Research ArticleEvaluation of Underground Zinc Mine InvestmentBased on Fuzzy-Interval Grey System Theory and GeometricBrownian Motion
Zoran Gligoric Lazar Kricak Cedomir Beljic Suzana Lutovac and Jelena Milojevic
Faculty of Mining and Geology Djusina 7 11000 Belgrade Serbia
Correspondence should be addressed to Zoran Gligoric zorangligoricrgfbgacrs
Received 28 April 2014 Revised 4 July 2014 Accepted 6 July 2014 Published 23 July 2014
Academic Editor Baolin Wang
Copyright copy 2014 Zoran Gligoric et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Underground mine projects are often associated with diverse sources of uncertainties Having the ability to plan for theseuncertainties plays a key role in the process of project evaluation and is increasingly recognized as critical tomining project successTo make the best decision based on the information available it is necessary to develop an adequate model incorporating theuncertainty of the input parameters The model is developed on the basis of full discounted cash flow analysis of an undergroundzinc mine project The relationships between input variables and economic outcomes are complex and often nonlinear Fuzzy-interval grey system theory is used to forecast zinc metal prices while geometric Brownian motion is used to forecast operatingcosts over the time frame of the project To quantify the uncertainty in the parameters within a project such as capital investmentore grade mill recovery metal content of concentrate and discount rate we have applied the concept of interval numbersThe finaldecision related to project acceptance is based on the net present value of the cash flows generated by the simulation over the timeproject horizon
1 Introduction
If we take into consideration that underground miningprojects are planned and constructed in an uncertain physicaland economic environment then evaluation of such projectsis truly interdisciplinary in nature
Mine investments provide a good example of irre-versible investment under uncertainty Irreversible invest-ment requires more careful analysis because once the invest-ment takes place it cannot be recouped without a significantloss of value Engineering economics is a widely used eco-nomic technique for the evaluation of engineering projectsWithin it different methods can be used to make the bestdecision that is whether to accept a project or not
There is a considerable literature dedicated to the problemof mining project evaluation Samis et al use Real OptionsMonte Carlo Simulation to examine the valuation of a multi-phase copper-gold project in the presence of a windfall profitstax [1] Dimitrakopoulos applies the Monte Carlo technique(conditional simulation) to quantify geological uncertaintysuch as ore grade and tonnage [2] Topal uses different
techniques to estimate the value of the mining project Themajor challenge of project evaluation is how to deal withthe uncertainty involved in capital investment Discountedcash flow (DCF) methods decision trees (DT) Monte Carlosimulation (MCS) and real options (RO) are commonly usedfor evaluating mining projects [3] Dessureault et al use realoptions pricing as a method for the flexible valuation of amining project This paper presents the methods that can beused for the calculation of process and project volatility inoperations and provides practical applications from miningoperations inUSA andCanada [4] Elkington et al noted thatuncertainty is intrinsic to all mining projects and should beplanned for by providing operating and strategic flexibility[5] Trigeorgis presents a decision-tree model for a miningproject in which the present value of the remaining cash flowsis uncertain [6] Samis and Poulin provide a related decision-tree model where mineral price is the underlying source ofuncertainty [7 8]
Prior to initialization a mining project is often evaluatedby calculating its net present value (NPV) The NPV isdefined as the discounted difference between the expected
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 914643 12 pageshttpdxdoiorg1011552014914643
2 Journal of Applied Mathematics
value of project revenues and costs over the life of the projectThe NPV is the preferred criterion of project profitabilitysince it reflects the net contribution to the ownerrsquos equityconsidering his cost of capital We propose a simulationapproach to incorporate uncertainty in the NPV calculationsSimulation of future zinc metal prices is performed by fuzzy-interval grey system theoryThe dynamic nature of operatingcosts is described by the stochastic process called geometricBrownian motion In this way we obtain the probabilitydistribution of operating costs for every year of the projectand after that we transform them into adequate intervalnumbers The remaining risk factors such as capital invest-ment ore grade mill recovery metal content of concentrateand discount rate are also quantified by interval numbersusing expert knowledge (estimation) When these intervalnumbers are incorporated in the NPV calculation we obtainthe interval-valued NPV that is the project value at risk
The main purpose of this study is to provide an efficientand easy way of strategic decision making particularly insmall underground mining companies We were motivatedby the fact that in our country as one of the many develop-ing countries there are mainly small underground miningcompanies employing just two or three mining engineerswho are responsible for both the production maintenanceand strategic planning In such an environment miningengineers do not have time to create adequate proceduresfor decisionmaking particularly for the decisions influencedby highly volatile parameters such as metal price There aremany stochasticmethods for treating the uncertainty ofmetalprices (eg Mean Reversion Process) but if we want to applythem it is necessary to collect a lot of historical data and runcomplex regression analysis in order to define the parametersof the simulation process Interval grey theory can handleproblems with unclear information very precisely Its conceptis intuitive and simple to understand for mining engineersIn order to build the forecasting model only a few data areneeded
The proposed model is tested on a hypothetical examplewhich is similar tomany real case studies and the experimentresults verify the rationality and effectiveness of the method
2 Preliminaries of Interval-ValuedDifferential Equations
21 Basic Concepts of Fuzzy Set Theory Various theoriesexist for describing uncertainty in the modelling of realphenomena and the most popular one is fuzzy set theory [9]In this paper we applied the concept of the interval-valuedpossibilistic mean of fuzzy number [10ndash12]
In the classical set theory an element either belongs ordoes not belong to a given set By contrast in fuzzy set theorya fuzzy subset
119860 defined on a universe of discourse 119883 ischaracterized by a membership function 120583
119860
(119909) which mapseach element in
119860 with a real number in the unit intervalGenerally this can be expressed as 120583
119860
(119909) 119883 rarr [0 1]where the value 120583
119860
(119909) is called the degree of membership ofthe element 119909 in the fuzzy set 119860 If the universal set119883 is fixed
a membership function fully determines a fuzzy set In fuzzyset theory classical sets are usually called crisp sets
Definition 1 Let 1198861
1198862
and 119886
3
be real numbers such that 1198861
lt
119886
2
lt 119886
3
A set
119860 with membership function
120583
119860
(119909) =
119909 minus 119886
1
119886
2
minus 119886
1
119886
1
le 119909 le 119886
2
119909 minus 119886
3
119886
3
minus 119886
2
119886
2
le 119909 le 119886
3
0 otherwise
(1)
is called a fuzzy triangular number and is denoted as
119860 =
(119886
1
119886
2
119886
3
) In geometric interpretations the graph of
119860(119909) isa triangle with its base on the interval [119886
1
119886
3
] and vertex at119909 = 119886
2
Fuzzy sets can also be represented via their 120574-levels
Definition 2 A 120574-level set of a fuzzy set 119860 is defined by [119860]
120574
=
119909 isin 119883 | 120583
119860
(119909) ge 120574 if 120574 gt 0 and [
119860]
120574
= cl119909 isin 119883 | 120583
119860
(119909) gt
0 (the closure of the support of
119860) if 120574 = 0
In particular a fuzzy set
119860 is a fuzzy number if andonly if the 120574-levels are nested nonempty compact intervals[119860
120574
lowast
119860
lowast120574
] This property is the basis for the lower-upperrepresentation of values of the 120574-levels [13] A fuzzy number
119860 is completely defined by a pair of functions 119860
lowast
119860
lowast
[0 1] rarr 119883 defining the end-points of 120574-levels (119860
120574
lowast
=
119860
lowast
(120574) 119860
lowast120574
= 119860
lowast
(120574)) and satisfying the following conditions
(1) 119860
lowast
is a bounded nondecreasing left-continuous func-tion on [0 1]
(2) 119860
lowast is a bounded nonincreasing left-continuous func-tion on [0 1]
(3) 119860
lowast
(120574) le 119860
lowast
(120574) for all 0 le 120574 le 1
According to Dubois and Prade [14] the interval-valuedprobabilistic mean of a fuzzy number
119860 with 120574 levels
119860
120574
=
[119886
120574
119887
120574
] 120574 isin [0 1] (see Figure 1) is the interval 119864(
119860) =
[119864
lowast
(
119860) 119864
lowast
(
119860)] where
119864
lowast
(
119860) = int
1
0
119886
120574
119889120574
119864
lowast
(
119860) = int
1
0
119887
120574
119889120574
(2)
Carlsson and Fuller [11] introduced the interval-valued pos-sibilistic mean of a fuzzy number
119860 as the interval 119872(
119860) =
[119872
lowast
(
119860)119872
lowast
(
119860)] First we note that from the equality
119872(
119860) = int
1
0
120574 (119886
120574
+ 119887
120574
) 119889120574 =
int
1
0
120574 ((119886
120574
+ 119887
120574
) 2) 119889120574
int
1
0
120574119889120574
(3)
it follows that 119872(
119860) is nothing else but the level-weightedaverage of the arithmetic means of all 120574-sets that is the
Journal of Applied Mathematics 3
0
1
120583A(x)
120574
a1 a2 a3X
120572 120573
120574-level cut
a120574
b120574
A120574
Figure 1 Fuzzy number and 120574-level cut
weight of the arithmetic mean of 119886120574 and 119887
120574 is just 120574 Secondwe can rewrite119872(
119860) as
119872(
119860) = int
1
0
120574 (119886
120574
+ 119887
120574
) 119889120574 =
2 int
1
0
120574119886
120574
119889120574 + 2 int
1
0
120574119887
120574
119889120574
2
=
1
2
(
int
1
0
120574119886
120574
119889120574
12
+
int
1
0
120574119887
120574
119889120574
12
)
=
1
2
(
int
1
0
120574119886
120574
119889120574
int
1
0
120574119889120574
+
int
1
0
120574119887
120574
119889120574
int
1
0
120574119889120574
)
(4)
Third let us take a closer look at the right-hand side of theequation for119872(
119860)The first quantity denoted by119872
lowast
(
119860) canbe reformulated as
119872
lowast
(
119860) = 2int
1
0
120574119886
120574
119889120574 =
int
1
0
120574119886
120574
119889120574
int
1
0
120574119889120574
=
int
1
0
Pos [119860 le 119886
120574
] 119886
120574
119889120574
int
1
0
Pos [119860 le 119886
120574
] 119889120574
=
int
1
0
Pos [119860 le 119886
120574
] timesmin [119860]
120574
119889120574
int
1
0
Pos [119860 le 119886
120574
] 119889120574
(5)
where Pos denotes possibility that is
Pos [119860 le 119886
120574
] = prod((minusinfin 119886
120574
] ) = sup⏟⏟⏟⏟⏟⏟⏟
119906le119886
120574
119860 (119906) = 120574 (6)
So 119872
lowast
(
119860) is nothing else but the lower possibility-weightedaverage of the minima of the 120574-sets and this is why we call itthe lower possibilistic mean value of
119860
In a similar manner we introduce 119872
lowast
(
119860) the upperpossibilistic mean value of
119860
119872
lowast
(
119860) = 2int
1
0
120574119887
120574
119889120574 =
int
1
0
120574119887
120574
119889120574
int
1
0
120574119889120574
=
int
1
0
Pos [119860 ge 119887
120574
] 119887
120574
119889120574
int
1
0
Pos [119860 ge 119887
120574
] 119889120574
=
int
1
0
Pos [119860 ge 119887
120574
] timesmax [119860]
120574
119889120574
int
1
0
Pos [119860 ge 119887
120574
] 119889120574
(7)
where we have used equality
Pos [119860 ge 119887
120574
] = prod([119887
120574
infin)) = sup⏟⏟⏟⏟⏟⏟⏟
119906ge119887
120574
119860 (119906) = 120574 (8)
The lower possibilistic mean 119872
lowast
(
119860) is the weighted averageof the minima of the 120574-levels of
119860 Similarly the upperpossibilistic mean 119872
lowast
(
119860) is the weighted average of themaxima of the 120574-levels of 119860 According to Carlsson the fuzzynumber can now be expressed as follows
119872
lowast
(
119860) = 2int
1
0
120574119886
120574
119889120574
119872
lowast
(
119860) = 2int
1
0
120574119887
120574
119889120574
(9)
Definition 3 Let
119860 = (119886
2
120572 120573) be a triangular fuzzy numberwith centre 119886
2
left-width 120572 gt 0 and right-width 120573 gt 0 (seeFigure 1) then 119886
120574 and 119887
120574 are computed as follows
1 120574 = 120572 (119886
120574
minus (119886
2
minus 120572)) 997888rarr 119886
120574
= 119886
2
minus (1 minus 120574) 120572
1 120574 = 120573 ((119886
2
) minus 119887
120574
) 997888rarr 119887
120574
= 119886
2
+ (1 minus 120574) 120573
(10)
Now the 120574-level of
119860 is computed by
[
119860]
120574
= [119886
2
minus (1 minus 120574) 120572 119886
2
+ (1 minus 120574) 120573] forall120574 isin [0 1] (11)
That is
119872
lowast
(
119860) = 2int
1
0
120574 [119886
2
minus (1 minus 120574)] 120572119889120574 = 119886
2
minus
120572
3
119872
lowast
(
119860) = 2int
1
0
120574 [119886
2
+ (1 minus 120574)] 120573119889120574 = 119886
2
+
120573
3
(12)
and therefore
119872(
119860) = [119872
lowast
(
119860) 119872
lowast
(
119860)] = [119886
2
minus
120572
3
119886
2
+
120573
3
] (13)
Obviously 119872(
119860) is a closed interval bounded by the lowerand upper possibilistic mean values of 119860The crisp possibilis-tic mean value of
119860 is defined as the arithmetic mean of its
4 Journal of Applied Mathematics
lower possibilistic and upper possibilistic mean values thatis
119872(
119860) =
119872
lowast
(
119860) + 119872
lowast
(
119860)
2
(14)
According to the above way of transformation (see (13)) weobtain an interval number having both a lower bound 119886
119897 andupper bound 119886
119906 119860 = [119886
119897
119886
119906
] where 119886
119897
= 119886
2
minus 1205723 and 119886
119906
=
119886
2
+ 1205733
From interval arithmetic the following operations ofinterval numbers are defined as follows
Definition 4 For forall119896 gt 0 and a given interval number 119860 =
[119886
119897
119886
119906
] 119886
119897
119886
119906
isin 119877
119896 times [119886
119897
119886
119906
] = [119896 times 119886
119897
119896times119886
119906
] (15)
and for forall119896 lt 0
minus119896 times [119886
119897
119886
119906
] = [minus119896 times 119886
119906
minus119896 times 119886
119897
] (16)
Definition 5 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
]
[119886
119897
119886
119906
] + [119887
119897
119887
119906
] = [119886
119897
+ 119887
119897
119886
119906
+ 119887
119906
]
[119886
119897
119886
119906
] minus [119887
119897
119887
119906
] = [119886
119897
minus 119887
119906
119886
119906
minus 119887
119897
]
(17)
Definition 6 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the multiplication is defined as follows
[119886
119897
119886
119906
] times [119887
119897
119887
119906
] = [min (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
(18)
Definition 7 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the division is defined as follows
[119886
119897
119886
119906
] divide [119887
119897
119887
119906
] = [min(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
0 notin [119887
119897
119887
119906
]
(19)
22 Interval-Valued Differential Equations In this sectionwe consider an interval-valued differential equation of thefollowing form
119910 = 119891 (119905 119910 (119905)) 119910 (119905
0
) = 119910
0
(20)
where 119891 [119886 119887] times 119864 rarr 119864 with 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119909
119910(119905))] for 119910(119905) isin 119864 119910(119905) = [119910
119897
(119905) 119910
119906
(119905)] 119910
0
= [119910
119897
0
119910
119906
0
] Notethat we consider only Hukuhara differentiable solutions thatis there exists 120575 gt 0 such that there are no switching pointsin [119905
0
119905
0
+ 120575] [15 16]
Definition 8 Let 119891 [119886 119887] rarr 119864 be Hukuhara differentiableat 1199050
isin ]119886 119887[ We say that 119891 is (i)-Hukuhara differentiable at119905
0
if
119891 (119905
0
) = [
119891
119897
(119905
0
)
119891
119906
(119905
0
)] (21)
and that 119891 is (ii)-Hukuhara differentiable at 1199050
if
119891 (119905
0
) = [
119891
119906
(119905
0
)
119891
119897
(119905
0
)] (22)
The solution of the differential equation (20) depends onthe choice of the Hukuhara derivative ((i) or (ii)) To solvethe interval-valued differential equation it is necessary toreduce the interval-valued differential equation to a systemof ordinary differential equations [17ndash20]
Let [119910(119905)] = [119910
119897
(119905) 119910
119906
(119905)] If 119910(119905) is (i)-Hukuhara differ-entiable then119863
1
119910(119905) = [ 119910
119897
(119905) 119910
119906
(119905)] transforms (20) into thefollowing system of ordinary differential equations
119910
119897
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(23)
Also if 119910(119905) is (ii)-Hukuhara differentiable then 119863
2
119910(119905) =
[ 119910
119906
(119905) 119910
119897
(119905)] transforms (20) into the following system ofordinary differential equations
119910
119897
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(24)
where 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119905 119910(119905))]
3 Model of the Evaluation
31 The Concept of Evaluation Economic evaluation of amine project requires estimation of the revenues and coststhroughout the defined lifetime of the mine Such evaluationcan be treated as strategic decision making under multiplesources of uncertaintiesTherefore tomake the best decisionbased on the information available it is necessary to developan adequatemodel incorporating the uncertainty of the inputparameters The model should be able to involve a commontime horizon taking the characteristics of the input variablesthat directly affect the value of the proposed project
The model is developed on the basis of full discountedcash flow analysis of an underground zinc mine project Theoperating discounted cash flows are usually estimated on anannual basis Net present value of investment is used as akey criterion in the process of mine project estimation Theexpected net present value of the project is a function of thevariables as
119864 (NPV | 119876 119875 119866119872119862 119868 119903 119905) ge 0 (25)
where 119876 denotes the production rate (capacity) 119875 denotesthe zinc metal price 119866 is the grade 119872 is mill recovery 119862 isoperating costs 119868 is capital investment 119903 is discount rate and119905 is the lifetime of the project that is the period in which thecash flow is generated
Journal of Applied Mathematics 5
In this paper we treat in detail only the variability ofmetalprices and operating costs without intending to decrease thesignificance of the remaining parameters These parametersare taken into account on the basis of expert knowledge(estimation)
32 Forecasting the Revenue of the Mine Most mining com-panies realize their revenues by sellingmetal concentrates as afinal product Estimatingmineral project revenue is indeed adifficult and risky activity Annual mine revenue is calculatedbymultiplying the number of units produced and sold duringthe year by the sales price per unit
The value of the metal concentrate can be expressed asfollows
119881con = 119875 sdot (119898con minus 119898mr) (26)
where 119875 is metal price ($119905) 119898con is metal content ofconcentrate ()119898mr is metal recovery ratio () and
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8) sdot 100
119898congt 85 85
(27)
Annual mine revenue is calculated according to the followingequation
119877year = 119876year sdot 119881con sdot
119866 sdot 119872
119898con (28)
where 119876year is annual ore production (119905year) 119866 is grade ofthe ore mined ()119872 is mill recovery ()
Annual ore production is derived from themining projectschedule and is defined as crisp value The concept of grade(119866) is defined as the ratio of useful mass of metal tothe total mass of ore and its critical value fluctuates overdeposit space and can be estimated by experts and defined asinterval number 119866 = [119866
