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Robust, flexible and operational mine designstrategies
B. Groeneveld*1, E. Topal2 and B. Leenders3
Strategic planning in mining is an important value accretive process. One of the most essential
aspects during the planning phase is determining the correct system design. A traditional mine
design process develops a fixed system for one set of conditions or expected values. An
alternative is to develop a robust system that deals with variation, by handling a range of
conditions within the optimisation process. Conversely, a flexible design can be generated which
changes the system dynamically over time in response to change. It is hypothesised that a flexible
design generates more value than a robust design which in turn generates more value than a
traditional design. However, due to constant change, a fully flexible design is not practical.
Ideally, a hybrid of the two methods would be optimal. An operational design is proposed as a
manual solution to this problem. This paper compares these different new design methodologies.
Keywords: Strategic planning, Decision making, Mine design, Robust mine design, Flexible mine design
IntroductionDecision making in mining operations can take manyyears due to the period of time required to explore anddevelop a large deposit. During the study and construc-tion period, many uncertainties can unfold and multipleeconomic cycles may occur. Making decisions based onsingle point estimates of the future limits flexibility,potentially resulting in premature foreclosure of anoperation. By considering changing conditions (bothupwards and downwards movements) in the decisionmaking process, a company is able to include flexibilityin an operation allowing value to be maximised. Currentmethodologies impair a decision maker’s ability tojustify this additional flexibility.
Building flexibility into an operation provides acompany with the ability to quickly respond to change;however, flexibility comes at a price. For example, anoperation of building a crusher to feed a processingplant has an initial plan to produce at 6 Mtpa. Anoption exists to build the allowance for a plantexpansion to 8 Mtpa (by increasing the size of thefoundations and footings to allow physical room forlarger equipment). However, when the decision to buildthe plant is made, to minimise initial capital investment,this flexibility is removed. One year into the operation,the sale price of the product doubles and costs slightlyincrease, while all other variables hold. In this environ-ment, it is considered favourable to expand theoperation; however, due to the cost cutting decision,this flexibility was removed from the plant and to make
the matter worse, production needs to be maintained sothe only option is to build a new crusher. Unfortunately,this will come at a significant capital cost and time tobuild; reducing the overall value of the operation hadthis flexibility initially been incorporated. This upfrontflexibility is difficult to justify with current decisionmaking tools. Advances in an area broadly known asreal options ‘in’ projects (ROIP) are beginning toaddress this gap (de Neufville et al., 2005; de Neu-fville, 2006; Wang and de Neufville, 2005, 2006; Cardinet al., 2008; Groeneveld et al., 2009).
Real options ‘in’ projects are located midway betweenfinancial real options analysis (which does not deal withengineering system flexibility) and traditional engineeringapproaches (which do not deal with financial flexibility).An analysis done using ROIP methods allows the designof a system under uncertainty to be studied. Through theanalysis process, the value of each design option can betested. Having this information allows the decision makerto make an informed decision on what flexibility to in-corporate in the final design.
Previous papers using ROIP methods have shown thevalue of this technique (Cardin et al., 2008; Groeneveldet al., 2009; Groeneveld and Topal, 2011). Cardin et al.(2008) implemented this technique for mining projectswith a Chilean mine in the ‘Cluster Toki’ region. In thepaper, a methodology is implemented where operatingplans are varied by truck fleet capacity and crusher sizein response to changing prices. For each price scenario,the optimal operational plan is selected. The applicationof this method resulted in y30% more project valuethan current estimates. This paper provides a strongbasis on which to grow ROIP theory in mining. Thoughthere are several deficiencies in the model. The approachlimits the flexibility by only including a handful of staticoperational plans; it assumes that the schedule ofmaterial moved is fixed, fails to deal with variation in
1Telfer Mine, Newcrest Mining Limited, Melbourne, Vic., Australia2Mining Engineering Department, Western Australia School of Mines,Curtin University of Technology, Kalgoorlie, WA, Australia3Strategic Development, Rio Tinto Iron Ore, Perth, WA, Australia
*Corresponding author, email [email protected]
20
� 2012 Institute of Materials, Minerals and MiningPublished by Maney on behalf of the Institute and The AusIMMReceived 20 March 2011; accepted 29 October 2011DOI 10.1179/1743286311Y.0000000018 Mining Technology 2012 VOL 121 NO 1
ore grade and recovery and fails to consider options inthe main value adding stages of a mining operation.
