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548zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
Table 6.18. Incremental capital investments and annual cash flows.
Scenario A Capital investment ($103) ACash flow ($103)
7-7 3332 773
7-6 3334 628
7-5 2666 402
4000 374 7-3
Repeating this process for the others one finds
Scenario i
7-7 7-6 23.1%
7-6 —> 7-5 18.6%
7-5 - » 7-4 14.6%
7 - 4 - * 7-3 8.9%
Now it can be seen that the optimum scenario is 7-4 rather than 7-3. The milling rate would
be 15,300 tpd.
16. Examine all of the other scenarios with respect to the best current one. For pit 7, the
best scenario is 7-4. For pit 6, scenario 6-6 has a higher total ore tonnage than 7-4 with
an acceptable rate of return on the additional capital expenditure. It thus becomes the best
current alternative. Plan 6-5 is better than 6-6 and 6-4 is better than 6-5. Plan 6-3 is not
acceptable. Pit 5 alternatives are compared against 6-4. As can be seen from the table, the
overall best plan is between Plan 5-4 and Plan 5-3. The economic cutoff (mill cutoff) for
the pit is 0.34% Cu.
This leads to the following operation
Ore reserve = 253,000,000 tons
Mill rate = 33,580 tpd
Mining rate = 116,600 tpd
Average ore grade = 0.864% Cu
Mine life = 25 years
Capital investment = $95,500,000
6.7 LANE'S ALGORITHM
6.7.1 Introduction
In 1964, K.F. Lane (Lane, 1964) presented what has become a classic paper entitled 'Choos-
ing the Optimum Cut-off Grade'. This section will describe his approach and illustrate it
with an example. As has been discussed earlier cutoff grade is the criterion normally used in
mining to discriminate between ore and waste in the body of a deposit. Waste may either be
left in place or sent to waste dumps. Ore is sent to the treatment plant for further processing
and eventual sale.
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 549 yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA
Maximum Unit Capacity Costs
M m
C c
R r
f
s ywusgaI
y
Figure 6.21. The model described by the Lane algorithm (Lane, 1964).
The choice of cutoff grade can directly affect profits. This chapter will examine the
principles which determine the best choice of a cutoff grade under different circumstances.
A mining operation is considered to consist of three stages:
- mining,
- concentrating, and
- refining.
Each stage has its own associated costs and a limiting capacity. The operation as a whole
will incur continuing fixed costs. The three most important economic criteria which can be
applied are:
Case I: Maximum present value.
Case II: Maximum total profits.
Case DOE: Maximum immediate profit.
The maximum present value gives the economic optimum and is that generally applied
lacking special circumstances. It is the one which will be used in this book. As has been
shown by Lane (1964) the second and third correspond to the application of special discount
rates in the first. Case EE, maximum total profits, corresponds to a discount rate of zero percent
whereas Case III is for a high value.
In this chapter, attention is focussed on choosing a cutoff grade to maximize the present
value of the cash flow from the operation.
6.7.2 Model definition
Figure 6.21 is a diagrammatic representation of the elements and symbols used in the
model.
Definitions of the maximum capacities, unit costs and quantities involved in the evaluation
are presented below.
Material
Ore
Concentrate
Product
MINE
CONCENTRATOR
REFINERY
Market
Other Factor;
Fixed costs
Selling prici
Recovery
550zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
1. Maximum capacity: M is the maximum amount of material (ore and waste) that the mine
can produce in a given time period (for example 1000 tons/year). It is therefore a restriction
on the maximum rate of progress through the orebody.
C is the maximum amount of ore which can be put through the concentrator in a given
time period (for example 500 tons/year), assuming unrestricted availability of input ore from
the mine. A concentrate of fixed grade is produced.
R is the maximum amount of final product produced in the time period (for example
500 lbs/year), assuming unrestricted availability of concentrate from the concentrator. The
maximum can be due to a restriction on refinery throughput or a market limitation.
2. Costs: m are the mining costs expressed in $/ton of material moved. These are assumed
to be the same irrespective as to whether the material is classified as ore or waste. The unit
mining costs include drilling, blasting, loading, hauling, etc.
c are the concentrating costs expressed in $/ton of material milled. The unit cost c includes
crushing, grinding, floating, leaching, etc. It also includes some haulage if ore is trucked
farther than waste (if not, this can become a credit item in calculating c).
r includes all costs incurred at the product and selling stages such as smelting, refining,
packaging, freight, insurance, etc. These are expressed in terms of $ per unit of product. For
copper it would be $/lb.
/ , the fixed cost, includes all costs such as rent, administration, maintenance of roads
and buildings, etc. which are independent of production levels (within normal limits of
variation) but which would cease were the mine to be closed. It is expressed in terms of a
fixed cost over the production period considered (for example 1 year). Other costs such as
head office charges, depreciation, etc. are not included.
s, the selling price, is expressed in terms of selling price per unit of product. It is a gross
figure provided all selling charges are included in r. If not they must be subtracted from i.
)', the recovery, is an overall figure for the concentrator and the refinery. It is that proportion
of the mineral contained in the original ore feed retained in the final product.
3. Quantities: T is the length of the production period being considered (for example 1 year);
Q,„ is the quantity of material to be mined, Qc is the quantity of ore sent to the concentrator
and Qr is the amount of product actually produced over this production period.
6.7.3 The basic equations
Using the definitions given in the preceding section, the basic equations can be developed.
The total costs Tc are
Tc = inQm + cQc + rQr +fT (6.7)
Since the revenue R is
R = sOr (6.8)
the profit P is given by
P = R — Tc = sOr — (niOm + cQc + rQr + f T ) (6.9)
Combining terms, yields
P = {s-r)Qr-cQc-mQm-fT (6.10)
This is the basic profit expression. It can be used to calculate the profit from the next Qm of
material mined.
Table 6.19. Initial mineral inventory for the Lane
example. zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA
Production planning 551
Grade (lbs/ton) Quantity (tons)
0.0-0.1 100 0.1-0.2 100
0.2-0.3 100 0.3-0.4 100
0.4-0.5 100
0.5-0.6 100 0.6-0.7 100
0.7-0.8 100
0.8-0.9 100
0.9-1.0 100
Qm = 1000
6.7.4 An illustrative example
To introduce the reader in a soft way to the problem being explored in detail in this section
consider the following example. A final pit has been superimposed on a mineral inventory.
