open pit optimisation – modelling time and opportunity costs

Open pit optimisation – modelling time andopportunity costs

T. Elkington* and R. Durham

Strategic mine planning is the process of determining the configuration that will optimise project

objectives. Current methods for ensuring that objectives are optimised, for a given project

configuration, contain a number of limitations. In particular, the strategic mine planning process for

a given configuration is often completed by the sequential optimisation of key decisions. This

approach does not allow for relationships between decisions to be measured accurately. As such,

suboptimal mine plans are often produced. The ability to model and optimise key decisions

simultaneously, so as to achieve greater value, is investigated in this paper. Some of the areas in which

significant advances are made include time cost modelling, cutoff grade and stockpile optimisation for

open pit operations. A case study was used to benchmark the proposed model against a marginal

cutoff grade scheduling method, resulting in significant net present value increases.

Keywords: Open pit optimisation, Scheduling, Cutoff grades, Mixed integer programming, Time costs, Opportunity costs

List of symbols

Decision variablesThere are six main types of decision variables used inthis model. All non-binary variables are real numbersbetween 0 and 1.

mcbt Portion of bench b in pushback c that ismined in period t

pcbgt Portion of group g within bench b inpushback c that is mined in period t

xcbgt Portion of group g within bench b inpushback c that is stockpiled in period t

ycbgt Portion of group g within bench b inpushback c that is processed from thestockpile in period t

dcbt Binary indicating whether bench b in push-back c can be extracted in period t (1) ornot (0)

ft Maximum portion of mining, processing orsales capacity utilised in period t

CoefficientsMcbt Discounted cost of mining bench b in

pushback c in period t

Phbgt Discounted net revenue (revenue minus orecosts) of processing group g in bench bwithin pushback c in period t

Xcbt Discounted cost of stockpiling group g inbench b within pushback c in period t

Ycbgt Discounted net revenue (revenue minus orecost and reclaiming cost) of reclaiming and

processing group g in bench b withinpushback c in period t

Ft Discounted time cost in period t

Ocbg Ore tonnage in group g of pushback c inbench b

Qcb Total tonnage of pushback c in bench b

Shcbe Total product in group g of pushback c inbench b for element e

Qmint ,Qmax

t Minimum and maximum tonnage mined inperiod t

Bminct ,Bmax

ct Minimum and maximum bench advancefor pushback c in period t

Sminet ,Smax

et Minimum and maximum selling limit forelement e in period t

Gminet ,Gmax

et Minimum and maximum average gradelimit for element e in period t

Amint ,Amax

t Minimum and maximum tonnage added tostockpile in period t

Rmint ,Rmax

t Minimum and maximum tonnagereclaimed from stockpile in period t

Imint ,Imax

t Minimum and maximum tonnage in stock-pile at the end of period t

Lminc ,Lmax

c Minimum and maximum lead (in benches)of pushback c

IntroductionOptimisation in strategic mine planning is the process ofconverging upon a project configuration and plan thatmaximises the ‘potential’ of the resource under con-sideration. This ‘potential’ can be measured in a numberof ways, although it is often measured by the net presentvalue (NPV) of expected future cash flows. The NPVobjective encourages the mine planner to configure theoperation in such a way to defer negative cash flows andbring forward positive cash flows.

School of Civil and Resource Engineering, The University of WesternAustralia, 35 Stirling Highway, Crawley, WA 6009, Australia

*Corresponding author, email [email protected]

� 2009 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the InstituteReceived 9 February 2009; accepted 29 March 2009DOI 10.1179/174328609X446619 Mining Technology 2009 VOL 118 NO 1 25

Time costs are costs that are driven by the mine liferather than the amount of material produced. From aplanning perspective, these costs represent a deterrent tomine life extensions and emphasise the need to maximiseutilisation of available capacity. Time costs are oftenapportioned to either the mining cost or processing costdepending on which capacity is limiting the operation.When the limiting capacity changes over the life of theoperation, the full time cost will not be incorporated inthe optimisation. As such, the penalty to mine lifeextensions will not be reflected.

