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An integrated artificial neural network and fuzzy clustering algorithm for performance assessment of decision making units

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An Integrated Artiﬁcial Neural Network Fuzzy C-Means-NormalizationAlgorithm for performance assessment of decision-making units: The casesof auto industry and power plant
q
A. Azadeh
a,
⇑
, M. Saberi
b
, M. Anvari
c
a
Department of Industrial Engineering, College of Engineering, University of Tehran, P.O. Box 11365-4563, Iran
b
Member of Young Researcher Club, Islamic Azad University, Tafresh Branch, P.O. Box 19585-466, Tafresh , Iran
c
Department of Industrial Engineering, Iran University of Science and Technology, P.C. 16844, Narmak, Tehran, Iran
a r t i c l e i n f o
Article history:
Received 13 April 2009Receivedinrevisedform16November 2010Accepted 21 November 2010Available online 1 December 2010
Keywords:
Performance assessmentArtiﬁcial Neural Network (ANN)Fuzzy C-MeansNormalizationAuto industryPower plant
a b s t r a c t
Efﬁciency frontier analysis has been an important approach of evaluating ﬁrms’ performance in privateand public sectors. There have been many efﬁciency frontier analysis methods reported in the literature.However, the assumptions made for each of these methods are restrictive. Each of these methodologieshas its strength as well as major limitations. This study proposes two non-parametric efﬁciency frontieranalysis sub-algorithms based on (1) Artiﬁcial Neural Network (ANN) technique and (2) ANN and FuzzyC-Means for measuring efﬁciency as a complementary tool for the common techniques of the efﬁciencystudies in the previous studies. Normal probability plot is used to ﬁnd the outliers and select from thesetwomethods. Theproposedcomputational algorithms areabletoﬁndastochastic frontier basedonasetof input–output observational data and do not require explicit assumptions about the functional struc-tureofthestochasticfrontier. Inthesealgorithms, forcalculatingtheefﬁciencyscores, asimilarapproachto econometric methods has been used. Moreover, the effect of the return to scale of decision-makingunit (DMU) on its efﬁciency is included and the unit used for the correction is selected by notice of itsscale (under constant return to scale assumption). Also in the second algorithm, for increasing DMUs’homogeneousness, Fuzzy C-Means method is used to cluster DMUs. Two examples using real data arepresented for illustrative purposes. First example which deals with power generation sector shows thesuperiority of Algorithm 2 while the second example dealing auto industries of various developed coun-tries shows the superiority of Algorithm1. Overall, we ﬁnd that the proposed integrated algorithmbasedon ANN, Fuzzy C-Means and Normalization approach provides more robust results and identiﬁes moreefﬁcient units than the conventional methods since better performance patterns are explored.
2010 Elsevier Ltd. All rights reserved.
1. Introduction
Theappropriateuseoffewresourceswiththeavailabletechnol-ogyisreferredtoastechnicalefﬁciency.Efﬁciencyfrontieranalysishas been an important approach of evaluating ﬁrms’ performancein private and public sectors. As alternatives to determine the efﬁ-ciency boundaries, the international experience reports a signiﬁ-cant number of methodologies with different approaches andmethods to characterize such efﬁciency ( Jamasb & Pollitt, 2001).In rough terms, these methodologies can be classiﬁed accordingto how the frontier is estimated. There are two competing para-digms on efﬁciency analysis. Parametric and non-parametric ap-proaches are widely-used in the efﬁciency measurement. Theﬁrst include the estimation of both deterministic and stochasticfrontier functions (SFF) which is based on the econometric regres-sion theory and has been widely accepted in the econometricsﬁeld. The latter include DEA and Free Disposal Hull (FDH) whichare based on mathematical programming approaches. Each of these two methodologies has its strength as well as major limita-tions. In all of these methodologies, the frontier is deﬁned by themost efﬁcient DMU of the sample. Mathematically, the frontiermethods are introduced as a high-reliability analysis tool and havebeen largely used for studies in the electrical ﬁeld (Pollit, 1995;Sanhueza, Rudnick, & Lagunas, 2004).
