A New Measure for Detecting Influential DMUs in DEA

Research ArticleA New Measure for Detecting Influential DMUs in DEA

Irmak Acarlar1 Harun KJnacJ1 and Vadoud Najjari2

1 Statistics Department Faculty of Science Gazi University Ankara Turkey2Department of Mathematics Islamic Azad University Maragheh Branch Maragheh Iran

Correspondence should be addressed to Vadoud Najjari vnajjarigaziedutr

Received 28 June 2014 Accepted 2 October 2014 Published 16 October 2014

Academic Editor Farhad Hosseinzadeh Lotfi

Copyright copy 2014 Irmak Acarlar et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We discuss about influential DMUs and also we review some related studies in the literature Then we propose a new method todetect influential DMUs which is based on Euclidean distance and omitting the efficient DMUs by using single case deletion Ourmethod will be explained with an empirical example

1 Introduction

Data envelopment analysis (DEA) is a nonparametricmethodfor evaluating the relative efficiency of decision-making units(DMUs) on the basis ofmultiple inputs and outputs In recentyears DEA has had important role in application of manyfields such as energy [1 2] banking [3] and sport [4 5]

DEA is a useful technique to evaluate the performanceof DMUs meanwhile if a data set contains one or moreinfluential DMUs obviously calculated results (by DEA) ofthe performance are changed Influential DMUs are atypi-cal observations Some of them are result of recording ormeasurement errors and should be corrected (if possible) ordeleted from data So detecting influential observations hasan important role in DEA [6]

Influential observations for the first time were introducedby Cook [7] in regression analysis as follows an influentialobservation causes noticeable effect on the estimation ofparameters and fitted values in the regression He alsoproposed a practical statistic that is called Cook distance andit is based on Mahalanobis distance Then several methodsand statistics were proposed to detect influential observationsin the regression Most of these methods are based on casedeletion approach Some of these methods and statistics aregiven by Belsley et al [8] Cook and Weisberg [9] andChatterjee and Hadi [10]

General approach for detecting influential observationsis the case deletion technique This technique is applied by

single and multiple casesrsquo deletions [8] In the single casedeletion 119901th observation is eliminated from data and thenthe result of computation is compared by the result which iscomputed using all data Multiple cases are the generalizedform of the single case deletion namely these cases areapplied by eliminating 119896 observations where 1 lt 119896 lt 1198992

and 119899 is the number of observationsThe main idea about influential observations in DEA is

similar to the regression analysis Indeed an influential DMUis an efficient DMU which basically extends the productionpossibility set according to its own coordinate and thereforeit may cause several problems as follows

(1) The influential DMU may cause that one DMU to beinefficient while by omitting the influential DMU itcan be an efficient one

(2) The influential DMU may result in decreasing thesuperefficiency scores of some efficient DMUs

(3) The influential DMU may result in decreasing theefficiency scores of some inefficient DMUs

Particularly the mentioned last item is significantbecause one of the main objectives of DEA is identifyingthe efficient DMUs and then expressing several suggestionsto improve the efficiency of inefficient DMUs Clearly theseinfluential DMUs may cause wrong suggestions for improv-ing the efficiency of inefficient DMUs

One of the first propositions about detecting influentialDMUs in DEA was given by Wilson [6] He proposed

Hindawi Publishing CorporationJournal of OptimizationVolume 2014 Article ID 567692 7 pageshttpdxdoiorg1011552014567692

2 Journal of Optimization

a method that is based on the superefficiency scores bymodified DEA which contains case deletion technique Thismethod allows researcher to prioritize observations in theefficient subset for the future scrutiny This prioritizationdepends on the number of efficiency scores that are influ-enced by a given observation

Pastor et al [11] propose a method for detecting influ-ential DMUs which is based on Uniformly Most PowerfulTest They consider BCC model and they define a ratio thatis calculated by division 119895th efficiency score obtained from allDMUs and 119895th efficiency score obtained by elimination of119901thefficient DMU Then they define a binary variable accordingto 119895th ratio either smaller than 095 or larger than 095Hence they obtain a binomial variable by sum of these binaryvariables Ruiz and Sirvent [12] use similar approach andthey propose an alternative method to identify the influentialDMUs in radial and nonradial DEA

A method proposed by Jahanshahloo et al [13] aimsto detect the influential DMUs by the way of deteriorationefficiency scores of inefficient DMUs in the radial DEATheyfocused on BCCmodel but they pointed out that the methodalso can be used in the CCR model This method is based ona specific ratio like the proposed ratio by Pastor et al [11]

In this study we propose a new method for detecting theinfluential DMUsThis newmethod is based on the Euclideandistance and excluding the efficient DMUs by using the singlecase deletion

The structure of this study is as follows The next sectionpresents some basic concepts ofDEA In Section 3 we discusson influential observation in DEA and we propose a newapproach for detecting influential DMUs Section 4 illustratesthe newmethod by an example Finally conclusions are givenin Section 5

