Net present value maximization model for optimum cut-off grade policy of open pit mining operations

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1 Syopsis Net peset vlue mximiztio model fo optimum cut-off gde policy of ope pit miig opetios by M.W.A. Asd* d E. Topl The optimum cut-off gde policy mximizes the et peset vlue (NPV) of ope pit miig opetio subject to the miig, pocessig, d efiig cpcity costits. The tditiol ppoches to cut-off gde detemitio igoe the escltio of the ecoomic pmetes such s metl pice d opetig costs duig life of opetio, d cosequetly led to uelisticlly highe vlues of the objective fuctio. Futhe, the NPV of miig opetio declies due to the depletio of the vilble eseves, cusig declie i the optimum cut-off gde, i.e. highe cut-off gdes i the ely yes of opetio d lowe cut-off gdes duig the lte yes. Hece, low gde mteil mied i the elie yes my be stockpiled fo pocessig duig lte yes to offset the effect of escltig ecoomic pmetes o NPV. This ppe demosttes the combied impct of itoducig ecoomic pmetes, escltio d stockpilig optios ito the cut-off gde optimiztio model. The model pomises ehcemet i NPV s illustted i cse study icopotig pcticl spects of ope pit miig opetio. Keywods miig, modellig, cut-off gde, stockpilig, optimiztio. Itoductio Cut-off gde is the citeio tht discimites betwee oe d wste withi give miel deposit1,2. If mteil gde i the miel deposit is bove cut-off gde it is clssified s oe, d if mteil gde is below cut-off gde, it is clssified s wste. Oe, beig the ecoomiclly exploitble potio of the miel deposit, is set to the pocessig plt fo cushig, gidig, d cocettio of the metl cotet. The poduct of the pocessig plt is clled cocette, which is fed to the efiey fo poductio of efied metl. Hece, idel ope pit miig opetio cosists of thee stges i.e. mie, pocessig plt, d efiey3,4. Log-ge poductio plig of ope pit miig opetio is depedet upo sevel fctos; howeve, cut-off gde is the most sigifict spect, s it povides bsis fo the detemitio of the qutity of oe d wste i give peiod5. Evetully, the pofit ove time my be ehced oly by flow of high gde mteil to the pocessig plt. This sttegy suppots the objective fuctio d, depedig upo the gde-toge distibutio of the deposit, highe NPV my be elized duig elie yes to ecove the iitil ivestmet6,7. Howeve, s the deposit becomes depleted, the NPV s well s the cutoff gde declie; hece, cut-off gde policy d the poductio pl defied s esult of this policy dictte pheomel ifluece o the ovell ecoomics of the miig opetio8,9. The optimum cut-off gdes, which e dymic due to the decliig effect of NPV, ot oly deped o the metl pice d csh costs of miig, pocessig, d efiig stges, but lso tke ito ccout the limitig cpcities of these stges d gde-toge distibutio of the deposit. Theefoe, the techique tht detemies the optimum cut-off gde policy cosides the oppotuity cost of ot eceivig futue csh flows elie duig mie life, due to the limitig cpcities of y of miig, pocessig, o efiig stges10,11. Howeve, metl pice d opetig costs of miig, pocessig, d efiig chge duig mie life, d this hppes quite ofte, due to the loge life of most of the ope pit miig opetios. Igoig the effect of these chges i the ecoomic pmetes o the optimum cut-off gde policy would led to uelistic poductio pls12,13. Futhe, the decliig effect of NPV llows highe cut-off gdes i the ely yes of mie life d lowe cut-off gdes i the lte yes, * Miig d Mteils Egieeig, McGill Uivesity, Cd. Miig Egieeig d Mie Suveyig, Weste Austli School of Mies, Cuti Uivesity of Techology The Southe Afic Istitute of Miig d Metllugy, SA ISSN X/ Ppe eceived M. 2010; evised ppe eceived Feb T s c t i o P p e The Joul of The Southe Afic Istitute of Miig d Metllugy VOLUME 111 NOVEMBER

