Fuzzy financial analyses of demand-side management alternatives

Fuzzy financial analyses of demand-sidemanagement alternatives

J.N. Sheen

Abstract: Fuzzy financial models are derived for the profitability analyses of demand-sidemanagement (DSM) alternatives. The present value of cost and equivalent uniform annual costmodels are selected to determine the least-cost solution, while the net present value, payback yearand benefit/cost ratio models are proposed for the execution of cost–benefit analyses. The meansand variances of the fuzzy financial indexes associated with DSM alternatives are evaluated byMellin transform in order to determine their relative ranking in a decision-making process. Theperformance of the proposed models is verified through the simulation of a numerical example andby considering their application to two practical DSM programmes: the choice of a suitable air-conditioning system for an office building and the evaluation of different cogeneration alternativesfor a synthetic rubber corporation in Taiwan. These investigations confirm not only that the resultsof the proposed fuzzy financial models are consistent with those of the conventional crisp models,but they also demonstrate that the proposed methods represent readily implemented possibilityanalysis tools for use in the arena of uncertain financial decision-making.

1 Introduction

The term demand-side management (DSM) refers to theinduced changes in the timing and volume of electricitydemand caused by the implementation of such measures asload control, improved energy efficiency, on-site energystorage, and the promotion of off-peak uses of electricity.However, a DSM initiative will only meet with success if theactors who must implement the initiative, and those whohave the opportunity to participate in it, are both able toperceive a benefit from doing so [1].

Prior to adopting a DSM program, potential participantsmust explore the soundness of the project by performing afeasibility study that investigates all aspects of the project,including its anticipated future financial and economicperformance. The financial evaluation aspect mainlyconcerns the monetary aspects of the project and itsfinancial rewards and profitability from the investors’perspectives. Meanwhile, the economic evaluation goesbeyond this by attempting to relate the project to thenational economy, in terms of its economic, social andenvironmental implications. However, the typical DSMprogrammes adopted by power utilities are not of asufficient scope to require such an economic evaluation,and a financial study that addresses the profitability of theproject together with a consideration of its technological,financial, cost and benefits aspects usually suffices [2].

The cash-flow models applied in many financial decision-making problems often involve some degree of uncertainty.In the case of deficient data, most decision-makers tend to

rely on an expert’s knowledge of financial information whencarrying out their financial modelling activities. Since thenature of this knowledge often tends to be vague ratherthan random, Dr. Zadeh introduced fuzzy set theory, whichaimed to rationalise the uncertainty caused by vagueness orimprecision. However, practical applications of fuzzy settheory in the profitability arena require two laborious tasks,namely fuzzy mathematical operations and the comparisonor ranking of the resultant complex fuzzy numbers. Fuzzymathematics is based on the extended principles presentedin [3–5], in which the traditional addition, subtraction,multiplication, division, power, logarithmic and exponentmathematical operations are applied to fuzzy numbers.Dubois and Prade [4] demonstrated that, when performingthe binary manipulation of fuzzy numbers, the resultantincreasing (decreasing) part arose from binary operationson the non-decreasing (non-increasing) parts of the twofuzzy numbers. The extended operations ensured that theresultant fuzzy number continuously maintained its fuzzyproperties during the arithmetic operating procedure. It isfound that fuzzy mathematics tends to be cumbersome foreven the more straightforward operations such as additionand subtraction. Unfortunately, financial and engineeringapplications involving fuzzy sets typically require the morecomplex nonlinear mathematical operations such as pro-duct, division, power and logarithmic manipulations [6]. Insome cases, fuzzy operations of this type may require aninsurmountable computational effort. Consequently, it hasbeen proposed that approximated triangular fuzzy numbersbe used to examine the resultant fuzzy profitability indexes[7].

Following the manipulation of the approximated fuzzyfinancial function by fuzzy mathematics, the task ofcomparing or ranking the resultant complex fuzzy numberscan invoke another problem because fuzzy numbers do notalways yield a totally ordered set in the same way that crispnumbers do. Many authors have investigated the use ofalternative fuzzy set ranking methods, and these methodshave been reviewed and compared by Chen and Hwang [8].

The author is with the Department of Electrical Engineering, Cheng-ShiuUniversity, 840 Cheng-Ching Road, Neau-Song Country, Kaohsiung, Taiwan,Republic of China

E-mail: [email protected]

r IEE, 2005

IEE Proceedings online no. 20041301

doi:10.1049/ip-gtd:20041301

Paper first received 2nd December 2003 and in revised form 6th August 2004

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005 215

TheMellin Transform [9, 10] has been proposed as a meansof calculating the mean and variance values of theapproximated fuzzy resulted indexes. A rigorous rankingof the fuzzy numbers can then be obtained by simplycomparing the means and variances of the fuzzy numbers.

2 Fuzzy numbers

When dealing with uncertainty, decision-makers arecommonly provided with information that is characterisedby vague linguistic descriptions such as ‘high risk’, ‘lowprofit’, ‘high annual interest rate’ etc. The principalobjective of fuzzy set theory is to quantify these vaguedescriptive terms. Dr. Zadeh proposed a membershipfunction that accords each object a grade (or degree) ofmembership within the interval [0,1]. A fuzzy set isdesignated as 8xAX, mA(x)A [0,1], where mA(x) is the gradeof membership, ranging from 0 to 1, of a vague predicate,A, over the universe of objects, X. The closer the objectmatches the vague predicate, the higher its grade ofmembership. The membership function may be viewed asrepresenting an opinion poll of human thought or as anexpert’s opinion. A fuzzy number is a normal and a convexfuzzy set, and its membership function can be denoted as:mAðxÞ ¼ ða1; fA1

ðaÞ=a2; a3=fA2ðaÞ; a4Þ, where fA1

ðaÞ is acontinuous monotonically increasing function of a for0 � a � 1, fA2

ðaÞ is a continuous monotonically decreasingfunction of a for, 0 � a � 1; fA1

ð0Þ ¼ a1, fA1ð1Þ ¼ a2,

fA2ð1Þ ¼ a3, fA2

ð0Þ ¼ a4, and a1oa2 � a3oa4[11]. Thetrapezoidal fuzzy number (TrFN) is a particular form offuzzy number in which fA1

and fA2are both straight-line

segments, and in the case where a2¼ a3, this TrFN becomesa triangular fuzzy number (TFN). Implementing the TFN ismathematically straightforward and, more importantly, itrepresents a rational basis for quantifying the vagueknowledge associated with most decision-making problems[6, 7, 12, 13].

