Int. J.IndustrialMathematisVol. 3,No. 3(2011)181-191
Ranking Units in DEA by Using the Voting
System
M. Khanmohammadi a
,R. Fallahnejad b
(a)Department ofMathematis,SieneandResearh Branh, IslamiAzadUniversity,Eslamshar,
Iran.
(b) Departmentof Mathematis,KhorramAbadBranh,IslamiAzadUniversity, Khorram Abad,
Iran.
Reeived 29Otober2010;revised 2June2011; aepted 13June2011.
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One of the main problems in Data Envelopment Analysis (DEA) is ranking Deision
MakingUnits (DMUs). Thereexist manydierent DEAmodels. Thesemodelsoftendo
nothaveanytheoretialproblemsdealingwithmostdata. However,beauseeahofthese
modelsonsidersa ertain theoryforranking,they may givedierent ranks. So, oftenin
pratie,hoosingarankingmodel,theresultsofwhihtheDeisionMaker(DM)wouldbe
ableto trustisan importantissue. Inthisartile, rankingisdonebyproposinga method
in whih the ranks of dierent ranking modelsare used, eah of whih is important and
signiant. Thismethod isbasedon thevoting system. Invoting systems,one andidate
may reeive dierent votes in dierent ranking plaes. The total sore of eah andidate
is the weighted sum of the votes that the andidate reeives in dierent plaes. The
andidate that has the highest total sore has the best rank. In thispaper, we onsider
thevarious rankingmodelsasvoterswhihan rankDMUs from the topto theendand
DMUs asandidatestry to obtainafullrank fromtheirvotes.
Keywords: Dataenvelopmentanalysis;Ranking;Rankvotingsystems.
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1 Introdution
Data envelopment analysis (DEA) wasoriginated in1978 byCharnes et al.[5 ℄and the
rst DEA model was alled the CCR (Charnes, Cooper and Rhodes) model. DEA is a
linearprogrammingbasedtehniqueformeasuringtherelativeeÆienyofafairly
homo-geneous set of deisionmaking units (DMUs) in their use of multiple inputs to produe
multiple outputs. It identies a subset of eÆient 'best pratie' DMUs and for the
re-mainingDMUs,themagnitudeoftheirineÆienyisderived byomparison to afrontier
onstrutedfrom the'bestpraties'. EÆientDMUs areidentiedbyan eÆienysore
equal to 1, and ineÆient DMUs have eÆieny sores less than 1. Although eÆieny
sore an be a riterion for ranking ineÆient DMUs, thisriterion annot rank eÆient
DMUs. Inthelastdeade,a varietyof modelsweredevelopedto rankDMUs. Adleret.al
[1 ℄ dividedthe ranking methods into some areas. The rst area involves the evaluation
of a ross-eÆieny matrix, in whih the unitsare self- and peer-evaluated. The seond
area, generally knownas thesuper-eÆieny method,ranks throughtheexlusion of the
unit from the produtionpossibilityset and analyzingthe hangein thePareto frontier.
The third grouping is based on benhmarking, in whih a unit is highly ranked if it is
hosen as a useful target for many other units. The fourth group utilizes multivariate
statistialtehniques,whiharegenerally appliedaftertheDEAdihotomylassiation.
The fth researh area ranks ineÆient units through proportional measures of
ineÆ-ieny. The last approah requires the olletion of additional, preferential information
fromrelevant deision-makersandombinesmultiple-riteriadeisionmethodologieswith
the DEA approah. However, whilst eah tehnique is useful ina speialist area, no one
methodology an bepresribed here astheomplete solutionto the questionof ranking.
