European Journal of Operational Research 210 (2011) 310–317

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Efficiency measurement using independent component analysisand data envelopment analysis

Ling-Jing Kao a, Chi-Jie Lu b, Chih-Chou Chiu a,⇑a Department of Business Management, National Taipei University of Technology, Taiwan, ROCb Department of Industrial Engineering and Management, Ching Yun University, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 5 September 2009Accepted 10 September 2010Available online 18 September 2010

Keywords:Independent component analysisData envelopment analysisEfficiency measurement

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.09.016

⇑ Corresponding author. Tel.: +886 2 2771 2171/34E-mail address: chih3c@ntut.edu.tw (C.-C. Chiu).

a b s t r a c t

Efficiency measurement is an important issue for any firm or organization. Efficiency measurementallows organizations to compare their performance with their competitors’ and then develop correspond-ing plans to improve performance. Various efficiency measurement tools, such as conventional statisticalmethods and non-parametric methods, have been successfully developed in the literature. Among thesetools, the data envelopment analysis (DEA) approach is one of the most widely discussed. However, prob-lems of discrimination between efficient and inefficient decision-making units also exist in the DEA con-text (Adler and Yazhemsky, 2010). In this paper, a two-stage approach of integrating independentcomponent analysis (ICA) and data envelopment analysis (DEA) is proposed to overcome this issue.We suggest using ICA first to extract the input variables for generating independent components, thenselecting the ICs representing the independent sources of input variables, and finally, inputting theselected ICs as new variables in the DEA model. A simulated dataset and a hospital dataset providedby the Office of Statistics in Taiwan’s Department of Health are used to demonstrate the validity of theproposed two-stage approach. The results show that the proposed method can not only separate perfor-mance differences between the DMUs but also improve the discriminatory capability of the DEA’s effi-ciency measurement.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Efficiency measurement is an important issue for any type ofbusiness or organization. Efficiency measurement allows busi-nesses or organizations to compare their performance with theircompetition and develop a corresponding strategy to improve per-formance. Among various efficiency measurement tools, such asconventional statistical methods, non-parametric methods, andartificial intelligence methods, developed in the literature, it hasbeen affirmed that DEA can effectively measure the relative effi-ciencies of multiple decision making units (DMUs) with similargoals and objectives. For example, DEA has been widely appliedin different types of businesses or organizations, such as banks(Kao and Liu, 2009; Sahoo and Tone, 2009; Cooper et al., 2008),schools (Hu et al., 2009; Ray and Jeon, 2008; Mancebóon andMuuiz, 2008) and hospitals (Hua et al., 2009; Ancarani et al.,2009; Kirigia et al., 2008).

However, DMU has two shortcomings. First, when the inputs ofDMU are strongly correlated, the efficiency estimates of DMUobtained from the slack variable analysis of DEA can be biased. This

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is caused by the fact that DEA techniques mainly use the method ofweighting to calculate the ratio between the inputs and outputs ofeach DMU. Second, when the model is incorrectly specified or thenumber of units is too small, DEA may fail in its discriminationcapability. To solve these problems, Adler and Golany (2001,2002) suggested using the principal component analysis (PCA) toproduce uncorrelated linear combinations of original inputs andoutputs. Adler and Yazhemsky (2010) concluded that PCA–DEAoutperforms PCA-variable reduction by comparing their discrimi-nation performance in a simulation exercise.

Independent component analysis (ICA) is the other solution tothe problem of input correlation. Essentially, ICA is a novel statis-tical signal processing technique used to extract independentsources from observed multivariate statistical data where no rele-vant data mixture mechanisms are available (Hyvärinen et al.,2001; Hyvärinen and Oja, 2000). It is a methodology for capturingboth second and higher order statistics, and it projects the inputdata onto the basis vectors that are as statistically independentas possible (Bartlett et al., 2002; Draper et al., 2003). These charac-teristics of ICA distinguish ICA from PCA which is used to find a setof the most representative projection vectors such that theprojected samples retain the most information about the originalsamples (Turk and Pentland, 1991). The literature has applied

L.-J. Kao et al. / European Journal of Operational Research 210 (2011) 310–317 311

ICA in human face recognition on FERET database (Bartlett et al.,2002; Liu and Wechsler, 1999) and the Olivetti and Yale databases(Yuen and Lai, 2002). The latter study, Liu and Wechsler (1999),and Bartlett et al. (2002) have shown that ICA outperforms PCA.However, Moghaddam (2002) states that the performances of ICAand PCA have no significant difference.