Opportunity costs measure the cost of an opportunityforegone. When incorporating discounting, a block’svalue will decrease over time (for constant economic andtechnical parameters). A decision to extract particularblock in an earlier time period will displace the possiblevalue gained from extracting a different block in thatperiod, causing it to lose value in discounted terms. Thisforegone opportunity must be incorporated in decisionmaking process to achieve optimal NPV results.However, opportunity cost is often neglected from thecutoff grade decision through the enforcement ofmarginal cutoff grades. The marginal cutoff grade isthe grade at which the undiscounted value of a block isindifferent between being processed and being waste. Toallow for the possibility of achieving higher NPVsolutions, cutoff grade should be an optimised outputrather than a rigidly imposed input.

This paper outlines the development of a mixedinteger programming (MIP) formulation that supportsoptimal decision making for open pit mine planning. Inparticular, the method simultaneously optimises thestrategic ‘life of mine’ schedule and cutoff grades. Indoing so, the impact of time costs and opportunity costsare accurately reflected in the optimisation.

Strategic optimisation of open pitprojectsOptimisation techniques in open pit mine planning haveadvanced significantly over recent years. Perhaps themost widely recognised application of optimisation isthe Lerchs–Grossman algorithm,1 which applies graphtheory to determine the mining outline that maximisesundiscounted cash flows. Whittle and Rozman2 recog-nised that mining companies were more likely to want tomaximise discounted, rather than undiscounted, cashflows. In order to maximise discounted cash flows it isnecessary to know not just what material is mined, butalso the timing of each block’s extraction. A parametricmethod was proposed to identify ‘nested pit shells’: pitoutlines contained within the final pit. These pit shellscan be used to guide the scheduling of material withinthe final pit.

The problem of generating optimal mine scheduleswas raised by Johnson,3 who developed a linear pro-gramming (LP) formulation for determining the timingof extraction of regularised resource blocks within themine, with a maximum NPV objective. Johnson’s LPformulation was found to mine partial blocks.4 Owingto the fractional solution provided by the LP formula-tions, integer constraints were needed to producefeasible solutions. Ramazan and Dimitrakopoulos5

examined the integer programming alternative andfound that even small problems took a long time to

solve. Caccetta and Hill6 described a different integerprogramming formulation and branch and cut solutionstrategy to improve solution times. However, the abilityto find timely solutions for complex problems remainsan issue.

In order to reduce the size of the problem, a numberof approaches have aggregated blocks together forscheduling. These aggregations are typically known as‘panels’. King7 outlined a dynamic programmingmethod to simultaneously optimise cutoff grade andmining sequence. Stone et al.8 incorporated a grade andlocation based clustering approach within an MIPformulation to optimise pits for iron ore deposits inthe presence of blending requirements. Menabde et al.9

provided a stochastic integer programming formulationto optimise cutoff grade and mining sequence in thepresence of uncertainty. Ramazan10 implemented afundamental tree approach to creating productionpanels to schedule within an MIP formulation.

The previously mentioned approaches have an objec-tive of maximising NPV. However, a number of authorshave explored other options. Dimitrakopoulos andRamazan11 described a method to maximise the robust-ness of a schedule for multiple orebody realisations.Dimitrakopoulos and Ramazan12 developed a stochasticinteger programming formulation to minimise deviationfrom specified tonnage and grade constraints givenorebody uncertainty.

Methodology

Activity resolutionA panel based approach is taken for the timing of theextraction of material. This panel is typically taken to bea bench within a pushback. It is assumed that all blockswithin a panel are extracted at the same rate. However,several ‘groups’ can be created within each panel. Eachgroup is able to be sent to any destination (Fig. 1). Thiscould be to waste, to the mill(s) or to the stockpile(s).The challenge lies in the method of grouping blockstogether within the panels.

There are two key areas that require optimisation.First, the timing of extraction of each panel is optimised.This will determine the optimal mining sequence.Second, the destination of the material for each groupwithin each panel is optimised. These areas areoptimised under the applied constraints. The character-istics of each panel and group, as well as the assumedparameter of the operation, will impact on the optimalsolution.

There is no cutoff grade explicitly described in thismethod. The cutoff grade is a result of the destination towhich material is sent in a given period. As such, themethod can be used for blending.

Grouping dataFor this approach, blocks are grouped by gradecharacteristics and rock type. Tonnages of the blockswithin each group are summed, and a tonnes weightedaverage of grade is allocated. Effective grouping willresult in each group having its own unique ‘group’characteristics that are similar to each of the blockscontained within the group. In this way, the loss in theresolution of data is minimised.

