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Integration of real options into short-term mine planning and production scheduling
M INING SCIENCE AND TECHNOLOGY Mining Science and Technology 19 (2009) 0674–0678
www.elsevier.com/locate/jcumt
Integration of real options into short-term mine planning and production scheduling LI Shu-xing, KNIGHTS Peter CRC Mining, University of Queensland, Brisbane Qld 4072, Australia Abstract: Commodity prices have fallen sharply due to the global financial crisis. This has adversely affected the viability of some mining projects, including leading to the possibility of bankruptcy for some companies. These price falls reflect uncertainties and risks associated with mining projects. In recent years, much work has been published related to the application of real options pricing theory to value life-of-mine plans in response to long term financial uncertainty and risk. However, there are uncertainties and risks associated with medium/short-term mining operations. Real options theory can also be applied to tactical decisions involving uncertainties and risks. This paper will investigate the application of real options in the mining industry and present a methodology developed at University of Queensland, Australia, for integrating real options into medium/short-term mine planning and production scheduling. A case study will demonstrate the validity and usefulness of the methodology and techniques developed. Keywords: real options; mine planning; production scheduling; economic uncertainty; dump truck dispatch
1
Introduction
The mining industry has gone from boom to bust rapidly with the current economic crisis. This is a typical example of economic uncertainty associated with mining industry. To survive during this period of economic uncertainty, mining companies should pursue alternative strategies in their mine development. The strategies in real options, such as deferring, scaling down or abandoning a mining project, may be applicable to current economic situations. These real options can convert economic uncertainty into opportunities. Indeed many mining companies are implementing those real options’ strategies in their life-of-mine/long-term mining plan in order to minimise economic losses. A real option is a right, but not an obligation to introduce a contingency plan into a project within a specific period of time allowing for predetermined conditions. Real options’ principles and strategies are applicable to operating mines in medium/short-term mine planning. Implementing alternative mine designs and production schedules can reduce the overall mining cost and improve the overall mine performance in response to the various uncertainties, such as economic, geological and geotechnical uncertainties[1–2]. This paper will introduce a new form of real opReceived 11 February 2009; accepted 25 April 2009 Corresponding author. E-mail address: [email protected]
tions, or mine planning flexibility and contingency, into medium/short-term mine planing and production scheduling process and demonstrate the method for valuating these flexibilities based on the concept of real options. The uncertainty parameter addressed in this paper is fuel price uncertainty. Real options’ strategies are proposed for medium/short-term mine plan in response to uncertainties in fuel price. The methods for estimating the values of the real options are developed. The case study demonstrates that real options’ principles and strategies are useful for medium/short-term mine planning in response to the uncertain situations.
2
Applications of real options in mining industry
Real options strategies originated from finance and stock market investment analysis. In corporate finance, real options, such as put and call options, are applied in capital budgeting decisions. Put option is the right to sell the underlying asset to receive the exercise price. Call option is the right to buy the underlying asset by paying the exercise price. In the mining industry, there are a limited research and case studies applying real options to valuate mining projects. The real options techniques identified for mining project include cancellation, expan-
LI Shu-xing et al
Integration of real options into short-term mine planning and production scheduling
sion, deferral and abandonment. These flexibilities are mainly identified and applied for life-of-mine (LOM) planning in response to the uncertainty of commodity price. For instance, Lemelin et al. apply real options techniques to evaluate Mine 2 at Raglan[3]. It is a complex mining project, which consists of a number of mineralized zones and produces many payable metals, and as such, the valuation process becomes very complicated. A least-squares Monte Carlo (LSM) approach is used for valuing real options. The valuation of the mineralised zones takes into account the uncertainty associated with the prices of all payable metals simultaneously, and the management flexibility to switch among the different operating alternatives. An economic evaluation of Mine 2 at Raglan is improved in both precision and accuracy using the real options analysis (ROA) methodology. The business model includes the flexibility of choosing different production options throughout the mine life depending on variables such as metal prices, foreign exchange rates and fuel prices. It is a dynamic process that forecasts the value of a project should expansion or abandonment be decided. Shafiee and Topal apply real options valuation in a conceptual mining project. In this application, a hypothetical gold mine is used as a case to demonstrate the project evaluation using DCF and ROV methods[4]. DCF method assumes that time and risks are constant in evaluation. ROV assumes that time is flexible and also manages the risk. ROV gives flexibility in choosing the project time, controlling extraction rates, cut-off grade, expanding capacity, temporary closing and eventual abandonment. The ROV method produces a higher NPV than that of DCF method. The operating flexibility is also incorporated into the design of a mineral processing plant[5]. To maximize the profitability of a mineral processing plant, possible future variations in plant performance due to the uncertainty should be considered before the plant is constructed. The plant performance can be improved either by re-adjusting some operating parameters or by adding new capacity. The study shows that the capital investment is necessary in order to improve the plant performance. The study suggests a method for valuing the operating flexibility of mineral processing plants based on the real options techniques.