In this paper, we propose a new ICA–DEA approach of efficiencymeasurement not only to address the problem of input correlationbut also to improve discrimination capability. Our proposed ap-proach consists of two stages. In the first stage, we use ICA to gen-erate independent components (ICs) to ensure statisticalindependence among the input variables. In the second stage, theestimated ICs containing the key factors that affect efficiency mea-surement are applied in DEA as the new input variables. To evalu-ate the performance of the proposed method, a simulated datasetand a hospital dataset provided by the Office of Statistics in Tai-wan’s Department of Health are used in this study. We also com-pare the discrimination capability of ICA–DEA with otherapproaches such as PCA–DEA and variable reduction (VR). The re-sult shows that the efficiency analysis performed by the proposedICA–DEA approach can avoid efficiency misjudgment. Moreover,the optimal level of input and output for each inefficient hospitalcan be estimated successfully.

The rest of this paper is organized as follows: Sections 2 and 3give a brief introduction to independent component analysis anddata envelopment analysis, respectively. In Section 4, we beginby developing the proposed two-stage model and comparing it toPCA–DEA and VR with one simulated dataset. In Section 5, oneempirical application is provided. The hospital data from the Officeof Statistics in Taiwan’s Department of Health is analyzed bythe proposed approach. And concluding remarks are offered inSection 6.

2. Independent component analysis

Let X ¼ ~x1; ~x2; . . . ; ~xm½ �T be a multivariate data matrix of sizem � n, m 6 n, consisting of observed random variables ~xi of size1 � n, i = 1,2, . . . ,m. In the basic ICA model, the matrix X can bemodeled as

X ¼ AS ¼Xm

i¼1

~ai~si; ð1Þ

where ~ai is the ith column of unknown mixing matrix A of sizem �m; ~si is the ith row of source matrix S of size m � n. The vectors~si are unknown latent sources (variables) that cannot be directly ob-served from the observed variables ~xi. The ICA model aims at findingan m �m de-mixing matrix W such that

B ¼ ~bi

h i¼WX ¼ ~wiX½ �; ð2Þ

where ~bi is the ith row of the matrix B, i = 1,2, . . . ,m. The vectors ~bi

must be as statistically independent as possible, and are called inde-pendent components (ICs). The ICs are used to estimate the latentvariables ~si. The vector ~wi in Eq. (2) is the ith row of the de-mixingmatrix W, i = 1,2, . . . ,m. It is used to transform the observed multi-variate matrix X to generate the corresponding IC, i.e., ~bi ¼ ~wiX,i = 1,2, . . . ,m.

The ICA modeling is formulated as an optimization problem bysetting up the measure of independence of ICs as an objectivefunction and using some optimization techniques to solve forthe de-mixing matrix W (Bell and Sejnowski, 1995; Sánchez,2002). In general, the ICs are obtained by using the de-mixingmatrix W to multiply the matrix X. The de-mixing matrix Wcan be determined using an unsupervised learning algorithm withthe objective of maximizing the statistical independence of ICs.

And the statistical independence of ICs can be measured interms of their non-Gaussian properties (Hyvärinen et al., 2001;Hyvärinen and Oja, 2000).

Normally, non-Gaussianity can be verified by two common sta-tistics: kurtosis and negentropy. The kurtosis of a random variable~b, fourth-order cumulant, is classically defined by

kurtð~bÞ ¼ E ~b4� �

� 3 E ~b2� �� �2

: ð3Þ

If variable ~b is assumed to be zero mean and unit variance, theright-hand side simplifies to Eð~b4Þ � 3. This shows that kurtosis is

simply a normalized version of the fourth moment Eð~b4Þ. For a

Gaussian eb, the fourth moment equals 3 E ~b2� �� �2

. Thus, kurtosis

is zero for a Gaussian random variable and non-zero for mostnon-Gaussian random variables.

Unlike kurtosis, negentropy is determined according to theinformation quantity of (differential) entropy. Entropy is a mea-sure of the average uncertainty in a random variable. The differ-ential entropy H of random variable ~b with density f ð~bÞ isdefined as Hð~bÞ ¼ �

Rpð~bÞ log pð~bÞd~b. According to a fundamental

result of information theory, a Gaussian variable will have thehighest entropy value among a set of random variables withequal variance (Hyvärinen and Oja, 2000). For obtaining a mea-sure of non-Gaussianity, the negentropy J is defined as follows:

Jð~bÞ ¼ H ~bgauss

� �� Hð~bÞ; ð4Þ

where ~bgauss is a Gaussian random vector of the same covariance ma-trix as ~b.

