Underground mine projects are often associated with diverse sources of uncertainties. Having the ability to plan for these uncertainties plays a key role in the process of project evaluation and is increasingly recognized as critical to mining project success. To make the best decision, based on the information available, it is necessary to develop an adequate model incorporating the uncertainty of the input parameters. The model is developed on the basis of full discounted cash flow analysis of an underground zinc mine project. The relationships between input variables and economic outcomes are complex and often nonlinear. Fuzzyinterval grey system theory is used to forecast zinc metal prices while geometric Brownian motion is used to forecast operating costs over the time frame of the project. To quantify the uncertainty in the parameters within a project, such as capital investment, ore grade, mill recovery, metal content of concentrate, and discount rate, we have applied the concept of interval numbers. The final decision related to project acceptance is based on the net present value of the cash flows generated by the simulation over the time project horizon.
If we take into consideration that underground mining projects are planned and constructed in an uncertain physical and economic environment, then evaluation of such projects is truly interdisciplinary in nature.
Mine investments provide a good example of irreversible investment under uncertainty. Irreversible investment requires more careful analysis because, once the investment takes place, it cannot be recouped without a significant loss of value. Engineering economics is a widely used economic technique for the evaluation of engineering projects. Within it, different methods can be used to make the best decision, that is, whether to accept a project or not.
There is a considerable literature dedicated to the problem of mining project evaluation. Samis et al. use Real Options Monte Carlo Simulation to examine the valuation of a multiphase coppergold project in the presence of a windfall profits tax [
Prior to initialization, a mining project is often evaluated by calculating its net present value (NPV). The NPV is defined as the discounted difference between the expected value of project revenues and costs over the life of the project. The NPV is the preferred criterion of project profitability since it reflects the net contribution to the owner’s equity considering his cost of capital. We propose a simulation approach to incorporate uncertainty in the NPV calculations. Simulation of future zinc metal prices is performed by fuzzyinterval grey system theory. The dynamic nature of operating costs is described by the stochastic process called geometric Brownian motion. In this way, we obtain the probability distribution of operating costs for every year of the project and after that we transform them into adequate interval numbers. The remaining risk factors such as capital investment, ore grade, mill recovery, metal content of concentrate, and discount rate are also quantified by interval numbers using expert knowledge (estimation). When these interval numbers are incorporated in the NPV calculation, we obtain the intervalvalued NPV, that is, the project value at risk.
The main purpose of this study is to provide an efficient and easy way of strategic decision making, particularly in small underground mining companies. We were motivated by the fact that, in our country, as one of the many developing countries, there are mainly small underground mining companies employing just two or three mining engineers who are responsible for both the production maintenance and strategic planning. In such an environment, mining engineers do not have time to create adequate procedures for decision making, particularly for the decisions influenced by highly volatile parameters such as metal price. There are many stochastic methods for treating the uncertainty of metal prices (e.g., Mean Reversion Process), but if we want to apply them, it is necessary to collect a lot of historical data and run complex regression analysis in order to define the parameters of the simulation process. Interval grey theory can handle problems with unclear information very precisely. Its concept is intuitive and simple to understand for mining engineers. In order to build the forecasting model, only a few data are needed.
The proposed model is tested on a hypothetical example, which is similar to many real case studies, and the experiment results verify the rationality and effectiveness of the method.
Various theories exist for describing uncertainty in the modelling of real phenomena and the most popular one is fuzzy set theory [
In the classical set theory, an element either belongs or does not belong to a given set. By contrast, in fuzzy set theory, a fuzzy subset
Let
Fuzzy sets can also be represented via their
A
In particular, a fuzzy set
According to Dubois and Prade [
Fuzzy number and
In a similar manner, we introduce
Let
Now, the
From interval arithmetic, the following operations of interval numbers are defined as follows.
For
For any two interval numbers
For any two interval numbers
For any two interval numbers
In this section we consider an intervalvalued differential equation of the following form:
Let
Let
Economic evaluation of a mine project requires estimation of the revenues and costs throughout the defined lifetime of the mine. Such evaluation can be treated as strategic decision making under multiple sources of uncertainties. Therefore, to make the best decision, based on the information available, it is necessary to develop an adequate model incorporating the uncertainty of the input parameters. The model should be able to involve a common time horizon, taking the characteristics of the input variables that directly affect the value of the proposed project.
The model is developed on the basis of full discounted cash flow analysis of an underground zinc mine project. The operating discounted cash flows are usually estimated on an annual basis. Net present value of investment is used as a key criterion in the process of mine project estimation. The expected net present value of the project is a function of the variables as