The TFN of the vague predicate A can be expressedsimply as A¼ (a1, a2, a2), where the vertexes a1, a2, and a3denote the smallest possible value, the most promisingvalue, and the largest possible value describing a fuzzyevent, respectively. Of these values, the most promisingvalue can be considered as the conventional (classic) crispnumber. It is noted that these parameters are analogous tothe lower, medium, and higher values in the domain of thetriangular probability distribution. However, the para-meters in a TFN represent the values accorded by humanthought to the possibility of an event occurring, while theparameters in a triangular probability distribution representthe values associated with the probabilistic occurrence ofthat event. The membership function of the vague predicateA shown in Fig. 1 is described by the following linearrelationships:

mAðxÞ ¼mA1ðxÞ ¼ x� a1

a2 � a1a1 � x � a2

mA2ðxÞ ¼ a3 � x

a3 � a2a2 � x � a3

8>><>>: ð1Þ

x ¼ fA1ðaÞ ¼ m�1A1

¼ a1 þ ða2 � a1Þa 0 � a � 1

fA2ðaÞ ¼ m�1A2

¼ a3 � ða3 � a2Þa 0 � a � 1

�ð2Þ

Equation (1) represents a mapping from any given value ofx to its corresponding grade of membership a, whereas (2)provides the inverse mapping from any given value of a toits corresponding value of x. As shown in (3), the TFN canalso be designated in an a-cut form. The a-cut of a fuzzy setA is a crisp set containing all the elements of the universal

set X, whose membership grades in A are greater than, orequal to, the specified value of a. The a-cut of the fuzzy setA is given by:

Aa ¼ ½fA1ðaÞ; fA2

ðaÞ�¼ a1 þ ða2 � a1Þa; a3 � ða3 � a2Þa½ � ð3Þ

Possibility (or confidence level) analyses is performed byusing the membership function of the fuzzy number given in(1)–(3). In this analysis, if x lies between a1 and a2, then thepossibility of X can be obtained by substituting x intomA1ðxÞ. Similarly, if x lies between a2 and a3, then the

possibility of x can be obtained by substituting x intomA2ðxÞ. At a specific membership grade or at a specific

possibility a the range of x can be calculated from the a-cutgiven in (3).

3 Fuzzy number comparsion

3.1 Converting a fuzzy number into itscorresponding probabilistic density functionAccording to Chen and Hwang [8], the published fuzzyranking methods may be classified as belonging to one oftwo categories. The first category refers to the possibilisticmethod, and is based on possibility theory, while the secondcategory groups probabilistic methods, which are foundedon well-known probability theory. The conversion of themembership function of a fuzzy number, m(x), into aprobability density function, f(x), can be achieved by usingone of the two linear transformations presented in [5], i.e.

1 Proportional probability density function ðppdf Þ :pðxÞ ¼ hpmAðxÞ

ð4Þ

2 Uniformprobability density function ðupdf Þ :uðxÞ ¼ mAðxÞ þ ðhu � 1Þ ð5Þ

where hp and hu are the conversion constants, which ensurethat the area under the continuous probability function isequal to one.

Figures 2a and 2b show the conversions of a TFN intothe corresponding ppdf and updf functions, respectively. Inthe case of the proportional conversion, the domain and thevertexes of the triangular ppdf are the same as the convertedTFN, but its height changes from 1 to hp¼ 2/(a3–a1).Regarding the updf function, the height of the triangular

updf is given by hu ¼ ð2=ða3 � a1ÞÞ1=2, and its vertexes areexpressed as:

a01 ¼ a2 � huða2 � a1Þ; a02 ¼ a2;a03 ¼ a2 þ huða3 � a2Þ

ð6Þ

Figure 3 shows the conversion of a TrFN into thecorresponding ppdf and updf functions. The height of thetrapezoidal ppdf is given by hp¼ 2/[(a4+a3)�(a2+a1)].

)((x) = f11

αµ −1AA

)((x) = f22

αµ −1AA

)(, α fx

αµ (x),

1a 2a 3a

1.0

Fig. 1 Membership function of triangular fuzzy numbera Proportional conversionb Uniform conversion

216 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005

Meanwhile, the height and the vertexes of the trapezoidalupdf are expressed, respectively, as:

hu ¼ ða2 � a3Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 � a3Þ2 þ 2ða4 � a3 þ a2 � a1Þ

q� �=ða4 � a3 þ a2 � a1Þ

ð7Þ

a01 ¼a2 � huða2 � a1Þ; a02 ¼ a2; a03 ¼ a3;

a04 ¼a3 þ huða4 � a3Þð8Þ

The conversion of a fuzzy number into its correspondingproportional distribution is computationally straightfor-ward. However, when the uniform distribution conversionmethod is used, both the domain and the range of theresulting distribution are reduced (or increased). Thereduced (or increased) domain indicates the partial rejection(or addition) of some members from (or to) the set. Hence,this particular conversion method is not entirely suitable.Furthermore, when applying the proportional probabilitydensity distribution to convert the fuzzy number, it is notedthat the range of the membership grade of the resultantproportional distribution is greatly reduced when the fuzzynumber has a wide domain. Consequently, the ability of themembership function to discriminate precisely amongmembers of the fuzzy set is impaired. Fortunately, however,the domain of fuzzy numbers used to denote an expert’sknowledge is always sufficiently narrow to avoid thisbecoming a problem. Therefore, and in view of thelimitations of the uniform distribution method, it isrecommended that the proportional density conversionfunction be applied when comparing fuzzy numbers.

3.2 Mellin transform and ranking of fuzzynumbersOperational calculus techniques are particularly useful whenanalysing probabilistic models as part of a decision-makingprocess. In the probabilistic modelling context, it is oftenpossible to reduce complex operations involving differentia-tion and integration to simple algebraic manipulations inthe transform domain. The Mellin transform is a useful tool

for studying the distributions of certain combinations ofrandom variables, especially for those concerned with therandom variables associated with products and quotients.The Mellin transform, Mx(s), of a function f(x), where x ispositive, is defined in [9, 10] as follows:

MxðsÞ ¼Z 10

xs�1f ðxÞdx 0oxo1 ð9Þ

The Mellin transform has a unique one-to-one correspon-dence with the transformed function, i.e. f(x)2Mx(s) [14].The moments of a distribution represent the expected valuesof the power of a random variable with an f(x) distribution.In general, the pth moment of a random variable, X, abouta real number, c, is defined as:

MpðxÞ ¼ E½ðX � cÞp� ¼ZX

ðx� cÞpf ðxÞdx ð10Þ

The moments of interest in economic analyses are thoseabout the origin (c¼ 0) and those about the mean (c¼ m),typically for p¼ 1,2,3 and 4. If the pth moments about theorigin and the mean are denoted by E[Xp] and mp,respectively, then:

mp ¼ E½ðX � mÞp� ¼ZX

ðx� mÞpf ðxÞdx ð11Þ

The first moment about the origin represents the mean ofthe distribution, m¼E[X], and the second moment aboutthe mean represents the variance, s2. Meanwhile, the skewand the kurtosis of the distribution are denoted by m3 andm4, respectively. Comparing (9) with (10) shows that Mx(s)is a special case of Mp(x), where c¼ 0 and p¼ s�1. In otherwords, if f(x) is viewed as a probability density function,the Mellin transform, Mx(s)¼E[Xs�1], provides a means ofestablishing a series of moments of the distribution [15].Comparing the first two moments of a distribution with theMellin transform, allows the mean and variance to beexpressed as (12) and (13), respectively:

m ¼ E½X � ¼ Mxð2Þ ð12Þ

m2 ¼ s2 ¼ Var½X � ¼ Mxð3Þ � ðMxð2ÞÞ2 ð13Þ

curve 1: fuzzy numbercurve 2: ppdf as (a

3−a

1) > 2

curve 3: ppdf as (a3−a

1) > 2

curve 1: fuzzy numbercurve 2: uppdf as (a

3−a

1) > 2

curve 3: uppdf as (a3−a

1) > 2

1a

2a

3a

ph

ph

1 1

2

3

''1a 2a 3a

uh

uh

1

3a ′ 3a′′1a′

1a

1

2

3

a b

Fig. 2 TFN conversion into probability density functiona Proportional Conversionb Uniform Conversion

1a 1a 3a 4a′1a ′

1a ′4a ′

4a'''

uh

uh

1

'

1

2

3

curve 1: fuzzy number curve 2: ppdf as

curve 3: ppdf as [(a4 − a3) + (a2 − a1)] < 2

[(a4 − a3) + (a2 − a1)] < 2 [(a4 − a3) + (a2 − a1)] < 2

[(a4 − a3) + (a2 − a1)] < 2

curve 1: fuzzy number curve 2: uppdf as

curve 3: ppdf as

1a 2a 3a 4a

ph

ph

1 1

2

3

a b

Fig. 3 TrFN conversion into probability density function


The close correlation between the Mellin transform and theexpected values enables certain important operating proper-ties involving products, quotients and powers of randomvariables to be readily determined. Its operational proper-ties, including linearity, translation, scaling and, mostimportantly, convolution, yield the important Mellin trans-form operations, which are summarised in Table 2 in [15].Figure 4 of the present study presents seven typicaltriangular and trapezoidal fuzzy numbers, and theirrespective Mellin transforms are summarised in Table 1.Computing Mx(s) at s¼ 1, 2 and 3, gives the mean andvariance of the triangular fuzzy number A(a1, a2, a3) as:

mA ¼ Mxð2Þ ¼a1 þ a2 þ a3

3ð14Þ

s2A ¼1

18ða2

1 þ a22 þ a2

3 � a1a2 � a2a3 � a3a1Þ ð15Þ

The mean and variance of a TFN is determined only by itsvertexes, i.e. they are independent of its height. Theconsistent domain properties between fuzzy numbers andtheir converted probability density functions are animportant advantage when using the Mellin transformmethod to calculate these values. Generally, in theprobabilistic approach, the means and variances of the

fuzzy numbers are computed and a preferred ranking offuzzy numbers is obtained by comparing the means andvariances of the individual numbers. In the particular caseof fuzzy financial evaluations, it is important to note that afuzzy number with a lower mean value is ranked higherthan a fuzzy number with a higher mean in cost evaluationmodels, whereas in benefit evaluation models, a fuzzynumber with a higher mean is placed above a number witha lower mean. If the means of two fuzzy numbers happen tobe equal, the number with a smaller spread (variance) isjudged to be better.

4 Fuzzy financial evaluation

Previous researchers, including Kaufmann and Gupta [6]and Ward [12], have conducted fuzzy discounted cash flowanalyses in which either the periodic cash flow or thediscount rate was specified as a fuzzy number. Furthermore,Buckley [11], Chiu and Park [7] and Kahraman et al. [13]addressed problems in which both the periodic cash flowand the discount rate were expressed by fuzzy numbers.These studies also developed various economic equivalenceformulae for use in rudimentary financial calculations.However, these models have only limited application in thefinancial decision-making arena since they consider only asingle payment or, at best, a few payments, when derivingtheir financial indexes. In real-world applications, it is morerealistic to consider that the periodic cash flow may besubject to occasional uncertain variations. Accordingly, thepresent study adopts a parameter, d, to represent theinflation rate [16]. This parameter is specified as a fuzzynumber and is used to reflect an uncertain geometric seriesof cash flows.

At the planning stage, a decision-maker is seldom inpossession of all the information necessary to make anaccurate assessment of the initial investment, I, and the firstyear’s cash flow in (or out), A. Therefore, the fuzzy initialinvestment, I, the fuzzy first year’s cash flow in, A, the fuzzyinflation rate, d%, and the fuzzy interest rate, r%, should allbe denoted as TFN.

1

2a 3a 4a1a

1

2a 3a 4a 1a

1

2a 3a

1

1a 2a

1

2a 3a1a 2a 3a

1

a b

a b c d

gfe

Fig. 4 Selected fuzzy numbers (in conjunction with Table 1)

Table 1: Mellin transforms for selected fuzzy numbers

Fuzzy number shape Conversion constant of ppdf, hp Mellin transform, Mx(s)