The negentropy is always non-negative and is zero if and only if~b has a Gaussian distribution. Since negentropy is very difficult tocompute, an approximation of negentropy is proposed as follows(Hyvärinen and Oja, 2000):

Jð~bÞ � E Gð~bÞn o

� E Gð~oÞf gh i2

; ð5Þ

where ~o is a Gaussian variable of zero mean and unit variance, and ~bis a random variable with zero mean and unit variance. G is a non-

quadratic function, and is given by Gð~bÞ ¼ exp �~b2=2� �

in this

study. The FastICA algorithm proposed by Hyvärinen et al. (2001)is adopted in this paper to solve for the de-mixing matrix W.

The ICA modeling usually consists of two preprocessing steps:centering and whitening (Hyvärinen and Oja, 2000). In the center-ing step, the matrix X is centered by subtracting the row means ofthe matrix, i.e., ~xi ~xi � Eð~xiÞð Þ. The matrix X with zero mean isthen passed through the whitening matrix Q to remove the secondorder statistic of the input matrix, i.e., Z = QX. The whitening ma-trix Q is twice the inverse square root of the covariance matrix ofthe input matrix, i.e., Q ¼ 2 C~xð Þ�ð1=2Þ, where C~x ¼ E ~x~xT

� �is the

covariance matrix of X. The rows of the whitened input matrix Z,denoted by ~z, are uncorrelated and have unit variance, i.e.,C~x ¼ Eð~z~zTÞ.

ICA can be viewed as an extension of principal component anal-ysis (PCA) (Hyvärinen and Oja, 2000). However, the objective of ICAis different from that of PCA. PCA is a dimensionality reductiontechnique that reduces the data dimension by projecting the corre-lated variables into a smaller set of new variables that are uncorre-lated and retain most of the original variance. But the objective ofPCA is only to de-correlate variables, not to make them indepen-dent. PCA can only impose independence up to second order statis-tical information while constraining the direction vectors to beorthogonal, whereas ICA has no orthogonal constraint and involveshigh-order statistics, i.e., it not only de-correlates the data (secondorder statistics) but also reduces high order statistical dependen-cies (Lee, 1998). Hence, ICs reveal more useful information from

312 L.-J. Kao et al. / European Journal of Operational Research 210 (2011) 310–317

observed data than principal components (PCs) do. The differencesbetween PCA and ICA also exist between ICA and principal axis fac-toring (PAF) which is a well-known exploratory factor analyticalmodel. Basically, PAF extracts factors based on second order statis-tical information, and the factors are assumed to be uncorrelatedand Gaussian distributed (Hyvärinen et al., 2001).

A simple comparison between PCA and ICA is shown in Fig. 1.Fig. 1(a) shows three original variables that are two different typesof sinusoidal variables (~s1and ~s2Þ and a random variable (~s3Þ. Theseoriginal variables are transformed into observed mixing variables(~x1; ~x2and ~x3Þ using unknown mixing matrix A, i.e., X=AS. WhenPCA is applied to these mixing variables, it gives three principalcomponents as observed in Fig. 1(c). Fig. 1(c) shows that the PCAresults are different from the original variables in Fig. 1(a). TheICA solution is depicted in Fig. 1(d). Fig. 1(d) clearly reveals thatthe original variables can be reconstructed by ICA without anyknowledge of the original variables and mixing matrix.

The ICA model in Eq. (1) and the simple ICA example in Fig. 1show that the ICA model contains two limitations (Hyvärinenand Oja, 2000). The first of them is that the variances (energies)of the independent components cannot be determined becauseboth S and A are unknown. In other words, S and A are unidentifiedunless one of them is fixed. To alleviate this problem, mostresearchers assume that each IC has zero mean and unit variancein the ICA model (Hyvärinen and Oja, 2000). However, that stillleaves the ambiguity of the sign because one could multiply theIC by �1 without affecting the model.

The other limitation of the ICA model is that one cannot deter-mine the order of the ICs because of the simultaneous unknowns ofS and A. A number of methods have been suggested to determine

Fig. 1. A simple comparison

the component order (Cardoso and Souloumiac, 1993; Back andWeigend, 1997; Cheung and Xu, 2001). Hyvärinen (1999) sug-gested that ICs can be sorted according to their non-Gaussianity.In this paper, the ICs are sorted based on their kurtosis values.