1.1. Problem formulation
There have been several efﬁciency frontier analysis methodsreported in the literature (for instance only about power plantsCook & Green, 2005; Golany, Yaakov, & Rybak, 1994; Goto &Tsutsui,1997;Knittel,2002;Lam&Shiu,2001;Olatfubi&Dismukes,
2000; Park & Lesourd, 2000; Sanhueza et al., 2004; Sueyoshi &
0360-8352/$ - see front matter
2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2010.11.016
q
This manuscript was processed by Area Editor Ibrahim H. Osman.
⇑
Corresponding author. Tel.: +98 21 88967810; fax: +98 21 66461680.
E-mail addresses:
aazadeh@ut.ac.ir, ali@azadeh.com (A. Azadeh).
Computers & Industrial Engineering 60 (2011) 328–340
Contents lists available at ScienceDirect
Computers & Industrial Engineering
journal homepage: www.elsevier.com/locate/caie
Goto, 2001). But, the assumptions made for each of these methodsare restrictive. Conﬂicting conclusions of efﬁciency are often re-sulted by using the different methods due to the unsuitability of the assumptions. Their frontier sensitive to outliers and will bedeterministic. The non-parametric approach makes no assumptionabout the functional form of the frontier. Instead, it speciﬁes cer-tain assumptions about the underlying technology that in combi-nation with the data set allow the construction of the productionset. For instance, the DEA frontier is very sensitive to the presenceoftheoutliersandstatisticalnoisewhichindicatesthatthefrontierderived from DEA analysis may be warped if the data are contam-inated by statistical noise (Bauer, 1990). On the other hand, DEAcan hardly be used to predict the performance of other decision-making units. In summary, it should be mentioned that previousstudies have the following drawbacks:(1) Have assumptions about the functional form.(2) Are sensitive to outliers and statistical noise.(3) Do not have forecasting capability.(4) Do not consider a method for increasing the homogenousstatus of DMUs.
1.2. Objective
The main objective of the present study is to propose an Inte-grated Artiﬁcial Neural Network Fuzzy C-Means algorithm thathas the following features:(1) Contributes to the use of neuro-fuzzy models in the area of efﬁciency measurement.(2) Handles environmental complexity.(3) Requires no assumption about the functional form.(4) Handles outliers and statistical noise.(5) Forecasts the future behaviour.(6) Increases the homogenous status of DMUs.One of the main objectives of this paper is to contribute to theuse of neural networks in the efﬁciency measurement. To this end,for estimating production (cost) function, ANN method is appliedand for calculating the efﬁciency scores, a similar approach toeconometric methods is used and two distinct algorithms are pro-posed.UsingANNguidesthealgorithmtoreachtheﬁrstﬁveobjec-tives. Also, Fuzzy C-Means is used to increase the homogenousstatus of DMUs as stated in the sixth objective. Previous studiesbyAzadeh, Ghaderi, Anvari, andSaberi (2006a, 2006c)andAzadeh,Ghaderi, Anvari, Saberi, and Izadbakhsh (2006b) showthe applica-bility approach of ANN and ANN-Fuzzy C-Means in selectedpowerplants. However, this study presents an algorithm which is notonlybasedonANN-FuzzyC-MeansbutalsoitisbasedANNandse-lects either ANN-Fuzzy C-Means or ANN algorithm based on thestructure of data and existence of outliers by using the normalprobability plot. In addition, ANN-Fuzzy C-Means which is pro-posed in previous studies (Azadeh, Ghaderi, Anvari, & Saberi,2006a, 2006c; Azadeh, Ghaderi, Anvari, Saberi, & Izadbakhsh,
2006b) is expanded for output oriented models.The paper is organized as follows. Section 2 provides an intro-duction to ANNs in efﬁciency analysis, where neural networksform a promising analysis tool together with known econometricmodels and non-parametric methods. This section concludes witha review of recent published papers about ANN and efﬁciency.Section 3 is dedicated to ANN in efﬁciency analysis and two algo-rithms and a selection mechanism are proposed in this sectionfor assessing efﬁciency of DMUs. Two empirical illustrations formeasuring performance of thermal power plants and auto indus-tries are carried out in Section 4. The ﬁnal section of the paper of-fers conclusions and suggests areas for the future research.
