias Palma and Saquib Saki from a later stage. I wish to thank
my friend and colleague Canberk Aras, I brilliant young researcher, and a brother from many
anxious sleepless nights at mines who I’ve been fortunate to have met.
Finally, I thank my brothers (and their spouses): N’vula and Edgar, José and Carlota, Telmo
and Lay, for being so incredibly supportive and for having always sacrificed so much for me. I
owe them so much; my debt can never be paid. My gratitude extends to my nephews: Maura, Yana,
Muary, Muana, Jacinto, Nico and Kiary for the light-heartedness and inspiration. Your love
represents the one immaterial dimension of life which gives everything its meaning. I thank my
wife Euridice, for being a constant champion from day one; for the endless source of steadfast
support, and for the plans and dreams I have asked she forfeit so that my own could come to
fruition. Mostly, I thank her for her love.
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CHAPTER 1.
INTRODUCTION
This thesis examines the problem of determining an optimal open pit mine production
schedule which allows the quantification of grade uncertainty, i.e., a precise determination of the
production volumes and respective sequencing in time, that is optimal with respect to the objective
of maximizing the net present value NPV of the potential cash flows to be generated over the life-
of-mine of the project for a given risk tolerance. This is an important and difficult problem, with a
multidisciplinary nature that might call for the simultaneous consideration of aspects from the
economic, statistical and optimization sciences. The economic dimension of production scheduling
directly reflects consideration of the time-value of money effects, i.e., the relative preference on
the part of management to realize income, sooner rather than later, so as to capitalize on potential
investment opportunities presently available.
However, there exist limits to how soon profits can be realized, that is, since most mining
operations have limited resources available, the large majority of economically viable mineral
deposits cannot realistically be mined “instantaneously.” Instead, mining likely spans over a set of
time periods, and this characteristic makes considering the time-value of money an important
deliberation. By defining when cash flows (from the sales of the metal commodity) can be realized,
the specific production schedule adopted is crucial in determining a mining project’s .
Furthermore, any profit-maximizing production schedule must necessarily be feasible, and the
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meaning of feasibility is not dissociable from the manner in which the problem is modeled.
However, independently of the particular modeling choice, it is widely accepted that the open pit
mine production scheduling problem is, in its essence, a large-scale linear
mathematical constrained optimization problem (Johnson, 1968).
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Aiming at taking advantage of the tools from optimization theory, the problem is most
commonly tackled by initially discretizing the ore deposit into individual three-dimensional blocks
whose technical and economic characteristics are assumed known. These might include
quantitative variables such as the block’s total ore content, the block’s ore recovery at the
processing plant, or qualitative variables including (among others) a block’s lithological group or
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rock type. Subdividing the mineral deposit into discrete blocks is important, because it allows the
to be expressed formally in terms of either binary or continuous decision variables
which determine (in the case of discrete variables) when, if ever, the block should be mined and,
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if mined, the destination to which the block should be sent. Alternatively, in the case of continuous
decision variables, it is possible to mine fractions of a given block, that is, a given block might not
be extracted fully. Since the total number of blocks in the deposit may be large (possibly reaching
tens of millions for realistic orebodies), the total number of variables - which increases
geometrically with the number of time periods - can potentially be very large as well.
One additional complicating aspect of is the fact that all mining problems are
subject, in different forms and degrees, to operational requirements that limit the set of feasible
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decision paths to be considered. Within the formalism of Operations Research, the problem
requirements are expressed in terms of linear combinations of the problem’s decision variables,
and modeled in the form of inequalities in the set of problem constraints. The combination of a
large number of decision variables and the possible existence of a large variety of problem
requirements - although typically in smaller number - confers a combinatorial nature to
optimization problems which makes them tendentiously hard to solve, even in deterministic terms.
Furthermore, there exists growing recognition of the fact that a true optimal solution to
cannot be found under a deterministic framework. In effect, it has become increasingly
noted that, in various dimensions, including: confidence in estimated grades, confidence in the
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defined geologic boundaries of the orebody, confidence in the evolution of ore prices or confidence
in the performance and availability of mine equipment, among others, uncertainty pervades
mining, and realistic modeling of must take into account the possible economic impact
of the risks associated with it. Because of uncertainty it is possible one might solve a decision
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problem incorporating input parameters which are, in effect, quite different from the true
(unknown) problem parameters, that is, one might generate an “optimal” solution to the wrong
problem.
This dissertation addresses only grade uncertainty and, although we refer to and geological
uncertainty interchangeably, it is intended that both terms allude to the degree of confidence in
some estimated grade. The field of geostatistics offers the best quantitative resource modeling tools
for incorporating grade uncertainty into mine production schedules (Isaaks & Srivastava, 1989;
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Godoy, 2004), as well as the formal theoretical framework supporting the vast majority of grade
estimation techniques (Matheron, 1968; Journel, 1990; Delfiner & Chilés, 1997). One such method
- that of geostatistical conditional simulations (GCS) - has emerged as one of the most powerful
tools for adequate integration of grade risk into mine production plans. GCS allow for the
generation of sets of equally likely representations (realizations) of the deposit, which honor the
hard data at those locations of the ore deposit which have been sampled, and are conditioned on
both the sampled and newly simulated grade values. GCS are desirable because they reproduce
fairly the local grade variability, its spatial continuity, and by replacing a single grade estimate
with a set of possible grades, they present a viable framework for probabilistic analysis in mine
planning.
It is quite common to model geological (grade) uncertainty by assigning mineral resources
to three distinct resource classification categories: Inferred, Indicated, and Measured (in increasing
order of confidence), which reflect objective (as well as subjective) considerations as to the degree
of risk involved in achieving estimated grades. Indeed, given a particular level of risk-tolerance, a
decision maker will strive to define an appropriate “mix” of the proportions of material falling into
each of the resource categories, making up the composition of the mill feed for any given time
period. An example composition might be one in which upper bounds on the proportion of Inferred
material require it to be below 30%, and lower bounds on the proportions of Indicated and
Measured material, to be at least 20% and 50% of the total mill feed ore, respectively. The natural
approach to solving such problems is by directly formulating them as stochastic linear
programming models, in which decision variables are indexed to each of the individual,
alternative scenarios. However, this is not truly a viable approach in the context of mine production
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scheduling, because inherently difficult-to-solve problems are made intractable by the addition of
scenario-dependent variables.
In this dissertation, a solution methodology is presented whereby different categories of
ore resource requirements (i.e., proportions of inferred, indicated and measured material) are
enforced in the form of “ore risk constraints. A risk-quantified open pit mine production scheduling
problem is formulated which includes ore-risk constraints, together with other common
capacity or blending and sequencing constraints. Next, an iterative large-scale decomposition
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algorithm, tailored to solve to proven optimality the , is used to solve an LP relaxation
of said problem.
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The solution generated is confronted with a set of available geostatistical simulations of
the orebody in order to verify that, to a decision maker’s level of risk tolerance, it can be deemed
acceptable. If the present solution meets the risk requirements, it is considered optimal; however,
in the event that it does not, and assuming our concern focuses only on satisfying risk requirements,
then three distinct courses of action are possible: (a) adjust the current required proportions of each
of the resource categories (Inferred, Indicated or Measured) to new equally acceptable levels and
resolve the problem; (b) keep the current risk requirement levels (required material proportions)
and adopt the current “risky” schedule; or (c) determine whether additional drilling is justified
given the tradeoff between the expected benefit from reduced uncertainty versus the costs of
additional data gathering.
According to this framework, a mine production schedule is considered optimal not simply
because it corresponds to the solution of an optimization problem, but only after is has been shaped
by a decision maker’s risk tolerance level. In this sense, our problem formulation reflects an
alternative view regarding the best approach to integrating grade uncertainty into production
schedules, as well as a restatement of the problem where mineral resource classification categories
are incorporated. This clearly is a very simple view of the problem. However, it is also very
flexible, transparent, and importantly, very practical.
1.1 Dissertation Contents
This thesis dissertation assumes the reader is familiar with the basic principles of open pit
mine planning and design, such as outlined in Hustrulid and Kuchta (2013). The following is a
chapter-by-chapter description of the contents in the dissertation:
Chapter 1: Briefly introduces the Open Pit Mine Production Scheduling Problem
together with a description of how, traditionally, the problem has been understood and modelled
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within a deterministic framework. We hint at some of the characteristics which make this a
challenging problem, and emphasize the need for considering uncertainty in its multiple
dimensions including the one most relevant to our research: grade uncertainty. We give indication
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on how mineral resource classification categories, which have been widely used in industry, can
be used as a useful proxy for confidence in grade estimation. Finally, we outline a solution
methodology striving to include said classification categories, together with a mine planner’s
tolerance to risk, in a viable and practical framework.
Chapter 2: Deals with some of the basic foundational concepts in mine planning and open
pit production scheduling. It briefly reviews a subset of the mine planning literature relevant to the
research, focusing on distinct solution methodologies and classifying these into the following
traditional categories: (i) exact deterministic methods and (ii) suboptimal block aggregation
methods.
Chapter 3: Provides the strict minimum background in geostatistics that is required for the
reader to comfortably follow some of the discussions in the subsequent sections. Emphasis is
placed on those concepts which have a specific earth sciences nexus and may depart slightly from
more general statistical methods. It is the author’s conviction that the following topics merit said
distinction: data declustering, usage of scatterplots as well as their relation to measures of spatial
continuity such as the correlogram, covariogram or the variogram. We discuss estimation criteria
to illuminate the specific sense in which geostatistical estimation methods are considered “best”
or “unbiased.” We introduce kriging, highlight its basic mechanics by way of a small numerical
example and point to some of its shortcomings. This chapter assumes no prior knowledge of
geostatistics and reads as a tutorial on the subject. A reader more versed in the topic of geostatistics
might skip this chapter.
Chapter 4: Provides a simple introduction to the topic of geostatistical sequential
(conditional) simulations and, similar to kriging, a small numerical example of the method
highlighting the basic steps involved is included. Conditional simulations constitute the
“workhorse” for stochastic mine planning, and are often used as a valid framework for mineral
resource classification. This leads the way to stochastic production scheduling in Chapter 5.
Chapter 5: Presents the current stochastic mine production scheduling framework,
in particular, its use of multiple geostatistical conditional simulations as input, which contrasts
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with the deterministic case, in which a single estimated model constitutes the sole support for mine
planning. Two distinct models are included as paradigmatic examples of . Finally, we give
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indication to some of the challenges with said models and - to an important extent as a consequence
of the challenges mentioned - we introduce the concept of mineral resource classification
categories.
Chapter 6: Provides a description of the proposed problem formulation and solution
methodology. The chapter is organized as an argument motivating the solution framework
adopted, covering the defining characteristics of the problem to be solved, the nature of uncertainty
in mine planning, the distinction between an exact and a heuristic starting point as a basis for mine
planning, and it concludes with a flowchart of the solution methodology. It describes how the
research presented integrates two powerful exact solution algorithms into a framework that takes
into account uncertainties as conveyed by mineral resource classification categories and risk
measured through conditional simulations.
Chapter 7: Contains a description of the general mathematical models adopted for the
methodology upon which this research is focused. It states the problem, its underlying
assumptions, and presents general and detailed mathematical model formulations of open pit mine
production scheduling problems that consider grade uncertainties associated with the resource
model.
Chapter 8: Examines the two exact solution algorithms adopted for the proposed solution
methodology, namely: the Bienstock-Zuckerberg and the PseudoFlow algorithms.
Although fairly technical, the discussions are not intended as a comprehensive treatment of either
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algorithm, and we refer the reader to the original work by Bienstock & Zuckerberg (2009), as well
as to Hochbaum (2001, 2008, and 2012) for the detailed proofs on complexity, convergence and
optimality. The related work in Munoz et al. (2016) extends that of Bienstock & Zuckerberg
(2009), shows the applicability of BZ to the broader class of Resource Constrained Project
Scheduling problems (RCPSP), and is insightful in presenting the problem in the context of
delayed column generation algorithms. Nonetheless, the discussions therein are sophisticated and
written for an operations research audience. In our presentation, a considerable effort is directed
to helping the technical practitioner understand the links between these theoretical concepts and
mining practice. The chapter includes small conceptual examples of both the BZ and HPF
algorithms. Knowledge of operations research techniques would be helpful, but not absolutely
necessary.
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Chapter 9: Presents the results from the application of the proposed methodology to a two-
dimensional and a three-dimensional synthetic case study. Throughout, the presentation is framed
in the context of comparing our results to those obtained in a scenario in which risk constraints are
not present, and discussing some of the implications of the results obtained.
Chapter 10: Concludes the dissertation by summarizing its key findings and listing
suggestions for future work.
1.2 Original Contributions
The research work in this dissertation provides the technical practitioner with a tool, in the
form of an integrated solution methodology, which allows the user to impart a specific degree of
risk tolerance into mine production planning. Differently from other approaches reported in the
literature, in the methodology developed in this dissertation, incorporation of management’s
tolerance to grade uncertainty is transparent, and this is achieved in two specific ways: (i) by
explicitly soliciting from the decision maker a threshold reflecting preferences regarding risk, and
(ii) by explicitly incorporating well-established industry metrics for uncertainty in classification
of mineral resources into the optimization models. Said industry metrics are commonly referred to
as the Inferred, Indicated and Measured resource classification categories, and compared to a direct
block-by-block consideration of scenario-based grades, resource classification categories offer the
following advantages:
(i) They can be included into the optimization models similarly to traditional blending
constraints, thereby circumventing the “dimensionality curse” prevalent in most
stochastic optimization models for the .
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(ii) They constitute, a priori, the most common tool by which geologists and
geostatisticians, responsible for ore resource modeling, communicate a degree of
confidence in estimated mineral resource grades. This enhances transparency,
increases the chances for adoption of the proposed methods in practice, and helps to
bridge the gap between production schedules ignorant of uncertainty and “uncertainty-
conscious” commodity markets.
The solution methodology developed relies importantly on our implementation of the
Bienstock-Zuckerberg (BZ) algorithm and the integration, within it, of the PseudoFlow
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algorithm (HPF), both of which constitute state-of-the-art exact solution algorithms. Indeed, a
considerable effort has been dedicated to investigating the principles underlying both
algorithms, in an effort to make these more transparent to a technical mining audience. Our
methods demonstrate, both in two dimensional and three dimensional case studies, how said
algorithms can be used to obtain mine production schedules satisfying traditional, lower and
upper bounding, mining and/or milling capacity constraints, grade blending constraints, as well
as less traditional (unpublished at the time of this writing) mineral resource risk constraints. In
addition, by allowing for the specification of user-defined levels of risk, the tool developed
empowers the user to operate what amounts to a “dial” whereby its tolerance to risk is able to
shape the production schedules. It should be noted that this is not the same as simplistic
sensitivity analysis on an individual operational or economic parameter. In effect, in the
framework developed in this research, new requirements on either the proportions of material
belonging to a given resource category, or changes in the risk tolerance threshold, imply that a
completely new production schedule need be generated and confronted against the decision
maker’s preferences.
Potentially, one of the future directions for the research reported on in this dissertation
consists of the generation of optimal pushback designs under uncertainty. This idea follows
naturally from the work developed in this dissertation, and this exact concept is demonstrated
in one of the examples presented in Section 9.2.
Our methodology also provides an incremental contribution to the problem of determining
the optimal amount of infill drilling (OID) for a mining operator. This is accomplished in two
ways: (i) by exposing those regions of the ore deposit whose (grade) uncertainty most directly
impacts mining in any given period, and (ii) by indicating an upper bound on how much
management might be willing to invest on further drilling in return for greater conversion of
mineral resources from inferred to a higher confidence class.
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CHAPTER 2.
TRADITIONAL MINE PLANNING
Broad agreement exists within mine planning practitioners as to what solving the long-term
open pit mine production scheduling problem entails. To this effect, Dagdelen (1985)
outlines the following necessary steps:
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1. Development of a block model (geological and economical).
2. Determination of the final economic pit limits.
3. Generating mineable pushbacks within the final pit limits.
4. Generating a production schedule spanning the life of the mine project.
At the completion of the initial exploration stage, the biggest challenge is to successfully
generate a reliable, geologic block model of the ore deposit. Drawing from a limited set of collected
data samples, practitioners attempt to estimate the distribution of grades throughout the deposit by
using one or more of an assortment of geostatistical interpolation techniques. Typical practice
consists of discretizing the ore deposit into three-dimensional blocks, compositing the drillhole
sample data and, finally, using some preferred interpolation technique assigning average grades to
unsampled blocks (David, 1977). At completion, every individual block in the block model is
assigned a corresponding grade value. However, due to the limited number of drillhole samples
within a given deposit, a certain degree of uncertainty exists in the estimated block grades.