2 Data Envelopment Analysis

Thefirst introduction onDEAwas practiced by Charnes et al[14]They proposedCCRmodel which is also called ConstantReturn to Scale (CRS) The CCR model evaluates bothtechnical and scale efficiencies via optimal value of the ratioform The modified version of CCR model is BCC modelwhich is also called Variable Returns to Scale proposed byBanker et al [15]The BCCmodel is used to estimate the puretechnical efficiency of DMUs by reference to the efficiencyfrontier

DEA can be applied in twomodels which are called input-and output-oriented models The primal form of input-oriented BCC (VRS) model is considered in this paper andit is given as follows

min 120579119900

subject to 120579119900119909119894119900minus

119899

sum

119895=1

120582119895119909119894119895ge 0

119899

sum

119895=1

120582119895119910119903119895minus 119910119903119900ge 0

119899

sum

119895=1

120582119895= 1

120582119895ge 0

119894 = 1 2 119898 119895 = 1 2 119899

119903 = 1 2 119904

(1)

where 120579119900

is efficiency score of DMU119900 119909119894119900

and 119910119903119900

(all nonnegative) are 119894th input and 119903th output of theDMU

119900 respectively and 120582

119895is intensity of DMU

119895 If

the 120579119900is equal to one then DMU

119900is called an efficient

DMUIn DEA a large number of efficient DMUs usually occur

in the results of analysis Therefore efficient DMUs cannotevaluate each other since their scores are equal to oneTo overcome this problem the analyst can either add newDMUs to data set or order the efficient DMUs by usingsome criterion Andersen andPetersen [16] proposed amodelto obtain superefficiency scores and these scores are usefulin both ordering the efficient DMUs and comparing thembetween one another in DEA

The primal form of input-oriented BCC (VRS) superef-ficiency model is considered in this paper and it is given asfollows

min 120579lowast

119900

subject to 120579lowast

119900119909119894119900minus

119899

sum

119895=1119895 =119900

120582119895119909119894119895ge 0

119899

sum

119895=1119895 =119900

120582119895119910119903119895minus 119910119903119900ge 0

119899

sum

119895=1119895 =119900

120582119895= 1

120582119895ge 0

119894 = 1 2 119898 119895 = 1 2 119899 119895 = 119900

119903 = 1 2 119904

(2)

where 120579lowast119900is the superefficiency score of DMU

119900 In the input-

oriented BCC model the superefficiency scores of efficientDMUs are greater or equal to one However superefficiencyscores of the inefficient DMUs are the same as their efficiencyscores that are obtained by BCC model in (1)

3 A New Measure for DetectingInfluential DMUs

DMUs consist of two groups which are influential DMUs andnoninfluential DMUs An influential DMU is defined as aDMU which affects the efficiency scores of some inefficientDMUs [6]This DMU also changes production possibility set

Journal of Optimization 3

Table 1 Artificial data with efficiency (120579) and superefficiency (120579lowast)scores by input-oriented BCC model

DMU 119883 119884 120579 120579lowast

119860 1 03 100 150119861 15 11 100 105119862 2 17 100 109119863 33 27 100 113119864 6 4 100 160119865 53 31 078 078119866 4 2 060 060119867 29 17 069 069119868 21 05 054 054119869 31 1 046 046119870 44 16 044 044119871 47 05 024 024

and extends this set to its own coordinate In this study weclassify noninfluential DMUs in three groups as follows

(1) The first group consists of efficient DMUs such thatincluding or excluding the influential DMU has notany effect on the efficiency scores of these DMUs

(2) The second group consists of inefficient DMUs suchthat including or excluding the influential DMU hasnot any effects on the inefficiency of these DMUs

(3) The third group consists of inefficient DMUs suchthat excluding the influential DMUs make themefficient DMUs

Clearly DMUs in the first type are on the efficiency frontierby the BCC model so their efficiency scores are equal toone

Table 1 consists of 12 artificial DMUs and also theirefficiency and superefficiency scores by the input-orientedBCC model These data consist of one input (119883) and oneoutput (119884) variable Let 120595

1and 120595

2be sets of the efficient

and inefficient DMUs respectively Evidently if data involvesan influential DMU since the influential DMU is also anefficient DMU it becomes an element of the set 120595

