Risk Model of LongTerm Production Scheduling in Open Pit Gold Mining


 Bernice Welch
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1 Rsk Model of LongTerm Producton Schedulng n Open Pt Gold Mnng R Halatchev 1 and P Lever 2 ABSTRACT Open pt gold mnng s an mportant sector of the Australan mnng ndustry. It uses large amounts of nvestments, whch need to be managed effcently under condtons of great uncertanty that are nherent for any mne project. The sources of uncertanty are prce volatlty and varablty of geologcal, technologcal, techncal and economc parameters. These specfc features lay down the requrement for developng an adequate probablstc approach to provde accurate estmates of the assocated rsk n the decsonmakng process. Ths paper offers a rsk model of longterm producton schedulng of open pt gold mnes. The model s developed on the bass of full dscounted cash flow analyss (DCFA) to the level of assessng the net operatng dscounted cash flows. It accounts for all varables of the DCFA, whch exhbt a stochastc behavour and contrbute to the overall mne project uncertanty. The Monte Carlo smulaton technque s used for modellng the rsk of not achevng the planned dscounted proft over any pont of the tme dscretsaton of the cash flows. The rsk model also provdes a soluton for the rsk estmate of mne nvestments over the payback perod, whch s of a partcular nterest to mne nvestors. A case study s presented to llustrate the applcablty of the model developed. INTRODUCTION Rsk analyss of open pt mne projects n the Australan gold mnng ndustry s a requred procedure n the decsonmakng process. It deals wth the problem of a quanttatve estmaton of uncertanty n nformaton about producton schedulng parameters. The soluton to ths problem s of great mportance for achevng sustanable explotaton of the avalable resources of gold ore. At present, the most popular mnng practce for assessng the varablty of producton schedulng parameters on the economc forecasts of mnng ventures s by usng senstvty analyss. Unfortunately, ths analyss cannot provde quanttatve estmates of the rsk of a mne project as t s a mathematcally smplstc procedure. There have been a lmted number of research developments on rsk models, whch have paved the way for a new theoretcal approach. Major attenton n these models has been dedcated to the geologcal factor, such as, the uncertanty n the reserve evaluaton. The uncertanty due to the geologcal factor, however, represents only a percentage of the overall uncertanty that a mne project could have. There are other mportant factors, such as, economc, technologcal, and techncal ones, whch are assocated wth mne projects. Ths gves grounds to regard the rsk models of mne projects developed usng only the geologcal factor as naccurate and unrelable for mnng purposes. A strong argument for makng ths concluson s, for example, the recent collapse of Pasmnco Century Mne, whch was due to the underestmaton of the uncertanty of 1. CRCMnng, The Unversty of Queensland, Brsbane Qld Emal: 2. CRCMnng, The Unversty of Queensland, Brsbane Qld Emal: CRC Mnng Technology Conference Fremantle, WA, September
2 R HALATCHEV and P LEVER both the economc and technologcal factors. There are also some research developments n rsk models that take nto account a restrcted number of technologcal, techncal and economc parameters. Ths ndcates that the problem of rsk modellng of mne projects has not yet been solved successfully. Ths paper offers a rsk model that s developed on the bass of a full dscounted cash flow analyss of a gold mne project and takes nto account the varablty of all possble parameters. The model s defned as a probablty of not achevng the planned dscounted proft over any pont of the tme dscretsaton of the dscounted cash flow analyss, ncludng the payback perod. In ths way t provdes ntegral estmates of the rsk over the lfeofmne. From the poston of mne nvestors, whose major concern s the return of ther nvestments, ths rsk model would be of nterest. Assumptons THEORETICAL FRAMEWORK The development of a rsk model of longterm producton schedulng of gold mnng s possble only wth the ntroducton of some assumptons. Ths s due to the complexty of the problem. The current rsk model s developed under the followng assumptons: The model deals only wth open pt gold mnng. The model s based on an example dscounted cash flow analyss of a mne project wth the objectve of achevng the annual net dscounted cash flows. The mne has three technologcal flows basc ore, secondary ore and waste supply. Waste goes to external dumps. Basc ore has a