3
Incorporation of real options into medium/short-term mine planning and production scheduling in response to the fuel price uncertainty
3.1
Proposed real options in dump truck dispatch plan
With the increase of fuel price, the fuel cost for
675
haul trucks in overburden removal operation accounts for a significant portion of operating cost in a surface mine. In a medium/short-term mine plan and production schedule, the optional plan for haulage operation and the usage of trucks in response to the ups and downs of fuel price may significantly reduce fuel consumption and the operating cost of overburden removal, and therefore improve the overall performance of the mine. Integrating real options into dump truck dispatch plan in response to the fuel price uncertainty is a solution to the operating cost increase. A real option for dispatching dump truck plan is proposed in terms of the fuel price uncertainty. In this proposed real option, the following assumptions must be held: 1) the production level should be maintained; 2) the volume constraint of each overburden dump should be observed; 3) trucks and shovels should be fully utilised; 4) the number of operating trucks on site can be scheduled based on the fuel price variations; 5) the cost for maintaining extra trucks and haul roads should be accounted for. The proposed real option is to dispatch the trucks to closest dumps during the period of high fuel price in order to reduce the number of trucks used. Therefore the fuel cost of dump trucks will not blow out and can be kept under control. On the other hand, the destinations of trucks will be reversed during the period of low fuel price in order to balance the storage volume of dumps. More specifically, if the fuel price increases to a certain level, which may be called an upper trigger level, the real option of dump truck dispatch plan will be implemented. The real option of dump truck dispatch plan during this high fuel price period means to alter the dump trucks destinations to the closest dumps in order to lower the number of trucks used, reduce the amount of fuel consumed and save fuel costs. On the other hand, if the fuel price decreases to a level lower than the upper trigger level, the dump truck dispatch plan will be back to normal. Furthermore if the fuel price continues to decrease and reach a lower trigger level, a dump truck dispatch plan will be implemented that makes up the imbalance of dump storage capacities. 3.2
Implementation and evaluation of real options proposed
Proposed real options are implemented in two stages. The first stage is implemented during the period of fuel price higher than the upper trigger level. The purpose is to reduce fuel consumption and cost during this period. The second stage is implemented in the period of fuel price lower than the lower trigger level. The purpose is to rebalance the dump storage capacities during this low fuel cost period. The method proposed for implementing real options in dump truck dispatch plan are as follows:
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1) simulate fuel price using mean-reverting model for each time interval; 2) obtain the model of the fuel price at each time interval; 3) select the fuel price assumed in the original mine plan as an upper trigger level; 4) if the fuel price increases over the upper trigger level and will last for a prolonged period, implement the first stage of the real option for dump truck dispatch plan; 5) if the fuel price decreases below the upper trigger level, the dump truck dispatch plan is reversed back to normal; 6) if the fuel price decreases to the lower trigger level, the dump truck dispatch plan will implement the second stage of the real options, i.e., to move overburden materials to the under-filled dumps in order to balance the dump storage capacity until the balance is achieved; 7) once the balance is achieved, or the fuel price increase above the lower trigger level, the truck dispatch plan is reversed back to normal; 8) as the variations of fuel price, repeat the process from 4 to 7. The real option implemented needs to be valued to determine its worthiness and effect on the overall performance of a mine operation. The value of a real option implemented in mine planning is the value difference between the original plan and the real option plan. The valuation method of a real option follows the steps below: 1) collect the data for overburden removal operation; 2) review the original dump truck dispatch plan (O plan); 3) set the upper trigger of fuel price. Once the fuel price increases to the level above an upper trigger: 4) calculate the cost of O plan; 5) implement the first stage of real option plan (R1 plan); 6) calculate the cost of R1 plan. Once the fuel price decreases to the level below a lower trigger: 7) calculate the cost of O plan; 8) implement the second part of real option plan (R2 plan); 9) calculate the cost of R2 plan; 10) sum the costs of O plan during the periods of both fuel price increases and decreases; 11) sum the costs of both R1 and R2 plan; 12) calculate the maintenance cost of trucks and haul roads; 13) subtract the costs of R1+R2 plan and maintenance cost from O plan to obtain the value of real option implemented at each time interval; 14) extend to whole period when the real option implemented;
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15) the final result is the value of real option implemented.