1 In

Table 1 these sets are 1205951

= 119860 119861 119862119863 119864 and 1205952

=

119865 119866119867 119868 119869 119870 119871 For details we investigate scatter plotsof the variables (119883 and 119884) to identify influential DMUsin these data Figure 1 provides a pattern of the DMUs inTable 1 also it displays efficiency frontier by the BCC modelClearly DMU 119864 is an influential DMU since this DMUextends the production possibility set to its coordinate (seeFigure 1) It also affects on the superefficiency scores of someefficient DMUs and the efficiency scores of some inefficientDMUs With excluding DMU 119864 DMUs 119860 119861 119862 and 119863

in the set 1205951are stable on their efficiency score however

their superefficiency scores are affected by the influentialDMU These DMUs are classified as the first type DMUs By

y

45

4

35

3

25

2

15

1

05

00 1 2 3 5 64 7 8

A

B

C

D

E

F

GH

I

J

K

L

Figure 1 Efficiency frontier and production possibility set ofartificial data

excluding the DMU 119864 the DMUs 119866119867 119868 119869 119870 and 119871 in theset 1205952save their inefficiency but efficiency scores of some

of them may be affected These DMUs are classified as thesecond type DMUs Finally 119865 is a particular DMU in the set1205952 it becomes an efficient DMU by omitting the DMU 119864

Namely this point locates on the frontier after omitting theinfluential DMU and the production possibility set becomessmaller DMU 119865 is classified as the third type DMU Ofcourse the discussion above is only based on visualization ofdata and we need a reliable method in detecting influentialDMUs

31 Detecting Influential DMUs We are preparing to presenta new method in identifying the influential DMUs Supposedata consists of 119899 observations Φ = 1 2 119899 whereany 119894 = 1 2 119899 points on the 119894th DMU and 120595

1and 120595

2

are sets of the efficient and inefficient DMUs respectivelyLet 120601all be an 119899 times 1 vector consisting of efficiency scoresin these data which is obtained by the input-oriented BCCmodel

120601all = [12057911205792 sdot sdot sdot 120579119899minus1120579119899]119879

119899times1 (3)

where for 119894 = 1 2 119899 120579119894is the efficiency score of

119894th DMU Let 120579119894119901

be the efficiency score of 119894th DMUthat is obtained by the input-oriented BCC model afteromitting 119901th DMU from data (120579

119901119901cannot be calcu-

lated due to omitting 119901th DMU) and 120601119901

is an (119899 minus

1) times 1 vector consisting of these efficiency scores asbelow

120601119901= [12057911199011205792119901sdot sdot sdot 120579(119901minus1)119901

120579(119901+1)119901

sdot sdot sdot 120579(119899minus1)119901

120579119899119901]119879

(119899minus1)times1

(4)


Table 2 Results of calculations in detecting influence of DMUs

DMU 120579119894

120579119894119860

120579119894119861

120579119894119862

120579119894119863

120579119894119864

119863

119860 1 mdash 1 1 1 1 00387

119861 1 1 mdash 1 1 1 00005

119862 1 1 1 mdash 1 1 00057

119863 1 1 1 1 mdash 1 00044

119864 1 1 1 1 1 mdash 00487

119865 07794 07794 07794 07794 08368 1 mdash

119866 05975 05975 05975 06281 06304 05975 mdash

119867 06897 06897 06897 075 06897 06897 mdash

119868 05357 07143 05442 05357 05357 05357 mdash

119869 04637 04839 04839 04637 04637 04637 mdash

119870 04356 04356 04383 04688 04356 04356 mdash

119871 02394 03191 02432 02394 02394 02394 mdash119880 = 00211

In order to generate a measure to compare 120601all and 120601119901 (thesevectors dimensions must be the same) in the 120601

119901 let 120579119901119901= 1

Then 120601119901can be rewritten as follows

120601lowast


1 120579(119901+1)119901


120579119899119901]119879

119899times1

(5)

For any 119901 isin 1205951to calculate influence of the 119901th efficient

DMU we propose to use Euclidean distance measure There-fore square of the Euclidean distance between 120601all and 120601

lowast

119901is

given as below

119863119901=10038171003817100381710038171003817120601all minus 120601

lowast

119901

10038171003817100381710038171003817

2

2

=

119899

sum

119894=1

(120579119894minus 120579119894119901)2

(6)

where sdot indicates 1198712normThis distance is only calculated

to investigate the influence of efficient DMUs Evidentlyif 119901th DMU has less influence on inefficient DMUs thenthe value of 119863

119901is smaller than the other efficient DMUs

distances On the other hand if 119901th DMUhasmore influenceon inefficient DMUs then the value of119863

119901increases Hence if

the biggest distance pertains to119863119901 it indicates the 119901th DMU

has the most influence on the inefficient DMUs We have todetermine an upper bound (cut-off point) such as 119880 wherefor any 119901 isin 120595

1 119863119901gt 119880 means that the 119901th DMU is an

influential oneLet119863 = 119863

119901| 119901 isin 120595

1 then we define an upper bound in

detection the influential DMUs as follows

119880 = mean (119863) + 3radicVar (119863) (7)

Therefore for any 119901 isin 1205951 DMUp is an influential DMU if

119863119901gt 119880 The main problem with this cut-off point however

is that both the mean and variance are nonrobust Extreme

values inflate the mean and variance yielding a high cut-offpoint This problem can be avoided by replacing mean andvariance by more robust estimators such as the median andthe median absolute deviation respectively as follows