3. Data envelopment analysis basis

DEA is also known as the efficient frontier approach (Cooperet al., 2000, 2004). The term ‘‘envelopment” refers to the idea thatinefficient DMUs are located inside an area enveloped by the effi-cient DMUs. DEA is constructed based on the concept of relativeefficiency, which is defined as the ratio of the weighted sum ofoutputs to the weighted sum of inputs (Cooper et al., 2004). Thesolution of DEA requires that the weights for inputs and outputsof each unit are selected to maximize its efficiency under certainconstraints. Thus, the mathematical programming form of theCCR model is formulated as follows (Cooper et al., 2004):

Maximize h ¼u1y1p þ u2y2p þ � � � þ udydp

v1x1p þ v2x2p þ � � � þ vmxmp

Subject tou1y1j þ u2y2j þ . . .þ udydj

v1x1j þ v2x2j þ . . .þ vmxmj6 1; j ¼ 1 to n

v i > 0 for i ¼ 1;2; . . . ;m

ur > 0 for r ¼ 1;2; . . . ;d

ð6Þ

where x1j,x2j, . . . ,xmj are the m inputs, y1j ,y2j, . . . ,ydj are the d outputsof the unit j,v1,v2, . . . ,vm are weights for the inputs, and u1,u2, . . .,us

are the weights for the outputs, p is the designated unit for an opti-mization run, and n is the total number of units in the study.Because each decision unit has its own preferences in input and

between PCA and ICA.

L.-J. Kao et al. / European Journal of Operational Research 210 (2011) 310–317 313

output variables, the implementation of the model in Eq. (6) canderive the values of vi(i = 1,2, . . . ,m) and ur(r = 1,2, . . . ,d) for eachdecision unit.

4. Research methodology and simulation study

The proposed ICA–DEA approach is illustrated in Fig. 2. Afterdeciding the input variables, we used ICA to convert observed in-put data into separate independent signals. Then, these separatedsignals were integrated into the DEA approach in the second stageof our technique to construct a model that could measureefficiency.

The performance of the proposed ICA–DEA model relative to theperformance of PCA–DEA and VR is investigated by a simulationanalysis. To be consistent with other studies, we used the datasetdetermined by simulated Cobb-Douglas production functions~y ¼ ~x0:25

1 ~x0:22 ð~x3~x4~x5~x6~x7~x8Þ0:09e�s with a half normal inefficiency dis-

tribution (Adler and Yazhemsky, 2010) for comparison. There are50 DMUs in total, and 8 inputs are included in the dataset. Follow-ing (Adler and Yazhemsky, 2010), we first generated 10,000 posi-tive observations of the variables ~x from a normal distributionwith mean 10 and variation 1. A single output was then chosento permit the utilization of standard production function to com-pute the output values. Consequently, over the entire simulatedpopulation of 10,000 observations and a sub-sample size of50 DMUs, we can have the average percentage reported from10,000/50 = 200 samples. In the data generation process, we as-sumed no probability mass along the frontier, and only one singleinefficiency (s) was simulated from each DMU. In this study, wedraw the inefficiency independently from a half normal distribu-tion with l = 0 and r = 1.

The simulated data was then used to compare results from thePCA–DEA model, the VR model, and our ICA–DEA model. To inves-tigate the influence of the percentage of retained information onthe number of inputs in the analysis, we followed Adler andYazhemsky (2010) and reduced the percentage of retained infor-mation from 100% to 80% in 4 percent steps. As in the work doneby Adler and Yazhemsky (2010), we used the percentage ofretained information to decide the number of PCs or variables in-cluded in the DEA model. In other words, we manually determinedthe number of ICs, PCs, or variables to retain, such that the percent-age of information remaining is greater or equal to the setup level.

Table 1 provides the incorrect efficiency classification usingICA–DEA, PCA–DEA and VR with different input dimensions.According to Table 1, the level of information reduction has appre-ciable influence when the three approaches are used to classify theinefficiency. Moreover, the ICA–DEA model gives the lowest incor-rect classification result, and the classification results made by thePCA–DEA model are slightly better than those generated by VR.They are consistent with the conclusion made by Lee et al.(2004). (The ICA solution extracts the original source signal to a

observations of inputs

Stage I

data conversion

by ICA

Fig. 2. Research

much greater extent than the PCA solution if the latent variablesfollow non-Gaussian distribution.)