Initially, those blocks that cannot possibly beeconomic are identified and allocated to one ‘waste’

Elkington and Durham Open pit optimisation – modelling time and opportunity costs

26 Mining Technology 2009 VOL 118 NO 1

group. A clustering algorithm is applied to group theremaining ‘potential ore’ blocks.

FormulationObjective function

The objective of the optimisation model is to maximiseNPV (Eq. 1). This incorporates the discounted netrevenue from material that is directly processed (Pcbgt)and processed off the stockpile (Ycbgt), as well as costsfrom time costs (Ft), and material added to the stockpile(Xcbt), and mining (Mcbt).

MaximiseX

c,b,g,t

Pcbgt phbgtzX

c,b,g,t

Ycbgt yhbgt{X

c,b,g,t

Xcbgt Xcbgt{

X

c,c,b,t

Mcbt mcbt{X

t

Ftft (1)

Constraints

Quantity constraints

These constraints relate to the capacity of mining (Eq. 2)and processing (Eq. 3) in each period. These capacitiesare non-decreasing over time and each additional unit ofcapacity incurs a capital establishment cost.

Qmint ¡

X

c,b,g

Qcbmcbt¡Qmaxt Vt (2)

Omint ¡

X

c,b,g

Ocbg pcbgt¡Omaxt Vt (3)

Product constraints

These constraints relate to the minimum and maximumtotal output (Eq. 4) and average head grade feed to themill (Eq. 5) for each element.

Sminet ¡

X

c,b,g

Scbge pcbgt¡Smaxet Ve,t (4)

X

c,b,g

Ocbg pcbgtGminet ¡

X

c,b,g

Scbge pcbgt¡

X

c,b,g

Ocbg pcbgtGmaxet Ve,t (5)

Advance constraints

A bench within a pushback can only be mined once (Eq.6). Also, the maximum number of benches mined withina pushback can be controlled (Eq. 7).X

t

mcbt¡1 Vc,b (6)

Bminct ¡

X

b

mcbt¡Bmaxct Vc,t (7)

Material flow constraints

It is important that material flow equilibrium is achievedthroughout the optimisation from period to period.Firstly it is important that the amount of materialprocessed directly and added to the stockpile from aparcel progresses at the same rate as the mining of theparcel (Eq. 8). Secondly, material cannot be removedfrom the stockpile until it has been placed there (Eq. 9).

pcbgtzxcbgt{X

c

mcbt¡0 Vc,b,t (8)

Xt

q~1

ycbgq{Xt

q~1

xcbgq¡0 Vc,b,g,t (9)

Sequencing constraints

In order to optimise the sequencing of panels, a series ofconstraints are used to ensure that practical schedulingrequirements are met i.e. that the material is uncoveredbefore it is mined (Eq. 10, 11) and that minimum (Eq.12) and maximum (Eq. 13) lead times are enforced.

1 Open pit optimisation concept


Mining Technology 2009 VOL 118 NO 1 27

http://www.maneyonline.com/action/showImage?doi=10.1179/174328609X446619&iName=master.img-000.jpg&w=376&h=227

mcbt{dcbt¡0 Vc,b,t (10)

dcbt{Xt

a~1

mc(b{1)a¡0 Vc,b,t (11)

dcbt{Xt

a~1

m(c{1)(bzLminc )a¡0 Vc,b,t (12)

dcbt{Xt

a~1

m(cz1)(b{Lmaxc )a¡0 Vc,b,t (13)

Stockpile constraints

The stockpile constraints are used to control theaddition (Eq. 14) and reclamation (Eq. 15) of materialfrom the stockpile over time, as well as imposing overallsize constraints (Eq. 16). These constraints allow anymaterial on the stockpile to be reclaimed at any timeafter being placed there. There is no application of gradeaveraging, or inventory control. In this way, theoptimisation is able to strategically guide the engineeras to how stockpiles may be designed.