Uniform a1

ðb � aÞhpðbs � asÞ

s

Regular TFN b2

ða3 � a1Þhp

sðs þ 1Þa3ðas

3 � as2Þ

a3 � a2

� a1ðas2 � as

1Þa2 � a1

� �

Left-skewed TFN c2

ða2 � a1Þhp

sðs þ 1Þ sas2 �

a1ðas2 � as

1Þa2 � a1

� �

Right-skewed TFN d2

ða3 � a2Þhp

sðs þ 1Þa3ðas

3 � as2Þ

a3 � a2

� sas2

� �

Regular TrFN e2

ða4 þ a3Þ � ða2 þ a1Þhp

sðs þ 1Þðasþ1

4 � asþ13 Þ

a4 � a3� ða

sþ12 � asþ1

1 Þa2 � a1

� �

Left-skewed TrFN f2

ð2a3 � a2 � a1Þhp

sðs þ 1Þ ðs þ 1Þas3 �ðasþ1

2 � asþ11 Þ

a2 � a1

� �

Right-skewed TrFN g2

ða4 þ a3 � 2a2Þhp

sðs þ 1Þðasþ1

4 � asþ13 Þ

a4 � a3� ðs þ 1Þas

2

� �


4.1 Least-cost solutionDSM participants in power utilities are frequently requiredto conduct some form of financial evaluation process inorder to identify the least-cost solution that still satisfies therequired demand or provides the required service. Ingeneral, the choice of the least-cost solution is concernedonly with differences in the present values of the costs of thevarious alternatives. However, if differences exist in thevolume of the physical output of the various proposals, theevaluation process must be based on per unit of outputbasis. Various DSM alternatives exist, including direct loadcontrol, thermal (hot or cold) energy storage, time-of-usepricing etc. These initiatives share common objectives andresults, i.e. clipping the peak demand, filling the off-peakvalley, shifting the electricity consumption from peak-timeto the off-peak period in utilities, meeting the productionschedule etc. Figure 5 shows the cash flow of a typical DSMalternative. The Figure makes the assumption that allparameters are end-of-year (end-period) predicates, andthat the initial investment, Ao¼ I, and the annual operatingcost, At, are both positive. Additionally, n represents theyear number over the timescale of the project evaluation. Aparticular DSM initiative is only evaluated and comparedagainst other iniatives in terms of the costs of thealternatives. If the initiative possesses any salvage value,this value is deducted from its cost after it has beendiscounted to the base year. The two most commonlyapplied methods of doing this are the present value of cost(PVC) method and the equivalent uniform annual cost(EUAC) method.

The choice of using either the PVC or the EUACmethodto evaluate the least-cost alternative is largely a matter ofpersonal preference. In the USA, the EUAC method tendsto be preferred since its implications are more readilyunderstood in the context of business decisions. Further-more, it is more straightforward to compute than the PVCmethod for a regular annual series of disbursements,particularly if the capital is obtained through loans.However, the PVC method allows for a more detailedevaluation of future costs [2], and is consequently themethod adopted within this current study.

4.1.1 Present value of cost (PVC) meth-od: The present value of the costs illustrated in Fig. 5can be represented as:

PVC ¼ I þXn

t¼1

At

ð1þ rÞt¼ I þ

Xn

t¼1

Að1þ dÞt�1

ð1þ rÞt

¼ I þ A1� 1þd

1þr

� �n

ðr � dÞ ð16Þ

In the fuzzy case, the membership function of the fuzzyPVC is represented as:

mPVCðxÞ ¼ðPVC1; fPVC1ðaÞ=PVC2;

PVC2=fPVC2ðaÞ; PVC3Þ

ð17Þ

where fPVCiðaÞ ¼ fIiðaÞ þ fAiðaÞ1�

1þfdiðaÞ

1þfrð3�iÞ ðaÞ

� �n

frð3�iÞ ðaÞ�fdi ðaÞ; i ¼ 1; 2

and fIiðaÞ; fAiðaÞ; fdiðaÞ; friðaÞ; i ¼ 1; 2 are the member-ship functions of the initial investment, I, the first year’scash flow, A, the inflation rate, d, and the interest rate r,respectively.

The PVC of a project is a complicated nonlinearrepresentation, whose solution requires tedious computa-tional effort. For simplicity, the corresponding approxi-mated triangular fuzzy present value of cost, APVC, can bederived by linearising fPVC1

ðaÞ and fPVC2ðaÞ of the exact

PVC. The present study adopts computer simulation toexplore the divergence between the PVC and the APVC ateach membership grade a for different ranges of A, I, d%and r%. The main conclusions from this simulation can besummarised as follows:

1 The divergence between the PVC and the APVC is notsignificantly influenced by changes in the values of A and I.However, it is strongly influenced by the values of d% andr% due to presence of the nth power of d and r within thePVC formula. It is determined that the approximationbecomes increasingly poorer as the value of n increases.

2 The divergence is good on the left-hand side, but gives poorresults on the right-hand side. The divergence ranges from1.2% on the left-hand side to 3.5% on the right-hand side.

3 When the confidence ranges of d% and r% are within710%, the maximum divergence percentage is relativelysmall, which indicates that the APVC can substitute for thePVC.

On the basis of these simulation results, the financialindexes applied throughout the remainder of the currentstudy are based on approximated fuzzy numbers ratherthan exact fuzzy numbers.

If all the fuzzy economic numbers, i.e. A, d, r and I, arerepresented in the form of TFNs, i.e. A¼ (A1, A2, A3),d¼ (d1, d2, d3)%, r¼ (r1, r2, r3)% and I¼ (I1, I2, I3),respectively, then the a-cut and the vertexes of the APVCcan be represented as (18), the detailed derivations areshowing in the Appendix.

APVC ¼ ½fAPVC1ðaÞ; fAPVC2

ðaÞ� ð18Þwhere: fAPVC1

ðaÞ ¼ PVC1 þ ðPVC2 � PVC1Þa, fAPVC2ðaÞ

¼ PVC3� ðPVC3 � PVC2Þa, and

PVC1 ¼ I1 þ A1

1� 1þ d11þ r3

� �n

r3 � d1;

PVC2 ¼ I2 þ A2

1� 1þ d21þ r2

� �n

r2 � d2;

PVC3 ¼ I3 þ A3

1� 1þ d31þ r1

� �n

r1 � d3

4.1.2 Equivalent uniform annual cost (EUAC)method: The EUAC represents the summation of theannual capital cost and the annual operating cost, and is

t, year

0

Ao = I

AA(1+d )

A(1+d )2A(1+d )n

1 2 3 n

Fig. 5 Cash flows of DSM alternative (the alternatives havingsame demands, only costs are concerned)


expressed in (19), where CRF represents the capital recoveryfactor.