Frequently, ore resource modelers choose to classify mineral resources into distinct risk categories
as a “risk mitigation strategy” against grade uncertainty. Specifically, standard practice defines
three distinct mineral resource classification categories, namely, the Inferred, Indicated and
Measured categories, which are defined as follows (CIM, 2006):
(i) An Inferred mineral resource is that part of a mineral resource for which quantity and
grade or quality are estimated on the basis of limited geological evidence and sampling.
Geological evidence is sufficient to imply, but not verify, geological and grade or
quality continuity.
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(ii) An Indicated mineral resource is that part of a mineral resource for which quantity,
grade or quality, densities, shape and physical characteristics are estimated with
sufficient confidence to allow the application of modifying factors in sufficient detail
to support mine planning and evaluation of the economic viability of the deposit.
(iii) A Measured mineral resource is that part of a mineral resource for which quantity,
grade or quality, densities, shape, and physical characteristics are estimated with
confidence sufficient to allow the application of Modifying Factors to support detailed
mine planning and final evaluation of the economic viability of the deposit.
Once the construction of a geological block model is completed, the subsequent task is to
translate geological information, such as grades, into economic block values. This requires
examining an important set of economic and operational parameters. The economic value of
individual blocks reflects the net result of calculating the revenue generated from extracting the
block’s “metal” element and subtracting the cost of mining, hauling and processing or stockpiling
it (Johnson, 1968; Lane, 1988). Block values are thus determined using a single estimate of grades
from the geologic model despite the existence of a significant degree of uncertainty surrounding
the estimates of the block grades. It is then possible to build a sophisticated economic model for
mine valuation including consideration of minimum internal rates of return (Armstrong et al.,
2007). It is also possible to assign other attributes to each block, including a material type,
equipment hours required to mine it and plant hours required to process it, among others.
2.1 The Ultimate Pit Limit Problem
In traditional mine planning, the economic block model is the basis for the determination of
ultimate pit limits, which contain all the blocks deemed economical to mine. This optimization
problem is widely known as the Ultimate Pit Limit (UPL) problem, and includes constraints
regarding the sequencing of blocks ensuring all blocks are mined only after all of its predecessor
(overlying) blocks have been mined. In addition, since the (UPL) problem assumes unlimited
resources are available, then no system constraints prevent a feasible solution from mining a given
block as early as necessary, thereby excluding the need for consideration of multiple time periods,
and allowing the (UPL) to be expressed as a single-time-period problem. In effect, it is common
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Hochbaum (2008) introduces a new algorithm for the maximum flow problem which the
author refers to as the PseudoFlow algorithm (HPF). The algorithm uses so-called pseudoflows –
a vector of flows that “floods” the network - to solve a problem that is equivalent to the maximum
flow problem and which the author names the maximum blocking cut problem. Once said problem
is solved, a solution to the original maximum flow problem can be determined efficiently.
Development of the PseudoFlow algorithm is strongly influenced by the previous work presented
in Lerchs and Grossman (1965). Accordingly, the PseudoFlow algorithm adopts some of the
concepts present in LG (e.g., strong and weak tree branches) and, importantly, it adopts a data
structure (a normalized tree) that allows it to quickly reach an optimal solution, provided the
corresponding pseudoflows are of good quality. The author proves the finite convergence of the
algorithm, as well as its optimality. The author presents a parametric PseudoFlow algorithm in
addition to a simplex variant of the PseudoFlow algorithm, and show that the complexity of said
algorithm is linear in the number of arcs and nodes of the network. HPF is widely considered the
fastest algorithm solving the maximum problem and, given its important role in the research
developed in this dissertation, it is discussed in greater detail in Section (8.2).
2.2 The Open Pit Mine Production Scheduling Problem (OPMPSP)
After obtaining the ultimate pit limits, the next step consists of designing pushbacks, i.e.,
to include haul road access contouring the limits of each nested pit, and act as a guide during the
short-term scheduling of yearly production. It often occurs that (i) more than one possible
destination exists to which the extracted material can be sent, and (ii) the destinations to which the
individual production blocks can be sent are predetermined in advance of the actual production
schedules being generated. This is achieved by determining a break-even grade that results from
equating the profit generated from sending a given production block to one destination versus
sending it to an alternative destination. Once all of the alternative destinations are compared, a list
of breakeven cutoff grades is generated which serves to determine the best possible destination for
any individual production block. Additionally, it is common to determine cutoff grades which, to
some extent at least, recognize and attempt to take into account some of the mining system’s
potential operational constraints, which can include a milling capacity bottleneck or a mine
transportation capacity bottleneck (Lane, 1988). The combination of these break-even cutoff
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The traditional approach to mine planning has been deterministic in nature. This
characterization is the result of the following assumptions being satisfied (Froyland et al., 2004):
A deterministic block model is given as input data,
Mining and processing costs, the selling price of the product and future discount rates are
perfectly known into the future,
Grade control is assumed to be perfect; i.e., once a block has been blasted its content is
precisely known and,
The infrastructure is fixed throughout the life of the mine (e.g., mining and processing
capacities).
Ideally, a comprehensive approach to the mine production scheduling problem
would allow one to simply formulate and solve the problem, independently of the determination
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of ultimate pit limits and phases or pushbacks. However, using exact optimization methods such
as mixed integer linear programming leads to formulations which are intractable (very
hard to solve) due to the combinatorial nature of such problems. To our knowledge, there are no
(cid:4666)(cid:1839)(cid:1835)(cid:1838)(cid:1842)(cid:4667)
known optimum open-pit mine production scheduling techniques presently able to tackle very
large-scale realistic problems, i.e., problems which might include over a million variables and
constraints. In discussing the steps for effective mine planning previously outlined, Dagdelen
(1992) stresses the circular nature of the problem and, thus, the resulting iterative characteristics
of any optimal solution algorithm.
As interest in the solution of the widened, a larger swath of researchers studied
the problem, and the breadth of solution algorithms constructed also increased. This is reflected in
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the diversity and depth of the technical literature on the subject. Nonetheless, the published
academic literature contains work which reviews and systematizes some of the most important
contributions in the field. Out of these, two particular references stand out: Osanloo et al., 2008
and Newman et al., 2010 which provide a thorough discussion of the application of Operations
Research techniques to a wide spectrum and classes of mine planning problems. It is common to
catalog solution algorithms for production scheduling problems (both open-pit and underground)
into one of the following categories; (a) exact methods which may or may not resort to block
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aggregation, (b) heuristic (Ramazan 2001, Cullenbine et al. 2011, Somrit 2011) and metaheuristic
methods (Lamghari et al., 2012; Silva et al., 2014).
Exact methods are mathematically proven to solve the OPMPSP to true optimality. Johnson
(Johnson, 1968) is credited with pioneering the application of exact mathematical optimization
methods to solve the production scheduling problem. The author uses Dantzig-Wolfe
decomposition principles to solve the OPMPSP by separating the multi-time period subproblem
into single-time period problems which are solved as ultimate pit limit problems. The master
problem enforces operational constraints (i.e., production and processing requirements) while the
subproblems enforce sequencing constraints. However, the optimal solutions obtained are not
necessarily integer and therefore may not be feasible in practice.
Dagdelen (1985) uses an LP formulation to model the production scheduling problem
including mining capacity, blending and processing capacity. The multi-time period OPMPSP is
notoriously difficult to solve (i.e., it belongs to the class of NP-hard problems). Dagdelen then
applies a large-scale decomposition technique - the Lagrangian relaxation method - to solve the
problem by decomposing the larger and more complex multi-time period problem into smaller
single time period problems which can be solved using optimal ultimate pit limit algorithms such
as maximum flow. The subgradient method is applied to find the Lagrangian parameters in his
formulation. He states that despite generating integer solutions, the proposed approach cannot
avoid the so-called “gap problem” and hence the solutions obtained may not always be feasible.
Akaike (1999) extends the work on Lagrangian relaxation methods by Dagdelen (1985). A
new scheme for updating the Lagrangian multipliers is proposed, and instead of directly solving
the OPMPSP as an LP, the author converts the problem into a network relaxation formulation so
that the problem can be solved expeditiously by Lerchs-Grossman (LG) or Max-Flow algorithms.
The network flow problem is solved as multi-time-period sequencing maximum flow problem. All
constraints other than sequencing are relaxed (i.e., moved to the objective function and multiplied
by a penalty). By decreasing the overall solution time the author states this increases the
applicability of the Lagrangian approach to large-scale production scheduling problems. However,
the author concedes that due to the “gap problem,” the algorithm is not able to consistently find
feasible solutions.
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Cacceta and Hill (2003), building on an idea first presented by Johnson (1968), modify the
conventional integer programming (IP) formulation in which the decision variables specify that
blocks are to be mined “at” a specific time period t, to a formulation where blocks are mined “by”
a specified time period t. Their model accounts for inventory variables and, since the model
determines whether a block is to be sent to the mill to be processed as ore, or to be sent to the waste
dump (or an alternative destination), it also is an attempt at cutoff grade optimization. It should be
noted that if the could be solved to true integer optimality, the resulting cutoff grades
would also be optimal (Johnson, 1968; Dagdelen, 1985).
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Gaupp (2008) employs three complementary strategies for expediting the solution of a
Mixed Integer Programming (MILP) formulation of the OPMPSP: (1) A variable reduction
technique (implementing earliest starts and latest starts) to reduce the number of decision variables
considered in the model; (2) Strengthening the solution procedure through the application of cuts;
and (3) Using Lagrangian relaxation techniques. The author states that the approach allows for
optimal (or near optimal) solutions to be found more quickly than by solving the original problem.
The size of the problems solved by the proposed technique - in the order of tens of thousands of
variables – can be considered relatively small when compared to some of the test instances reported
in the literature (Espinoza et al., 2012).
Bienstock & Zuckerberg (2009) present an exact Lagrangian-based algorithm to solve the
LP relaxation of the Precedence Constrained Production Scheduling Problem (RCPSP); henceforth
referred to as BZ algorithm. The authors frame the problem in terms of sets of jobs to be scheduled
observing a set of precedence rules, and processed according to a predefined set of processing
options which have limited resources. This is a problem most commonly referred to as the Mine
Production Scheduling Problem (MPSP) within the technical mining literature, and a parallel exists
between jobs and mine blocks, as well as between processing options and mine destinations. The
BZ is a decomposition algorithm which solves an LP relaxation of the RCPSP by dividing the
problem into a master problem and a subproblem which are interdependent and communicate with
one another. A subset of the constraints in the original problem - considered “complicating
constraints” because they cause the problem to become harder to solve – are dualized and penalized
(i.e., removed from the constraint set and moved to the objective function objective with an
associated penalty) thus defining a new problem in which only less complicating constraints
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remain. Said problem is defined in the space of the original decision variables and constitutes the
pricing subproblem of the BZ algorithm. Conversely, BZ’s master problem is defined on a new
space of aggregated variables generated from the solutions of the subproblem - similar to a column
generation algorithm, (see Munoz et al., 2016) - which the authors define as partitions. Iterations
of the BZ algorithm can start from either an initial set of partitions or an initial set of dual vector
values. Assuming an initial set of partitions is readily available, a master problem is solved, which
is expressed in terms of the individual partitions. Next, a vector of duals associated with said set
of hard constraints is used to update the objective function of BZ’s pricing subproblem. This is
followed by a new iteration of the subproblem which generates a new partition. Importantly, BZ
defines the sets of partitions at any iteration to be composed of orthogonal partitions and this
requires that a newly obtained solution to the subproblem be made orthogonal with the set of
preexisting partitions. The procedure of generating orthogonal partitions out of an initial set of
partitions is referred to as partitioning and constitutes a crucial step to be taken before any
individual partition is used in the master problem. The authors prove finite convergence and
optimality of the BZ algorithm while also reporting solving efficiently very large instances of
RCPSP (including millions of variables and tens of millions of constraints) in only a few seconds.
Bley, Boland, Fricke and Froyland (2010) propose a solution methodology for the
production scheduling model using cutting planes. An original integer programming (IP)
formulation is strengthened by adding inequalities (cuts) derived by combining precedence and
capacity constraints to eliminate a substantial number of decision variables from the model as a
preprocessing step preceding optimization. Testing suggests that significant reductions in the
computation time required to achieve optimal integer solutions can be realized. Unfortunately, the
instances tested were relatively small as they contained only a few hundred blocks and 5 to 10 time
periods.
Somrit (2011) develops a fast solution methodology to solve the ultimate pit limit problem,
maximizing the total dollar value for a given block model of an open pit project. The author
presents a method for open pit phase design that follows in the tradition of Lagrangian relaxation
methods using subgradient parameterization for long-term open pit mine production scheduling
(Dagdelen, 1985; Kawahata, 2006). Similar to Dagdelen and Kawhata earlier, this methodology
also encounters the so-called “gap problem.” However, the author proposes a method to obtain an
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exact solution to the problem when a gap exists. In addition, the author presents a revised
maximum flow formulation which is developed to solve each sub-problem with a new open pit
mine phase design solution methodology, based on the maximum flow problem proposed by
Johnson (1968). Subproblems are solved iteratively to determine individual phases in a
“backward” ordering relative to time, i.e., if P represents the set of all phases to be considered,
then the first phase to be determined is phase P (the last phase to be mined), followed by phase P-
1 (the penultimate phase to be mined), successively until finally the first phase is determined. For
all instances studied, results indicate that the revised formulation of the maximum flow problem
is much faster than widely used commercial software packages that employ the Lerchs-Grossman
(LG) algorithm (Lerchs & Grossman, 1965). The author also proves that optimal solutions to the
ultimate pit limit problem are always obtained using the revised maximum flow problem
formulation, while some commercial software packages may not.
Cullenbine et al. (2011) propose a Sliding Time Window Heuristic (STWH), for problems
which include lower bounds on resource constraints. The authors emphasize the significance of
including such constraints and demonstrate that their presence can dramatically affect solution
times. In each iteration k of the STWH algorithm, a sub-problem is formed by partitioning the set
of time periods in the horizon (T) into three subsets (windows): T ≡ { }, within
(cid:3038) (cid:3038) (cid:3038)
which variables are fixed to integer values ( ), or their integrality (cid:1846) is e (cid:1515)ith (cid:1846)e r(cid:3015) (cid:3021)re (cid:1515)lax (cid:1846)e(cid:3019)d(cid:3039)(cid:3051) ( ) or
(cid:3038) (cid:3038)
enforced ( ). The algorithm fixes all varia (cid:1846)b le s in the first window to feasible integer va (cid:1846)r (cid:3019)ia (cid:3039)(cid:3051)bles,
(cid:3038)
defines the current second window in which full integrality is enforced and defines the resulting
(cid:1846) (cid:3015)(cid:3021)
(current) third window in which only a relaxed version of the model is enforced. In each successive
iteration, new sub-problems are produced in which a new partition of T is constructed by sliding
all three windows to an adjacent set of time periods, and the procedure is repeated. Finally, linking
the partial, integer feasible solutions from each sub-problem creates a complete solution. Clearly,
the window represented by can be enlarged enough so as to encompass the entire set “T” and,
(cid:3038)
in this “trivial” case, the pro (cid:1846)b (cid:3015)le(cid:3021)m resolves to a pure integer programming problem. However, for
non-trivial instances, STWH’s myopic strategy may not guarantee that an optimal integer solution
is always found.
Lambert & Newman (2013) expand the previous work by Cullenbine to solve larger
instances of the open pit block sequencing problem than previously possible with the STWH. Their
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method employs three methodologies to reduce solution times: (i) eliminate variables which must
assume a value of 0 in the optimal solution; (ii) use heuristics to generate an initial integer feasible
solution for use by the branch-and-bound algorithm; and (iii) employ Lagrangian relaxation, using
information obtained while generating the initial solution to select a dualization scheme for the
resource constraints. The combination of these techniques allows for the determination of near-
optimal solutions more quickly than solving the original problem. In addition, the method is
guaranteed to always find a feasible integer solution. The authors end by demonstrating their
techniques on instances containing 25,000 blocks and 10 time periods, and 10,000 blocks and 15
time periods, which are solved to near-optimality.
Munoz et al. (2016) study the Lagrangian decomposition algorithm recently developed by
Daniel Bienstock and Mark Zuckerberg (2009) for solving the LP relaxation of a class of open pit
mine production scheduling problem . The authors show that the algorithm is, in fact,
applicable to a broader class of scheduling problems, the Resource Constrained Project Scheduling
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Problem , of which the is a special case, including multi-model variants of
the that consider batch processing jobs. The authors present the BZ in the context of a
(cid:4666)(cid:1844)(cid:1829)(cid:1842)(cid:1845)(cid:1842)(cid:4667) (cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
Delayed Column Generation Algorithm and draw parallels with the Dantzig-Wolfe algorithm
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(DW). In addition, algorithmic speedups are proposed that can result in increased overall efficiency
for the BZ algorithm. The authors run computational experiments and compare the performance
of both BZ and DW confirming that, for the problems tested, the BZ outperforms the DW
algorithm. Finally, it is shown that algorithmic tuning can improve the performance of the BZ
algorithm. Given the relevance of the discussions in Munoz et al., 2016 to the present research,
this reference will be revisited in greater detail in Section 8.1.