2. Ann and efﬁciency
ArtiﬁcialNeuralNetworks(ANNs)areapromisingalternativetoeconometric models and they are information processing para-digms that are inspired by the way biological nervous systems,such as the brain, process information. ANNs are conﬁgured forspeciﬁc applications, such as pattern recognition, function approx-imator, through learning process. ANNs are made up simple pro-cessing units which are linked by weighted connections to formstructures that are able to learn relationships between sets of vari-ables (Appendix A reviews ANNs).In this study application of ANNs in efﬁciency analysis is dis-cussed.Withintheefﬁciencyliterature,fewapplicationshavebeenmade in this ﬁeld. Commonly, neural network technique is usedasacomplementarytool for parametricandnon-parametricmethodssuch as DEA, to ﬁt production functions and measure efﬁciencyunder non-linear contexts. In fact, applying ANNs can reduce therestrictive assumptions each of these methods. This heuristicmethod can be useful for non-linear process that has an unknownfunctional form(Enders, 2004) andTherehasbeenavastliteratureabout ANNs, basically in the empirical ﬁeld, showed that ANNscomparabilityorsuperioritytoconventionalmethodsforestimatingfunctions (Tang, Almeida, & Fishwick, 1991; Tang and Fishwick,1993; Chiang, Urban, & Baldridge, 1996; Hill, O’Connor, & Remus,1996; Indro, Jiang, Patuwo, & Zhang, 1999; Jhee & Lee, 1993;Kohzadi, Boyd, Kermanshahi, & Kaastra, 1996; Stern, 1996; BrianHwarng, 2001; Azadeh, Ghaderi, & Sohrabkhani, 2006d; Azadeh,Ghaderi,Tarverdian,&Saberi,2006e,2006f ).Sointhisstudy,ANNsisselectedforestimatingproductionfunctionandthenperformanceevaluation.For instance consider DEA approach, a basic principle to useANNs is for generalizing efﬁciency frontier functions which con-cavity is an important characteristic of them and they may be ap-plied to frontier analysis (Wang, 2003). Moreover, the efﬁciency
predictionpowerof ANNsis uniqueandtheﬂexibilityof it tosolvecomplex problems, where the main information lies implicitly inthe data, is very applicable (Wu, Yang, & Liang, 2006).
The idea of combination of neural networks and DEA for classi-ﬁcationand/or predictionwas ﬁrst introducedbyAthanassopoulosand Curram (1996). They treated DEA as a preprocessing method-ology to screen training cases in a study. Their application is bankwithmulti-output:4inputs,3outputs.Afterselectingsamples,theANNs are then trained as tools to learn a non-linear forecastingmodel. They assume that inefﬁciency distributions are semi-nor-mal and exponential and conclude that DEA is superior to ANNfor measurement purpose. Their study indicates that ANN resultsare more similar with the constant returns to scale and less withthe variable returns to scale results. The latter, is a consequenceof the implicit assumption of constant returns to scale adoptedby the ANN models.Costa and Markellos (1997) analysed the London undergroundefﬁciencywithtimeseriesdatafor1970–1994,wherethereare2in-puts–ﬂeetandworkers–and1output–kms.TheyexplainhowtheANNsresultsaresimilartoCOLSandDEA.Theyproposedtwoproce-dures: (a) similar way to COLS after neural training and (b) by anoversizednetworkuntilsomesignaltonoiseratioisreached.Then,inefﬁciency is determined as observation-frontier distance. How-ever, ANNs offer advantages in the decision making, the impact of constantversusvariablereturnstoscaleorcongestionareas(Costa& Markellos, 1997). Santin and Valin (2000) study on education
efﬁciency by a two-level model: student–production function isestimated by ANNs and school. They infer that ANN is superior toeconometric approach at frontier estimation. Pendharkar andRodger (2003) used DEA as a data screening approach to create asub sample training data set that is ‘approximately’ monotonic,
A. Azadeh et al./Computers & Industrial Engineering 60 (2011) 328–340
329
which is a key property assumed in certain forecasting problems.Their results indicate that the predictive power of an ANN trainedonthe‘efﬁcient’trainingdatasubsetisstrongerthanthepredictiveperformanceofanANNtrainedonthe‘inefﬁcient’trainingdatasub-set. Santin, Delgado, and Valino (2004) used a neural network ap-proach for a simulated non-linear production function andcompared its performance with conventional alternatives such asstochasticfrontierandDEAindifferentobservationsandnoisesce-narios.TheresultssuggestedthatANNsareapromisingalternativetoconventionalapproaches,toﬁtproductionfunctionsandmeasureefﬁciency under non-linear contexts. Wu et al. (2006) presented aDEA–NN
1
study for performance assessment of branches of a largeCanadian bank. The results are operable to the normal DEA resultsonthewhole.TheyconcludedthattheDEA–NNapproachproducesa more robust frontier and identiﬁes more efﬁcient units becausebetter performance patterns are explored. Furthermore, for worseperformers,itprovidestheguidanceonhowtoimprovetheirperfor-mancetodifferentefﬁciencyratings.Ultimately,theyconcludedtheneural network approach requires no assumptions about the pro-duction function(the major drawbackof the parametric approach)and it is highly ﬂexible. ANN has been viewed as a good tool toapproximate numerous non-parametric and non-linear problems.Therefore, the proposed algorithm estimates more robust resultsand more efﬁcient units than the conventional approach becausebetterperformancepatternsareexplored.
