ical

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concept of cut-o 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-o grade, in multiple metal deposits an innite number

any grade that, for any specic reason, is used to sep- deposit. Whittle and Wharton added the idea of using

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*

Minerals Engineering 16 (2Corresponding author. Tel.: +98-21-64542929; fax: +98-21-

6413969.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-o grade

criteria to dene 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-o grade of ore to maximize the net present

value (NPV). The determination of the optimum cut-ograde of single metal deposit can be very complex even

when price and cost are assumed constant, but it in-

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-o grade (Lane, 1988), in

multiple metal deposits an innite number points arepossible candidates for the optimum cut-o grades and

the objective function evaluation of these innite pointsof points are possible candidates for the optimum cut-o grades. The objective function evaluation of these innite points is im-

possible. In this paper, the equivalent grade factor is used to nd optimum cut-o grade of multiple metal deposits. First, the

objective function is dened for multiple metal deposits and then objective function is converted to one variable function by using

equivalent factors. The optimum equivalent cut-o grade of main metal can be found by the optimization techniques such as the

Lane algorithm or elimination methods. At nal step, the optimum cut-o grades will be determined by interpolation of grade-

tonnage distribution of deposit.

2003 Elsevier Ltd. All rights reserved.

Keywords: Modelling; Optimization

1. Introduction

One of the most critical parameters in mining oper-

ation is cut-o grade. Taylor presents one of the best

denitions of cut-o grade. He dened cut-o grade as

volves the costs and capacities of the several stages of

the mining operations, the waste/ore ratios, average

grades of dierent increments of the ore body and so on.

Lane (1964, 1988) has developed a comprehensive

theory of cut-o grade calculation for a single metalTechn

Using equivalent grade factors tof multiple

M. Osanlo

Department of Mining, Metallurgy and Petroleum Enginee

424 Hafez Ave., PO Box 1

Received 11 September

Abstract

One of the most important aspects of mine design is to deter

the cut-o is directed to dierent destinations. Optimization o

studies. The most commonly criteria used in cut-o grade opE-mail address: mosanloo@hotmail.com (M. Osanloo).

0892-6875/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0892-6875(03)00163-8Note

nd the optimum cut-o gradestal deposits

M. Ataei

Amirkabir University of Technology, Tehran Polytechnic,

4413, Tehran, 15914 Iran

accepted 14 April 2003

the optimum cut-o grades. Material grading above and below

-o grade is now an accepted principle for open pit planning

ation is to maximize net present value. Lane formulated the

003) 771776This article is also available online at:

www.elsevier.com/locate/minengsmelting (Annels, 1991). In this method, it is possible

sent to the concentrator, Qr1: the amount of product 1

C

772 M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771776actually produced over this production period, Qr2: theamount of product 2 actually produced over this pro-duction period.

If d is discount rate, the dierence m between thepresent values of the remaining reserves at times t 0and t T is (Hustrulid and Kuchta, 1995)m P VTd 2where V is the present values at time t 0. SubstitutingEq. (1) into Eq. (2) yields

m s1 r1Qr1 s2 r2Qr2 mQm cQc f VdT 3

The quantities of rened metals Qr1 and Qr2 are re-lated to that send from the mine to concentrator (Qc),thereforethat 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-o approach,

dollar value cut-o approach (Annels, 1991; Barid and

Satchwell, 2001).

All of these methods are associated with some aws: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-o grades cal-

culated by these methods are not optimum.

This paper describes the use of equivalent grade fac-

tors to optimum the cut-o 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 rening

stage, where the concentrate is smelted and rened to a

nal 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 prot is determined by using the following equation:

P s1 r1Qr1 s2 r2Qr2 mQm cQc fT 1where P : prot ($), m: mining cost ($/ton of materialmoved), c: concentrating cost ($/ton of material con-centrated), r1: renery cost ($/unit of product 1), r2: re-nery cost ($/unit of product 2), f : xed cost ($), s1:selling price ($/unit of product 1), s2: selling price ($/unitof product 2), T : the length of the production period,Qm: quantity of material to be mined, Qc: quantity of oreQr1 gg1y1Qc 4Qr2 gg2y2Qc 5where 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 ofmetal 2 from the ore.