2 Net peset vlue mximiztio model fo optimum cut-off gde policy due to depletio of high gde mteil. Theefoe, depedig upo the existig cicumstces i ope pit miig opetio, the poductio pls my lso iclude the flexibility of povidig stockpiles of low gde oe mied i the elie yes to be pocessed lte s it becomes ecoomicl to do so. This ehces ot oly the life, but lso the NPV of miig opetio. The mgemet of stockpiles of low gde oe is possible usig the followig two optios14: 1. The stockpile is utilized pllel to the miig opetio. This mes tht mteil is set to the pocessig plt eithe fom mie o stockpile. This decisio is bsed o the ovell ecoomy/pofitbility of the opetio. 2. The stockpile is utilized fte the mie is exhusted. This simplifies the decisio-mkig, sice the stockpile cts s dditiol potio of the deposit, whee ll vilble mteil is ecoomicl. Howeve, the high gde mteil i the stockpile is scheduled to be utilized elie th the low gde mteil. I this study, the secod cse is chose owig to ese of opetio. Theefoe, keepig i view the pospect of cotibutio to the miig idusty, we popose extesio i the estblished Le s theoy of optimum cut-off gdes15,16. The poposed cut-off gde optimiztio model cosides ot oly dymic metl pice d cost escltio, but lso esults i the cetio of stockpile of low gde oe duig the mie life d its utiliztio s oe fte the exhustio of the deposit. Le s oigil theoy hs bee modified by Dgdele3,4, Dgdele d Kwht6, Bsceti9, Osloo d Atei11, Asd12,13,14, Dgdele d Asd17, Atei d Osloo18, Osloo et l.19, d Kig20, but these studies did ot ttempt to lyse the combied impct of ecoomic pmetes escltio d stockpilig o NPV. We implemet the itetive lgoithmic steps of the modified model i Visul C++ pogmmig lguge to develop ltetive cut-off gde policies i cse study of hypotheticl coppe deposit. The esults demostte the effect of the chge i ecoomic pmetes d the stockpilig optio o mie plig with icese/decese i NPV. The model Peequisites fo the pplictio of cut-off gde optimiztio model iclude the developmet of ultimte pit limit o pit extet d pushbck ( mgeble potio of the deposit iside the ultimte pit limit tht my be mied, pocessed, d efied i umbe of yes/peiods) desig, oe eseves i tems of miel gde d toge distibutio i ech pushbck, d miig, pocessig, d efiig stge cpcities, the opetig costs of these stges, d the cuet metl pice. The objective fuctio of cut-off gde optimiztio model is to mximize the NPV of the opetio subject to miig, pocessig, efiig, d stockpile cpcity costits, which my be epeseted mthemticlly s follows: [1] Subject to: Hee, [2] [3] [4] [5] [6] whee = peiod (ye) idicto, N = totl life of opetio (yes), Nm = mie/deposit life (yes), P = pofit ($/ye), d = discout te (%), M = miig cpcity (tos/ye), C = cocettig o millig cpcity (tos/ye), R = efiig cpcity (tos /ye), S = stockpile cpcity (tos), p = metl sellig pice ($/to of poduct), m = miig cost ($/to of mteil mied), c = cocettig o millig cost ($/to of oe), = efiig cost ($/to of poduct), f = dmiisttive/ fixed cost ($/ye), Qm = qutity of mteil mied (tos/ye), Qc = qutity of oe pocessed (tos/ye), Q = qutity of cocette efied (tos /ye), Qs = qutity of mteil stockpiled (tos/ye). The model elies o the fct tht the cpcities of the miig, pocessig, d efiig stges limit the opetio eithe idepedetly o joitly. While idividul stge cuses costied poductio, it leds to the detemitio of efiey limitig ecoomic cut-off gdes fo miig, pocessig, d efiig, epeseted s γ m, γ c, d γ, espectively. Howeve, if pi of stges is limitig the opetio, the the output fom ech costiig stge must be blced to utilize the mximum cpcity of these stges. This equies the detemitio of thee blcig cut-off gdes piig mie pocessig plt, mie efiey, d pocessig plt efiey, epeseted s γ mc, γ m, d γ c, espectively. Ultimtely, the