Within the pit outline are contained 1000 tons of material. The grade distribution is shown
in Table 6.19. The associated costs, price, capacities, quantities and recovery are:
Costs
m — mining = $ l / ton
c = concentrating = $2/ton
r — refining = $5/lb
/ = fixed cost = $300/year
Price
s = $25/lb
Capacities
M — lOOtons/year
C = 50tons/year
R = 40 lbs/year
Quantities
Qm — amount to be mined (tons)
Oc = amount sent to the concentrator (tons)
Qr = amount of concentrator product sent for refining (lbs)
Recovery (Yield)
y — 1.0(100%recovery is assumed).
552zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
There are a great number of possible mine, concentrator and refinery operating combi-
nations. Which is the optimum? In this section the basic equations will be developed in
addition to demonstrating the process. However, prior to beginning the theoretical treat-
ment, it is considered useful to briefly consider just one of these operating combinations.
The total amount of material to be mined Qm is 1000 tons. If the mine is operated at capacity
(100 tons/year) then the pit would be mined out over a time period of 10 years. Assuming
that the grades (Table 6.19) are equally distributed throughout the pit and a concentrator
cutoff grade of 0.50 lbs/ton is used, then 50 tons of material having an average grade of
0.75 lbs/ton would be sent to the concentrator every year. The other 50 tons would be sent
to the waste dump. Since the concentrator capacity C is 50 tons/year, this is an accept-
able situation. The concentrator product becomes the refinery feed. In this case it would
be 37.5 lbs/year (0.75 lbs/ton x 50 tons/year). Since the refinery can handle 40 lbs/year, it
would be operating at below its rated capacity R. This combination of mining rate and cutoff
grade would yield a yearly profit Py of
Py = (25 - 5)37.5 - 2 x 50 - 1 x 100 - 300 = $250
These profits would continue for 10 years and hence the total profit would be $2500. The
NPV assuming an interest rate of 15% would be
The first question to be asked is whether some other combination of mine production rate
and concentrator cutoff grade would yield a better profit from this deposit? The larger
question is whether the various plant capacities (with their associated costs) are optimum?
The procedure described in this section is a way of determining the combination yielding the
maximum profit for a given set of operating constraints. The constraints may then be changed
(mine, concentrator and refining capacities, for example) and the profit corresponding to
this new combination determined as well as how the various capacities should be utilized
over the life of the pit.
6.7.5 Cutoff grade for maximum profit
Step 1. Determination of the economic cutoff grade - one operation constraining the total
capacity
As indicated, the basic profit expression (6.10) is
P = {s- r)Qr - cOc - mOm - fT
Calculate cutoff grade assuming that the mining rate is the governing constraint. If the
mining capacity M is the applicable constraint, then the time needed to mine material Qm is
Qm xroligfSOMC
M
Equation (6.10) becomes
P = ( s - r)Qr - cOc - (m + Qm (6.12)
To find the grade which maximizes the profit under this constraint one first takes the derivative
of (6.12) with respect to g.
r )d Q r / Q c f n j f \ d Q m
dg dg dg V MJ dg
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 553
However the quantity to be mined is independent of the grade, hence
d Qm = 0 (6.14)
dg
Equation (6.13) becomes
d P . dQr d Qc T = 0 - r ) ~ c — (6.15) dg d g dg
The quantity refined Qr is related to that sent by the mine for concentration Qc by
Qr —yrmgedcSRQ gyQcywusgaI ( 6. 16)
where g is the average grade sent for concentration, and y is the recovery.
Taking the derivative of Qr with respect to grade one finds that
~r~ = gy-7- (6.17 dg dg
Substituting Equation (6.17) into (6.15) yields
dP r n d Oc
- T = [(s - r)gy ~ c] " f 1 (6-18) dg dg
The lowest acceptable value of g is that which makes
dg
Thus the cutoff grade gm based upon mining constraints is the value of g which makes
(s - r)gy - c = 0
Thus g m = g = 7 C 7 (6.19)
y(s - r)
Substituting the values from the example yields MH
$2 g m = l . Q ( $ 2 5 - $ 5 ) = 0 - 1 0 1 b s / t O n
Calculate cutoff grade assuming that the concentrating rate is the governing constraint. If
the concentrator capacity C is the controlling factor in the system, then the time required to
mine and process a Qc block of material (considering that mining continues simultaneously
with processing) is
Qc
Tc = ~ (6-20)
Substituting Equation (6.20) into (6.10) gives
P = (s-r)Qr-cQc-mQm-f^ (6.21)
Rearranging terms one finds that
P = ( s - r)Qr - U + Q Qc- >nQm
554zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
Differentiating with respect to g and setting the result equal to zero yields
— =(s- r)- c + - m—— = 0 dg dg V C) dg dg
As before
dg
dQr
d Qc
Thus
, dQr ( f \ dOc
d Qc s - r The cutoff grade when the concentrator is the constraint is xroligfSOMCC+
fr
gc = — ^ r (6.22)
y(s - r)
For the example, this becomes
ec „.. 0.40 s 1.0(525 - $5)
Calculate cutoff grade assuming that the refining rate is the governing constraint. If the
capacity of the refinery (or the ability to sell the product) is the controlling factor then the
time is given by
Qr Tr = ^ r (6-23)
K Substituting (6.23) into Equation (6.10) yields
P = (s- r)Qr-f%- - cQc - rnQm
K
P= ( s - r - t \ Or-cQc- rnQm (6.24)
or
P= ( x - r -
R
Differentiating with respect to g and setting the result equal to zero gives