2.1. Fuzzy C-Means
In Fuzzy C-Means clustering, each data point belongs to a clus-ter with membership value that indicates the grade of member-ship. Fuzzy C-Means was proposed as an improvement ontraditional clusteringmethodsbyBezdek(1981). UsingFuzzylogic
in the proposed clustering method improves its ﬂexibility and istherefore ideal for non-linear situations. Overlapping as one of important problemin clustering is also decreased by the proposedFuzzy C-Means approach.
3. Methodology
TwodistinctANNalgorithmsareproposedtomeasuretheunits’efﬁciency in current period. These algorithms can estimate efﬁ-ciency by considering input (output) oriented by ﬁnding produc-tion (cost) function by using ANN approach, same as econometricmethods. Also for simplicity, we consider one output (input), butit is easy to extend to various outputs (inputs).
3.1. The proposed algorithm
1. Determination of input (
s
) and output (P) variables under inputoriented assumption (input (
C
) and output (
s
) variables underoutput oriented assumption) of the model.2. Collectdataset
S
inallavailablepreviousperiodswhichdescribestheinput–outputrelationshipforDMUs.Assumethatthereare
n
DMUstobeevaluated.Notethatthecurrentperioddata(
S
c
:testdata
2
)doesnotbelongto
S
.Then,obtainthepreprocesseddataset(
S
and
S
c
) after the data are scaled between 0 and 1.
3. Divide
S
into two subsets: training (
S
1
) and valid (
S
2
) data. Theﬁrst subset is the training set, which is used for computing thegradientandupdatingthenetworkweightsandbiases.Thesec-ond subset is the validation set. The error on the validation setis monitored during the training process. The validation and(also training errors) will normally decrease during the initialphase of training. However, when the network begins to overﬁt the data, the error on the validation set will typically begintorise. Whenthevalidationerrorincreasesforaspeciﬁednum-ber of iterations, the training is stopped, and the weights andbiasesatlowestscalesarereturned.Thevalidationdataistakenout from the training data and it should be representativeacrosstherangeofoutcomes.
3
It isnecessarytostrikeabalancebetween the size of training and validation data sets. Whenlarge amounts of data are available, the selection of validationdata can be done using a simple random choice. But accordingto our problem, extrapolation ability of ANN should be calcu-lated. Therefore, the data for validation is chosen for the periodwhich is closer to
S
c
.4. Use ANN method to estimate relation between input (
s
) andoutput (
s
). For this reason select architecture and trainingparameters. All networks used in this study have a single hid-den layer because the single hidden layer network is found tobe sufﬁcient to model any function (Cybenko, 1989; Patuwo,Hu, & Hung, 1993). To ﬁnd the appropriate number of hiddennodes, following steps are performed for networks with oneto
q
nodes intheir hiddenlayer. Whenthe valueof
q
is optionaland should be changed if after following next steps, the goalerror has not met.
4
Train the model using the training data (
S
). In this studyLevenberg–Marquardt (LM) training algorithm is used.