Groeneveld et al. (2009) outlined a methodology fordetermining a flexible mine design by dynamicallyincluding design options in the optimisation in orderto maximise value. Multiple ‘states of the world’ weresimulated and the optimal design for each state wasdetermined. A dataset of results was formed from theoutput of each simulated scenario. From this dataset, acumulative probability graph was produced, commonlyknown as a value at risk graph. This resulting curverepresents a theoretical maximum achievable value.However, this assumes that a decision maker can makeperfect decisions. In reality, this is impracticable.
An alternative is to produce a single fixed design whichhandles optimally handles change. This is achieved bytaking the full set of uncertainties into the optimisationand developing a fixed design that optimally handles thechanging uncertainties. The resultant design would be arobust design as it would best handle variation. Since theoptimisation includes high price and low price scenarios,it will attempt to produce a design that minimises anylosses and maximise any potential upside, while consider-ing that these are extreme scenarios and the mainscenarios are around the average. Note that this designis fundamentally different to a design just generated basedon the average value and this will be shown in theillustrative case study.
However, a robust design proposes that a ‘set andwatch’ approach is taken by management. Therefore, itfails to value active management of an operation. So, aflexible model proposes constant change which is notpracticable and a robust model does not allow manage-ment decisions. To overcome these limitations, it isproposed that an operational design be generated wherethe initial periods have a fixed ‘robust’ design and thelater years have a flexible design where management hasthe ability to have multiple options in the pipeline.
This paper compares these different design methodol-ogies. A summary of the methodology used for generat-ing the designs is outlined followed by the mathematicalformulation of the robust design under uncertainty, usingmixed integer programming (MIP) and Monte Carlosimulation (MCS) techniques. Furthermore, an opera-tional design is proposed as a solution to the limitationsof the robust and flexible design methodologies. Finally,the different design methodologies are compared againsteach other and a traditional design approach, in an illu-strative case study.
MethodologyIt is proposed that for these different design scenarios, acombination of MCS and MIP techniques be utilised.Uncertainties (or stochastic parameters) are simulatedthrough MCS as inputs to the MIP model. The MIPmodel allows for ‘go’ or ‘no go’ decisions to bemodelled. An optimised MIP model therefore deter-mines the optimal execution timing of design options fora set of uncertainties (‘states of the world’).
Design optionsFour categories of system design options are incorpo-rated in the model. These are mine options, preproces-sing stockpile options, processing plant options andcapacity constraint options. Mine options represent the
physical extraction capacity that is required to movematerial from the ground. This constraint may be anannual tonnage constraint or an effective flat haulconstraint which considers the time required to movematerial. Preprocessing stockpiles are stores of materialafter extraction from the ground, either for long termlow grade scenarios, fluctuating demand scenarios or forwaste material storage. Processing plant options repre-sent the physical and/or chemical process that isundertaken to ‘recover’ ore from the gangue material.Capacity constraint options represent physical capacityconstraints at any point in the network. These mayrepresent attributes such as port capacity, loadingfacilities, crusher capacities or conveyor capacities.
Resource representationThe representation of the resource in the model is byparcels of material which contain multiple grade bins.These parcels are designed to represent a physicalconstraint on the resource, such that they must be fullymined before mining a parcel lower in the physicalsequence. A parcel may be made up of one or moregrade bins. The average grade of a parcel is the weightedaverage of the grade bins. A grade bin represents aquantity of material at a specified grade. The size of theparcels can be determined by the modeller, but eachparcel requires a binary variable in the model forscheduling which increases the solution time of themodel. However, the grade bins within a parcel are anattempt to provide a level of detail that maintains theselectivity of the model.
Flow pathsA flow path represents a route for material to travel.Multiple processing plants/routes can be included andproducts can be generated at any point in the network.Different routes through the network are termed ‘flowpaths’. To explain this concept, further consider Fig. 1.