EUAC ¼ I þ A1� 1þ d

1þ r

� �n

r � d

0BB@

1CCA � CRF;

CRF ¼ rð1þ rÞn

ð1þ rÞn � 1

ð19Þ

The a-cut and the ‘ fuzzy EUAC, i.e. the AEUAC, can bederived in the same way as APVC, and are:

AEUAC ¼ ½fAEUAC1ðaÞ; fAEUAC2

ðaÞ� ð20Þwhere

fAEUAC1ðaÞ ¼ EUAC1 þ ðEUAC2 � EUAC1Þa;

fAEUAC2ðaÞ ¼ EUAC3 � ðEUAC3 � EUAC2Þa;

EUAC1 ¼ I1 þ A1

1� 1þ d11þ r3

� �n

r3 � d1

0BB@

1CCA r1ð1þ r1Þn

ð1þ r1Þn � 1

� �;

EUAC2 ¼ I2 þ A2

1� 1þ d21þ r2

� �n

r2 � d2

0BB@

1CCA r2ð1þ r2Þn

ð1þ r2Þn � 1

� �;

EUAC3 ¼ I3 þ A3

1� 1þ d31þ r1

� �n

r1 � d3

0BB@

1CCA r3ð1þ r3Þn

ð1þ r3Þn � 1

� �:

.4.2 Cost-benefit analysesIn evaluating some DSM programmes, including thecogeneration case, some participants represent the cashflow out by the initial capital investment, Ao¼ I, andconsider the cash flow in to be given by the annual netbenefit, At, which is determined as the difference betweenthe annual operating cost and the annual productionrevenue. These cash flows are shown in Fig. 6. The presentstudy develops three fuzzy financial evaluation models, i.e.net present value (NPV), payback year (PBY) and benefit/cost ratio (BCR), to assess the profitability of a DSMinitiative. Although the internal rate of return (IRR)indicator is commonly used in conventional crisp cost–benefit analyses, it has been noted by previous researchersthat it is not applicable to the fuzzy case [11, 13]. The IRRon an investment is defined as the rate of interest earned onthe uncovered balance of an investment. It is also the ratethat would make the NPV of the project equal to zero, i.e.Xn

t¼1At=ð1þ r�Þt� �

� I ¼ 0: ð21Þ

Since the right hand side of (21) is fuzzy, but 0 is a crispnumber, achieving equality is impossible [13]. Buckley [11]has also confirmed that the IRR technique does not extendto fuzzy cash flows.

4.2.1 Net present value (NPV) method: TheNPV of the investment and the costs illustrated in Fig. 6 canbe represented as:

NPV ¼ �I þ A1þ r

þ Að1þ dÞð1þ rÞ2

þ . . .

þ Að1þ dÞn�1

ð1þ rÞn

¼ �I þ A1� 1þd

1þr

� �n

ðr � dÞ ð22Þ

In general, when evaluating different investment alterna-tives, a positive NPV during the service lifetime indicatesthat the investment has a positive effectiveness. Further-more, the greater the value of the NPV, the more effective itis. The a-cut and the vertexes of the Approximated fuzzyNPV, i.e. the ANPV, can be represented as

ANPV ¼ ½fANPV1ðaÞ; fANPV2

ðaÞ� ð23Þwhere fANPV1

ðaÞ ¼ NPV1 þ ðNPV2 �NPV1Þa; fANPV2ðaÞ

¼ NPV3 � ðNPV3 �NPV2Þa ; and

NPV1 ¼ �I3þ A1

1�1þ d11þ r3

� �n

r3�d1;

NPV2 ¼ �I2 þ A2

1�1þ d21þ r2

� �n

r2�d2;

NPV3 ¼ �I1 þ A3

1�1þ d31þ r1

� �n

r1�d3

4.2.2 Pay back year (PBY) method: The PBYmeasure indicates the number of years required to recoverthe initial investment, and represents an alternative meansof evaluating the cost–benefit of a project. It is noted that amore effective investment is represented by a smaller PBYvalue. Basically, the PBY represents the smallest value of nsuch that the value of the NPV in (22) is equal to zero. ThePBY is determined from:

PBY ¼ln 1� ðr � dÞ I0

A

� �ln 1þd

1þr

� � ð24Þ

The a-cut and the vertexes of the ANPV can be representedas:

APBY ¼ ½fAPBY1ðaÞ; fAPBY2

ðaÞ� ð25Þwhere fAPBY1

ðaÞ ¼ PBY1 þ ðPBY2 � PBY1Þa, fAPBY2ðaÞ

¼ PBY3 � ðPBY3 � PBY2Þa, and

PBY1 ¼ln 1� ðr1 � d3Þ

I1A3

� �

ln ð1þ d31þ r1

Þ

0BB@

1CCA;

PBY2 ¼ln 1�ðr2�d2Þ

I2A2

� �

lnð1þ d21þ r2

Þ

0BB@

1CCA;PBY3 ¼

ln 1�ðr3�d1ÞI3A1

� �

ln ð1þ d11þ r3

Þ

0BB@

1CCA

t, year

Ao = I

A(1+d ) A(1+d )2

A(1+d )n

0

A

1 2 3 n

Fig. 6 Cash flows of DSM alternative (the alternatives having netbenefits)


4.2.3 Benefit/cost ratio (BCR) method: Thismethod compares the discounted total net benefit of theproject with the initial investment, I. The ratio may beregarded as either the profit/investment ratio or the netpresent value per investment. It is noted that a moreeffective investment is represented by a higher BCR value.

BCR ¼

Pnt¼1

Að1þ dÞt�1

ð1þ rÞt

I¼ A

I

Xn

t¼1

ð1þ dÞt�1

ð1þ rÞt

!ð26Þ

The a-cut and the vertexes of the ANPV can be representedas:

ABCR ¼ ½fABCR1ðaÞ; fABCR2

ðaÞ� ð27Þ

where fABCR1ðaÞ ¼ BCR1 þ ðBCR2 � BCR1Þa, fABCR2

ðaÞ¼ BCR3� ðBCR3 � BCR2Þa, and

BCR1 ¼A1

I3

1� 1þ d11þ r3

� �n

ðr3 � d1Þ;

BCR2 ¼A2

I2

1� 1þ d21þ r2

� �n

ðr2 � d2Þ;

BCR3 ¼A3

I1

1� 1þ d31þ r1

� �n

ðr1 � d3Þ

5 Case studies

5.1 Simple numerical example: comparisonof four mutually exclusive alternativesUsing Example 2 from Chiu and Park [7], the present valuesof four mutually exclusive alternatives, A1¼ (2350,2725,2850), A2¼ (2250,2650,2800), A3¼ (2325,2600,2900) andA4¼ (2200,2425,2725) are modeled as triangular fuzzynumbers. The membership functions of these alternativesare shown in Fig. 7. Since it is difficult to establish apreference for one of the four projects over the others on thebasis of their present values, their expected values arecalculated using several conventional dominance methodsand the Mellin Transform. The corresponding preferencerankings are listed in Table 2.