119897
119866
119906
] Mill recovery (119872) is relatedto the flotation as the most widely used method for theconcentration of fine grained minerals It can also be definedas interval number 119872 = [119872
119897
119872
119906
] Metal content representsthe quality of concentrate and we also apply the concept of aninterval number to define it 119898con = [119898
119897
con 119898119906
con]The major external source of risk affecting mine revenue
is related to the uncertainty about market behaviour of metalprices Forecasting the precise future state of themetal price isa very difficult task formine planners To predict futuremetalprices we apply the concept based on the transformation ofhistorical metal prices into adequate fuzzy-interval numbersand grey system theory
The forecasting model of metal prices is composed of thefollowing steps
Step 1 Create the set PDF = pdf119895
119895 = 1 2 119873 wherepdf119895
is the probability density function of metal prices forevery historical year The minimum number of elements ofthe set is four 119895min = 1 2 3 4
Step 2 Transform the set PDF into the set TFN = (119886
119895
119887
119895
119888
119895
)where (119886
119895
119887
119895
119888
119895
) is an adequate fuzzy triangular number
Step 3 Transform the set TFN into the set INT = [119886
119897
119895
119886
119906
119895
]where [119886
119897
119895
119886
119906
119895
] is an adequate interval number
Step 4 Using grey system prediction theory create a greydifferential equation of type 119866119872(1 1) that is the first-ordervariable grey derivative
Step 5 Testing of 119866119872(1 1) by residual error testing and theposterior error detection method
321 Analysis of Historical Metal Prices For every historicalyear it is necessary to define a probability density functionwith the following characteristics shape by histogram meanvalue (120583
119895
) and standard deviation (120590
119895
) In this way we obtainthe sequence of probability density functions of 119875 119875
119895
sim
(pdf119895
120583
119895
120590
119895
) 119895 = 1 2 119873 where 119873 is the total numberof historical years
322 Fuzzification of Metal Prices The sequence of obtainedpdf119895
of 119875119895
can be transformed into a sequence of triangularfuzzy numbers of 119875
119895
119875119895
sim TFN119895
119895 = 1 2 119873 that is119875
1
sim pdf1
rarr 119875
1
sim TFN1
1198752
sim pdf2
rarr 119875
2
sim TFN2
119875
119873
sim pdf119873
rarr 119875
119873
sim TFN119873
The method of transformationis based on the following facts the support of themembershipfunction and the pdf are the same and the point with thehighest probability (likelihood) has the highest possibilityFor more details see Swishchuk et al [21] The uncertaintyin the 119875 parameter is modelled by a triangular fuzzy numberwith the membership function which has the support of 120583
119895
minus
2120590
119895
lt 119875
119895
lt 120583
119895
+ 2120590
119895
119895 = 1 2 119873 set up for around 95confidence interval of distribution function If we take intoconsideration that the triangular fuzzy number is defined asa triplet (119886
1119895
119886
2119895
119886
3119895
) then 119886
1119895
and 119886
3119895
are the lower boundand upper bound obtained from the lower and upper boundof 5 of the distribution and the most promising value 119886
2119895
isequal to the mean value of the distribution For more detailssee Do et al [22]
323 Metal Prices as Interval Numbers The sequence ofobtainedTFN
119895
of119875119895
is transformed into a sequence of intervalnumbers of 119875
119895
119875
119895
sim INT119895
119895 = 1 2 119873 that is 1198751
sim
TFN1
rarr 119875
1
sim INT1
119875
2
sim TFN2
rarr 119875
2
sim INT2
119875
119873
sim TFN119873
rarr 119875
119873
sim INT119873
The method of transformationis described in Section 21 (basic concepts of fuzzy set theory)According to this method of transformation we obtain set
119875
119895
sim INT119895
119875
119895
= [119875
119897
119895
119875
119906
119895
] = [119886
2119895
minus
120572
119895
3
119886
2119895
+
120573
119895
3
]
119895 = 1 2 119873
(29)
324 Forecasting Model of Metal Prices The grey model is apowerful tool for forecasting the behaviour of the system in
6 Journal of Applied Mathematics
the future It has been successfully applied to various fieldssince it was proposed by Deng [23ndash31] In this paper we use aone-variable first-order differential grey equation 119866119872(1 1)The essence of 119866119872(1 1) is to accumulate the original data(historical metal prices) in order to obtain regular data Bysetting up the grey differential equation we obtain the fittedcurve in order to predict the future states of the system
Definition 9 Assume that
119875
(0)
(119905
119895
) = [119875
(0)119897
(119905
119895
) 119875
(0)119906
(119905
119895
)]
= [119875
(0)119897
(119905
1
) 119875
(0)119906
(119905
1
)]
[119875
(0)119897
(119905
2
) 119875
(0)119906
(119905
2
)]
[119875
(0)119897
(119905
119873
) 119875
(0)119906
(119905
119873
)]
(30)
is the original series of interval metal prices obtained bytransformation Sampling interval is Δ119905
119895
= 119905
119895
minus 119905
119895minus1
= 1 year
Definition 10 Let 119875(1)(119905119895
) = [119875
(1)119897
(119905
119895
) 119875
(1)119906
(119905
119895
)] be a newsequence generated by the accumulated generating operation(AGO) where
119875
(1)119897
(119905
119895
) =
119873
sum
119895=1
119875
(0)119897
(119905
119895
)
119875
(1)119906
(119905
119895
) =
119873
sum
119895=1
119875
(0)119906
(119905
119895
)
(31)
In the process of forecasting metal prices 119875
(1)119897
(119905
119895
) and119875
(1)119906
(119905
119895
) are the solutions of the following grey differentialequation
119889 [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
119889119905
+ [119902
119897
119902
119906
] sdot [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
= [119908
119897
119908
119906
]
(32)
Obviously (32) is an interval-valued differential equation (see(20))
To get the values of parameters 119902 = [119902
119897
119902
119906
] and 119908 =
[119908
119897
119908
119906
] the least square method is used as follows
otimes[
119902
119908
] = otimes ([119861
119879
119861]
minus1
119861
119879
119884
(0)
) (33)
where
otimes119861 =
[
[
[
[
[
[
[
[
[
[
minus
1
2
(119875
(0)
(1) + 119875
(0)
(2)) 1
minus
1
2
(119875
(0)
(2) + 119875
(0)
(3))
1
minus
1
2
(119875
(0)
(119873 minus 1) + 119875
(0)
(119873)) 1
]
]
]
]
]
]
]
]
]
]
otimes119884
(0)
= [119875
(0)
(2) 119875
(0)
(3) sdot sdot sdot 119875
(0)
(119873)]
119879
(34)
Note that the sign otimes indicates the interval number
If
119875
(1)
(119905) is considered as (i)-Hukuhara differentiablethen the following system of ordinary differential equationsis as follows
119875
(1)119897
(119905) = 119908
119897
minus 119902
119906
sdot 119875
(1)119906
(119905) 119875
119897
(119905
0
) = 119875
(0)119897
(1)
119875
(1)119906
(119905) = 119908
119906
minus 119902
119897
sdot 119875
(1)119897
(119905) 119875
119906
(119905
0
) = 119875
(0)119906
(1)
(35)
The solution of this system (forecasted equation) is as follows
119875
(1)119897
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
1
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
2
+
119908
119906
119902
119897
119895 = 1 2 119873
119875
(1)119906
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
3
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
4
+
119908
119897
119902
119906
119895 = 1 2 119873
(36)
where
119885
1
= 119875
(0)119897
(1) minus
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
+
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
2
= 119875
(0)119897
(1) +
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
minus
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
3
= 119875
(0)119906
(1) minus
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
+
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
119885
4
= 119875
(0)119906
(1) +
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
minus
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
(37)
To obtain the forecasted value of the primitive (original)metal price data at time (119905
119895
+ 1) the inverse accumulatedgenerating operation (IAGO) is used as follows
otimes
119875
(0)
(119905
1
) = otimes119875
(0)
(119905
1
) otimes
119875
(0)
(119905
119895
+ 1)
= otimes
119875
(1)
(119905
119895
+ 1)
minus otimes
119875
(1)
(119905
119895
) 119895 = 1 2 119873
(38)
325 The Model Accuracy
Definition 11 For a given interval grey number otimes([119886119897 119886119906]) itis common to take a whitening value
otimes([119886
119897
119886
119906
]) = 120596 sdot119886
119897
+(1minus
120596) sdot 119886
119906 for forall120596 isin [0 1] Furthermore if 120596 = 05 it is calledequal weight average whitenization [23]
Journal of Applied Mathematics 7
Residual error testing is composed of the calculationof relative error and absolute error between
otimes119875
(0)
(119905
119895
) andotimes
119875
(0)
(119905
119895
) based on the following formulas
Δ120576 (119905
119895
) =otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
Δ120576 (119905
119895
)
1003816
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
(39)
The posterior error detection method means calculation ofthe standard deviation of original metal price series (119878
1
) andstandard deviation of absolute error (119878
2
)
119878
1
=
radic
sum
119873
119895=1
(otimes119875
(0)
(119905
119895
) minusotimes119875
(0)
(119905
119895
))
2
119873 minus 1
119878
2
=
radic
sum
119873
119895=1
(Δ120576 (119905
119895
) minus Δ120576 (119905
119895
))
2
119873 minus 1
(40)
The variance ratio is equal to
119862vr =119878
2
119878
1
(41)
The standard of judgment is represented as follows [24]
119862vr =
lt035 Excellencelt060 Passlt065 Reluctant passge065 No pass
(42)
33 Volatility of Costs Capital development in an under-ground mine consists of shafts ramps raises and lateraltransport drifts required to access ore deposits with expectedutility greater than one yearThis is the development requiredto start up the ore production and to haul the ore to thesurface Experience with investments in capital developmentmight show that such expenditures can run considerablyhigher than the estimates but it is quite unlikely that actualcosts will be lower than estimatedThus the interval numbermight represent capital investment for the project 119868 = [119868
119897
119868
119906
]Operating costs are incurred directly in the production
process These costs include the ore and waste developmentof individual stopes the actual stoping activities the mineservices providing logistical support to the miners and themilling and processing of the ore at the plant These costsare generally more difficult to estimate than capital costsfor most mining ventures If we take into consideration thatproduction will be carried out for many years then it isvery important to predict the future states of operating costsAlthough there is some intention to create a correlationbetween metal price and operating cost it is very hard todefine it since price and cost vary continuously and are differ-ent over time At the project level there will not be a perfectcorrelation between price and cost because of adjustments tovariables such as labour energy explosives and fuel as wellas other material expenditures that are supplied by industries
that are not directly linked to metal price fluctuations Inorder to protect themselves suppliers are offering short-termcontracts to mines that are the opposite of traditional long-term contracts Some components of the operating cost suchas inputs used for mineral processing are usually purchasedatmarket prices that fluctuatemonthly annually or even overshorter periods
The uncertainties related to the future states of operatingcosts are modelled with a special stochastic process geo-metric Brownian motion Certain stochastic processes arefunctions of a Brownianmotion process and these havemanyapplications in finance engineering and the sciences Somespecial processes are solutions of Ito-Doob type stochasticdifferential equations (Ladde and Sambandham [32])
In this model we apply a continuous time process usingthe Ito-Doob type stochastic differential equation to describethe movement of operating costs A general stochastic differ-ential equation takes the following form
119889119862
119905
= 120583 (119862
119905
119905) 119889119905 + 120590 (119862
119905
119905) 119889119882
119905
119862
1199050= 119862
0
(43)
Here 119905 ge 119905
0
119882119905
is a Brownian motion and 119862
119905
gt 0 this isthe cost process 119862
119905
is called the geometric Brownian motion(GBM) which is the solution of the following linear Ito-Doobtype stochastic differential equation
119889119862
119905
= 120583119862
119905
119889119905 + 120590119862
119905
119889119882
119905
(44)
where 120583 and 120590 are some constants (120583 is called the drift and120590 is called the volatility) and 119882
119905
is a normalized BrownianmotionUsing the Ito-Doob formula applied to119891(119862
119905
) = ln119862
119905
we can solve (44) The solution of (44) is given by the exactdiscrete-time equation for 119862
119905
119862
119905
= 119862
119905minus1
sdot 119890
(120583minus120590
22)Δ119905+119873(01)120590
radicΔ119905
(45)
where 119873(0 1) is the normally distributed random variableand Δ119905 = 1 (year) Equation (45) describes an operatingcost scenario with spot costs 119862
119905
By simulating 119862
119905
we obtainoperating costs for every year Simulated values of the costsare obtained by performing the following calculations
119862 = 119862
119904
119905
=
[
[
[
[
[
[
[
119862
1
1
119862
1
2
119862
1
119879
119862
2
1
119862
2
2
119862
2
119879
119862
119878
1
119862
119878
2
119862
119878
119879
]
]
]
]
]
]
]
119904 = 1 2 119878 119905 = 1 2 119879
(46)
where 119878 denotes the number of simulations and119879 the numberof project years In the space 119862
119904
119905
each row represents onesimulated path of costs over the project time while eachcolumn represents simulated values of costs for every yearThe main objective of using simulation is to determine thedistribution of the 119862 for every year of the project In this waywe obtain the sequence of probability density functions of 119862119862
119905
sim (pdf119905
120583
119905
120590
119905
) 119905 = 1 2 119879Applying the same concept of metal prices transfor-
mation we obtain the future sequence of operating costsexpressed by interval numbers 119862(119905
119894
) = [119862
119897
(119905
119894
) 119862
119906
(119905
119894
)] 119894 =
1 2 119879 where 119879 is the total project time
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
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 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
value of project revenues and costs over the life of the projectThe NPV is the preferred criterion of project profitabilitysince it reflects the net contribution to the ownerrsquos equityconsidering his cost of capital We propose a simulationapproach to incorporate uncertainty in the NPV calculationsSimulation of future zinc metal prices is performed by fuzzy-interval grey system theoryThe dynamic nature of operatingcosts is described by the stochastic process called geometricBrownian motion In this way we obtain the probabilitydistribution of operating costs for every year of the projectand after that we transform them into adequate intervalnumbers The remaining risk factors such as capital invest-ment ore grade mill recovery metal content of concentrateand discount rate are also quantified by interval numbersusing expert knowledge (estimation) When these intervalnumbers are incorporated in the NPV calculation we obtainthe interval-valued NPV that is the project value at risk
The main purpose of this study is to provide an efficientand easy way of strategic decision making particularly insmall underground mining companies We were motivatedby the fact that in our country as one of the many develop-ing countries there are mainly small underground miningcompanies employing just two or three mining engineerswho are responsible for both the production maintenanceand strategic planning In such an environment miningengineers do not have time to create adequate proceduresfor decisionmaking particularly for the decisions influencedby highly volatile parameters such as metal price There aremany stochasticmethods for treating the uncertainty ofmetalprices (eg Mean Reversion Process) but if we want to applythem it is necessary to collect a lot of historical data and runcomplex regression analysis in order to define the parametersof the simulation process Interval grey theory can handleproblems with unclear information very precisely Its conceptis intuitive and simple to understand for mining engineersIn order to build the forecasting model only a few data areneeded
The proposed model is tested on a hypothetical examplewhich is similar tomany real case studies and the experimentresults verify the rationality and effectiveness of the method
2 Preliminaries of Interval-ValuedDifferential Equations
21 Basic Concepts of Fuzzy Set Theory Various theoriesexist for describing uncertainty in the modelling of realphenomena and the most popular one is fuzzy set theory [9]In this paper we applied the concept of the interval-valuedpossibilistic mean of fuzzy number [10ndash12]
In the classical set theory an element either belongs ordoes not belong to a given set By contrast in fuzzy set theorya fuzzy subset
119860 defined on a universe of discourse 119883 ischaracterized by a membership function 120583
119860
(119909) which mapseach element in
119860 with a real number in the unit intervalGenerally this can be expressed as 120583
119860
(119909) 119883 rarr [0 1]where the value 120583
119860
(119909) is called the degree of membership ofthe element 119909 in the fuzzy set 119860 If the universal set119883 is fixed
a membership function fully determines a fuzzy set In fuzzyset theory classical sets are usually called crisp sets
Definition 1 Let 1198861
1198862
and 119886
3
be real numbers such that 1198861
lt
119886
2
lt 119886
3
A set
119860 with membership function
120583
119860
(119909) =
119909 minus 119886
1
119886
2
minus 119886
1
119886
1
le 119909 le 119886
2
119909 minus 119886
3
119886
3
minus 119886
2
119886
2
le 119909 le 119886
3
0 otherwise
(1)
is called a fuzzy triangular number and is denoted as
119860 =
(119886
1
119886
2
119886
3
) In geometric interpretations the graph of
119860(119909) isa triangle with its base on the interval [119886
1
119886
3
] and vertex at119909 = 119886
2
Fuzzy sets can also be represented via their 120574-levels
Definition 2 A 120574-level set of a fuzzy set 119860 is defined by [119860]
120574
=
119909 isin 119883 | 120583
119860
(119909) ge 120574 if 120574 gt 0 and [
119860]
120574
= cl119909 isin 119883 | 120583
119860
(119909) gt
0 (the closure of the support of
119860) if 120574 = 0
In particular a fuzzy set
119860 is a fuzzy number if andonly if the 120574-levels are nested nonempty compact intervals[119860
120574
lowast
119860
lowast120574
] This property is the basis for the lower-upperrepresentation of values of the 120574-levels [13] A fuzzy number
119860 is completely defined by a pair of functions 119860
lowast
119860
lowast
[0 1] rarr 119883 defining the end-points of 120574-levels (119860
120574
lowast
=
119860
lowast
(120574) 119860
lowast120574
= 119860
lowast
(120574)) and satisfying the following conditions
(1) 119860
lowast
is a bounded nondecreasing left-continuous func-tion on [0 1]
(2) 119860
lowast is a bounded nonincreasing left-continuous func-tion on [0 1]
(3) 119860
lowast
(120574) le 119860
lowast
(120574) for all 0 le 120574 le 1
According to Dubois and Prade [14] the interval-valuedprobabilistic mean of a fuzzy number
119860 with 120574 levels
119860
120574
=
[119886
120574
119887
120574
] 120574 isin [0 1] (see Figure 1) is the interval 119864(
119860) =
[119864
lowast
(
119860) 119864
lowast
(
119860)] where
119864
lowast
(
119860) = int
1
0
119886
120574
119889120574
119864
lowast
(
119860) = int
1
0
119887
120574
119889120574
(2)
Carlsson and Fuller [11] introduced the interval-valued pos-sibilistic mean of a fuzzy number
119860 as the interval 119872(
119860) =
[119872
lowast
(
119860)119872
lowast
(
119860)] First we note that from the equality
119872(
119860) = int
1
0
120574 (119886
120574
+ 119887
120574
) 119889120574 =
int
1
0
120574 ((119886
120574
+ 119887
120574
) 2) 119889120574
int