4
Case study and results
4.1 Description of mine site A case study has been conducted in a hypothetical surface mine site. The overburden removal areas and dumps are shown in Fig. 1. There are two working areas and two dumps. Overburden in working area 1 will be dumped at Dump 1 while overburden in working area 2 will be dumped at Dump 2. The storage capacity of Dump 1 can only hold the overburden materials from working area 1 while the storage capacity of Dump 2 is only large enough to store the overburden materials from working area 2. Distances of various routes are listed in Table 1.
Fig. 1
Layout of working areas and dumps
Table 1
Distances of various routes
Route
Distance (m)
Dump capacity
Working area 1 -> Dump 1
3000
Dump 1 for working area 1
Working area 2 -> Dump 2
1000
Dump 2 for working area 2
Working area 1 -> Dump 2
2000
Working area 2 -> Dump 1
4000
4.2 Real options In response to the volatility of fuel price, the real options are proposed to be implemented in overburden removal operation in this case study. The purpose of implementing real options in dump truck dispatch plan is to reduce the number of trucks used in overburden removal in the event of fuel price becoming very high. In this case study, the upper trigger level of fuel price is assumed as Aus$1.20 per litre. Conditional to the same production level, the reduction of the truck number is achieved by shortening the haulage distance which means to change the dumping destinations to nearest dumps. The trucks leaving idled can be scheduled for maintenance. Also, scheduled new trucks purchase can be re-scheduled for a later time. When the fuel price decreases to a lower
Integration of real options into short-term mine planning and production scheduling
trigger level, the truck dumping destinations are reversed to balance the dumps’ storage capacities. The lower fuel price trigger level of Aus$0.80 per litre is assumed in this case study. During the period of fuel price falling into the range between the upper and lower trigger levels, the original dump truck dispatch plan is implemented. 4.3 Fuel price simulation It is widely recognised that the price of oil and some commodities are modelled using Brownian motion models and mean reverting models. The common stochastic process describing the behaviour of commodity prices are Brownian motion models, meanreverting models based on the Ornstein-Uhlenbeck process and jump processes. Geometric Brownian motion (GBM) and mean-reverting process (MRP) can be easily estimated from historical data of commodity prices. The mean-reverting model is adopted in this case study to simulate the fuel price. Mean-reverting process (MRP) is modelled using Eq.(1)[6]: dS = Ș(ȝ–lnS)Sdt + ıSdz
(1)
where S is commodity price; Ș the reversion speed; ȝ the logarithm of the equilibrium price; ı the standard deviation; dz the increment of a Wiener process, which equals İtdt. The random variable İt has zero mean and unit standard deviation. According to this process, the expected instantaneous drift is not constant, but depends on the difference between the logarithms of the spot price S and the equilibrium price ȝ. If S is lower than the equilibrium price ȝ, the expected drift for the next period is positive. If S is higher than the equilibrium price, the expected drift will be negative. Therefore the stochastic process reflects well the forces of supply and demand that control the commodity price. Fig. 2 shows the simulation of fuel price for a 5 year period. In this realisation, the highest fuel price reaches Aus$1.40 per litre and lowest Aus$0.60 per litre. The equilibrium price is Aus$1.00 per litre. There is a period of 10 months during which the fuel prices are above the upper trigger level. Interestingly there is also a period during which the fuel prices are below the lower trigger level. Therefore the two stage real options can be implemented during these two up and down periods. 4.4 Valuation of real options To estimate the value of the real options implemented in this case study, various parameter assumptions on dump trucks and haulage operations are made. Table 2 lists the parameters assumed in calculating costing hours for dump truck.
677
)XHOSULFH
LI Shu-xing et al
Fig. 2
Upper and lower trigger levels and opportunity windows for real options
Table 2
Parameters assumed and calculations of costing hours
Basic data Distance (m)
Area 1 -> Dump 1
Area 2 -> Dump 2
3000
1000
Truck numbers
10
10
Truck payload (t) Fuel price upper trigger (Aus$/Litre) Fuel price lower trigger (Aus$/Litre) Times and trips
220
220
1.2
1.2
0.8
0.8
Manoeuvring (min)
1
1
Loading (min)
3
3
Accelerating (min)
1
1 600
Haulage speed (m/min)
600
Decelerate and dump (min)
1.5
1.5
Return speed (m/min)
600
600
Circle time (min)
16.5
9.8
Trips per hour Tonnage removed and costing hours per truck Tons per hour continuous
3.6
6.1
800
1342
Overall job efficiency (%)
75
75
Mechanical availability (%)
85
85
Annual outage factor (%)
95
95
Production utilisation (%)
0.6
0.6
Scheduled hours per year
7248
7248
Productive hours per year
4389
4389
3511656
5892439
483
483
Annual production (t) Costing hours per month
Based on the parameters assumed and the fuel price simulations, the real option proposed in this study for dump truck dispatch plan is valuated. The value of real option is calculated using Eq.(2). Vro = Cor – Cro – Cm
(2)
where Vro is the value of real option implemented; Cor is the fuel cost of original plan; Cro is the fuel cost of real option plan; Cm is the cost of maintenance or exercise.