optimum cut-off gde γ is selected betwee the limitig ecoomic d blcig cut-off gdes. As the gde d mout of low gde stockpile mteil i peiod is lso depedet upo the detemitio of optimum cut-off gde, the solutio to this poblem my be peseted i two sequetil steps. The fist step detemies the optimum cut-off gde, d the secod step defies the gde d mout of stockpile mteil. Optimum cut-off gde Dymic metl pices d opetig costs ifluece limitig ecoomic cut-off gdes, while the gde-toge distibutio of the deposit is the oly fcto ffectig blcig cut-off gdes13. The optimum cut-off gde mog six limitig ecoomic d blcig cut-off gdes is clculted s follows: Assumig tht the gde-toge distibutio of pushbck cosists of K gde icemets i.e. (γ 1, γ 2 ), (γ 2, γ 3 ), (γ 3, γ 4 ),, (γ K 1, γ K ), d, fo ech gde icemet, thee exist t k tos of mteil. I geel, if k * epesets gde icemet (γ k, γ k+1 ) d the lowe gde i k * i.e. γ k is 742 NOVEMBER 2011 VOLUME 111 The Joul of The Southe Afic Istitute of Miig d Metllugy

3 Net peset vlue mximiztio model fo optimum cut-off gde policy cosideed s the cut-off gde, the qutity of oe t o, qutity of wste t w, d the vege gde of oe γ e the give i Equtios [7], [8], d [9]: [7] [8] [9] [13] Substitutig Equtio [10] ito Equtio [13] yields the bsic peset vlue expessio tht dicttes the clcultio of the limitig ecoomic cut-off gdes: Miig, pocessig, o efiig cpcities defie time τ, ledig to thee vlues depedig upo the ctul costiig cpcity i.e. o, espectively. Qm, Qc, Qc γ y M C R Substitutig these vlues ito Equtio [14] geetes the bsic equtios fo limitig ecoomic cut-off gdes: [14] T s c t i o If y is the metllugicl ecovey, the Qm, Qc, d Q e sequetilly detemied ccodig to y oe of the followig thee coditios: 1. Set: 2. If Qc > C o Q > R fom coditio 1, the set: [15] [16] [17] P p e 3. If Q > o Qm > M fom coditio 2, the set: Miig the ext Qm mout of mteil my equie time τ. Fo clcultig the pofit geeted fom Qm t the ed of time τ, Equtio [6] my be updted s: [10] Sice the objective fuctio is to mximize the NPV of futue pofits, ssumig tht ς is the mximum possible et peset vlue of futue pofits t time zeo (i.e. ow) d Ω is the mximum possible et peset vlue of futue pofits (P τ+1 to P N ) t time τ, the the sceio my be peseted s show o the time digm i Figue 121. Kowig the discout te d: [11] [12] I Equtio [15], the mie hs bottleeck tht limits the opetio d theefoe delys the oppotuity of chievig futue positive csh flows. Hece, the oppotuity f cost + dς is distibuted pe to of mteil mied. I this M sceio, oe my be pocessed d efied s soo s mteil is mied. Theefoe, cut-off gde should be such tht the pocessig d efiig costs e coveed. This shows tht evey uit of mteil fo which [(p )γ m y ] is gete th the pocessig cost c, should be clssified s oe. Thus, the mie limitig cut-off gde, which ivokes costit 1 (Equtio [2]), becomes: [18] Similly, i Equtio [16] the pocessig plt hs bottle-eck tht delys the opetio, d the oppotuity f + dς cost is distibuted pe to of oe pocessed. The cut-off C gde is chose such tht i dditio to pocessig d efiig costs, it pys the oppotuity cost of ot eceivig the futue csh flows. Thus, the pocessig plt limitig cutoff gde, which ivokes the secod costit (Equtio [3]), becomes: The icese i peset vlue ν is elized though miig the ext Qm of mteil d the diffeece of ς d Ω epesets this icese. Kowig tht τ is the shot itevl of time, Equtio [12] my be witte s: ς 0 P + Ω P τ+1 τ τ+1 Figue 1 Time digm of peset vlue of futue pofits t time zeo d τ P N N [19] Also, i Equtio [17] the efiey is esposible fo delyig the futue csh flows, d the oppotuity cost f + dς is distibuted pe uit of cocette efied. R Theefoe, the efiey limitig cut-off gde, which ivokes the thid costit (Equtio [4]), becomes: The Joul of The Southe Afic Istitute of Miig d Metllugy VOLUME 111 NOVEMBER

