Um bom trabalho

1. Technical Note
Using equivalent grade factors to find the optimum cut-off grades
of multiple metal deposits
M. Osanloo *, M. Ataei
Department of Mining, Metallurgy and Petroleum Engineering, Amirkabir University of Technology, Tehran Polytechnic,
424 Hafez Ave., PO Box 15875-4413, Tehran, 15914 Iran
Received 11 September 2002; accepted 14 April 2003
Abstract
One of the most important aspects of mine design is to determine the optimum cut-off grades. Material grading above and below
the cut-off is directed to different destinations. Optimization of cut-off grade is now an accepted principle for open pit planning
studies. The most commonly criteria used in cut-off grade optimization is to maximize net present value. Lane formulated the
concept of cut-off grade optimization for single metal deposit but this method cannot be use in multiple metal deposits. Because in
single metal deposits six points are possible candidates for the optimum cut-off grade, in multiple metal deposits an infinite number
of points are possible candidates for the optimum cut-off grades. The objective function evaluation of these infinite points is im-
possible. In this paper, the equivalent grade factor is used to find optimum cut-off grade of multiple metal deposits. First, the
objective function is defined for multiple metal deposits and then objective function is converted to one variable function by using
equivalent factors. The optimum equivalent cut-off grade of main metal can be found by the optimization techniques such as the
Lane algorithm or elimination methods. At final step, the optimum cut-off grades will be determined by interpolation of grade-
tonnage distribution of deposit.
Ó 2003 Elsevier Ltd. All rights reserved.
Keywords: Modelling; Optimization
1. Introduction
One of the most critical parameters in mining oper-
ation is cut-off grade. Taylor presents one of the best
definitions of cut-off grade. He defined cut-off grade as
‘‘any grade that, for any specific reason, is used to sep-
arate two courses of action, e.g. to mine or to leave, to
mill or to dump . . .’’ (Taylor, 1972, 1985).
Most researchers have used break-even cut-off grade
criteria to define ore as a material that just will pay
mining and processing costs. These methods are not
optimum but the mine planner often seeks to optimize
the cut-off grade of ore to maximize the net present
value (NPV). The determination of the optimum cut-off
grade of single metal deposit can be very complex even
when price and cost are assumed constant, but it in-
volves the costs and capacities of the several stages of
the mining operations, the waste/ore ratios, average
grades of different increments of the ore body and so on.
Lane (1964, 1988) has developed a comprehensive
theory of cut-off grade calculation for a single metal
deposit. Whittle and Wharton added the idea of using
opportunity cost. They introduced two pseudo costs,
which are also important. They are referred to as delay
cost and the change cost (Whittle and Wharton,
1995a,b) but this algorithm cannot be use in multiple
metal deposits. The reason is due to the fact that, while
in single metal deposits six points are possible candi-
dates for the optimum cut-off grade (Lane, 1988), in
multiple metal deposits an infinite number points are
possible candidates for the optimum cut-off grades and
the objective function evaluation of these infinite points
is impossible.
These types of deposits can be evaluated based on a
value per ton of ore calculated from the net smelter re-
turn (NSR). NSR represented the total value of metals
recovered from each ton of ore minus the cost of
smelting (Annels, 1991). In this method, it is possible
*
Corresponding author. Tel.: +98-21-64542929; fax: +98-21-
6413969.
E-mail address: mosanloo@hotmail.com (M. Osanloo).
0892-6875/03/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0892-6875(03)00163-8
Minerals Engineering 16 (2003) 771–776
This article is also available online at:
www.elsevier.com/locate/mineng

2. that to express the grade of one metal in term of another
(Zhang, 1998; Liimatainen, 1998). Other methods for
ore/waste discrimination in multiple metal deposits are
critical level method, single grade cut-off approach,
dollar value cut-off approach (Annels, 1991; Barid and
Satchwell, 2001).
All of these methods are associated with some flaws:
none of these methods consider the grade distribution of
the deposits and do not take into account time value of
money. Furthermore, they completely ignore the ca-
pacities of the mining system, so the cut-off grades cal-
culated by these methods are not optimum.
This paper describes the use of equivalent grade fac-
tors to optimum the cut-off grades of multiple metal
deposits.
2. Objective function
In large open pit mines, there are typically three
stages of operations: (i) the mining stage, where units of
various grade are mined up to some capacity, (ii) the
treatment stage, where ore is milled and concentrated,
again up to some capacity constraint and (ii) the refining
stage, where the concentrate is smelted and refined to a
final product which is shipped and sold; the latest stage
is also subject to capacity constraints. Each stage has its
own associated costs and a limiting capacity.