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Mining Science and Technology
In Eq.(2), the cost of original plan (Cor) is divided into two parts: 1) the cost of original plan during the period of fuel price above upper trigger level (Corh); 2) the cost of original plan during the period of fuel price below lower trigger level (Corl). Similarly, the cost of real option plan (Cro) is also calculated in two parts: 1) the cost of real option plan during the period of fuel price above upper trigger level (Croh); 2) the cost of real option plan during the period of fuel price below lower trigger level (Crol). The results of valuation on real option implemented in this study are shown in Table 3. If the fuel price varies as shown in Fig. 2, the proposed real option for a 10 months period is worth approximately Aus$0.7 m, which accounts for 2.5% cost savings. Table 3
Value of real option implemented (costs in Aus$)
Original cost during 10 months period
Area 1 -> Dump 1
Area 2 -> Dump 2
Total
Above upper trigger price ($1.2)
9207339
9207339
18414678
Below lower trigger price ($0.8)
4573843
4573843
9147686
Area 1 -> Dump 2 7347270 Area 1 -> Dump 1 4573843
Area 2 -> Dump 2 9207339 Area 2 -> Dump 1 5497852
Total cost of original plan Real options cost during 10 months period Above upper trigger price ($1.2)
Below lower trigger price ($0.8)
27562364
16554609
10071695
Cost of implementing options
250000
Total cost of real options plan
26876304
Final value of real options
686059
4.5 Discussions In implementing the real option proposed for dump truck dispatch, an opportunity window is required. Two variables, fuel price (P) and time span (T) of fuel price variations, define an opportunity window. To enable the implementation of a real option proposed, the opportunity window should be large enough to allow the gains to offset costs. In other words, the fuel price needs to increase above upper trigger level and decrease to below lower trigger level and also last for a substantial length in both cases. If the fuel price varies substantially, the time span required could be shorter, otherwise, the time span required must be in a prolong time. The relationship between fuel price variations and time span required can be described using Eq.(3). This equation is derived from the Eq.(2). Cm T= a[(Trn − Trnh ) Ph + (Trn − Trnl ) Pl ]
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lower trigger level; Cm is the cost of maintenance; a is a constant related to fuel cost each truck. Based on Eq.(3), if fuel price is well above the upper trigger level, the time span required can be shorten. Also if the fuel price is well below the lower trigger level, the time span can also be shorten. On the other hand, if the time span is large, the upper or lower fuel price trigger level can be narrowed. To offset the dump storage capacity, the time span for lower opportunity window should be equal to that for upper opportunity window. The cost arisen from implementing the proposed real options must be accounted for in real options valuation. This is because the cost never occurs if the real options are not implemented. Due to the time span is not very long, it is not necessary to apply discounting on the real option values obtained at each time interval.
5
Conclusions
A real options strategy is designed and implemented in response to fuel price uncertainties. In a mining project evaluation, the value of a mining project can be better evaluated and improved by incorporating real options into various life-of-mine/longterm planning strategy. Furthermore, real options can also be applied to medium/short mine plan to improve the performance of a mining operation. The proposed incorporation of real options into dump truck dispatch plan and case study shows that the real options can be a useful strategy to overcome uncertainty in fuel price variations. A significant economic benefit can be gained from the implementation of real options in medium/short-term mine plan.
References [1]
[2]
[3]
(3)
where T is the time span for real options; Ph is fuel price above the upper trigger level; Pl is fuel price below the lower trigger level; Trn is the truck number used in original plan; Trnh is the truck number used in real option plan during the period fuel price above the upper trigger level; Trnl is the truck number used in real option plan during the period fuel price below the
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[4] [5] [6]
Li S, Dimitrakopoulos R, Scott J, Dunn D. Quantification of geological uncertainty and risk using stochastic simulation and applications in the coal mining industry. In: Orebody Modelling and Strategic Mine Planning, Spectrum Series 14. AusIMM, 2007: 233–240. Li S, Scott J, Dimitrakopoulos R. Stochastic simulation-based risk assessment for drilling optimisation. In: Application of Computers and Operations Research in the Mineral Industry. Tucson: A. A. Balkema Publishers, 2005. Lemelin B, Abdel S S A, Poulin R. Valuing mine 2 at Raglan using real options. International Journal of Mining, Reclamation and Environment, 2006, 20(1): 46–56. Shafiee A, Topal E. Applied real option valuation in a conceptual mining project. In: Australian Mining Technology Conference. 2008(16-18): 173–187. Abdel S S A. Valuing the operating flexibility of mineral processing plants. CIM Bulletin, 2007, 100(1099): 7–14. Schwartz E S. The stochastic behaviour of commodity prices: implications for valuation and hedging. J Finance, 1997, 52(3): 923–973.