5. Empirical application

5.1. Data

Hospital data (from 557 hospitals in 2005) provided by theOffice of Statistics in Taiwan’s Department of Health is used in thisstudy to illustrate the proposed ICA–DEA approach and to comparethe performance of the proposed method with alternative ap-proaches. These hospitals can be categorized into three differentclasses according to their functional complexity: (1) medical cen-ters and regional hospitals (highest level); (2) district hospitals(medium and low level); and (3) primary clinics (basic level).Usually, hospitals with different sizes provide different levels ofmedical treatment and service. According to the governmentaccreditation definition, a medical center has at least 500 beds,and a regional hospital has at least 250 beds. In order to controlthe difference in sizes and make the data sample more homoge-neous, we selected 21 hospitals (i.e., 21 DMUs) with more than500 beds for analysis.

Since the results of efficiency measurement are related to theselection of input and output variables, we chose variables byreviewing the existing literature on hospital efficiency (Hu et al.,2009; Puig-Junoy, 2000; Sahin and Ozcan, 2000; Parkin andHollingsworth, 1997; Hu and Huang, 2004). In our empirical study,five input and three output variables were chosen to calculate theefficiency of each DMU. These five input variables are four labor-related variables (including doctors, nurses, paramedical persons,and administrative staffers) and one capital-related variable(beds). A patient’s health improvement is the most commonly usedoutput indicator for a hospital, but measuring the level of im-proved health is very difficult. One possible solution is to adoptthe immediate products of a hospital, such as a hospital’s servicevolume, as a proxy for medical outputs. Therefore, in this empiricalstudy, we selected outpatient visits, emergency visits, and opera-tions as our three output variables. Table 2 illustrates the defini-tions and explanations of input and output variables. Table 3provides summary statistics of input and output variables.

Table 3 also shows that the variance of each input or output var-iable is still large. The variables having the largest standard devia-tion of input and output variables are beds and outpatient visits,respectively.

As discussed above, the correlation between input and outputvariables will cause biased estimates in DEA. We report the resultof correlation analysis in Table 4. Table 4 shows that there are sig-nificant positive correlations among all of the variables. The low-est correlation coefficient is 0.508 between doctors andemergency visits. The highest coefficient 0.972 exists betweennurses and doctors. Practically, subsets of the inputs or outputs

Stage II

construction of efficiency

measurement model by DEA

efficiency measurement

structure.

Table 1Incorrect efficiency classification under varying inputs reduction for ~y ¼ ~x0:25

1 ~x0:22 ð~x3~x4~x5~x6~x7~x8Þ0:09e�s function.

Incorrect classification Incorrectly defined inefficient Incorrectly defined efficient

Method ICA- DEAa PCA–DEAa VR ICA- DEAa PCA–DEAa VR

Percentage information retained P 100 0 0 0 7.623 7.621 7.62596 0 0 0 5.012 5.552 6.86292 0 0 0 3.897 4.316 6.55288 0 0 0.021 2.768 3.953 5.97584 0.018 0.028 0.035 2.139 3.272 5.44680 0.055 0.076 0.423 1.815 2.763 4.812

The number of samples containing m inputs as a function of information retainedMethod ICA–DEA PCA–DEA VR

m 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1

Percentage information retained P100 200 200 20096 10 148 42 10 152 38 105 93 292 55 122 23 55 121 24 21 153 2688 3 30 120 43 4 4 26 120 45 5 32 122 31 1584 1 32 118 49 3 35 110 52 3 28 140 2980 6 46 148 6 50 144 5 50 133 12

a The DEA model is under a constant return-to-scale case (CRS).

Table 2Definition and explanation of variables.

Variables Definition and explanation

Inputs Beds The total number of registered beds within thehospital, including acute, chronic, and specialbeds

Doctors The total number of physicians who are full-timeemployees, including dentists and Chinesemedicine doctors

Nurses The total number of nurses employed inhospitals (including midwives)

Administrativepersons

The total number of health service providersemployed in hospitals, including pharmacists,dietitians, physiotherapists, occupationaltherapy technologists, and radiologicaltechnologists

Administrativestaffers

The total number of full-time equivalentpersonnel (including social workers, researchers,and non-professionals)

Outputs Outpatientvisits

The total number of patients to outpatientdepartments within a year

Emergencyvisits

The total number of patients to emergency roomwithin a year

Operations The total number of inpatient and outpatientsurgeries within a year

Table 3Summary statistics of variables.

Variables Mean Std.Dev.

Min. Max.

Inputs Beds (~x1) 1,451 796 340 3,236Doctors (~x2Þ 489 267 130 1,106Nurses (~x3Þ 1,229 641 360 2,723Administrativepersons (~x4Þ

294 143 34 629

Administrativestaffers (~x5Þ

843 415 121 1,624

Outputs Outpatient visits(~y1Þ

1,270,154 566,016 286,623 2,514,534

Emergency visits(~y2Þ

83,094 30,404 40,615 132,550

Operations (~y3Þ 28,260 15,750 5,614 75,348

Table 4Correlation coefficients between variables.