By considering costs and revenues in this operation,
the profit is determined by using the following equation:
P ¼ ðs1 À r1ÞQr1 þ ðs2 À r2ÞQr2 À mQm À cQc À fT ð1Þ
where P: profit ($), m: mining cost ($/ton of material
moved), c: concentrating cost ($/ton of material con-
centrated), r1: refinery cost ($/unit of product 1), r2: re-
finery cost ($/unit of product 2), f : fixed cost ($), s1:
selling price ($/unit of product 1), s2: selling price ($/unit
of product 2), T: the length of the production period,
Qm: quantity of material to be mined, Qc: quantity of ore
sent to the concentrator, Qr1: the amount of product 1
actually produced over this production period, Qr2: the
amount of product 2 actually produced over this pro-
duction period.
If d is discount rate, the difference m between the
present values of the remaining reserves at times t ¼ 0
and t ¼ T is (Hustrulid and Kuchta, 1995)
m ¼ P À VTd ð2Þ
where V is the present values at time t ¼ 0. Substituting
Eq. (1) into Eq. (2) yields
m ¼ ðs1 À r1ÞQr1 þ ðs2 À r2ÞQr2 À mQm
À cQc À ðf þ VdÞT ð3Þ
The quantities of refined metals Qr1 and Qr2 are re-
lated to that send from the mine to concentrator (Qc),
therefore
Qr1 ¼ gg1y1Qc ð4Þ
Qr2 ¼ gg2y2Qc ð5Þ
where gg1: average grade of metal 1 sent for concentra-
tion, gg2: average grade of metal 2 sent for concentration,
y1: recovery of metal 1 from the ore, y2: recovery of
metal 2 from the ore.
Substituting Eqs. (4) and (5) into Eq. (3) yields
m ¼ ½ðs1 À r1Þgg1y1 þ ðs2 À r2Þgg2y2 À cŠQc
À mQm À ðf þ VdÞT ð6Þ
One would now like to schedule the mining operation
in such a way that the depreciation in the present value
takes place sooner rather than later. This is because later
profits are discounted more than those captured earlier.
In examining Eq. (6), this means that m has to be max-
imized. m is a function of two variables: grade of metal 1
and grade of metal 2.
In Eq. (6), the grade of metal 2 is converting to grade
of metal 1 by using equivalent factor. Therefore m will be
function of grade of metal 1 and Eq. (6) yields
m ¼ ðs1
À r1Þy1 gg1
þ
ðs2 À r2Þy2
ðs1 À r1Þy1
gg2
À c
Qc
À mQm À ðf þ VdÞT ð7Þ
Equivalent factor is equal to
Feq ¼
ðs2 À r2Þy2
ðs1 À r1Þy1
ð8Þ
Substituting Eq. (8) into Eq. (7) yields
m ¼ ½ðs1 À r1Þy1ðgg1 þ Feqgg2Þ À cŠQc À mQm À ðf þ VdÞT
ð9Þ
To calculate the average equivalent grade of ore based
upon equivalent factor and average grade of each metal,
the following equation can be used:
ggeq ¼ gg1 þ Feqgg2 ð10Þ
Substituting Eq. (10) into Eq. (9) yields
m ¼ ½ðs1 À r1Þy1ggeq À cŠQc À mQm À ðf þ VdÞT ð11Þ
Eq. (11) is the fundamental formula for calculation of
optimum cut-off grades of ore. The time taken T is re-
lated to the constrain capacity. Three cases arise de-
pending upon which of the three capacities is actually
limiting factor.
• If the mining capacity (M) is the limiting factor then
the time T is given by
T ¼
Qm
M
ð12Þ
• If the concentrating capacity (C) is the limiting factor
then the time T is controlled by the concentrator
T ¼
Qc
C
ð13Þ
772 M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771–776

3. • If the refinery output of main metal (Rm) is the limit-
ing factor then the time T is controlled by the refining
of main metal
T ¼
Qr1
Rm
¼
gg1y1Qc
Rm
ð14Þ
Substituting Eqs. (12)–(14) into Eq. (11) yields the fol-
lowing equations:
mm ¼ ½ðs1 À r1Þggeqy1 À cŠQc À m
þ
f þ Vd
M
Qm ð15Þ
mc ¼ ðs1
À r1Þggeqy1 À c
þ
f þ Vd
C
Qc À mQm ð16Þ
mr ¼ s1
À r1 À
f þ Vd
Rm
ggeqy1 À c
Qc À mQm ð17Þ
Now for any pair of cut-off grades, it is possible to
calculate the corresponding mm, mc and mr. The control-
ling capacity is always the one corresponding to the least
of these three equations. Therefore
max me ¼ max½minðmm; mc; mrÞŠ ð18Þ
In Eqs. (15)–(17), V is unknown value because it
depends upon the cut-off grade. Since the unknown V
appears in the equation thus iterative process must be
used.