www.elsevier.com/locate/jcumt
Integration of real options into short-term mine planning and production scheduling LI Shu-xing, KNIGHTS Peter CRC Mining, University of Queensland, Brisbane Qld 4072, Australia Abstract: Commodity prices have fallen sharply due to the global financial crisis. This has adversely affected the viability of some mining projects, including leading to the possibility of bankruptcy for some companies. These price falls reflect uncertainties and risks associated with mining projects. In recent years, much work has been published related to the application of real options pricing theory to value life-of-mine plans in response to long term financial uncertainty and risk. However, there are uncertainties and risks associated with medium/short-term mining operations. Real options theory can also be applied to tactical decisions involving uncertainties and risks. This paper will investigate the application of real options in the mining industry and present a methodology developed at University of Queensland, Australia, for integrating real options into medium/short-term mine planning and production scheduling. A case study will demonstrate the validity and usefulness of the methodology and techniques developed. Keywords: real options; mine planning; production scheduling; economic uncertainty; dump truck dispatch
1
Introduction
The mining industry has gone from boom to bust rapidly with the current economic crisis. This is a typical example of economic uncertainty associated with mining industry. To survive during this period of economic uncertainty, mining companies should pursue alternative strategies in their mine development. The strategies in real options, such as deferring, scaling down or abandoning a mining project, may be applicable to current economic situations. These real options can convert economic uncertainty into opportunities. Indeed many mining companies are implementing those real options’ strategies in their life-of-mine/long-term mining plan in order to minimise economic losses. A real option is a right, but not an obligation to introduce a contingency plan into a project within a specific period of time allowing for predetermined conditions. Real options’ principles and strategies are applicable to operating mines in medium/short-term mine planning. Implementing alternative mine designs and production schedules can reduce the overall mining cost and improve the overall mine performance in response to the various uncertainties, such as economic, geological and geotechnical uncertainties[1–2]. This paper will introduce a new form of real opReceived 11 February 2009; accepted 25 April 2009 Corresponding author. E-mail address: [email protected]
tions, or mine planning flexibility and contingency, into medium/short-term mine planing and production scheduling process and demonstrate the method for valuating these flexibilities based on the concept of real options. The uncertainty parameter addressed in this paper is fuel price uncertainty. Real options’ strategies are proposed for medium/short-term mine plan in response to uncertainties in fuel price. The methods for estimating the values of the real options are developed. The case study demonstrates that real options’ principles and strategies are useful for medium/short-term mine planning in response to the uncertain situations.
2
Applications of real options in mining industry
Real options strategies originated from finance and stock market investment analysis. In corporate finance, real options, such as put and call options, are applied in capital budgeting decisions. Put option is the right to sell the underlying asset to receive the exercise price. Call option is the right to buy the underlying asset by paying the exercise price. In the mining industry, there are a limited research and case studies applying real options to valuate mining projects. The real options techniques identified for mining project include cancellation, expan-
LI Shu-xing et al
Integration of real options into short-term mine planning and production scheduling
sion, deferral and abandonment. These flexibilities are mainly identified and applied for life-of-mine (LOM) planning in response to the uncertainty of commodity price. For instance, Lemelin et al. apply real options techniques to evaluate Mine 2 at Raglan[3]. It is a complex mining project, which consists of a number of mineralized zones and produces many payable metals, and as such, the valuation process becomes very complicated. A least-squares Monte Carlo (LSM) approach is used for valuing real options. The valuation of the mineralised zones takes into account the uncertainty associated with the prices of all payable metals simultaneously, and the management flexibility to switch among the different operating alternatives. An economic evaluation of Mine 2 at Raglan is improved in both precision and accuracy using the real options analysis (ROA) methodology. The business model includes the flexibility of choosing different production options throughout the mine life depending on variables such as metal prices, foreign exchange rates and fuel prices. It is a dynamic process that forecasts the value of a project should expansion or abandonment be decided. Shafiee and Topal apply real options valuation in a conceptual mining project. In this application, a hypothetical gold mine is used as a case to demonstrate the project evaluation using DCF and ROV methods[4]. DCF method assumes that time and risks are constant in evaluation. ROV assumes that time is flexible and also manages the risk. ROV gives flexibility in choosing the project time, controlling extraction rates, cut-off grade, expanding capacity, temporary closing and eventual abandonment. The ROV method produces a higher NPV than that of DCF method. The operating flexibility is also incorporated into the design of a mineral processing plant[5]. To maximize the profitability of a mineral processing plant, possible future variations in plant performance due to the uncertainty should be considered before the plant is constructed. The plant performance can be improved either by re-adjusting some operating parameters or by adding new capacity. The study shows that the capital investment is necessary in order to improve the plant performance. The study suggests a method for valuing the operating flexibility of mineral processing plants based on the real options techniques.