119880 = med (119863) + 3radicMAD (119863) (8)

where

MAD (119863) = med |119863 minusmed (119863)| (9)

This criterion at first was proposed by Hadi [17] to detectinfluential observations in linear regression also for detailssee [10 18]

To illustrate this method we use the artificial data inTable 1 and results are shown in Table 2 In these data the cut-off point value is 119880 = 00211 DMU 119860 and DMU 119864 violatethis cut-off point therefore they are influential DMUs Letus be precise on these influential DMUs It is seen that thereare no changes on the efficiency scores of DMUs 119861 119862 119863and 119864 in the case of omitting DMU 119860 and their efficiencyscores are equal to one However some efficiency scores ofthe inefficient DMUs such as 119868 119869 and 119871 are increased Theefficiency score of DMU 119868 increases to 120579

119868119860= 07143 This

rise is noticeable As it can be seen in Figure 1 that omittingDMU 119860 makes a new frontier that is closer to the point 119868(the doted frontier) and it causes the increasing of efficiencyscores in the DMUs 119868 119869 and 119871 Besides by omitting DMU119864 the efficiency score of DMU 119865 becomes 1 This rise isalso noticeable since omitting DMU 119864makes a new frontierwhich crosses the point 119865 (the doted frontier)

Finally since 119863119860= 0 0387 and 119863

119864= 00487 so DMU 119864

is the most influential DMU


Table 3 Meteorological data

DMU 119884 1198831

1198832

1 025 122 292 029 202 0743 035 189 2144 032 176 2915 027 142 3556 027 103 1437 034 165 128 035 262 2179 026 062 14110 047 3 22411 034 215 10112 030 121 48513 029 131 26514 037 212 23615 042 288 10716 047 365 12917 046 293 24918 044 242 19519 041 286 22620 044 307 11921 100 287 16522 051 295 22223 052 372 21924 052 369 21625 046 351 21526 042 322 46427 042 295 16128 039 149 15829 042 306 18930 029 263 16331 043 306 22932 044 329 18533 042 204 18434 045 31 14635 047 365 15436 031 189 19137 043 316 28438 045 312 20939 040 261 16740 040 266 17241 024 145 07742 048 297 43543 091 335 3244 033 246 13945 041 345 23546 037 255 24447 049 393 35148 047 363 26549 059 402 14250 054 329 229119884 average solar radiation 119883

1 average duration of exposure sunlight and

1198832 average velocity of the wind

4 Empirical Example

In this section an empirical example is presented to examinethe proposed new method in Section 31 We provide mete-orological data of 50 regions in January 2010 (from TurkishStateMeteorological Service) which are shown inTable 3Thedata consist of one output variable 119884 and two input variables1198831 1198832 where 119884 is average solar radiation (wattm2) 119883

1is

average duration of exposure sunlight (hours) and 1198832is the

average velocity of wind (msec)Table 4 provides efficiency scores (120579) and superefficiency

scores (120579lowast) of these data It also provides diagnostic resultssuch as the efficiency scores (120579

119894119901) obtained by omitting 119901th

efficient DMU and square of Euclidian distance119863 discussedin Section 31

Using Table 4 it is seen that DMU2 DMU

9 DMU

21

and DMU41are efficient and their superefficiency scores are

sorted as 120579lowast21gt 120579lowast

9gt 120579lowast

41gt 120579lowast

2

Let us be precise on the case omitting DMU9and DMU

21

in Table 4 By omitting DMU9 inefficient DMU

6becomes

an efficient DMU and also efficiency scores of the inefficientDMU

1 DMU

5 and DMU

12have a salient increase DMU

12

is a remarkable example for this case and its efficiencyscore rises from 06008 to 09016 In the case of excludingDMU

21 there are salient increases on the scores of DMU

43

and DMU49

such that these two DMUs become efficientDMUs Furthermore the efficiency scores of all inefficientDMUs (except DMU

1) increase while these DMUs save

their inefficiency Evidently omitting an efficient DMU hasinfluence on the efficiency scores of inefficient DMUs whichare referenced by the efficient one

The119863distance for the efficientDMUs is presented on thelast column of Table 4 as follows 119863

2= 00147 119863

9= 04528

11986321

= 08841 and 11986341

= 00268 also the cut-off pointvalue is 119880 = 08572 With relying to the discussed measurein Section 31 it is clearly seen that only 119863