3. Determination of optimum cut-off grades
As previously mentioned, using equivalent factor, the
objective function must be converted to one variable
function. Then optimization techniques such as Lane
algorithm or elimination methods can be use to find
optimum cut-off grades.
According to Lane algorithm, there are three limiting
cut-off grades and three balancing cut-off grades. If only
the capacity of one operation is limited factor then the
break-even cut-off grade for that stage will be the opti-
mum cut-off grade. To find the grades that maximize the
NPV under different constraints, one first takes the de-
rivative of Eqs. (15)–(17) with respect to grade. In next
step, setting derivative of Eqs. (15)–(17) equal zero, it
will obtain three economic optimum cut-off grades.
When mining operations are constrained by more
than one capacity, the optimum cut-off grade calculated
by conventional method may not necessarily be a break-
even cut-off grade. In such a case, the balancing cut-off
grade for each pair of stage needs to be considered as
well. A balancing grade is one that which allows both
stages of the pair being considered to achieve maximum
capacities jointly. Therefore, the balancing cut-off
grades are independent of economics and being deter-
mined by using the grade distribution and the capacities
of each of the different system. Based on these consid-
erations, now six cut-off grades are candidate for overall
optimum cut off grade. The optimum cut-off grade will
be one of the six cut-off grades consisting of the three
limiting economic cut-off grades and the three balancing
cut-off grades. Fig. 1 shows six candidate cut-off grades.
Lane has presented a graphical method to determine
overall optimum cut-off of ore among of these six cut-off
grades.
The optimum cut-off grade for a particular pair of
stages is the balancing grade limited by both stages. If
only one of the stages in the pair is a bottleneck then the
optimum cut-off grade for the pair is the breakeven cut-
off grade for the limiting stage. The overall optimum
cut-off grade is the middle value of the optimum cut-offs
for the three stages.
In this case, objective function is a unimodal func-
tion. So elimination methods such as dichotomous
search method, Fibonacci search method and Golden
section search method will be used to find optimum cut-
off grades. These methods require only objective func-
tion evaluations and do not use the derivative of the
function to find the optimum point (Rardin, 1998). In
these methods at first step, the uncertainty space of the
problem is estimated. In next step by selecting test points
in uncertainty space and evaluating and comparing ob-
jective function at these test points, a part of uncertainty
space will be eliminated. This procedure is repeated until
uncertainty interval in each direction is less than a small-
specified positive value e. Where e is desirable accuracy
to determine the optimum cut-off grades. Ratio of re-
mained length after elimination process to initial length
in each dimension is called reduction ratio. Among of
these methods, the reduction ratio of golden section
search method is optimum and equal to 0.618 (this
number called the golden number). In this method, ratio
of eliminated length to initial length will be equal 0.382.
Using the golden section rule means that for every stage
of the uncertainty range reduction (except the first one),
Fig. 1. mm, mc mr and me curves and six candidate cut-off grades.
M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771–776 773

4. the objective function will be evaluated at one new point
(Chong and Zak, 1996; Rao, 1996).
Fig. 2 shows flowchart to calculate the optimum cut-
off grade of ore by the golden section search method. In
first step, assume ½L; UŠ be the initial interval of uncer-
tainty and note that the initial interval included the
optimum point. Then, select two test points, g1 and g2.
Locations of these points are
g1 ¼ L þ ðU À LÞ Â 0:382 ð19Þ
g2 ¼ L þ ðU À LÞ Â 0:618 ð20Þ
In next step, the objective function of points g1 and g2
will be calculated. Depending on the objective function
value of these points, the length of the new interval of
uncertainty successively is reduced. By placing a new
observation, the process is repeated until the optimum
point with desirable accuracy is found.
4. Example
Consider a hypothetical situation wherein final pit
limits of Cu/Mo deposit has been superimposed on a
mineral inventory. The pit outline contains 14.4 million
tons of materials. The grade–tonnage distribution and
average grades of ore for each metal are shown in Tables
1–3 and associated costs, prices, capacities, quantities
and recoveries are given in Table 4.