d P ( f\dQr d Oc dQm — = j — r — — — c— m—-— = 0 dg V RJ dg dg dg
Simplifying and rearranging gives
gr = 7 ~~ TT (6.25)
( s - r - f y y
For this example yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA
$2
( $ 2 5 - $ 5 - ^ ) 1.0 = 0.16 (6.26)
Production planning 555 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
One can now calculate the amount of material which would be concentrated and refined
under the various constraints as well as the time required. When the mining rate of 100
tons/year is the constraint
1000 tons T m = — 7 — = 1 Q y e a r s
lOOtons/year
Since the cutoff grade gm is 0.10 lbs/ton, a quantity Qc of 900 tons having an average grade
of 0.55 lbs/ton would be sent to the concentrator. The total amount of product refined and
sold Qr is
Q r = 900 x 0.55 = 495 lbs
Substituting these values into the profit equation gives
Pm = ($25 - $5)495 - $2 x 900 - $1 x 1000 - $300 x 10 = $4100
The same procedure can be followed with the other two limiting situations. The results are
given below:
Concentrator limit:
gc = 0.401bs/ton
Qc = 0.60 x 1000 = 600 tons
C = 50 tons/year
^ 6 0 0 , „ Tc = — = 12years
1000 Qm - - ¡ y = 83.3 tons/year
g — 0.71bs/tons
Qr = 600 x 0.7 x 1.0 = 420 lbs
Pc = (25 - 5)420 - 2 x 600 - 1 x 1000 - 300 x 12
= $2600
Refinery limit:
gr = 0.16 lbs/ton
g = 0.581bs/tons
Qc = 0.84 x 1000 = 840 tons
Q r = 840 x 0.58 x 1.0 = 487.2 lbs
487.2 Tr = = 12.18years
40
1000 M — = 82.1 tons/year
12.18 n
840 C = = 69 tons/year
12.18 '
Pr = (25 - 5)487.2 - 2 x 840 - 1 x 1000 - 300 x 12.18
= $ 3 4 1 0
556zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
Table 6.20. Total profits as a function of concentrator cutoff with mine
operating at capacity. g
g Profits ($)
Pm Pc Pr
0.0 4000 1000 3250
0.1 4100 1700 3386
0.16 4064 2024 3410
0.2 4000 2200 3400
0.3 3700 2500 3287.50
0.4 3200 2600 3050
0.5 2500 2500 2687.50
0.6 1600 2200 2200
0.7 500 1700 1587.50
0.8 - 8 0 0 1000 850
0.9 - 2 3 0 0 100 -12 .50
0.95 - 3 1 2 5 - 4 2 5 -511.25
In summary, for each operation taken as a single constraint, the optimum cutoff grades are: yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA
g m = 0 . 1 0
gc = 0.40
gr = 0.16
The total profits assuming the single constraint of mining, concentrating or refining are
given as a function of cutoff grade in Table 6.20. The values have been plotted in
Figure 6.22.
Step 2. Determination of the economic cutoff grade by balancing the operations
In the first step, it was assumed that only one of the operations was the limiting factor to
production capacity.
A second type of cutoff is based simply on material balance. To be able to calculate this
one needs to know the distribution of grades of the mined material. The average grade of
the treated material can be found as a function of the chosen cutoff. The average grade, and
the number of units involved are given as a function of cutoff grade in Table 6.21.
For both the mine and mill to be at their respective capacities, then
Qm = 100 tons
Qc = 50 tons
As can be seen from Table 6.21, the cutoff grade should be 0.5 lbs/ton. This balancing cutoff
between mine and concentrator is expressed as gmc. For the concentrator and the refinery to
be at full capacity
Qc = 50 tons
Qr = 40 tons
The relationship between Qc and Or is shown in Table 6.22.
In examining Table 6.22, it can be seen that the required balancing cutoff grade gcr is
gcr = 0.60
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 557
Figure 6.22. Total profit as a function of cutoff grade under different constraints.
Table 6.21. Concentrator feed as a function of concentrator cutoff with mine operating at capacity.
Mined amount
(2m) (tons)
Concentrator cutoff
grade (gc) (lbs/ton)
Feed going to the
concentrator (<2c)(tons)
100 0 100
100 0.1 90
100 0.2 80
100 0.3 70
100 0.4 60
100 0.5 50
100 0.6 40
100 0.7 30
100 0.8 20
100 0.9 10
The final balancing cutoff is between the mine and the refinery. As seen in Table 6.23 (assum-
ing 100% concentrator recovery) a cutoff grade of 0.4 yields 42 lbs of product whereas 0.5
yields 37.5 lbs. The desired level of 40 lbs lies between these two. Interpolating one finds that
gmr = 0.456 lbs/ton
558zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
Table 6.22. Refinery product as a function of concentrator cutoff with concentrator
operating at capacity.
Amount to be Concentrator Avg. conc. Refinery
concentrated (Qc) cutoff grade (gc) feed grade (gc) product (Qr)
(tons) (lbs/ton) (lb/ton) (lbs)
50 0 0.5 25
50 0.1 0.55 27.5
50 0.2 0.5 30
50 0.3 0.65 32.5
50 0.4 0.7 35
50 0.5 0.75 37.5
50 0.6 0.8 40
50 0.7 0.85 42.5
50 0.8 0.9 45
50 0.9 0.95 47.5
Table 6.23. Refinery feed as a function of mine cutoff with the mine operating at
capacity (assuming 100% concentratory recovery).
Mined amount (Qm)
(tons)
Mine cutoff grade (gm)
(lbs/ton)
Refinery product (Qr)
(lbs)
100 0 50
100 0.1 49.5
100 0.2 48
100 0.3 45.5
100 0.4 42
100 0.5 37.5
100 0.6 32
100 0.7 25.5
100 0.8 18
100 0.9 9.5
In summary, when the operations are taken in combination, the optimum cutoff grades are:
gmc = 0.50
gcr = 0.60
gmr = 0-456
Step 3. Determining the overall optimum of the six cutoff grades
There are six possible cutoff grades. Three (gmc, gcr, and gmr) are based simply upon the grade
distribution of the mined material and capacities. The other three (gm, gc, and gr) are based
upon capacities, costs and the price. The objective is to find the cutoff grade which produces
the overall maximum profit in hght of the mining, concentrating and refining constraints.