Hene,seletingthebestrankingmodelorthewayof ombiningdierentrankingmodels
for ranking DMUs is an important point in ranking DMUs in DEA. In this paper, we
propose a methodology, based uponthe voting system, forranking DMUs. This method
is espeially appliable if we annot prefer any ranking model on the others. In voting
systems, one andidate mayreeive dierent votes in dierent ranking plaes. The total
sore of eah andidate is the weighted sum of the votes that the andidate reeives in
dierent plaes. The one that has the highest total sore has the best rank. Although
the andidatesinranked voting systems are regardedas DMUs inDEA, and eah DMU
is onsideredto have t outputs(ranked votes) and onlyone inputwith amount unity, in
ourapproah we onsider therankingmodelsasvotersand DMUsasandidates.
Thispaperhasbeenorganized asfollows. Insetion 2wereview some rankingmodelsin
the voting system. Setion 3 introdues our proposed method. A numerial example is
given insetion4, and setion5 ontainsouronlusions.
2 Ranking models for voting systems
Inthissetionwebrieydesribesomeof theexistingrankedvoting systemswhihan
beseenintheliterature. Inrankedvotingsystemseahvoterseletsandranksandidates
inorderofpreferene. Itisassumedthattherearenotiesineahvoter'sranking. Thetotal
soreofeahandidateistheweightedsumofthevoteshe/shereeivesindierentplaes.
The problem is to determine an ordering of all n andidates by obtaining a total sore.
Some ofthevoting systemsassign xedweightsto thedierentranks andtheandidates
with the highest sore are the winners. But it is lear that the winning andidate an
varyaordingtotheweightsused. Toavoidthisproblem,CookandKress [6 ℄proposeda
DEAmodeltoassesseahandidatewiththemostfavorableweights. DEAoftensuggests
thatmorethanoneunitisequallyeÆient,i.e.,theyahievethemaximumsore. Forthis
[9 ℄ have proposed a disrimination method using a ross-evaluation matrix. Noguhi et
al. [19 ℄ haveproposedthefollowingmodeltoselet thewinnerinwhihastrongordering
onstraint onditionisimposedon weights:
max Z i = m X j=1 v ij w j s:t: m X j=1 v ij w j
1; i=1;:::;n;
w
1 2w
2
:::mw
m ; w m = 2 Nm(m+1) : (2.1)
in whih it is assumed that eah voter selets andidates among andidates and ranks
them from topto the plae, whih is the relative importaneweight assoiated with the
plae , denotesthe numberof the th plae votes earned byandidate and is the number
of voters. In this paper we use this model for ranking. Hashimoto [10 ℄ has proposed a
super-eÆieny model to Cook and Kress's ranked voting model. Obata and Ishii [21 ℄
have suggested exludingnon-DEA eÆient andidatesand usingnormalized weightsfor
disrimination. Wang andChin[28 ℄haveproposedamethodthatdisriminatestheDEA
eÆientandidatesbyonsideringtheirbestandleastrelativetotalsores. Sinetheleast
relative total sores and the best relative total sores are notmeasured withinthe same
range in[28 ℄,Wang et al. [29 ℄ have proposeda method forsolvingsuh aase.
3 Our proposed method
3.1 Illustration
As wasmentioned earlier, there is a great variety of DEA ranking modelsfor ranking
DMUs, e.g., ross-eÆieny, super-eÆieny, benhmarking,statistial tehniquesand so
on. Also, there exist dierent viewpoints for utilizing the DEA tehnique; for instane,
input-orientedandoutput-orientedviewsinsomeofthemsuhasAndersenandPetersen's
modelinvariablereturns tosaletehnologies. Now, themainproblemistheseletion of
themostsuitablemodelandviewpoint,whihisbeausetherankingsofunitsobtainedby
variousmodelsmaynotbethesame. Forexample,one ranking modelmayassigna rank
A to a DMU whileanother one assigns a rank Bto the same DMU. Thus, one may not
trusttherankobtainedbyaertainrankingmodel. Ifweanpreferarankingmodelover
others,then thereisnotanyproblem. Butwe annotusuallyseletthebest. Eah ofthe
above-mentionedmodelsandviewpointshassome advantages whihwewould likenotto
ignore. So,itseemslogialto trydierentmodelsandombinetheresultsofthedierent
models and viewpoints. In this paper, we onsider the various ranking modelsas voters
whihanrankDMUsfromthetoptotheendandtrytoobtainafullrankfromtheirvotes.