According to Eq. (8), equivalent factor for this situ-
ation is
Feq ¼
ð6797:15 À 190Þ Â 0:8
ð1674:5 À 63Þ Â 0:82
¼
4
1
Fig. 2. Flowchart for finding the optimum cut-off grade by the golden
section search method.
Table 1
Grade–tonnage distribution of copper and molybdenum
Copper (%) Molybdenum (%)
0–0.025 0.025–0.05 0.05–0.075 0.075–0.1 0.1
0–0.1 1,320,000 900,000 285,000 315,000 510,000
0.1–0.2 360,000 300,000 240,000 135,000 60,000
0.2–0.3 735,000 525,000 300,000 210,000 30,000
0.3–0.4 1,110,000 570,000 375,000 135,000 30,000
0.4–0.5 525,000 255,000 75,000 60,000 90,000
0.5–0.6 510,000 300,000 210,000 105,000 30,000
0.6–0.7 375,000 270,000 210,000 90,000 90,000
0.7 645,000 690,000 570,000 495,000 360,000
Table 2
Average grade of copper of different copper and molybdenum intervals
Copper (%) Molybdenum (%)
0–0.025 0.025–0.05 0.05–0.075 0.075–0.1 0.1
0–0.1 0.02 0.03 0.02 0.03 0.05
0.1–0.2 0.12 0.17 0.16 0.19 0.14
0.2–0.3 0.25 0.27 0.25 0.22 0.26
0.3–0.4 0.33 0.32 0.35 0.34 0.37
0.4–0.5 0.44 0.47 0.45 0.48 0.46
0.5–0.6 0.53 0.55 0.57 0.54 0.55
0.6–0.7 0.67 0.63 0.65 0.64 0.66
0.7 0.98 1.04 1.02 1.09 1.01
774 M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771–776

5. Now using the equivalent factor and average grade of
each metal, the equivalent copper grade of different
copper grade is calculated (Table 5).
Converting molybdenum grade into copper grade, the
grade–tonnage distribution of two metal deposits is
converted into one-dimensional grade tonnage distri-
bution and cut-off grade optimization method of single
metal deposit such as Lane method or elimination
method was used to calculate the optimum cut-off
grades in year by year. Then the grade–tonnage curve of
deposit is adjusted for each year of mine life. To do this,
tonnage of ore in first year of mine life from the grade
distribution intervals above optimum cut-off grades and
tonnage of waste in first year of mine life from the grade
distribution intervals below optimum cut-off grades was
subtracted. These calculations are repeated until the end
of mine life. Table 6 shows the optimum cut-off grades
of ore for different years of mine life.
5. Justification of the proposed method
To justify proposed method, the NPV of break-even
equivalent cut-off grade was calculated and compared
with NPV calculated by proposed method. By defini-
tion, the break-even equivalent cut-off grade is grade
that revenue is equal to costs. So
ð1674:5 À 63Þ Â
geq
100
 0:82 ¼ 1:06 þ 3:52
) geq ¼ 0:3466
Thus, break-even equivalent cut-off grade of copper is
0.3466%. Using interpolation technique and Table 5, the
copper grade is calculated to be 0.1263%. For this grade
of copper: Tonnage of ore ¼ 10779464.65, Tonnage of
waste ¼ 3620536.45, Waste: ore ¼ 0.3358, Average equiv-
alent grade ¼ 0.6238%.