2.3 Production Scheduling Model Based On Block Aggregation Methods
Typically, mathematical formulations of realistic instances of the Open Pit Mine
Production Scheduling Problem require that the variables include a substantial number
of integer (binary) variables, which allow for the modeling of features such as (i) blocks being
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mined fully or not at all, and/or (ii) scheduling “activities” which can only be started once a
predecessor activity has been completely fulfilled (e.g., fully finish mining a bench before starting
one underneath it). If models include integer-valued variables alone, then they belong to the realm
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of integer programing (IP) or, for circumstances in which both linear and integer variables are
included, they are considered mixed integer linear programs . Due to the typically large
size of mining ore deposits, IP or MILP formulations for tend to be very large, leading
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some researchers to explore the possibility of developing block aggregation techniques that have
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the potential of expediting solutions. While still preserving feasibility, the majority of block
aggregation techniques are not guaranteed to achieve proven optimality. Indeed, most block
aggregation techniques tend to be both “static” in the sense that, once blocks are initially combined
into aggregated units, they remain as part of the same unit throughout the life of the mine, and
naïve in the sense that the criteria used to determine how individual blocks are to be aggregated
are ignorant (or rather indifferent) to the implications of aggregation to the optimality of the
solutions subsequently obtained.
In the context of aggregation techniques, Ramazan (2001) proposes the Fundamental Tree
algorithm and models problems with fixed cutoff grades, blending constraints, and production and
processing constraints. The model includes lower-bounding constraints on processing although not
on production. Ramazan’s aggregated “fundamental trees” are built according to three properties:
1) the value of the aggregated blocks is positive, 2) extraction of a Fundamental Tree does not
violate allowable pit slopes, and 3) a tree cannot be subdivided into smaller trees that possess
properties 1) and 2). The multi-time-period production schedules are generated by a mixed integer
linear programming (MILP) formulation using the fundamental trees. The algorithm can generate
multiple solutions and, in certain instances, may not produce the true optimal solution. In addition,
the case study presented is also relatively small – considering twelve thousand (12,000) blocks
which are then reduced to 1,600 trees.
Kawahata (2006) builds on the work pioneered by Dagdelen (1985) and extends it to
include dynamic cutoff grade optimization. Additionally, his algorithm addresses multiple
destinations, multiple pits, stockpiles and upper and lower bounds on capacity constraints. The
author defines two subproblems, such that one represents “aggressive” mine sequencing while the
other corresponds to “conservative” mine sequencing. The aggressive sequencing problem
consists of mining only the strict minimum volume of waste required to reach the ore in any given
bench, while the conservative schedule attempts to mine the totality of the waste present in a given
bench. Variable reduction is achieved by applying two Lagrangian relaxation sub-problems, one
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for the most aggressive mine sequencing case and the other for the most conservative block
sequencing. Although the Lagrangian Relaxation method used constitutes an exact technique, the
methodology presented in Kawahata is classified as an aggregation methodology because the
algorithm groups blocks on a bench-by-bench basis, seeking to adhere to predefined phases. The
algorithm is then implemented in such fashion that optimality of the final schedules generated may
not always be guaranteed.
Boland, Dumitrescu, Froyland and Gleixner (2009) develop a model which, the authors
state, incorporates cutoff grade optimization, but with no blending constraints and no lower bounds
on resource consumption. Their methodology aggregates blocks according to precedence rules to
form bins. Bins (sets of blocks) are used to schedule mine production while individual blocks
determine processing decisions at the mill. Their aggregation and disaggregation techniques solve
large problems (up to 96,821 blocks) in reasonable time (in as little as 420 seconds) since the
aggregation of sets of blocks into bins results in a much smaller number of variables. The authors
report solving instances of 96,000 and up to 25 time periods in a few hundred seconds. In general,
however, the inclusion of lower bounds on resource constraints in the problem formulation can
dramatically increase the required solution time for the OPMPSP, so it would be interesting to
confirm if the algorithms would still fare as well as it did, had the lower bounding constraints been
incorporated.
2.4 Production Scheduling Problem Based on Heuristic Methods
Heuristic methods are attractive for their ability to potentially solve large, realistic
problems. Unfortunately, these methods are not guaranteed to solve every problem instance to
optimality. To be clear, the description of heuristic solution methodologies provided includes
many for which key components rely on exact solution methods. In this sense, the only approaches
grouped together are those tailored techniques in which the presence of some specific algorithmic
feature results in a departure from a strict exact solution method. Some methodologies based on
some paradigmatic examples in the literature are presented below:
Gershon (1987) develops a mine scheduling heuristic based on a block’s positional weight,
“the sum of the ore qualities within the cone generated downward from a block within the ultimate
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pit,” to determine when a block should be mined. The positional weight of a block defines the
desirability of removing that block at a particular point in time such that higher positional weights
are more desirable. An accessible block with the highest rank is extracted and then the entire
procedure, starting from determining the positional weight of the remaining blocks in the ultimate
pit, is conducted again until all the blocks in the ultimate pit have been removed.
Elevli, Dagdelen and Salamon (1989) build an algorithm based on Lagrangian relaxation
to improve the NPV of an open pit mine. The algorithm converts production scheduling for a single
tonnage constraint into a series of ultimate pit limit problems which can be solved by the LG
algorithm. The algorithm improved the cash flows generated compared to previous manual
schedules. However, decomposing the original multi-period problem into a series of single-time
period problems (solved one at a time) may not generate the optimal production schedule
consistently.
Chicoisne et al. (2012) propose an algorithm to solve a linear relaxation of the OPMPSP
and an LP-based heuristic to obtain integer feasible solutions. The authors use their algorithm to
improve on the optimal LP solution obtained in a first stage and generate an integer feasible
solution, and discuss the capabilities of alternative heuristic methods which are able to identify a
topological ordering (sorting) of the blocks in the mine. These methods are shown to be
computationally inexpensive, and, therefore, can be employed effectively to determine an initial
feasible integer solution (IFIS). Additionally, the authors discuss a local search heuristic which is
able to improve the quality of the incumbent IFIS so that it is very close to optimality, albeit
requiring slightly more computational time. Their method is able to solve instances of up to 5
million blocks and 15 time periods. The authors discuss an extension of the algorithm which
accommodates two side constraints, although not considering lower bounding constraints.
Lamghari, Dimitrakopoulos and Ferland (2014) propose a two-phase hybrid solution
method to the mine production scheduling problem (OPMPSP). The first phase relies on solving
a series of linear programming problems to generate an initial solution. In the second phase, a
variable neighborhood descent procedure is applied to improve the solution. Upper bounds
provided by CPLEX (IBM, 2012) are used to evaluate the efficiency and optimality of the proposed
method. Computational experiments show that the method finds excellent solutions (less than
3.2% optimality gap) within a few seconds or up to a few minutes.
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CHAPTER 3.
GEOSTATISTICAL QUANTIFICATION OF GRADES IN
MINERAL RESOURCE MODELS
One of the most challenging tasks facing geologists and engineers alike consists of
determining how to address the uncertainty associated with the estimated grades in resource
models and the consequent impact such uncertainty has on mine production schedules. In general,
this challenge is met by the execution of a sequence of steps beginning with the collection of
drillhole samples aimed at providing the largest possible amount of information regarding the
orebody’s “concentration” of some mineralogical element of interest (e.g., gold, copper or silver).
However, due to the typically large dimensions of viable mineral deposits, and the monetary and
time expenditures associated with exploration work, actual sampling of the orebody can never be
exhaustive. This, in turn, means that given the relatively small number of data samples available
at the exploration stage, geologists are forced to contend with imperfect information in building
geological models of the mineral deposit. Hence, building such models necessarily entails
estimating grades at locations previously not sampled, resulting in a significant degree of
uncertainty.
3.1 Background Concepts - Statistical Data Analysis in the Earth Sciences
Given the multidisciplinary nature of the research contained in this thesis, it is convenient
to revisit and briefly describe a small set of elementary statistical concepts which are relevant to
the discussions in the ensuing chapters. Although most of the traditional statistical tools remain
applicable, their use within the realm of ore resource estimation requires significant
“customization,” as well as an adequate theoretical framework capable of providing sound formal
support. One important example of the need for customization in ore resource estimation is the
prevalence of “clustering,” i.e., a tendency for the collected grade samples to be concentrated in
specific areas of the orebody rather than uniformly distributed across its extension. This often
justifies the fact that traditional central tendency measures such as the arithmetic average produce
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very misleading results whenever a prior account of “clustering” is not taken. Likewise, the need
to remain within the confines of a sound theoretical foundation is reflected, in practice, in the use
of concepts such as “geostatistical domains” which help ensure that important conditions for the
validity of geostatistical models are verified. One such condition is the presumption of
“stationarity” which is a crucial tenet of the theory of regionalized random variables which
undergirds most of geostatistics.
Clustering and consideration for “geostatistical domains” are two significant features of
statistical data analysis specific to the earth sciences fields (although others have adopted some of
the same concepts). However, the aspect that is likely the most distinctive component of
geostatistics and which marks its sharpest boundary with traditional statistics is the nullification
of the premise on independence in the data samples. In effect, it is precisely the assumption of
dependence between (nearby) samples that geostatistical methods seek to exploit and leverage in
drawing conclusions about the distribution of grades at locations yet unsampled. This is contrary
to the traditional statistical analysis framework in which the premise of “independent, identically
distributed” random variables is a central assumption. An additional challenge in ore
resource modeling includes the treatment to give to distribution outliers, which typically play a
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disproportionate role in the economic valuation of the ore deposit.
Finally, introducing these concepts offers two important advantages, namely:
contextualizing those discussions in which it is assumed enough geostatistical background exists
and improving the clarity and readability of the document.
3.2 Data Declustering
Whenever a drilling campaign occurs, it is desirable that standards of accuracy,
representativeness, efficiency (non-redundancy of the samples collected) and statistical
significance be observed. In the practice of ore resource modeling, these goals are prosecuted
concurrently with the need to delimit, as closely and with as much detail as possible, those portions
of the deposit in which the highest grades are found. This is an inherent feature of resource
estimation, as well as a sensible approach since it is legitimate to concentrate limited exploration
resources on those areas of the orebody offering the biggest potential for high economic payoffs,
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rather than dispersing them on delineating “barren” sections. However, the emphasis on data
collection centered on high-grade domains can lead to a biased set of samples which overrepresents
the high-grade sample population, and, as a consequence, may lead to overestimation of the
deposit’s average grade. This is the reason why traditional univariate summary statistics such as
the histogram, mean, median, variance, for example, all lead to flawed outcomes if clustering is
not addressed.
A number of different techniques exist which allow the resource modeler to tackle
clustering by determining sample weights which permit the declustering of a set of drillhole
samples. One such method is the polygonal declustering method (Isaaks & Srivastava, 1989) which
assigns each sample with a weight proportional to the area or volume of interest of the respective
sample (see Figure 3.1).
Figure 3.1: Polygonal declustering. Samples and their respective area of influence.
The cell declustering technique (Journel, 1983; Deutsch, 1989) offers a popular alternative
to polygonal declustering by accounting for the sampling density inside predefined grid cells (see
Figure 3.2).
The method works as follows:
1. Divide the volume of interest into uniform grid cells
2. Determine the number of occupied cells , and the number of data in each occupied cell
(cid:1863) = (cid:883),…,(cid:1837)
; k 0 = 1,…, (cid:1837)(cid:2868)
3. Weight each data point (inversely) according to the number of data falling in the same cell:
(cid:1866)(cid:3038)(cid:2868) (cid:1837)(cid:2868)
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(3.1)
(cid:883)
(cid:1875)(cid:3036) =
(cid:1866)(cid:3038)(cid:1837)(cid:2868)
(3.2)
(cid:3015)
(cid:882)(cid:3409)∑(cid:1875)(cid:3036) (cid:3409)(cid:883) ∀(cid:1861)
This method is reliant on the speci(cid:3036)f=i(cid:2869)c choice of cell size for the declustering procedure. It
is common to experiment with a number of different cell sizes and to choose that which minimizes
the declustered weighted average.
Figure 3.2: Cell declustering method illustration. On the right, declustered weights for a specific
cell size; on the left, plot of cell size vs. declustered mean (Rossi & Deutsch, 2014)
Owing to its ease of application, cell declustering by the nearest neighbor technique is also
very popular although the specifics of its implementation are not discussed in the present thesis.
3.3 Graphical Methods - Scatterplots
Best practice in the context of reserve estimation recommends that all consequential
geostatistical analysis be preceded by detailed exploratory data analysis (EDA) studies covering
most of the univariate central tendency statistics, as well as common distribution dispersion
measures. These tools do not obviate, however, the need for a sound geological knowledge of the
deposit and its combination with the extensive use of adequate graphical or “visualization”
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statistical tools. Among others, such tools include so-called “indicator maps,” moving window
statistics, contour and symbol maps, but also, crucially, the scatterplot. These are important
bivariate statistical tools which allow for visualizing the strength of the relationship between two
random variables but, perhaps more importantly in the context of geostatistics, they facilitate the
detection of anomalies and clusters in the data. The dots on a scatterplot correspond to (x, y) pairs
in which each of the coordinates is associated with one of the variables (see Figure 3.3). On the
left of said figure, points that fall in quadrants I and III make the correlation negative, while
quadrants II and IV make it positive. In the center, uncorrelated data show strong relationship due
to outliers of estimates. On the right, strongly correlated data show low correlation due to outlier
data Also, the strength of the relationship between the variables, as measured by the correlation
coefficient, is very sensitive to aberrant data, increasing the importance of being able to visualize
the data and checking that correlation values are not unduly influenced by a small set of data values
which depart markedly from the general trend in the sample set.
Figure 3.3: Scatterplots highlighting the impact of outlier data to the strength of the correlation
coefficient. (Rossi & Deutsch, 2014).
Importantly, scatterplots also denounce the mixing of more than one statistical population
in the data set. This is usually viewed in the form of separate clusters on the graphs; and it is
important that these have a concrete physical justification, for example, combining more than one
geostatistical domain or data originating from different deposits, among others. For instance,
Figure 3.4 shows the existence of two distinct populations whose nature of association is also
different.
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Figure 3.4: Scatterplots. Bivariate data set showing two distinct populations (Caers, 2011).
3.4 Measuring Spatial Continuity – Correlograms, Covariograms and Variograms
The concept of spatial continuity is central to geostatistics. In essence, it establishes that
all else being equal, data samples that are closer in space tend to have characteristics which are
more similar than samples further apart in space. The singular exception being when there exist
directions of major and minor geological continuity. The principle of spatial continuity
incorporates the existence of strong geological continuity in most ore deposits. This results from
the specific nature of the geological processes responsible for the genesis of most ore deposits, and
which impart on them physical patterns which allow for reasonable continuity in space, and
consequently, increased predictability as well (Isaaks & Srivastava, 1989). For instance, a
sedimentary deposit will tend to display greater geologic (spatial) continuity along the direction of
its depositional layers, and much smaller continuity (greater variability) across the said layers (see
Figure 3.5).
Figure 3.5: Euclidean distance vs. statistical distance. (Caers, 2011).
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This implies that in the context of geostatistical data analysis the Euclidean distance differs
from the “statistical distance,” and, in fact, given the prevailing direction of major continuity,
samples aligned along this direction such as samples 1 and 2 in Figure 3.5 might be more closely
related than samples aligned in its transversal direction (e.g., samples 1 and 3), despite the latter
being further apart physically than the former.
Perhaps the most relevant application of scatterplots in geostatistics is in the form of so-
called “h-scatterplots.” Considering that “u” stands for a location in space, an h-scatterplot gives
a measure of the association between two samples which are “ ” (distance units) apart: and
, respectively.
ℎ (cid:1877)(cid:4666)(cid:1873)(cid:4667)
(cid:1877)(cid:4666)(cid:1873)+ℎ(cid:4667)
Figure 3.6: H-Scatterplots. On the left, pairing of the data points based on separation (lag) distance
“h” and direction Ɵ. On the right the corresponding h-scatterplot. (Caers, 2011)
H-scatterplots are generated from pairing all the data points which are separated by some
lag distance “h” along a certain direction. If, for a given predetermined direction and separation
distance, the correlation coefficient ( ) associated to the pairs of data is calculated, then, since
the association between points decreases as they become further apart, so too do the respective
(cid:2025)(cid:4666)ℎ(cid:4667)
values of . For any direction of choice Ɵ it is thus possible to plot the value of vs lag
distance. Such a graphical display has the advantage of communicating the strength of the
(cid:2025)(cid:4666)ℎ(cid:4667) (cid:2025)(cid:4666)ℎ(cid:4667)
association between the variables across space and is known as the correlogram or autocorrelation
plot (see Figure 3.7).