3
Incorporation of real options into medium/short-term mine planning and production scheduling in response to the fuel price uncertainty
3.1
Proposed real options in dump truck dispatch plan
With the increase of fuel price, the fuel cost for
675
haul trucks in overburden removal operation accounts for a significant portion of operating cost in a surface mine. In a medium/short-term mine plan and production schedule, the optional plan for haulage operation and the usage of trucks in response to the ups and downs of fuel price may significantly reduce fuel consumption and the operating cost of overburden removal, and therefore improve the overall performance of the mine. Integrating real options into dump truck dispatch plan in response to the fuel price uncertainty is a solution to the operating cost increase. A real option for dispatching dump truck plan is proposed in terms of the fuel price uncertainty. In this proposed real option, the following assumptions must be held: 1) the production level should be maintained; 2) the volume constraint of each overburden dump should be observed; 3) trucks and shovels should be fully utilised; 4) the number of operating trucks on site can be scheduled based on the fuel price variations; 5) the cost for maintaining extra trucks and haul roads should be accounted for. The proposed real option is to dispatch the trucks to closest dumps during the period of high fuel price in order to reduce the number of trucks used. Therefore the fuel cost of dump trucks will not blow out and can be kept under control. On the other hand, the destinations of trucks will be reversed during the period of low fuel price in order to balance the storage volume of dumps. More specifically, if the fuel price increases to a certain level, which may be called an upper trigger level, the real option of dump truck dispatch plan will be implemented. The real option of dump truck dispatch plan during this high fuel price period means to alter the dump trucks destinations to the closest dumps in order to lower the number of trucks used, reduce the amount of fuel consumed and save fuel costs. On the other hand, if the fuel price decreases to a level lower than the upper trigger level, the dump truck dispatch plan will be back to normal. Furthermore if the fuel price continues to decrease and reach a lower trigger level, a dump truck dispatch plan will be implemented that makes up the imbalance of dump storage capacities. 3.2
Implementation and evaluation of real options proposed
Proposed real options are implemented in two stages. The first stage is implemented during the period of fuel price higher than the upper trigger level. The purpose is to reduce fuel consumption and cost during this period. The second stage is implemented in the period of fuel price lower than the lower trigger level. The purpose is to rebalance the dump storage capacities during this low fuel cost period. The method proposed for implementing real options in dump truck dispatch plan are as follows:
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Mining Science and Technology
1) simulate fuel price using mean-reverting model for each time interval; 2) obtain the model of the fuel price at each time interval; 3) select the fuel price assumed in the original mine plan as an upper trigger level; 4) if the fuel price increases over the upper trigger level and will last for a prolonged period, implement the first stage of the real option for dump truck dispatch plan; 5) if the fuel price decreases below the upper trigger level, the dump truck dispatch plan is reversed back to normal; 6) if the fuel price decreases to the lower trigger level, the dump truck dispatch plan will implement the second stage of the real options, i.e., to move overburden materials to the under-filled dumps in order to balance the dump storage capacity until the balance is achieved; 7) once the balance is achieved, or the fuel price increase above the lower trigger level, the truck dispatch plan is reversed back to normal; 8) as the variations of fuel price, repeat the process from 4 to 7. The real option implemented needs to be valued to determine its worthiness and effect on the overall performance of a mine operation. The value of a real option implemented in mine planning is the value difference between the original plan and the real option plan. The valuation method of a real option follows the steps below: 1) collect the data for overburden removal operation; 2) review the original dump truck dispatch plan (O plan); 3) set the upper trigger of fuel price. Once the fuel price increases to the level above an upper trigger: 4) calculate the cost of O plan; 5) implement the first stage of real option plan (R1 plan); 6) calculate the cost of R1 plan. Once the fuel price decreases to the level below a lower trigger: 7) calculate the cost of O plan; 8) implement the second part of real option plan (R2 plan); 9) calculate the cost of R2 plan; 10) sum the costs of O plan during the periods of both fuel price increases and decreases; 11) sum the costs of both R1 and R2 plan; 12) calculate the maintenance cost of trucks and haul roads; 13) subtract the costs of R1+R2 plan and maintenance cost from O plan to obtain the value of real option implemented at each time interval; 14) extend to whole period when the real option implemented;
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15) the final result is the value of real option implemented.