If concentrator capacity is controlling factor, then the
mine life is equal to
10779463:65
750; 000
¼ 14:37 year
Yearly revenue will be equal to
Table 5
Equivalent copper grade of different copper grade
Copper grade (%) Average grade Equivalent
copper grade (%)
Tonnage
Copper (%) Molybdenum (%)
0–0.1 0.0282 0.0368 0.1754 3,330,000
0.1–0.2 0.1522 0.044 0.3282 1,095,000
0.2–0.3 0.2525 0.0364 0.3981 1,800,000
0.3–0.4 0.332 0.0437 0.5068 2,220,000
0.4–0.5 0.4521 0.0266 0.5585 945,000
0.5–0.6 0.5442 0.0451 0.7246 1,215,000
0.6–0.7 0.652 0.043 0.824 1,035,000
0.7–2 1.0269 0.0567 1.2537 3,760,000
Table 3
Average grade of molybdenum of different copper and molybdenum intervals
Copper (%) Molybdenum (%)
0–0.025 0.025–0.05 0.05–0.075 0.075–0.1 0.1
0–0.1 0.002 0.026 0.052 0.076 0.113
0.1–0.2 0.017 0.031 0.06 0.085 0.114
0.2–0.3 0.011 0.028 0.066 0.091 0.137
0.3–0.4 0.031 0.042 0.058 0.094 0.138
0.4–0.5 0.006 0.035 0.054 0.091 0.119
0.5–0.6 0.012 0.039 0.07 0.082 0.152
0.6–0.7 0.014 0.029 0.062 0.085 0.12
0.7 0.009 0.038 0.063 0.086 0.128
Table 4
Economic parameters for a manual example
Parameter Unit Quantity
Mine capacity Tons per year 2,500,000
Mill capacity Tons per year 750,000
Refining capacity (copper) Tons per year 5000
Refining capacity (molybde-
num)
Tons per year 1000
Mining cost Dollars per ton 1.06
Milling cost Dollars per ton 3.52
Refining cost (copper) Dollars per ton 63
Refining cost (molybdenum) Dollars per ton 190
Fixed costs Dollars per
year
790,000
Price (copper) Dollars per ton 1674.5
Price (molybdenum) Dollars per ton 6797.15
Recovery (copper) % 82
Recovery (molybdenum) % 80
Discount rate % 20
M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771–776 775

6. Yearly revenue ¼ ð1674:5 À 63Þ Â
0:6238
100
 0:82
 750; 000
¼ 6182964:363
Moreover, yearly cost will be equal to
Yearly cost ¼ 750; 000 Â ð1 þ 0:3358Þ Â 1:06 þ 750; 000
 3:52
¼ 4492019:035
Yearly cash flow and NPV of mining operation under
break-even equivalent cut-off grade found to be
6182964:363 À 4492019:035 ¼ 1690945:325 $
NPV ¼
1690945:325½ð1 þ 0:2Þ
14:37
À 1Š
0:2 Â ð1 þ 0:2Þ
14:37
¼ 7839174:188
Therefore, NPV of mining operation under break-
even equivalent cut-off grade is $ 7839174.188 and NPV
of mining operation under proposed method according
to Table 6 is $ 16,355,000. Thus, if mining operation is
operated under proposed method, NPV is more than
twice of NPV of mining operation under break-even
equivalent cut-off grade.
6. Conclusion
One of the important aspects of mining is deciding
what material in a deposit is worth mining and pro-
cessing, and on the contrary, what material is waste.
This decision-making is summarized by the cut-off grade
policy. Cut-off grades of multiple metal deposits are
evaluated by several methods such as NSR method,
critical level method, single grade cut-off approach,
dollar value cut-off approach. None of these methods is
optimum. In this paper, proposed that minable ore is
ranked based on metals contribution of the mine reve-
nue and equivalent grade of main metal is determined
using equivalent factors. Objective function is expressed
to one variable function. Then optimization techniques
such as Lane algorithm or elimination methods must be
use to find optimum equivalent cut-off grade for main
metal (caused more revenue). Optimum cut-off grades
are determined by interpolation of grades–tonnage dis-
tribution. A verification example is presenting for con-
firming the approach proposed in this study. The
comparison of results are shown the NPV of mining
operation under proposed method is more than twice of
NPV of mining operation under break-even equivalent
cut-off grade.
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Table 6
The optimum cut-off grades of different years of mine life
Year Copper cut-off grade (%) Qm (Ton) Qc (Ton) Qr1 (Ton) Profit ($) NPV ($)
1 0.4617 2,010,300 750,000 4976.9 4,425,600 16,355,000
2 0.3725 1,345,100 750,000 4546.2 4,013,100 15,201,000
3 0.3645 1,601,700 750,000 4487.3 3,961,200 14,228,000
4 0.3555 1,555,600 750,000 4421 3,899,900 13,112,000
5 0.3455 1,507,300 750,000 4347.4 3,828,700 11,834,000
6 0.3355 1,462,000 750,000 4273.8 3,754,300 10,373,000
7 0.3255 1,419,200 750,000 4200.2 3,677,200 8,693,000
8 0.2634 1,222,500 750,000 3850.5 3,286,600 6,754,000
9 0.2464 1,181,500 750,000 3774.9 3,196,200 4,818,000
10 0.2293 794,700 521,330 2571.4 3,103,000 2,586,000
776 M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771–776