4 Net peset vlue mximiztio model fo optimum cut-off gde policy [20] The blcig cut-off gdes deped upo the gde toge distibutio of idividul pushbck. Theefoe, these cut-off gdes e deduced fom the gde-toge distibutio cuves epesetig qutity of oe pe uit of mteil mied, ecoveble metl cotet pe uit of mteil mied, d the ecoveble metl cotet pe uit of oe, s give i Figues 2, 3, d 4, espectively. The mie d pocessig plt blcig cut-off gde is the oe which ivokes the fist d secod costits (Equtios [2] d [3]). The mie d pocessig plt will be i blce whe qutity of oe pe uit of mteil mied equls the tio C/M. Fo the gde ctegoy k*, the tio of oe tos to totl tos mied, epeseted s mc(k*) is: Fo the gde ctegoy k*, the ecoveble metl, epeseted s c(k*) is: Kowig c(k*), the pocessig plt d efiey blcig cut-off gde is detemied fom the cuve peseted i Figue 4. As tio R/C lies betwee c(k*) d [25] [21] Kowig this tio, the mie d pocessig plt blcig cut-off gde is detemied fom the cuve peseted i Figue 2. As tio C/M lies betwee mc(k * ) d mc(k * +1) o the y-xis, the coespodig vlue o the x- xis epesetig the mie d pocessig plt blcig cut-off gde γ mc is detemied by lie ppoximtio s follows: Figue 2 Gde-toge cuve fo mie d pocessig plt blcig cut-off gde [22] Similly, the mie d efiey blcig cut-off gde is the oe tht ivokes the fist d thid costits (Equtios [2] d [4]). The mie d efiey will be i blce whe the ecoveble metl cotet pe uit of mied mteil equls the tio R/M. Fo the gde ctegoy k*, the tio of ecoveble metl cotet to the totl tos mied, epeseted s m(k*) is: [23] Kowig this tio, the mie d efiey blcig cutoff gde is detemied fom the cuve peseted i Figue 3. As tio R/M lies betwee m(k*) d m(k*+1) o the y- xis, the coespodig vlue o the x-xis epesetig the mie d efiey blcig cut-off gde γ m is detemied by lie ppoximtio s follows: Figue 3 Gde-toge cuve fo mie d efiey blcig cut-off gde [24] Also, the pocessig plt d efiey blcig cut-off gde is the oe tht ivokes the secod d thid costits (Equtios [3] d [4]). Theefoe, the pocessig plt d efiey will be i blce whe the ecoveble miel cotet pe uit of oe equls the tio R/C. Figue 4 Gde-toge cuve fo pocessig plt d efiey blcig cut-off gde 744 NOVEMBER 2011 VOLUME 111 The Joul of The Southe Afic Istitute of Miig d Metllugy

5 Net peset vlue mximiztio model fo optimum cut-off gde policy c(k*+1) o the y-xis, the coespodig vlue o x-xis epesetig the pocessig plt d efiey blcig cutoff gde γ c is detemied by lie ppoximtio s follows: [26] Oce the thee limitig ecoomic cut-off gdes i.e. γ m, γ c, d γ, d thee blcig cut-off gdes i.e. γ mc, γ m, d γ c e detemied, the optimum cut-off gde γ is selected fom mog them. The equtios of limitig ecoomic cut-off gdes evel tht fo miig opetio, the optimum cutoff gde my eve be less th γ m, sice it epesets the lowest (bek eve) cut-off gde. Also, the optimum cut-off gde my eve exceed γ c, sice this my schedule some of the vluble oe to the wste dumps. Theefoe, the optimum cut-off gde γ lies betwee γ m d γ c i.e. γ m γ γ c. If m ~ epesets the medi vlue, the the followig citeio dicttes the selectio of the optimum cut-off gde: [27] Cetio of stockpiles The cetio of stockpiles follows the detemitio of the optimum cut-off gde γ. The optimum cut-off gde clssifies the followig: 1. The mteil bove optimum cut-off gde i.e. tos of oe t o (γ). This mteil is set to the pocessig plt 2. The itemedite gde stockpile mteil i.e. tos of potetil oe betwee the lowest cut-off gde γ 1 d the optimum cut-off gde γ, epeseted s t s (γ 1, γ) 3. The mteil below the lowest cut-off gde γ 1 i.e. tos of wste t w (γ 1 ). This mteil is set to the wste dumps. As descibed i the pevious sectio, the gde-toge