Examples of flow paths in Fig. 1 include the pathfrom the resource (R) to mine 1 (M1) to stockpile 1 (S1)to plant 1 (P1) through circuit 1 (C1) to product A whichwould be RM1S1P1C1A, the path from the resource (R)to mine 1 (M1) to waste stockpile 1 (W1) which wouldbe RM1W1, the path from the resource (R) to mine 1(M1) to stockpile 1 (S1) to plant 4 (P4) through circuit 2(C2) to product B (B) which would be RM1S1P4C2Band the path from the resource to mine 3 (M3) tostockpile 1 (S1) to plant 3 (P3) through circuit 1 (C1) toproduct A (A) which would be RM3S1P3C1A. This isonly a small number of the potential paths through thenetwork.
StockpilingStockpiling is used in mine operations for many reasonsincluding blending of material, storage of excess mineproduction, storage of waste material and storage of lowgrade ore for future production. When material isstockpiled, the grade and tonnage of the material isknown. However, as the material is mixed on thestockpile, the grade and the tonnage become unknown.Since the quantity of material in the stockpile isunknown before the optimisation, this gives rise to anon-linear constraint. To solve this problem, virtualgrade bins are created in the stockpile. These grade binshave a maximum and minimum grade of material whichcan enter the bin. As material is added to a stockpile, it
Groeneveld et al. Robust, flexible and operational mine design strategies
Mining Technology 2012 VOL 121 NO 1 21
will be fed into a grade bin that has a suitable graderange. Many grade bins can be created without adver-sely affecting the performance of the model which limitsthe averaging effect.
Stochastic parametersThe model incorporates uncertainty around the inputparameters by MCS. Each simulation of values repre-sents a ‘state of the world’ that is equally probable in thefuture. Various parameters can be incorporated in themodel, for example price, capital cost, operating cost,equipment utilisation, recovery and time to build.Running a set of simulations is intended to give arepresentative sample of the future ‘states of the world’.
ModelsThree different models have been proposed in order tojustify increased flexibility to determine a flexible design,a robust design and an operational design. All modelsuse MCS and MIP techniques to determine a systemdesign. The fundamental difference between the modelsis that under a robust design, multiple ‘states of theworld’ are considered together, while a flexible designconsiders just one ‘state of the world’ at a time. Anoperational design is developed by determining a fixeddesign for the first couple of periods and having aflexible design for periods after that. The flexible designmodel has been published previously in (Groeneveldet al., 2009; Groeneveld and Topal, 2011). The robustmodel is outlined in this paper and the operational planis a new hybrid of these two models with the maindifference that the design in the initial years is fixed.That is no binary value exits for design options in theinitial periods.
Flexible designSome researchers (Groeneveld et al., 2009; Groeneveldand Topal, 2011) have previously outlined a methodol-ogy for undertaking flexible mine design. The basis ofthese models was to optimise a design for a given singlestate of the world. This was achieved by includingmining, plant and capacity constraint design options inthe system and allowing the MIP model to internallyhandle the design options. MCS was used to generatethese different ‘states of the world.’
This methodology assumes that a decision makermakes optimal decisions based on the knowledge of allstates of the project over time (i.e. what price and costsoccurred over time). In reality, forecasting the exact final
state of a project is difficult if not impossible. Theproposed methodology provides information and insightthat can be used by the decision maker in conjunctionwith other tools to make timely, informed and valueadding decisions.
Robust designA new robust design methodology is proposed in thispaper to develop a design that better handles all ‘statesof the world’ as opposed to just a single design. It usesthe same concepts and assumption developed in theflexible design model but differs by considering numer-ous states at once. A robust design is achieved by solvingone ‘large’ MIP model that generates one design frommultiple possible options. In essence, this design is theone which handles a range of conditions the best out ofall possible options.
Operational designAn operational design methodology is proposed as apractical alternative to overcome the limitations of therobust and flexible methods. Essentially, an operationaldesign is where the initial years of a fully flexible designhave been fixed by the modeller. This allows manage-ment to make decisions today to enable the business tobe an ‘ongoing’ concern. By not fixing future decisions,management can maintain flexibility in order to benefitfrom any upside potential and minimise any downsiderisk. As this method incorporates future flexibility intothe analysis, the impact of the initial fixed decisions canbe tested and tweaked in order to maximise value.
Robust model formulationThe developed MIP model optimises the system designfor a risk neutral investor for all simulated ‘states of theworld’. Each design option can impact capital commit-ment, revenue generated and operating expenses. Theoptimisation process seeks to determine the design withthe highest equally weighted net present value for allgiven financial and technical scenarios. An outline of themathematical formulation is provided below.