Substituting Eqs. (4) and (5) into Eq. (3) yields

m s1 r1gg1y1 s2 r2gg2y2 cQc mQm f VdT 6

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

prots 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 1and 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 befunction of grade of metal 1 and Eq. (6) yields

m s1

r1y1 gg1

s2 r2y2s1 r1y1 gg2 c

Qc

mQm f VdT 7Equivalent factor is equal to

Feq s2 r2y2s1 r1y1 8

Substituting Eq. (8) into Eq. (7) yields

m s1 r1y1gg1 Feqgg2 cQc mQm f VdT9

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 gg1 Feqgg2 10Substituting Eq. (10) into Eq. (9) yields

m s1 r1y1ggeq cQc mQm f VdT 11Eq. (11) is the fundamental formula for calculation of

optimum cut-o 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 thenthe time T is given by

T QmM

12 If the concentrating capacity (C) is the limiting factor

then the time T is controlled by the concentrator

T Qc 13

o grades. These methods require only objective func-

tion evaluations and do not use the derivative of the

function to nd the optimum point (Rardin, 1998). In

these methods at rst step, the uncertainty space of theproblem 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-

specied positive value e. Where e is desirable accuracyto determine the optimum cut-o grades. Ratio of re-

mained length after elimination process to initial lengthin 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 rst one),

M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771776 7733. Determination of optimum cut-o 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 nd

optimum cut-o grades.

According to Lane algorithm, there are three limiting

cut-o grades and three balancing cut-o grades. If only

the capacity of one operation is limited factor then thebreak-even cut-o grade for that stage will be the opti-

mum cut-o grade. To nd the grades that maximize the

NPV under dierent constraints, one rst 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-o grades.

When mining operations are constrained by more

than one capacity, the optimum cut-o grade calculatedby conventional method may not necessarily be a break-

even cut-o grade. In such a case, the balancing cut-o

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-o

grades are independent of economics and being deter-

mined by using the grade distribution and the capacitiesof each of the dierent system. Based on these consid- If the renery output of main metal (Rm) is the limit-ing factor then the time T is controlled by the reningof main metal

T Qr1Rm

gg1y1QcRm

14

Substituting Eqs. (12)(14) into Eq. (11) yields the fol-

lowing equations:

mm s1 r1ggeqy1 cQc m

f VdM

Qm 15

mc s1

r1ggeqy1 c

f VdC

Qc mQm 16

mr s1

r1 f VdRm

ggeqy1 c

Qc mQm 17

Now for any pair of cut-o grades, it is possible to

calculate the corresponding mm, mc and mr. The control-ling capacity is always the one corresponding to the leastof these three equations. Therefore

max me maxminmm; mc; mr 18In Eqs. (15)(17), V is unknown value because it

depends upon the cut-o grade. Since the unknown Vappears in the equation thus iterative process must be

used.erations, now six cut-o grades are candidate for overall

optimum cut o grade. The optimum cut-o grade will

be one of the six cut-o grades consisting of the three

limiting economic cut-o grades and the three balancing

cut-o grades. Fig. 1 shows six candidate cut-o grades.

Lane has presented a graphical method to determine

overall optimum cut-o of ore among of these six cut-ogrades.

The optimum cut-o 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-o grade for the pair is the breakeven cut-

o grade for the limiting stage. The overall optimum

cut-o grade is the middle value of the optimum cut-os

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 nd optimum cut-

Fig. 1. mm, mc mr and me curves and six candidate cut-o grades.

rst step, assume L;U 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 L U L 0:382 19g2 L U L 0:618 20In next step, the objective function of points g1 and g2will 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 nal pit

limits of Cu/Mo deposit has been superimposed on a

mineral inventory. The pit outline contains 14.4 million

tons of materials. The gradetonnage distribution and

average grades of ore for each metal are shown in Tables

13 and associated costs, prices, capacities, quantities

and recoveries are given in Table 4.According to Eq. (8), equivalent factor for this situ-

Fig. 2. Flowchart for nding the optimum cut-o grade by the golden

section search method.

774 M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771776the objective function will be evaluated at one new point

(Chong and Zak, 1996; Rao, 1996).