1
0
120574119889120574
(3)
it follows that 119872(
119860) is nothing else but the level-weightedaverage of the arithmetic means of all 120574-sets that is the
Journal of Applied Mathematics 3
0
1
120583A(x)
120574
a1 a2 a3X
120572 120573
120574-level cut
a120574
b120574
A120574
Figure 1 Fuzzy number and 120574-level cut
weight of the arithmetic mean of 119886120574 and 119887
120574 is just 120574 Secondwe can rewrite119872(
119860) as
119872(
119860) = int
1
0
120574 (119886
120574
+ 119887
120574
) 119889120574 =
2 int
1
0
120574119886
120574
119889120574 + 2 int
1
0
120574119887
120574
119889120574
2
=
1
2
(
int
1
0
120574119886
120574
119889120574
12
+
int
1
0
120574119887
120574
119889120574
12
)
=
1
2
(
int
1
0
120574119886
120574
119889120574
int
1
0
120574119889120574
+
int
1
0
120574119887
120574
119889120574
int
1
0
120574119889120574
)
(4)
Third let us take a closer look at the right-hand side of theequation for119872(
119860)The first quantity denoted by119872
lowast
(
119860) canbe reformulated as
119872
lowast
(
119860) = 2int
1
0
120574119886
120574
119889120574 =
int
1
0
120574119886
120574
119889120574
int
1
0
120574119889120574
=
int
1
0
Pos [119860 le 119886
120574
] 119886
120574
119889120574
int
1
0
Pos [119860 le 119886
120574
] 119889120574
=
int
1
0
Pos [119860 le 119886
120574
] timesmin [119860]
120574
119889120574
int
1
0
Pos [119860 le 119886
120574
] 119889120574
(5)
where Pos denotes possibility that is
Pos [119860 le 119886
120574
] = prod((minusinfin 119886
120574
] ) = sup⏟⏟⏟⏟⏟⏟⏟
119906le119886
120574
119860 (119906) = 120574 (6)
So 119872
lowast
(
119860) is nothing else but the lower possibility-weightedaverage of the minima of the 120574-sets and this is why we call itthe lower possibilistic mean value of
119860
In a similar manner we introduce 119872
lowast
(
119860) the upperpossibilistic mean value of
119860
119872
lowast
(
119860) = 2int
1
0
120574119887
120574
119889120574 =
int
1
0
120574119887
120574
119889120574
int
1
0
120574119889120574
=
int
1
0
Pos [119860 ge 119887
120574
] 119887
120574
119889120574
int
1
0
Pos [119860 ge 119887
120574
] 119889120574
=
int
1
0
Pos [119860 ge 119887
120574
] timesmax [119860]
120574
119889120574
int
1
0
Pos [119860 ge 119887
120574
] 119889120574
(7)
where we have used equality
Pos [119860 ge 119887
120574
] = prod([119887
120574
infin)) = sup⏟⏟⏟⏟⏟⏟⏟
119906ge119887
120574
119860 (119906) = 120574 (8)
The lower possibilistic mean 119872
lowast
(
119860) is the weighted averageof the minima of the 120574-levels of
119860 Similarly the upperpossibilistic mean 119872
lowast
(
119860) is the weighted average of themaxima of the 120574-levels of 119860 According to Carlsson the fuzzynumber can now be expressed as follows
119872
lowast
(
119860) = 2int
1
0
120574119886
120574
119889120574
119872
lowast
(
119860) = 2int
1
0
120574119887
120574
119889120574
(9)
Definition 3 Let
119860 = (119886
2
120572 120573) be a triangular fuzzy numberwith centre 119886
2
left-width 120572 gt 0 and right-width 120573 gt 0 (seeFigure 1) then 119886
120574 and 119887
120574 are computed as follows
1 120574 = 120572 (119886
120574
minus (119886
2
minus 120572)) 997888rarr 119886
120574
= 119886
2
minus (1 minus 120574) 120572
1 120574 = 120573 ((119886
2
) minus 119887
120574
) 997888rarr 119887
120574
= 119886
2
+ (1 minus 120574) 120573
(10)
Now the 120574-level of
119860 is computed by
[
119860]
120574
= [119886
2
minus (1 minus 120574) 120572 119886
2
+ (1 minus 120574) 120573] forall120574 isin [0 1] (11)
That is
119872
lowast
(
119860) = 2int
1
0
120574 [119886
2
minus (1 minus 120574)] 120572119889120574 = 119886
2
minus
120572
3
119872
lowast
(
119860) = 2int
1
0
120574 [119886
2
+ (1 minus 120574)] 120573119889120574 = 119886
2
+
120573
3
(12)
and therefore
119872(
119860) = [119872
lowast
(
119860) 119872
lowast
(
119860)] = [119886
2
minus
120572
3
119886
2
+
120573
3
] (13)
Obviously 119872(
119860) is a closed interval bounded by the lowerand upper possibilistic mean values of 119860The crisp possibilis-tic mean value of
119860 is defined as the arithmetic mean of its
4 Journal of Applied Mathematics
lower possibilistic and upper possibilistic mean values thatis
119872(
119860) =
119872
lowast
(
119860) + 119872
lowast
(
119860)
2
(14)
According to the above way of transformation (see (13)) weobtain an interval number having both a lower bound 119886
119897 andupper bound 119886
119906 119860 = [119886
119897
119886
119906
] where 119886
119897
= 119886
2
minus 1205723 and 119886
119906
=
119886
2
+ 1205733
From interval arithmetic the following operations ofinterval numbers are defined as follows
Definition 4 For forall119896 gt 0 and a given interval number 119860 =
[119886
119897
119886
119906
] 119886
119897
119886
119906
isin 119877
119896 times [119886
119897
119886
119906
] = [119896 times 119886
119897
119896times119886
119906
] (15)
and for forall119896 lt 0
minus119896 times [119886
119897
119886
119906
] = [minus119896 times 119886
119906
minus119896 times 119886
119897
] (16)
Definition 5 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
]
[119886
119897
119886
119906
] + [119887
119897
119887
119906
] = [119886
119897
+ 119887
119897
119886
119906
+ 119887
119906
]
[119886
119897
119886
119906
] minus [119887
119897
119887
119906
] = [119886
119897
minus 119887
119906
119886
119906
minus 119887
119897
]
(17)
Definition 6 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the multiplication is defined as follows
[119886
119897
119886
119906
] times [119887
119897
119887
119906
] = [min (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
(18)
Definition 7 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the division is defined as follows
[119886
119897
119886
119906
] divide [119887
119897
119887
119906
] = [min(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
0 notin [119887
119897
119887
119906
]
(19)
22 Interval-Valued Differential Equations In this sectionwe consider an interval-valued differential equation of thefollowing form
119910 = 119891 (119905 119910 (119905)) 119910 (119905
0
) = 119910
0
(20)
where 119891 [119886 119887] times 119864 rarr 119864 with 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119909
119910(119905))] for 119910(119905) isin 119864 119910(119905) = [119910
119897
(119905) 119910
119906
(119905)] 119910
0
= [119910
119897
0
119910
119906
0
] Notethat we consider only Hukuhara differentiable solutions thatis there exists 120575 gt 0 such that there are no switching pointsin [119905
0
119905
0
+ 120575] [15 16]
Definition 8 Let 119891 [119886 119887] rarr 119864 be Hukuhara differentiableat 1199050
isin ]119886 119887[ We say that 119891 is (i)-Hukuhara differentiable at119905
0
if
119891 (119905
0
) = [
119891
119897
(119905
0
)
119891
119906
(119905
0
)] (21)
and that 119891 is (ii)-Hukuhara differentiable at 1199050
if
119891 (119905
0
) = [
119891
119906
(119905
0
)
119891
119897
(119905
0
)] (22)
The solution of the differential equation (20) depends onthe choice of the Hukuhara derivative ((i) or (ii)) To solvethe interval-valued differential equation it is necessary toreduce the interval-valued differential equation to a systemof ordinary differential equations [17ndash20]
Let [119910(119905)] = [119910
119897
(119905) 119910
119906
(119905)] If 119910(119905) is (i)-Hukuhara differ-entiable then119863
1
119910(119905) = [ 119910
119897
(119905) 119910
119906
(119905)] transforms (20) into thefollowing system of ordinary differential equations
119910
119897
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(23)
Also if 119910(119905) is (ii)-Hukuhara differentiable then 119863
2
119910(119905) =
[ 119910
119906
(119905) 119910
119897
(119905)] transforms (20) into the following system ofordinary differential equations
119910
119897
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(24)
where 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119905 119910(119905))]
3 Model of the Evaluation
31 The Concept of Evaluation Economic evaluation of amine project requires estimation of the revenues and coststhroughout the defined lifetime of the mine Such evaluationcan be treated as strategic decision making under multiplesources of uncertaintiesTherefore tomake the best decisionbased on the information available it is necessary to developan adequatemodel incorporating the uncertainty of the inputparameters The model should be able to involve a commontime horizon taking the characteristics of the input variablesthat directly affect the value of the proposed project
The model is developed on the basis of full discountedcash flow analysis of an underground zinc mine project Theoperating discounted cash flows are usually estimated on anannual basis Net present value of investment is used as akey criterion in the process of mine project estimation Theexpected net present value of the project is a function of thevariables as
119864 (NPV | 119876 119875 119866119872119862 119868 119903 119905) ge 0 (25)
where 119876 denotes the production rate (capacity) 119875 denotesthe zinc metal price 119866 is the grade 119872 is mill recovery 119862 isoperating costs 119868 is capital investment 119903 is discount rate and119905 is the lifetime of the project that is the period in which thecash flow is generated
Journal of Applied Mathematics 5
In this paper we treat in detail only the variability ofmetalprices and operating costs without intending to decrease thesignificance of the remaining parameters These parametersare taken into account on the basis of expert knowledge(estimation)
32 Forecasting the Revenue of the Mine Most mining com-panies realize their revenues by sellingmetal concentrates as afinal product Estimatingmineral project revenue is indeed adifficult and risky activity Annual mine revenue is calculatedbymultiplying the number of units produced and sold duringthe year by the sales price per unit
The value of the metal concentrate can be expressed asfollows
119881con = 119875 sdot (119898con minus 119898mr) (26)
where 119875 is metal price ($119905) 119898con is metal content ofconcentrate ()119898mr is metal recovery ratio () and
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8) sdot 100
119898congt 85 85
(27)
Annual mine revenue is calculated according to the followingequation
119877year = 119876year sdot 119881con sdot
119866 sdot 119872
119898con (28)
where 119876year is annual ore production (119905year) 119866 is grade ofthe ore mined ()119872 is mill recovery ()
Annual ore production is derived from themining projectschedule and is defined as crisp value The concept of grade(119866) is defined as the ratio of useful mass of metal tothe total mass of ore and its critical value fluctuates overdeposit space and can be estimated by experts and defined asinterval number 119866 = [119866
119897
119866
119906
] Mill recovery (119872) is relatedto the flotation as the most widely used method for theconcentration of fine grained minerals It can also be definedas interval number 119872 = [119872
119897
119872
119906
] Metal content representsthe quality of concentrate and we also apply the concept of aninterval number to define it 119898con = [119898
119897
con 119898119906
con]The major external source of risk affecting mine revenue
is related to the uncertainty about market behaviour of metalprices Forecasting the precise future state of themetal price isa very difficult task formine planners To predict futuremetalprices we apply the concept based on the transformation ofhistorical metal prices into adequate fuzzy-interval numbersand grey system theory
The forecasting model of metal prices is composed of thefollowing steps
Step 1 Create the set PDF = pdf119895
119895 = 1 2 119873 wherepdf119895
is the probability density function of metal prices forevery historical year The minimum number of elements ofthe set is four 119895min = 1 2 3 4
Step 2 Transform the set PDF into the set TFN = (119886
119895
119887
119895
119888
119895
)where (119886
119895
119887
119895
119888
119895
) is an adequate fuzzy triangular number
Step 3 Transform the set TFN into the set INT = [119886
119897
119895
119886
119906
119895
]where [119886
119897
119895
119886
119906
119895
] is an adequate interval number
Step 4 Using grey system prediction theory create a greydifferential equation of type 119866119872(1 1) that is the first-ordervariable grey derivative
Step 5 Testing of 119866119872(1 1) by residual error testing and theposterior error detection method
321 Analysis of Historical Metal Prices For every historicalyear it is necessary to define a probability density functionwith the following characteristics shape by histogram meanvalue (120583
119895
) and standard deviation (120590
119895
) In this way we obtainthe sequence of probability density functions of 119875 119875
119895
sim
(pdf119895
120583
119895
120590
119895
) 119895 = 1 2 119873 where 119873 is the total numberof historical years
322 Fuzzification of Metal Prices The sequence of obtainedpdf119895
of 119875119895
can be transformed into a sequence of triangularfuzzy numbers of 119875
119895
119875119895
sim TFN119895
119895 = 1 2 119873 that is119875
1
sim pdf1
rarr 119875
1
sim TFN1
1198752
sim pdf2
rarr 119875
2
sim TFN2
119875
119873
sim pdf119873
rarr 119875
119873
sim TFN119873
The method of transformationis based on the following facts the support of themembershipfunction and the pdf are the same and the point with thehighest probability (likelihood) has the highest possibilityFor more details see Swishchuk et al [21] The uncertaintyin the 119875 parameter is modelled by a triangular fuzzy numberwith the membership function which has the support of 120583
119895
minus
2120590
119895
lt 119875
119895
lt 120583
119895
+ 2120590
119895
119895 = 1 2 119873 set up for around 95confidence interval of distribution function If we take intoconsideration that the triangular fuzzy number is defined asa triplet (119886
1119895
119886
2119895
119886
3119895
) then 119886
1119895
and 119886
3119895
are the lower boundand upper bound obtained from the lower and upper boundof 5 of the distribution and the most promising value 119886
2119895
isequal to the mean value of the distribution For more detailssee Do et al [22]
323 Metal Prices as Interval Numbers The sequence ofobtainedTFN
119895
of119875119895
is transformed into a sequence of intervalnumbers of 119875
119895
119875
119895
sim INT119895
119895 = 1 2 119873 that is 1198751
sim
TFN1
rarr 119875
1
sim INT1
119875
2
sim TFN2
rarr 119875
2
sim INT2
119875
119873
sim TFN119873
rarr 119875
119873
sim INT119873
The method of transformationis described in Section 21 (basic concepts of fuzzy set theory)According to this method of transformation we obtain set
119875
119895
sim INT119895
119875
119895
= [119875
119897
119895
119875
119906
119895
] = [119886
2119895
minus
120572
119895
3
119886
2119895
+
120573
119895
3
]
119895 = 1 2 119873
(29)
324 Forecasting Model of Metal Prices The grey model is apowerful tool for forecasting the behaviour of the system in
6 Journal of Applied Mathematics
the future It has been successfully applied to various fieldssince it was proposed by Deng [23ndash31] In this paper we use aone-variable first-order differential grey equation 119866119872(1 1)The essence of 119866119872(1 1) is to accumulate the original data(historical metal prices) in order to obtain regular data Bysetting up the grey differential equation we obtain the fittedcurve in order to predict the future states of the system
Definition 9 Assume that
119875
(0)
(119905
119895
) = [119875
(0)119897
(119905
119895
) 119875
(0)119906
(119905
119895
)]
= [119875
(0)119897
(119905
1
) 119875
(0)119906
(119905
1
)]
[119875
(0)119897
(119905
2
) 119875
(0)119906
(119905
2
)]
[119875
(0)119897
(119905
119873
) 119875
(0)119906
(119905
119873
)]
(30)
is the original series of interval metal prices obtained bytransformation Sampling interval is Δ119905
119895
= 119905
119895
minus 119905
119895minus1
= 1 year
Definition 10 Let 119875(1)(119905119895
) = [119875
(1)119897
(119905
119895
) 119875
(1)119906
(119905
119895
)] be a newsequence generated by the accumulated generating operation(AGO) where
119875
(1)119897
(119905
119895
) =
119873
sum
119895=1
119875
(0)119897
(119905
119895
)
119875
(1)119906
(119905
119895
) =
119873
sum
119895=1
119875
(0)119906
(119905
119895
)
(31)
In the process of forecasting metal prices 119875
(1)119897
(119905
119895
) and119875
(1)119906
(119905
119895
) are the solutions of the following grey differentialequation
119889 [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
119889119905
+ [119902
119897
119902
119906
] sdot [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
= [119908
119897
119908
119906
]
(32)
Obviously (32) is an interval-valued differential equation (see(20))
To get the values of parameters 119902 = [119902
119897
119902
119906
] and 119908 =
[119908
119897
119908
119906
] the least square method is used as follows
otimes[
119902
119908
] = otimes ([119861
119879
119861]
minus1
119861
119879
119884
(0)
) (33)
where
otimes119861 =
[
[
[
[
[
[
[
[
[
[
minus
1
2
(119875
(0)
(1) + 119875
(0)
(2)) 1
minus
1
2
(119875
(0)
(2) + 119875
(0)
(3))
1
minus
1
2
(119875
(0)
(119873 minus 1) + 119875
(0)
(119873)) 1
]
]
]
]
]
]
]
]
]
]
otimes119884
(0)
= [119875
(0)
(2) 119875
(0)
(3) sdot sdot sdot 119875
(0)
(119873)]
119879
(34)
Note that the sign otimes indicates the interval number
If
119875
(1)
(119905) is considered as (i)-Hukuhara differentiablethen the following system of ordinary differential equationsis as follows
119875
(1)119897
(119905) = 119908
119897
minus 119902
119906
sdot 119875
(1)119906
(119905) 119875
119897
(119905
0
) = 119875
(0)119897
(1)
119875
(1)119906
(119905) = 119908
119906
minus 119902
119897
sdot 119875
(1)119897
(119905) 119875
119906
(119905
0
) = 119875
(0)119906
(1)
(35)
The solution of this system (forecasted equation) is as follows
119875
(1)119897
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
1
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
2
+
119908
119906
119902
119897
119895 = 1 2 119873
119875
(1)119906
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
3
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
4
+
119908
119897
119902
119906
119895 = 1 2 119873
(36)
where
119885
1
= 119875
(0)119897
(1) minus
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
+
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
2
= 119875
(0)119897
(1) +
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
minus
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
3
= 119875
(0)119906
(1) minus
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
+
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
119885
4
= 119875
(0)119906
(1) +
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
minus
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
(37)
To obtain the forecasted value of the primitive (original)metal price data at time (119905
119895
+ 1) the inverse accumulatedgenerating operation (IAGO) is used as follows
otimes
119875
(0)
(119905
1
) = otimes119875
(0)
(119905
1
) otimes
119875
(0)
(119905
119895
+ 1)
= otimes
119875
(1)
(119905
119895
+ 1)
minus otimes
119875
(1)
(119905
119895
) 119895 = 1 2 119873
(38)
325 The Model Accuracy
Definition 11 For a given interval grey number otimes([119886119897 119886119906]) itis common to take a whitening value
otimes([119886
119897
119886
119906
]) = 120596 sdot119886
119897
+(1minus
120596) sdot 119886
119906 for forall120596 isin [0 1] Furthermore if 120596 = 05 it is calledequal weight average whitenization [23]
Journal of Applied Mathematics 7
Residual error testing is composed of the calculationof relative error and absolute error between
otimes119875
(0)
(119905
119895
) andotimes
119875
(0)
(119905
119895
) based on the following formulas
Δ120576 (119905
119895
) =otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
Δ120576 (119905
119895
)
1003816
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
(39)
The posterior error detection method means calculation ofthe standard deviation of original metal price series (119878
1
) andstandard deviation of absolute error (119878
2
)
119878
1
=
radic
sum
119873
119895=1
(otimes119875
(0)
(119905
119895
) minusotimes119875
(0)
(119905
119895
))
2
119873 minus 1
119878
2
=
radic
sum
119873
119895=1
(Δ120576 (119905
119895
) minus Δ120576 (119905
119895
))
2
119873 minus 1
(40)
The variance ratio is equal to
119862vr =119878
2
119878
1
(41)
The standard of judgment is represented as follows [24]
119862vr =
lt035 Excellencelt060 Passlt065 Reluctant passge065 No pass