distibutio of the pushbck cosists of K gde icemets i.e. [γ 1, γ 2 ], [γ 2, γ 3 ], [γ 3, γ 4 ],...,[γ K-1, γ K ], whee ech gde icemet of pushbck cosists of t k tos of mteil. If the optimum cut-off gde γ exists i the k gde icemet i.e. [γ k, γ k +1 ], d the lowest cut-off gde γ 1 exists i k gde icemet i.e. [γ k, γ k +1 ] d ssumig tht optimum cut-off gde γ = γ k d lowest cut-off gde γ l = γ k, the: [28] If T epesets the totl vilble tos i the pushbck, the: [32] Similly, the qutities mied Qm, pocessed Qc, d efied Q my be defied s fuctio of optimum cut-off gde γ usig the thee coditios give i the pevious sectio. The gde-toge distibutio of stockpiles i.e. the gde icemets d vilble tos i ech gde icemet is deduced fom the gde-toge distibutio of the pushbck. If α epesets the diffeece betwee the lowest cut-off gde icemet i.e. k d tht of optimum cut-off gde icemet i.e. k, the: [33] Now, if α > 0, the the stockpile tos fo espective gde icemets e detemied usig Equtios [34], [35], d [36]: 1. The tos of mteil i the fist stockpile gde icemet, which is sme s tht of lowest cut-off gde, i.e. k, my be detemied s: [34] 2. The tos of mteil i stockpile gde icemets fom (k +1) to (k -1), epeseted s k, e: [35] 3. The tos of mteil i the lst stockpile gde icemet, which is sme s tht of the optimum cutoff gde, i.e. k, my be detemied s: [36] Similly, if α = 0, i.e. the lowest cut-off gde d the optimum cut-off gde exist i the sme gde icemet, which my be epeseted s k, the the tos of mteil i the stockpile oly gde icemet e: T s c t i o P p e [37] [29] [30] [31] A demosttio of the computtios peseted i Equtios [34], [35], [36], d [37] is offeed though exmple i the ext sectio. Coppe deposit cse study Coside hypotheticl coppe deposit divided ito thee pushbcks15. Tble I pesets cpcities, pice of coppe, The Joul of The Southe Afic Istitute of Miig d Metllugy VOLUME 111 NOVEMBER

6 Net peset vlue mximiztio model fo optimum cut-off gde policy Tble I Ecoomic pmetes d opetiol cpcities Pmete Uit Qutity Mie cpcity tos/ye Mill cpcity tos/ye Coppe efiig cpcity tos/ye Stockpile cpcity tos Pice of coppe $/to Miig cost $/to 1.05 Millig cost $/to 2.66 Refiig cost $/to Fixed cost $/to Coppe pice escltio %/ye 0.80 Miig cost escltio %/ye 2.50 Millig cost escltio %/ye 3.00 Refiig cost escltio %/ye 2.50 Fixed cost escltio %/ye 2.50 Recovey of coppe % 90 Discout te % 15 opetig costs, d escltio tes fo this ope pit miig opetio. Tble II gives the gde-toge distibutio withi ultimte pit limits fo ll thee pushbcks. The pocess of detemitio of the optimum cut-off gde d the cetio of stockpiles peseted i the pevious sectios is computtio-itesive; theefoe dilogue-bsed pplictio i Visul C++ implemetig the itetive lgoithmic steps is used fo the developmet of optimum cut-off gde policies i this cse study. The lgoithmic itetios cotiue i ticiptio of the NPV covegece, i.e. the clcultio of the optimum cut-off gde fo peiod is epeted util o futhe impovemet i NPV is possible. A desciptio of these steps is s follows: 1. Set to 1 d itetio i to 1 2. Compute vilble eseves Q. If Q = 0, the go to step 10, othewise go to ext step 3. If i = 1, set V to 0 4. Set ς = V 5. Compute:. γ m, γ c, γ, γ mc, γ m, d γ c b. γ usig Equtio [27] c. t o (γ), t w (γ), d γ (γ) usig Equtios [7], [8], d [9], espectively d. Qm, Qc, d Q usig the coditios descibed i the model e. N bsed o the limitig cpcity idetified i step 5(d) f. P usig Equtio [10] P ((1+d) N -1) g. V = d(1+d)n. 6. If i = 1, check fo ς covegece (i.e. compe V (step 5(g)) with pevious V (step 4). If ς is coveged (withi some tolece, sy $ ), the go to step 7, othewise go to step 4 7. Kowig γ, compute stockpile gde-toge distibutio usig Equtios [34], [35], [36], d [37] 8. Kowig tht Qm is