Inthissetion,wedesribeourproposedmethodwithasimpleexample,taken from
Sexton et al. [23 ℄, Adler et al. [1 ℄ and Jahanshahloo et al. [14℄. There are six DMUs,
eah using two inputs to produe two outputs. The raw data are presented in Table 1.
Adleret al. [1 ℄ ranked these six DMUsusingsome ranking models. We usethefollowing
SBM[20 ℄,SA DEA[27 ℄, L-Innity[12 ℄ andDistane-based approah[2 ℄,all of whihan
belassiedassuper-eÆienymodels,andthree CommonSetofWeights(CSW)models
whihwe denotebyCSW1,CSW2andCSW3(CSW1[11 ℄,CSW2[15 ℄,CSW3[7℄),
Cross-EÆienymodel[23 ℄,and threestatistial methods: CCA(Canonialorrelation analysis
[8 ℄) , DDEA (linear disriminant analysis [28 ℄) and DR/DEA (Disriminant analysis of
ratios [26 ℄). In this example, we onsider onstant returns to sale. As an be seen in
thelast olumnof Table 1,DMUs A, B, Cand D areCCR-eÆient and DMUs E and F
are ineÆient. So, some of the ranking models an rank DMUs E and F by these CCR
eÆienysores. Inall ofsuhrankingmodelsDMUEobtainsrank5and DMUFreeives
rank 6. Sine the numberof suh models is more thanothers, usually, their ranking for
ineÆient DMUs an be appeared intherankingpresentedbyour ranking.
Table1
Rawdatafornumerialexample.
DMU Input Input Output Output CCReÆieny
A 150 0.2 14000 3500 1
B 400 0.7 14000 21000 1
C 320 1.2 42000 10500 1
D 520 2.0 28000 42000 1
E 350 1.2 19000 25000 0.978
F 320 0.7 14000 15000 0.868
Now, onsider Table 2,whihuses 17 rankingmodelsto rank thesix DMUs inTable
1. Table2showstheranksassignedtoDMUsbyrankingmodels. Theresults
orrespond-ing to the statistial-based modelsCCA, DDEA and DR/DEA are taken from [1 ℄. The
numberof rank j assigned to alternative i is easily alulated (see Table 3). In Table 4,
resultsobtainedbymodel1andtheproposedrankaregiven. Byusingthismethodwean
obtain the rank for eah DMU with more ertainty, beause this method has inherently
onsideredvariousviewpointsbasedonwhihtherankingmodelsusedinthemethodare
onstruted.
Table2
RankingDMUsby17RankingModels.
RankingModel A B C D E F
AP 1 2 3 4 5 6
L1 4 3 1 2 5 6
Ch-Re-Set 2 1 4 3 5 6
Maj 4 3 1 2 5 6
M-Maj 2 1 3 4 6 5
LJK 4 3 1 2 5 6
SBM 1 3 2 4 5 6
SADEA 1 2 3 4 5 6
LInnity 3 2 1 4 5 6
CSW1 2 5 1 3 4 6
CSW2 1 5 2 3 4 6
CSW3 1 4 3 2 5 6
CrossEÆieny 1 3 2 4 5 6
CCA 1 2 3 4 5 6
DDEA 3 1 2 4 5 6
DR/DEA 1 5 2 3 4 6
Table3
Thenumberofassigningeahrankto eahDMU.
DMU Plae1 Plae2 Plae3 Plae4 Plae5 Plae6
A 8 4 2 3 0 0
B 3 4 5 2 3 0
C 5 5 6 0 1 0
D 1 4 4 8 0 0
E 0 0 0 3 13 1
F 0 0 0 0 1 16
Table4
Proposed rankforDMUs.