The local optimums for each pair of operations are first considered. The corresponding
optimum grades for each pair (Gmc , Grc, and Gmr) are selected using the following rules:
gm if gmc < gm
Gmc = go if gmc > gc (6.26a)
gmc otherwise
Production planning 559
gr i f g r c < gr
gc i f g r c > gc zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA(6.26b)
grc otherwise
8m i f gmr < gm
gr if gmr > gr (6.26c)
§mr otherwise
The overall optimum cutoff grade G is just the middle value of Gmc, Gmr, and Grc. In our
example the six possible cutoff grades are:
gm = 0.10
gc = 0.40
gr ~ 0.16
gmc = 0.50
gmr = 0.456
gcr = 0.60
Consider them in groups of three:
g m = 0 . 1 0
gc = 0.40 Choose the middle value Gmc = 0.40
gmc = 0.50
gm = 0 . 1 0
gr = 0.16 Choose the middle value Gmr = 0.16
gmr = 0.456
g ryxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA = 0.10
gc — 0.40 Choose the middle value Gcr = 0.40
gcr = 0.60
Considering the three middle values
Gmc = 0.40
Gmr = 0.16
Gcr = 0.40
one chooses one numerically in the middle
G = 0.401bs/ton
From Table 6.24, the average grade gc of the material sent to the concentrator for a cutoff
of 0.40 lbs/ton would be
gc = 0 . 7 0 lbs/ton
For 100% recovery the quantities are
Qm = 1000 tons
Qc = 6 0 0 tons
Qr — 420 lbs
Group 1
Group2
Group3
560zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
Table 6.24. Grade distribution for the first 100 ton parcel mined.
Grade (lb/ton) Quantity (tons)
0.0-0.1 10
0.1-0.2 10
0.2-0.3 10
0.3-0.4 10
0.4-0.5 10
0.5-0.6 10
0.6-0.7 10
0.7-0.8 10
0.8-0.9 10
0.9-1.0 10
Qm = 100
Applying the respective capacities to these quantities one finds that
600
Tc= — = 12 years
4 2 0
Tr = — - = 10.5 years 40 1000
Tm = = 10 years 100 J
Since the concentrator requires the longest time, it controls the production capacity. The
total profit is
P = $2600
and the profit per year Py is
$2600 Px = = $216.70/year
12years
The net present value of these yearly profits assuming an interest rate of 15% is
(1 + f) 12 _ i i 1512 _ 1 N P V = ^ V r T p - = $ 2 1 6 - 7 0 o l 5 a l 5 F = $ 1 1 7 4 - 6 0
In summary: the concentrator is the controlling production limiter; concentrator
feed = 50 tons/year; optimum mining cutoff grade = 0.40 lbs/ton (constant); total con-
centrator feed = 600 tons; average concentrator feed grade = 0.70 lbs/ton; years of
production = 12 years; copper production/year = 35 lbs; total copper produced = 420 lbs;
total profits = $2600.40; net present value = $1174.60.
6.7.6 Net present value maximization
The previous section considered the selection of the cutoff grade with the objective being
to maximize profits. In most mining operations today, the objective is to maximize the net
present value NPV. In this section the Lane approach to selecting cutoff grades maximizing
NPV subject to mining, concentrating and refining constraints will be discussed.
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 561
For the example of the previous section a fixed mining cutoff grade of 0.40 lbs/ton was
used. One found that
Qm = 83.3 tons/y ear
Qc = 50tons/year
Q r = 351bs/year
/ = $300/year
Lifetime = 1 2 years
The yearly profit would be
Pj = 35.0 x $20 - 50 x $2 - 83.3 x $1 - $300
= $216.70
when finding the maximum total profit, the profits realized in the various years are simply
summed.
The total profit is, therefore
12
PT = Y^Pj = 1 2 X 216.7 = $2600.40
i= i
The net present value for this uniform series of profits (Chapter 2) is calculated using
NPV = P , ( 1 + I ' ) " - 1
1 ¿(1 + 0"
Assuming an interest (discount) rate of 15% one finds that
The question to be raised is 'Could the NPV be increased using a cutoff grade which, instead
of being fixed, varies throughout the life of the mine?' If so, then 'What should be the cutoff
grades as a function of mine life?' These questions have been addressed by Lane and are
the subject of this section.
Assume that just prior to mining increment Qm (shown to commence at time f = 0 in
Figure 6.23 for simplicity), the present value of all remaining profits is V. This is composed
of two parts. The first, PVp, is from the profit P realized at time T by mining the quantity
Qm. The second, PVW is obtained from profits realized by mining the material remaining
after time T. These profits are indicated as P\ occurring at time Tu P2 occurring at time
7?, etc., in Figure 6.23. The value of all these remaining profits for mining conducted after
t = T as expressed at time T is W. The present values of W and P, respectively, discounted
to time t = 0 are given by
pv~(f = 0 )=(IW (6-27)
pv'(i = 0) = a T ^ (6-28)
where d is the discount rate.
562zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
V - - V = PVP + PVW
PV, w
PVp ywusgaI
w
—i N
A y
Figure 6.23. A diagrammatic representation of the NPV calculation.
The present value at time t = 0 is therefore
W P
(6.30)
(6.31)
Since the present value at time t = T of the remaining reserves is W, the difference v
between the present values of the remaining reserves at times t = 0 and t = T is
v = V - W
Equation (6.29) can be rewritten as
W + P = V(l + d)T
Applying the binomial expansion to the term (1 + d)T one finds that
r (T - 1 )d2 T(T - 1 )(T - 2)d
3
(1 + d)T = 1 + Td + T - — + — 1 — + • • • (6.32)
For d small, (1 +d)T can be approximated by
(1 + d)T » 1 + Td (6.33)
Combining Equations (6.31) and (6.33) results in
W + P = V(1 + Td) = V + VTd
or
V -W = P- VTd (6.34)
Comparing Equations (6.30) and (6.34) one finds that
v — P — VTd (6.35)
The profit P obtained through the mining of Qm in time T is given as before by
P = {s — r)Qr — cQc — mQm —fT (6.36)
Combining Equations (6.35) and (6.36) yields
v = (s - r)Qr - cOc - mQm - T(f + Vd) (6.37)
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 563
One would now like to schedule the mining in such a way that the decline in remaining
present value takes place as rapidly as possible. This is because later profits get discounted
more than those captured earlier. In examining Equation (6.37), this means that v should
be maximized. As in the previous section one first takes the derivative of v with respect to
grade. Setting the derivative equal to zero, one can solve for the appropriate cutoff grades
subject to mining, concentrating and refining constraints.