NotationIndices
b a grade bin of material within a parceld product typee dependent optionsf flow path of material through the design
network
1 Example network of design options showing numerous flow paths
Groeneveld et al. Robust, flexible and operational mine design strategies
22 Mining Technology 2012 VOL 121 NO 1
g grade element of material within a resource
j bin of material within a stockpile: this binwill have a maximum and minimum grade ofmaterial which can enter
l design options
m mining options within the set of designoptions
n simulated ‘state of the world’
p parcel of material
r required rate of return on the project
t time period step (periods do not need to beequal)
s stockpiling options within the set of designoptions
y tolerance factor for the deviation of themining of a bin within a parcel
Capitalised indices are the maximum value or upperlimit of that index.
ParametersCl,t,n the capital cost of option l in time t for trial n
Dd,t,n the capacity of product d in time t for trial n
Dl,t,n the disposal cost of option l in time t for trialn
DT the lag time between these relationships (i.e.build option two, three periods after optionone)
FDl,t,n the fixed cost saved from not operatingoption l from time t for trial n to the end ofthe project life T
Fl,t,n the fixed cost of operating option l from timet for trial n to the end of the project life T
GLg,d the lower grade limit of grade g product d
GLj the lower grade limit of bin j
GUg,d the upper grade limit of grade g product d
GUj the upper grade limit of bin j
Gp,b the grade of parcel p bin b
Gg,l,n the grade g through plant l for trial n
Gg,s,j the calculated average, maximum or mini-mum metal units of grade g in stockpile s inbin j
Kl,t,n the capacity of option l in time t for trial n
Ks,t,n the stockpile capacity of stockpile s in time tfor trial n
Mp,b,l,t,n the mining cost from parcel p to bin bthrough mine option l in time t for trial n
Pg,d,t,n the sale price of grade element g for productd in time t for trial n (in $/metal unit)
Rl the recovery of material through circuit l
Rs,j the calculated average, maximum or mini-mum recovery for all material in stockpile sbin j
Rp,b the available resource of parcel p bin b
Rpz1 the available resource of the successor parcelpz1
Vl,t,n the variable cost of option l in time t for trial n
VariablesGg,d,t,n the metal units of grade g produced for
product d in time t for trial n
IDl,t1, if option l is disposed in time t;0, otherwise:
� �XIs,j,t,n the flow in from stockpile s bin j in time t for
trial n
XOs,j,t,n the flow out from stockpile s bin j in time tfor trial n
Xp,b,f,t,n the tonnage from parcel p bin b through flowpath f in time t for trial n
Xf,t,n the tonnage mined through flow path f intime t for trial n
Xp,b,t,n the tonnage mined from parcel p bin b intime t for trial n
Xpzl,b,t,n the tonnage mined from the successor parcelpz1 bin b in time t for trial n
Ye,t the dependent option e of Yl,t
Yl,t1, if option l is executed in time t;0, otherwise:
� �
Yp,t,n
1, if pushback p is fully mined in
time t for trial n;0, otherwise:
( )
Formulation
Objective functionThe objective function seeks to maximise the equallyweighted before tax net present value (NPV) for allsimulated ‘states of the world’
Xn
n~1
1
N
XT
t~1
1
1zrð ÞtXD,G
d~1,g~1
Pd,g,t,nGg,d,t,n{
"(
XP,B,F
p~1,b~1,f~1jl[f
Vl,t,nXp,b,f,t,n{XP,B,F
p~1,b~1,f~1 l[fj jl[m
Mp,b,l,t,nXp,b,f,t,n{
XL
l~1
Cl,t,nYl,t{XL
l~1
Fl,t,nYl,t{XL
l~1jt=1
Dl,t,nIDl,tzXL
l~1jt=1
FDl,t,nIDl,t
359=;
(1)
The constraints in the model can be divided into fivecategories: resource, option, stockpiling, product andflow balance constraints.