Amint ¡

X

c,b,g

xcbgt¡Amaxt Vc,b,t (14)

Rmint ¡

X

c,b,g

ycbgt¡Rmaxt Vc,b,t (15)

Imint ¡

X

c,b,g

Xt

q~1

ycbgt{X

c,b,g

Xt

q~1

xcbgt¡Imaxt

Vc,b,t (16)

Case studyA hypothetical copper orebody is used to demonstratethe importance of accurate time and opportunity costmodelling in open pit schedule and cutoff gradeoptimisation. A pit optimisation was completed andfive pushbacks were selected, according to commonpractice, with a strategic mine planning package.Figure 2 shows these pushbacks in plan and sectionview.

The configuration analysed in this case study involvesore being fed to a flotation circuit and the resultantconcentrate is sold. The base case configuration has amining capacity of 50 Mtpa and processing capacity of20 Mtpa. A discount rate of 10% was applied. Thissection examines how value is optimised for this

configuration under various assumptions and providesfurther applications.

Marginal cutoff grade analysisFor this case study, marginal cutoff grade scheduleswere generated for both mining limited (Fig. 3) andprocessing limited (Fig. 4) assumptions. The onlydifference in the input parameters for these schedulesis the application of time costs. All time costs areapportioned to the mining cost for the mining limitedassumption, and processing costs for the processinglimited assumption.

The results demonstrate the problems associated withpre-allocating time costs to determine marginal cutoffgrades.

Loading time costs onto mining costs results in a lowmarginal cutoff grade of 0?2% CuEq. With a low cutoffgrade, there are a relatively large number of blocksclassified as ore. Ironically, the operation is processinglimited in all but three periods (years 10, 12 and 14). Thelow cutoff grade also results in a long mine life of 16years, with a large amount of excess mining capacityavailable in many periods.

The opposite is true when loading time costs ontoprocessing cost. This case resulted in a higher marginalcutoff grade of 0?5% CuEq. At this cutoff grade, themine struggles to find enough material to feed the mill,making it mining limited in years 1, 6, 8, 10 and 12.

The traditional approach tends to provide contra-dictory results. However, the processing limited assump-tion seems better suited to the scenario and produces ahigher NPV ($184 million) than the mining limitedassumption ($23 million). The discrepancy between thesevalues is significant, and indicates the value of accurateand efficient strategic planning. A more appropriateapproach would be to allow the optimisation to determinehow time costs should be allocated in order to optimisevalue. This is explored in the next section.

Optimised schedule and cutoffThe MIP formulation in this paper allows for thesimultaneous optimisation of cutoff grade and schedul-ing. The basis of the approach is to create a number ofgrade groups within each panel that can be accumulated,and sent to the same destination. These groups aregenerated by applying a clustering algorithm to linkblocks with similar grade characteristics. As the number

2 Pushbacks in a plan view and b section view (not to

scale)

3 Marginal cutoff grade schedule (mining limited

assumption)





of groups increases, the size of each group reduces i.e.the grade intervals decrease. As a result, the resolutionof the model more closely resembles that of the inputblock model. In a practical sense, increasing the numberof groups increases the number of cutoff grade optionsto select from. Analysis showed that 20 groups providedadequate resolution for this case.

The resultant schedule for 20 groups is shown in Fig. 5.The NPV for this case is $348 million, a substantialincrease from the marginal cutoff grade cases. The increasein NPV is achieved through the ability to manipulatecutoff grade to bring forward revenue (by controllingaverage grade) and delay waste stripping. There is a lowcutoff grade in the first year while waste is being stripped.The strip ratio drops as the ore is accessed, and there isadditional mining capacity available to increase the cutoffgrade in years 2–5. The strip ratio starts to increase afteryear 5 as the mine deepens, and the cutoff grade declinesbetween years 6 and 9. This result is consistent with thetheory of Lane13 that suggests a declining cutoff strategyoften provides high NPV solutions.

Subsequent cutoff grade optimisation cases were run,with time costs loaded onto all blocks (Fig. 6) and onlyore blocks (Fig. 7) to establish the value associated withaccurately modelling time costs in the optimisation.

The mining limited (time cost loaded onto miningcost) case returned an NPV of $262 million. The cutoffgrade schedule shows a declining cutoff grade strategy,

albeit at a lower grade than the ‘no assumption’ case.However, Lane’s theory states that a declining cutoffgrade strategy is only appropriate when the operation isprocessing limited. As the operation is clearly notmining limited (as assumed) in years 2–5, the cutoffgrade increases. As mining capacity is not fully utilisedin many of the periods, the entire time cost is notincorporated in the optimisation. Consequently, the lifeof the operation is almost two years longer than the ‘noassumption’ optimisation.