(42)
33 Volatility of Costs Capital development in an under-ground mine consists of shafts ramps raises and lateraltransport drifts required to access ore deposits with expectedutility greater than one yearThis is the development requiredto start up the ore production and to haul the ore to thesurface Experience with investments in capital developmentmight show that such expenditures can run considerablyhigher than the estimates but it is quite unlikely that actualcosts will be lower than estimatedThus the interval numbermight represent capital investment for the project 119868 = [119868
119897
119868
119906
]Operating costs are incurred directly in the production
process These costs include the ore and waste developmentof individual stopes the actual stoping activities the mineservices providing logistical support to the miners and themilling and processing of the ore at the plant These costsare generally more difficult to estimate than capital costsfor most mining ventures If we take into consideration thatproduction will be carried out for many years then it isvery important to predict the future states of operating costsAlthough there is some intention to create a correlationbetween metal price and operating cost it is very hard todefine it since price and cost vary continuously and are differ-ent over time At the project level there will not be a perfectcorrelation between price and cost because of adjustments tovariables such as labour energy explosives and fuel as wellas other material expenditures that are supplied by industries
that are not directly linked to metal price fluctuations Inorder to protect themselves suppliers are offering short-termcontracts to mines that are the opposite of traditional long-term contracts Some components of the operating cost suchas inputs used for mineral processing are usually purchasedatmarket prices that fluctuatemonthly annually or even overshorter periods
The uncertainties related to the future states of operatingcosts are modelled with a special stochastic process geo-metric Brownian motion Certain stochastic processes arefunctions of a Brownianmotion process and these havemanyapplications in finance engineering and the sciences Somespecial processes are solutions of Ito-Doob type stochasticdifferential equations (Ladde and Sambandham [32])
In this model we apply a continuous time process usingthe Ito-Doob type stochastic differential equation to describethe movement of operating costs A general stochastic differ-ential equation takes the following form
119889119862
119905
= 120583 (119862
119905
119905) 119889119905 + 120590 (119862
119905
119905) 119889119882
119905
119862
1199050= 119862
0
(43)
Here 119905 ge 119905
0
119882119905
is a Brownian motion and 119862
119905
gt 0 this isthe cost process 119862
119905
is called the geometric Brownian motion(GBM) which is the solution of the following linear Ito-Doobtype stochastic differential equation
119889119862
119905
= 120583119862
119905
119889119905 + 120590119862
119905
119889119882
119905
(44)
where 120583 and 120590 are some constants (120583 is called the drift and120590 is called the volatility) and 119882
119905
is a normalized BrownianmotionUsing the Ito-Doob formula applied to119891(119862
119905
) = ln119862
119905
we can solve (44) The solution of (44) is given by the exactdiscrete-time equation for 119862
119905
119862
119905
= 119862
119905minus1
sdot 119890
(120583minus120590
22)Δ119905+119873(01)120590
radicΔ119905
(45)
where 119873(0 1) is the normally distributed random variableand Δ119905 = 1 (year) Equation (45) describes an operatingcost scenario with spot costs 119862
119905
By simulating 119862
119905
we obtainoperating costs for every year Simulated values of the costsare obtained by performing the following calculations
119862 = 119862
119904
119905
=
[
[
[
[
[
[
[
119862
1
1
119862
1
2
119862
1
119879
119862
2
1
119862
2
2
119862
2
119879
119862
119878
1
119862
119878
2
119862
119878
119879
]
]
]
]
]
]
]
119904 = 1 2 119878 119905 = 1 2 119879
(46)
where 119878 denotes the number of simulations and119879 the numberof project years In the space 119862
119904
119905
each row represents onesimulated path of costs over the project time while eachcolumn represents simulated values of costs for every yearThe main objective of using simulation is to determine thedistribution of the 119862 for every year of the project In this waywe obtain the sequence of probability density functions of 119862119862
119905
sim (pdf119905
120583
119905
120590
119905
) 119905 = 1 2 119879Applying the same concept of metal prices transfor-
mation we obtain the future sequence of operating costsexpressed by interval numbers 119862(119905
119894
) = [119862
119897
(119905
119894
) 119862
119906
(119905
119894
)] 119894 =
1 2 119879 where 119879 is the total project time
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
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 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
0
1
120583A(x)
120574
a1 a2 a3X
120572 120573
120574-level cut
a120574
b120574
A120574
Figure 1 Fuzzy number and 120574-level cut
weight of the arithmetic mean of 119886120574 and 119887
120574 is just 120574 Secondwe can rewrite119872(
119860) as
119872(
119860) = int
1
0
120574 (119886
120574
+ 119887
120574
) 119889120574 =
2 int
1
0
120574119886
120574
119889120574 + 2 int
1
0
120574119887
120574
119889120574
2
=
1
2
(
int
1
0
120574119886
120574
119889120574
12
+
int
1
0
120574119887
120574
119889120574
12
)
=
1
2
(
int
1
0
120574119886
120574
119889120574
int
1
0
120574119889120574
+
int
1
0
120574119887
120574
119889120574
int
1
0
120574119889120574
)
(4)
Third let us take a closer look at the right-hand side of theequation for119872(
119860)The first quantity denoted by119872
lowast
(
119860) canbe reformulated as
119872
lowast
(
119860) = 2int
1
0
120574119886
120574
119889120574 =
int
1
0
120574119886
120574
119889120574
int
1
0
120574119889120574
=
int
1
0
Pos [119860 le 119886
120574
] 119886
120574
119889120574
int
1
0
Pos [119860 le 119886
120574
] 119889120574
=
int
1
0
Pos [119860 le 119886
120574
] timesmin [119860]
120574
119889120574
int
1
0
Pos [119860 le 119886
120574
] 119889120574
(5)
where Pos denotes possibility that is
Pos [119860 le 119886
120574
] = prod((minusinfin 119886
120574
] ) = sup⏟⏟⏟⏟⏟⏟⏟
119906le119886
120574
119860 (119906) = 120574 (6)
So 119872
lowast
(
119860) is nothing else but the lower possibility-weightedaverage of the minima of the 120574-sets and this is why we call itthe lower possibilistic mean value of
119860
In a similar manner we introduce 119872
lowast
(
119860) the upperpossibilistic mean value of
119860
119872
lowast
(
119860) = 2int
1
0
120574119887
120574
119889120574 =
int
1
0
120574119887
120574
119889120574
int
1
0
120574119889120574
=
int
1
0
Pos [119860 ge 119887
120574
] 119887
120574
119889120574
int
1
0
Pos [119860 ge 119887
120574
] 119889120574
=
int
1
0
Pos [119860 ge 119887
120574
] timesmax [119860]
120574
119889120574
int
1
0
Pos [119860 ge 119887
120574
] 119889120574
(7)
where we have used equality
Pos [119860 ge 119887
120574
] = prod([119887
120574
infin)) = sup⏟⏟⏟⏟⏟⏟⏟
119906ge119887
120574
119860 (119906) = 120574 (8)
The lower possibilistic mean 119872
lowast
(
119860) is the weighted averageof the minima of the 120574-levels of
119860 Similarly the upperpossibilistic mean 119872
lowast
(
119860) is the weighted average of themaxima of the 120574-levels of 119860 According to Carlsson the fuzzynumber can now be expressed as follows
119872
lowast
(
119860) = 2int
1
0
120574119886
120574
119889120574
119872
lowast
(
119860) = 2int
1
0
120574119887
120574
119889120574
(9)
Definition 3 Let
119860 = (119886
2
120572 120573) be a triangular fuzzy numberwith centre 119886
2
left-width 120572 gt 0 and right-width 120573 gt 0 (seeFigure 1) then 119886
120574 and 119887
120574 are computed as follows
1 120574 = 120572 (119886
120574
minus (119886
2
minus 120572)) 997888rarr 119886
120574
= 119886
2
minus (1 minus 120574) 120572
1 120574 = 120573 ((119886
2
) minus 119887
120574
) 997888rarr 119887
120574
= 119886
2
+ (1 minus 120574) 120573
(10)
Now the 120574-level of
119860 is computed by
[
119860]
120574
= [119886
2
minus (1 minus 120574) 120572 119886
2
+ (1 minus 120574) 120573] forall120574 isin [0 1] (11)
That is
119872
lowast
(
119860) = 2int
1
0
120574 [119886
2
minus (1 minus 120574)] 120572119889120574 = 119886
2
minus
120572
3
119872
lowast
(
119860) = 2int
1
0
120574 [119886
2
+ (1 minus 120574)] 120573119889120574 = 119886
2
+
120573
3
(12)
and therefore
119872(
119860) = [119872
lowast
(
119860) 119872
lowast
(
119860)] = [119886
2
minus
120572
3
119886
2
+
120573
3
] (13)
Obviously 119872(
119860) is a closed interval bounded by the lowerand upper possibilistic mean values of 119860The crisp possibilis-tic mean value of
119860 is defined as the arithmetic mean of its
4 Journal of Applied Mathematics
lower possibilistic and upper possibilistic mean values thatis
119872(
119860) =
119872
lowast
(
119860) + 119872
lowast
(
119860)
2
(14)
According to the above way of transformation (see (13)) weobtain an interval number having both a lower bound 119886
119897 andupper bound 119886
119906 119860 = [119886
119897
119886
119906
] where 119886
119897
= 119886
2
minus 1205723 and 119886
119906
=
119886
2
+ 1205733
From interval arithmetic the following operations ofinterval numbers are defined as follows
Definition 4 For forall119896 gt 0 and a given interval number 119860 =
[119886
119897
119886
119906
] 119886
119897
119886
119906
isin 119877
119896 times [119886
119897
119886
119906
] = [119896 times 119886
119897
119896times119886
119906
] (15)
and for forall119896 lt 0
minus119896 times [119886
119897
119886
119906
] = [minus119896 times 119886
119906
minus119896 times 119886
119897
] (16)
Definition 5 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
]
[119886
119897
119886
119906
] + [119887
119897
119887
119906
] = [119886
119897
+ 119887
119897
119886
119906
+ 119887
119906
]
[119886
119897
119886
119906
] minus [119887
119897
119887
119906
] = [119886
119897
minus 119887
119906
119886
119906
minus 119887
119897
]
(17)
Definition 6 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the multiplication is defined as follows
[119886
119897
119886
119906
] times [119887
119897
119887
119906
] = [min (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
(18)
Definition 7 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the division is defined as follows
[119886
119897
119886
119906
] divide [119887
119897
119887
119906
] = [min(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
0 notin [119887
119897
119887
119906
]
(19)
22 Interval-Valued Differential Equations In this sectionwe consider an interval-valued differential equation of thefollowing form
119910 = 119891 (119905 119910 (119905)) 119910 (119905
0
) = 119910
0
(20)
where 119891 [119886 119887] times 119864 rarr 119864 with 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119909
119910(119905))] for 119910(119905) isin 119864 119910(119905) = [119910
119897
(119905) 119910
119906
(119905)] 119910
0
= [119910
119897
0
119910
119906
0
] Notethat we consider only Hukuhara differentiable solutions thatis there exists 120575 gt 0 such that there are no switching pointsin [119905
0
119905
0
+ 120575] [15 16]
Definition 8 Let 119891 [119886 119887] rarr 119864 be Hukuhara differentiableat 1199050
isin ]119886 119887[ We say that 119891 is (i)-Hukuhara differentiable at119905
0
if
119891 (119905
0
) = [
119891
119897
(119905
0
)
119891
119906
(119905
0
)] (21)
and that 119891 is (ii)-Hukuhara differentiable at 1199050
if
119891 (119905
0
) = [
119891
119906
(119905
0
)
119891
119897
(119905
0
)] (22)
The solution of the differential equation (20) depends onthe choice of the Hukuhara derivative ((i) or (ii)) To solvethe interval-valued differential equation it is necessary toreduce the interval-valued differential equation to a systemof ordinary differential equations [17ndash20]
Let [119910(119905)] = [119910
119897
(119905) 119910
119906
(119905)] If 119910(119905) is (i)-Hukuhara differ-entiable then119863
1
119910(119905) = [ 119910
119897
(119905) 119910
119906
(119905)] transforms (20) into thefollowing system of ordinary differential equations
119910
119897
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(23)
Also if 119910(119905) is (ii)-Hukuhara differentiable then 119863
2
119910(119905) =
[ 119910
119906
(119905) 119910
119897
(119905)] transforms (20) into the following system ofordinary differential equations
119910
119897
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(24)
where 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119905 119910(119905))]
3 Model of the Evaluation
31 The Concept of Evaluation Economic evaluation of amine project requires estimation of the revenues and coststhroughout the defined lifetime of the mine Such evaluationcan be treated as strategic decision making under multiplesources of uncertaintiesTherefore tomake the best decisionbased on the information available it is necessary to developan adequatemodel incorporating the uncertainty of the inputparameters The model should be able to involve a commontime horizon taking the characteristics of the input variablesthat directly affect the value of the proposed project
The model is developed on the basis of full discountedcash flow analysis of an underground zinc mine project Theoperating discounted cash flows are usually estimated on anannual basis Net present value of investment is used as akey criterion in the process of mine project estimation Theexpected net present value of the project is a function of thevariables as
119864 (NPV | 119876 119875 119866119872119862 119868 119903 119905) ge 0 (25)
where 119876 denotes the production rate (capacity) 119875 denotesthe zinc metal price 119866 is the grade 119872 is mill recovery 119862 isoperating costs 119868 is capital investment 119903 is discount rate and119905 is the lifetime of the project that is the period in which thecash flow is generated
Journal of Applied Mathematics 5
In this paper we treat in detail only the variability ofmetalprices and operating costs without intending to decrease thesignificance of the remaining parameters These parametersare taken into account on the basis of expert knowledge(estimation)
32 Forecasting the Revenue of the Mine Most mining com-panies realize their revenues by sellingmetal concentrates as afinal product Estimatingmineral project revenue is indeed adifficult and risky activity Annual mine revenue is calculatedbymultiplying the number of units produced and sold duringthe year by the sales price per unit
The value of the metal concentrate can be expressed asfollows
119881con = 119875 sdot (119898con minus 119898mr) (26)
where 119875 is metal price ($119905) 119898con is metal content ofconcentrate ()119898mr is metal recovery ratio () and
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8) sdot 100
119898congt 85 85
(27)
Annual mine revenue is calculated according to the followingequation
119877year = 119876year sdot 119881con sdot
119866 sdot 119872
119898con (28)
where 119876year is annual ore production (119905year) 119866 is grade ofthe ore mined ()119872 is mill recovery ()
Annual ore production is derived from themining projectschedule and is defined as crisp value The concept of grade(119866) is defined as the ratio of useful mass of metal tothe total mass of ore and its critical value fluctuates overdeposit space and can be estimated by experts and defined asinterval number 119866 = [119866
119897
119866
119906
] Mill recovery (119872) is relatedto the flotation as the most widely used method for theconcentration of fine grained minerals It can also be definedas interval number 119872 = [119872
119897
119872
119906
] Metal content representsthe quality of concentrate and we also apply the concept of aninterval number to define it 119898con = [119898
119897
con 119898119906
con]The major external source of risk affecting mine revenue
is related to the uncertainty about market behaviour of metalprices Forecasting the precise future state of themetal price isa very difficult task formine planners To predict futuremetalprices we apply the concept based on the transformation ofhistorical metal prices into adequate fuzzy-interval numbersand grey system theory
The forecasting model of metal prices is composed of thefollowing steps
Step 1 Create the set PDF = pdf119895
119895 = 1 2 119873 wherepdf119895
is the probability density function of metal prices forevery historical year The minimum number of elements ofthe set is four 119895min = 1 2 3 4
Step 2 Transform the set PDF into the set TFN = (119886
119895
119887
119895
119888
119895
)where (119886
119895
119887
119895
119888
119895
) is an adequate fuzzy triangular number
Step 3 Transform the set TFN into the set INT = [119886
119897
119895
119886
119906
119895
]where [119886
119897
119895
119886
119906
119895
] is an adequate interval number
Step 4 Using grey system prediction theory create a greydifferential equation of type 119866119872(1 1) that is the first-ordervariable grey derivative
Step 5 Testing of 119866119872(1 1) by residual error testing and theposterior error detection method
321 Analysis of Historical Metal Prices For every historicalyear it is necessary to define a probability density functionwith the following characteristics shape by histogram meanvalue (120583
119895
) and standard deviation (120590
119895
) In this way we obtainthe sequence of probability density functions of 119875 119875
119895
sim
(pdf119895
120583
119895
120590
119895
) 119895 = 1 2 119873 where 119873 is the total numberof historical years
322 Fuzzification of Metal Prices The sequence of obtainedpdf119895
of 119875119895
can be transformed into a sequence of triangularfuzzy numbers of 119875
119895
119875119895
sim TFN119895
119895 = 1 2 119873 that is119875
1
sim pdf1
rarr 119875
1
sim TFN1
1198752
sim pdf2
rarr 119875
2
sim TFN2
119875
119873
sim pdf119873
rarr 119875
119873
sim TFN119873
The method of transformationis based on the following facts the support of themembershipfunction and the pdf are the same and the point with thehighest probability (likelihood) has the highest possibilityFor more details see Swishchuk et al [21] The uncertaintyin the 119875 parameter is modelled by a triangular fuzzy numberwith the membership function which has the support of 120583
119895
minus
2120590
119895
lt 119875
119895
lt 120583
119895
+ 2120590
119895
119895 = 1 2 119873 set up for around 95confidence interval of distribution function If we take intoconsideration that the triangular fuzzy number is defined asa triplet (119886
1119895
119886
2119895
119886
3119895
) then 119886
1119895
and 119886
3119895
are the lower boundand upper bound obtained from the lower and upper boundof 5 of the distribution and the most promising value 119886
2119895
isequal to the mean value of the distribution For more detailssee Do et al [22]
323 Metal Prices as Interval Numbers The sequence ofobtainedTFN
119895
of119875119895
is transformed into a sequence of intervalnumbers of 119875
119895
119875
119895
sim INT119895
119895 = 1 2 119873 that is 1198751
sim
TFN1
rarr 119875
1
sim INT1
119875
2
sim TFN2
rarr 119875
2
sim INT2
119875
119873
sim TFN119873
rarr 119875
119873
sim INT119873
The method of transformationis described in Section 21 (basic concepts of fuzzy set theory)According to this method of transformation we obtain set
119875
119895
sim INT119895
119875
119895
= [119875
119897
119895
119875
119906
119895
] = [119886
2119895
minus
120572
119895
3
119886
2119895
+
120573
119895
3
]
119895 = 1 2 119873
(29)
324 Forecasting Model of Metal Prices The grey model is apowerful tool for forecasting the behaviour of the system in
6 Journal of Applied Mathematics
the future It has been successfully applied to various fieldssince it was proposed by Deng [23ndash31] In this paper we use aone-variable first-order differential grey equation 119866119872(1 1)The essence of 119866119872(1 1) is to accumulate the original data(historical metal prices) in order to obtain regular data Bysetting up the grey differential equation we obtain the fittedcurve in order to predict the future states of the system
Definition 9 Assume that
119875
(0)
(119905
119895
) = [119875
(0)119897
(119905
119895
) 119875
(0)119906
(119905
119895
)]
= [119875
(0)119897
(119905
1
) 119875
(0)119906
(119905
1
)]
[119875
(0)119897
(119905
2
) 119875
(0)119906
(119905
2
)]
[119875