A B C D E F
Model1results 1 0.680292 0.849635 0.5547445 0.3080292 0.2510949
ProposedRank 1 3 2 4 5 6
3.2 Some pratial points
A few pointsareworthmentioningwithrespetto theproposedmethod:
1: Although we an onsider a ertain group of ranking modelssuh as super-eÆieny
models, it is better to onsider theother ranking modelsinthe other groups, as well. It
is lear that ombining the results of dierent DEA ranking models in dierent groups
provides more realistirankingresultsformanagers. Almost allof themodelshave some
shortomings, but using the ombination of the resultsobtained from them may redue
theeets oftheir shortomings.
2: As mentioned before, in voting systems it is assumed that there are no ties in eah
voter's ranking. So we mustuseonlytherankingmodelswhihan givedierent ranking
sores to DMUs.
3:Someoftherankingmodelsaredependentonaspeitehnology,apointwhihmust
be onsidered while using them. For example, ross-eÆieny is used only for onstant
returns to sale tehnology, and the super-eÆieny model based on improved outputs
proposedbyKhodabakhshi[17℄ mustbe usedforvariable returnsto saletehnology,
be-ause using it foronstant returns to salewill give the same resultsas the LJK model.
Another point is about theranking modelswhih are dependent on the orientation, like
AP. Whenthey areusedforvariable returnsto sale,input-and output-orientedranking
modelsmay give dierent ranks, thuseahof them an bea voter.
5:Itmayhappenthatweonfronttheaseinwhihadeisionmakerpreferssomeranking
modelsto the others. Moreover, thevoters may nothave thesame value to thedeision
maker. In suh ases, we an onsider various weights for the ranking models as voters
and therefore their ranks. For example, deision makers an determine the importane
of the ranking models by using the pairwise omparison matries. The weights an be
6: Many of the introdued modelsmay not be able to rank in some ases in whih they
have rarelyhappenedto fall. Thisshouldnotause to ignoresuh modelsinother ases.
As an example, as we know, AP model has some problems and, therefore, after the
in-trodution of the model by Andersen and Petersen, many of the researhers introdued
modelsto overome these problems. However, inspiteof the problems, themodel isstill
inuseforreasons like simpliityintheimplementationorits theoretial basis.
7:Another pointis thatwe donot onlyonsiderranking eÆientunits. In some models,
like ommon set of weights model, the number of eÆient units is less than some other
DEA models and, therefore, some of the units that are eÆient in some models may be
ineÆientinsome othermodels. Moreover, thosemodelsthatan rankonlyeÆientunits
an beonsidered asmodelsthat rankall units, beause theeÆienysore of ineÆient
units may be onsidered as their ranking sore. In addition, beause in most ranking
models, it an be seen that ineÆient units have a lower rank, therefore, the result of
ranking allthe units,isinlinedto resultsof ranking onlyeÆient units.
3.3 Steps of the proposed method
Step1: Determine whih of the DMUs must be ranked, all of them or only a subset of
them suh astheeÆientDMUs.
Step2: Seletsuitableranking modelsand thenrank DMUswith them.
Step3:Considereahoftherankingmodelsasavoterand determinethenumberofeah
rank assignedto eahDMU.
Step4: Usemodel1or otherranking voting systemmodelsto rank DMUs.
4 Empirial Example
Letusrank20Iranianbankbranhesbyourproposedmethod. Thedataanbeseenin
[3 ℄and[14 ℄(seeTable5). AsanbeseeninthelastolumnofTable5,DMUs1,4,7,12,15,17
and 20 areCCReÆient.
Now, onsider Table 6,in whih 14 ranking modelsare used to rank these DMUs.
Mod-els used in this example an be seen under Table 6. Table 6 shows the ranks assigned
to DMUs by ranking models. The number of rank j assigned to DMUi is given inTable
Table5
DMUs'Data (EmpirialExample)andTheirCCREÆienies.