4
Case study and results
4.1 Description of mine site A case study has been conducted in a hypothetical surface mine site. The overburden removal areas and dumps are shown in Fig. 1. There are two working areas and two dumps. Overburden in working area 1 will be dumped at Dump 1 while overburden in working area 2 will be dumped at Dump 2. The storage capacity of Dump 1 can only hold the overburden materials from working area 1 while the storage capacity of Dump 2 is only large enough to store the overburden materials from working area 2. Distances of various routes are listed in Table 1.
Fig. 1
Layout of working areas and dumps
Table 1
Distances of various routes
Route
Distance (m)
Dump capacity
Working area 1 -> Dump 1
3000
Dump 1 for working area 1
Working area 2 -> Dump 2
1000
Dump 2 for working area 2
Working area 1 -> Dump 2
2000
Working area 2 -> Dump 1
4000
4.2 Real options In response to the volatility of fuel price, the real options are proposed to be implemented in overburden removal operation in this case study. The purpose of implementing real options in dump truck dispatch plan is to reduce the number of trucks used in overburden removal in the event of fuel price becoming very high. In this case study, the upper trigger level of fuel price is assumed as Aus$1.20 per litre. Conditional to the same production level, the reduction of the truck number is achieved by shortening the haulage distance which means to change the dumping destinations to nearest dumps. The trucks leaving idled can be scheduled for maintenance. Also, scheduled new trucks purchase can be re-scheduled for a later time. When the fuel price decreases to a lower
Integration of real options into short-term mine planning and production scheduling
trigger level, the truck dumping destinations are reversed to balance the dumps’ storage capacities. The lower fuel price trigger level of Aus$0.80 per litre is assumed in this case study. During the period of fuel price falling into the range between the upper and lower trigger levels, the original dump truck dispatch plan is implemented. 4.3 Fuel price simulation It is widely recognised that the price of oil and some commodities are modelled using Brownian motion models and mean reverting models. The common stochastic process describing the behaviour of commodity prices are Brownian motion models, meanreverting models based on the Ornstein-Uhlenbeck process and jump processes. Geometric Brownian motion (GBM) and mean-reverting process (MRP) can be easily estimated from historical data of commodity prices. The mean-reverting model is adopted in this case study to simulate the fuel price. Mean-reverting process (MRP) is modelled using Eq.(1)[6]: dS = Ș(ȝ–lnS)Sdt + ıSdz
(1)
where S is commodity price; Ș the reversion speed; ȝ the logarithm of the equilibrium price; ı the standard deviation; dz the increment of a Wiener process, which equals İtdt. The random variable İt has zero mean and unit standard deviation. According to this process, the expected instantaneous drift is not constant, but depends on the difference between the logarithms of the spot price S and the equilibrium price ȝ. If S is lower than the equilibrium price ȝ, the expected drift for the next period is positive. If S is higher than the equilibrium price, the expected drift will be negative. Therefore the stochastic process reflects well the forces of supply and demand that control the commodity price. Fig. 2 shows the simulation of fuel price for a 5 year period. In this realisation, the highest fuel price reaches Aus$1.40 per litre and lowest Aus$0.60 per litre. The equilibrium price is Aus$1.00 per litre. There is a period of 10 months during which the fuel prices are above the upper trigger level. Interestingly there is also a period during which the fuel prices are below the lower trigger level. Therefore the two stage real options can be implemented during these two up and down periods. 4.4 Valuation of real options To estimate the value of the real options implemented in this case study, various parameter assumptions on dump trucks and haulage operations are made. Table 2 lists the parameters assumed in calculating costing hours for dump truck.