grade hgher than the mllfeed cutoff grade and goes drectly to the processng plant. Secondary ore has a grade between mllfeed cutoff grade and breakeven cutoff grade. The technologcal scheme of the gold processng s descrbed n detal n (Halatchev, 2005). The model does not take nto account the envronmental consequences of the mnng actvty. The model treats only the mne project rsk as a component of the dscount rate (Smth, 1994). Determnstc model The operatng dscounted cash flow ODCF( t ) of a gold mne project at tme t can be expressed analytcally as follows: g ma p sp m ODCF( t ) f{ r, S, Ro, C, C, C, C, C m m = d t t t bt s b t t s C w αb αs γ t γ t mb ms t t t t t t Q bo t, Q so t w ag s ce wc, Q, n, n, n, C, D, T, C, C } t bo t so w t t t t t t t,,,,,,,, (1) The defnton of each parameter of Equaton 1 s gven n Table 1. All parameters partcpate n an optmsaton procedure of longterm producton schedulng that s base on the concept of mnemllmarket nteracton (Halatchev, 2005). The operatng dscounted cash flows are usually estmated on an annual bass. Equaton 1 wll be used for expressng the cumulatve dscounted cash flow CDCF( t )of the mne project over the tme perod t : CDCF( t ) = ODCF( t ), j= 1 j (2) 2 Fremantle, WA, September 2005 CRC Mnng Technology Conference
3 RISK MODEL OF LONGTERM PRODUCTION SCHEDULING IN OPEN PIT GOLD MINING TABLE 1 Defnton of the DCFA parameters. Parameter r d g S t Ro t ma C t p C t m C b t m C s t m b t m s t sp C s t C w t bo Q t so Q t w Q t α b t α s t m γ t γ t ag C t D t s T t ce C t wc C t n bot n sot n w t n Defnton Dscount rate of the mne project Prce of gold Royalty as per cent of the revenue Unt marketng cost of payable metal Unt processng cost of basc ore Unt mnng cost of basc ore Unt mnng cost of secondary ore Gold as a fnal product n basc ore producton Gold as a fnal product n secondary ore producton Unt cost of secondary ore stockplng Unt cost of waste removal Producton rate of a shovel n basc ore Producton rate of a shovel n secondary ore Producton rate of a shovel n waste Basc ore grade Secondary ore grade Mnng recovery of ore Total recovery of gold as a fnal product Admnstraton and general cost Deprecaton allowance State ncome tax Captal expendture Workng captal Number of shovels for basc ore excavaton Number of shovels for secondary ore excavaton Number of shovels for waste excavaton Number of years over the lfeofmne Equaton 2 can provde assessments of two general evaluaton crtera of the mne nvestment analyss. The frst crteron s the cumulatve dscounted cash flow over the payback perod (PBP), whch s one of the most common evaluaton crtera used by mnng companes at present. By defnton, the payback perod s the number of years requred for the cash ncome from a project to return the ntal nvestments n the project (Gentry et al, 1984). The analytcal expresson of ths mportant crteron under the condton that the tmet s equal to the payback perod t PBP s as follows: CDCF( t ) = ODCF( t ) PBP PBP j= 1 j (3) CRC Mnng Technology Conference Fremantle, WA, September
4 R HALATCHEV and P LEVER The second crteron s the net present value (NPV) of the mne project, whch s an ntegral evaluaton crteron that recognses the tme effect of money over the lfeofmne. In ths case, the tme s equal to the lfeofmne t n and the relevant analytcal expresson can be wrtten as: t NPV = CDCF( t ) = ODCF( t ) n n j= 1 j (4) The analyss of Equatons 24 ndcates that the cumulatve dscounted cash flows of a project are defned as a sum of all operatng dscounted cash flows over a partcular perod of tme. In other words, the cumulatve dscounted cash flows perform as a functon of the operatng dscounted cash flows. For the sake of convenence, the parameters of the dscounted cash flow analyss (DCFA) can be grouped as follows: frst group: economc parameters gold prce, royalty, unt marketng cost of gold, unt processng cost of basc ore, unt mnng cost of basc ore, unt mnng cost of secondary ore, unt cost of waste removal, unt cost of secondary ore stockplng, admnstraton and general cost, deprecaton allowance, state ncome tax, captal expendture, workng captal, and dscount rate of the project; second group: technologcal parameters mnng recovery of ore, total recovery of gold as a fnal product, mllfeed cutoff grade, breakeven cutoff grade, and the lfeofmne; thrd group: techncal parameters producton rate of a shovel n basc ore, producton rate of a shovel n secondary ore, producton rate of a shovel n waste, and number of shovels; and fourth group: geologcal parameters ore grade and tonnage. The analyss of all