Fig. 2 shows owchart to calculate the optimum cut-o grade of ore by the golden section search method. InTable 1

Gradetonnage distribution of copper and molybdenum

Copper (%) Molybdenum (%)

00.025 0.0250.05 0

00.1 1,320,000 900,000 2

0.10.2 360,000 300,000 2

0.20.3 735,000 525,000 3

0.30.4 1,110,000 570,000 3

0.40.5 525,000 255,000

0.50.6 510,000 300,000 210,000 105,000 30,000

0.60.7 375,000 270,000 210,000 90,000 90,000

>0.7 645,000 690,000 5

Table 2

Average grade of copper of dierent copper and molybdenum intervals

Copper (%) Molybdenum (%)

00.025 0.0250.05 0

00.1 0.02 0.03 0

0.10.2 0.12 0.17 0

0.20.3 0.25 0.27 0

0.30.4 0.33 0.32 0

0.40.5 0.44 0.47 0

0.50.6 0.53 0.55 0

0.60.7 0.67 0.63 0

>0.7 0.98 1.04 170,000 495,000 360,000

.050.075 0.0750.1 >0.1

.02 0.03 0.05

.16 0.19 0.14

.25 0.22 0.26

.35 0.34 0.37

.45 0.48 0.46

.57 0.54 0.55

.65 0.64 0.66

.02 1.09 1.01ation is

Feq 6797:15 190 0:81674:5 63 0:82 4

1

.050.075 0.0750.1 >0.1

85,000 315,000 510,000

40,000 135,000 60,000

00,000 210,000 30,000

75,000 135,000 30,000

75,000 60,000 90,000

Table 3

Average grade of molybdenum of dierent copper and molybdenum intervals

Copper (%) Molybdenum (%)

00.025 0.0250.05 0.050.075 0.0750.1 >0.1

00.1 0.002 0.026 0.052 0.076 0.113

0.10.2 0.017 0.031 0.06 0.085 0.114

0.20.3 0.011 0.028 0.066 0.091 0.137

0.30.4 0.031 0.042 0.058 0.094 0.138

0.40.5 0.006 0.035 0.054 0.091 0.119

0.50.6 0.012 0.039 0.07 0.082 0.152

0.60.7 0.014 0.029 0.062 0.085 0.12

>0.7 0.009 0.038 0.063 0.086 0.128

M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771776 775Table 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

Rening capacity (copper) Tons per year 5000

Rening capacity (molybde-

num)

Tons per year 1000

Mining cost Dollars per ton 1.06

Milling cost Dollars per ton 3.52

Rening cost (copper) Dollars per ton 63

Rening cost (molybdenum) Dollars per ton 190

Fixed costs Dollars per 790,000Now using the equivalent factor and average grade of

each metal, the equivalent copper grade of dierent

copper grade is calculated (Table 5).

Converting molybdenum grade into copper grade, the

gradetonnage distribution of two metal deposits is

converted into one-dimensional grade tonnage distri-

bution and cut-o grade optimization method of single

metal deposit such as Lane method or eliminationmethod was used to calculate the optimum cut-o

grades in year by year. Then the gradetonnage curve of

deposit is adjusted for each year of mine life. To do this,

tonnage of ore in rst year of mine life from the grade

of copper: Tonnage of ore 10779464.65, Tonnage ofwaste 3620536.45,Waste: ore 0.3358, Average equiv-

Table 5

Equivalent copper grade of dierent copper grade

Copper grade (%) Average grade

Copper (%) Molybdenum (

00.1 0.0282 0.0368

0.10.2 0.1522 0.044

0.20.3 0.2525 0.0364

0.30.4 0.332 0.0437

0.40.5 0.4521 0.0266

0.50.6 0.5442 0.0451

0.60.7 0.652 0.043

0.72 1.0269 0.0567

year

Price (copper) Dollars per ton 1674.5

Price (molybdenum) Dollars per ton 6797.15

Recovery (copper) % 82

Recovery (molybdenum) % 80

Discount rate % 20alent grade 0.6238%.If concentrator capacity is controlling factor, then the

mine life is equal to

10779463:65

750; 000 14:37 yeardistribution intervals above optimum cut-o grades and

tonnage of waste in rst year of mine life from the grade

distribution intervals below optimum cut-o grades was

subtracted. These calculations are repeated until the end

of mine life. Table 6 shows the optimum cut-o grades

of ore for dierent years of mine life.