Resource constraints
XT
t~1
Xp,b,t,n{Rp,bƒ0 V p,b,n (2)
XB,t
b~1,tt~1
Xp,b,tt,nƒ
XB
b~1
Rp,bYp,t,n V p,t,n (3)
Xpz1,b,t,nƒRpz1
Xt
tt~1
Yp,tt,n Vp,t,n (4)
XT
t~1
Yp,t,nƒ1 Vp,n (5)
XB
b~1
1
Rp,b
Xp,b,t,n{1
Rp,b
Xp,b,t,nƒc% V p,b,t,n (6)
XB
b~1
1
Rp,bXp,b,t,n{
1
Rp,bXp,b,t,n§{c% V p,b,t,n (7)
The resource constraints in the model limit whatquantities and grades of material can be produced bythe system. The amount of material extracted from amining grade bin in a pit has an upper bound based onthe resource model which is constrainted by equa-tion (2). Scheduling constraints are encorporated by
Groeneveld et al. Robust, flexible and operational mine design strategies
Mining Technology 2012 VOL 121 NO 1 23
equations (3) and (4). Equation (3) ensures that a parcelis fully mined by setting the binary value to 1 in the periodthat the parcel is fully mined. Equation (4) forces a parcelpredecessor to be fully mined before mining starting inthe sucessor. Equation (5) is a set packing constraintwhich forces a parcel to only be mined once. Two fur-ther optional constraints (equations (6) and (7)) try tominimise grade variability by restricting the model tomine low grade and high grade material within a parcel inan even ratio within a given tolerance value (y%).
Option constraints
XF
f~1jl[f
Xf,t,n{Xt
tt~1
Kl,t,nYl,ttzXt{1
tt~1
Kl,t,nIDl,ttƒ0 V l,t,n (8)
Yl,t{Xt{DT
tt~1je[l
Ye,ttƒ0 V l,t (9)
IDl,t{Xt{1
tt~1
Yl,tt{Xt{1
tt~2
IDl,ttƒ0 V l,t=1 (10)
Six constraints are used to model the various designoptions. Each option is reflected in the model by a binaryvariable. This binary variable reflects a decision aboutwhether this option is used or not. An upper capacitylimit for each option is set through equation (8). Optiondependence relationships can be modelled through theuse of equation (9). Equation (10) introduces a disposalbinary variable to model disposal or closure of an optionand this equation ensures that options cannot be disposedof unless they have previously been built.
Stockpiling constraints
XIs,j,t,n~XP,B,F
p~1,b~1,f~1js,j[f
Xp,b,f,t,n
V s,j,t,njGp,bwGLj and Gp,bvGUj (11)
XOs,j,t,n~XF
f~1js,j[f
Xf,t,n V s,j,t,n (12)
XOs,j,t,nƒ
Xt{1
tt~1
XIs,j,tt,n{Xt{1
tt~2
XOs,j,tt,n V s,j,t,n (13)
Ks,t,n§
XJ,t
j~1,tt~1
XIs,j,tt,n{XJ,t
j~1,tt~2
XOs,j,tt,n V s,t,n (14)
Stockpiling is handled in the model through the use ofvariables for each flow path to track material flowinginto a grade bin within a stockpile. A grade bin definesthe upper and lower grade limits of material which canflow into a bin. The material flowing into a stockpileequals the tonnage from all flow paths into the stockpile,as stated in equation (11). Each grade bin of material inthe resource has exactly one bin in the stockpile that itcan flow into based on the upper and lower grade limitsof the stockpile bin (this is handled preoptimisation inthe model formulation process). Likewise, the totaltonnage flowing out of a stockpile bin equals the sum ofthe tonnage in each flow path out of the stockpile.Furthermore, the material leaving the stockpile must not
exceed the material entering the stockpile, modelled byequation (13). Equation (14) applies an upper limit tothe stockpiling capacity.
Product constraints
Gg,d,t,n{GUg,d
XP,B,F
p~1,b~1,f~1jd,l[f js=[f
RlXp,b,f,t,n{
GUg,d
XF,J,S
f~1,j~1,s~1jd,j,s[fjt=1
Rs,jXOs,j,t,nƒ0 V g,d,t,n
(15)
Gg,d,t,n{GLg,d
XP,B,F
p~1,b~1,f~1jd,l[fjs=[f
RlXp,b,f,t,n{
GLg,d
XF,J,S
f~1,j~1,s~1jd,j,s[f jt=1
Rs,jXOs,j,t,n §0 V g,d,t,n
(16)
Dd,t,n{XP,B,F
p~1,b~1,f~1jd,l[f js=[f
RlXp,b,f,t,n{
XF,J,S
f~1,j~1,s~1jd,j,s[f jt=1
Rs,jXOs,j,t,n§0 V d,t,n (17)
The system can produce multiple products allowingdifferent marketing strategies to be analysed. Twoequations handle the grade limits for the variousproducts produced by equations (15) and (16).Equation (15) determines the grade of the materialthrough the flow path less the upper grade limitmultiplied by the material recovered from the plant.Equation (16) handles the lower grade limit. An uppercapacity limit for the amount of product produced isrestricted through equation (17).