groups of parameters shows that the economc parameters form the largest group. Ths means that the development of a rsk model, most of all, has to place a strong emphass on the accurate stochastc modellng of the economc parameters, whch s a complex problem. Ths s especally vald for the longterm forecasts of gold prce because the mnng companes n gold mnng are prce takers. The groups of technologcal and techncal parameters also play an mportant role n the development of the rsk model for longterm producton schedulng. These parameters requre a specfc modellng approach dealng wth the dynamcs of mne explotaton as well as the rgorous restrctons of the mnng and processng technologes and producton equpment used. The group of geologcal parameters ncludes ore grade and tonnage. The ore grade and tonnage are very mportant factors as ther estmates are the real bass of mne producton schedulng. For the objectves of a longterm schedulng and a feasblty study, the estmates of ore grade and tonnage are determned at the stage of depost exploraton. Rsk model The rsk model takes nto account all parameters lsted n Table 1, as they affect the stochastc nature of the dscounted cash flows of the mne project. Each parameter represents a varable havng ts own stochastc law of performance. Equaton 2 s used for assessng the cumulatve dscounted cash flows n the form of a sum of all operatng dscounted cash flows for a gven perod of the DCFA. Based on ths equaton the general rsk model can be formulated as the probablty of not achevng the planned dscounted proft over a gven perod of tme. Analytcally, the rsk model can be presented as follows: 4 Fremantle, WA, September 2005 CRC Mnng Technology Conference
5 RISK MODEL OF LONGTERM PRODUCTION SCHEDULING IN OPEN PIT GOLD MINING { } R( t ) = Pr CDCF( t ) < CDCF ( t ), (5) plan where: CDCFplan ( t ) s the planned cumulatve dscounted cash flow at tme t. Logcally the probablty or rsk of not achevng the planned dscounted proft over the PBP can be presented n the followng way: { } R( t ) = Pr CDCF( t ) < CDCF ( t ) (6) PBP PBP plan PBP where: CDCFplan ( t PBP )s the planned cumulatve dscounted cash flow at tme t PBP. Analogcally, the probablty or rsk of not achevng the planned dscounted proft over the lfeofmne can be presented as follows: { } R( t ) = Pr CDCF( t ) < CDCF ( t ) (7) n n plan n where: CDCFplan ( t n )s the planned cumulatve dscounted cash flow at tme t n, whch s the equvalent of the planned NPV of the mne project. The planned cumulatve dscounted cash flows n Equatons 57 are assessed wth a determnstc dscounted cash flow analyss. The evaluaton of the rsk model requres the determnaton of the statstcal characterstcs of cumulatve dscounted cash flows such as, mean and varance (or standard devaton). The latter determne the stochastc behavour of CDCF( t ) as a random quantty. The determnaton of the statstcal characterstcs as well as the assessment of the rsk deal wth the knowledge of the probablty densty functon (pdf) of CDCF( t ). Ths knowledge can be acqured by testng the emprcal dstrbuton wth relevant statstcal tests. The estmaton of the rsk based on models 79 represents a soluton of the followng ntegral: CDCF plan ( t ) R( t ) = f( x) dx, (8) where: fx ()s the probablty densty functon of CDCF( t ). The theoretcal lower lmt of the ntegral n Equaton 8 s chosen to be ( ) because the cumulatve dscounted cash flows of a mne project can take negatve values. An orgnal rsk model can also be formulated f the assessments of the rsk are made toward the operatng dscounted cash flows of the mne project. The general form of ths rsk model wll be: { } R( t ) = Pr ODCF( t ) < ODCF ( t ), (9) plan where: ODCFplan ( t )s the planned operatng dscounted cash flow at tme t. The planned operatng dscounted cash flows have to be equal zero, e: ODCF ( t ) = 0, (10) plan CRC Mnng Technology Conference Fremantle, WA, September