An inspection of Table 2 reveals that project A1 is rankedhighest by all the methods other than the method presentedby Chang, in which A3 has the largest expected value due toits wider range. Furthermore, it is observed that project A4

is ranked lowest by all the ranking methods. It can be seenthat the rankings of A2 and A3 vary from method tomethod as a result of the particular aspects emphasised byeach individual method. Although most of the methodsindicate a ranking of A34A2, it is noted that the methodput forward by Duducts and Prade indicates A24A3 withinsignificant variation.

As described in Bortolan and Degani [17], and in Yager[18], no unique optimal rankhing method exists for theconsistent ranking of triangular fuzzy numbers. Ultimately,it is the nature of the problem and the attitude of thedecision-maker that determines the most appropriateranking method. Nevertheless, it remains the intention ofthe current author to develop a tool for decision-making,which may be easily implemented and which is readilycomprehensible.

5.2 Cooling energy storage air-conditioning system for a target officebuildingA cooling energy storage (CES) air-conditioning systemfunctions by removing heat from a thermal storage mediumduring periods of low cooling demand, and then subse-quently releasing the stored cooling at a later time to meetan air-conditioning or process cooling load. A previousstudy has characterised the CES-system as a pumpingstorage operating resource [19]. Various forms of coolstorage media may be used, including chilled water, ice, or a

2200 2500 2900

1.0

1A

2A

3A

4A

µ (P

V)

PV

Fig. 7 Fuzzy present values of four mutually exclusive alternatives

Table 2: Preference of four mutually exclusive alternatives

Decision indexvalue of alternatives

Methods A1 A2 A3 A4 Preference

Chang[23] 660417 705833 749896 6431253�2�1�4

K&G[6] 2663 2588 2606 24441�3�2�4

Jain[24] 0.79 0.70 0.70 01�3 � 2�4

D&P[25] 1.00 0.85 0.81 0.551�2�3�4

C&P[7] 2914 2832 2868 26931�3�2�4

Mellin transform 2642 2567 2608 24501�3�2�4


eutectic salt phase-change material. Figure 8 below shows ablock diagram of a typical CES-system containing threedifferent coolant paths. Depending on the desired mode ofoperation, various combinations of these coolant paths canbe achieved through the appropriate adjustment of theCES-system control mode [20, 21]. Previous studies haveindicated that CES-systems can provide substantial operat-ing cost savings by generating cooling using cheaper off-peak energy, thereby reducing, or eliminating, on-peakdemand charges [20, 21, 22].

Using the models developed in [20], a partial load-levelling eutectic salt CES-system was designed andconstructed to meet the air-conditioning demands of atarget office building in Kaohsiung. The energy demands ofthis building are designed in accordance with the Energy-Saving Building Code and Regulations issued by theConstruction and Planning Administration of the InteriorMinistry, Taiwan. The intention is to design an environ-mentally friendly building, and hence it is referred to as the‘Green Building’. In terms of the demanded thermalperformance, the ENVOLPE (envelope load) of thebuilding was restricted to a maximum of 130kWh persquare metre of floor area per year. It is noted that thetarget building has a total floor area of 30000m2 spreadover its 22 floors.

To determine the appropriate type of CES-system for thisbuilding, it is first necessary to establish its dynamic coolingload profile by means of a precise yearly cooling loadsimulation program HASP (heating, air condition andsanitary engineering program) using the average weatheryearly (AWY) information for Kaohsiung, Taiwan. Figure9 shows the corresponding results for the typical dailycooling load of the target office in July 2000. The AWY-HASP simulation results indicate a 1050RT (refrigeratington) daily cooling peak demand, and a 9600RT-HR(refrigerating tonne–hour) maximum daily cooling energyrequirement. Using this cooling load profile and Taipower’soff-peak time periods, (4) and (5) from [20] can be used to

determine the optimal partial tank storage capacity ratioand the nominal chiller size.

The results indicate that the proposed CES-system shouldcomprise an optimal partial tank storage capacity of4600RT-HR (48% of daily cooling peak demand) andtwo chiller units with a total of 600RT cooling capacity.The storage tank, with a total volume of 782m3, wasconstructed in the basement of the building, together with118000 sets of eutectic salt encapsulated PE-containers andother circulation systems and assemblies. For comparisonpurposes, a conventional air-conditioning system (i.e. with-out a cooling energy storage function) with two 600RTchillers was also designed to satisfy the dynamic coolingload profile simulated by the AWY-HASP program.Analyses revealed that the peak demands of the targetoffice building with and without the CES-system were421kW and 790kW, respectively. Figures 10a and billustrate the typical annual energy consumptions of thetarget building for the cases of the conventional air-conditioning system and the CES-system, respectively.

Using the simulated cooling load profile of the targetbuilding and the Taiwan Power Company (Taipower) TOUtariff given in Table 3, the investment costs and profitabilityof the conventional air-conditioning system and the CES-system can be calculated. Table 4 provides the experts’opinions of the triangular fuzzy investments and annualoperating costs of the CES-system and the conventional air-conditioning system. The experts also assume that the fuzzyyearly interest rate r and the fuzzy inflation rate d can beexpressed as (5,6,7)% and (1,2,3)% in TFN form,respectively. Additionally, the plant life is assumed to be

A, B

chillers

storagetank

coolingspace

A

CB,C B

C

A,CA,B,C

pump

Fig. 8 Block diagram of the CES-system

5 10 15 200

200

400

600

800

1000

1200

time, h

cool

ing

dem

and,

r t

Fig. 9 Typical daily cooling load (AWY-HASP) of targetbuilding during July

1 2 3 4 5 6 7 8 9 10 11 120

50

100

150

200

montha

b

month

mon

th's

per

iodi

c en

ergy

co

nsum

ptio

n, M

Wh

off-peakshoulderpeak

1 2 3 4 5 6 7 8 9 10 11 120

50

100

150

200

Mon

th's

per

iodi

c en

ergy

co

nsum

ptio

n, M

Wh

off-peakshoulderpeak

Fig. 10 Typical annual energy consumption of conventional airconditioner and CES-systema Typical annual energy consumption conventional air conditionerb Typical annual energy consumption of CES-system


ten years. The triangular fuzzy APVC and AEUAC for theCES-system and the conventional air-conditioning systemare calculated from (18) and (20), respectively. Thecorresponding results are presented in Table 4. ThisTable also summarises the possible ranges of the APVCand AEUAC at a specific value of 80% confidence (i.e.a¼ 0.8).