(0)119897
(119905
119873
) 119875
(0)119906
(119905
119873
)]
(30)
is the original series of interval metal prices obtained bytransformation Sampling interval is Δ119905
119895
= 119905
119895
minus 119905
119895minus1
= 1 year
Definition 10 Let 119875(1)(119905119895
) = [119875
(1)119897
(119905
119895
) 119875
(1)119906
(119905
119895
)] be a newsequence generated by the accumulated generating operation(AGO) where
119875
(1)119897
(119905
119895
) =
119873
sum
119895=1
119875
(0)119897
(119905
119895
)
119875
(1)119906
(119905
119895
) =
119873
sum
119895=1
119875
(0)119906
(119905
119895
)
(31)
In the process of forecasting metal prices 119875
(1)119897
(119905
119895
) and119875
(1)119906
(119905
119895
) are the solutions of the following grey differentialequation
119889 [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
119889119905
+ [119902
119897
119902
119906
] sdot [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
= [119908
119897
119908
119906
]
(32)
Obviously (32) is an interval-valued differential equation (see(20))
To get the values of parameters 119902 = [119902
119897
119902
119906
] and 119908 =
[119908
119897
119908
119906
] the least square method is used as follows
otimes[
119902
119908
] = otimes ([119861
119879
119861]
minus1
119861
119879
119884
(0)
) (33)
where
otimes119861 =
[
[
[
[
[
[
[
[
[
[
minus
1
2
(119875
(0)
(1) + 119875
(0)
(2)) 1
minus
1
2
(119875
(0)
(2) + 119875
(0)
(3))
1
minus
1
2
(119875
(0)
(119873 minus 1) + 119875
(0)
(119873)) 1
]
]
]
]
]
]
]
]
]
]
otimes119884
(0)
= [119875
(0)
(2) 119875
(0)
(3) sdot sdot sdot 119875
(0)
(119873)]
119879
(34)
Note that the sign otimes indicates the interval number
If
119875
(1)
(119905) is considered as (i)-Hukuhara differentiablethen the following system of ordinary differential equationsis as follows
119875
(1)119897
(119905) = 119908
119897
minus 119902
119906
sdot 119875
(1)119906
(119905) 119875
119897
(119905
0
) = 119875
(0)119897
(1)
119875
(1)119906
(119905) = 119908
119906
minus 119902
119897
sdot 119875
(1)119897
(119905) 119875
119906
(119905
0
) = 119875
(0)119906
(1)
(35)
The solution of this system (forecasted equation) is as follows
119875
(1)119897
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
1
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
2
+
119908
119906
119902
119897
119895 = 1 2 119873
119875
(1)119906
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
3
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
4
+
119908
119897
119902
119906
119895 = 1 2 119873
(36)
where
119885
1
= 119875
(0)119897
(1) minus
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
+
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
2
= 119875
(0)119897
(1) +
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
minus
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
3
= 119875
(0)119906
(1) minus
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
+
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
119885
4
= 119875
(0)119906
(1) +
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
minus
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
(37)
To obtain the forecasted value of the primitive (original)metal price data at time (119905
119895
+ 1) the inverse accumulatedgenerating operation (IAGO) is used as follows
otimes
119875
(0)
(119905
1
) = otimes119875
(0)
(119905
1
) otimes
119875
(0)
(119905
119895
+ 1)
= otimes
119875
(1)
(119905
119895
+ 1)
minus otimes
119875
(1)
(119905
119895
) 119895 = 1 2 119873
(38)
325 The Model Accuracy
Definition 11 For a given interval grey number otimes([119886119897 119886119906]) itis common to take a whitening value
otimes([119886
119897
119886
119906
]) = 120596 sdot119886
119897
+(1minus
120596) sdot 119886
119906 for forall120596 isin [0 1] Furthermore if 120596 = 05 it is calledequal weight average whitenization [23]
Journal of Applied Mathematics 7
Residual error testing is composed of the calculationof relative error and absolute error between
otimes119875
(0)
(119905
119895
) andotimes
119875
(0)
(119905
119895
) based on the following formulas
Δ120576 (119905
119895
) =otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
Δ120576 (119905
119895
)
1003816
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
(39)
The posterior error detection method means calculation ofthe standard deviation of original metal price series (119878
1
) andstandard deviation of absolute error (119878
2
)
119878
1
=
radic
sum
119873
119895=1
(otimes119875
(0)
(119905
119895
) minusotimes119875
(0)
(119905
119895
))
2
119873 minus 1
119878
2
=
radic
sum
119873
119895=1
(Δ120576 (119905
119895
) minus Δ120576 (119905
119895
))
2
119873 minus 1
(40)
The variance ratio is equal to
119862vr =119878
2
119878
1
(41)
The standard of judgment is represented as follows [24]
119862vr =
lt035 Excellencelt060 Passlt065 Reluctant passge065 No pass
(42)
33 Volatility of Costs Capital development in an under-ground mine consists of shafts ramps raises and lateraltransport drifts required to access ore deposits with expectedutility greater than one yearThis is the development requiredto start up the ore production and to haul the ore to thesurface Experience with investments in capital developmentmight show that such expenditures can run considerablyhigher than the estimates but it is quite unlikely that actualcosts will be lower than estimatedThus the interval numbermight represent capital investment for the project 119868 = [119868
119897
119868
119906
]Operating costs are incurred directly in the production
process These costs include the ore and waste developmentof individual stopes the actual stoping activities the mineservices providing logistical support to the miners and themilling and processing of the ore at the plant These costsare generally more difficult to estimate than capital costsfor most mining ventures If we take into consideration thatproduction will be carried out for many years then it isvery important to predict the future states of operating costsAlthough there is some intention to create a correlationbetween metal price and operating cost it is very hard todefine it since price and cost vary continuously and are differ-ent over time At the project level there will not be a perfectcorrelation between price and cost because of adjustments tovariables such as labour energy explosives and fuel as wellas other material expenditures that are supplied by industries
that are not directly linked to metal price fluctuations Inorder to protect themselves suppliers are offering short-termcontracts to mines that are the opposite of traditional long-term contracts Some components of the operating cost suchas inputs used for mineral processing are usually purchasedatmarket prices that fluctuatemonthly annually or even overshorter periods
The uncertainties related to the future states of operatingcosts are modelled with a special stochastic process geo-metric Brownian motion Certain stochastic processes arefunctions of a Brownianmotion process and these havemanyapplications in finance engineering and the sciences Somespecial processes are solutions of Ito-Doob type stochasticdifferential equations (Ladde and Sambandham [32])
In this model we apply a continuous time process usingthe Ito-Doob type stochastic differential equation to describethe movement of operating costs A general stochastic differ-ential equation takes the following form
119889119862
119905
= 120583 (119862
119905
119905) 119889119905 + 120590 (119862
119905
119905) 119889119882
119905
119862
1199050= 119862
0
(43)
Here 119905 ge 119905
0
119882119905
is a Brownian motion and 119862
119905
gt 0 this isthe cost process 119862
119905
is called the geometric Brownian motion(GBM) which is the solution of the following linear Ito-Doobtype stochastic differential equation
119889119862
119905
= 120583119862
119905
119889119905 + 120590119862
119905
119889119882
119905
(44)
where 120583 and 120590 are some constants (120583 is called the drift and120590 is called the volatility) and 119882
119905
is a normalized BrownianmotionUsing the Ito-Doob formula applied to119891(119862
119905
) = ln119862
119905
we can solve (44) The solution of (44) is given by the exactdiscrete-time equation for 119862
119905
119862
119905
= 119862
119905minus1
sdot 119890
(120583minus120590
22)Δ119905+119873(01)120590
radicΔ119905
(45)
where 119873(0 1) is the normally distributed random variableand Δ119905 = 1 (year) Equation (45) describes an operatingcost scenario with spot costs 119862
119905
By simulating 119862
119905
we obtainoperating costs for every year Simulated values of the costsare obtained by performing the following calculations
119862 = 119862
119904
119905
=
[
[
[
[
[
[
[
119862
1
1
119862
1
2
119862
1
119879
119862
2
1
119862
2
2
119862
2
119879
119862
119878
1
119862
119878
2
119862
119878
119879
]
]
]
]
]
]
]
119904 = 1 2 119878 119905 = 1 2 119879
(46)
where 119878 denotes the number of simulations and119879 the numberof project years In the space 119862
119904
119905
each row represents onesimulated path of costs over the project time while eachcolumn represents simulated values of costs for every yearThe main objective of using simulation is to determine thedistribution of the 119862 for every year of the project In this waywe obtain the sequence of probability density functions of 119862119862
119905
sim (pdf119905
120583
119905
120590
119905
) 119905 = 1 2 119879Applying the same concept of metal prices transfor-
mation we obtain the future sequence of operating costsexpressed by interval numbers 119862(119905
119894
) = [119862
119897
(119905
119894
) 119862
119906
(119905
119894
)] 119894 =
1 2 119879 where 119879 is the total project time
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
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 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
lower possibilistic and upper possibilistic mean values thatis
119872(
119860) =
119872
lowast
(
119860) + 119872
lowast
(
119860)
2
(14)
According to the above way of transformation (see (13)) weobtain an interval number having both a lower bound 119886
119897 andupper bound 119886
119906 119860 = [119886
119897
119886
119906
] where 119886
119897
= 119886
2
minus 1205723 and 119886
119906
=
119886
2
+ 1205733
From interval arithmetic the following operations ofinterval numbers are defined as follows
Definition 4 For forall119896 gt 0 and a given interval number 119860 =
[119886
119897
119886
119906
] 119886
119897
119886
119906
isin 119877
119896 times [119886
119897
119886
119906
] = [119896 times 119886
119897
119896times119886
119906
] (15)
and for forall119896 lt 0
minus119896 times [119886
119897
119886
119906
] = [minus119896 times 119886
119906
minus119896 times 119886
119897
] (16)
Definition 5 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
]
[119886
119897
119886
119906
] + [119887
119897
119887
119906
] = [119886
119897
+ 119887
119897
119886
119906
+ 119887
119906
]
[119886
119897
119886
119906
] minus [119887
119897
119887
119906
] = [119886
119897
minus 119887
119906
119886
119906
minus 119887
119897
]
(17)
Definition 6 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the multiplication is defined as follows
[119886
119897
119886
119906
] times [119887
119897
119887
119906
] = [min (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max (119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
(18)
Definition 7 For any two interval numbers [119886
119897
119886
119906
] and[119887
119897
119887
119906
] the division is defined as follows
[119886
119897
119886
119906
] divide [119887
119897
119887
119906
] = [min(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)
max(
119886
119897
119887
119897
119886
119897
119887
119906
119886
119906
119887
119897
119886
119906
119887
119906
)]
0 notin [119887
119897
119887
119906
]
(19)
22 Interval-Valued Differential Equations In this sectionwe consider an interval-valued differential equation of thefollowing form
119910 = 119891 (119905 119910 (119905)) 119910 (119905
0
) = 119910
0
(20)
where 119891 [119886 119887] times 119864 rarr 119864 with 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119909
119910(119905))] for 119910(119905) isin 119864 119910(119905) = [119910
119897
(119905) 119910
119906
(119905)] 119910
0
= [119910
119897
0
119910
119906
0
] Notethat we consider only Hukuhara differentiable solutions thatis there exists 120575 gt 0 such that there are no switching pointsin [119905
0
119905
0
+ 120575] [15 16]
Definition 8 Let 119891 [119886 119887] rarr 119864 be Hukuhara differentiableat 1199050
isin ]119886 119887[ We say that 119891 is (i)-Hukuhara differentiable at119905
0
if
119891 (119905
0
) = [
119891
119897
(119905
0
)
119891
119906
(119905
0
)] (21)
and that 119891 is (ii)-Hukuhara differentiable at 1199050
if
119891 (119905
0
) = [
119891
119906
(119905
0
)
119891
119897
(119905
0
)] (22)
The solution of the differential equation (20) depends onthe choice of the Hukuhara derivative ((i) or (ii)) To solvethe interval-valued differential equation it is necessary toreduce the interval-valued differential equation to a systemof ordinary differential equations [17ndash20]
Let [119910(119905)] = [119910
119897
(119905) 119910
119906
(119905)] If 119910(119905) is (i)-Hukuhara differ-entiable then119863
1
119910(119905) = [ 119910
119897
(119905) 119910
119906
(119905)] transforms (20) into thefollowing system of ordinary differential equations
119910
119897
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(23)
Also if 119910(119905) is (ii)-Hukuhara differentiable then 119863
2
119910(119905) =
[ 119910
119906
(119905) 119910
119897
(119905)] transforms (20) into the following system ofordinary differential equations
119910
119897
(119905) = 119891
119906
(119905 119910 (119905)) 119910
119897
(119905
0
) = 119910
119897
0
119910
119906
(119905) = 119891
119897
(119905 119910 (119905)) 119910
119906
(119905
0
) = 119910
119906
0
(24)
where 119891(119905 119910(119905)) = [119891
119897
(119905 119910(119905)) 119891
119906
(119905 119910(119905))]
3 Model of the Evaluation
31 The Concept of Evaluation Economic evaluation of amine project requires estimation of the revenues and coststhroughout the defined lifetime of the mine Such evaluationcan be treated as strategic decision making under multiplesources of uncertaintiesTherefore tomake the best decisionbased on the information available it is necessary to developan adequatemodel incorporating the uncertainty of the inputparameters The model should be able to involve a commontime horizon taking the characteristics of the input variablesthat directly affect the value of the proposed project
The model is developed on the basis of full discountedcash flow analysis of an underground zinc mine project Theoperating discounted cash flows are usually estimated on anannual basis Net present value of investment is used as akey criterion in the process of mine project estimation Theexpected net present value of the project is a function of thevariables as
119864 (NPV | 119876 119875 119866119872119862 119868 119903 119905) ge 0 (25)
where 119876 denotes the production rate (capacity) 119875 denotesthe zinc metal price 119866 is the grade 119872 is mill recovery 119862 isoperating costs 119868 is capital investment 119903 is discount rate and119905 is the lifetime of the project that is the period in which thecash flow is generated
Journal of Applied Mathematics 5
In this paper we treat in detail only the variability ofmetalprices and operating costs without intending to decrease thesignificance of the remaining parameters These parametersare taken into account on the basis of expert knowledge(estimation)
32 Forecasting the Revenue of the Mine Most mining com-panies realize their revenues by sellingmetal concentrates as afinal product Estimatingmineral project revenue is indeed adifficult and risky activity Annual mine revenue is calculatedbymultiplying the number of units produced and sold duringthe year by the sales price per unit
The value of the metal concentrate can be expressed asfollows
119881con = 119875 sdot (119898con minus 119898mr) (26)
where 119875 is metal price ($119905) 119898con is metal content ofconcentrate ()119898mr is metal recovery ratio () and
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8) sdot 100
119898congt 85 85
(27)
Annual mine revenue is calculated according to the followingequation
119877year = 119876year sdot 119881con sdot
119866 sdot 119872
119898con (28)
where 119876year is annual ore production (119905year) 119866 is grade ofthe ore mined ()119872 is mill recovery ()
Annual ore production is derived from themining projectschedule and is defined as crisp value The concept of grade(119866) is defined as the ratio of useful mass of metal tothe total mass of ore and its critical value fluctuates overdeposit space and can be estimated by experts and defined asinterval number 119866 = [119866
119897
119866
119906
] Mill recovery (119872) is relatedto the flotation as the most widely used method for theconcentration of fine grained minerals It can also be definedas interval number 119872 = [119872
119897
119872
119906
] Metal content representsthe quality of concentrate and we also apply the concept of aninterval number to define it 119898con = [119898
119897
con 119898119906
con]The major external source of risk affecting mine revenue
is related to the uncertainty about market behaviour of metalprices Forecasting the precise future state of themetal price isa very difficult task formine planners To predict futuremetalprices we apply the concept based on the transformation ofhistorical metal prices into adequate fuzzy-interval numbersand grey system theory
The forecasting model of metal prices is composed of thefollowing steps
Step 1 Create the set PDF = pdf119895
119895 = 1 2 119873 wherepdf119895
is the probability density function of metal prices forevery historical year The minimum number of elements ofthe set is four 119895min = 1 2 3 4
Step 2 Transform the set PDF into the set TFN = (119886
119895
119887
119895
119888
119895
)where (119886
119895
119887
119895
119888
119895
) is an adequate fuzzy triangular number
Step 3 Transform the set TFN into the set INT = [119886
119897
119895
119886
119906
119895
]where [119886
119897
119895
119886
119906
119895
] is an adequate interval number
Step 4 Using grey system prediction theory create a greydifferential equation of type 119866119872(1 1) that is the first-ordervariable grey derivative
Step 5 Testing of 119866119872(1 1) by residual error testing and theposterior error detection method
321 Analysis of Historical Metal Prices For every historicalyear it is necessary to define a probability density functionwith the following characteristics shape by histogram meanvalue (120583
119895
) and standard deviation (120590
119895
) In this way we obtainthe sequence of probability density functions of 119875 119875
119895
sim
(pdf119895
120583
119895
120590
119895
) 119895 = 1 2 119873 where 119873 is the total numberof historical years
322 Fuzzification of Metal Prices The sequence of obtainedpdf119895
of 119875119895
can be transformed into a sequence of triangularfuzzy numbers of 119875
119895
119875119895
sim TFN119895
119895 = 1 2 119873 that is119875
1
sim pdf1
rarr 119875
1
sim TFN1
1198752
sim pdf2
rarr 119875
2
sim TFN2
119875
119873
sim pdf119873
rarr 119875
119873
sim TFN119873
The method of transformationis based on the following facts the support of themembershipfunction and the pdf are the same and the point with thehighest probability (likelihood) has the highest possibilityFor more details see Swishchuk et al [21] The uncertaintyin the 119875 parameter is modelled by a triangular fuzzy numberwith the membership function which has the support of 120583
119895
minus
2120590
119895
lt 119875
119895
lt 120583
119895
+ 2120590
119895
119895 = 1 2 119873 set up for around 95confidence interval of distribution function If we take intoconsideration that the triangular fuzzy number is defined asa triplet (119886
1119895
119886
2119895
119886
3119895
) then 119886
1119895
and 119886
3119895
are the lower boundand upper bound obtained from the lower and upper boundof 5 of the distribution and the most promising value 119886
2119895
isequal to the mean value of the distribution For more detailssee Do et al [22]
323 Metal Prices as Interval Numbers The sequence ofobtainedTFN
119895
of119875119895
is transformed into a sequence of intervalnumbers of 119875
119895
119875
119895
sim INT119895
119895 = 1 2 119873 that is 1198751
sim
TFN1
rarr 119875