6 R HALATCHEV and P LEVER Ths means that the mnng company sets a strategy for achevng only postve operatng cash flows over the lfeofmne. Ths approach s useful as t could provde another range of the rsk estmates of the mne project. A sutable technque for assessng the statstcal characterstcs of CDCF( t ) or ODCF( t ) s the mplementaton of the Monte Carlo smulaton technque. There are two prncpal reasons for that. The frst reason s that CDCF( t ) or ODCF( t ) are nonlnear functons of the varables of the operatng dscounted cash flows that do not allow a drect determnaton of the analytcal expressons of the statstcal characterstcs of the dependent varable. The second reason s that these functons comprse a great number of varables, whch adds to the complexty of the problem. It s worth notng that the rsk assessments of a mne project can be done n two dfferent ways dependng on the type of probablty dstrbutons used for the stochastc descrpton of the dscounted cash flow crtera. The frst method uses contnuos probablty dstrbutons whle the second method deals wth dscrete probablty dstrbutons. Varables modellng The rsk model uses the parameters provded n Table 1 as random varables, wth some exceptons. The exceptons are the followng parameters: dscount rate of the mne project, number of years over the lfeofmne, number of shovels, mllfeed cutoff grade and breakeven cutoff grade, whch are treated as constants. The dscount rate of the mne project has a complex nature. In the context of the rsk model 6, ts utlsaton for assessng the payback perod of the mne project does not need any stochastc nterpretaton (Davs, 1995). The other two parameters have estmates that are obtaned from a producton schedulng optmsaton procedure (Halatchev, 2005). These estmates determne a strongly defned varant of an optmum producton schedule to be used as a determnstc bass of the present rsk model. The parameters, mllfeed cutoff grade and breakeven cutoff grade, are ndrect varables of the present rsk analyss. They affect the dvson of ore tonnage nto basc ore and secondary ore. An assumpton s used that they are constants n order to smplfy the calculaton procedures of the rsk model. Every varable of the rsk model has a specfc stochastc behavour. In prncple, two mathematcal models can be used for the stochastc descrpton of the varables. These are the model of random quantty and the model of random functon. The model of random quantty s applcable for varables havng a statonary character of varaton. The statonary s admssble to be defned even n a wde sense. The mplementaton of the model of random quantty requres testng procedures to determne the probablty dstrbuton of the emprcal materal as well as the estmates of the mean and varance (or standard devaton) of each ndependent varable. The random functon model s applcable for varables havng a complex stochastc performance. Such a behavour assumes the presence of a trend component as well as a seasonal component n some cases. Sutable mathematcal technques for mplementng the model of random functon are the autoregressve movng average (ARMA) modellng and autoregressve ntegrated movng average (ARIMA) modellng of the BoxJenkns methodology (Halatchev, 1996), random walk modellng (Rudenno, 1982), and conventonal regresson analyss. The modellng of ore grade and tonnage, n prncple, s a specfc task n the rsk analyss of longterm producton schedules. It s usually suggested to be made wth the methods of geostatstcs. There are several geostatstcal methods but the most wdely used methods are the condtonal smulaton and krgng (Souza et al, 2004). Condtonal smulaton s consdered as a relable method as t provdes a quanttatve measure of uncertanty. Krgng, as known, provdes the best local estmate n the leastsquare sense. Both methods assess confdence ntervals around the mean and hence, they can be used for assessng the varance of each block of the mne sequence, whch s the requrement of the 6 Fremantle, WA, September 2005 CRC Mnng Technology Conference
7 RISK MODEL OF LONGTERM PRODUCTION SCHEDULING IN OPEN PIT GOLD MINING current rsk model. The geostatstcal modellng, ndeed, s the mplementaton of the model of random functon, where the source of uncertanty deals wth the spatal varablty of the ore grade and tonnage. The choce of a partcular model depends not only on the stochastc behavour of each varable of the rsk model but also depends on the amount and qualty of the avalable data. Dfferent stages of the mne project, such as, exploraton, prefeasblty, feasblty, and explotaton would provde dfferent number and qualty of the nput data necessary for modellng the rsk model varables. Ths means that the present rsk model can not defne exactly what varables have to be descrbed wth the model of random quantty and what varables have to be descrbed wth the model of random functon. CASE STUDY The case study s based on a hypothetcal open pt gold mne. The mne desgn as well as the relevant geologcal model s descrbed n detal n (Halatchev, 2005). The optmum producton schedule of the mne s shown n Fgure 1 and t s used as a determnstc bass of the current rsk analyss. The schedule has a multstage stablsaton of the mnng rate due to the mplementaton of the prncple of waste deferment. The lfeofmne s 35 years. 