Flow balance constraints
Xp,b,t,n~XF
f~1
Xp,b,f,t,n V p,b,t,n (18)
Xf,t,n~XP,B
p~1,b~1
Xp,b,f,t,n V f ,t,njf 6 [s (19)
Gg,d,t,n~XP,B,F
p~1,b~1,f~1jd,l[f js=[f
Gg,l,nRlXp,b,f,t,nz
XF,J,S
f~1,j~1,s~1jd,j,s,l[f jt=1
Gg,s,jRs,jXf,t,n V g,d,t,n (20)
Table 1 Summary of deposit
Parcel Bins TypeTonnage/Mtpa Copper/% Gold/g t21
1 37 Ore 73.8 1.3 0.11 1 Waste 118.3 … …2 29 Ore 57.5 1.4 0.22 1 Waste 244.5 … …3 9 Ore 14.5 1.4 0.13 1 Waste 74.1 … …4 18 Ore 25.6 1.2 0.14 1 Waste 190.86 … …
Groeneveld et al. Robust, flexible and operational mine design strategies
24 Mining Technology 2012 VOL 121 NO 1
Several flow constraints are used in the model to equatethe amount of material entering a node (or option) withthe amount of material exiting a node. Briefly, theseconstraints represent:
(i) equation (18): the tonnage from a bin equals thetonnage through all paths from that bin
(ii) equation (19): the tonnage mined through eachflow path that does not go through a stockpileequating to material mined from each parcel andbin through the flow path
(iii) equation (20): the total amount of metal unitsrecovered to a product equates to the metalcontent through all direct feed to a plant plus anymaterial treated after being stockpiled.
Non-negativity and integrality constraintThese constraints enforce non-negativity and integralityof the variables, as appropriate
Gg,d,t,n,Xp,b,f,t,n,Xp,b,t,n,Xf,t,n,XIs,j,t,n,XOs,j,t,n§0
V g,d,p,b,f ,s,j,t,n
IDl,t Integer V l,t and Yl,t, Yp,t binary V p,l,t
(21)
Illustrative case studyThis case study examines the use of several differentmining capacity and plant capacity options for adeposit. The deposit was divided into four parcelsgenerated by a single deterministic optimisation, suchas Gemcom Whittle pit shells. While this may beconsidered to be removing the optimality from themodel upfront, it was primarily used as a starting pointfor ‘shape’ generation. Refinement through an iterativeprocess could easily be undertaken to improve optim-ality. Additionally, the purpose of this case study is toexamine the execution of mining and plant options,which is not heavily reliant on schedule and/or sequenceof extraction.
Table 1 provides a summary of the deposit used inthis case study. The case study uses a single resourcemodel; however, multiple stochastic models could beincluded. For the purposes of this case study, a physicalconstraint exists between parcels in a sequential order,i.e. parcel two cannot start before parcel one is finished.
This case study will use four mining options, fourprocessing options and two stockpiling options toundertake an analysis of the deposit. A summary ofthe options included in the model is outlined in Table 2.Stochastic variables for the following items incorporatethe uncertainty into the model, commodity prices for
gold and copper, capital cost, operating cost and plantutilisation. No detailed analysis of the underlying natureof the stochastic variables was undertaken, as detailedresearch in this area has been undertaken in other papers(Dowd and Xu, 1995; Godoy and Dimitrakopoulos,2004; Dimitrakopoulos and Abdel Sabour, 2007; Topal,2008; Shafiee and Topal, 2010).