21gt 119880 therefore

DMU21is an influential DMU in these data

5 Conclusion

In this studywe classify noninfluentialDMUs in three groupswhich are as follows (1) the first group consists of efficientDMUs such that including or excluding the influential DMUhas not any effect on the efficiency scores of these DMUs(2) The second group consists of inefficient DMUs suchthat including or excluding the influential DMU has notany effects on the inefficiency of these DMUs (3) The thirdgroup consists of inefficient DMUs such that excluding theinfluential DMUsmakes them efficientDMUs ClearlyDMUsin the first type are on the efficiency frontier by the BCCmodel so their efficiency scores are equal to one Then wepropose a new method to detect influential DMUs whichis based on Euclidean distance and omitting the efficientDMUs by using single case deletion We apply this methodon meteorological data of 50 regions in January 2010 (fromTurkish State Meteorological Service) The method is alsomore practical than some of other similar methods since themeasure is based on the single case deletion


Table 4 Efficiency scores superefficiency scores and diagnostic results in the meteorological data

DMU 120579 120579lowast

1205791198942

1205791198949

12057911989421

12057911989441

119863

1 05082 05082 05082 08443 05082 05082 mdash2 1 11162 mdash 1 1 1 001473 05970 05970 05970 06660 06882 06113 mdash4 04810 04810 04810 06567 05291 04832 mdash5 04609 04609 04609 07259 04700 04609 mdash6 08678 08678 08678 1 08915 08920 mdash7 08598 08598 08598 08974 09834 09138 mdash8 05149 05149 05149 05479 05973 05401 mdash9 1 16613 1 mdash 1 1 0452810 05406 05406 05406 05650 07034 05599 mdash11 08420 08420 08737 08420 09556 08849 mdash12 06008 06008 06008 09016 06340 06008 mdash13 05446 05446 05446 08242 05714 05446 mdash14 05479 05479 05479 06094 06432 05600 mdash15 08496 08496 09150 08496 09926 08496 mdash16 07555 07555 08050 07555 08998 07555 mdash17 05107 05107 05107 05443 06553 05245 mdash18 06237 06237 06237 06597 07888 06453 mdash19 05161 05161 05161 05446 06357 05375 mdash20 07859 07859 08426 07859 09532 07859 mdash21 1 57643833 1 1 mdash 1 0884122 05698 05698 05698 05958 07649 05863 mdash23 05167 05167 05167 05211 07255 05410 mdash24 05217 05217 05217 05257 07326 05465 mdash25 05045 05045 05045 05122 06656 05315 mdash26 03380 03380 03380 04320 03943 03380 mdash27 06041 06041 06045 06041 07738 06464 mdash28 08228 08228 08228 09081 09887 08406 mdash29 05487 05487 05487 05582 06924 05822 mdash30 05451 05451 05451 05565 05827 05938 mdash31 05097 05097 05097 05333 06430 05309 mdash32 05445 05445 05445 05464 07081 05800 mdash33 06830 06830 06830 07350 08485 07025 mdash34 06655 06655 06929 06655 08630 06820 mdash35 06298 06298 06715 06298 08113 06298 mdash36 06055 06055 06055 06646 06624 06312 mdash37 04439 04439 04439 04775 05538 04564 mdash38 05359 05359 05359 05513 06913 05619 mdash39 06186 06186 06186 06328 07613 06569 mdash40 06097 06097 06097 06246 07562 06459 mdash41 1 11658 1 1 1 mdash 0026842 04304 04304 04304 05214 05137 04304 mdash43 07772 07772 07772 07907 1 07772 mdash44 06433 06433 06433 06474 07403 06988 mdash45 04637 04637 04637 04786 05782 04883 mdash46 04995 04995 04995 05430 05899 05161 mdash47 03826 03826 03826 04106 05024 03896 mdash48 04538 04538 04538 04726 05939 04705 mdash49 07881 07881 08235 07881 1 07881 mdash50 05468 05468 05468 05649 07559 05638 mdash

119880 = 08572


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees andeditors whose deep comments helped to improve the originalversion of this contribution

References

[1] I Alp and A Sozen ldquoEfficiency assessment of Turkeyrsquos car-bonization indexrdquo Energy Sources Part A Recovery Utilizationand Environmental Effects vol 33 no 18 pp 1678ndash1691 2011

[2] A Sozen I Alp and A Ozdemir ldquoAssessment of operationaland environmental performance of the thermal power plants inTurkey by using data envelopment analysisrdquo Energy Policy vol38 no 10 pp 6194ndash6203 2010

[3] M Mercan A Reisman R Yolalan and A B Emel ldquoThe effectof scale and mode of ownership on the financial performanceof the Turkish banking sector results of a DEA-based analysisrdquoSocio-Economic Planning Sciences vol 37 no 3 pp 185ndash2022003

[4] I Alp ldquoPerformance of evaluation of Goalkeepers of WorldCuprdquo Gazi University Journal of Science vol 19 no 2 pp 119ndash125 2006

[5] T R Anderson and G P Sharp ldquoA new measure of baseballbatters using DEArdquo Annals of Operations Research vol 73 pp141ndash155 1997

[6] P W Wilson ldquoDetecting influential observations in data envel-opment analysisrdquo Journal of Productivity Analysis vol 6 no 1pp 27ndash45 1995