Step 1. Deteimination of the economic cutoff grades - one operation constraining the total
capacity
(a) Calculate cutoff grade assuming that the mining rate is the governing constraint.
The time Tm is given by
ryrmgedcSRQ -9m M
Substituting this into Equation (6.37) yields
f+Vd vm = ( s - r)Qr - cQc - m + "
M Q,r
Differentiating (6.39) with respect to grade g gives
dv , d Qr
dg dg
d 2 c
dg yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA772 +
dQrr f + Vd
M J dg
However the quantity mined Qm does not depend upon the grade:
d Qn
dg = 0
Hence
dvm d Qr
— = (J - 7-) — dg dg
d2c
' d g
(6.38)
(6.39)
(6.40)
(6.41)
(6.42)
The relationship between the quantities refined Qr and those sent for concentration Oc is
Qr = Qcgcy (6.43)
where gc is the average grade of ore sent for concentration and y is the recovery in
concentration.
Thus
d2 r _ d2c -r~ = gcy-j-dg dg
Substituting Equation (6.44) into (6.42) yields
dv,„ r -, dOc — = [(i - r)gcy - cj —— dg L dg
The average grade gc is defined as the mining cutoff (breakeven) grade g„, when
dg
Setting Equation (6.45) equal to zero and solving for gc = gm one finds that
c gm — (s - r)y
(6.44)
(6.45)
(6.46)
(6.47)
564zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
Substituting the values from the example yields
$2 gm = = 0.10 lbs/ton S m ( $ 2 5 - $ 5 ) 1 . 0 1
(b) Calculate cutoff grade assuming that the concentrating rate is the governing constraint.
If the concentrator throughput rate is the limiting factor then the time T is controlled by the
concentrator.
Oc T = = £ (6.48)
where Qc is the total number of tons which will be sent to the concentrator, and C is the
tons/year capacity.
Equation (6.37) becomes
vc = (yxutsronljiebXVUTSONLKJIHFECAJ - r)Or - cQc - rnQm - ( / + dV)^ (6.49)
c+f + dV vc = ( s - r)Or
J— Qc-mQm-rn (6.50)
Since the total amount of material Om is fixed,
rnQm — const
Thus the cutoff grade affects only Qr and Qc.
Substituting as before
Qr = Qc'gcy
one finds that
f + dV
C
To make vc as large as possible the term
Vr = (s - r)gcy - c + Qc-mQm (6.51)
C
should be as large as possible. At breakeven (the cutoff grade), the term is zero. Thus
(s - r)gcy - c +
c + f+dV
gc = ' ; (6.52) y(s - r)
(c) Calculate cutoff grade assuming the refitting rate is the governing constraint. If the
refinery output is the limiting factor then the time T is controlled by the refinery,
O r
T = y (6.53)
where Qr is output of the refinery and R is the refining/sales capacity per year.
Substituting into Equation (6.37) yields yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA
v r = (J - r)Qr - cQc - mQm - (/ + dV) % (6.54) K
( f + dV\ v r = i s - r - 1 Or - cQc - rnQm (6.55)
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 565
As before
Qr = grl'Qc
Thus
/ f + dV\ vr = I s - r — I gryQc - cQc - mQm
The total amount of material in the pit is fixed, therefore
mQm = const
Maximizing the expression for Vr one finds that
f + dV\ s~r — I gryQc = cQc
Solving yields
^ = 7 j ^ y - (6.56)
In summary, this first type of cutoff grade determination is based upon finding the grade for
which the net increase in overall present value is zero. The expressions are as in formulas
(6.47), (6.52) and (6.56):
8m yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA(5 - r)y
gc =
ctreUSODC + f-±f
y(s - r)
c gr =
As can be seen, the expressions for gc and gr contain the unknown value of V.
Step 2. Determination of the economic cutoff grade by balancing the operations
This step is exactly the same as that discussed in the previous section. Hence only the results
will be presented here.
gmo = 0.50
gar = 0.60
gmr = 0.456
Step 3. Deteimining the optimum of the six cutoff grades
There are six possible cutoff grades. Three are based simply upon the grade distribution of
the mined material and capacities.
gmc = 0.50
gcr = 0.60
gmr = 0.456
566zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit inine planning and design-. Fundamentals
The other three are based upon capacities and cost/price. Substituting in the known values
one finds that
gm — 0.10
_ I f + dV O , 300 +0.15V = c _ 50
8c y(s-r) 1 . 0 ( 2 5 - 5 )
_ 8 +0 .003V
~~ 2 0
( 2 5 - 5 -300 + 0.15y
40 ) 1.0 12.5 — 0.003757
Of these, two of the limiting economic cut-off grades are not known initially since they
depend upon knowing the overall present value. This in turn depends upon the cutoff grade.
Since the unknown V appears in the equations an iterative process must be used.
An optimum grade will be determined for each of the three pairs of operations. This will
be followed by finding the optimum of the three final candidates. For the mine and the
concentrator considered as a pair, there are three possible candidates for the optimum cutoff
grade Gmc. These are gm, gc, and gnw. The following rules are used to select Gmc.
Gmc =
gm if gmc — gm
gc if gmc > gc
gmc otherwise
This simple sorting algorithm yields the middle value. Treating the concentrator and refinery
as a pair, the optimum Grc is found from
gr
gc
grc
if grc < gr
if grc > gc
otherwise
Finally the optimum cutoff grade Gmr when the mine and refinery are treated as a pair is
gm
Gmr =
if gmr < gm
if gmr > gr
otherwise
As the first step in the iteration process, it will be assumed that V = 0.
Applying these rules to the example values
g m = 0 . 1 0
gc = 0.40
gr = 0.16
gmc = 0.50
grc = 0.60
gmr = 0.456
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 567
Table 6.25. Concentrator product as a function of concentrator cutoff grade with mine operating at
capacity.