5. Justication of the proposed method

To justify proposed method, the NPV of break-even

equivalent cut-o grade was calculated and compared

with NPV calculated by proposed method. By deni-

tion, the break-even equivalent cut-o grade is grade

that revenue is equal to costs. So

1674:5 63 geq100

0:82 1:06 3:52) geq 0:3466

Thus, break-even equivalent cut-o grade of copper is

0.3466%. Using interpolation technique and Table 5, thecopper grade is calculated to be 0.1263%. For this gradeYearly revenue will be equal to

Equivalent

copper grade (%)Tonnage

%)

0.1754 3,330,000

0.3282 1,095,000

0.3981 1,800,000

0.5068 2,220,000

0.5585 945,000

0.7246 1,215,000

0.824 1,035,000

1.2537 3,760,000

6182964:363

Hustrulid, W., Kuchta, M., 1995. Open-pit Mine Planning and Design,

n)

776 M. Osanloo, M. Ataei / Minerals Engineering 16 (2003) 771776Moreover, yearly cost will be equal to

Yearly cost 750; 000 1 0:3358 1:06 750; 000 3:52

4492019:035Yearly cash ow and NPV of mining operation under

break-even equivalent cut-o grade found to be

6182964:363 4492019:035 1690945:325 $

NPV 1690945:3251 0:214:37 1

0:2 1 0:214:37 7839174:188

Therefore, NPV of mining operation under break-

even equivalent cut-o 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-o grade.

6. Conclusion

One of the important aspects of mining is decidingYearly revenue 1674:5 63 0:6238100

0:82 750; 000

Table 6

The optimum cut-o grades of dierent years of mine life

Year Copper cut-o grade (%) Qm (Ton) Qc (To

1 0.4617 2,010,300 750,000

2 0.3725 1,345,100 750,000

3 0.3645 1,601,700 750,000

4 0.3555 1,555,600 750,000

5 0.3455 1,507,300 750,000

6 0.3355 1,462,000 750,000

7 0.3255 1,419,200 750,000

8 0.2634 1,222,500 750,000

9 0.2464 1,181,500 750,000

10 0.2293 794,700 521,330what 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-o gradepolicy. Cut-o grades of multiple metal deposits are

evaluated by several methods such as NSR method,

critical level method, single grade cut-o approach,

dollar value cut-o 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 expressedto one variable function. Then optimization techniques

such as Lane algorithm or elimination methods must bevol. 1. A.A. Balkema, Rotterdam, Brookled (pp. 512544).

Lane, K.F., 1964. Choosing the optimum cut-o grade. Quarterly of

the Colorado School of Mines 59 (4), 811829.

Lane, K.F., 1988. The Economic Denition of Ore-cut-o Grades in

Theory and Practice. Mining Journal Books Limited, London.

p. 145.

Liimatainen, J., 1998. Valuation model and equivalence factors for

base metal ores. In: Singhal, J. (Ed.), Proceeding of Mine Planning

and Equipment Selection. A.A. Balkema, Rotterdam, Brookled,

pp. 317322.

Rao, S.S., 1996. Engineering optimization (Theory and Practice), third

ed. A Wiley-Interscience Publication, John Wiley and Sons Inc,use to nd optimum equivalent cut-o grade for main

metal (caused more revenue). Optimum cut-o grades

are determined by interpolation of gradestonnage dis-

tribution. A verication example is presenting for con-

rming 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 equivalentcut-o grade.

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Annels, A.E., 1991. Mineral Deposit EvaluationA Partial Approach.

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Qr1 (Ton) Prot ($) NPV ($)

4976.9 4,425,600 16,355,000

4546.2 4,013,100 15,201,000

4487.3 3,961,200 14,228,000

4421 3,899,900 13,112,000

4347.4 3,828,700 11,834,000

4273.8 3,754,300 10,373,000

4200.2 3,677,200 8,693,000

3850.5 3,286,600 6,754,000

3774.9 3,196,200 4,818,000

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77.

Using equivalent grade factors to find the optimum cut-off grades of multiple metal depositsIntroductionObjective functionDetermination of optimum cut-off gradesExampleJustification of the proposed methodConclusionReferences