1
sim INT1
119875
2
sim TFN2
rarr 119875
2
sim INT2
119875
119873
sim TFN119873
rarr 119875
119873
sim INT119873
The method of transformationis described in Section 21 (basic concepts of fuzzy set theory)According to this method of transformation we obtain set
119875
119895
sim INT119895
119875
119895
= [119875
119897
119895
119875
119906
119895
] = [119886
2119895
minus
120572
119895
3
119886
2119895
+
120573
119895
3
]
119895 = 1 2 119873
(29)
324 Forecasting Model of Metal Prices The grey model is apowerful tool for forecasting the behaviour of the system in
6 Journal of Applied Mathematics
the future It has been successfully applied to various fieldssince it was proposed by Deng [23ndash31] In this paper we use aone-variable first-order differential grey equation 119866119872(1 1)The essence of 119866119872(1 1) is to accumulate the original data(historical metal prices) in order to obtain regular data Bysetting up the grey differential equation we obtain the fittedcurve in order to predict the future states of the system
Definition 9 Assume that
119875
(0)
(119905
119895
) = [119875
(0)119897
(119905
119895
) 119875
(0)119906
(119905
119895
)]
= [119875
(0)119897
(119905
1
) 119875
(0)119906
(119905
1
)]
[119875
(0)119897
(119905
2
) 119875
(0)119906
(119905
2
)]
[119875
(0)119897
(119905
119873
) 119875
(0)119906
(119905
119873
)]
(30)
is the original series of interval metal prices obtained bytransformation Sampling interval is Δ119905
119895
= 119905
119895
minus 119905
119895minus1
= 1 year
Definition 10 Let 119875(1)(119905119895
) = [119875
(1)119897
(119905
119895
) 119875
(1)119906
(119905
119895
)] be a newsequence generated by the accumulated generating operation(AGO) where
119875
(1)119897
(119905
119895
) =
119873
sum
119895=1
119875
(0)119897
(119905
119895
)
119875
(1)119906
(119905
119895
) =
119873
sum
119895=1
119875
(0)119906
(119905
119895
)
(31)
In the process of forecasting metal prices 119875
(1)119897
(119905
119895
) and119875
(1)119906
(119905
119895
) are the solutions of the following grey differentialequation
119889 [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
119889119905
+ [119902
119897
119902
119906
] sdot [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
= [119908
119897
119908
119906
]
(32)
Obviously (32) is an interval-valued differential equation (see(20))
To get the values of parameters 119902 = [119902
119897
119902
119906
] and 119908 =
[119908
119897
119908
119906
] the least square method is used as follows
otimes[
119902
119908
] = otimes ([119861
119879
119861]
minus1
119861
119879
119884
(0)
) (33)
where
otimes119861 =
[
[
[
[
[
[
[
[
[
[
minus
1
2
(119875
(0)
(1) + 119875
(0)
(2)) 1
minus
1
2
(119875
(0)
(2) + 119875
(0)
(3))
1
minus
1
2
(119875
(0)
(119873 minus 1) + 119875
(0)
(119873)) 1
]
]
]
]
]
]
]
]
]
]
otimes119884
(0)
= [119875
(0)
(2) 119875
(0)
(3) sdot sdot sdot 119875
(0)
(119873)]
119879
(34)
Note that the sign otimes indicates the interval number
If
119875
(1)
(119905) is considered as (i)-Hukuhara differentiablethen the following system of ordinary differential equationsis as follows
119875
(1)119897
(119905) = 119908
119897
minus 119902
119906
sdot 119875
(1)119906
(119905) 119875
119897
(119905
0
) = 119875
(0)119897
(1)
119875
(1)119906
(119905) = 119908
119906
minus 119902
119897
sdot 119875
(1)119897
(119905) 119875
119906
(119905
0
) = 119875
(0)119906
(1)
(35)
The solution of this system (forecasted equation) is as follows
119875
(1)119897
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
1
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
2
+
119908
119906
119902
119897
119895 = 1 2 119873
119875
(1)119906
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
3
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
4
+
119908
119897
119902
119906
119895 = 1 2 119873
(36)
where
119885
1
= 119875
(0)119897
(1) minus
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
+
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
2
= 119875
(0)119897
(1) +
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
minus
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
3
= 119875
(0)119906
(1) minus
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
+
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
119885
4
= 119875
(0)119906
(1) +
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
minus
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
(37)
To obtain the forecasted value of the primitive (original)metal price data at time (119905
119895
+ 1) the inverse accumulatedgenerating operation (IAGO) is used as follows
otimes
119875
(0)
(119905
1
) = otimes119875
(0)
(119905
1
) otimes
119875
(0)
(119905
119895
+ 1)
= otimes
119875
(1)
(119905
119895
+ 1)
minus otimes
119875
(1)
(119905
119895
) 119895 = 1 2 119873
(38)
325 The Model Accuracy
Definition 11 For a given interval grey number otimes([119886119897 119886119906]) itis common to take a whitening value
otimes([119886
119897
119886
119906
]) = 120596 sdot119886
119897
+(1minus
120596) sdot 119886
119906 for forall120596 isin [0 1] Furthermore if 120596 = 05 it is calledequal weight average whitenization [23]
Journal of Applied Mathematics 7
Residual error testing is composed of the calculationof relative error and absolute error between
otimes119875
(0)
(119905
119895
) andotimes
119875
(0)
(119905
119895
) based on the following formulas
Δ120576 (119905
119895
) =otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
Δ120576 (119905
119895
)
1003816
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
(39)
The posterior error detection method means calculation ofthe standard deviation of original metal price series (119878
1
) andstandard deviation of absolute error (119878
2
)
119878
1
=
radic
sum
119873
119895=1
(otimes119875
(0)
(119905
119895
) minusotimes119875
(0)
(119905
119895
))
2
119873 minus 1
119878
2
=
radic
sum
119873
119895=1
(Δ120576 (119905
119895
) minus Δ120576 (119905
119895
))
2
119873 minus 1
(40)
The variance ratio is equal to
119862vr =119878
2
119878
1
(41)
The standard of judgment is represented as follows [24]
119862vr =
lt035 Excellencelt060 Passlt065 Reluctant passge065 No pass
(42)
33 Volatility of Costs Capital development in an under-ground mine consists of shafts ramps raises and lateraltransport drifts required to access ore deposits with expectedutility greater than one yearThis is the development requiredto start up the ore production and to haul the ore to thesurface Experience with investments in capital developmentmight show that such expenditures can run considerablyhigher than the estimates but it is quite unlikely that actualcosts will be lower than estimatedThus the interval numbermight represent capital investment for the project 119868 = [119868
119897
119868
119906
]Operating costs are incurred directly in the production
process These costs include the ore and waste developmentof individual stopes the actual stoping activities the mineservices providing logistical support to the miners and themilling and processing of the ore at the plant These costsare generally more difficult to estimate than capital costsfor most mining ventures If we take into consideration thatproduction will be carried out for many years then it isvery important to predict the future states of operating costsAlthough there is some intention to create a correlationbetween metal price and operating cost it is very hard todefine it since price and cost vary continuously and are differ-ent over time At the project level there will not be a perfectcorrelation between price and cost because of adjustments tovariables such as labour energy explosives and fuel as wellas other material expenditures that are supplied by industries
that are not directly linked to metal price fluctuations Inorder to protect themselves suppliers are offering short-termcontracts to mines that are the opposite of traditional long-term contracts Some components of the operating cost suchas inputs used for mineral processing are usually purchasedatmarket prices that fluctuatemonthly annually or even overshorter periods
The uncertainties related to the future states of operatingcosts are modelled with a special stochastic process geo-metric Brownian motion Certain stochastic processes arefunctions of a Brownianmotion process and these havemanyapplications in finance engineering and the sciences Somespecial processes are solutions of Ito-Doob type stochasticdifferential equations (Ladde and Sambandham [32])
In this model we apply a continuous time process usingthe Ito-Doob type stochastic differential equation to describethe movement of operating costs A general stochastic differ-ential equation takes the following form
119889119862
119905
= 120583 (119862
119905
119905) 119889119905 + 120590 (119862
119905
119905) 119889119882
119905
119862
1199050= 119862
0
(43)
Here 119905 ge 119905
0
119882119905
is a Brownian motion and 119862
119905
gt 0 this isthe cost process 119862
119905
is called the geometric Brownian motion(GBM) which is the solution of the following linear Ito-Doobtype stochastic differential equation
119889119862
119905
= 120583119862
119905
119889119905 + 120590119862
119905
119889119882
119905
(44)
where 120583 and 120590 are some constants (120583 is called the drift and120590 is called the volatility) and 119882
119905
is a normalized BrownianmotionUsing the Ito-Doob formula applied to119891(119862
119905
) = ln119862
119905
we can solve (44) The solution of (44) is given by the exactdiscrete-time equation for 119862
119905
119862
119905
= 119862
119905minus1
sdot 119890
(120583minus120590
22)Δ119905+119873(01)120590
radicΔ119905
(45)
where 119873(0 1) is the normally distributed random variableand Δ119905 = 1 (year) Equation (45) describes an operatingcost scenario with spot costs 119862
119905
By simulating 119862
119905
we obtainoperating costs for every year Simulated values of the costsare obtained by performing the following calculations
119862 = 119862
119904
119905
=
[
[
[
[
[
[
[
119862
1
1
119862
1
2
119862
1
119879
119862
2
1
119862
2
2
119862
2
119879
119862
119878
1
119862
119878
2
119862
119878
119879
]
]
]
]
]
]
]
119904 = 1 2 119878 119905 = 1 2 119879
(46)
where 119878 denotes the number of simulations and119879 the numberof project years In the space 119862
119904
119905
each row represents onesimulated path of costs over the project time while eachcolumn represents simulated values of costs for every yearThe main objective of using simulation is to determine thedistribution of the 119862 for every year of the project In this waywe obtain the sequence of probability density functions of 119862119862
119905
sim (pdf119905
120583
119905
120590
119905
) 119905 = 1 2 119879Applying the same concept of metal prices transfor-
mation we obtain the future sequence of operating costsexpressed by interval numbers 119862(119905
119894
) = [119862
119897
(119905
119894
) 119862
119906
(119905
119894
)] 119894 =
1 2 119879 where 119879 is the total project time
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
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 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
In this paper we treat in detail only the variability ofmetalprices and operating costs without intending to decrease thesignificance of the remaining parameters These parametersare taken into account on the basis of expert knowledge(estimation)
32 Forecasting the Revenue of the Mine Most mining com-panies realize their revenues by sellingmetal concentrates as afinal product Estimatingmineral project revenue is indeed adifficult and risky activity Annual mine revenue is calculatedbymultiplying the number of units produced and sold duringthe year by the sales price per unit
The value of the metal concentrate can be expressed asfollows
119881con = 119875 sdot (119898con minus 119898mr) (26)
where 119875 is metal price ($119905) 119898con is metal content ofconcentrate ()119898mr is metal recovery ratio () and
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8) sdot 100
119898congt 85 85
(27)
Annual mine revenue is calculated according to the followingequation
119877year = 119876year sdot 119881con sdot
119866 sdot 119872
119898con (28)
where 119876year is annual ore production (119905year) 119866 is grade ofthe ore mined ()119872 is mill recovery ()
Annual ore production is derived from themining projectschedule and is defined as crisp value The concept of grade(119866) is defined as the ratio of useful mass of metal tothe total mass of ore and its critical value fluctuates overdeposit space and can be estimated by experts and defined asinterval number 119866 = [119866
119897
119866
119906
] Mill recovery (119872) is relatedto the flotation as the most widely used method for theconcentration of fine grained minerals It can also be definedas interval number 119872 = [119872
119897
119872
119906
] Metal content representsthe quality of concentrate and we also apply the concept of aninterval number to define it 119898con = [119898
119897
con 119898119906
con]The major external source of risk affecting mine revenue
is related to the uncertainty about market behaviour of metalprices Forecasting the precise future state of themetal price isa very difficult task formine planners To predict futuremetalprices we apply the concept based on the transformation ofhistorical metal prices into adequate fuzzy-interval numbersand grey system theory
The forecasting model of metal prices is composed of thefollowing steps
Step 1 Create the set PDF = pdf119895
119895 = 1 2 119873 wherepdf119895
is the probability density function of metal prices forevery historical year The minimum number of elements ofthe set is four 119895min = 1 2 3 4
Step 2 Transform the set PDF into the set TFN = (119886
119895
119887
119895
119888
119895
)where (119886
119895
119887
119895
119888
119895
) is an adequate fuzzy triangular number
Step 3 Transform the set TFN into the set INT = [119886
119897
119895
119886
119906
119895
]where [119886
119897
119895
119886
119906
119895
] is an adequate interval number
Step 4 Using grey system prediction theory create a greydifferential equation of type 119866119872(1 1) that is the first-ordervariable grey derivative
Step 5 Testing of 119866119872(1 1) by residual error testing and theposterior error detection method
321 Analysis of Historical Metal Prices For every historicalyear it is necessary to define a probability density functionwith the following characteristics shape by histogram meanvalue (120583
119895
) and standard deviation (120590
119895
) In this way we obtainthe sequence of probability density functions of 119875 119875
119895
sim
(pdf119895
120583
119895
120590
119895
) 119895 = 1 2 119873 where 119873 is the total numberof historical years
322 Fuzzification of Metal Prices The sequence of obtainedpdf119895
of 119875119895
can be transformed into a sequence of triangularfuzzy numbers of 119875
119895
119875119895
sim TFN119895
119895 = 1 2 119873 that is119875
1
sim pdf1
rarr 119875
1
sim TFN1
1198752
sim pdf2
rarr 119875
2
sim TFN2
119875
119873
sim pdf119873
rarr 119875
119873
sim TFN119873
The method of transformationis based on the following facts the support of themembershipfunction and the pdf are the same and the point with thehighest probability (likelihood) has the highest possibilityFor more details see Swishchuk et al [21] The uncertaintyin the 119875 parameter is modelled by a triangular fuzzy numberwith the membership function which has the support of 120583
119895
minus
2120590
119895
lt 119875
119895
lt 120583
119895
+ 2120590
119895
119895 = 1 2 119873 set up for around 95confidence interval of distribution function If we take intoconsideration that the triangular fuzzy number is defined asa triplet (119886
1119895
119886
2119895
119886
3119895
) then 119886
1119895
and 119886
3119895
are the lower boundand upper bound obtained from the lower and upper boundof 5 of the distribution and the most promising value 119886
2119895
isequal to the mean value of the distribution For more detailssee Do et al [22]
323 Metal Prices as Interval Numbers The sequence ofobtainedTFN
119895
of119875119895
is transformed into a sequence of intervalnumbers of 119875
119895
119875
119895
sim INT119895
119895 = 1 2 119873 that is 1198751
sim
TFN1
rarr 119875
1
sim INT1
119875
2
sim TFN2
rarr 119875
2
sim INT2
119875
119873
sim TFN119873
rarr 119875
119873
sim INT119873
The method of transformationis described in Section 21 (basic concepts of fuzzy set theory)According to this method of transformation we obtain set
119875
119895
sim INT119895
119875
119895
= [119875
119897
119895
119875
119906
119895
] = [119886
2119895
minus
120572
119895
3
119886
2119895
+
120573
119895
3
]
119895 = 1 2 119873
(29)
324 Forecasting Model of Metal Prices The grey model is apowerful tool for forecasting the behaviour of the system in
6 Journal of Applied Mathematics
the future It has been successfully applied to various fieldssince it was proposed by Deng [23ndash31] In this paper we use aone-variable first-order differential grey equation 119866119872(1 1)The essence of 119866119872(1 1) is to accumulate the original data(historical metal prices) in order to obtain regular data Bysetting up the grey differential equation we obtain the fittedcurve in order to predict the future states of the system
Definition 9 Assume that
119875
(0)
(119905
119895
) = [119875
(0)119897
(119905
119895
) 119875
(0)119906
(119905
119895
)]
= [119875
(0)119897
(119905
1
) 119875
(0)119906
(119905
1
)]
[119875
(0)119897
(119905
2
) 119875
(0)119906
(119905
2
)]
[119875
(0)119897
(119905
119873
) 119875
(0)119906
(119905
119873
)]
(30)
is the original series of interval metal prices obtained bytransformation Sampling interval is Δ119905
119895
= 119905
119895
minus 119905
119895minus1
= 1 year
Definition 10 Let 119875(1)(119905119895
) = [119875
(1)119897
(119905
119895
) 119875
(1)119906
(119905
119895
)] be a newsequence generated by the accumulated generating operation(AGO) where
119875
(1)119897
(119905
119895
) =
119873
sum
119895=1
119875
(0)119897
(119905
119895
)
119875
(1)119906
(119905
119895
) =
119873
sum
119895=1
119875
(0)119906
(119905
119895
)
(31)
In the process of forecasting metal prices 119875
(1)119897
(119905
119895
) and119875
(1)119906
(119905
119895
) are the solutions of the following grey differentialequation
119889 [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
119889119905
+ [119902
119897
119902
119906
] sdot [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
= [119908
119897
119908
119906
]
(32)
Obviously (32) is an interval-valued differential equation (see(20))
To get the values of parameters 119902 = [119902
119897
119902
119906
] and 119908 =
[119908
119897
119908
119906
] the least square method is used as follows
otimes[
119902
119908
] = otimes ([119861
119879
119861]
minus1
119861
119879
119884
(0)
) (33)
where
otimes119861 =
[
[
[
[
[
[
[
[
[
[
minus
1
2
(119875
(0)
(1) + 119875
(0)
(2)) 1
minus
1
2
(119875
(0)
(2) + 119875
(0)
(3))
1
minus
1
2
(119875
(0)
(119873 minus 1) + 119875
(0)
(119873)) 1
]
]
]
]
]
]
]
]
]
]
otimes119884
(0)
= [119875
(0)
(2) 119875
(0)
(3) sdot sdot sdot 119875
(0)
(119873)]
119879
(34)
Note that the sign otimes indicates the interval number
If
119875
(1)
(119905) is considered as (i)-Hukuhara differentiablethen the following system of ordinary differential equationsis as follows
119875
(1)119897
(119905) = 119908
119897
minus 119902
119906
sdot 119875
(1)119906
(119905) 119875
119897
(119905
0
) = 119875
(0)119897
(1)
119875
(1)119906
(119905) = 119908
119906
minus 119902
119897
sdot 119875
(1)119897
(119905) 119875
119906
(119905
0
) = 119875
(0)119906
(1)
(35)
The solution of this system (forecasted equation) is as follows
119875
(1)119897
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
1
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
2
+
119908
119906
119902
119897
119895 = 1 2 119873
119875
(1)119906
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
3
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
4
+
119908
119897
119902
119906
119895 = 1 2 119873
(36)
where
119885
1
= 119875
(0)119897
(1) minus
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
+
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
2
= 119875
(0)119897
(1) +
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
minus
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
3
= 119875
(0)119906
(1) minus
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
+
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
119885
4
= 119875
(0)119906