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Fig. 2
Upper and lower trigger levels and opportunity windows for real options
Table 2
Parameters assumed and calculations of costing hours
Basic data Distance (m)
Area 1 -> Dump 1
Area 2 -> Dump 2
3000
1000
Truck numbers
10
10
Truck payload (t) Fuel price upper trigger (Aus$/Litre) Fuel price lower trigger (Aus$/Litre) Times and trips
220
220
1.2
1.2
0.8
0.8
Manoeuvring (min)
1
1
Loading (min)
3
3
Accelerating (min)
1
1 600
Haulage speed (m/min)
600
Decelerate and dump (min)
1.5
1.5
Return speed (m/min)
600
600
Circle time (min)
16.5
9.8
Trips per hour Tonnage removed and costing hours per truck Tons per hour continuous
3.6
6.1
800
1342
Overall job efficiency (%)
75
75
Mechanical availability (%)
85
85
Annual outage factor (%)
95
95
Production utilisation (%)
0.6
0.6
Scheduled hours per year
7248
7248
Productive hours per year
4389
4389
3511656
5892439
483
483
Annual production (t) Costing hours per month
Based on the parameters assumed and the fuel price simulations, the real option proposed in this study for dump truck dispatch plan is valuated. The value of real option is calculated using Eq.(2). Vro = Cor – Cro – Cm
(2)
where Vro is the value of real option implemented; Cor is the fuel cost of original plan; Cro is the fuel cost of real option plan; Cm is the cost of maintenance or exercise.
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In Eq.(2), the cost of original plan (Cor) is divided into two parts: 1) the cost of original plan during the period of fuel price above upper trigger level (Corh); 2) the cost of original plan during the period of fuel price below lower trigger level (Corl). Similarly, the cost of real option plan (Cro) is also calculated in two parts: 1) the cost of real option plan during the period of fuel price above upper trigger level (Croh); 2) the cost of real option plan during the period of fuel price below lower trigger level (Crol). The results of valuation on real option implemented in this study are shown in Table 3. If the fuel price varies as shown in Fig. 2, the proposed real option for a 10 months period is worth approximately Aus$0.7 m, which accounts for 2.5% cost savings. Table 3
Value of real option implemented (costs in Aus$)
Original cost during 10 months period
Area 1 -> Dump 1
Area 2 -> Dump 2
Total
Above upper trigger price ($1.2)
9207339
9207339
18414678
Below lower trigger price ($0.8)
4573843
4573843
9147686
Area 1 -> Dump 2 7347270 Area 1 -> Dump 1 4573843
Area 2 -> Dump 2 9207339 Area 2 -> Dump 1 5497852
Total cost of original plan Real options cost during 10 months period Above upper trigger price ($1.2)
Below lower trigger price ($0.8)
27562364
16554609
10071695
Cost of implementing options
250000
Total cost of real options plan
26876304
Final value of real options
686059
4.5 Discussions In implementing the real option proposed for dump truck dispatch, an opportunity window is required. Two variables, fuel price (P) and time span (T) of fuel price variations, define an opportunity window. To enable the implementation of a real option proposed, the opportunity window should be large enough to allow the gains to offset costs. In other words, the fuel price needs to increase above upper trigger level and decrease to below lower trigger level and also last for a substantial length in both cases. If the fuel price varies substantially, the time span required could be shorter, otherwise, the time span required must be in a prolong time. The relationship between fuel price variations and time span required can be described using Eq.(3). This equation is derived from the Eq.(2). Cm T= a[(Trn − Trnh ) Ph + (Trn − Trnl ) Pl ]
No.5
lower trigger level; Cm is the cost of maintenance; a is a constant related to fuel cost each truck. Based on Eq.(3), if fuel price is well above the upper trigger level, the time span required can be shorten. Also if the fuel price is well below the lower trigger level, the time span can also be shorten. On the other hand, if the time span is large, the upper or lower fuel price trigger level can be narrowed. To offset the dump storage capacity, the time span for lower opportunity window should be equal to that for upper opportunity window. The cost arisen from implementing the proposed real options must be accounted for in real options valuation. This is because the cost never occurs if the real options are not implemented. Due to the time span is not very long, it is not necessary to apply discounting on the real option values obtained at each time interval.
5
Conclusions
A real options strategy is designed and implemented in response to fuel price uncertainties. In a mining project evaluation, the value of a mining project can be better evaluated and improved by incorporating real options into various life-of-mine/longterm planning strategy. Furthermore, real options can also be applied to medium/short mine plan to improve the performance of a mining operation. The proposed incorporation of real options into dump truck dispatch plan and case study shows that the real options can be a useful strategy to overcome uncertainty in fuel price variations. A significant economic benefit can be gained from the implementation of real options in medium/short-term mine plan.
References [1]
[2]
[3]
(3)
where T is the time span for real options; Ph is fuel price above the upper trigger level; Pl is fuel price below the lower trigger level; Trn is the truck number used in original plan; Trnh is the truck number used in real option plan during the period fuel price above the upper trigger level; Trnl is the truck number used in real option plan during the period fuel price below the
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[4] [5] [6]
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