~x1 ~x2 ~x3 ~x4 ~x5 ~y1 ~y2 ~y3

~x1 1.000~x2 0.957a 1.000~x3 0.972a 0.915a 1.000~x4 0.950 a 0.931 a 0.963 a 1.000~x5 0.850 a 0.850 a 0.865 a 0.879a 1.000~y1 0.769 a 0.776 a 0.781 a 0.847 a 0.781 a 1.000~y2 0.611 a 0.508 a 0.669 a 0.611 a 0.767 a 0.642a 1.000~y3 0.825 a 0.780 a 0.854a 0.871 a 0.804a 0.714 a 0.546a 1.000

a p-value < 0.05.

314 L.-J. Kao et al. / European Journal of Operational Research 210 (2011) 310–317

are always correlated. The high correlations between variablescould cause issues with the distribution of the weights. Droppingsome from the assessment sometimes could reduce the efficiency

ratings for some DMUs (Nunamaker, 1985). However, it also occa-sionally leads to significant changes in efficiencies (Dyson et al.,2001). Therefore, there is a need to convert the observed inputdata into separate independent signals by the ICA approach beforeconducting DEA.

5.2. Efficient score computations

After the ICs are determined, efficiency scores are computed bythe CCR input orienting model, proposed by Charnes et al. (1978),in the DEA-PRO software. To demonstrate the validity of the pro-posed model, the performance of the proposed ICA–DEA methodis compared to the single DEA model and the PCA–DEA model.The single DEA model simply applies the DEA model to input vari-ables to measure the efficiency of hospitals without using ICA orPCA as preprocessing tools. The PCA–DEA method first appliesPCA to the input variables for generating principal components(PCs) and then conducts DEA analysis based on the generated PCs.

In the single DEA model, the overall efficiency scores of 21 hos-pitals (DMUs) are summarized in Table 5. From Table 5, it can befound that the single DEA model produces a high average efficiencyscore (0.948) and a small standard deviation (0.122). The numberof efficient DMUs in the single DEA model is 12. The single DEAmodel results in too many efficient DMUs and cannot distinguishthe difference in performance between the DMUs very well.

In the PCA–DEA method, component loadings and the variancesregarding the components were computed for the five input vari-ables first. The proportion of the total variance explained by eachprincipal component is additive, with each new component con-tributing less than the preceding one to the explained variance.

Table 5Summary of the results of single DEA, ICA–DEA and PCA–DEA models.

Single DEAmodel

PCA–DEAmodel

ICA–DEAmodel

Average score 0.948 0.482 0.216Standard deviation 0.122 0.195 0.254Maximum efficiency

score1 1 1

Minimum efficiencyscore

0.761 0.322 0.153

Number of efficientDMUs

12 8 4

Total number of DMUs 21 21 21% of efficient DMUs 57.1 38.1 19.1

Table 7The Kurtosis values and de-mixing matrix (W) corresponding to the ICs.

~b1~b2

~b3~b4

~b5

~x1 �0.00281 0.01196 0.00143 �0.01273 0.00129~x2 0.00051 �0.00185 0.00318 0.00773 �0.00246~x3 0.00630 �0.00838 �0.00281 0.00969 0.00282~x4 0.00213 �0.00363 0.00343 �0.02473 0.00187~x5 �0.00334 �0.00245 0.00246 0.00550 0.00310Kurtosis

values10.200 0.898 1.802 �0.511 �0.297

L.-J. Kao et al. / European Journal of Operational Research 210 (2011) 310–317 315

Subsequently, the components were rotated to eliminate medium-range loadings to make the interpretation of the components easier(Johnson and Wichern, 2007). An interpretation of the rotated fiveprincipal components in Table 6 is made by examining the compo-nent loadings, noting the relationship to the original variables.Among the five principal components, the first component appearsto measure capital and partial labor (nurse) investment, with bedand nurse increasing with positive values. In the second compo-nent, doctors and administrative staffers are important; the effectof medical expertise is demonstrated by the positive loading ofdoctors in this component. According to Table 6, we found thatPC1 can explain 95.6476% of data variation, and PC2 can explain2.8061% of sample variation. Because the two main PCs can totallyexplain data variation, these two PCs would be enough to representthe features of input data and are used as new input variables forthe DEA model.