2.50E E+7 mnng rate waste Rock,tonnes 1.50E E E+6 basc ore secondary ore 0.00E t n Tme,years FIG 1  Optmum producton schedule of the mne. The transformaton of the optmum producton schedule nto an optmum schedule of the excavaton equpment s shown n Fgure 2. The basc and secondary ore are excavated wth two Lebherr 994 shovels. The waste schedule s served by two O&K shovels. The Lebherr 994 shovel annual producton rate s modelled wth a mean estmate of tonnes and a standard devaton of tonnes. The O&K shovel producton rate has a mean of tonnes and standard devaton of tonnes. The modellng of the standard devaton for varable shovel producton rates s based on a method developed by Halatchev and Lever (2004). CRC Mnng Technology Conference Fremantle, WA, September
8 R HALATCHEV and P LEVER 2.50E E+7 mnng rate Rock, tonnes 1.50E E+7 O&K_1 5.00E+6 O&K_2 Lebherr_1 Lebherr_2 0.00E t n Tme,years FIG 2  Optmum schedule of shovels. The grade dstrbuton over the lfeofmne of the basc and secondary ore of the optmum producton schedules are graphcally shown n Fgure 3. The grades are presented as average estmates for each perod of the schedule tmng. The nput data of the rsk modellng s summarsed n Table 2. The data represents statstcal estmates of the rsk model varables, such as, mean and standard devaton. An assumpton s used for modellng the ndependent varables of Equaton 1 wth the model of random quantty as well as usng a normal probablty dstrbuton functon n order to smplfy the calculaton procedures. It s worth notng that the standard devaton of each varable s assessed as ±10 per cent varaton of the mean estmate. Ths s vald even for those varables, whch have tme dependant mean estmates over the lfeofmne. These varables are the basc and secondary ore grades, the dstrbuton of whch s shown n Fgure 3. The tme dependant varables are also the admnstraton and general cost, deprecaton allowance, state ncome tax, captal expendture, and workng captal. The admnstraton and general cost s assessed condtonally as $0.67 per tonne of basc ore producton (Stermole, 1979). The captal cost of $ made at year zero ncludes: $ for mne equpment, $ for mll shell and buldngs, $ for mll equpment and $ for talngs dam. There are two other captal costs of $ made at year 11 and year 22 for equpment replacement and spare parts. The deprecaton allowance s made toward any type of captal cost usng the method of straghtlne deprecaton. The amount of workng captal s $ , whch s allocated to year zero. The results of the DCFA are graphcally llustrated n Fgure 4, whch shows the dstrbuton of the determnstc operatng (ODCF) and cumulatve (CDCF) dscounted cash flows over the lfeofmne. The estmate of the payback perod s 2.90 years. At ths pont of tme the cumulatve dscounted cash flows become equal to zero. Fgure 4 also shows the stochastc ODCF and CDCF, whch are modelled wth the Monte Carlo smulaton technque. The determnstc estmate of the NPV of the project s 8 Fremantle, WA, September 2005 CRC Mnng Technology Conference
9 RISK MODEL OF LONGTERM PRODUCTION SCHEDULING IN OPEN PIT GOLD MINING basc ore grade Grade, grams/tonne 2.00 secondary oregrade t n Tme, years FIG 3  Optmum producton schedule of ore grades. TABLE 2 Statstcal estmates of parameters modelled as random quantty. Parameter Unts Mean Standard devaton Prce of gold $A per gram Royalty $A per gram Unt marketng cost of payable metal $A per gram Unt processng cost of basc ore $A per tonne Unt mnng cost of basc ore $A per tonne Unt mnng cost of secondary ore $A per tonne Unt cost of waste removal $A per tonne Unt cost of secondary ore stockplng $A per tonne Basc ore grade Grams per tonne Varable Varable Secondary ore grade Grams per tonne Varable Varable Mnng recovery of ore Relatve unts Total recovery of gold Relatve unts Lebherr_994 shovel producton rate Tonnes per year Varable Varable O&K shovel producton rate Tonnes per year Varable Varable Admnstraton and general cost $A per year Varable Varable Deprecaton allowance $A per year Varable Varable State ncome tax $A per year 30% 1% Captal expendture $A per year Varable Varable Workng captal $A per year Varable Varable CRC Mnng Technology Conference Fremantle, WA, September
