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

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

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

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 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

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 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

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