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3.5 Estimation Criteria
For clarity of exposition, it is convenient to discuss in advance of describing the strengths
and weaknesses of kriging, how to decide on the performance of a given estimation method. In
what follows, we borrow from the discussion found in Isaaks & Srivastava (1989). It is natural to
expect an estimation method to be considered adequate if the estimates it produces ( ) are
generally very close to the true grade values ( ) at the locations studied. Also, in the event that the
(cid:1874)̂
predictions generated deviate from true grade values, it is desirable that they be: (i) as small as
(cid:1874)
possible and (ii) on average, the number of overestimates be approximately equal to the number
of underestimates (unbiasedness condition). Often, the deviations between true and estimated
grades at a given location are referred to as “residuals” in the geostatistical literature and are
defined as:
(3.8)
(cid:1870) = (cid:1874)̂−(cid:1874)
Hence, positive values for “ ” represent overestimation while negative values for “ ”
represent underestimation. To aid in the assessment of the efficacy of estimation methods
(cid:1870) (cid:1870)
traditional univariate and bivariate statistical tools are mobilized, and typically include histograms
of the calculated “residual” values and scatterplots of estimated vs true grades. The distribution of
residuals should be centered about a mean value of “ ” so that no clear estimations bias can
be evidenced.
(cid:1865) = (cid:882)
Figure 3.12: Hypothetical distributions of residuals. On the left, the mean of the distribution of
residuals is negative which indicates an underestimation bias. On the right, a positive mean
indicates an overestimation bias and, in the center, no indication of bias exists.
Analysis of the mean of the distribution might be complemented by the inspection of other
measures of central tendency, such as the median and the mode. That is because when a mean
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value equals 0, it can easily be the result of a very large number of small underestimates being
balanced by a small number of very large overestimates, or vice-versa. This results in skewed
distributions of residuals similar to those shown in Figure 3.13 (left). It is desirable, however, that
the shape of the distribution be approximately symmetric, signifying that not only the proportion
of underestimates and overestimates balance out, but also that the magnitude of said deviations is
also fairly similar. Such an ideal is closer to being met when the mean, median and mode of the
distribution all coincide at 0.
Figure 3.13: Two different distributions of residuals. On the left, the distribution is (positively)
skewed indicating the magnitude of overestimates is significantly larger than that of the
underestimates. On the right, an almost-symmetric distribution indicates similar magnitudes of
underestimates and overestimates.
An equally desirable feature of the residuals distribution is that it have the smallest possible
spread. This gives a measure of how large is the range of values across which the deviations span.
Evidently, the variability of the residuals can be adequately quantified by the variance (or standard
deviation) statistic.
Figure 3.14: Spread of two different distributions of residuals. The variance of the distribution is
greater for the one on the left than it is for one on the right.
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In addition to analyzing the center and spread of the distribution in its entirety, it is
important to also investigate the performance of a given estimator when only specific segments of
the full spectrum of estimates are considered. In most ore deposits, for example, it is common that
the variance of the population of high-grade samples be much higher than the variance of low-
grade samples (“proportional effect”), and this might reasonably be expected to influence the
quality of the estimates made. By subdividing the set of estimates into two subgroups: high-grade
and low-grade ore, it is possible to check for the existence of any bias in each of the individual
subgroups and, if such a bias exists, it is said that the estimator is “conditionally biased” (the
“condition” being the cutoff threshold chosen).
Conditional bias can be checked by generating scatterplots of estimated grades versus
estimation errors. In Figure 3.15, examples of such scatterplots are shown illustrating the
differences in the estimation performance for two different sets of estimates:
Figure 3.15: Conditional bias. On the left, a clear tendency for overestimation of high grades and
underestimation of lower exists. On the right, the propensity for underestimation or overestimation
is essentially the same, regardless of grade range under consideration.
In the case of the scatterplot on the right, it is possible to choose any range of grades for
inspection and the estimation remains unbiased, i.e., the proportion of underestimates and
overestimates approximately compensate each other, meaning that the estimation processes is
conditionally unbiased. Also, if the full spectrum of possible grades is inspected as a single unit,
the same conclusion is reached, meaning that the estimation processes is also globally unbiased.
By the same argument, it can be seen that in the case of the scatterplot on the left, the estimation
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process is conditionally biased despite being globally unbiased (see Figure 3.15, left). Finally,
scatterplots of true vs estimated grades can be informative of the quality of the estimation by
showing the extent to which the cloud of plotted points cluster close to the diagonal 45-degree line
passing through the origin (see Figure 3.16).
Figure 3.16: Scatterplots of estimated vs. true values, can provide a good sense of the quality of
the estimation process.
The diagonal 45-degree line represents the ideal of perfect estimation (true grades =
estimated grades) which is unattainable in practice. Nonetheless, the closer the cloud of points is
to it, the better the quality of the estimates. Figure 3.17 concludes the discussion regarding the
basic criteria underlying judgments on the quality of estimation, by conveying the sense in which
the concepts of accuracy and precision are understood in the context of mineral ore reserve
estimation.
Figure 3.17: Accuracy and precision. Left is precise but inaccurate; center is accurate but
imprecise, and right is both precise and accurate.
Aiming at generating probability distributions of residuals centered about zero is a proxy for
accuracy in the same way as aiming at minimizing their spread is a proxy for precision.
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3.6.1 Small Numerical Example of Estimation by Kriging
In what follows, we draw from the discussions in Hoffman (2014) and provide a succinct
description of the (ordinary) kriging estimation which likely helps clarify some of the concepts
explored in the prior sections. The algorithm starts with an initial set of measurements of the value
of some property of interest (e.g., gold concentration or reservoir porosity) at three locations;
and the estimation goal is to determine the value of the same property at unsampled location
(cid:1878)(cid:3036)
(represented as a question mark in Figure 3.18).
(cid:1878)(cid:2868) (cid:1876)(cid:2868)
? ? = ?
Figure 3.18: Initial set of three sampled locations and the desired estimation location x
0
(represented by the blue square with an interrogation mark).
STEP 1
Calculate the distances between the individual samples themselves and, the distance
between each sample and the estimation location . Next, store sai (cid:1856)d(cid:3036) (cid:3037)values in matrices
and , respectively. (cid:1856)(cid:3036)(cid:2868)
(cid:1830)(cid:3036)(cid:3037) (cid:1830)(cid:3036)(cid:2868)
STEP 2
Determine a spatial continuity function: correlogram, covariogram or variogram function,
relating Euclidean separation distances to the degree of (statistical) spatial continuity in the
deposit. Recall that kriging takes into account “geological distance” which means that
samples which are geologically closer (similar) get larger weights. Likewise, redundant or
geologically distant (dissimilar) data get smaller weights (Figure 3.19).
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3.6.2 Limitations of the Kriging Estimator
The advent of ever more powerful computers made it possible for kriging to enjoy great
popularity in the field of geostatistical resource estimation. However, the method has well-known
limitations, one of which being its “smoothing effect.” Smoothing occurs because of three key
aspects of the kriging estimator, namely: (i) the kriging weights are all positive, (ii) the kriging
weights must add to one, and finally, (iii) a linear relationship is assumed between the value to be
estimated at a given unknown location, and the known surrounding data. The combined effect of
these features leads to estimated values at unsampled locations not reproducing the extreme values
(high or low) in the dataset, as they tend to fall somewhere close to the center of the range
(spectrum) of sampled values. Alternatively, kriging estimation can be thought of as attempting to
approximate a non-linear “grade surface” by a linear interpolation method. This results in kriging’s
inability to adequately characterize the full spatial variability of the grade distribution in a given
deposit (see Figure 3.23 below).
Linear Interpolator
e
d
a
r Grade surface
G
e
r
O
A C B Location
LEGEND
Known grade at sampled location
Estimated grade at unsampled location
Figure 3.23: Linear interpolation principle underlying kriging. Applying any convex linear
estimator to estimate the grade value at location “C” g c from known locations “A” and “B” will
produce an estimate located somewhere in the line segment connecting the grade values at “A”
and “B,” g a and g b respectively. By construction, g c will not be lower than g a or higher than g b.
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One important reason why Kriging’s inability to capture the deposit’s variability is
consequential is the fact that it could lead to significant misclassification of the material into ore
and waste categories at the mine planning stage. In the context of mine planning it is common to
define one (or more) operating cutoff grades, below which the total metal contained in the material
to be mined is too small to justify the cost of its extraction. The decision on how to classify rock
material (e.g., ore or waste), and thus the decision on whether or not to extract and process rock
material is contingent on how estimated (rather than true) block grades compare to such
predetermined cutoff grades. Rock material with an estimated grade above a specific cutoff grade
is classified as ore and potentially mined and sent to the mill plant for processing whereas material
below the cutoff grade is (ideally) not mined or, if mined, then sent to a waste dump destination.
Within the realm of geostatistics and ore resource estimation, the challenges posed by
incomplete information and its subsequent impact on kriging estimates are commonly represented
in the form of schematic “material classification diagrams” such as the one presented in Figure
3.24.
True grades True grade = Estimated grade
(cid:1859)
Ore estimated
to be waste
Ore blocks
300
Waste (cid:1859)
blocks
Waste estimated to be ore
300 Estimated
grades
Figure 3.24: Material classification diagram showing possible misclassification due to selection
based on estimated, rather than true, values. Misclassification occurs in the upper left and lower
right quadrants. All other quadrants represent locations correctly classified, despite the
classification not being perfect.
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Graphically, the figure reflects a hypothetical scenario in which for every spatial location
under study, both true and estimated grades are available. Each point in the graph corresponds to
a physical location in the deposit which has as its coordinates (x, y) = (estimated grade, true grade).
Also, an example cutoff grade (300 in the present case) is defined, together with the 45-degree line
intersecting the origin which represents the unattainable case of perfect estimation (estimated
grade = true grade).
Misclassification is said to occur in two distinct circumstances: in the first case, the
predicted grade at location is smaller than the mining threshold (cutoff grade) while the
actual true grade at said location is greater than the threshold. Such occurrences are depicted in the
(cid:1873)(cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)
upper left quadrant of the graph and configure what the statistical literature calls “false positive”
errors. The second case of misclassification is also formally referred to as “false negative” error
and corresponds to occurrences in which, for a given location, the estimated grade is greater than
the threshold while the actual true grade is not. Likewise, it is common to construct “loss-
functions” that assign penalties to the deviations between estimated and true grades; and to
illustrate schematically the influence of imperfect selection on the economic performance of
operating mines in the form of graphs such as presented in Figure 3.25 below (Isaaks & Srivastava,
1989; Rossi & Deutsch, 2014):
Figure 3.25: Scatterplot of material misclassification showing hypothetical true vs estimated SMU
values (Rossi & Deutsch, 2014).
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From a mining perspective, portions of the deposit associated with the occurrence of false
positive errors are mined when they should not have been; areas corresponding to false negative
errors will be left unmined when they should, in fact, have been.
The challenges with misclassification resulting from the smoothing effects of kriging are
compounded by a different, though related, aspect of geostatistical ore resource estimation: the
“support effect.” When making technical and operational decisions or, when reporting ore
resources and reserves, mine planners need to consider the scale (or “support”) in terms of volume
or tonnage at which the grade estimates are made. For instance, operational mining decisions
cannot be based on grade estimates made at the scale of composited drillhole samples because one
cannot realistically mine at the degree of selectivity represented by such “point estimates.” It is
thus necessary to define a selective mining unit (SMU), which corresponds to some compromise
between factors such as: the expected bench height, the equipment dimensions or the expected ore
dilution, and which better corresponds to the true scale at which mining is expected to occur. Since
the volumes associated with SMUs are typically much larger than those of drillhole samples, it is
natural that a significant degree of “averaging” of extremely low (as well as extremely high) grades
take place, leading to a distribution of SMU grades which underrepresents these categories of the
original (point-scale) drillhole distribution (see Figure 3.26).
Figure 3.26: Grade distribution of SMUs vs. the grade distribution of composites. Shaded areas
represent the proportion of rock material in the extremities (both top and low) of the original
composites distribution which is absent from the SMU distribution, (as it “migrates” towards its
center).
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If the level at which a cutoff grade is placed corresponds to a very high or very low quantile
in the grade probability distribution function (pdf), then the smoothing characteristics of the
kriging estimator can be particularly impactful, possibly leading to a significant underestimation
of both the proportion of (very) high-grade and (very) low-grade material present in the deposit
Correspondingly, it can be theoretically shown that a probability distribution of kriging
estimates (e.g., SMU blocks) has mean equal to that of the original sampled data (e.g., drillhole
composites), but significantly smaller variability in grades, as measured by the SMU distribution
variance (David, 1977; Isaaks & Srivastava, 1989; Goovaerts, 1997). Figure 3.27 shows the
general expected differences between original grade pdfs versus estimated grades pdfs. The
average grade of the distribution does not change in both cases. However, the spread evidenced in
the original composite grade pdf is reduced when compared to the SMU grade distribution, as the
tails of the distribution of original grades move to the center (towards the mean).
Figure 3.27: Comparison of pdfs for original drillhole sample data (composited) versus
estimated (kriged) SMU block data (adapted from Barnes, 2014). Grade units are purposely
omitted, for generality of interpretation. Both grade distributions have identical means,
approximately 201 “grade units;” however, the composited data has significantly larger
variability: 164 vs. 115 for SMUs.
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Hence, for circumstances in which preserving the sample data variability is crucial,
traditional estimation methods such as kriging (or other convex linear estimator as IDS) do not
suffice because, as a result of the smoothing effect, the probability distribution of the set of
estimated grades displays a smaller variance than the probability distribution of the original
sampled grades. In effect, often times a small fraction of the deposit containing very high grades
is critical to its valuation and will tend to be underestimated as a result of smoothing. In the case
of gold deposits, for example, it is not uncommon that only 10% of the total deposit tonnage
represent 40% (or more) of its total metal content (Parker, 2015). This is one important aspect why
a “smoothed” image of the ore deposit, such as the one provided by kriging, might not be fully
adequate for mine planning purposes. Additionally, for many operational aspects of mining, there
is interest in recognizing the extent to which, uncertainty in the estimated geological models
translates into uncertainty in meeting predefined operational or economic targets. These may
include ore tons sent to the mill, yearly cash flows to be realized, average grade in the mill feed
among others.
Because missing any of the aforementioned operational targets can result in significant
financial consequences for mining operators, the ability to quantify uncertainty is of paramount
importance. This is the context in which the mineral resource classification methodology is
discussed in the subsequent Section.
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CHAPTER 4.
GEOSTATISTICAL QUANTIFICATION OF GRADE
UNCERTAINTY THROUGH MINERAL RESOURCE
CLASSIFICATION
For most mining companies, economic and market valuation is reliant, to an important
extent, on the total mineral resources it is able to report as assets. Furthermore, in the case of
publicly traded companies, most mining jurisdictions require that any event reasonably considered
to have a “material” impact on the company’s valuation be disclosed to the public.
The exist a variety of reporting codes originated at some of the world’s leading mining
jurisdictions, namely, the SME code (US), CIM (Canada), SAIM (South Africa) and JORC
(Australia). Said codes define Inferred, Indicated and Measured categories for mineral resource
classification, which characterize the degree of uncertainty associated with the estimated grade of
a block in a given ore deposit.
The following are the internationally accepted definitions for mineral resource together
with the distinct mineral resource classification categories:
Mineral Resource
A mineral resource is a concentration or occurrence of solid material of economic interest
in or on the Earth’s crust in such form, grade, or quality and quantity that there are reasonable
prospects for eventual economic extraction. The location, quantity, grade or quality, continuity
and other geological characteristics of a mineral resource are known, estimated or interpreted from
specific geological evidence and knowledge, including sampling.
Measured Mineral Resource
A Measured mineral resource is that part of a mineral resource for which quantity, grade
or quality, densities, shape, and physical characteristics are estimated with confidence sufficient
to allow the application of Modifying Factors to support detailed mine planning and final
evaluation of the economic viability of the deposit. Geological evidence is derived from detailed
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and reliable exploration, sampling and testing and is sufficient to confirm geologically and grade
or quality continuity between points of observation. A Measured mineral resource has a higher
level of confidence than that applying to either an Indicated mineral resource or an Inferred mineral
resource. It may be converted to a Proved Mineral Reserve or to a Probable Mineral Reserve.
Indicated Mineral Resource
An Indicated mineral resource is that part of a mineral resource for which quantity, grade
or quality, densities, shape and physical characteristics are estimated with sufficient confidence to
allow the application of modifying factors in sufficient detail to support mine planning and
evaluation of the economic viability of the deposit. Geological evidence is derived from adequately
detailed and reliable exploration, sampling and testing and is sufficient to assume geological and
grade or quality continuity between points of observation. An Indicated mineral resource has a
lower level of confidence than that applying to a Measured mineral resource and may only be
converted to a Probable Mineral Reserve.