Concentrator Concentrator Average feed Concentrator/refinery cutoff feed grade product (lbs/ton) (tons) (lbs/ton) (lbs)
0.0 100 0.50 50 0.1 90 0.55 49.5 0.2 80 0.60 48
0.3 70 0.65 45.5 0.4 60 0.70 42 0.5 50 0.75 37.5
0.6 40 0.80 32
0.7 30 0.85 25.5
0.8 20 0.90 18
0.9 10 0.95 9.5
1.0 0 1.00 0
one finds that
Gmc = 0.40
Grc = 0.40
Gmr = 0.16
The overall optimum cutoff grade G is the middle value of Gmc, Gmr, and Grc.
G — middle value (Gmc, Gmr, Grc)
In this case
G = 0.40
Step 4. Calculation of quantities
The next step in the procedure is to determine the maximum quantities Qm, Qc and Qr
which could be produced and not violate the capacities. Assume that 100 tons are mined
(Qm — 100). The grade distribution of this material is as shown in Table 6.24.
From Table 6.25, a cutoff grade of 0.40 would yield an average feed grade of 0.70. Each of
the capacities must, however, be considered. A mining capacity (Qm = M) of 100 tons would
mean 60 tons to the concentrator and 42 product units. Both the concentrator and refinery
capacities are exceeded. In meeting the concentrator capacity (Qc = 50), the required mining
and refinery capacities are:
5 Qm = - X 100 = 83.3
6
Qr = 50 x 0.70 = 35
These are both less than the maximum values. In meeting the refinery capacity of Qr = 40,
the required concentrating and mining capacities are:
40 Qc = — = 57.1
0.7
57.1 Qm = — 1 0 0 = 95.2
568zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit inine planning and design-. Fundamentals
Thus the concentrating capacity is violated. The result is that the concentrator is the
bottleneck.
In the further calculations
Qm = 83.3
Qc = 50
Qr = 35
The profit from time period T is expressed as
P = (s — r)Qr — cQc — mQm —fT
For T = 1 year one finds that
P = (25 - 5) x 35 - 2 x 50 - 1 x 83.3 - 300 x 1 = $216.7
Since the total amount of material to be mined from the pit is Q — 1000 units, the number
n of years required is
1000 n = = 12 years
83.3
The present value V corresponding to 12 equally spaced payments of P — $216.7 using an
interest rate of 15 percent is
216.7[1.1512 — 1]
0 .15(1.15)" = $ 1 1 7 4 - 6
This value of V becomes the second approximation of V (the first was V = 0) for use in the
formulas to calculate gc and gr.
c | f+dV 2 | 300+0.15x1174.6
c ' 50 = 0.576 oc
y(s - r) 2 5 - 5
c 2 _ S r = 7 7 = on 300+0.15x1174.6 =
S ~ ryrmgedcSRQ -R-y 40
The new six choices become
gm = 0.10
gc = 0.576
= 0.247
gmc = 0.50
grc =zywvutsrqponmlkjihgfedcbaZYXVUTSRQPONMLKJIHFEDCBA 0.60
gmr = 0.456
Applying the rules to select the overall pair optimum yields
Gmc = 0.50
Grc --- 0.576
Gmr = 0.247
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 569
The overall optimum is the middle value
G = 0.50
Returning to the grade distribution Table 6.24 one finds that the average grade is 0.75. If the
mining rate Qm = 100, then Oc = 50 and Or = 37.5. Both the mine and the concentrator are
at their rated capacities.
The profit in a given year is
P ~ (s — r)Qr — cQc — mQm —fT
= 20 x 37.5 - 2 x 50 - 1 x 100 - 300
= $250
The number of years is
Q 1000 n = —— = — — = 10 years
Qm 100
The present value becomes
1.1510 — 1 V = 2 5 ° 0 . 1 5 x 1 . 1 5 - 0 - $ 1 2 5 4 - 7
This becomes the third estimate for V to be used in calculating gc and gr.
2 300+0.15x1254.7
= ¿ f = 0.588 zywvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLJIHGFEDCBA
2
2 0 - ™ 8 r 300+0.15x1254.7 — 0 . 2 5 7 40
The six possible values are:
Em = 0.10
gc = 0.588
g r = 0.257
gmc = 0.50
grc =zywvutsrqponmlkjihgfedcbaZYXVUTSRQPONMLKJIHFEDCBA 0.60
gmr = 0.456
The optimum pairs are:
Gmc = 0.50
Grc = 0-588
Gmr - 0.257
and the overall optimum (G = 0.50) is the same as found with the previous estimate. Hence
in year 1
Optimum cutoff grade = 0.501bs/ton
Quantity mined = 100 tons
Quantity concentrated = 50 tons
Quantity refined = 37.5 lbs
Profit = $250
570zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit inine planning and design-. Fundamentals
Table 6.26. Reserve distribution at the end of year 1.
Grade (lbs/ton) Quantity (tons)
0.0-0.1 90
0.1-0.2 90
0.2-0.3 90
0.3-0.4 90
0.4-0.5 90
0.5-0.6 90
0.6-0.7 90
0.7-0.8 90
0.8-0.9 90
0.9-1.0 90
Total = 900
Table 6.27. Reserve distribution at the start of year 8.
Grade (lbs/ton) Quantity (tons)
0.0-0.1 30
0.1-0.2 30
0.2-0.3 30
0.3-0.4 30
0.4-0.5 30
0.5-0.6 30
0.6-0.7 30
0.7-0.8 30
0.8-0.9 30
0.9-1.0 30
Total = 300
The reserves must now be adjusted to those given in Table 6.25 and the process is repeated
assuming V = 0, calculating gc and gr, etc.