From (16), it is clear that the present value of cost isproportional to the initial investment, annual cost andinflation rate, and is inversely proportional to the interestrate. Therefore, for the CES-system, the smallest possiblevalue of the APVC, i.e. $1329.6k, occurs at the smallestpossible values of the investment ($650k), annualcost ($93k) and inflation rate (1%), and at the largestpossible interest rate (7%). Similarly, the most promisingvalue of the APVC for the CES-system is determined tobe $1522.3k, which corresponds to an investment of $740k,an annual cost of $98k, an inflation rate of 2%, andan interest rate of 6%. It is noted that this fuzzyresult corresponds to the result generated by the conven-tional mathematical approach. Finally, the largest valueof the APVC for the CES-system is calculated to be$1711.0k, and occurs for an investment of $810k,an annual cost of $103k, an inflation rate of 3%, andan interest rate of 5%. The possibility or sensitivity of theCES-system can be analysed from the data presented inTable 4.

The means and standard deviations of the APVC for theCES-system and the conventional air-conditioning systemcan be obtained by substituting the vertexes of the fuzzyAPVC calculated from (18) into (14) and (15). Thecorresponding results are shown in Table 4. The resultsindicate that the proposed CES-system has a smaller APVCmean value. Consequently, the CES-system is the preferredchoice to meet the cooling requirements of the target officebuilding. It is noted that the fuzzy AEUAC model yields aconsistent result.

5.3 Co-generation of selected syntheticrubber corporationIn a second case study, the developed fuzzy decision modelsare used to evaluate a co-generation project for a selectedsynthetic rubber corporation in Kaohsiung, Taiwan. Thiscorporation has a wide spread ratio between steam andelectricity demands, in which the steam demand variesbetween 10 and 65 tones/h, while the power demand variesfrom 2000kW to 11000kW. Consequently, the electricity/steam (E/S) ratio lies between 95.03 and 246.75. In thestudied corporation, the deficient electricity requirementscan be supplied on a complementary basis from theTaipower grid, but the corporation generates its own steam.The corporation’s steam and electricity demands areselected at random in order to estimate the E/S ratio, andthe results are summarised in Table 5.

Two cogeneration alternatives are suggested by theconsultant company, Pacific Engineers & Constructors, awell known consultant in the field of power engineering inTaiwan, namely extracted condensing steam turbine gen-erator system (EC-TG) and backpressure steam turbinegenerator system (BP-TG). The EC-TG system has aflexible operating characteristic, which enables it to meetvarious steam and electricity demands, but involves a higherinitial investment and operating cost. By contrast, the BP-TG system requires less initial investment and has reducedfuel costs, but the electricity supplied follows the steamdemand, and any deficient electricity must be supplied bythe utility’s grid, which decreases the net benefit of thesystem. According to the proposal supplied by theconsultant company for this case, the cogeneration alter-natives require the following major items of apparatus:

� Alternative 1: Extracted condensing steam turbinegenerator system

a High pressure boiler: steam outlet: 95 tonne/h, 93kg/cm2-G,5051C

Table 3: Two-season, three-section TOU tariff for HV customers in Taipower

Season Demand charge($/kW-month)

Effective energy charge (cent/kWh)

Peak Shoulder Off-peak

Summer season (June to September) 6.086 8.743 5.257 2.000

Other season 4.543 - 5.086 1.857

Table 4: Least-cost study of conventional A/C and CES-system

Conventional A/C CES-system

Investment cost (490,560,620) (650,740,810)

Operating cost (140,145,150) (93,98,103)

PVC method APVC (1513.1,1717.5,1932.1) (1329.6,1522.3,1711.0)

mAPVC 1720.9 1520.9

sAPVC 85.5 77.9

APVC range (as m¼ 0.8) [1675.1,1758.7] [1482.6,1558.7]

EUAC method AEUAC (195.9,233.4,275.1) (172.2,206.8,243.6)

mAEUAC 234.8 207.5

sAEUAC 16.2 14.6

AEUAC range (as m¼0.8) [225.4,241.2] [199.5,213.8]

Note: Values in kilo US$


b Steam turbine: steam inlet: 90kg/cm2-G, 5001C

Process steam supplies: medium pressure: 65 tonne/h,12.2kg/cm2-G

low pressure: 5 tonne/h, 3.5kg/cm2-G

Exhaust backpressure: 3.5 inch Hg-A

c Condenser: cooling water inlet: 1500m3/h, 341C, out-let:441C

d Generator: rated output 12500kW

� Alternative 2: Backpressure steam turbine generatorsystem

a High-pressure boiler: 85 tonne/h, 93kg/cm2-G, 5051C

b Steam turbine: steam inlet: 90kg/cm2-G, 5001C

Process steam supplies: medium pressure: 65 tonne/h,12.2kg/cm2-G

low pressure: 5 tonne/hour/hour, 3.5kg/cm2-G

c Generator: rated output 12500kW

Table 6 gives the corresponding triangular fuzzy invest-ments, operating revenues and operating costs of the twoalternatives as determined by consultant experts’ opinions.Experts assume the fuzzy yearly interest rate, r, and the

fuzzy inflation rate, d, to be (5,6,7)% and (1,2,3)% in TFNform, respectively. Additionally, the plant life is assumed tobe 15 years.

Table 7 indicates the triangular fuzzy ANPV, ABCR andAPBY results for the two mutually exclusive alternatives,together with their respective means and standard devia-tions. The table also presents the possible ranges of thesemodels at a specific value of 80% confidence (i.e. a¼ 0.8).The results indicate that the proposed EC-TG system has asmaller PBY mean value, and larger NPV and BCR meanvalues. Consequently, the EC-TG system is the preferredchoice in this particular case.

6 Conclusions

This paper has derived fuzzy profitability models, whichenable the DSM participants to perform a financialevaluation of DSM alternatives. The least-cost solutioncan be determined from either the present value of cost(PVC) model or from the equivalent uniform annual cost(EUAC) model, while cost–benefit analysis is performedusing the net present value (NPV), payback year (PBY) andbenefit/cost ratio (BCR). In an approach consistent withthat presented in previous studies, the financial parametersare represented in the form of approximated triangularfuzzy numbers, and the Mellin transform is then applied toderive the means and variances of these numbers in order toperform fuzzy ranking within the decision-making process.