In this paper, we treat in detail only the variability of metal prices and operating costs, without intending to decrease the significance of the remaining parameters. These parameters are taken into account on the basis of expert knowledge (estimation).
Most mining companies realize their revenues by selling metal concentrates as a final product. Estimating mineral project revenue is, indeed, a difficult and risky activity. Annual mine revenue is calculated by multiplying the number of units produced and sold during the year by the sales price per unit.
The value of the metal concentrate can be expressed as follows:
Annual ore production is derived from the mining project schedule and is defined as crisp value. The concept of grade
The major external source of risk affecting mine revenue is related to the uncertainty about market behaviour of metal prices. Forecasting the precise future state of the metal price is a very difficult task for mine planners. To predict future metal prices, we apply the concept based on the transformation of historical metal prices into adequate fuzzyinterval numbers and grey system theory.
The forecasting model of metal prices is composed of the following steps.
Create the set
Transform the set PDF into the set
Transform the set TFN into the set
Using grey system prediction theory, create a grey differential equation of type
Testing of
For every historical year it is necessary to define a probability density function with the following characteristics: shape by histogram, mean value
The sequence of obtained
The sequence of obtained
The grey model is a powerful tool for forecasting the behaviour of the system in the future. It has been successfully applied to various fields since it was proposed by Deng [
Assume that
Let
To get the values of parameters
If
For a given interval grey number
Residual error testing is composed of the calculation of relative error and absolute error between
Capital development in an underground mine consists of shafts, ramps, raises, and lateral transport drifts required to access ore deposits with expected utility greater than one year. This is the development required to start up the ore production and to haul the ore to the surface. Experience with investments in capital development might show that such expenditures can run considerably higher than the estimates, but it is quite unlikely that actual costs will be lower than estimated. Thus, the interval number might represent capital investment for the project:
Operating costs are incurred directly in the production process. These costs include the ore and waste development of individual stopes, the actual stoping activities, the mine services providing logistical support to the miners, and the milling and processing of the ore at the plant. These costs are generally more difficult to estimate than capital costs for most mining ventures. If we take into consideration that production will be carried out for many years, then it is very important to predict the future states of operating costs. Although there is some intention to create a correlation between metal price and operating cost, it is very hard to define it, since price and cost vary continuously and are different over time. At the project level, there will not be a perfect correlation between price and cost because of adjustments to variables such as labour, energy, explosives, and fuel, as well as other material expenditures that are supplied by industries that are not directly linked to metal price fluctuations. In order to protect themselves, suppliers are offering shortterm contracts to mines that are the opposite of traditional longterm contracts. Some components of the operating cost such as inputs used for mineral processing are usually purchased at market prices that fluctuate monthly, annually, or even over shorter periods.
The uncertainties related to the future states of operating costs are modelled with a special stochastic process, geometric Brownian motion. Certain stochastic processes are functions of a Brownian motion process and these have many applications in finance, engineering, and the sciences. Some special processes are solutions of ItôDoob type stochastic differential equations (Ladde and Sambandham [
In this model, we apply a continuous time process using the ItôDoob type stochastic differential equation to describe the movement of operating costs. A general stochastic differential equation takes the following form:
Applying the same concept of metal prices transformation, we obtain the future sequence of operating costs expressed by interval numbers;
The net present value (NPV) of the mine project is an integral evaluation criterion that recognizes the time effect of money over the lifeofmine. It is calculated as a difference between the sum of discounted values of estimated future cash flows and the initial investment and can be defined as follows:
Finally, the last parameter that can be expressed by an interval number is the discount rate. Discounted cash flow methods are widely used in capital budgeting; however, determining the discount rate as a crisp value can lead to erroneous results in most mine project applications. A discount rate range can be established in a way which is either just acceptable (maximum value) or reasonable (minimum value);
A positive NPV will lead to the acceptance of the project and a negative NPV rejects it; that is,
The management of a small mining company is evaluating the opening of a new zinc deposit. The recommendations from the prefeasibility study suggest the following:
the underground mine development system connecting the ore body to the surface is based on the combination of ramp and horizontal drives. This system is used for the purpose of ore haulage by dump trucks and conduct intake fresh air. Contaminated air is conducted to the surface by horizontal drives and declines,
they suggest purchasing new mining equipment.
The completion of this project will cost the company about 3500 000 USD over three years. At the beginning of the fourth year, when construction is completed, the new mine will produce 100,000
The input parameters required for the project evaluation are given in Table
Input parameters.
Mine ore production (t/year)  100 000  
Zinc grade (%) 
 