Inferred Mineral Resource
An Inferred mineral resource is that part of a mineral resource for which quantity and grade
or quality are estimated on the basis of limited geological evidence and sampling. Geological
evidence is sufficient to imply but not verify geological and grade or quality continuity. An
Inferred Resource has a lower level of confidence than that applying to an Indicated mineral
resource and must not be converted to a Mineral Reserve. It is reasonably expected that the
majority of Inferred mineral resources could be upgraded to Indicated mineral resources with
continued exploration.
Depending on a set of modifying factors and on the judgment of the “Qualified Person,”
Indicated and Measured mineral resources can be converted to Probable and Proved Reserves.
Modifying factors involve changes in technical dimensions such as mining, metallurgical, and
economic parameters, or societal dimensions such as the legal (regulatory) environment, the
requirements for compliance with environmental standards, social perception and governmental
policy. It is noted that Inferred resources are not directly convertible into reserves.
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In practice, there exists no formal definition for the precise meaning of qualitative
statements such as: “sufficient level of confidence to allow mine planning and the evaluation of
economic viability…” as stated in the definition of Indicated resources. Nonetheless, industry has
converged on a common set of procedures and metrics which have become accepted best practice.
Some important mining consultants (e.g., AMEC) recommend that appropriate resource
classification analysis focus on three dominant aspects: (i) spatial continuity, (ii) sensitivity of
economic parameters to changes in grade and (iii) establishing a target for relative accuracy in
grade estimates (Parker et al., 2014).
Assessments regarding the spatial continuity of the mineralization entail the following steps:
1. Looking through cross-sectional and plan views of the orebody to get a clearer idea of the
degree of connectivity in the ore mineralization across the orebody.
2. Re-evaluating classification (possibly to downgrade) where structural geological features
markedly increase local uncertainty. For instance, these features might include geological
faults responsible for corridors of Inferred material, or dikes, containing barren or
deleterious alteration material (such as talc in iron deposits).
3. Avoiding spottiness, that is, relatively small patches resources belonging to one category
to be “mixed” inside a larger pool of resources belonging to a different category.
Figure 4.1: Spottiness in mineral resource classification. On the left, a spotted image of the ore
deposit is smoothed so as to look like the image on the right. (Stephenson et al., 2006).
Assessments regarding the sensitivity of economic parameters to changes in estimated
average grade are obviously important because, for most operations, a small error can have a large
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impact on cash flows and NPV, particularly if there exists a bias that persists through the life-of-
mine.
For illustrative purposes, one real case example of a sensitivity table is extracted from a
larger study on a copper deposit by Parker et al. (2014) and presented below:
Table 4.1: Sensitivity analysis of economic parameters to deviations in grade estimates
Additionally, practitioners doing ore resource modeling frequently target a specific level of
estimation accuracy referred to as the “relative accuracy” metric and calculated as:
(4.1)
(cid:4666)(cid:1874)̂−(cid:1874)(cid:4667)
(cid:1870)(cid:3028) =
(cid:1874)
where , represents the grade estimate and represents the true (unknown) grade. The relative
accuracy metric is accompanied by the definition of a set of complementary factors which are
(cid:1874)̂ (cid:1874)
important to consider, namely, the time period or production increment and the confidence interval
(CI) for the average grade estimate.
Different approaches to constructing CI’s for mineral resource classification have become
popularized, including by using the estimation variance or by the implementing of simulation-
based techniques which consist of constructing a bootstrap confidence interval for the estimated
grade (Boisvert & Silva, 2014). For methods based on estimation variance, it is of importance to
examine what factors affect it most directly. It can be shown formally (Matheron, 1968, Journel &
Huijbregts, 1978; Isaaks & Srivastava, 1989) that for any estimator expressed as a convex
weighted linear combination of some random variable (sample grades), the estimation variance is
expressed as:
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This motivates the development of new methods replacing kriging variance with bootstrap
confidence intervals constructed from incorporating sequential conditional simulations in
quantifying risk. Within the conditional simulation framework, the single grade of an individual
block is replaced by a set of possible equally-likely grades, each of which associated to an
individual simulation of the ore deposit. Hence, instead of a single estimated value, each block is
associated to a histogram representing a probability distribution which is expected to fully capture
the grade variability at the block’s location. It is important to note that, unlike kriging, conditional
simulation values are dependent on the actual local grade values (i.e., conditional simulations
honor the hard data and are conditioned on both hard data and simulated values) which is desirable
for addressing the challenges with the proportional effect. Confidence intervals are constructed
centered on the kriged estimate, by counting the number of times the target relative accuracy
threshold can be met and dividing it by the total number of simulations generated . For instance,
let be an indicator variable taking the value 1 if the estimated grade value is contained in
| |
the (cid:1835)r(cid:3049)̂elative accuracy interval defined as ; 0 otherwise. Then i (cid:4666)s
(cid:1874)
̂a
(cid:4667)
90% CI for the
mean , if and only if, the following co n(cid:3045)d it =ion (cid:1874)̂ i ±s m (cid:1870)(cid:3028)et: (cid:3045)
(cid:4666)(cid:1874)̂(cid:4667)
(4.8)
(cid:883)
(cid:1829)(cid:1835)(cid:2877)(cid:2868) = (cid:3045) (cid:1861)(cid:1858)(cid:1858): ∑(cid:1835)(cid:3049)̂ (cid:3410) (cid:882).(cid:891)(cid:882)
As a summary, the general quantitative| gu|id elines for mineral resource classification most
widely adopted by the mining industry are presented below (Parker & Dohm, 2014):
Inferred Resources: mineral resources for which insufficient geological information
exists to establish confidence levels.
Indicated Resources: mineral resources for which a 90% confidence interval
corresponding to a ± 15% relative accuracy over an annual production increment can be defined.
This implies that the true (unknown) average grade will be within 85 and 115% of the estimate
90% of the time. Annual production increments are typically used for Pre-feasibility and
Feasibility cash flows. Typically, operating mines can withstand deviations from plan
corresponding to one year in 20 as being below 85% of the estimate – normal business risk. Actual
realized grades consistently below the 85% threshold, often result in the mine incurring in a loss.
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Measured Resources: mineral resources for which a 90% confidence interval
corresponding to a ± 15% relative accuracy over a quarterly or monthly production increment can
be defined. Similar to Indicated resources, this implies that the true average grade is expected to
be within 85 and 115% of the estimate 90% of the time. Quarterly or monthly production
increments are typically used for determining Operating Budget cash flows.
4.1 Geostatistical Conditional Simulations
Geostatistical conditional simulation techniques provide a viable framework for the
quantification of grade variability in mineral deposits. Using this approach instead of a single
estimated model, a set of equally likely realizations of the orebody are generated. These
simulations are such that each single realization honors the hard data at previously sampled
locations, reproduce first-order statistics of the dataset (such as data histograms) and second-order
statistics including the bivariate spatial distribution of the data (e.g., by reproducing data
variograms) (Journel, 1989).
Figure 4.3: Samples collected from the orebody (black dots) are used in exploratory data analysis
to build histograms and experimental variograms of the data.
Both the original data variograms and histograms should be honored by the subsequent
conditional simulation realizations (see Figure 4.3 and Figure 4.4).
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In essence, exploratory data analysis pertains to traditional first order statistics, including
central tendency statistics such as the (declustered) mean, median mode and the construction of
histograms of the grade distribution. Spatial data analysis is represented by the computation of
variograms, which are used at a later stage (together with data histograms), as an element for cross-
validation of the generated simulations. In the next section, a small numerical example of SGS is
presented to illustrate the core steps of said methods.
4.2 Numerical Example of Sequential Gaussian Conditional Simulation
Similar to ordinary kriging, a succinct description of Sequential Conditional (Gaussian)
Simulation is presented to help enlighten some of the key concepts of SGS. Recall that the
simulation paradigm is most useful in contexts in which a measure of local uncertainty (other than
the kriging variance) is important. The steps of SGS given in Figure 4.5 are illustrated through the
example shown in Figure 4.6 to Figure 4.10.
STEP 1
Transfer all data to Standard Gaussian (normal) space by previously ranking the data and
converting it to standard normal scores (ns) [-3, 3]
≈
Figure 4.6: Mapping the sample data from the original data into the Gaussian space.
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conditional simulation is thought of as representing an alternative, equally probable, representation
of the mineral deposit.
After passing through a transfer function (such as an optimization model solved using
CPLEX), the ensemble of conditionally simulated realizations yields a distribution of possible
values for key operational indicators. Said performance metrics might include, among others,
yearly discounted cash flows, total ore tonnes mined and average grade of the material sent to the
milling plant(s). This is in contrast to the traditional view which ignores uncertainty and assumes
a single, perfectly accurate, although usually wrong, estimate of the metric of interest.
Figure 5.2 complements Figure 5.1, and illustrates part of the process of building “risk
profiles” for independent production schedules generated for each of the individual conditional
simulation realizations.
(cid:1845)(cid:1861)(cid:1865)(cid:1873)(cid:1864)(cid:1853)(cid:1872)(cid:1861)(cid:1867)(cid:1866) (cid:1845)(cid:1855)ℎ(cid:1857)(cid:1856)(cid:1873)(cid:1864)(cid:1857)
Figure 5.2: Risk-oriented view of open pit optimization (adapted from Leite et al., 2007).
Godoy et al. (2003) provide an extensive discussion on the role of risk profiles in capturing
geological uncertainty and their applicability in incorporating uncertainty into strategic mine
production scheduling. In their discussion, the authors cite a variety of case studies which serve as
examples. Some of the most effective uses of geostatistical conditional simulations (GCS) include,
but are not limited to (Godoy & Dimitrakopoulos, 2007):
Quantification of uncertainty and risk analysis of an ultimate pit limit, including net value,
costs, tonnage, grade and metal,
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Identification of areas of upside potential and downside risk in terms of ultimate pit limits,
Trade-off analysis for pushback depletion strategies,
Quantification and assessment of uncertainty related to ore blocks driving the increment
between successive pit shells.
Dimitrakopoulos, Martinez and Ramazan (2007) developed a methodology for maximizing the
potential upside while also minimizing the downside associated with geological uncertainty using
the instances of grade realization resulting from geostatistical simulations. In their approach, the
authors first generate a set of conditional simulations of the orebody intended to capture the
deposit’s geological uncertainty.
All of the mining decisions corresponding to a specific production schedule are then
“imposed” on all other orebody realizations, and compliance with a number of pre-specified
operational performance metrics - such as average grade or total milling requirements - is verified.
These results are recorded and used to compare the expected performance of an individual
mine plan, under the existence of geological uncertainty (which is represented by the previously
generated set of orebody realizations). The performance of an individual mining plan regarding
each of the chosen operational metrics is summarized in distinct plots which are gathered, and their
ensemble is said to correspond to a so-called “risk profile” for the respective plan. From risk
profiles, upside and downside statistics are created and added to the objective function to be
optimized.
Many interesting discussions on the very latest advances in strategic mine planning can be
found in Dimitrakopoulos (2014). The original conceptual framework has been extended and
brought to a stage in which researchers use Stochastic Linear Programming methods to
formulate two-stage (and less frequently multi-stage) stochastic models of the mine production
(cid:4666)(cid:1845)(cid:1838)(cid:1842)(cid:4667)
scheduling problem. In the most recent formulations of the mine production scheduling
problem, researchers draw more explicitly from the formalism presented by authors such as Birge
(cid:1845)(cid:1838)(cid:1842)
and Louveaux (2011).
Lamghari and Dimitrakopoulos (2013a) provide a paradigmatic formulation of a stochastic
rendition of , which is representative of the popular SLP modeling approach. The
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
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authors model a problem which assumes mining decisions can be made separately from processing
decisions, and occur under grade uncertainty, that is, mining decisions take place before the true
metal content of the ore blocks can be known with certainty. In order to characterize the uncertainty
associated with the grades, a finite number of potential outcomes or scenarios are generated, such
that each block is assigned a possible grade value per scenario. The problem includes limits on the
maximum yearly mining capacity, multiple ore processing destinations with limited milling
capacity, as well as a stockpiling option allowing ore material to be stored (whenever more ore is
mined than can be processed in a given period), or reclaimed (whenever less ore is mined than
must be processed in a given period). In general, the option to stockpile ore material adds flexibility
to mine production schedules, relative to the circumstance in which some of the ore mined must
be wasted so that richer ore can be processed. However, typically the use of stockpiles is associated
with additional re-handling costs resulting from the need to load and haul the previously stored
ore. Hence, in an ideal scenario, scheduled ore blocks are sent directly to the processing plant
without visiting the stockpiles, so that said re-handling costs can be minimized. The model
presented in Lamghari (2013a) can be seen as the deterministic equivalent of a two-stage SLP
rendition of the OPMPSP, in which decision variables are divided into first-stage (scenario-
independent) variables corresponding to mining decisions, and second stage (scenario-indexed)
variables corresponding to sending or reclaiming material to and from the stockpile (recourse
decisions). First-stage and second-stage decision variables are linked by the fact that, for each
scenario considered, initial mining decisions (taken without to regard to the scenario realization)
might lead to either excessive or insufficient ore tonnage being sent to the milling plant, and
consequently, to the need for sending or reclaiming ore from the stockpile.
The objective function of the model aims at maximizing the expected life-of-mine NPV
for the project by: (i) minimizing total (ore and waste) mining costs, together with the losses from
sending material to the stockpile, and (ii) maximizing the expected revenue from ore sent directly
to the mill, together with the (delayed) gains accrued from the ore reclaimed from the stockpile.
Independently of the solution method selected for generating the actual production
schedules, the structure of the formulation model tends to lead to the deferment of mining riskier
blocks to later stages of the life of the mine, regardless of whether these blocks could have
potentially been mined sooner. Deviations from target production level at the mill plant is
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quantified by “slack” variables (in the operational constraints) which are added to the objective
function and associated with monetary costs. Said monetary costs could be interpreted as
“penalties” for the use of the stockpile.
In practice, the values assigned to the penalty terms in the objective function are adjusted
empirically so as to ensure maximum adherence to operational targets. However, by selecting a
destination for each individual block a priori the models cannot account for cutoff grade
optimization, thus, generating suboptimal schedules. Likewise, the models in both this approach
and the one discussed in Martinez et al. (2007), are representative of the broad spectrum of mining
stochastic optimization models and tend, by design, to push the riskier blocks to the latest possible
periods in the project’s life. Finally, it is noted that the schedules generated by the model in
Lamghari (2013), may only be considered optimal in “an expected-value sense.”
5.1 Shortcomings of Current Deterministic and Stochastic Production Scheduling
Models
The traditional, deterministic production scheduling models discussed earlier assume that
the estimated grade values in the geologic block model are known with certainty. As such, the
corresponding mine schedules are assumed to be achievable during the mining operations when,
in fact, due to the existence of uncertainty surrounding the estimation of each block, this is not
necessarily true. It is often very difficult to consistently meet the desired production requirements
in every singular period throughout the project’s mine life. The optimized schedules may not be
optimum if one cannot achieve them in practice.
Currently, the framework better suited to address the inevitable geologic uncertainty
underlying ore reserve estimates is that of stochastic mine planning. Geostatistical techniques, such
as conditional simulations, allow for the generation of a set of equally probable realizations of the
orebody which are used as input to a stochastic production scheduling optimizer in place of a single
estimated model (Dimitrakopoulos, 2000). The current approach to stochastic mine planning
focuses on determining schedules that minimize the risk of not meeting pre-established goals while
optimizing financial and operational targets (Whittle, 2014). However, for its many advantages
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over a traditional production scheduling optimization framework, the present stochastic paradigm
still has weaknesses. One important challenge resides in the fact that one equates variability of
grades with risk, such that, the schedules adhere very strictly to operational targets. In the current
stochastic mine production scheduling approach, risk is seen as something to be avoided at all
costs, leading researchers to design models and solvers that seek to minimize the risk of not
meeting predetermined operational targets at the hindrance of realizing potentially higher profits
(Ramazan & Dimitrakopoulos, 2007). In particular, it should be noted that the equivalence
between risk and uncertainty is faulty because risk is only that fraction of uncertainty which results
in unfavorable outcomes. Indeed, risk constitutes part, and not the totality of potential outcomes
arising from the underlying uncertainty in ore reserve estimation.
Similarly, there exists the fact that, for fully stochastic optimization models, (including
scenario-indexed variables), each of the individual geostatistical simulations represents a new set
of decision variables to be added to the model, obviously resulting in dramatic increase in problem
size and leading to attempts to solve the models using heuristic (or metaheuristic) solvers. This
latest shortcoming is the principal driver for the search of alternative approaches to incorporating
grade uncertainty.