[7] R D Cook ldquoDetection of influential observation in linearregressionrdquo Technometrics vol 19 no 1 pp 15ndash18 1977

[8] D A Belsley E Kuh and R E Welsch Regression DiagnosticsIdentifying Influential Data and Sources of Collinearity WilleySeries in Probability and Mathematical Statistics JohnWiley ampSons New York NY USA 1980

[9] R D Cook and S Weisberg Residuals and Influence in Regres-sion Chapman amp Hall New York NY USA 1982

[10] S Chatterjee and A S Hadi Regression Analysis by ExampleWilley Series in Probability and Mathematical Statistics JohnWiley amp Sons Hoboken NJ USA 2006

[11] J T Pastor J L Ruiz and I Sirvent ldquoStatistical test fordetecting influential observations in DEArdquo European Journal ofOperational Research vol 115 no 3 pp 542ndash554 1999

[12] J L Ruiz and I Sirvent ldquoTechniques for the assessment ofinfluence in DEArdquo European Journal of Operational Researchvol 132 no 2 pp 390ndash399 2001

[13] G R Jahanshahloo F Hosseinzadeh N Shoja G Tohidi andS Razavyan ldquoA method for detecting influential observationin radial DEA modelsrdquo Applied Mathematics and Computationvol 147 no 2 pp 415ndash421 2004

[14] A Charnes W W Cooper and E Rhodes ldquoMeasuring theefficiency of decision making unitsrdquo European Journal of Oper-ational Research vol 2 no 6 pp 429ndash444 1978

[15] R D Banker A Charnes and W W Cooper ldquoSome modelsfor estimating technical and scale inefficiencies in data envel-opment analysisrdquoManagement Science vol 30 no 9 pp 1078ndash1092 1984

[16] P Andersen and N C Petersen ldquoA procedure for rankingefficient units in data envelopment analysisrdquo ManagementScience vol 39 no 10 pp 1261ndash1264 1993

[17] A S Hadi ldquoA new measure of overall potential influence inlinear regressionrdquo Computational Statistics and Data Analysisvol 14 no 1 pp 1ndash27 1992

[18] P J Rousseeuw and C Croux ldquoAlternatives to the median abso-lute deviationrdquo Journal of the American Statistical Associationvol 88 no 424 pp 1273ndash1283 1993

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Stochastic AnalysisInternational Journal of


a method that is based on the superefficiency scores bymodified DEA which contains case deletion technique Thismethod allows researcher to prioritize observations in theefficient subset for the future scrutiny This prioritizationdepends on the number of efficiency scores that are influ-enced by a given observation

Pastor et al [11] propose a method for detecting influ-ential DMUs which is based on Uniformly Most PowerfulTest They consider BCC model and they define a ratio thatis calculated by division 119895th efficiency score obtained from allDMUs and 119895th efficiency score obtained by elimination of119901thefficient DMU Then they define a binary variable accordingto 119895th ratio either smaller than 095 or larger than 095Hence they obtain a binomial variable by sum of these binaryvariables Ruiz and Sirvent [12] use similar approach andthey propose an alternative method to identify the influentialDMUs in radial and nonradial DEA

A method proposed by Jahanshahloo et al [13] aimsto detect the influential DMUs by the way of deteriorationefficiency scores of inefficient DMUs in the radial DEATheyfocused on BCCmodel but they pointed out that the methodalso can be used in the CCR model This method is based ona specific ratio like the proposed ratio by Pastor et al [11]

In this study we propose a new method for detecting theinfluential DMUsThis newmethod is based on the Euclideandistance and excluding the efficient DMUs by using the singlecase deletion

The structure of this study is as follows The next sectionpresents some basic concepts ofDEA In Section 3 we discusson influential observation in DEA and we propose a newapproach for detecting influential DMUs Section 4 illustratesthe newmethod by an example Finally conclusions are givenin Section 5

2 Data Envelopment Analysis

Thefirst introduction onDEAwas practiced by Charnes et al[14]They proposedCCRmodel which is also called ConstantReturn to Scale (CRS) The CCR model evaluates bothtechnical and scale efficiencies via optimal value of the ratioform The modified version of CCR model is BCC modelwhich is also called Variable Returns to Scale proposed byBanker et al [15]The BCCmodel is used to estimate the puretechnical efficiency of DMUs by reference to the efficiencyfrontier