5
Evaluate the model using the test data (
S
c
) and obtainingMAPE
6
error.
Then the model which has the lowest error is selected forestimating production function.5. Run ANN
⁄
for
S
c
.6. Apply normal probability plot for checking existence of out-liers. For this, at ﬁrst for eachindex (output (s) andinput(s))(
xk
,
k
=1,
. . .
,
m
) normalize the data:
^
x
ik
¼ ð
x
ik
l
k
Þ
=
r
k
i
¼
1
;
. . .
;
n
where
l
k
is mean and
r
k
is standard deviation for data of in-dex
x
k
in
S
c
. Then use normal probability plot for {
x
ik
}
n
⁄
m
, if
p
-value<0.05andyoucanseesomepointsfarfromthelinegotostep 10 Otherwise, do steps 7–9.7. Calculate the error between the real output (
P
real(i)
for inputoriented model and
C
real(i)
for output oriented model) andANN model output (
P
ANN
⁄
(i)
for input oriented model and
C
ANN
⁄
(i)
for output oriented model) in the period which youwant to assess the efﬁciency of its DMUs (
S
c
):
E
i
¼
P
real
ð
i
Þ
P
ANN
ð
i
Þ
i
¼
1
;
. . .
;
n
for input oriented model
E
i
¼
C
ANN
ð
i
Þ
C
real
ð
i
Þ
i
¼
1
;
. . .
;
n
for output oriented model
ð
1
Þ
1
To implement the DEA–ANN model, they deﬁned an algorithm. In this algorithmafter collecting a data set, CCR method (Charnes, Cooper, & Rhodes, 1978) is used tocalculate efﬁciency score of DMUs. The preprocessed data set is obtained and isgrouped into four categories based on the efﬁciency scores and the neural network istrained with some groups of data subset until the pre-speciﬁed epochs or accuracy issatisﬁed. Then the trained neural network model is applied to calculate efﬁciencyscores of all DMUs and post process the calculated efﬁciency scores by regressanalysis between DEA–NN results and CCR DEA results.
2
The test set error is not used during the training, but it is used to comparedifferent ANN models.
3
In our problem, each period have all range of outcomes. Therefore, all of the dataof this period can be selected for test or the data of this period can be sorted by thevalue of the output variable, partitioned and one validation data point is chosen atrandom from each partition. In this way the stratiﬁcation tries to ensure thatvalidation data is chosen across the range of outcomes.
4
In this study the value of the desired minimum error has been deﬁned between2% and 4% (96–98% conﬁdence) and the value of q has been deﬁned 20. The error isestimated by Mean Absolute Percentage Error (MAPE).
5
LM algorithm is selected by noting of reaching the goal error in appropriate timeand using of this algorithm in previous similar studies.
6
Mean absolute percentage error
MAPE
¼
1
N
P
N i
¼
1Actualvalue
i
Setpointvalue
i
Setpointvalue
i
ð
N
:
the number of rows
Þ
.
330
A. Azadeh et al./Computers & Industrial Engineering 60 (2011) 328–340
8. Shift frontier function fromneural network for obtaining theeffect of the largest positive error which is one of the uniquefeature of this algorithm:
E
0
i
¼
E
i
=
P
ð
ANN
Þ
i
i
¼
1
;
. . .
;
n
for input oriented model
E
0
i
¼
E
i
=
C
ð
ANN
Þ
i
i
¼
1
;
. . .
;
n
for output oriented model
ð
2
Þ
This option consists of not considering the largest error, butcalculatesbynotingtheDMUscaleConstantReturnstoScale(CRS)). To this end ﬁnd the largest
E
0
i
indicate the DMU withthebestperformance.SupposethatDMU
k
havetheLargest
E
0
i
and we have:
E
0
k
¼
max
ð
E
0
i
Þ ð
3
Þ
Therefore, the value of the shift for each of the DMUs is dif-ferent and is calculated by:
Sh
i
¼
E
k
P
ð
ANN
Þ
i
=
P
ð
ANN
Þ
k
i
¼
1
;
...
;
n
forinputorientedmodel
ð
4
Þ
Sh
i
¼
E
k
C
ð
ANN
Þ
i
=
C
ð
ANN
Þ
k
i
¼
1
;
...