The processing limited assumption (time costs loadedonto processing cost) optimisation has a resultant NPVof $271 million, a slight improvement over the mininglimited assumption case. This improvement can beattributed to the operation being processing limitedmore often than mining limited. Interestingly, the cutoffgrade for this scenario is relatively constant over the lifeof the operation, not demonstrating the typical decliningcutoff grade strategy for processing limited operations.

As with the marginal cutoff case, the prior assumptionof the operation’s limiting component tends to becontradictory, i.e. the execution of this assumptionbiases the result towards the other assumption. The onlyway to avoid this problem is to omit the priorassumption and allow the optimisation to determinethe limiting component, as has been shown. The NPVgains due to the application of time costs with this

5 Optimised cutoff grade and schedule

6 Optimised cutoff grade schedule (mining limited

assumption)

4 Marginal cutoff grade schedule (processing limited

assumption)

7 Optimised cutoff grade schedule (processing limited

assumption)







method, in this case, were up to 30%. This represents asubstantial improvement over traditional approaches.

Optimised schedule, cutoff and stockpileStockpile utilisation is an additional degree of freedomthat can be added to the optimisation. The impact ofstockpiles on cutoff grade has yet to be optimised in theliterature. However, the presence of a stockpile allowsexcess mining capacity to be economically utilised, aslow grade material can still be processed at a later date.The stockpile acts to redistribute grade throughout thelife of the operation, typically bringing high gradematerial forward and delaying the processing of lowgrade material. Stockpiles do have associated costs(stockpile addition and rehandling costs) which havebeen incorporated. For demonstration purposes, limitson stockpile size and rehandling were not considered,other than the requirement that stockpile addition andrehandling quantities be included in the overall mininglimit.

The resulting NPV for this scenario is $437 million,representing a 26% improvement due to the presenceof the stockpile. Analysis of the schedule (Fig. 8)shows that mining rate is more evenly distributed thanin previous cases, as the stockpile is able to providebalance. This is achieved by utilising periods withexcess mining capacity (years 2, 3, 5, 7, 9 and 11) tostockpile lower grade economic material with the viewto processing it in periods when there is not enoughore coming from the mine (years 4, 6, 8, 10 and 12).

The direct ore cutoff grades for the stockpile case aresignificantly higher than the without stockpile case(Fig. 9). This is demonstrated in every period. Thereare two key reasons for this. For the no stockpile case,there is excess mining capacity in a number of periods.With the option of stockpiling, some or all of this excesscapacity can be utilised to stockpile material. Only the20 Mt of highest grade material will be sent directly tothe mill. Mining material faster generally uncovershigher grade ore sooner. As a result, the cutoff gradeincreases, with the low grade economic ore being sent tothe stockpile. The second key reason is linked to thematerial that is already in the stockpile. A high directore cutoff grade in the early periods can lead to somemoderate to high grade ore being placed on thestockpile. Potential ore in the pit competes with thematerial in the stockpile for processing. The direct orecutoff must have a higher grade than the stockpiled

material in order to be processed directly, pushing up thedirect ore cutoff grade.

These issues are very difficult to resolve manually.There are discounting implications of optimising cutoffgrade for stockpiling scenarios. This multiperiod opti-misation formulation was able to resolve the graderedistribution and discounting issues associated with thestockpiling of material.

Blend for better recoveriesThe second project configuration explores the potentialto limit the percentage of soluble copper in the flotationprocess to improve the metallurgical recovery of bothgold and copper. By limiting the average percentage ofsoluble copper (CuSol) in each time period to amaximum of 25%, an improvement in metallurgicalrecovery for both copper and gold of y7% is assumed tobe possible. Increasing the copper and gold recoveries bythe same percentage ensures that the equivalent coppergrade calculation is still valid. The blending limitsrepresent average throughput, high CuSol blocks canbe blended with low CuSol blocks to increase processedtonnes. Limits are not applied to individual blocks. As inthe base case configuration, the mining capacity is50 Mtpa and processing capacity is 20 Mtpa. Thissection examines how value is optimised for thisconfiguration under various assumptions.