10 R HALATCHEV and P LEVER 3.00E+8 determnstc flow CDCF 2.00E+8 stochastc flow Dscounted cash flows,$au 1.00E E+0 determnstc flow cp.1 cp.2 cp.3 stochastc flow ODCF 1.00E E t PBP t n Tme, years FIG 4  Dstrbuton of dscounted cash flows. $ , whle the stochastc estmate s $ The dfference of $ ndcates that there s overestmaton of the determnstc NPV, whch s calculated only wth mean estmates of the ndependent varables. The Monte Carlo smulaton was performed wth a computer module wrtten n C++. The number of generatons of NPV s The results obtaned from the rsk modellng are llustrated graphcally n Fgure 5. There are two types of rsk estmates that are related to the ODCF and CDCF respectvely. The analyss of Fgure 5 shows that the CDCF related rsk estmates over the lfeofmne have a slght trend of ncreasng by the end of the mne project. The last estmate s, ndeed, the rsk of not achevng the NPV of the mne project, whch s per cent. The rsk of not achevng the planned CDCF over the payback perod s per cent. Generally all rsk estmates regardng the CDCF exceed 50 per cent as there s an overestmaton of the determnstc cash flows wth regard to the stochastc cash flows. The ODCF related rsk estmates show a very useful pcture. The rsk of not achevng the planned ODCF n the year that determnes the payback perod s 0.26 per cent, whle the rsk related to the last year of the project s 2.62 per cent. There are three crtcal ponts marked on the graph. The crtcal ponts 1 (cp.1) and 2 (cp.2) are related to the rsk estmates of per cent and per cent that are obtaned n the year 11 and year 22, respectvely. These estmates represent a sgnfcant ncrease of the rsk due to the captal costs spent n the years n questons. The thrd crtcal pont (cp.3) s related to the rsk estmate of per cent obtaned n the year 26. Most lkely the rsk ncrease at ths pont s due to the ncrease of waste and decrease of basc ore grade, whch can be seen n Fgure 1 and Fgure 3, respectvely. It s worth notng that all rsk estmates regardng the ODCF and CDCF are obtaned usng a normal probablty densty functon. Ths s llustrated n Fgure 6 that shows a hstogram of the NPV sample of 1000 relevant Monte Carlo generatons. 10 Fremantle, WA, September 2005 CRC Mnng Technology Conference
11 RISK MODEL OF LONGTERM PRODUCTION SCHEDULING IN OPEN PIT GOLD MINING Rsk,% CDCF cp.2 cp cp ODCF 0.00 t PBP t n Tme, years FIG 5  Rsk estmates of ODCF and CDCF. 100 Frequency E+08 NPV ($AU) 5.00E+08 FIG 6  Hstogram of NPV wth normal probablty densty functon. CONCLUSIONS The followng conclusons can be made from the present nvestgaton: the results presented are a research attempt for defnng the theoretcal framework for a general rsk model of the projects n open pt gold mnng; CRC Mnng Technology Conference Fremantle, WA, September
12 R HALATCHEV and P LEVER the rsk model developed takes nto account the uncertanty of all varables of an open pt gold mne project by usng the dscounted cash flow analyss tll the level of assessng the planned net operatng dscounted cash flows over the lfeofmne; the soluton presented for assessng the rsk of not achevng the planned dscounted proft over the lfeofmne and, n partcular, at any tme step of the dscounted cash flow analyss, has the potental to sgnfcantly mprove the decsonmakng process. further research s requred for achevng an effectve rsk model to account for detals of the stochastc performance of the rsk model varables and ther mutual correlaton on dfferent levels of modellng. ACKNOWLEDGEMENTS The results presented n the paper were obtaned as part of the Case Study Model Development project funded by CRCMnng under the Smart Mnng Systems Research Program. REFERENCES Davs, G A, (Ms)use of Monte Carlo smulatons n NPV analyss, Mnng Engneerng, January, pp Gentry, D W and O Nel, T J, Mne Investment Analyss (AIME: New York). Halatchev, R, Rsk model of planned proft from a surface mnng venture, Trans Instn Mn Metal, (Secton A: Mnng Industry) 105, pp A137A142. Halatchev, R, A model of dscounted proft varaton of open pt producton sequencng optmzaton, n Proceedngs APCOM 2005 Symposum, pp Halatchev, R and Lever, P, Realtme stochastc modellng of shovel waste producton rates n open pt mnes, n Proceedngs 2004 CRCMnng Annual Conference, [CD ROM] (CRCMnng: Pnjarra Hlls). Rudenno, V, Random walk models of future metal prces, Trans Instn Mn Metall, (Secton A: Mnng Industry) 91, pp A71A74. Smth, L D, Dscount rates and rsk assessment n mneral project evaluatons, Trans Instn Mn Metal, (Secton A: Mnng Industry) 103, pp A137A147. Souza, L E, Costa, J F and Koppe, J C, Uncertanty estmate n resources assessment: a geostatstcal contrbuton, Natural Resources Research, 13(1):115. Stermole, F J, Economc evaluaton of open pt mnes, n Proceedngs Open Pt Mne Plannng and Desgn Conference (eds: Crawford and Hustruld) pp (AIME: New York). 12 Fremantle, WA, September 2005 CRC Mnng Technology Conference
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