(1) +
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
minus
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
(37)
To obtain the forecasted value of the primitive (original)metal price data at time (119905
119895
+ 1) the inverse accumulatedgenerating operation (IAGO) is used as follows
otimes
119875
(0)
(119905
1
) = otimes119875
(0)
(119905
1
) otimes
119875
(0)
(119905
119895
+ 1)
= otimes
119875
(1)
(119905
119895
+ 1)
minus otimes
119875
(1)
(119905
119895
) 119895 = 1 2 119873
(38)
325 The Model Accuracy
Definition 11 For a given interval grey number otimes([119886119897 119886119906]) itis common to take a whitening value
otimes([119886
119897
119886
119906
]) = 120596 sdot119886
119897
+(1minus
120596) sdot 119886
119906 for forall120596 isin [0 1] Furthermore if 120596 = 05 it is calledequal weight average whitenization [23]
Journal of Applied Mathematics 7
Residual error testing is composed of the calculationof relative error and absolute error between
otimes119875
(0)
(119905
119895
) andotimes
119875
(0)
(119905
119895
) based on the following formulas
Δ120576 (119905
119895
) =otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
Δ120576 (119905
119895
)
1003816
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
(39)
The posterior error detection method means calculation ofthe standard deviation of original metal price series (119878
1
) andstandard deviation of absolute error (119878
2
)
119878
1
=
radic
sum
119873
119895=1
(otimes119875
(0)
(119905
119895
) minusotimes119875
(0)
(119905
119895
))
2
119873 minus 1
119878
2
=
radic
sum
119873
119895=1
(Δ120576 (119905
119895
) minus Δ120576 (119905
119895
))
2
119873 minus 1
(40)
The variance ratio is equal to
119862vr =119878
2
119878
1
(41)
The standard of judgment is represented as follows [24]
119862vr =
lt035 Excellencelt060 Passlt065 Reluctant passge065 No pass
(42)
33 Volatility of Costs Capital development in an under-ground mine consists of shafts ramps raises and lateraltransport drifts required to access ore deposits with expectedutility greater than one yearThis is the development requiredto start up the ore production and to haul the ore to thesurface Experience with investments in capital developmentmight show that such expenditures can run considerablyhigher than the estimates but it is quite unlikely that actualcosts will be lower than estimatedThus the interval numbermight represent capital investment for the project 119868 = [119868
119897
119868
119906
]Operating costs are incurred directly in the production
process These costs include the ore and waste developmentof individual stopes the actual stoping activities the mineservices providing logistical support to the miners and themilling and processing of the ore at the plant These costsare generally more difficult to estimate than capital costsfor most mining ventures If we take into consideration thatproduction will be carried out for many years then it isvery important to predict the future states of operating costsAlthough there is some intention to create a correlationbetween metal price and operating cost it is very hard todefine it since price and cost vary continuously and are differ-ent over time At the project level there will not be a perfectcorrelation between price and cost because of adjustments tovariables such as labour energy explosives and fuel as wellas other material expenditures that are supplied by industries
that are not directly linked to metal price fluctuations Inorder to protect themselves suppliers are offering short-termcontracts to mines that are the opposite of traditional long-term contracts Some components of the operating cost suchas inputs used for mineral processing are usually purchasedatmarket prices that fluctuatemonthly annually or even overshorter periods
The uncertainties related to the future states of operatingcosts are modelled with a special stochastic process geo-metric Brownian motion Certain stochastic processes arefunctions of a Brownianmotion process and these havemanyapplications in finance engineering and the sciences Somespecial processes are solutions of Ito-Doob type stochasticdifferential equations (Ladde and Sambandham [32])
In this model we apply a continuous time process usingthe Ito-Doob type stochastic differential equation to describethe movement of operating costs A general stochastic differ-ential equation takes the following form
119889119862
119905
= 120583 (119862
119905
119905) 119889119905 + 120590 (119862
119905
119905) 119889119882
119905
119862
1199050= 119862
0
(43)
Here 119905 ge 119905
0
119882119905
is a Brownian motion and 119862
119905
gt 0 this isthe cost process 119862
119905
is called the geometric Brownian motion(GBM) which is the solution of the following linear Ito-Doobtype stochastic differential equation
119889119862
119905
= 120583119862
119905
119889119905 + 120590119862
119905
119889119882
119905
(44)
where 120583 and 120590 are some constants (120583 is called the drift and120590 is called the volatility) and 119882
119905
is a normalized BrownianmotionUsing the Ito-Doob formula applied to119891(119862
119905
) = ln119862
119905
we can solve (44) The solution of (44) is given by the exactdiscrete-time equation for 119862
119905
119862
119905
= 119862
119905minus1
sdot 119890
(120583minus120590
22)Δ119905+119873(01)120590
radicΔ119905
(45)
where 119873(0 1) is the normally distributed random variableand Δ119905 = 1 (year) Equation (45) describes an operatingcost scenario with spot costs 119862
119905
By simulating 119862
119905
we obtainoperating costs for every year Simulated values of the costsare obtained by performing the following calculations
119862 = 119862
119904
119905
=
[
[
[
[
[
[
[
119862
1
1
119862
1
2
119862
1
119879
119862
2
1
119862
2
2
119862
2
119879
119862
119878
1
119862
119878
2
119862
119878
119879
]
]
]
]
]
]
]
119904 = 1 2 119878 119905 = 1 2 119879
(46)
where 119878 denotes the number of simulations and119879 the numberof project years In the space 119862
119904
119905
each row represents onesimulated path of costs over the project time while eachcolumn represents simulated values of costs for every yearThe main objective of using simulation is to determine thedistribution of the 119862 for every year of the project In this waywe obtain the sequence of probability density functions of 119862119862
119905
sim (pdf119905
120583
119905
120590
119905
) 119905 = 1 2 119879Applying the same concept of metal prices transfor-
mation we obtain the future sequence of operating costsexpressed by interval numbers 119862(119905
119894
) = [119862
119897
(119905
119894
) 119862
119906
(119905
119894
)] 119894 =
1 2 119879 where 119879 is the total project time
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
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 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
the future It has been successfully applied to various fieldssince it was proposed by Deng [23ndash31] In this paper we use aone-variable first-order differential grey equation 119866119872(1 1)The essence of 119866119872(1 1) is to accumulate the original data(historical metal prices) in order to obtain regular data Bysetting up the grey differential equation we obtain the fittedcurve in order to predict the future states of the system
Definition 9 Assume that
119875
(0)
(119905
119895
) = [119875
(0)119897
(119905
119895
) 119875
(0)119906
(119905
119895
)]
= [119875
(0)119897
(119905
1
) 119875
(0)119906
(119905
1
)]
[119875
(0)119897
(119905
2
) 119875
(0)119906
(119905
2
)]
[119875
(0)119897
(119905
119873
) 119875
(0)119906
(119905
119873
)]
(30)
is the original series of interval metal prices obtained bytransformation Sampling interval is Δ119905
119895
= 119905
119895
minus 119905
119895minus1
= 1 year
Definition 10 Let 119875(1)(119905119895
) = [119875
(1)119897
(119905
119895
) 119875
(1)119906
(119905
119895
)] be a newsequence generated by the accumulated generating operation(AGO) where
119875
(1)119897
(119905
119895
) =
119873
sum
119895=1
119875
(0)119897
(119905
119895
)
119875
(1)119906
(119905
119895
) =
119873
sum
119895=1
119875
(0)119906
(119905
119895
)
(31)
In the process of forecasting metal prices 119875
(1)119897
(119905
119895
) and119875
(1)119906
(119905
119895
) are the solutions of the following grey differentialequation
119889 [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
119889119905
+ [119902
119897
119902
119906
] sdot [119875
(1)119897
(119905) 119875
(1)119906
(119905)]
= [119908
119897
119908
119906
]
(32)
Obviously (32) is an interval-valued differential equation (see(20))
To get the values of parameters 119902 = [119902
119897
119902
119906
] and 119908 =
[119908
119897
119908
119906
] the least square method is used as follows
otimes[
119902
119908
] = otimes ([119861
119879
119861]
minus1
119861
119879
119884
(0)
) (33)
where
otimes119861 =
[
[
[
[
[
[
[
[
[
[
minus
1
2
(119875
(0)
(1) + 119875
(0)
(2)) 1
minus
1
2
(119875
(0)
(2) + 119875
(0)
(3))
1
minus
1
2
(119875
(0)
(119873 minus 1) + 119875
(0)
(119873)) 1
]
]
]
]
]
]
]
]
]
]
otimes119884
(0)
= [119875
(0)
(2) 119875
(0)
(3) sdot sdot sdot 119875
(0)
(119873)]
119879
(34)
Note that the sign otimes indicates the interval number
If
119875
(1)
(119905) is considered as (i)-Hukuhara differentiablethen the following system of ordinary differential equationsis as follows
119875
(1)119897
(119905) = 119908
119897
minus 119902
119906
sdot 119875
(1)119906
(119905) 119875
119897
(119905
0
) = 119875
(0)119897
(1)
119875
(1)119906
(119905) = 119908
119906
minus 119902
119897
sdot 119875
(1)119897
(119905) 119875
119906
(119905
0
) = 119875
(0)119906
(1)
(35)
The solution of this system (forecasted equation) is as follows
119875
(1)119897
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
1
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
2
+
119908
119906
119902
119897
119895 = 1 2 119873
119875
(1)119906
(119905
119895
+ 1)
=
1
2
sdot 119890
radic119902
119897sdot119902
119906sdot119905
sdot 119885
3
+
1
2
sdot 119890
minusradic119902
119897sdot119902
119906sdot119905
sdot 119885
4
+
119908
119897
119902
119906
119895 = 1 2 119873
(36)
where
119885
1
= 119875
(0)119897
(1) minus
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
+
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
2
= 119875
(0)119897
(1) +
119902
119906
sdot119875
(0)119906
(1)
radic119902
119897
sdot 119902
119906
minus
119908
119897
radic119902
119897
sdot 119902
119906
minus
119908
119906
119902
119897
119885
3
= 119875
(0)119906
(1) minus
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
+
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
119885
4
= 119875
(0)119906
(1) +
119875
(0)119897
(1) sdotradic119902
119897
sdot 119902
119906
119902
119906
minus
119908
119906
radic119902
119897
sdot 119902
119906
minus
119908
119897
119902
119906
(37)
To obtain the forecasted value of the primitive (original)metal price data at time (119905
119895
+ 1) the inverse accumulatedgenerating operation (IAGO) is used as follows
otimes
119875
(0)
(119905
1
) = otimes119875
(0)
(119905
1
) otimes
119875
(0)
(119905
119895
+ 1)
= otimes
119875
(1)
(119905
119895
+ 1)
minus otimes
119875
(1)
(119905
119895
) 119895 = 1 2 119873
(38)
325 The Model Accuracy
Definition 11 For a given interval grey number otimes([119886119897 119886119906]) itis common to take a whitening value
otimes([119886
119897
119886
119906
]) = 120596 sdot119886
119897
+(1minus
120596) sdot 119886
119906 for forall120596 isin [0 1] Furthermore if 120596 = 05 it is calledequal weight average whitenization [23]
Journal of Applied Mathematics 7
Residual error testing is composed of the calculationof relative error and absolute error between
otimes119875
(0)
(119905
119895
) andotimes
119875
(0)
(119905
119895
) based on the following formulas
Δ120576 (119905
119895
) =otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
Δ120576 (119905
119895
)
1003816
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
(39)
The posterior error detection method means calculation ofthe standard deviation of original metal price series (119878
1
) andstandard deviation of absolute error (119878
2
)
119878
1
=
radic
sum
119873
119895=1
(otimes119875
(0)
(119905
119895
) minusotimes119875
(0)
(119905
119895
))
2
119873 minus 1
119878
2
=
radic
sum
119873
119895=1
(Δ120576 (119905
119895
) minus Δ120576 (119905
119895
))
2
119873 minus 1
(40)
The variance ratio is equal to
119862vr =119878
2
119878
1
(41)
The standard of judgment is represented as follows [24]
119862vr =
lt035 Excellencelt060 Passlt065 Reluctant passge065 No pass
(42)
33 Volatility of Costs Capital development in an under-ground mine consists of shafts ramps raises and lateraltransport drifts required to access ore deposits with expectedutility greater than one yearThis is the development requiredto start up the ore production and to haul the ore to thesurface Experience with investments in capital developmentmight show that such expenditures can run considerablyhigher than the estimates but it is quite unlikely that actualcosts will be lower than estimatedThus the interval numbermight represent capital investment for the project 119868 = [119868
119897
119868
119906
]Operating costs are incurred directly in the production
process These costs include the ore and waste developmentof individual stopes the actual stoping activities the mineservices providing logistical support to the miners and themilling and processing of the ore at the plant These costsare generally more difficult to estimate than capital costsfor most mining ventures If we take into consideration thatproduction will be carried out for many years then it isvery important to predict the future states of operating costsAlthough there is some intention to create a correlationbetween metal price and operating cost it is very hard todefine it since price and cost vary continuously and are differ-ent over time At the project level there will not be a perfectcorrelation between price and cost because of adjustments tovariables such as labour energy explosives and fuel as wellas other material expenditures that are supplied by industries
that are not directly linked to metal price fluctuations Inorder to protect themselves suppliers are offering short-termcontracts to mines that are the opposite of traditional long-term contracts Some components of the operating cost suchas inputs used for mineral processing are usually purchasedatmarket prices that fluctuatemonthly annually or even overshorter periods
The uncertainties related to the future states of operatingcosts are modelled with a special stochastic process geo-metric Brownian motion Certain stochastic processes arefunctions of a Brownianmotion process and these havemanyapplications in finance engineering and the sciences Somespecial processes are solutions of Ito-Doob type stochasticdifferential equations (Ladde and Sambandham [32])
In this model we apply a continuous time process usingthe Ito-Doob type stochastic differential equation to describethe movement of operating costs A general stochastic differ-ential equation takes the following form
119889119862
119905
= 120583 (119862
119905
119905) 119889119905 + 120590 (119862
119905
119905) 119889119882
119905
119862
1199050= 119862
0
(43)
Here 119905 ge 119905
0
119882119905
is a Brownian motion and 119862
119905
gt 0 this isthe cost process 119862
119905
is called the geometric Brownian motion(GBM) which is the solution of the following linear Ito-Doobtype stochastic differential equation
119889119862
119905
= 120583119862
119905
119889119905 + 120590119862
119905
119889119882
119905
(44)
where 120583 and 120590 are some constants (120583 is called the drift and120590 is called the volatility) and 119882
119905
is a normalized BrownianmotionUsing the Ito-Doob formula applied to119891(119862
119905
) = ln119862
119905
we can solve (44) The solution of (44) is given by the exactdiscrete-time equation for 119862
119905
119862
119905
= 119862
119905minus1
sdot 119890
(120583minus120590
22)Δ119905+119873(01)120590
radicΔ119905
(45)
where 119873(0 1) is the normally distributed random variableand Δ119905 = 1 (year) Equation (45) describes an operatingcost scenario with spot costs 119862
119905
By simulating 119862
119905
we obtainoperating costs for every year Simulated values of the costsare obtained by performing the following calculations
119862 = 119862
119904
119905
=
[
[
[
[
[
[
[
119862
1
1
119862
1
2
119862
1
119879
119862
2
1
119862
2
2
119862
2
119879
119862
119878
1
119862
119878
2
119862
119878
119879
]
]
]
]
]
]
]
119904 = 1 2 119878 119905 = 1 2 119879
(46)
where 119878 denotes the number of simulations and119879 the numberof project years In the space 119862
119904
119905
each row represents onesimulated path of costs over the project time while eachcolumn represents simulated values of costs for every yearThe main objective of using simulation is to determine thedistribution of the 119862 for every year of the project In this waywe obtain the sequence of probability density functions of 119862119862
119905
sim (pdf119905
120583
119905
120590
119905
) 119905 = 1 2 119879Applying the same concept of metal prices transfor-
mation we obtain the future sequence of operating costsexpressed by interval numbers 119862(119905
119894
) = [119862
119897
(119905
119894
) 119862
119906
(119905
119894
)] 119894 =
1 2 119879 where 119879 is the total project time
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
Metal price ($t)
2009 2010 2011 2012 2013
January 1203 2415 2376 1989 2031
February 1118 2159 2473 2058 2129
March 1223 2277 2341 2036 1929
April 1388 2368 2371 2003 1856
May 1492 1970 2160 1928 1831
June 1555 1747 2234 1856 1839
July 1583 1847 2398 1848 1838
August 1818 2047 2199 1816 1896
September 1879 2151 2075 2010 1847
October 2071 2374 1871 1904 1885
November 2197 2283 1935 1912 1866
December 2374 2287 1911 2040 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
Residual error testing is composed of the calculationof relative error and absolute error between
otimes119875
(0)
(119905
119895
) andotimes
119875
(0)
(119905
119895
) based on the following formulas
Δ120576 (119905
119895
) =otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
Δ120576 (119905
119895
)
1003816
1003816
1003816
1003816
1003816
=
1003816
1003816
1003816
1003816
1003816
otimes119875
(0)
(119905
119895
) minusotimes
119875
(0)
(119905
119895
)
1003816
1003816
1003816
1003816
1003816
(39)
The posterior error detection method means calculation ofthe standard deviation of original metal price series (119878
1
) andstandard deviation of absolute error (119878
2
)
119878
1
=
radic
sum
119873
119895=1
(otimes119875
(0)
(119905
119895
) minusotimes119875
(0)
(119905
119895
))
2
119873 minus 1
119878
2
=
radic
sum
119873
119895=1
(Δ120576 (119905
119895
) minus Δ120576 (119905
119895
))
2
119873 minus 1
(40)
The variance ratio is equal to
119862vr =119878
2
119878
1
(41)
The standard of judgment is represented as follows [24]
119862vr =
lt035 Excellencelt060 Passlt065 Reluctant passge065 No pass
(42)
33 Volatility of Costs Capital development in an under-ground mine consists of shafts ramps raises and lateraltransport drifts required to access ore deposits with expectedutility greater than one yearThis is the development requiredto start up the ore production and to haul the ore to thesurface Experience with investments in capital developmentmight show that such expenditures can run considerablyhigher than the estimates but it is quite unlikely that actualcosts will be lower than estimatedThus the interval numbermight represent capital investment for the project 119868 = [119868
119897
119868
119906
]Operating costs are incurred directly in the production
process These costs include the ore and waste developmentof individual stopes the actual stoping activities the mineservices providing logistical support to the miners and themilling and processing of the ore at the plant These costsare generally more difficult to estimate than capital costsfor most mining ventures If we take into consideration thatproduction will be carried out for many years then it isvery important to predict the future states of operating costsAlthough there is some intention to create a correlationbetween metal price and operating cost it is very hard todefine it since price and cost vary continuously and are differ-ent over time At the project level there will not be a perfectcorrelation between price and cost because of adjustments tovariables such as labour energy explosives and fuel as wellas other material expenditures that are supplied by industries