The performance of the proposed fuzzy models has beenverified by simulating a simple numerical example, and byapplying the models to two practical DSM proposals inTaiwan. The results have shown that the CES-system is aneffective means of clipping peak demand and shifting theparticipant’s electricity consumption from the peak to off-peak period. This generates a significant improvement in thesystem efficiency of Taipower. The simulation has alsoshown that the CES-system provides economic benefits tothe DSM participant. The proposed fuzzy models havebeen applied to the evaluation of an extracted condensingsteam turbine generator system and a back-pressure steam

Table 6: Revenues and costs of cogeneration alternatives

EC-TG system BP-TG system

Capital investment (13140,13710,14280) (9140,10000,10850)

Annual revenue (6000,7140,8000) (4570,5570,6280)

Annual cost (3710,4280,4850) (3140,3570,4000)

Annual profit (2290,2860,3150) (1430,2000,2280)

Note: unit in kilo US$

Table 5: Electricity/steam ratio analyses of the selected company

Electricity/steam ratio (E/S)

Steamdemand(tonne)

No. of data Weighting, % Max. value Min. value Averagevalue

Standarddeviation

Below10.0

0 0

10.1–15.0 12 1.15 412.91 136.67 324.03 73.39

15.1–20.0 36 3.46 373.60 109.90 231.17 45.23

20.1–25.0 51 4.90 306.32 117.20 216.00 25.20

25.1–30.0 68 6.53 278.37 183.14 225.44 21.69

30.1–35.0 73 7.01 270.13 96.29 208.05 24.41

35.1–40.0 160 15.37 246.74 95.03 192.36 24.92

40.1–45.0 188 18.06 239.41 115.84 182.96 23.72

45.1–50.0 218 20.94 215.27 116.04 179.13 17.44

50.1–55.0 141 13.54 211.33 125.20 173.04 12.71

55.1–60.0 68 8.26 188.36 144.52 164.62 9.46

60.1–65.0 8 0.77 165.72 137.05 150.12 7.82

1041 100 412.91 95.03 189.94(weightedaverage)

32.71


turbine generator in a synthetic rubber corporation with awide spread ratio between steam and electricity demands.The results generated using the proposed fuzzy models havebeen shown to be consistent with those yielded by theconventional crisp models. The results of this present studyhave confirmed that the proposed methods serve as readilyimplemented possibility analysis tools for use in the arena offinancial uncertain decision-making.

7 References

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4 Dubois, D., and Prade, H.: ‘Fuzzy Sets and Systems: Theory andApplications’, (Academic Press, New York, USA, 1980)

5 Kaufmann, A., and Gupta, M.M.: ‘Introduction to fuzzy arithmetic:theory and applications’, (Van Nostrand Reinhold, New York, USA,1985)

6 Kaufmann, A., and Gupta, M.M.: ‘Fuzzy mathematical models inengineering and management science’, (Elsevier Science Publishers,BV, 1988)

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8 Chen, S.J., and Hwang, C.L.: ‘Fuzzy multiple attribute decisionmaking methods and applications’, Lect. Notes Econ. Math. Syst.,(Springer, New York, 1992)

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10 Debnath, L.: ‘Integral transforms and their application’, (CRC Press,New York, USA, 1995)

11 Buckley, J.J.: ‘The fuzzy mathematics of finance’, Fuzzy Sets Syst.,1987, 21, pp. 257–273

12 Ward, T.L.: ‘Discounted fuzzy cash flow analysis’. Proc. 1985 FallIndustrial Eng, Conf., Institute of Industrial Engineers, 1985, pp. 476–481

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8 Appendix

From (16), the membership function of the fuzzy PVC canbe represented as:

mPVCðxÞ ¼ðPVC1; fPVC1ðaÞ=PVC2;

PVC2=fPVC2ðaÞ; PVC3Þ

ð28Þ

where fPVCiðaÞ ¼fIiðaÞ þ fAiðaÞ1� 1þ fdiðaÞ

1þ frð3�iÞ ðaÞ

!n

frð3�iÞ ðaÞ � fdiðaÞ;

i ¼ 1; 2and fIiðaÞ; fAiðaÞ; fdiðaÞ; friðaÞ; i ¼ 1; 2 are the member-ship functions of the initial investment, I, the first year’scash flow, A, the inflation rate, d, and the interest rate r,respectively. If all the fuzzy economic numbers, i.e. A, d, rand I, are represented in the form of TFNs, i.e. A¼ (A1, A2,A3), d¼ (d1, d2, d3)%, r¼ (r1, r2, r3)% and I¼ (I1, I2, I3),respectively, then the a-cut and the vertexes of the PVC canbe represented as:

PVC ¼ ½fPVC1ðaÞ; fPVC2

ðaÞ�¼ ðPVC1; PVC2; PVC3Þ ð29Þ

fPVC1ðaÞ¼ I1 þ ðI2 � I1Það Þ þ A1 þ ðA2 � A1Það Þ

1� 1þ d1 þ ðd2 � d1Þa1þ r3 � ðr3 � r2Þa

� �n

ðr3 � d1Þ � ðr3 � r2 þ d2 � d1Þa

ð30Þ

Table 7: Cost-benefit analyses of two cogeneration alternatives

EC-TG BP-TG

NPV method ANPV (7826.6,17636.7,26328.2) (2954.5,11920.8,19427.5)

mANPV 17263.8 11434.3

sANPV 3778.9 3366.9

ANPV range (as a¼0.8) [15438.7,19235.8] 9972.1,13305.0]

BCR method ABCR (1.55,2.29,3.00) (1.27,2.19,3.13)

mABCR 2.28 2.20

sABCR 0.30 0.38

ABCR range (as a¼0.8) [2.12,2.29] [2.03,2.35]

PBY method APBY(years) (4.53,5.53,8.12) (4.35,5.80,10.53)

mAPBY 6.06 6.89

sAPBY 1.14 1.32

APBY range (as a¼0.8) [5.31,5.90] [5.45,6.40]

Note: unit in kilo US$


fPVC2ðaÞ¼ I3 � ðI3 � I2Það Þ þ A3 � ðA3 � A2Það Þ

1� 1þ d3 � ðd3 � d2Þa1þ r1 þ ðr2 � r1Þa

� �n

ðr1 � d3Þ þ ðr2 � r1 þ d3 � d2Þa

ð31Þ

PVC1 ¼ fPVC1ð0Þ ¼ I1 þ A1

1� 1þ d11þ r3

� �n

r3 � d1ð32Þ

PVC2 ¼ fPVC1ð1Þ ¼ fPVC2

ð1Þ

¼ I2 þ A2

1� 1þ d21þ r2

� �n

r2 � d2ð33Þ

PVC3 ¼ fPVC2ð0Þ ¼ I3 þ A3

1� 1þ d31þ r1

� �n

r1 � d3ð34Þ

By linearizing (30) and (31), the a-cut and the vertexes of theAPVC can be represented as (18). The vertexes of theAPVC are the same as the vertexes of the exact PVC.


Documents

Fuzzy financial analyses of demand-side management alternatives