;
n
foroutputorientedmodel
In this approach in spite of the previous studies (Athanasso-poulos & Curram, 1996 called this measure ‘‘standardizedefﬁciency’’) the effect of the scale of DMU on its efﬁciencyis considered and the unit used for the correction is selectedbynoticeof itsscale(CRS)(Costa&Markellos, 1997–Delgado,
2005).9. Calculate efﬁciency scores: The efﬁciency scores take valuesbetween0and1. Thismaximumscoreisassignedtotheunitused for the correction.
F
i
¼
P
i
=
ð
P
ð
ANN
Þ
i
þ
Sh
i
Þ
i
¼
1
;
...
;
n
forinputorientedmodel
ð
5
Þ
F
i
¼ð
C
ð
ANN
Þ
i
Sh
i
Þ
=
C
i
i
¼
1
;
...
;
n
foroutputorientedmodel
10. Cluster DMUs by Fuzzy C-Means method (
x
cluster areobtained after running Fuzzy C-Means method which isdeveloped by Dunn and improved by Bezdek) and for eachcluster do steps 11–14 (Bezdek, 1981; Dunn, 1973;Windham, 1981).11. Calculate weigh of DMU
i
:
W
i
V
i
¼
P
i
=
A
v
e
ð
P
1
;
. . .
;
P
i
1
;
P
i
þ
1
;
. . .
;
P
cj
Þ
i
¼
1
;
. . .
;
n
for input oriented model
ð
6
Þ
V
i
¼
C
i
=
A
v
e
ð
C
1
;
. . .
;
C
i
1
;
C
i
þ
1
;
. . .
;
C
cj
Þ
i
¼
1
;
. . .
;
n
for output oriented model
W
i
¼
V
i
=
Sum
ð
V
i
Þ
where
c
j
(1
6
j
6
x
) is the number of DMUs which belongs to
j
th cluster of
x
clusters.12. Calculate the error between the real output (
P
real(i)
) and ANNmodel output (
P
ANN
⁄
(i)
) in the period which the efﬁciency isto be assessed for its DMU (
S
c
):
E
i
¼
P
real
ð
i
Þ
P
ANN
ð
i
Þ
i
¼
1
;
. . .
;
n
for input oriented model
E
i
¼
C
ANN
ð
i
Þ
C
real
ð
i
Þ
i
¼
1
;
. . .
;
n
for output oriented model
ð
7
Þ
13. Shift frontier function fromneural network for obtaining theeffect of the largest positive error which is one of the uniquefeatures of this algorithm:
E
0
i
¼
E
i
=
W
i
i
¼
1
;
. . .
;
n
ð
8
Þ
This option consists of not considering the largest error, butcalculates by noting the DMU scale (Constant Returns to Scale(CRS)). To this end ﬁnd:The largest
E
0
i
which indicates the DMU with best perfor-mance. Suppose that DMUk have the Largest
E
0
i
and we have:
E
0
k
¼
max
ð
E
0
i
Þ ð
9
Þ
Thus,thevalueoftheshiftforeachoftheDMUsisdifferentandis calculated by:
Sh
i
¼
E
0
k
W
i
=
W
k
i
¼
1
;
. . .
;
n
ð
10
Þ
In this approach in spite of the previous studies (Athanasso-poulos &Curram, 1996calledthis measure‘‘standardizedefﬁ-ciency’’) the effect of the scale of DMUs on its efﬁciency isconsidered and the unit used for the correction is selectedby notice of its scale (CRS).14. Calculate efﬁciency scores. The efﬁciency scores take valuesbetween0and1.Thismaximumscoreisassignedtotheunitused for the correction in each cluster:
F
i
¼
P
i
=
ð
P
ð
ANN
Þ
i
þ
Sh
i
Þ
i
¼
1
;
. . .
;
n
for input oriented model
ð
11
Þ
F
i
¼ ð
C
ð
ANN
Þ
i
Sh
i
Þ
=
C
i
i
¼
1
;
. . .