5.2 The Nature of Uncertainty and the Appeal and Limits of Stochastic Models
An additional, related, though distinct facet of the problem studied, concerns the nature
and limits of the most widely used stochastic models, at least as reported in the most recent
literature (Dimitrakopoulos, 2014). Increasingly, these models are involved and are characterized
by a considerably high degree of mathematical sophistication. This occurs because intuitively there
exists a (justified) perception that the higher the degree of complexity in the models, the larger the
level of detail that can be captured and, therefore, the more realistic the models are. It is the author’s
high conviction, however, that in the great majority of the case-studies presented it is still difficult
to specify objectively the added insight gained by the increased complexity, or whether it actually
exists. Examples include holistic stochastic modeling of mining complexes, encompassing
multiple mines, processing streams, transportation schemes, supply contracts, as well as
accounting simultaneously for grade and price uncertainty (Zhang & Dimitrakopoulos, 2015). It is
stressed that at present, for many real case mine projects, it is still not possible to solve, to proven
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(5.5)
(cid:1871).(cid:1872). (cid:1849)(cid:1877)+(cid:1846)(cid:1876) = ℎ(cid:4666) (cid:4667)
(5.6)
(cid:1877) (cid:3410) (cid:882)
With the first-stage decision variables, the second-stage decision variables,
(cid:3041) (cid:3040)
the vector corresponds to a random vector containing the data parameters for the second-
(cid:1876) ∈ (cid:1844) (cid:1877) ∈ (cid:1844)
stage problem, that is, those parameters about which a certain degree of uncertainty exists, and
(cid:4666)(cid:1869),ℎ(cid:4667)
finally is the expectation operator over the uncertainty space defined by .
(cid:1831)
The standard formulation makes clear that the appropriate way to interpret the
meaning of an optimal solution obtained under uncertainty is to acknowledge that it is optimal
(cid:4666)(cid:1845)(cid:1838)(cid:1842)(cid:4667)
only in an expected value sense, i.e., since one can never fully eliminate uncertainty, there remains
a possibility that the solution generated from the model be different than the true (“wait-and-see”)
optimal. Indeed, the true optimal solution is never truly known until uncertainty is resolved, which
in the context of grade uncertainty in mining, means only after one mines the deposit, verifies what
is in effect “on the ground” and is in a position to confirm that the solution is truly optimal.
This crucial feature of stochastic programs is often lost on many of the discussions found
in the stochastic mine planning literature in which, on the contrary, researchers state confidently
that a “true optimal’ solution has been found. Although models are theoretically superior to
naïve deterministic models, it is important to recognize the probabilistic nature of statements about
(cid:4666)(cid:1845)(cid:1838)(cid:1842)(cid:4667)
optimality. In particular, given the complexity, the difficulty in solving fully stochastic models
using exact methods, as well as the inferior transparency, consideration should be given to
alternative, more practical modeling approaches.
The solution methodology proposed in this dissertation also provides a contribution to the
important problem of the optimal amount of infill drilling . Briefly, the problem
consists of maximizing the benefit (information) derived from a set of drillhole samples while
(cid:4666)(cid:1841)(cid:1835)(cid:1830)(cid:4667) (cid:4666)(cid:1841)(cid:1835)(cid:1830)(cid:4667)
minimizing the costs associated with drilling, i.e., such that the number of drillholes produced does
not exceed the minimum necessary to ensure that the continuity of the mineralization can be
adequately determined. However, it is evident that “adequate characterization…” is a subjective
statement that is directly related to the level of risk that management is willing to accept which, in
turn, determines how extensive and rigorous the data collection efforts are. The is
fundamentally a Value of Information problem, in which one must weigh the tradeoff between the
(cid:4666)(cid:1841)(cid:1835)(cid:1830)(cid:4667)
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potential benefits from a reduction in grade uncertainty against the costs associated with additional
infill drilling (Froyland et al., 2007).
By indicating which specific periods violate the pre-defined ore risk requirements, our
methodology exposes those areas of the deposit which do require additional infill drilling and
provides an effective tool for risk management. Moreover, it is a well-established proposition from
decision analysis theory, that information has value only to the extent to which future decisions
are affected by it (Clemen, 1997). Therefore it must be emphasized that the cannot be solved
independent of an assessment of the impact of additional information to mine scheduling decisions.
(cid:4666)(cid:1841)(cid:1835)(cid:1830)(cid:4667)
By directly linking mineral resource uncertainty to mining decisions, our method is well suited to
providing an estimate of an upper bound to the value of additional drilling, in other words, the
method helps identify, objectively, the most a decision maker might be willing to invest in
additional information gathering given his degree of risk tolerance.
In the solution methodology described in this dissertation, we show how in the context of
grade uncertainty, the “curse of dimensionality” can be circumvented by the adoption of mineral
resource classification categories, in place of approaching the problem from the perspective of a
fully stochastic optimization paradigm. Despite accounting for grade risk, our methodology
nonetheless achieves the complementary goals of practicality, transparency and flexibility.
Resource classification categories are central to our solution framework and are presented
in greater detail in the subsequent Section.
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CHAPTER 6.
NEW APPROACH TO OPEN PIT MINE PRODUCTION
SCHEDULING UNDER UNCERTAINTY
6.1 Defining the Problem
One aspect of mine planning which is of particular concern for mine managers, relates to
assessments of the degree of confidence in the estimated average grade of a panel of production,
typically corresponding to some period (or volume) of production larger than that of individual
mine blocks (see Figure 6.1). The fundamental mine planning problem confronting the mining
industry which this dissertation addresses can be stated very simply, it is that, during mine
production scheduling optimization, mine planners are often oblivious to the economic
implications or do not even consider ore resource risk classification. Indeed, some of the
consequences of said omission are rather forcefully illustrated by the events at a mining operation
in the Western US (whose name we omit for confidentiality reasons). In this project, the mine
planning team developed a mine production schedule which was determined to be “optimal” in a
deterministic sense and was thus scrupulously followed by the engineers and operators in the field.
c
c c c
c cc YEAR 1 NORTHPANEL
`
cc c c c
c
c
c c c c
c
c c cc c
YEAR 2 CENTRALPANEL PLANNED DRILLING
c c
c CURRENT DRILLING
c YEAR 3 SOUTHPANEL
c
Figure 6.1: Example plan view of the progression of a deterministic mine production scheduling
plan. Each of the individual production panels corresponds to 12 months’ worth of production.
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Despite being considered optimal, however, the plan had ignored any consideration
regarding ore resource risk classification, and the result was that by the end of business year 2015
approximately 50% of the total material mined belonged to the Inferred resource category.
Furthermore, the mining company concluded that it was missing close to 60 thousand ounces of
gold because, although some of the mined material contained metal, the bulk majority did not and
was instead barren (as is often the case with Inferred resources).
From an initial stance in which managers relied deeply on qualitative or empirical
assessments on the part of geologists, the mining industry has evolved to a position in which the
subjectivity in qualitative assessments is increasingly replaced (or complemented) by the
objectivity given by quantitative geostatistical methods. Within the realm of geostatistics, the
process of assigning categories to mineral resources as a function of the degree of confidence in
the grade estimates (and their corresponding spatial continuity) is referred to as “mineral resource
classification,” and constitutes one of the most important tools for managing grade risk.
Traditionally, three mineral resource classification categories are used: Inferred, Indicated and
Measured (as discussed in CHAPTER 4). Since each resource classification category has a
corresponding level of risk associated with it, it seems natural to generate mine production plans
in which the desired composition of the mined material is specified, a priori, so that a target mix
of the proportion of Inferred, Indicated, and Measured material is known (Figure 6.2).
YearlyRisk Composition of the Mined
Material
Production
Year
Figure 6.2: Notional depiction of desired “risk mix” in a yearly mine production scheduling plan.
Characteristically, Measured and Indicated resources are combined (grouped) and displayed
separately from Inferred resources.
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For instance, a mine manager might determine that the yearly composition of the ore
material in the mill feed, for the first five years of the mine project, must not contain more than
20% of Inferred material, 40% of Indicated ore and at least 40% of Measured material. Moreover,
it is often a stated risk management strategy that later years of the mine life accommodate a greater
fraction of Inferred material while earlier years be composed mostly of Indicated and Measured
material. In addition, Inferred resources do not qualify for reserves (cannot be reported as assets
in the financial accounting of any mining company), only Measured and Indicated resources do.
However, it is inevitable to mine through some proportion of Inferred material because it is too
expensive to convert all resources into Indicated and Measured categories.
In any event, one heuristic (although possibly plausible) mitigation strategy available to
mine planners consists of generating a geostatistical risk map where the riskiest portions of the
deposit are highlighted, so that once mine plans are generated, it is possible to visualize and avoid
particularly risky portions of the orebody. This would allow deferring (sharing) the risk associated
with the resources across the different time periods and increased predictability in meeting
production targets. Note that these are important objectives for operational and financial motives
as well. Whenever a (publicly listed) mining company reports to the market, it is typically expected
to produce an estimate of the amount of metal to be realized; for example, a company might
publicly state that it expects to produce 400 thousand ounces of gold and if such pre-established
targets are not met, it may suffer severe financial penalization, be it in the form of a decrease in its
stock price, a lowering of its creditworthiness leading to increased difficulty in financing, or others.
Therefore, one key goal of geostatistical modeling of ore resources (and production planning) is
not to “overcharge” any single period with an excessive volume of Inferred material.
The drawback of relying solely on a geostatistical map of grade uncertainty lies in the fact
that uncertainty is only influenced by factors such as the number of available samples, their relative
spatial distribution, and proximity (drilling pattern) as well as the underlying nature of the
mineralization (e.g., very erratic in the case of gold, or, very continuous in the case of coal). It does
not, a priori, have any association with the constraints shaping mining decisions to be taken in an
optimized mining plan. It is, therefore, desirable to bridge the disconnect between geostatistical
modeling on the one hand, and deterministic optimization on the other. This is achieved by means
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of explicitly including constraints enforcing the requirement for a certain pre-specified
composition of the mill feed, which accounts for mineral resource uncertainty.
The mine production scheduling model described in this thesis sheds light on and increases
the visibility of the consequences resulting from the absence of ore resource classification
parameters in current state-of-the-art stochastic (or deterministic) mine planning models. In the
same spirit, a solution methodology is developed that is simple, but capable of addressing grade
uncertainty as expressed in the form of various levels of risk in mine production plans.
6.2 Flowchart of Solution Methodology
The goal of the solution methodology presented in this dissertation is to provide an
effective and practical tool for realistic mine production scheduling, that incorporates geological
uncertainty, and also avoids the known “curse of dimensionality” associated with the traditional
integer , mixed integer linear programming , or even linear formulations of the
. The approach is distinct from others in the literature in that it explicitly incorporates
(cid:4666)(cid:1835)(cid:1842)(cid:4667) (cid:4666)(cid:1839)(cid:1835)(cid:1838)(cid:1842)(cid:4667) (cid:4666)(cid:1838)(cid:1842)(cid:4667)
traditional, industry accepted, ore resource classification categories, namely, the Inferred,
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
Indicated and Measured categories in describing uncertainties associated with block grade
estimation.
Similar to other related research (Dimitrakopoulos et al., 2000), the prior generation of a
set of geostatistical conditional simulations (GCS) is considered an important indispensable first
task. This is because GCS provide a necessary account of the grade variability at the local scale,
as well as the best current framework for the classification of ore resources into the traditional risk
classification categories (although this task can be achieved by means of traditional geostatistical
methods alone). Assuming GCS can be obtained from the ore modeling team or generated by
mining engineers, each of the individual blocks is classified into an appropriate resource
classification category. Although the specific details related to the process and techniques needed
for generating GCS can be decisive (Journel & Kyriakidis, 2004), their in-depth treatment is
considered beyond the scope of the present discussion (although a small qualitative example is
provided in Section 4.2.
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practice: by allowing for the aforementioned modifications the new plans incorporate explicitly
the decision maker’s preferences and are more likely to meet the required risk criteria.
A condensed schematic diagram of the general steps involved in the proposed methodology
is provided in Figure 6.3:
Figure 6.3: Schematic diagram of the steps of the proposed solution methodology. The dashed
lines connected to GCS indicate distinct stages of the methodology in which these might be used.
This iterative process is the basis of the proposed methodology. In brief summary, the
method proposed allows for controlling of the proportion (a user defined proportion) of material
in each of the resource classification categories, exactly as a blending constraint, then running
through the simulations and verifying if the derived mining schedule meets predefined risk criteria.
The resource risk classification criteria are a proxy for the variability associated with the grade of
the block, and when utilized according to the framework proposed herein, they are thought to be a
definite improvement compared to the alternative of inspecting the grade of every block
individually as is the case with fully stochastic solvers.
The generic stages of the proposed solution methodology, presented in condensed form in
the schematic diagram of Figure 6.3, are further expanded and presented in Figure 6.4, in the view
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of providing a detailed presentation of the individual steps in the methodology. The flowchart is
in effect comprised of two distinct blocks; one which is referred to as the “optimization block,”
whose constituent elements are depicted in black, and a second block referred to as the “validation
block,” whose constituent elements are depicted in purple. The optimization block terminates once
the entirety of the BZ algorithm is concluded and a deterministic optimal solution is obtained.
However, said solution is considered provisional until it has met all the validation criteria in the
second block, that is, until it can be verified that the plan satisfies some predefined risk threshold.
Once an optimal plan passes satisfactorily through the validation block, it is accepted as an optimal
plan and the overall solution procedure terminates.
The list of steps illustrated in the flowchart (see Figure 6.4) are described as follows:
STEP 1:
Read initial set of partitions
STEP 2: (cid:4666)(cid:1848)(cid:4667)
Solve the master problem and obtain the corresponding vector of duals
(cid:2868)
STEP 3: (cid:4666)(cid:1839)(cid:1842)(cid:4667) (cid:4666)(cid:2020) (cid:4667)
Check if the iteration counter (variable defining the current iteration number) is greater than
one. If so, proceed to step 4 otherwise proceed to step 5
(cid:1863)
STEP 4:
Check if the current dual vector is equal to the previous iteration’s dual vector . If true,
(cid:3038) (cid:3038)−(cid:2869)
terminate the algorithm (optimization block) and proceed to step 9, otherwise, proceed to step
(cid:2020) (cid:2020)
STEP 5:
Adjust the block values using dual vector extracted from the side constraints in the master
problem and, setup the pricing subproblem
STEP 6: (cid:4666)(cid:1839)(cid:1842)(cid:4667) (cid:4666)(cid:1845)(cid:1842)(cid:4667)
Solve the pricing subproblem using the PseudoFlow algorithm
(cid:4666)(cid:1845)(cid:1842)(cid:4667)
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CHAPTER 7.
RISK-QUANTIFIED OPEN-PIT MINE PRODUCTION
SCHEDULING UNDER UNCERTAINTY
.
The problem addressed in this research consists of a rendition of the Open Pit Mine
Production Scheduling Problem , which seeks to maximize NPV by determining when,
if ever, an open pit mine block should be mined, and if mined, what is the most profitable
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
destination where to send the previously mined block. The models include (per period): (i)
minimum requirement and maximum resource capacity constraints, (ii) minimum and maximum
average grade requirements for the mill feed, (iii) minimum and maximum average ore risk
requirements for the mill feed, expressed in the form of specific targets for the proportions of
Inferred, Indicated and Measured material. A deterministic risk-quantified formulation of
is presented, which excludes scenario-indexed decision variables. Furthermore,
(cid:3045)
fractional solutions are allowed, as decision variables are modeled as continuous. Note, however,
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
that said (deterministic) models include ore resource risk classification constraints and are solved
under a probabilistic framework that allows for management to incorporate its degree of risk
tolerance by specifying whether the schedules generated meet some desired (predefined) risk
threshold.
In addition, the models are general enough that multiple block destinations can be
considered, allowing the optimal block destination to be selected during the solution procedure, as
a function of the state of the overall mining system at a certain point in time. Moreover, the solution
algorithms adopted produce exact provably optimal solutions.
The models formulated incorporate the following underlying assumptions:
- Ore blocks can be mined over a continuum of time periods, rather than being mined fully
in a single time period.
- The operation holds no stockpiles.
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The first model assumption is an indispensable requirement for modeling the
as a linear program, with continuous variables. The second assumption has implications regarding
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
the importance of stockpiles and their respective modeling in mine production scheduling.
Ideally, mining operations should avoid adopting stockpiles since these necessarily involve
additional maintenance and re-handling costs that add to the overall operational cost structure.
However, practice shows that, as a result of operational and economic uncertainty, specifically,
the challenges with determining future fleet downturn and future evolution of commodity prices,
most mines do choose to hold stockpiles as a hedging strategy against risk. For example, mine
stockpiles can be used to sustain the mill plant operation, in the event that ore excavators break
down, and reallocation of transportation equipment is required. Similarly, mine stockpiles allow
for lower grade (marginal) ore material to be stored, rather than sent to the waste dump, if it is
believed that future commodity prices will rise significantly enough that currently marginal ore
might be considered profitable in the near future. Exclusion of ore material stockpiling (or
destinations other than the mill plant and the Waste Dump) in the models is that such hedging
strategies cannot be captured. However, it is reasonable to consider that the model formulation
could include stockpiles with some additional research.