DEA can be applied in twomodels which are called input-and output-oriented models The primal form of input-oriented BCC (VRS) model is considered in this paper andit is given as follows

min 120579119900

subject to 120579119900119909119894119900minus

119899

sum

119895=1

120582119895119909119894119895ge 0

119899

sum

119895=1

120582119895119910119903119895minus 119910119903119900ge 0

119899

sum

119895=1

120582119895= 1

120582119895ge 0

119894 = 1 2 119898 119895 = 1 2 119899

119903 = 1 2 119904

(1)

where 120579119900

is efficiency score of DMU119900 119909119894119900

and 119910119903119900

(all nonnegative) are 119894th input and 119903th output of theDMU

119900 respectively and 120582

119895is intensity of DMU

119895 If

the 120579119900is equal to one then DMU

119900is called an efficient

DMUIn DEA a large number of efficient DMUs usually occur

in the results of analysis Therefore efficient DMUs cannotevaluate each other since their scores are equal to oneTo overcome this problem the analyst can either add newDMUs to data set or order the efficient DMUs by usingsome criterion Andersen andPetersen [16] proposed amodelto obtain superefficiency scores and these scores are usefulin both ordering the efficient DMUs and comparing thembetween one another in DEA

The primal form of input-oriented BCC (VRS) superef-ficiency model is considered in this paper and it is given asfollows

min 120579lowast

119900

subject to 120579lowast

119900119909119894119900minus

119899

sum

119895=1119895 =119900

120582119895119909119894119895ge 0

119899

sum

119895=1119895 =119900

120582119895119910119903119895minus 119910119903119900ge 0

119899

sum

119895=1119895 =119900

120582119895= 1

120582119895ge 0

119894 = 1 2 119898 119895 = 1 2 119899 119895 = 119900

119903 = 1 2 119904

(2)

where 120579lowast119900is the superefficiency score of DMU

119900 In the input-

oriented BCC model the superefficiency scores of efficientDMUs are greater or equal to one However superefficiencyscores of the inefficient DMUs are the same as their efficiencyscores that are obtained by BCC model in (1)

3 A New Measure for DetectingInfluential DMUs

DMUs consist of two groups which are influential DMUs andnoninfluential DMUs An influential DMU is defined as aDMU which affects the efficiency scores of some inefficientDMUs [6]This DMU also changes production possibility set



DMU 119883 119884 120579 120579lowast

119860 1 03 100 150119861 15 11 100 105119862 2 17 100 109119863 33 27 100 113119864 6 4 100 160119865 53 31 078 078119866 4 2 060 060119867 29 17 069 069119868 21 05 054 054119869 31 1 046 046119870 44 16 044 044119871 47 05 024 024







1and 120595



1 In


= 119860 119861 119862119863 119864 and 1205952

=




y

45

4

35

3

25

2

15

1

05

00 1 2 3 5 64 7 8

A

B

C

D

E

F

GH

I

J

K

L






1and 120595

2



119899times1 (3)


119894th DMU Let 120579119894119901




is an (119899 minus



120579(119901+1)119901


120579119899119901]119879


(4)



DMU 120579119894

120579119894119860

120579119894119861

120579119894119862

120579119894119863

120579119894119864

119863

119860 1 mdash 1 1 1 1 00387

119861 1 1 mdash 1 1 1 00005

119862 1 1 1 mdash 1 1 00057

119863 1 1 1 1 mdash 1 00044

119864 1 1 1 1 1 mdash 00487

119865 07794 07794 07794 07794 08368 1 mdash

119866 05975 05975 05975 06281 06304 05975 mdash

119867 06897 06897 06897 075 06897 06897 mdash

119868 05357 07143 05442 05357 05357 05357 mdash

119869 04637 04839 04839 04637 04637 04637 mdash

119870 04356 04356 04383 04688 04356 04356 mdash

119871 02394 03191 02432 02394 02394 02394 mdash119880 = 00211


119901 let 120579119901119901= 1


120601lowast


1 120579(119901+1)119901


120579119899119901]119879

119899times1

(5)



lowast

119901is

given as below

119863119901=10038171003817100381710038171003817120601all minus 120601

lowast

119901

10038171003817100381710038171003817

2

2

=

119899

sum

119894=1

(120579119894minus 120579119894119901)2

(6)










119901| 119901 isin 120595



119880 = mean (119863) + 3radicVar (119863) (7)





119880 = med (119863) + 3radicMAD (119863) (8)

where

MAD (119863) = med |119863 minusmed (119863)| (9)



119868119860= 07143 This



119864= 00487 so DMU 119864




DMU 119884 1198831

1198832




4 Empirical Example


1is







9 DMU

21



9gt 120579lowast

41gt 120579lowast

2


21


6becomes


1 DMU

5 and DMU


12



43

and DMU49





2= 00147 119863

9= 04528

11986321

= 08841 and 11986341




5 Conclusion




DMU 120579 120579lowast

1205791198942

1205791198949

12057911989421

12057911989441

119863


119880 = 08572




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











DMU 119883 119884 120579 120579lowast

119860 1 03 100 150119861 15 11 100 105119862 2 17 100 109119863 33 27 100 113119864 6 4 100 160119865 53 31 078 078119866 4 2 060 060119867 29 17 069 069119868 21 05 054 054119869 31 1 046 046119870 44 16 044 044119871 47 05 024 024