Metal content of concentrate (%) 
 
Mill recovery (%) 
 


Metal price ($/t)  
2009  2010  2011  2012  2013  


January  1203  2415  2376  1989  2031  
February  1118  2159  2473  2058  2129  
March  1223  2277  2341  2036  1929  
April  1388  2368  2371  2003  1856  
May  1492  1970  2160  1928  1831  
June  1555  1747  2234  1856  1839  
July  1583  1847  2398  1848  1838  
August  1818  2047  2199  1816  1896  
September  1879  2151  2075  2010  1847  
October  2071  2374  1871  1904  1885  
November  2197  2283  1935  1912  1866  
December  2374  2287  1911  2040  1975  


Initial (capital) investment ($) 
 
Operating costs, geometric Brownian motion, 
Spot value 32; drift 0.020; 

Construction period (year)  3 

Mine life (year)  5 

Discount rate (%) 

The interval values of historical metal prices are calculated by Steps
Transformation
2009  2010  2011  2012  2013  




1658  2160  2195  1950  1910 

410  216  207  83  92 



838  1728  1781  1784  1726 

1658  2160  2195  1950  1910 

2478  2592  2609  2116  2094 

820  432  414  166  184 



1385  2016  2057  1895  1849 

1931  2304  2333  2005  1971 
Historical zinc metal prices expressed as triangular fuzzy numbers.
Based on data in Table
Relative and absolute error of the model.
Year  Original values ($/t)  Simulated values ($/t) 
Whitenization 
Relative error  Absolute error  Relative error (%)  

Original  Simulated  
2009 


1658  1658  0  0  0 
2010 


2160  2204  −44  44  −2.03 
2011 


2195  2098  97  97  +4.42 
2012 


1950  1997  −47  47  −2.41 
2013 


1910  1899  11  11  +0.57 
The standard deviation of the original metal price series (
According to (
Mine revenues over production period 2017–2021.
2017  2018  2019  2020  2021  







Uncertainty related to the unit operating costs is quantified according to (
Seven simulated operating cost paths on a yearly time resolution.
The results of the simulations and transformations are represented in Table
Simulation of the unit operating costs.
2013  2014  2015  2016  2017  2018  2019  2020  2021  

Simulation 1  32  35.24  36.47  40.67  46.07  55.00  54.14  55.69  48.59 














Simulation 500  32  31.17  35.17  37.91  37.66  36.27  37.15  34.34  36.08 





32  32.41  33.12  33.51  34.05  34.56  35.16  35.80  36.35 

0.0  3.16  4.55  5.70  6.60  7.49  8.45  9.42  10.00 



32  26.09  24.02  22.11  20.85  19.58  18.26  16.96  16.35 

32  32.41  33.12  33.51  34.05  34.56  35.16  35.80  36.35 

32  38.73  42.22  44.91  47.25  49.54  52.06  54.64  56.35 

0.0  6.32  9.10  11.40  13.20  14.98  16.90  18.84  20.00 



32  30.30  30.08  29.71  29.65  29.56  29.52  29.52  29.68 

32  34.52  36.16  37.32  38.45  39.56  42.09  42.08  43.01 
Annual operating costs are represented in Table
Operating costs over production period 2017–2021.
2017  2018  2019  2020  2021  







The discounted cash flow of the project is represented in Table
Discounted cash flow over production period 2017–2021.
2017  2018  2019  2020  2021  



















Discount factor 





Present value 





According to (
The combined effect of market volatility and uncertainty about future commodity prices is posing higher risks to mining businesses across the globe. In such times, knowing how to unlock value by maximizing the value of resources and reserves through strategic mine planning is essential. In our country, small mining companies are faced with many problems but the primary problem is related to the shortage of capital for investment. In such an environment every mining venture must be treated as a strategic decision supported by adequate analysis. The developed economic model is a mathematical representation of project evaluation reality and allows management to see the impact of key parameters on the project value. The interaction between production, costs, and capital is highly complex and changes over time, but needs to be accurately modelled so as to provide insights around capital configurations of that business.
The evaluation of a mining venture is made very difficult by uncertainty on the input variables in the project. Metal prices, costs, grades, discount rates, and countless other variables create a high risk environment to operate in. The incorporation of risk into modelling will provide management with better means to deal with uncertainty and the identification and quantification of those factors that most contribute to risk, which will then allow mitigation strategies to be tested. The model brings forth an issue that has the dynamic nature of the assessment of investment profitability. With the fuzzyinterval model, the future forecast can be done from the beginning of the process until the end.
From the results obtained by numerical example, it is shown that fuzzyinterval grey system theory can be incorporated into mine project evaluation. The variance ratio
With interval numbers, the end result will be interval NPV, which is the payoff interval for the project. Using the weight whitenization of the interval NPV, we obtain the payoff crisp value for the project. This value is the value at risk, helping the management of the company to make the right decision.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper is part of research conducted on scientific projects TR 33003 funded by the Ministry of Education, Science and Technological Development, Republic of Serbia.