mied d Qc is pocessed, djust the gde-toge distibutio of the deposit 9. Set = + 1, go to step If i = 1, the kowig P fom peiod 1 to N, fid the Ω i.e. peset vlue of futue csh flows t peiod, d go to step 11. If i = 2, the stop 11. Compute the optimum cut-off gdes policy usig ς = Ω fo coespodig ye, d go to step 4. The steps i the lgoithm geete ltetive policies peseted i Tbles III, d IV. Tble III shows the optimum policy without escltio d stockpile cosidetio. As idicted i Tble III, the optimum cut-off gde i ye 1 is 0.50%. At this cut-off gde, millio tos of mteil is mied, d 10 millio tos of oe is pocessed which esults i tos of efied coppe. Hece, the opetio hs excess miig cpcity, while pocessig plt d efiey e limitig the opetio. Theefoe, 0.50% optimum cut-off gde efes to the pocessig plt d efiey blcig cut-off gde (Equtio [26]). This ptte cotiues util ye 6, d it is woth metioig tht i the sme ye the eseves i pushbck 1 e exhusted d miig fom pushbck 2 is commeced. Fom ye 7 though ye 10, mie d pocessig plt e limitig the opetio d the 0.53% optimum cut-off gde efes to mie d pocessig plt blcig cut-off gde (Equtio [22]). It is woth clifyig tht fom oe ye to ext, the gde-toge distibutio dicttig the blcig cut-off gdes is djusted uifomly (i.e. without y chge i the stuctue of distibutio) mog the itevls. Cosequetly, the optimum cutoff gde coespodig to the limittio of simil pi of stges emis costt s obseved fom yes 1 to 6 d yes 7 to 10. Similly, fom ye 11 to the life of opetio, i.e. ye 17, the flow of mteil fom the mie to efiey is limited due to full utiliztio of the pocessig plt cpcity. Hece, the optimum cut-off gde duig these yes efes to the pocessig plt limitig ecoomic cut-off gde (Equtio [19]), d it is decliig with exhustio of the eseves d cosequet declie i the peset vlue of the emiig eseves. The objective fuctio, i.e. mximum NPV of the ope pit miig opetio, is pedicted to be $ s show i the optimum policy i Tble III. Tble IV pesets the optimum policy llowig escltio of ecoomic pmetes without the stockpilig optio. The pice escltio is 0.80% pe ye, which idictes tht i ye 1 the metl pice is $ pe to of coppe. Howeve, it escltes to $ pe to of coppe i ye Tble II Gde-toge distibutio of coppe deposit Coppe (%) Tos Pushbck 1 Pushbck 2 Pushbck > NOVEMBER 2011 VOLUME 111 The Joul of The Southe Afic Istitute of Miig d Metllugy

7 Net peset vlue mximiztio model fo optimum cut-off gde policy Tble III Life of opetio optimum poductio schedule without escltio d stockpilig optio Ye Pushbck Cut-off Gde (%) Avege Gde (%) Qm (tos) Qc (tos) Q (tos) Pofit ($ millio) NPV = $ millio T s c t i o P p e Tble IV Life of opetio optimum poductio schedule cosideig escltio without stockpilig optio Ye Pushbck Cut-off Gde (%) Avege Gde (%) Qm (tos) Qc (tos) Q (tos) Pofit ($ millio) , NPV = $ Millio 17. Similly, opetig d fixed costs esclte fom ye 1 though ye 17. As specified i Tble IV, the ted of optimum cut-off gdes fom yes 1 though 17 follows the sme model s peseted i the pevious policy. Howeve, the miimum optimum cut-off gde hs icesed fom 0.21% to 0.27% i ye 17. This shows tht the opetio esues flow of comptively high gde mteil eve i the fil yes to py off the esclted opetig d fixed costs. The impct of pice d cost escltio leds to 1.7% decese i mximum NPV i ye 1, i.e. fom $ to $ Tble V shows compehesive bekdow of the optimum policy, llowig both escltio of ecoomic pmetes d stockpilig optio. Owig to the simil gde-toge distibutio d iitil vlues fo the ecoomic pmetes, equivlet ted is followed fo selectio of the optimum cut-off gdes d the esultt flow of mteil fom mie to efiey. Tble V idictes tht NPV (lst colum) is decliig with exhustio of eseves (colum 2 d 3) fom ye 1 to ye 22. It descibes the pocess of discoutig the ul csh flows to clculte Ω (lst colum coespods to step 11 of the lgoithm) fo The Joul of The Southe Afic Istitute of Miig d Metllugy VOLUME 111 NOVEMBER