1and 120595



1 In


= 119860 119861 119862119863 119864 and 1205952

=




y

45

4

35

3

25

2

15

1

05

00 1 2 3 5 64 7 8

A

B

C

D

E

F

GH

I

J

K

L






1and 120595

2



119899times1 (3)


119894th DMU Let 120579119894119901




is an (119899 minus



120579(119901+1)119901


120579119899119901]119879


(4)



DMU 120579119894

120579119894119860

120579119894119861

120579119894119862

120579119894119863

120579119894119864

119863

119860 1 mdash 1 1 1 1 00387

119861 1 1 mdash 1 1 1 00005

119862 1 1 1 mdash 1 1 00057

119863 1 1 1 1 mdash 1 00044

119864 1 1 1 1 1 mdash 00487

119865 07794 07794 07794 07794 08368 1 mdash

119866 05975 05975 05975 06281 06304 05975 mdash

119867 06897 06897 06897 075 06897 06897 mdash

119868 05357 07143 05442 05357 05357 05357 mdash

119869 04637 04839 04839 04637 04637 04637 mdash

119870 04356 04356 04383 04688 04356 04356 mdash

119871 02394 03191 02432 02394 02394 02394 mdash119880 = 00211


119901 let 120579119901119901= 1


120601lowast


1 120579(119901+1)119901


120579119899119901]119879

119899times1

(5)



lowast

119901is

given as below

119863119901=10038171003817100381710038171003817120601all minus 120601

lowast

119901

10038171003817100381710038171003817

2

2

=

119899

sum

119894=1

(120579119894minus 120579119894119901)2

(6)










119901| 119901 isin 120595



119880 = mean (119863) + 3radicVar (119863) (7)





119880 = med (119863) + 3radicMAD (119863) (8)

where

MAD (119863) = med |119863 minusmed (119863)| (9)



119868119860= 07143 This



119864= 00487 so DMU 119864




DMU 119884 1198831

1198832




4 Empirical Example


1is







9 DMU

21



9gt 120579lowast

41gt 120579lowast

2


21


6becomes


1 DMU

5 and DMU


12



43

and DMU49





2= 00147 119863

9= 04528

11986321

= 08841 and 11986341




5 Conclusion




DMU 120579 120579lowast

1205791198942

1205791198949

12057911989421

12057911989441

119863


119880 = 08572




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











DMU 120579119894

120579119894119860

120579119894119861

120579119894119862

120579119894119863

120579119894119864

119863

119860 1 mdash 1 1 1 1 00387

119861 1 1 mdash 1 1 1 00005

119862 1 1 1 mdash 1 1 00057

119863 1 1 1 1 mdash 1 00044

119864 1 1 1 1 1 mdash 00487

119865 07794 07794 07794 07794 08368 1 mdash

119866 05975 05975 05975 06281 06304 05975 mdash

119867 06897 06897 06897 075 06897 06897 mdash

119868 05357 07143 05442 05357 05357 05357 mdash

119869 04637 04839 04839 04637 04637 04637 mdash

119870 04356 04356 04383 04688 04356 04356 mdash

119871 02394 03191 02432 02394 02394 02394 mdash119880 = 00211


119901 let 120579119901119901= 1


120601lowast


1 120579(119901+1)119901


120579119899119901]119879

119899times1

(5)



lowast

119901is

given as below

119863119901=10038171003817100381710038171003817120601all minus 120601

lowast

119901

10038171003817100381710038171003817

2

2

=

119899

sum

119894=1

(120579119894minus 120579119894119901)2

(6)










119901| 119901 isin 120595



119880 = mean (119863) + 3radicVar (119863) (7)





119880 = med (119863) + 3radicMAD (119863) (8)

where

MAD (119863) = med |119863 minusmed (119863)| (9)



119868119860= 07143 This



119864= 00487 so DMU 119864




DMU 119884 1198831

1198832




4 Empirical Example


1is







9 DMU

21



9gt 120579lowast

41gt 120579lowast

2


21


6becomes


1 DMU

5 and DMU


12



43

and DMU49





2= 00147 119863

9= 04528

11986321

= 08841 and 11986341




5 Conclusion




DMU 120579 120579lowast

1205791198942

1205791198949

12057911989421

12057911989441

119863


119880 = 08572




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











DMU 119884 1198831

1198832




4 Empirical Example


1is







9 DMU

21



9gt 120579lowast

41gt 120579lowast

2


21


6becomes


1 DMU

5 and DMU


12



43

and DMU49





2= 00147 119863

9= 04528

11986321

= 08841 and 11986341




5 Conclusion




DMU 120579 120579lowast

1205791198942

1205791198949

12057911989421

12057911989441

119863


119880 = 08572




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











DMU 120579 120579lowast

1205791198942

1205791198949

12057911989421

12057911989441

119863


119880 = 08572




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









A New Measure for Detecting Influential DMUs in DEA

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