that are not directly linked to metal price fluctuations Inorder to protect themselves suppliers are offering short-termcontracts to mines that are the opposite of traditional long-term contracts Some components of the operating cost suchas inputs used for mineral processing are usually purchasedatmarket prices that fluctuatemonthly annually or even overshorter periods
The uncertainties related to the future states of operatingcosts are modelled with a special stochastic process geo-metric Brownian motion Certain stochastic processes arefunctions of a Brownianmotion process and these havemanyapplications in finance engineering and the sciences Somespecial processes are solutions of Ito-Doob type stochasticdifferential equations (Ladde and Sambandham [32])
In this model we apply a continuous time process usingthe Ito-Doob type stochastic differential equation to describethe movement of operating costs A general stochastic differ-ential equation takes the following form
119889119862
119905
= 120583 (119862
119905
119905) 119889119905 + 120590 (119862
119905
119905) 119889119882
119905
119862
1199050= 119862
0
(43)
Here 119905 ge 119905
0
119882119905
is a Brownian motion and 119862
119905
gt 0 this isthe cost process 119862
119905
is called the geometric Brownian motion(GBM) which is the solution of the following linear Ito-Doobtype stochastic differential equation
119889119862
119905
= 120583119862
119905
119889119905 + 120590119862
119905
119889119882
119905
(44)
where 120583 and 120590 are some constants (120583 is called the drift and120590 is called the volatility) and 119882
119905
is a normalized BrownianmotionUsing the Ito-Doob formula applied to119891(119862
119905
) = ln119862
119905
we can solve (44) The solution of (44) is given by the exactdiscrete-time equation for 119862
119905
119862
119905
= 119862
119905minus1
sdot 119890
(120583minus120590
22)Δ119905+119873(01)120590
radicΔ119905
(45)
where 119873(0 1) is the normally distributed random variableand Δ119905 = 1 (year) Equation (45) describes an operatingcost scenario with spot costs 119862
119905
By simulating 119862
119905
we obtainoperating costs for every year Simulated values of the costsare obtained by performing the following calculations
119862 = 119862
119904
119905
=
[
[
[
[
[
[
[
119862
1
1
119862
1
2
119862
1
119879
119862
2
1
119862
2
2
119862
2
119879
119862
119878
1
119862
119878
2
119862
119878
119879
]
]
]
]
]
]
]
119904 = 1 2 119878 119905 = 1 2 119879
(46)
where 119878 denotes the number of simulations and119879 the numberof project years In the space 119862
119904
119905
each row represents onesimulated path of costs over the project time while eachcolumn represents simulated values of costs for every yearThe main objective of using simulation is to determine thedistribution of the 119862 for every year of the project In this waywe obtain the sequence of probability density functions of 119862119862
119905
sim (pdf119905
120583
119905
120590
119905
) 119905 = 1 2 119879Applying the same concept of metal prices transfor-
mation we obtain the future sequence of operating costsexpressed by interval numbers 119862(119905
119894
) = [119862
119897
(119905
119894
) 119862
119906
(119905
119894
)] 119894 =
1 2 119879 where 119879 is the total project time
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
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 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
34 Criterion of the Evaluation The net present value (NPV)of the mine project is an integral evaluation criterion thatrecognizes the time effect of money over the life-of-mine It iscalculated as a difference between the sum of discounted val-ues of estimated future cash flows and the initial investmentand can be defined as follows
otimesNPV
=
119879
sum
119905=1
((119876 sdot otimes119875
(119905119901+119905119894)sdot (119898con minus 119898mr) sdot (119866 sdot 119872119898con)
minusotimes119862
(119905119901+119905119894)) times ((1 + otimes119903)
119905119894)
minus1
) minus otimes119868
(47)
where 119876 is annual ore production (119905year) 119903 is discount rate119879 is the number of periods for the life of the investment 119868 isinitial (capital) investment and 119905
119901
is preproduction time timeneeded to prepare deposit to be mined (construction time)
Finally the last parameter that can be expressed by aninterval number is the discount rate Discounted cash flowmethods are widely used in capital budgeting howeverdetermining the discount rate as a crisp value can leadto erroneous results in most mine project applications Adiscount rate range can be established in a way which is eitherjust acceptable (maximum value) or reasonable (minimumvalue) 119903 = [119903
119897
119903
119906
]A positive NPV will lead to the acceptance of the project
and a negative NPV rejects it that is otimesNPV ge 0 Consider
otimes ([NPV119897NPV119906]) = 120596 sdotNPV119897 + (1 minus 120596) sdotNPV119906
120596 isin [0 1]
(48)
Weight whitenization of interval NPV is obtained by 120596NPV =
05
4 Numerical Example
Themanagement of a smallmining company is evaluating theopening of a new zinc deposit The recommendations fromthe prefeasibility study suggest the following
(i) the undergroundmine development system connect-ing the ore body to the surface is based on thecombination of ramp and horizontal drives Thissystem is used for the purpose of ore haulage by dumptrucks and conduct intake fresh air Contaminated airis conducted to the surface by horizontal drives anddeclines
(ii) they suggest purchasing new mining equipment
The completion of this project will cost the company about3500 000USDover three years At the beginning of the fourthyear when construction is completed the new mine willproduce 100000 119905year during 5 years of production
The input parameters required for the project evaluationare given in Table 1 Note that the situation is hypothetical andthe numbers used are to permit calculation
0010203040506070809
1
500 1000 1500 2000 2500
200920102011
20122013
Figure 2 Historical zinc metal prices expressed as triangular fuzzynumbers
The interval values of historical metal prices are calcu-lated by Steps 1 2 and 3 of themetal prices forecastingmodelThe values obtained are as represented inTable 2 and Figure 2
Based on data in Table 2 and (36) the fuzzy-intervalAGOGM(1 1)model of zinc metal prices is set up as follows
119875
(1)119897
(119905
119895
+ 1) = minus1490863 sdot 119890
00454sdot119905
minus 5932737 sdot 119890
minus00454sdot119905
+ 7562082
119875
(1)119906
(119905
119895
+ 1) = 1105573 sdot 119890
00454119905
minus 4399513 sdot 119890
minus00454sdot119905
+ 3487136
(49)
To test the precision of the model relative error and absoluteerror of the model are calculated and the results are repre-sented in Table 3
The standard deviation of the original metal price series(1198781
) and standard deviation of absolute error (1198782
) are 21677and 3793 respectively The variance ratio of the model is119862vr = 0175 These show that the obtained model has goodforecasting precision to predict the zinc metal prices
According to (28) annual mine revenues are representedin Table 4
Uncertainty related to the unit operating costs is quanti-fied according to (45) that is by geometric BrownianmotionFigure 3 represents seven possible paths (scenarios) of theunit operating costs over the project time
The results of the simulations and transformations arerepresented in Table 5
Annual operating costs are represented in Table 6The discounted cash flow of the project is represented in
Table 7According to (47) and data in Table 7 the net present
value of the project is as follows
otimesNPV = [minus8775 17784] minus [3000 4000]
= [minus11775 13784] millrsquos USD(50)
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
Metal price ($t)
2009 2010 2011 2012 2013
January 1203 2415 2376 1989 2031
February 1118 2159 2473 2058 2129
March 1223 2277 2341 2036 1929
April 1388 2368 2371 2003 1856
May 1492 1970 2160 1928 1831
June 1555 1747 2234 1856 1839
July 1583 1847 2398 1848 1838
August 1818 2047 2199 1816 1896
September 1879 2151 2075 2010 1847
October 2071 2374 1871 1904 1885
November 2197 2283 1935 1912 1866
December 2374 2287 1911 2040 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
Table 1 Input parameters
Mine ore production (tyear) 100 000Zinc grade ()
[42 50] = [0042 0050]Metal content of concentrate ()
119898con minus 119898mr =
(119898con minus 8) sdot 100
119898conle 85 119898con minus 8
(119898con minus 8 sdot 100)
119898congt 85 85
[47 52] = [047 052]
[39 42] = [039 042]
Mill recovery ()[75 80] = [075 080]
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 1975Initial (capital) investment ($)
[3000000 4000000]Operating costs geometric Brownian motionyearly time resolution ($t) equation (39)Number of simulations
Spot value 32 drift 0020cost volatility rate 010119878 = 500
Construction period (year) 3 2014 2015 2016
Mine life (year) 5 2017 2018 2021
Discount rate ()[7 10] = [007 010]
Table 2 Transformation pdf119895
rarr TFN119895
rarr INT119895
2009 2010 2011 2012 2013pdf119895
120583
119895
1658 2160 2195 1950 1910120590
119895
410 216 207 83 92TFN119895
119886
1119895
= 120583
119895
minus 2120590
119895
838 1728 1781 1784 1726119886
2119895
= 120583
119895
1658 2160 2195 1950 1910119886
3119895
= 120583
119895
+ 2120590
119895
2478 2592 2609 2116 2094120572
119895
= 120573
119895
820 432 414 166 184INT119895
119875
119897
119895
1385 2016 2057 1895 1849119875
119906
119895
1931 2304 2333 2005 1971
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
Table 3 Relative and absolute error of the model
Year Original values($t)
Simulatedvalues ($t)
Whitenization 120596 = 05 Relative error Absolute error Relative error ()Original Simulated
2009[1385 1931] [1385 1931] 1658 1658 0 0 0
2010[2016 2304] [1941 2466] 2160 2204 minus44 44 minus203
2011[2057 2333] [1792 2404] 2195 2098 97 97 +442
2012[1895 2005] [1647 2346] 1950 1997 minus47 47 minus241
2013[1849 1971] [1505 2293] 1910 1899 11 11 +057
Table 4 Mine revenues over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]
Table 5 Simulation of the unit operating costs
2013 2014 2015 2016 2017 2018 2019 2020 2021Simulation 1 32 3524 3647 4067 4607 5500 5414 5569 4859
Simulation 500 32 3117 3517 3791 3766 3627 3715 3434 3608pdf119894
120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635120590
119894
00 316 455 570 660 749 845 942 1000TFN119894
119886
1119894
= 120583
119894
minus 2120590
119894
32 2609 2402 2211 2085 1958 1826 1696 1635119886
2119894
= 120583
119894
32 3241 3312 3351 3405 3456 3516 3580 3635119886
3119894
= 120583
119894
+ 2120590
119894
32 3873 4222 4491 4725 4954 5206 5464 5635120572
119894
= 120573
119894
00 632 910 1140 1320 1498 1690 1884 2000INT119894
119862
119897
119894
($t) 32 3030 3008 2971 2965 2956 2952 2952 2968119862
119906
119894
($t) 32 3452 3616 3732 3845 3956 4209 4208 4301
Table 6 Operating costs over production period 2017ndash2021
2017 2018 2019 2020 2021119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]
Table 7 Discounted cash flow over production period 2017ndash2021
2017 2018 2019 2020 2021119877year (millrsquos USD) [2279 7466] [1974 7360] [1672 7269] [1374 7192] [1079 7131]119862year (millrsquos USD) [2965 3845] [2956 3956] [2956 4029] [2952 4208] [2968 4301]119877year minus 119862year [minus1566 4501] [minus1982 4404] [minus2357 4313] [minus2834 4240] [minus3222 4163]Discount factor [107 110] [114 121] [123 133] [131 146] [140 161]Present value [minus1423 4206] [minus1638 3863] [minus1772 3506] [minus1941 3236] [minus2001 2973]
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
000
1000
2000
3000
4000
5000
6000
2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Path 1Path 2Path 3Path 4
Path 5Path 6Path 7
Figure 3 Seven simulated operating cost paths on a yearly timeresolution
Weight whitenization of interval NPV is obtained by 120596NPV =
05 and the white value is NPV = 1004 million USD Thismeans the project is accepted
5 Conclusion
The combined effect of market volatility and uncertaintyabout future commodity prices is posing higher risks tomining businesses across the globe In such times knowinghow to unlock value by maximizing the value of resourcesand reserves through strategic mine planning is essential Inour country small mining companies are faced with manyproblems but the primary problem is related to the shortage ofcapital for investment In such an environment every miningventure must be treated as a strategic decision supportedby adequate analysis The developed economic model is amathematical representation of project evaluation reality andallows management to see the impact of key parameters onthe project value The interaction between production costsand capital is highly complex and changes over time butneeds to be accurately modelled so as to provide insightsaround capital configurations of that business
The evaluation of a mining venture is made very dif-ficult by uncertainty on the input variables in the projectMetal prices costs grades discount rates and countlessother variables create a high risk environment to operatein The incorporation of risk into modelling will providemanagement with better means to deal with uncertainty andthe identification andquantification of those factors thatmostcontribute to risk which will then allow mitigation strategiesto be tested The model brings forth an issue that has thedynamic nature of the assessment of investment profitabilityWith the fuzzy-interval model the future forecast can bedone from the beginning of the process until the end
From the results obtained by numerical example itis shown that fuzzy-interval grey system theory can beincorporated intomine project evaluationThe variance ratio119862vr = 0175 shows that the metal prices forecasting model iscredible to predict the future values of the most importantexternal parameter The operating costs prediction modelbased on geometric Brownian motion gives the same resultthat we get if we use scenarios however it does not requireus to simplify the future to the limited number of alternativescenarios
With interval numbers the end result will be intervalNPV which is the payoff interval for the project Using theweight whitenization of the interval NPV we obtain thepayoff crisp value for the project This value is the value atrisk helping the management of the company to make theright decision
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This paper is part of research conducted on scientific projectsTR 33003 funded by the Ministry of Education Science andTechnological Development Republic of Serbia
References
[1] M R Samis G A Davis andD G Laughton ldquoUsing stochasticdiscounted cash flow and real option monte carlo simulationto analyse the impacts of contingent taxes on mining projectsrdquoin Proceedings of the Project Evaluation Conference pp 127ndash137Melbourne Australia June 2007
[2] R Dimitrakopoulos ldquoApplied risk assessment for ore reservesand mine planningrdquo in Professional Development Short CourseNotes p 350 Australasian Institute of Mining and MetallurgyMelbourne Australia 2007
[3] E Topal ldquoEvaluation of a mining project using discounted cashflow analysis decision tree analysis Monte Carlo simulationand real options using an examplerdquo International Journal ofMining and Mineral Engineering vol 1 no 1 pp 62ndash76 2008
[4] S Dessureault V N Kazakidis and Z Mayer ldquoFlexibility val-uation in operating mine decisions using real options pricingrdquoInternational Journal of Risk Assessment and Management vol7 no 5 pp 656ndash674 2007
[5] T Elkington J Barret and J Elkington ldquoPractical applicationof real option concepts to open pit projectsrdquo in Proceedingsof the 2nd International Seminar on Strategic versus TacticalApproaches in Mining pp S261ndashS2612 Australian Centre forGeomechanics Perth Australia March 2006
[6] L Trigeorgis ldquoA real-options application in natural-resourceinvestmentsrdquo in Advances in Futures and Options Research vol4 pp 153ndash164 JAI Press 1990
[7] M Samis and R Poulin ldquoValuing management flexibility byderivative asset valuationrdquo in Proceedings of the 98th CIMAnnual General Meeting Edmonton Canada 1996
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Applied Mathematics
[8] M Samis and R Poulin ldquoValuing management flexibilitya basis to compare the standard DCF and MAP valuationframeworksrdquo CIM Bulletin vol 91 no 1019 pp 69ndash74 1998
[9] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965
[10] R Saneifard ldquoCorrelation coefficient between fuzzy numbersbased on central intervalrdquo Journal of Fuzzy Set Valued Analysisvol 2012 Article ID jfsva-00108 9 pages 2012
[11] C Carlsson and R Fuller ldquoOn possibilistic mean value andvariance of fuzzy numbersrdquo Tech Rep 299 Turku Centre forComputer Science 1999
[12] J W Park Y S Yun and K H Kang ldquoThe mean value andvariance of one-sided fuzzy setsrdquo Journal of the ChungcheongMathematical Society vol 23 no 3 pp 511ndash521 2010
[13] N Gasilov S E Amrahov and A G Fatullayev ldquoSolution oflinear differential equations with fuzzy boundary valuesrdquo FuzzySets and Systems 2013
[14] D Dubois and H Prade ldquoThe mean value of a fuzzy numberrdquoFuzzy Sets and Systems vol 24 no 3 pp 279ndash300 1987
[15] M Hukuhara ldquoIntegration des applications mesurables dont lavaleur est un compact convexerdquo Funkcialaj Ekvacioj vol 10 pp205ndash223 1967
[16] B Bede and L Stefanini ldquoNumerical solution of intervaldifferential equations with generalized hukuhara differentia-bilityrdquo in Proceedings of the Joint International Fuzzy SystemsAssociationWorld Congress and European Society of Fuzzy Logicand Technology Conference (IFSA-EUSFLAT rsquo09) pp 730ndash735July 2009
[17] S Mehrkanoon M Suleiman and Z A Majid ldquoBlock methodfor numerical solution of fuzzy differential equationsrdquo Interna-tional Mathematical Forum vol 4 no 45ndash48 pp 2269ndash22802009
[18] O Kaleva ldquoA note on fuzzy differential equationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 64 no 5 pp 895ndash900 2006
[19] Y C Cano and H R Flores ldquoOn new solutions of fuzzydifferential equationsrdquo Chaos Solitons amp Fractals vol 38 pp112ndash119 2008
[20] M Mosleh ldquoNumerical solution of fuzzy differential equationsunder generalized differentiability by fuzzy neural networkrdquoInternational Journal of Industrial Mathematics vol 5 no 4 pp281ndash297 2013
[21] A Swishchuk A Ware and H Li ldquoOption pricing with sto-chastic volatility using fuzzy sets theoryrdquo Northern FinanceAssociation Conference Paper 2008 httpwwwnorthernfina-nceorg2008papers182pdf
[22] J Do H Song S So and Y Soh ldquoComparison of deterministiccalculation and fuzzy arithmetic for two prediction modelequations of corrosion initiationrdquo Journal of Asian Architectureand Building Engineering vol 4 no 2 pp 447ndash454 2005
[23] R Guo and C E Love ldquoGrey repairable system analysisrdquoInternational Journal of Automation and Computing vol 2 pp131ndash144 2006
[24] S Hui F Yang Z Li Q Liu and J Dong ldquoApplication of greysystem theory to forecast the growth of larchrdquo InternationalJournal of Information and Systems Sciences vol 5 no 3-4 pp522ndash527 2009
[25] J L Deng ldquoIntroduction to grey system theoryrdquoThe Journal ofGrey System vol 1 no 1 pp 1ndash24 1989
[26] Y- Wang ldquoPredicting stock price using fuzzy grey predictionsystemrdquo Expert Systems with Applications vol 22 no 1 pp 33ndash38 2002
[27] Y W Yang W H Chen and H P Lu ldquoA fuzzy-grey model fornon-stationary time series predictionrdquo Applied Mathematics ampInformation Sciences vol 6 no 2 pp 445Sndash451S 2012
[28] D Ma Q Zhang Y Peng and S Liu A Particle Swarm Opti-mization Based Grey ForecastModel of Underground Pressure forWorking Surface 2011 httpwwwpapereducn
[29] F Rahimnia M Moghadasian and E Mashreghi ldquoApplicationof grey theory approach to evaluation of organizational visionrdquoGrey Systems Theory and Application vol 1 no 1 pp 33ndash462011
[30] N Slavek and A Jovıc ldquoApplication of grey system theory tosoftware projects rankingrdquo Automatika vol 53 no 3 2012
[31] C Zhu and W Hao ldquoGroundwater analysis and numericalsimulation based on grey theoryrdquo Research Journal of AppliedSciences Engineering and Technology vol 6 no 12 pp 2251ndash2256 2013
[32] G S Ladde andM Sambandham Stochastic versus Determinis-tic Systems of Differential Equations Marcel Dekker New YorkNY USA 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of