The results of the PCA–DEA model are also summarized in Table5. From the table, we found that the average efficiency score, stan-dard deviation, and number of efficient DMUs in the PCA–DEAmodel are 0.482, 0.195, and 8, respectively. Compared to the singleDEA model, the PCA–DEA method has a lower average efficiencyscore, a smaller number of efficient DMUs, and a larger standarddeviation.

In the proposed ICA–DEA model, we first applied the basic ICAapproach to estimate a de-mixing matrix W and five independentcomponents (~b1,~b2,. . ., ~b5Þ. In order to select the more meaningfulICs, the statistical independence of ICs is evaluated by computingthe kurtosis values of the ICs herein. The estimated de-mixing ma-trix W and the kurtosis value for each IC are summarized in Table7. Because the IC with the larger kurtosis value can be consideredthe more important IC (Hyvärinen, 1999), IC1, IC2 and IC3 (i.e., ~b1,~b2, ~b3Þ are regarded as key factors affecting the results of efficiencymeasurement and used as three new input variables for the DEAmodel. Note that the extracted IC might have negative valueswhich violate the semi-positive assumption for the DEA model,i.e., all inputs and all outputs are non negative, and at least one in-put and one output are positive. To solve this problem, we simply

Table 6Varimax rotated five components.

Rotatingloadings

PC1 PC2 PC3 PC4 PC5

~x1 0.679145 �0.615730 0.389697 0.087591 0.010291~x2 0.238138 0.131664 0.003925 �0.953530 0.129294~x3 0.584563 0.198768 �0.750860 0.189139 0.138593~x4 0.134505 0.067684 �0.081240 �0.090430 �0.981100~x5 0.349638 0.747966 0.526998 0.197875 0.037661Proportion

of Var.0.956476 0.028061 0.010451 0.004505 0.000507

CumulativeVar.

0.956476 0.984537 0.994988 0.999493 1

subtract each ICi from its corresponding minimum value, i.e., min(ICi).

From Eq. (2), we know that ~bi can be obtained by multiplying Xby its corresponding row vector of the de-mixing matrix (i.e., ~wiÞ.Thus, the values in the de-mixing matrix can be used to explainthe relationship between the selected ~bi and the original inputvariables X as follows:

~b1 ¼ �0:00281~x1 þ 0:00051~x2 þ 0:00630~x3 þ 0:00213~x4

� 0:00334~x5; ð7Þ

~b2 ¼ þ0:01196~x1 � 0:00185~x2 � 0:00838~x3 � 0:00363~x4

� 0:00245~x5; ð8Þ

~b3 ¼ þ0:00143~x1 þ 0:00318~x2 � 0:00281~x3 þ 0:00343~x4

þ 0:00246~x5: ð9Þ

From Eq. (7), we understand that ~x3 and ~x5 have a relatively signif-icant impact on ~b1. This implies that ~b1 is mainly affected by nurses(~x3Þ and administrative staffers (~x5Þ. Thus, ~b1 can be used to repre-sent the features of nurses and administrative staffers. As seen fromEqs. (8) and (9), ~b2 is mainly influenced by beds (~x1Þ;~b3containsmore information about doctors (~x2Þ and administrative persons(~x4Þ. Therefore, ~b2 and ~b3 can be used to represent the informationabout beds and the features of doctors and administrative persons,respectively.

The efficiency measurement results of the proposed ICA–DEAmodel are included in Table 5. It can be seen that the average effi-ciency score, standard deviation, and number of efficient DMUs inthe proposed method are 0.216, 0.254 and 4, respectively. Com-pared to the single DEA and PCA–DEA models, the proposed meth-od has the lowest average efficiency score, the least number ofefficient DMUs, and the highest standard deviation. This indicatesthat the proposed method has more ability to distinguish betweenthe performances of DMUs. In other words, the ICA approach canimprove the discriminatory capability of the DEA model in perfor-mance measurement.

5.3. Slack analysis

DEA provides not only efficiency results but also slack analysis.Slack analysis can provide guidelines to derive the optimal level ofinput and output resources for each DMU. As a result, each DMUcould have its input and output resources set at the optimal level– the original level minus the inefficient and slack amounts fromthe DEA results (Luo and Donthu, 2001).

Table 7 reports the results of slack analysis for DMU1 by usingthe single DEA model and ICA–DEA approach. The slack entries ofthe single DEA model are all positive. This implies that, comparedto the efficient hospitals, the investment in various inputs isexcessive, The slack analysis suggests that many of the expendi-tures could have been reduced while maintaining the same out-puts – Outpatient visits, Emergency visits, and Operations in

Table 8Slack analysis of single DEA method and ICA–DEA method using DMU1 as example.