8 Net peset vlue mximiztio model fo optimum cut-off gde policy Tble V Life of opetio optimum poductio schedule cosideig escltio d stockpilig optio Ye Pushbck Avilble mteil (tos) γ (%) Avilble mteil γ Mteil hdled (tos) Csh Flow NPV ($) Pushbck Pit Oe Wste/ γ (%) Qm Qs Qc Q 15% stockpile , , Stockpile Stockpile Stockpile Stockpile Stockpile Stockpile pticul peiod, which is the used to compute the optimum cut-off gde. Fo exmple, pocessig cpcity is limitig the opetio duig ye 15, hece γ = γ c, d keepig the esclted vlues of ecoomic pmetes, γ my be clculted s follows: Tble V lso demosttes the ccumultio of stockpile mteil fom yes 1 to 17. As the lowest cut-off gde emis 0.27% (fom Tble IV) d the optimum cut-off gde duig ye 1 is %, they exist i 4th d 9th gde icemets of the Pushbck 1 (see Tble II), espectively. Theefoe, the stockpile mteil cosists of six gde icemets, peseted s: [0.27, 0.30], [0.30, 0.35], [0.35, 0.40], [0.40, 0.45], [0.45, 0.50], [0.50, ] The tos of mteil fom the 2d to 5th stockpile gde icemets e detemied usig Equtio [35]: The mout of mteil i the fist stockpile gde icemet is detemied usig Equtio [34]: 748 NOVEMBER 2011 VOLUME 111 The Joul of The Southe Afic Istitute of Miig d Metllugy

9 Net peset vlue mximiztio model fo optimum cut-off gde policy Similly, the tos of mteil i the 6th stockpile gde icemet e detemied usig Equtio [36]: ecoomic situtios. As such, it is cotibutio to the mie plig commuity i tems of fcilittig the evlutio of diffeet ecoomic ltetives, ultimtely esuig the optimum utiliztio of esouces coupled with ppopite policy fomultio fo mkig mjo miig ivestmets. The model does ot coside ucetity i ecoomic pmetes, especilly, the ucetity ssocited with the metl pice. Also, it is limited to the cetio of log-tem stockpiles. Theefoe, the developmet of cut-off gde optimiztio models tkig ito ccout metl pice ucetity d llowig the pocessig of stockpile mteil duig mie life e some of the es fo futue esech. T s c t i o Theefoe, ppoximtely 3.4 millio tos of mteil is scheduled to stockpiles i ye 1. A totl of millio tos of w mteil is ccumulted i stockpiles fom yes 1 though 17, which othewise is scheduled to wste dumps i the pevious policies. Stockpilig optio pomises icese of five yes i the opetio s life log with 1% icese i NPV fom $ to $ Coclusios The optimum cut-off gde policies idicte tht the impct of escltio o the objective fuctio could be eomous i cses whee opetig d fixed costs e escltig t highe tes. This my chge some of the ecoomic ope pit opetios to uecoomic sceio. It is obseved tht the opetio peseted i the cse study becomes upofitble duig lte yes t escltio te of 6 pe cet pe ye i the opetig d fixed costs. Theefoe, log tem miig pls should lwys iclude escltio of ecoomic pmetes to estblish the fesibility of the miig vetue. The esults lso eflect tht the cetio of stockpiles scheduled ppoximtely 55 millio tos of dditiol oe fo pocessig, which fcilitted i eutlizig the effect of escltig ecoomic pmetes though ehcemet of the life of opetios log with NPV. Howeve, it is cle tht llowig the cetio of log-tem stockpiles is sttegic decisio, d execisig this optio depeds exclusively upo the opetig coditios of ope pit miig opetio. The poposed methodology is limited i its pplictio to the metllic oes; theefoe, while mkig this impott decisio, oe must give seious cosidetio to issues such s mteil deteiotio d compctio duig log exposue to the eviomet. Similly, loss of vlues my tke plce due to lechig, d oxidtio my itoduce pocessig complexities d esult i educed ecoveies. The poposed cut-off gde optimiztio