;
n
for output oriented model
ð
12
Þ
15. Calculate unique efﬁciency score for all DMUs. It should benoted that some units may belong to two or more clustersand their efﬁciency scores in the cluster with large scaleunits are less than the cluster with smaller scale units. Forcalculating a unique efﬁciency score for theses units bymeans of proposed algorithm, the degree of membership ineach of the clusters is used.
F
i
¼ ð
R
D
ij
F
ij
Þ
=
R
D
ij
1
P
j
P
x
if DMU
i
belong to more than one clusters
ð
13
Þ
where
D
ij
is the DMU
i
’s degree of membership in
j
th clusterand
F
ij
is the DMU
i
’s efﬁciency score in
j
th cluster.This point is important that the efﬁcient unit in cluster A is notmoreefﬁcientthantheefﬁcientunitinclusterBalthoughitmaybebetterthanit.Itisnotedthat,insomecases,perhapsinaparticularcluster, the obtained error
E
i
for all DMUs is negative. In this situ-ation, by notice of the proposed algorithm, frontier function fromneural network is shifted to lower level of production. In suchcases, the best unit is the DMU that has the lowest loss with re-spect to its scale.As it can be seen the proposed algorithm is more complicatedthan that of traditional DEA approach. There are however severaladvantages for using this approach. There have been several efﬁ-ciency frontier analysis methods reported in the literature. But,the assumptions made for each of these methods are restrictive.The non-parametric approach makes no assumption about thefunctional formof thefrontier. Instead, itspeciﬁescertainassump-tions that allow the construction of the production set. For in-stance, DEA frontier is very sensitive to the presence of theoutliers and statistical noise which indicates that the frontier de-rived from DEA analysis may be warped if the data are contami-nated by statistical noise. On the other hand, DEA can hardly beused to predict the performance of other decision-making units.In fact, applying ANNs can reduce the restrictive assumptions of each of these methods. Moreover, the efﬁciency prediction powerof ANN is unique and its ﬂexibility mechanism facilitates decisionmakers to solve complex problems, where the main informationlies implicitly in the data.
4. Case studies
Theproposedmethodisappliedtodatasetsoftwoactualcases:steam power generation and auto industry. Implementation of Algorithms 1 and2 for steampower generations andautoindustryareillustratedstepbystepinSections4.1and4.2,respectively.Theresults of running algorithms for the two actual case studies are
A. Azadeh et al./Computers & Industrial Engineering 60 (2011) 328–340
331
also discussed in each section. Section 4.3 presents discussions onthe superiority and robustness of the proposed algorithm.
4.1. Performance assessment of steam power generations
The performance and efﬁciency of power generation industry isof great importance to researchers and experts considering itscomplexity and particular requirements. This study deals with aninvestigation into technical efﬁciency of the Iranian generationelectricity industry. This study presents an Artiﬁcial NeuralNetwork approach for performance evaluating of steam powerplants by noting their important role in electricity generationand that the DMUs essentially perform the same tasks and arehomogeneous. In fact, this selection ensures that plants in thesample constitute a homogenous technology, thus forming a suit-able sample for applying the model. in this section the proposedmethodology is used to efﬁciency measurement for evaluating 19steam power plants in Iran for 2004.In these two algorithms it is assumed that the model is inputoriented because of the selected application which DMUs (powerplants) have particular orders to ﬁll (e.g. electricity generation)and hence the input quantities appear to be the primary decisionvariables.