Finally, although the models proposed are intended to address long-term mine production
schedules, it is anticipated that, given appropriate adjustments to time resolution and per-period
constraint parameters, the models can be applicable to short-term production planning as well.
Risk Quantified Open Pit Mine Production Scheduling Problem Formulation
We present a detailed formulation of the Risk Quantified Open Pit Mine Production
Scheduling Problem which adheres accurately to the set of problems modeled and solved in this
research. Indeed, apart from expanding the constraint sets, the sole distinction from the generic
variant of resides in the inclusion of risk constraints defining, a priori, the risk
composition of the mill plant feed. Owing to the fact that risk uncertainty constraints are present,
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
we refer to the new problem as with the “ ” superscript standing for “risk.” In our
(cid:3045)
notation we keep an additional parameter index to better differentiate the treatment of the
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667) (cid:1870)
"(cid:1870)"
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(7.16)
(cid:1876)(cid:3029)(cid:3031)(cid:3047) ∈ [(cid:882),(cid:883)] ∀(cid:1854) ∈ (cid:1828),(cid:1872) ∈ (cid:1846),(cid:1856) ∈ (cid:1830)
The objective function (7.1) seeks to maximize the discounted value of all extracted blocks
across all destinations and time periods. Constraints (7.2) enforce block precedence requirements
between a block and its predecessor blocks by ensuring that the cumulative proportion of
′
block (mined in period or earlier), across all destinations is not greater than the cumulative
(cid:1854) (cid:1854)
proportion of any of its predecessor blocks mined in time period or earlier. Constraints (7.3)
(cid:1854) (cid:1872) (cid:1856)
′
enforce an upper bound on the maximum milling capacity at the mill plant. Constraints (7.4)
(cid:1854) (cid:1872)
enforce lower bounds on the yearly minimum milling requirement at the mill plant. Note that for
implementation reasons (i.e., to conform to the format expected by the BZ algorithm) the
formulation slightly modifies lower bounding constraints so as to expresses them in the form of
upper bounding constraints.
Constraints (7.5) enforce yearly maximum total (ore plus waste) mining capacity.
Constraints (7.6) and (7.7) enforce yearly maximum and minimum limits on the proportion of
Inferred material in the mill feed. Constraints (7.8) and (7.9) enforce maximum and minimum
limits on the proportion of Indicated material in the mill feed. Similarly, constraints (7.10) and
(7.11) enforce maximum and minimum limits on the proportion of Measured material in the mill
feed. Constraints (7.12) and (7.13) enforce upper and lower bounds on the average grade in the
mill feed. Constraints (7.14) restrict the total mineable proportion of any given block across all
destinations and time periods, not to exceed the totality of the block. Constraints (7.16) ensure all
(cid:1854)
decision variables assume continuous values between zero and one.
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CHAPTER 8.
SOLUTION ALGORITHMS ADOPTED FOR RISK-QUANTIFIED
OPEN-PIT MINE PRODUCTION SCHEDULING UNDER
UNCERTAINTY
Due to the “curse of dimensionality” and the large scale of realistic instances of
the , many researchers adopt heurist or metaheuristic methods to solve their models.
Part of said techniques include: Tabu Search algorithms (Lamghari et al., 2012, 2014), Simulated
(cid:4666)(cid:1841)(cid:1842)(cid:1839)(cid:1842)(cid:1845)(cid:1842)(cid:4667)
Annealing algorithms (Godoy et al., 2002; Montiel et al., 2015), and a hybrid of Simulated
Annealing and Particle Swarm (Goodfellow et al., 2015) among others.
Although Heuristic solvers have the advantage of producing good quality solutions in
reasonable amounts of time (Kirkpatrick, 1983; Sattarvand & Niemann-Delius, 2013), they
introduce an additional layer of uncertainty, because these methods are highly sensitive on (user-
defined) initial and boundary conditions, so that one cannot be certain that the solution obtained is
truly optimal, even if all problem parameters are known with absolute certainty. This poses the
question of how much should one care about modeling and adequately characterizing uncertainty
in cases in which the deviations from optimality caused by grade uncertainty may be small
compared to the ones resulting from a heuristic modeling approach?
Figure 8.1: Comparison of the characteristics of solutions to proposed methodology vs. current
heuristic stochastic solution methods.
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Using terminology from set algebra, Bienstock and Zuckerberg refer to the columns as
(cid:3037)
“partitions” (as in set partitions), and indicate that for each such individual partition the non-zero
(cid:1874)
member elements of the set all assume the same value, i.e., . Hence, a
(cid:3038)
potentially large number of individual variables can be elimina (cid:1876)te(cid:3036)d =, b (cid:1876)e(cid:3037) i (cid:1861)n (cid:1858)g (cid:1876)r(cid:3036)e ,p (cid:1876)l(cid:3037)a ∈ced
(cid:1874)
by a single
variable ( ) representing the entire set, resulting in dramatic reductions on the size of decision
variables and originating large sets of redundant problem constraints which can also be eliminated.
(cid:2019)(cid:3036)
It is important to note, however, that this observation could already have been made with regards
to the DW algorithm and, therefore, in our view it does not in fact constitute a differentiating
factor.
The key advantage of the BZ algorithm relative to DW resides in the fashion in which
consecutive solutions to the pricing subproblems are used in the general algorithmic scheme.
Although both algorithms include and define the same exact class of subproblems, in the case of
the DW algorithm, the solutions generated at each iteration “ ” are mapped “directly” (exactly
(cid:3038)
as they are, coming from the subproblem) into the space of the restricted master problem; resulting
(cid:1874) (cid:1863)
in a newly generated data point on the space defined on “ .” It is from the convex combination of
said data points that the DW algorithm attempts at reaching optimality, but, by the same token,
(cid:2019)
there exists an inherent risk that the total number of data points to be enumerated until optimality
is reached be very large. For instance, in Figure 8.8 a notional example polyhedron is provided to
further illustrate this point. It can be seen that in a pathological case, the number of extreme point
solutions potentially visited (the set of corners along the surface) can be quite large.
Figure 8.8: Enumeration of extreme point solutions in the DW algorithm. The algorithm travels
across the extreme points of the feasibility polyhedron until an optimal solution x* maximizing the
objective function Z* is reached.
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finitude constraints are imposed by equation (8.94), and finally, constraint (8.95) ensures all
decision variables are bounded to be between 0 and 1, i.e.; .
The algorithmic steps start by defining the initial matri(cid:1876)x (cid:3036) ∈, a[(cid:882)ls,o(cid:883) ]r e∀f(cid:1861)erred to, originally, as the
“partition set” by Bienstock & Zuckerberg. The correspondence between the presentation in
(cid:1848)
Muñoz et al. (2016) and Bienstock & Zuckerberg’s (2009) original presentation, is that the
columns in the matrix are equivalent to the partition sets of the discussion in Bienstock &
Zuckerberg.
(cid:1848)
STEP 1:
Defining the initial partition set (“ ” matrix)
(cid:1848)
As alluded to in Section 8.1, it is a strict requirement for the application of the BZ algorithm
that the partition set be comprised of disjoint columns . This same condition is met in the
column generation formalism of Muñoz et al. by ensuring that all the columns comprising the
(cid:1848) (cid:1874)
matrix are orthogonal to each other. The intuitive physical interpretation of this condition in the
(cid:1848)
case of mining is that, at any given time period, none of individual partitions shares any blocks in
common.
Although any feasible orthogonal set of partitions could have been used as the starting partition
set, for simplicity, the initial set chosen is one whose individual columns consist of mining all the
blocks in the pit in a single time period. Again, since all the mine blocks contained in a single set
can be set equal to each other, this means that the initial individual decision variables for the
distinct blocks in one set can be replaced by a single variable representing the entire set which will
be referred to as “ .” Since the considered life of mine spans three time periods, then three starting
sets are adopted, , , and each representing a single partition set for periods 1, 2 and 3
(cid:2019)(cid:3036)
respectively.
(cid:2019)(cid:2869) (cid:2019)(cid:2870) (cid:2019)(cid:2871)
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These considerations are modeled in the form of block precedence constraints which
enforce allowable “mine slope” constraints. Figure 8.35 presents a schematic view of a directed
graph over which the maximum-closure is solved:
.
(A) (B)
1 1
(C)
Figure 8.35: Maximum-closure problem. (A) Initial directed graph; (B) the set of blue nodes is
identified as forming a closure; (C) Closure identified is one whose the total weight is maximum
among all possible closures, therefore it is the maximum-closure.
The goal of maximizing total economic value in the is therefore conditional
(constrained) on ensuring these physical requirements are met, but disregards any other operational
(cid:1847)(cid:1842)(cid:1838)
considerations, including limits on the total resources available, (typically mining and
transportation or processing capacities), as well as common mill-feed blending and stockpiling
requirements.
The equivalence between maximum-closure problems and the lies in the fact that, for
a given discretized orebody, a related directed graph can be constructed such that individual mine
(cid:1847)(cid:1842)(cid:1838)
blocks can be compared to nodes in the graph, precedence constraints can be compared to arcs in
the graph, and block values can be converted into node weights (Figure 8.36).
Hence, in the discussions that follow, mine blocks will be referred to as network nodes
interchangeably, depending on whether the context of the exposition relates to mine scheduling
(cid:4666)(cid:1854)(cid:4667)
problems or network flow problems, respectively.
(cid:4666)(cid:1866)(cid:4667)
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For any given node “ ” in the network there exist only two possible ways by which, flow
that has been induced into the network, is able to leave it: 1) through the arcs or 2) by routing
(cid:1866)
flow to all other outwardly pointing arcs. This implies that the answer to the network problem
(cid:1868)(cid:3041)
when the sum of the flow through arcs is minimized, is equivalent to solving a max flow
problem in which we push as much flow through the network as possible without considering the
(cid:1868)(cid:3041)
arcs. The specific modifications to the original graph transforming it into a graph consistent
with finding a maximum flow are:
(cid:1868)(cid:3041)
1. The addition of a source ( ) and sink node ( )
2. The connecting of all nodes with positive gains to the source node
(cid:1871) (cid:1872)
3. The connecting of all nodes with negative gains to the sink node and
4. The elimination of all the remaining arcs connecting to the sink node.
(cid:1868)(cid:3041)
The transformed graph of the problem depicted in Figure 8.38 is shown in Figure 8.39:
t=1 t=2 t=T
C
1T
1, 1 1, 2 1, T
C 11 - C12 U 211 C 12 - C13
C
2T
2, 1 2, 2 2, T
C 22 - C23
C 21 - C22
C C
n-1,T nT
C n2 - Cn3 n-1, 1 n-1, 2 n-1, T
U n,n-1 C n-1,2 - Cn-1,3
C n1 - Cn2
n, 1 n, 2 n, T
LEGEND Flow incident into node (1, 1)
1, 1
1, 3 Flow incident out of node (1, 3)
Figure 8.39: Multi-time-period maximum closure problem.
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4. Push the entire excess of the respective strong branch along the unique path from the root
of the strong branch to the root of the weak branch.
5. Split any arc encountered along the path described in 3) which does not have sufficient
residual capacity to accommodate the amount pushed. The tail node of that arc becomes a
root of a new strong branch with excess equal to the difference between the amount pushed
and the residual capacity.
6. The process of pushing excess and splitting is called normalization. The residual capacity
of the split arc is pushed further until it either reaches another arc to split or the deficit arc
adjacent to the root of the weak branch.
8.3.3 Numerical Example of the PseudoFlow Algorithm
Consider the problem described in figure below:
Decision Variables Cost Coefficients
1 2 3 4 -5 -1 -1 -1
5 6 4 4
Legend
C: Economic value of mining block b
Cb b
Figure 8.49: Single time period maximum closure problem.
We refer to the steps of the HPF outlined in the previous Section.
First iteration
For the first iteration source (s) and sink (t) adjacent arcs are saturated and a feasible pre-
assignment of flow is decided upon. Four units of flow are induced into the network by the arc
connecting nodes (s) and (6); all 4 units of flow reach node (4), only one unit of flow reaches the
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sink node (t) from the node (4) and the remainder 3 units of flow are re-routed from node (4) to
the source via the excess arc (shown in blue in Figure 8.50). Similar to the arc connecting (s) to
(6), the arc from (s) to (5) induces four units of flow into the network, two units of flow originating
from (5) are distributed through the arcs connecting node (5) to nodes (2) and (4) by assigning one
unit of flow per each of the arcs two units are assigned to node (1) and, from there reach the sink
node.
Since the demand in arc connecting node (1) to node “t” is greater than the flow on the arc
(i.e., 5>2) then, a flow deficit results and flow must be re-routed from “t” to node (1). The resulting
graph and tree structure are shown in Figure 8.50.
ROOT NODE
3
WEAK BRANCH
(cid:883)
2
5
5
4
1
1
(cid:887) 1 (cid:884) 1
4
1
4 1
1
4
(cid:885)
1
(cid:888)
4
(cid:886)
ROOT NODE (r)
s
STRONG BRANCH
3
Figure 8.50: An initial normalized tree for the HPF algorithm. The set of arcs A(s) and A(t) are all
saturated. An initial arbitrary flow assignment is chosen in which the strong and the weak group
of nodes are circled and the respective root nodes are highlighted as well.
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CHAPTER 9.
APPLICATION OF THE PROPOSED METHODOLOGY
9.1 Two-Dimensional Synthetic Case Studies
The model formulations for the two-dimensional synthetic case studies (as well as for the
three-dimensional examples) are coded using the C++ programming language and solved using
CPLEX 12.6.0.1 (IBM ILOG CPLEX Optimization Studio, 2015), preserving all of the default
settings. The cases are run on a MacBook Pro machine with one 2.5 GHz quad-core Intel Core I7
processor and 16GB of 1600MHz DDR3L onboard memory. For demonstration purposes, the
results from the application of the proposed methodology to a synthetic two-dimensional
conceptual model of an iron ore mineral deposit are presented and discussed. It is assumed that all
blocks have equal tonnage and that only two destinations are available: an ore processing plant
and a waste dump. Mine production will occur in three time periods, and time value of money
effects are modeled by adopting a 10% discount rate for the economic block values. In order to
highlight the importance of incorporating geological uncertainty, two alternative scenarios are
tested, one representing a traditional deterministic approach and the other risk-based approach.
Specifically, in the case of scenario #1, ore risk constraints are disregarded while in the case of
scenario #2 such constraints are enforced. The problem statement also includes distinct constraints
for total ore milling capacity, as well as lower and upper bounds on the average grade of material
entering the processing plant which are the same in both scenarios.
Figure 9.1 depicts the conceptual grade block model (% Fe) underlying the economic block
model of Figure 9.2.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 W W 67 66 W W 64 69 68 68 W W W W
2 W 62 W W 40 60 63 63 62 66 66 W
1 64 65 68 68 68 69 67 66 65 65
Figure 9.1: Two-dimensional conceptual cross section of the grade model corresponding to the
iron ore deposit used in the computations (% Fe). (The letter “W” standing for waste blocks).
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The economic block model corresponding to the ore deposit to be used as input to the
proposed solution methodology is shown in Figure 9.2.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 -4 -4 9.3 3.2 -0.1 -4 5 14.1 10.8 11.2 -4 -1.7 -4 -4
2 -4.2 2.9 -1.7 -4.3 4.1 4.2 1.7 1.9 4.9 2.1 1.2 -4.1
1 0.8 1 2.5 9.6 12.9 13.3 9.2 4.3 8.3 8
Figure 9.2: Two-dimensional conceptual cross section of an economic block model depicting 1st
period undiscounted economic block values ($/ton). (The values in red represent the economic
value of waste blocks while values in black correspond to ore blocks).
It is assumed that a significant number of geostatistical conditional simulations can be
obtained or generated and that their results may be used in classifying each block into its
corresponding resource classification category. For more detailed discussions on the use of
geostatistical conditional simulations for mineral resource classification - together with a
comparison of the performance of different geostatistical methods - refer to Silva and Boisvert
(2014).
Figure 9.3 depicts a cross-sectional view of the classification of the individual blocks into their
corresponding ore resource categories.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 W W IND IND W W MEAS MEAS INF INF W W W W
2 W IND W W MEAS MEAS INF INF INF MEAS INF W
1 MEAS INF IND INF IND MEAS MEAS INF INF INF
INERRED INDICATED MEASURED WASTE
LEGEND INF IND MEAS W
Figure 9.3: Two-dimensional conceptual cross section of an ore resource classification model.
Table 9.1 provides a summary of the mineral deposit showing proportions of Inferred,
Indicated and Measured material for the total (ore and waste), as well as for the economic ore
fraction (as a separate unit).
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Table 9.1: Proportions of Inferred, Indicated and Measured material considering the entirety of
the mineral deposit (ore and waste), as well as ore tonnes exclusively.
RESOURCE CLASSIFCATION CATEGORY ORE + WASTE ORE
INFERRED (%) 44.4 45.5
INDICATED (%) 19.4 22.7
MEASURED (%) 36.1 40.9
A summary of the operational requirements for each period is shown in Table 9.2
Table 9.2: milling capacity, ore blending and mineral resource risk requirements considered for
the production scheduling problem.