Marginal cutoff grades are no longer valid with theincorporation of blending CuSol. Nor is the definition of asingle cutoff grade. The selection of material for proces-sing will depend on the percentage of CuSol as well as thegrade of the block (CuEq). As selection for processing isdependent on the both CuEq and CuSol, the grouping ofblocks must consider both. The optimisation will generallyprefer high CuEq and low CuSol material. Depending onthe grade distribution, one of these elements may becomemore important. This may change over time. Figure 10shows the block allocation with 10 groups. A clusteringalgorithm attempts to allocate an equivalent number ofblocks to each group based on CuSol and CuEq proper-ties. As the number of groups increased, the range in gradecharacteristics of the each group reduces to more closelyreflect the resolution of the input block model. Analysisfound that 40 groups provided sufficient resolution forthis case, with the additional groups being necessary to

8 Optimised with stockpile schedule9 Direct ore cutoff grades for optimised with stockpile

and without stockpile cases





provide adequate resolution for the two-dimensionalgrade space.

The schedule for the 40 group case (Fig. 11) shows thatthe CuSol constraint is limiting in each of the first threeyears. In the first year, this constraint results in asignificant processing shortfall, as material in the upperbenches tends to have high CuSol grades. After year 3, theCuSol constraint has little impact on the solution, as thematerial in these areas has a lower CuSol grade.Significant revenue is foregone in the first three years, inreturn for greater metallurgical recovery. This scenarioreturns an NPV of $310 million, $38 million less than theequivalent base case.

Figure 12 demonstrates how ore and waste aredifferentiated in the presence of the CuSol constraintin year 2. There is a lower bound CuEq grade associatedwith the value of the block (CuSol does not generaterevenue). However, there is also a limit to the CuSolgrade accepted for any block to processing. Themaximum CuSol grade increases as the CuEq gradeincreases. The maximum CuSol grade accepted forprocessing in year 2 is y40%. For low CuEq grades,the maximum CuSol grade is 25%.

The ability to stockpile allows high CuSol materialthat is discarded from early periods in the previousoptimisation to be retained for later processing with lowCuSol material. As a result, the average CuSol gradeincreases from 19?5 to 24?5%, maximising utilisation ofhigh CuEq, high CuSol blocks. The NPV of this solutionis $467 million, and represents a 51% increase over the

no stockpile case. The majority of stockpiled material isgathered in years 1, 2, 3 and 5 when the average CuSolpercentage is high (Fig. 13). This allows the operation tobetter utilise mining capacity and continue for almosttwo years longer than the ‘no stockpile’ case.

The utilisation of the stockpile in year 2 is shown inFig. 14. The cutoff area for processing is similar to thatof the no stockpile case. However, the high CuEq, highCuSol material is now stockpiled rather than discardedas waste. This material is still able to be extracted for

10 Example of grouping with 10 groups for blended case

11 Optimised no stockpile schedule for blend case

12 Grades and recommended destination for no stockpile

case, year 2

13 Optimised with stockpile schedule for blend case

14 Grades and recommended destination for stockpile

case, year 2








high value at a point in time when it can be blended withlow CuSol material.

The cumulative discounted cashflow of the fourprominent scenarios (Fig. 15) shows that the ‘with stock-pile’ and ‘no stockpile’ scenarios for each processingconfiguration return similar total discounted value. For the‘no stockpile’ case, the base case processing configurationhas a higher NPV. This result is reversed for the ‘withstockpile’ case. The blend case processing configurationhas lower expected revenue in the early periods (particu-larly year 1) than the base case configuration, due to strictgrade constraints on CuSol. This gap is closed over theremaining periods, as low CuSol grade material becomesavailable and is able to be processed with improvedrecovery. While the NPV results for the two processingconfigurations are similar, the lower initial revenue of theblended case results in a longer payback period.

ConclusionsThe method demonstrated in this paper shows how tooptimise an open pit schedule incorporating cutoffgrades, blending, and stockpiling whilst accurately

modelling time and opportunity costs. The case studyshowed that, for this example, the method is able is ableto add substantial value over and above traditionaloptimisation methods.

This analysis does not guarantee to find the optimalproject configuration. Rather, it shows how a singleconfiguration can be optimised. It is the role of theengineer to determine the configurations they considerto have the potential to provide the highest value.

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15 Cumulative NPV comparison between cases




open pit optimisation – modelling time and opportunity costs

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