Distributions were set-up based on the parametersbelow. An assumption of constant growth in values wasmade and values were generated following the Markovproperty. For illustrating the techniques proposed, howthe variables are simulated does not have a materialimpact. The distributions are:
(i) gold price: starting value of $1200/oz, with anormal growth factor of 4% with standarddeviation of 20%
(ii) copper price: starting value of $5000/t, with anormal growth factor of 5% with standarddeviation of 10% and a correlation of 60% tothe gold price
(iii) capital cost multiple: with a normal growthfactor of 3% and a standard deviation 5% and acorrelation to the gold price and copper price of10%
(iv) operating cost multiple: with a normal growthfactor of 4% and a standard deviation 10% anda correlation to the gold price and copper priceof 30% and the capital cost index of 40%
(v) plant utilisation: triangular distribution with alower limit of 40%, midpoint of 85% and upperlimit of 95%.
Four hundred simulations or ‘states of the world’ wereused for the case study. Summaries of the ranges of
2 Gold price distribution over time
Table 2 Summary of design options
Option type Capacity/Mtpa Capital/$m Fixed cost/$/year Variable cost/$/t Disposal cost/$m
Mine A 20.0 38.5 2.60 1.22 3.5Mine B 25.0 42.5 3.25 1.21 4.0Mine C 30.0 45.0 3.90 1.12 4.5Mine D 50.0 55.0 6.50 1.10 5.5Processing A 7.5 575 13.0 1.75 5.0Processing B 10.0 700 16.8 1.31 7.0Processing C 12.5 850 20.0 1.05 8.5Processing D 15.0 975 22.5 0.87 9.5Stockpile waste 3600.0 … … 0.20 …Stockpile low grade 20.0 … 1.00 0.60 …
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Mining Technology 2012 VOL 121 NO 1 25
values for gold and capital cost index are shown inFigs. 2 and 3 respectively.
ResultsFive different design scenarios were modelled based onthe three different design methodologies. A robust, aflexible and two operational designs, along with atraditional scenario were generated. All models wereprocessed on a Quad core Ubuntu server with 2?66 GHzand 4 Gb RAM using the Gurboi 4?5?2 software.Solution times varied; however, for a fully flexiblemodel, the average solution time was 30 s. A summaryof the results is outlined in Table 3 with a discussionbelow.
A traditional plan for operating a mine was generatedby selecting the optimal design for a single deterministicscenario. The value for the stochastic parameters wasgenerated by taking the starting values and applying theaverage growth factors only over time, i.e. no randomvariation was included. This produces a value which issimilar to current industry practice for generating adesign. For this scenario, the design generated was tobuild mine D, plant B and plant D in period one and
mine A in period three with a net present value of$4050m. This design was then evaluated under thedifferent simulated ‘states of the world’ to test itsperformance under uncertainty.
A flexible design model was used to produce anoptimal design for each ‘state of the world’. Thisproduced a result which provided an upper bound forthe achievable value from the deposit, as the assumptionis made that decision makers have perfect informa-tion and can therefore make perfect decision into thefuture. To analyse the options used in the model, afrequency of execution table and graph (Table 4 andFig. 4 respectively) were generated. These show howoften a particular option is used, calculated by dividingthe number of times that an option is used by number ofsimulations. The available number of times that anoption could be executed was 400. This showed that inevery scenario, mine D and plant D were built. Mine Cwas built in most ‘states of the world’ and plant C wasbuilt in roughly half the ‘states of the world’. It alsoshows that around period five, there is increased needfor mining capacity, as a handful of scenarios expandcapacity in this period.
Next, a robust design was generated by optimising thedesign for multiple ‘states of the world’ in one model. Theresulting design was to build mine option D, plants C andD in period one and mine option C in period two. Itshould be noted that processing of the model with over 40‘states of the world’ is slow. This is because the number oflinear variables doubles with each additional ‘state of theworld’. For the model with 40 ‘states of the world’, therewere 22 million variables. Gurobi 4?5?2. solves this modelin 9 h on a Quad core 2?66 GHz Ubuntu Server with4 GB of RAM. In order to test the sensitivity of themodel, three scenarios were run with 25 ‘states of the-world’ and two scenarios with 30 ‘states of the world’.