PERIOD ORE MINED (blk) MIN GRADE (%Fe) MAX GRADE (%Fe) INFERRED (%) INDICATED (%) MEASURED (%)
1 3 64 66 0.35 0.15 0.40
2 10 64 66 0.35 0.15 0.40
3 8 64 66 0.35 0.15 0.40
For the computations carried out in scenario #2, that is, the scenario in which ore risk
constraints are considered, it is assumed that in designing the risk plan, the mine planner likely
desires to enforce upper bounds on the proportion of Inferred ore material, while choosing to
enforce lower bounds on both the proportions of Indicated and Measured ore. Note that different
bounds on the risk constraints might be enforced for different time periods, thus allowing for
potential risk deferment strategies that postpone the mining of riskier blocks for the later stages of
the mineral deposit’s mine life. Note that, initially the results are computed and presented on an
integer programming basis (i.e., if mined, blocks must be mined fully, as a single unit) so that key
differences between the production plans can be more easily communicated. However, this does
not alter the central insights about the impact of grade uncertainty on production schedules. The
production plans obtained for both scenarios are summarized in Table 9.3 and Table 9.4.
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Table 9.3: Mine production schedule corresponding to scenario #1 (risk-free scenario).
TIME PERIOD ORE MINED (blk) AVG GRADE (%Fe) INFERRED (%) INDICATED (%) MEASURED (%)
1 3 64.3 0 0 100
2 10 64.0 50 30 20
3 7 65.7 57 14 29
NPV ($) 103.65
Table 9.4: Mine production schedule corresponding to scenario #2 (risk-based scenario).
TIME PERIOD ORE MINED (blk) AVG GRADE (%Fe) INFERRED (%) INDICATED (%) MEASURED (%)
1 2 65.5 0 50 50
2 9 64.0 33 22 44
3 4 65.8 25 25 50
NPV ($) 90
From the results in said tables the following observations stand out: (i) the ideal plan of
scenario #1, (i.e., the one in which no grade uncertainty exists), shows a much larger presence of
Inferred material in its yearly composition of ore sent to the processing plant; (ii) for the first year
of mining, enough ore in the Indicated and Measured categories exists (is accessible), that mining
can avoid incorporating inferred material in the composition of the mill feed in both scenarios; (iii)
comparing the first year of mining in both scenarios, one sees that the inclusion of risk
requirements forces production to be redistributed so that a minimum threshold of indicated
material is met and the risk composition of the subsequent periods is not jeopardized. This leads
to lowering of average grades and decrease of generated cash flow; (iv) in the absence of risk
constraints, period two would have much higher proportions of Inferred and Indicated material,
indeed substantially higher than the proportion of Measured material for the same year; and finally,
(v) using the proportion of Inferred material as the key indicator of risk, period three emerges as
the riskiest year, that is, the one in which the greatest correction in this resource classification
category occurs (an initial proportion of Inferred material of 57 % in scenario #1 is altered to a
new corresponding value of 22 %)
The two alternative scenarios studied show the marked differences between traditional
mine production schedules, which disregard any risks associated with geological uncertainty in
grades, and mine plans which take due account of grade uncertainty. The physical extent of mining
is remarkably different in both cases, with the risk-considerate mine production plans extending
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far beyond the limits of the risk-inconsiderate mine production plans. Importantly, the sequencing
of production is itself different between the case-studies (see Figure 9.4 and Figure 9.5).
SCENARIO #1 NO RISK CONSTRAINTS NPV($) 90
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 8.5 2.9 -0.1 -4 5 14.1 9.8 10.2 -3.3 -1.4 -3.3 -3.3
2 -1.4 3.9 3.7 4.2 1.5 1.5 4.0 1.7 1.0 -3.4
1 2.1 8.7 11.7 12.1 7.6 3.6 6.9 6.6
LEGEND PER 1 PER 2 PER 3
RESOURCE MODEL
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 IND IND MEA MEA INF INF
2 MEA MEA INF INF INF MEA INF
1 IND INF IND MEA MEA INF INF INF
Figure 9.4: Mine production schedule for scenario #1, disregarding all risk associated with
grade uncertainty. Values depicted correspond to discounted economic block values.
SCENARIO #2 ENFORCING RISK CONSTRAINTS NPV($) 103.65
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 ` 9.3 2.9 -0.1 -4 5 12.8 9.8 10.2 -3.3 -1.4
2 -1.4 3.9 3.7 3.8 1.5 1.5 4.0 1.7
1 2.1 8.7 11.7 12.1 7.6
LEGEND PER 1 PER 2 PER 3 DUMP
RESOURCE MODEL
1 2 3 4 5 6 7 8 9 10 11 12 13 14
3 IND IND MEA MEA INF INF
2 MEA MEA INF INF INF MEA
1 IND INF IND MEA MEA
Figure 9.5: Mine production schedule enforcing ore risk constraint (scenario #2). Values
depicted correspond to discounted economic block values.
For the mine production plan corresponding to scenario #1 (no risk constraints), mining is
scheduled to start at roughly the center of the mineral deposit and then progress outwardly and in
depth in the subsequent periods 2 and 3. In the case of scenario #2 mining is more scattered about
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the ore deposit in the first period, where blocks positioned at coordinates and
are selected. Next, production evolves in depth and roughly to the center of the orebody in period
(cid:4666)(cid:1861),(cid:1862)(cid:4667) = (cid:4666)(cid:885),(cid:885)(cid:4667) (cid:4666)(cid:885),(cid:889)(cid:4667)
2. Eventually, in period 3 mining progresses further in depth and outwardly towards the western
flank of the orebody. These differences can be readily understood by considering the fact that in
the first scenario, the absence of ore risk constraints means that the solver is not considering the
need to meet these requirements in the composition of the material it chooses to send to the mill
plant. On the other hand, scenario #2 incorporates risk constraints enforcing limits on the
proportions of ore material belonging to each of the resource classification categories, causing the
solver to deviate from the path it would have otherwise taken, and reach for blocks outside the
physical boundary of the mine plan for scenario #1. Additionally, it should be noted that in the
case of scenario #2, in period 2, the positive-valued ore block positioned according to coordinates
is actually sent to the waste dump rather than the processing plant. This occurs
because the scheduler seeks to reach the largely more profitable ore blocks at the bottom of the
(cid:4666)(cid:1861),(cid:1862)(cid:4667) = (cid:4666)(cid:884),(cid:890)(cid:4667)
orebody while staying below the upper limits of total ore sent to the mill plant.
Clearly, NPV for scenario #1 will (necessarily) constitute an upper bound on the total NPV
that may be realized by mining according to the schedule associated with scenario #2, as it
corresponds to a less constrained optimization model. In the case shown in Figure 9.4 the NPV for
scenario #1 is approximately $111 while scenario #2 has an NPV of approximately $102.
Next, the proposed methodology is extended to a slightly larger case study emphasizing
the importance of allowing for the incorporation of risk tolerance into mine plans.
A cross-sectional view of the resource model for the deposit is as presented in Figure 9.6.
Figure 9.6: Two-dimensional conceptual cross section of the ore resource classification model.
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In constructing the conceptual resource model special note has been placed in reproducing
appropriately, the typical outcome from drillhole exploration campaigns. Commonly, such
campaigns place greater focus on the accurate delineation of those areas of the orebody which
display the highest grade results initially and, thus, hold the highest promise in terms of economic
profitability.
This results in the richest portions of the deposit being more densely sampled than its
poorer areas. In this example, the tendency to focus more on high-grade areas translates into the
outer parts of the deposit being less sampled due to their relatively lower grades. Given the lack
of detailed drillhole information, these areas have a higher degree of geological uncertainty and
are classified into the Inferred resource category. Conversely, the central parts of the deposit,
containing relatively higher grade material, are more densely sampled, accrue greater geological
information and are classified as Measured or Indicated material.
The set of estimated block grades for this iron ore example deposit are summarized in
Figure 9.7 below.
Figure 9.7: Two-dimensional conceptual cross section of the grade model corresponding to the
larger iron ore deposit used in the (% Fe). (The letter “W” standing for waste blocks).
It is important to note that in this model, it is possible for a small number of ore blocks to
contain significantly high grades, and to be classified in the Inferred category. That is the case of
blocks (3, 17) and (3, 19), and this is intended to induce an incentive for the solver to seek mining
blocks which, despite being profitable, can be considered potentially risky. The economic block
model for this example is presented in Figure 9.8.
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Figure 9.8: Two-dimensional conceptual cross section of an economic block model depicting 1st
period undiscounted economic block values ($/ton). (The values in red represent the economic
value of waste blocks while values in black correspond to ore blocks).
Two distinct cases in terms of risk preferences are tested. In one scenario, a mine
production plan is generated which does not include risk requirements for the composition of the
ore sent to the mill. This is called the “risk-free” scenario and it is equivalent to the decision maker
having a risk-neutral attitude, which implies indifference towards alternative investment options
providing equal economic payoffs, even as they represent very different levels of risk. Such a
scenario could be considered extreme or unrealistic but, it is in fact representative of much of the
current practice. In the alternative scenario, the decision maker is risk-averse and demands specific
requirements on the amount of Inferred, Indicated and Measured material in the ore sent to the
mill. Both plans consider three-year mine lives and specific operational constraints. The problem
requirements, as well as the summary of the realized mine plan for scenario #1, are presented in
Table 9.5 and Table 9.6, respectively.
Table 9.5: Mine production plan requirements for the “risk-free” plan, including lower and upper
bounds on the total milling capacity and average grade for the material sent to the mill plant. No
requirements placed in the proportions of Inferred, Indicated and Measured material.
PERIOD ORE MINED (blk) MIN GRADE (%Cu) MAX GRADE (%Cu) INFERRED (%) INDICATED (%) MEASURED (%)
1 10 64 66 NA NA NA
2 15 64 66 NA NA NA
3 15 64 66 NA NA NA
Table 9.6: Realized mine production plan including the previously unconstrained proportions of
Inferred, Indicated and Measured material.
PERIOD ORE MINED (blk) AVG GRADE (%Cu) INFERRED (%) INDICATED (%) MEASURED (%)
1 10 64.14 29 7 63
2 15 64 35 23 42
3 15 64 40 49 12
NPV ($M) 309.968
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Figure 9.9 depicts the mine production plan that results from solving for the operational
requirements of scenario 1. The decision variables in our optimization model are continuous and
allow for the mining of individual blocks across multiple periods. For clarity of exposition, rather
than displaying multiple periods in which a block is fractionally mined, we refer to the cumulative
proportion of the ore block mined by the end of a given time period t. For those blocks which are
fully mined by the end of time period t, the value displayed in the figure is precisely “t” (the
completion period). For blocks not fully mined by the end of time period “T” (the last time period)
the fraction of the block mined is displayed. Note that this is equivalent to presenting results in
terms of so-called “by variables.” However, since mine blocks can be mined in multiple periods it
might be more consistent to think of said periods not as yearly periods, but as representing a
“phase” which may be mined across a spectrum of time periods.
Figure 9.9: Mine production schedules for the “risk-free” scenarios.
For the first scenario, it is noted that the scheduler attempts to mine the center-west areas
of the deposit because these contain the highest valued ore blocks and realizes an NPV of
$(309.97*106). Also, both year 1 and year 2 productions are very small, while simultaneously, year
3 seems relatively large. Additionally, some of the blocks in the pit are mined only partially
(bottom left portion of the deposit). In practice, one would likely combine the advance
corresponding to both years 1 and 2 into a single pushback and, combine all of the blocks not fully
mined by the end of period 3 into another phase.
Next, it must be determined if the current plan meets the predefined risk criteria. In this
case, the adopted criterion was to generate a set of 10 orebody simulations, impose the current
schedule onto each of the simulations, and record the number of times the average grade realized
stayed within a ±15% accuracy interval around 64 % Fe. It is considered that a schedule can be
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accepted (and thus adopted as optimal) if, nine times out of ten, the average grade realized falls
inside the required accuracy interval.
Table 9.7 lists the results of testing the current schedule against all ten orebody realizations.
As expected, the “risk-free” plan fails to meet the objective risk requirement criteria in all three
time periods. Violations occur two times in period 1, three times in period 2 and most notably, four
times in time period 3 in which the mine plan includes large proportions of profitable Inferred
material.
Table 9.7: Uncertainty validation for current “risk-free” mine production plan
Clearly, the “risk-free” plan would be very difficult to accept as optimal under a
probabilistic framework, as all three time periods fail to meet the predefined risk threshold.
Contrasting the outcomes from the “risk-free” scenario, the results obtained in the case of
a risk-averse decision maker imposing limits on the proportions of material classified into each of
the distinct resource classification categories are presented next. First, Table 9.8 summarizes the
specific operational requirements for a feasible mining plan in the second scenario. Limits are
enforced on each of the resource classification categories per time period, and these include an
upper bound on the proportion of Inferred ore (5%) and a lower bound on both Indicated (10%)
and Measured (85%) resources. Next, the risk-based mine production schedule corresponding to
the updated risk requirements is presented in Table 9.9.
Table 9.8: Mine production plan requirements for the “risk-based” plan, including upper
bounds on the total ore tons, and both lower and upper bounds on the average grade of the ore
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material sent to the mill plant. Risk requirements are placed in the proportions of Inferred,
Indicated and Measured ore material.
PERIOD ORE MINED (blk) MIN GRADE (%Cu) MAX GRADE (%Cu) INFERRED (%) INDICATED (%) MEASURED (%)
1 10 64 66 5 10 85
2 15 64 66 5 10 85
3 15 64 66 5 10 85
Table 9.9: Realized mine production plan including the proportions of Inferred, Indicated and
Measured material.
TIME PERIOD ORE MINED (blk) AVG GRADE (%Cu) INFERRED (%) INDICATED (%) MEASURED (%)
1 10 64.10 0.0 10.0 90.0
2 14 66.0 0.0 14.3 85.7
3 7 65.0 0.0 14.3 85.7
NPV ($M) 210.854
The results for the risk-based schedule presented in Table 9.9 differ from the risk-free plan
(see Table 9.6) on multiple levels. First, there is a sharp decrease in NPV for the risk-based plan
[$(210.854*106)] relative to the risk-free plan [$(309.968*106)] which is consistent with the fact
that the latter corresponds to a more constrained problem than the former, but this is also due to
the scheduler being forced to avoid the riskier areas of the deposit which are rich in Inferred and
Indicated material. With the exception of period 1, the average grade of the mined material is
substantially higher for the risk-based schedule, although less overall tonnage is mined.
What is perhaps of most interest is the composition of the mill feed in terms of proportions
of the distinct resource classification categories. In the initial unconstrained production schedules,
the solver seeks to maximize profit by including a large fraction of Inferred and Indicated material
in the mill feed, thereby “loading” the plan with some of the riskier fractions of the deposit.
Specifically, periods 1 and 2 show proportions of Inferred material close to 30 %, and in period 3
this value raises to 40%. However, the central aspect to consider – one that is often missed - is that
this may (or may not) be the adequate plan for the mine depending on whether the particular risk
preferences of management are satisfied. As shown in Table 9.7, the initial plan obtained might
not always be the optimal (is a risk-tolerance sense), and therefore, what is crucial is that
management be able to impart its risk preferences on the schedules generated.
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The differences between scenario 1 (“risk-free”) and its alternative (“risk-averse”) extend
to the size and shape of the ultimate pit (see Figure 9.9 and Figure 9.10).
Figure 9.10: Mine production schedules for the “risk-averse” scenarios (scenario #2).
It can be seen (from Figure 9.10) that the production plan for scenario #1 results in a much
larger final pit, and with significantly different shape than the one for scenario #2, although there
is significant overlap between the two scenarios with regard to the starting location of mining and
its subsequent evolution. The differences in shape and size of the production plans also confirm
that risk analysis of the impact of uncertainty cannot rely solely on static sensitivity analysis of
individual economic parameters, because differences in levels of risk tolerance can result in
significantly different production schedules. Instead, it is important that new mine production be
generated when risk preference levels are not met, or are themselves updated.
Next, the performance of the “risk-averse” mining plan under conditions of uncertainty is
assessed, i.e., in order to be acceptable, it is required that nine times out of ten, the plan meets the
required ±15 % accuracy level around the grade of 64% Fe. As can be seen from Table 9.10, the
newly generated mine plan does meet the stated risk requirements for all three time periods. This
is in accordance with the expected result of enforcing the resource classification constraints. In
addition, it is noted that the portions of the pit limits from scenario 1 which display the greatest
changes relative to scenario 2 are precisely those being mined in the latest time period.
Incidentally, this is the period in which the violations are also the largest.
These small stylized models make clear that despite generating larger life-of-mine NPV,
the current level of geological knowledge of the deposit implies that scenario #1 plan is likely
unachievable in practice.
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