Variables Originalreal value

Slack entry by singleDEA model

Slack entry by ICA–DEA model

Beds (~x1) 2618 291 87Doctors (~x2Þ 1106 328 203Nurses (~x3Þ 2024 232 215Administrative

persons (~x4Þ481 52 120

Administrativestaffers (~x5Þ

1624 251 95

DMU1efficiencyscore

– 0.8673 0.9542

316 L.-J. Kao et al. / European Journal of Operational Research 210 (2011) 310–317

this case – thus improving the efficiency. For example, to be asefficient as 20 other hospitals, DMU1 can maintain the same out-put levels by cutting down 291 in beds, 328 in doctors, 232 innurses, 52 in administrative persons, and 251 in administrativestaffers.

Although the slack analysis can investigate the utilization of in-put or output resources to improve efficiency scores, the analysisresults from the single DEA method could be either underesti-mated or overestimated as long as the inputs of DMU are corre-lated. To solve this problem, we replaced the single DEA model’sinputs with the transformed inputs made by the ICA approach be-fore running the DEA analysis. The result of ICA–DEA model issummarized in Table 7.

Due to the adoption of the ICA technique, the slack entries gen-erated by the ICA–DEA model need to be re-transformed in order tounderstand exactly the utilization of input and output resources toimprove efficiency scores. In the re-transformed procedure, wefirst define the slack entries generated by the ICA–DEA model asDbi. Then, based on Eqs. (7)–(9) and correlation coefficients (qij)in Table 4, we can formulate our re-transforming procedure as anoptimization problem shown as follows:

Minimize e1 þ e2 þ e3;

Subject to Db1 þ 0:00281Dx1 � 0:00051Dx2 � 0:00630Dx3j�0:00213Dx4 þ 0:00334Dx5j 6 e1;

Db2 � 0:01196Dx1 þ 0:00185Dx2 þ 0:00838Dx3jþ0:00363Dx4 þ 0:00245Dx5j 6 e2;

Db3 � 0:00143Dx1 � 0:00318Dx2 þ 0:00281Dx3j�0:00343Dx4 � 0:00246Dx5j 6 e3;

~xi ¼ qij~xj i ¼ 1;2; . . . ;5; j ¼ 1;2; . . . ;5;

Dxi 2 Integer i ¼ 1;2; . . . ;5;

ð10Þ

where ~xi or ~xj is the original inputs; qij is the correlation coefficientbetween input variables ~xi and ~xj; Dbi is the slack entries generatedby the ICA–DEA model; andDxi is the re-transformed slack entriesof the ICA–DEA model.

Because the above optimization problem is in a simple linearprogramming format, solutions can be derived with the simplexmethod as in the usual linear programming approach. The 3rd col-umn in Table 8 shows the final slack analysis results of the pro-posed ICA–DEA method for DMU1. Similar to the results made bythe single DEA model, the slack entries are all positive comparedto the efficient hospitals. But the results made by the ICA–DEA ap-proach suggest fewer slack entries for beds, doctors, nurses, andadministrative staffers, and a greater number of slack entries foradministrative persons.

6. Conclusions

One of the most important managerial issues of any type ofinstitution is measuring the relative efficiency of its decision

making units. DEA is an efficiency comparison method based onlinear programming and has been used in a variety of contexts.In this study, a two-stage ICA–DEA approach is proposed to im-prove the discriminatory capability of DEA results. The proposedmethod first applies ICA to the input variables to generate ICs rep-resenting the independent sources of input variables. Then, theimportant ICs are identified and selected based on their kurtosisvalues. The selected ICs, regarded as the key factors affecting effi-ciency measurement, are utilized as new input variables in theDEA model. The proposed ICA–DEA approach is illustrated by Tai-wanese hospital data from 21 hospitals (DMUs) in 2005. Comparedto the single DEA and integrated PCA and DEA models, the pro-posed ICA–DEA method has the lowest average efficiency score,the fewest efficient DMUs, and the highest standard deviation. Thisresult provides evidence that the proposed ICA–DEA approach hasa superior ability to differentiate the performance of the DMUs andto overcome the shortcomings of DEA.

Acknowledgements

We thank the editor and the anonymous referee whose sugges-tions improved the quality of the paper significantly. This researchis in part supported by a research grant provided by the NationalScience Council of the Republic of China under Grant No. NSC 99-2221-E-027-024 -MY2.

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