4.1.1. Running algorithm
Step 1: Asshownbyseveralauthors,theproductionfunctionforconventional thermal steam-electric production may be describedconveniently withinanengineeringframework. Inthis framework,pertinent inputs are the fuel quantity consumed and installedpower,whichisthemaximumnominalpowertheplantsareinitiallydesignedfor. Ontheotherhand, laborinputscontributetoproduc-tion through control and maintenance services, which also requiresomecapital.Theoutputiselectricalenergyproduction.Butbynot-ingstudiesabout efﬁciencymeasurementof thermal powergener-ations in Iran which indicate that labor is not an effective factor(for instance, Emami Meibodi, 1998), in our study, electric power
(in megawatt hour) generated from thermal power plants in eachDMU (P) is used as the output variable, while capital (C), fuel (F),andinternalpower(Ic)arethreeinputsusedforpowergeneration.Capitalismeasuredintermsofinstalledthermalgeneratingcapac-ity in megawatt (MW) (Fare, Grosskopf, & Lovell, 1983; Hawdon,1997). Various natural elements have been used as fuel in theproductionofelectricpowerinvarioussteamplantsinIran(naturalgas,gasoilandMazute).Thechoiceoffueldependsonmanyfactorssuchasavailability,costandenvironmentalconcernsandeachfuelhasitslimitations.Ourﬁguresmeasurefuelconsumptionintermsof TeraJoule(TJ).Inotherwords,ourﬁgureshavealreadyadjustedforthe quality of fuel used in different plants. Internal power is theamount of energy consumed (in megawatt hour) within the site(forelectricallypoweredequipmentetc.).Step2:133rowsofdataarecollectedfrom1997to2003.Table1shows the real data for the inputs and output used in the model.Detailed information about power generation of thermal powerplants such as total output, generationcapacity and fuel consump-tion can be obtained from
‘‘Electric Power Industry in Iran’’
(includ-ing transmission and distribution) published by the TAVANIR Management Organization (1997–2004).Step3:
S
1
isdatafrom1997to2002(114rowsof data) and
S
2
is2003 data (19 rows of data).Step 4: In order to get the best ANN for the electricity produc-tion in steam power plants, 20 MLP–LM models are tested to ﬁndthe best architecture. The architecture of the stated MLP–LMmod-els and their MAPE error values are shown in Table 2. It seems the8thmodel(ithaseightneuronsinsinglehiddenlayer)hasthelow-est MAPE (relative error) and consequently is chosen as the pre-ferred model. In Fig. 1, the ANN architecture for the preferrednetwork (8th model) is shown. Fig. 2 present the MLP–LM perfor-manceof eachmodel.Examinationof Fig. 2suggeststhat8thmod-el is better than 4th model because stable state of 8–11th modelsindicate that their low errors are not casual.Step 5: Therefore, the preferred model fromprevious step is se-lected for estimating the electricity production in 2004.Step 6: Fig. 3 shows the Normal probability plot for powerplants in 2004. As seen from the plot some outliers exist and p-value is less than 0.005. Thus we should go to step10.Step 10: First we determine the best number of clusters. Thevalues of pc and pe are shown in Fig. 3. The ‘‘best’’ number of clus-ters is the point on the horizontal axis (c
⁄
) that the entropy value(pe) of c
⁄
lies belowthe risingtrendand the value for the partitioncoefﬁcient (pc) of c
⁄
lies above the falling trend. Fig. 4 shows that,according to these two criteria, the best partitioning of the data isachieved with 3 clusters.
Table 1
Real data indicators for evaluating performance of power plants in 2004.
Power plant name Install capacity (MW) Internal consumption (MWh) Fuel consumption (TJ) Gross production (MWh)Montazerghaem 625.88 241,139 30.95308 3,297,100Besat 247.5 139,505 17.00441 1,500,253Firoozi 50 13,039 3.412765 200,103Salimi 1760 311,276 117.1448 13,190,817Shazand 1300 642,909 57.30782 7,438,002Rajaei 1000 421,015 54.61188 6,342,203Beheshti 240 85,307 14.06063 1,435,991Tabriz 736 361,080 42.82585 4,341,330Mofatteh 1000 390,708 46.37235 5,134,547Bistoon 640 350,154 38.2554 4,210,280Ramin 1890 686,643 95.29557 12,561,867Madhaj 290 81,674 7.886569 992,587Bandarabbas 1280 588,855 73.2442 7,196,540Zarand 60 69,698 6.322743 691,402Esfahan 835 422,673 52.77859 5,621,431Montazeri 1600 796,262 112.4223 13,037,177Toos 600 271,901 37.45617 3,831,065Mashhad 120 73,050 7.955432 865,887Iranshahr 256 140,940 15.76105 1,492,847Average 764.7568 314,622.5 42.47745 4,914,812Standard deviation 583.8156 231,103.9 32.92167 4,209,986Min 50 13,039 3.412765 200,103Max 1890 796,262 117.1448 13,190,817332
A. Azadeh et al./Computers & Industrial Engineering 60 (2011) 328–340

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