model cosides the escltio of ecoomic pmetes, pticully, the opetig d fixed costs, with visio tht mie plig ctivity is focused o suvivl sttegies ude hsh Refeeces 1. TAYLOR, H.K. Geel bckgoud theoy of cut-off gdes. Tsctios of the Istitutio of Miig d Metllugy Sectio A, pp TAYLOR, H.K. Cut-off gdes some futhe eflectios. Tsctios of the Istitutio of Miig d Metllugy Tsctios Sectio A, pp DAGDELEN, K. Cut-off gde optimiztio. Poceedigs of the 23d Itetiol Symposium o the Applictio of Computes d Opetios Resech i the Miel Idusty, Tucso Aiso, pp DAGDELEN, K. A NPV optimiztio lgoithm fo ope pit mie desig. Poceedigs of the 24th Itetiol Symposium o the Applictio of Computes d Opetios Resech i the Miel Idusty, Motel, Cd, 1993.pp CETIN, E. d DOWD, P.A. The use of geetic lgoithms fo multiple cut-off gde optimiztio. Poceedigs of the 32d Itetiol Symposium o the Applictio of Computes d Opetios Resech i the Miel Idusty, Little, Colodo, pp DAGDELEN, K. d KAWAHATA, K. Vlue cetio though sttegic mie plig d cut-off gde optimiztio. Miig Egieeig, vol. 60, o. 1, pp HE, Y., ZHU, K., GAO, S., LIU, T., d LI, Y. Theoy d method of geeticeul optimizig cut-off gde d gde of cude oe. Expet Systems with Applictios, vol. 36, o. 4, pp AKAIKE, A. AND DAGDELEN, K. A sttegic poductio schedulig method fo ope pit mie. Poceedigs of the 28th Itetiol Symposium o the Applictio of Computes d Opetios Resech i the Miel Idusty, Golde, Colodo, pp BASCETIN, A. Detemitio of optiml cut-off gde policy to optimize NPV usig ew ppoch with optimiztio fcto. Joul of the Southe Afic Istitute of Miig d Metllugy, vol. 107, o. 2, pp ASAD, M.W.A. Developmet of geelized cut-off gde optimiztio lgoithm fo ope pit miig opetios. Joul of Egieeig d Applied Scieces, vol. 21, o. 2, pp OSANLOO, M. AND ATAEI, M. Usig equivlet gde fctos to fid the optimum cut-off gdes of multiple metl deposits. Miels Egieeig, vol. 16, o. 8, pp P p e The Joul of The Southe Afic Istitute of Miig d Metllugy VOLUME 111 NOVEMBER

10 Net peset vlue mximiztio model fo optimum cut-off gde policy 12. ASAD, M.W.A., Cut-off gde optimiztio lgoithm fo ope pit miig opetios with cosidetio of dymic metl pice d cost escltio duig mie life. Poceedigs of the 32d Itetiol Symposium o Applictio of Computes d Opetios Resech i the Miel Idusty (APCOM 2005), Tucso, Aizo, pp ASAD, M.W.A. Optimum cut-off gde policy fo ope pit miig opetios though et peset vlue lgoithm cosideig metl pice d cost escltio. Egieeig Computtios, vol. 24, o. 7, pp ASAD, M.W.A. Cut-off gde optimiztio lgoithm with stockpilig optio fo ope pit miig opetios of two ecoomic miels. Itetiol Joul of Sufce Miig, Reclmtio d Eviomet, vol. 19, o. 3, 2005b. pp LANE, K.F. Choosig the optimum cut-off gde. Colodo School of Mies Qutely, vol. 59, pp LANE, K.F. The Ecoomic Defiitio of Oe, Cut-off Gde i Theoy d Pctice. Miig Joul Books, Lodo, DAGDELEN, K. d ASAD, M.W.A. Multi-miel cut-off gde optimiztio with optio to stockpile. Society of Miig, Metllugy, d Explotio Egiees (SME) Aul Meetig, Pepit o ATAEI, M. d OSANLOO, M. Usig combitio of geetic lgoithm d gid sech method to detemie optimum cut-off gdes of multiple metl deposits. Itetiol Joul of Sufce Miig, Reclmtio d Eviomet, vol. 18, o. 1, pp OSANLOO, M., RASHIDINEJAD, F., d REZAI, B. Icopotig eviometl issues ito optimum cut-off gdes modelig t pophyy coppe deposits. Resouces Policy, vol. 33, o. 4, pp KING, B. Optiml miig piciples. Poceedigs of the Itetiol Symposium o Oebody Modellig d Sttegic Mie Plig. Austli Istitute of Miig d Metllugy, Peth, Weste Austli STERMOLE, F.J. d STERMOLE J.M. Ecoomic Evlutio d Ivestmet Decisio Methods, 9th Editio, Ivestmet Evlutio Copotio, Golde, Colodo, USA, NOVEMBER 2011 VOLUME 111 The Joul of The Southe Afic Istitute of Miig d Metllugy