Table 3 Summary of net present value results in $mfrom different models
AverageStandarddeviation Minimum Maximum
Traditional 3920 1050 1340 7640Flexible 4350 1140 1740 8070Robust 4260 1110 1560 8050Operational v1 4210 1150 1530 8055Operational v2 4290 1090 1670 8065
Table 4 Frequency of execution for flexible model by period
All periods 1 2 3 4 5 6 7 8 9 10
Mine A 40% 4% 3% 3% 4% 11% 7% 6% 2% 0% 0%Mine B 42% 1% 1% 3% 7% 19% 8% 4% 1% 0% 0%Mine C 97% 13% 7% 13% 19% 31% 11% 2% 2% 1% 0%Mine D 100% 86% 6% 2% 1% 1% 0% 4% 1% 0% 0%Plant A 13% 9% 1% 1% 0% 0% 0% 1% 0% 1% 0%Plant B 33% 28% 3% 2% 0% 0% 0% 0% 0% 0% 0%Plant C 51% 47% 2% 1% 0% 1% 0% 0% 0% 0% 0%Plant D 100% 90% 7% 2% 0% 1% 0% 0% 0% 0% 0%
4 Frequency of execution graph for flexible and opera-
tional design scenarios
3 Capital cost index distribution over time
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The states were all different scenarios generated fromthe MCS; however, the all generated the same designsolution. So, arguably for this model, the number of‘states of the world’ included was adequate at 25; how-ever, this may be different for other models and should betested for each deposit and set of options.
Based on the results of these results, two operationalmodels were developed. The design was fixed for the firsttwo periods. For period three onwards flexibility wasavailable so the model could turn on and off designoptions. The model formulation for this was the same asthe flexible model with the only difference being that thebinary values for the design options were fixed in periodsone and two. Two scenarios were constructed: versionone involved mine D being built in period one, plant Din period one and plant C in period two and version twoinvolved mine D in period one and mine C in periodtwo, plant D in period one and plant C in period one.Version one of the design has less capacity in the initialyears and produces less value than version two. Versiontwo produces more value as it has greater capacity in theinitial years (just having greater capacity earlier wouldnot always produce greater value). Interestingly, versiontwo design is exactly the same in the initial years as thedesign generated by the robust design. The difference isthat the operational design allows for flexibility in thelater years, thus the value of the operational design ishigher than the robust design. Also, version one under-performs when compared with the robust design, so thiswould indicate to the modeller that there is a betteroperational solution to be found. However, had therobust design and the flexible design not established thisbase line, this finding would have been missed.
A value of risk graph is a cumulative probabilitydistribution of project value in each ‘state of the world’.This is produced to highlight the differences betweenthe models, as shown in Fig. 5. It allows a decisionmaker to quickly and easily compare the different designscenarios.
In comparing the models, it can be seen that the flexiblemodel produces a higher expected value than any othermodel. This is as expected since the flexible model pro-duces an optimal design for each ‘state of the world’. Such
a flexible design may be impractical in reality. However, arobust design which does not incorporate any flexibilityproduces lower value than an operational and a flexibledesign as it is prohibited from reacting to change. Theoperational plan design has a lower expected value thanthe fully flexible design but higher than the robust designas it maintains flexibility in the later years and it is onlythe initial years that flexibility is limited. A traditionaldesign produces the lowest expected value overall sinceit has no flexibility and is not optimised to handleuncertainties.
The differences between the expected values of eachdesign approach can be attributed to two key aspects.First of all, actively managing the operation andallowing a flexible design (one that changes over time)will contribute significant additional value. The secondcomponent that contributes to additional value is beingable to develop a robust fixed design which can handle arange of conditions. Refinement of the initial designchosen for the operational plan may lead to the dis-covery of plans with a higher expected value.
ConclusionsThe paper has compared three different new designmethodologies: flexible, robust and operational andextended the application of real options ‘in’ designtheory to mining. It has been shown with clarity that allthese methods outperform an approach which usesaverage values (the current traditional design approach).A fully flexible design approach generates the mostvalue; however, it has practical implementation issuesdue to constant change. A robust design produces asingle design that handles variation the best; however,the design does not change over time and therefore doesnot value active management. An operational design isproposed to overcome the limitations of the other designmethodologies. While it does not produce as much valueas the flexible design (which is only a theoreticalmaximum), it does produce more value than a robustdesign. Finally, the worst performing approach was thetraditional approach which does not consider flexibili-ty and uncertainty. Flexible operations produce greater
5 Value at risk graph for design scenarios
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project value than fixed, rigid operations, thus activelymanaging an operation is imperative to maximisingvalue.
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