Through year 7, it will be found that the optimum cutoff grade remains at 0.50 with the
quantities mined, concentrated and refined being 100,50 and 37.5, respectively. The annual
profit is $250. The reserves going into year 8 are those given in Table 6.27. The balancing
grades remain at
gmc = 0.50 zywvutsrqponmlkjihgfedcbaZYXVUTSRQPONMLKJIHFEDCBA
= 0.60
gmr = 0.456
The first approximation for the economic cutoff grades (V = 0) is
gm
8c
gr
= 0.10
= 0.40
= 0.16
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 571
The optimum values of the pairs are
Gmc = GW = 0.40
Gmr — 0.16
The overall optimum is 0.40, and the quantities are
Qm = 83.3
Qc = 50
Qr = 35
The profit is $216.7 as before. The number of years becomes 3 0 0
n = 8 3 3 = y e a r s
The present value V becomes
f l 15-)3-6 - 1 V = 216 .7—— — = 571.2
0.15(1.15)3-6
Substituting this into the formulas for gc and gr yields
gc = 0.486
gr = 0.193
Combining them with the others
gm = 0.10
gmc = 0.50
grc =zywvutsrqponmlkjihgfedcbaZYXVUTSRQPONMLKJIHFEDCBA 0.60
gmr = 0.456
yields
Gmc = 0.486
Grc = 0.486
Gmr = 0.193
The overall optimum cutoff is G — 0.486 and the average grade above cutoff drops to 0.743:
Tons Grade Tons x Grade
4.2 0.493 2.07
30 0.55 16.50
30 0.65 19.50
30 0.75 22.50
30 0.85 25.50
30 0.95 28.50
Total =154 .2 Avg=: 0.743 Sum =114.57
There are 154.2 ore tons out of the 300 tons remaining to be mined. Since the concentrator
capacity is 50 tons/year, the mine life would be
154.2
50 = 3.08 years
572zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design: Fundamentals
Table 6.28. Reserve distribution at the start of year 9.
Grade (lbs/ton) Quantity (tons)
0.0-0.1 20.27
0.1-0.2 20.27
0.2-0.3 20.27
0.3-0.4 20.27
0.4-0.5 20.27
0.5-0.6 20.27
0.6-0.7 20.27
0.7-0.8 20.27
0.8-0.9 20.27
0.9-1.0 20.27
Total = 202.27
The yearly mine production becomes
300 Om - = 97.3
3.08
and Qr — 37.15. Calculating the profit one finds that
P = 20 x 37.15 - 2 x 50 - 1 x 97.3 - 300 = 245.7
The corresponding present value is
(1 15)3 08 - 1 V = 245.7— r— = $572.96
0.15 x (1.15)3 08
Repeating the process with this new estimate of V yields
gc = 0.486
g r = 0.193
These are the same as before. Hence the values for year 8 are
G = 0.486
2m = 9 7
Qc — 50
a - = 37.1
Profit = $245.7
The reserves are those given in Table 6.28. In year 9, the initial values for a cutoff of 0.4 are
Qm = 83.3
Qc = 50
Qr — 35
The profit would be $216.7.
Based upon this mining rate, the reserves would last
n = ^ y = 2.43 years
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 573
and the present value is
(1 15)2-43 - 1 V = 216.7——— — = $416.0
0.15 x (1.15)2-43
Recomputing gr and gc we obtain
gc = 0.462
gr = 0.183
The other possible values are:
gmyxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA — 0.10
gmc = 0.50
grczywvutsrqponmlkjihgfedcbaZYXVUTSRQPONMLKJIHFEDCBA = 0.60
gmr = 0.456
The optimum pair values are:
Gmc - 0.462
G^ = 0.462
Gmr = 0.183
The middle value of these is
G = 0.462
Examining the reserve distribution suggests that there are 109.05 tons out of the total
202.7 tons which are above cutoff. The average grade of this remaining ore is 0.731:
Tons Grade Tons x Grade
7.70 0.48 3.70
20.27 0.55 11.15
20.27 0.65 13.18
20.27 0.75 15.20
20.27 0.85 17.23
20.27 0.95 19.26
Total =109.05 Avg = 0.731 Sum = 79.72
Since the maximum concentrating rate is 50 tons/year the life is
109.05 „ i n = 2.18 years
50 J
The amount of product is
Qr = 50 x 0.731 = 36.55
202.7 _ Qm = = 93 * 2.18
The profit becomes
P = 20 x 36.55 - 2 x 50 - 1 x 93 - 300 = $238
574zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit inine planning and design-. Fundamentals
Table 6.29. Reserve distribution at the start of year 10.
Grade (lbs/ton) Quantity (tons)
0.0-0.1 11
0.1-0.2 11
0.2-0.3 11
0.3-0.4 11
0.4-0.5 11
0.5-0.6 11
0.6-0.7 11
0.7-0.8 11
0.8-0.9 11
0.9-1.0 11
Total =110
The present value is
(1.15)2'18 — 1 „ V = 2 3 8 0 . 1 5 x l . l 5 ^ X $ 4 1 7
Iterating again does not change the values. The new distribution is shown in Table 6.29.
In year 10, the initial values for a cutoff of 0.4 yields:
Qm = 83.3
Qc = 50
Q r = 35
Profit = $216.7
Based upon this mining rate, the reserves would last yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA
1 1 0 1 OO n = = 1.32 years 83.3
The present value is
(1 15)1-32 - 1 V = 2 1 6 - 7 0 T 5 x ( 1 . 1 5 ) . - 3 2 - $ 2 4 3 - 4
Calculating gc and gr and using this approximation for V yields
gc = 0.437
gr = 0.172
The other possible values are:
g m = 0 . 1 0
gmc = 0.50
grc - 0.60
gmr = 0.456
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 575
The optimum pair values are:
Gmc = 0.437
Grc = 0.437
Gmr = 0.172
The optimum value is G — 0.437. Examining the reserve distribution suggests that there are
62 tons of the 110 tons remaining which are above this cutoff.
Tons
7
Grade
0.469
0.55
0.65
0.75
0.85
0.95
Tons x Grade
3.28
6.05
7.15
8.25
9.35
10.45
Total = 62 Avg = 0.718 Sum = 44.53
The average grade is 0.718. The number of years would be
62 72 = — = 1.24 years
50 '
The rate of mining and refining would be
110 Qr, 89
1.24
Qr = 35.9
and the profit would become
P = 20 x 35.9 - 2 x 50 - 1 x 89 - 300
= $229
The present value is
(1.15)124 — 1 y = $229- = $243
0.15 x (1.15)1-24
Further iteration yields no change. In year 11, the grade distribution is shown in Table 6.30.
The initial values (V = 0), yield a cutoff of 0.4 and
Qm =zywvutsrqponmlkjihgfedcbaZYXVUTSRQPONMLKJIHFEDCBA 21
Qc = 12.6
QroUTSONMLIDCA = 8.8
The time would be the largest of
Tm = — = 0.21 yxwvutsrponmlkjihgfedcbaYWVUTSRQPONMLKJIHGFEDCBA100
12.6 T = = 0 25
c 50
576zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design: Fundamentals
Table 6.30. Reserve distribution at the start of year 11.
Grade (lbs/ton) Quantity (tons)
0.0-0.1 2.1
0.1-0.2 2.1
0.2-0.3 2.1
0.3-0.4 2.1
0.4-0.5 2.1
0.5-0.6 2.1
0.6-0.7 2.1
0.7-0.8 2.1
0.8-0.9 2.1
0.9-1.0 2.1
Total = 21.0
which is again controlled by the concentrator. The profit is
300 P = 20 x 8.8 - 12.6 x 2 - 21 x 1
4
= $54.8
The present value is
d 15)0-25 - 1 V = $ 5 4 - 8 0 C T 5 ^ = $ 1 2 - 5
Solving for gc and gr yields
= 0.402
gr = 0.161
Combining with the others
gmzywvutsrqponmlkjihgfedcbaZYXVUTSRQPONMLKJIHFEDCBA = 0.16
gmc = 0.50
g t c = 0.60
gmr = 0.456
one finds that
Gmc = 0.402
Grc = 0.402
Gmr = 0 . 1 6 1
The cutoff grade is
G = 0.402
The distribution is only slightly changed and farther iteration is not warranted.
Production planningzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 577
Table 6.31. The production schedule determined by the first pass.
Year Optimum Quantity Quantity Quantity Profit Net present
cutoff grade mined concentrated refined zywvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLJIHGFEDCBA($) value
(lbs/ton) (tons) (tons) (lbs)
1 0.5 100 50 37.5 250 1255 2 0.5 100 50 37.5 250 1193 3 0.5 100 50 37.5 250 1122 4 0.5 100 50 37.5 250 1040
5 0.5 100 50 37.5 250 946
6 0.5 100 50 37.5 250 838 7 0.5 100 50 37.5 250 714
8 0.486 97 50 37.1 245.7 574
9 0.462 93 50 36.55 238 417
10 0.437 89 50 35.9 229 243
11 0.40 21 12.6 8.8 55 53
Table 6.32. The final schedule for the manual example.
Year Optimum Quantity Quantity Quantity Profit Net present
cutoff grade mined concentrated refined ($) value
(lbs/ton) (tons) (tons) (lbs)
1 0.50 100.0 50.0 37.5 250.0 1257.8
2 0.50 100.0 50.0 37.5 250.0 1196.5
3 0.50 100.0 50.0 37.5 250.0 1126.0
4 0.50 100.0 50.0 37.5 250.0 1044.9
5 0.50 100.0 50.0 37.5 250.0 951.7
6 0.50 100.0 50.0 37.5 250.0 844.5
7 0.50 100.0 50.0 37.5 250.0 721.1
8 0.49 97.2 50.0 37.1 245.6 579.3
9 0.46 93.0 50.0 36.6 238.2 420.6
10 0.44 88.7 50.0 35.9 229.5 245.5
11 0.41 21.0 12.5 8.8 54.7 52.8
The net present value is calculated using the yearly profits.
_ (1.15)7 - 1 245.7 238 229 55 NPV — 250— 1 1 1 1
0.15 x (1.15)7 (1.15)8 (1.15)9 ^ ( L 1 5 ) i o ^ (i.i5)io.25
= 1040 + 80.32 + 67.70 + 56.61 + 13.12
= $ 1 2 5 8
Step 5. Repetition of the iteration process
In Table 6.31, the present value column reflects the current approximation to V as each years
cutoff grade was calculated. The present value of $1258 obtained using the yearly profits
should be the same as that shown in the table for year 1. Since the values are not the same
($1258 versus $1255), the process is repeated from the beginning using V = $1258 as the
initial estimate for V. Using a computer this iterative procedure is completed in fractions
of a second. The final results are shown in Table 6.32. The NPV is slightly higher than the
$1255 which would have been obtained by maintaining a constant cutoff grade of 0.5.
578zyxwvutsrqponmlkjihgfedcbaYXWVUTSRQPONMLKJIHGFEDCBA Open pit mine planning and design'. Fundamentals
In summary:
- Initially the mine and the concentrator are in balance, both operating at capacity. In the
last few years, the concentrator is the limiter.
- The cutoff grade begins at 0.50 lbs/ton and drops to 0.41 lbs/ton at the end of mine life.
- Mine life is slightly more than 10 years.
- Total copper produced = 380.9 lbs.
-Tota l profits = $2518.
- Net present value = $1257.80.
This net present value should be compared to that of $1174.60 obtained with the fixed
cutoff grade.
6.8 MATERIAL DESTINATION CONSIDERATIONS
6.8.1 Introduction
The term 'cutoff grade' is a rather poorly defined term in the mining literature. A major
reason for this is that there are many different cutoff grades. Furthermore the values change
with time, mining progress, etc. A cutoff grade is simply a grade used to assign a destination
label to a parcel of material.
The destination can change. During the evaluation of final pit limits, the destinations to
be assigned are:
- to the surface, and
- left in the ground.
Once the destination 'to the surface' has been assigned, then the destination label 'where
on the surface' must be assigned as well. In the distant past there were really only two
surface destinations:
- to the mill, and
- to the waste dump.
A grade was used to assign the location. The distinction between destinations was called
the mill cutoff grade. In more recent times, the potential future value of material carrying
values has been recognized. Hence the lean (low grade) ore dump has become a destination.
Thus the 3 destinations require 2 distinguishing grades:
Destination
- to the mill
- to the lean ore dump
- to the waste dump
Today there are many more possible destinations as our ability to handle and treat materials
have improved. Leach dumps/leach pads are a common destination. An active stockpile is
a less common destination.
This section will deal with alternate destinations to the mill and waste dump. These will
be discussed with respect to cutoff grade. However the reader should remember that these
simply are a way of assigning material destinations.
Assignment
mill cutoff grade
waste cutoff grade