Branh Inputs Outputs eÆieny
Sta Computerterminals Spae Deposits Loans Charge
1 0.950 0.700 0.155 0.190 0.521 0.293 1.000
2 0.796 0.600 1.000 0.227 0.627 0.462 0.833
3 0.798 0.750 0.513 0.228 0.970 0.261 0.991
4 0.865 0.550 0.210 0.193 0.632 1.000 1.000
5 0.815 0.850 0.268 0.233 0.722 0.246 0.899
6 0.842 0.650 0.500 0.207 0.603 0.569 0.748
7 0.719 0.600 0.350 0.182 0.900 0.716 1.000
8 0.785 0.750 0.120 0.125 0.234 0.298 0.798
9 0.476 0.600 0.135 0.080 0.364 0.244 0.789
10 0.678 0.550 0.510 0.082 0.184 0.049 0.289
11 0.711 1.000 0.305 0.212 0.318 0.403 0.604
12 0.811 0.650 0.255 0.123 0.923 0.628 1.000
13 0.659 0.850 0.340 0.176 0.645 0.261 0.817
14 0.976 0.800 0.540 0.144 0.514 0.243 0.470
15 0.685 0.950 0.450 1.000 0.262 0.098 1.000
16 0.613 0.900 0.525 0.115 0.402 0.464 0.639
17 1.000 0.600 0.205 0.090 1.000 0.161 1.000
18 0.634 0.650 0.235 0.059 0.349 0.068 0.473
19 0.372 0.700 0.238 0.039 0.190 0.111 0.408
20 0.583 0.550 0.500 0.110 0.615 0.764 1.000
Table6
RankingDMUsby14RankingModels
D DMU,M1AP [4℄,M2L1[13℄, M3 hangingtherefereneset[14℄,M4MAJ[4℄,M5 Modied
Maj2 [22℄, M6 LJK[24℄, M7 SBM[20℄, M8 SA DEA [27℄, M9 L Innity [12℄, M10 CSW1[11℄,
M11CSW2[15℄,M12CSW3[7℄,M13CrossEÆieny[23℄,M14 Distane-based[2℄.
DMU M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14
D1 7 7 7 7 7 7 7 7 7 13 13 7 6 7
D2 10 10 10 13 13 10 10 14 12 11 12 10 9 10
D3 8 8 8 8 8 8 8 8 8 6 5 8 7 8
D4 2 2 2 2 2 2 2 2 2 1 1 1 2 2
D5 9 9 9 11 11 9 9 9 11 8 9 9 8 9
D6 14 14 14 16 14 14 14 12 14 7 6 14 13 14
D7 5 3 3 3 5 3 4 3 5 1 2 2 4 3
D8 12 12 12 9 9 12 12 13 9 16 16 12 11 12
D9 13 13 13 10 10 13 13 10 10 12 11 13 12 13
D10 20 20 20 20 19 20 20 20 19 20 20 20 20 20
D11 16 16 16 15 17 16 16 16 17 14 15 16 16 16
D12 6 6 5 6 6 6 6 5 6 4 3 4 5 6
D13 11 11 11 12 12 11 11 11 13 10 10 11 10 11
D14 18 18 18 19 20 18 18 18 20 18 17 18 18 18
D15 1 1 1 1 1 1 1 1 1 3 8 3 1 1
D16 15 15 15 18 15 15 15 15 15 15 14 15 15 15
D17 3 4 6 5 3 4 3 4 3 9 7 6 14 4
D18 17 17 17 14 16 17 17 17 16 18 18 17 17 17
D19 19 19 19 17 18 19 19 19 18 19 19 19 19 19
Table 7
Thenumberofeahrankassignedto eahDMU.
DMU P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
D1 0 0 0 0 0 1 11 0 0 0
D2 0 0 0 0 0 0 0 0 1 7
D3 0 0 0 0 1 1 1 11 0 0
D4 3 11 0 0 0 0 0 0 0 0
D5 0 0 0 0 0 0 0 2 9 0
D6 0 0 0 0 0 1 1 0 0 0
D7 1 2 6 2 3 0 0 0 0 0
D8 0 0 0 0 0 0 0 0 3 0
D9 0 0 0 0 0 0 0 0 0 4
D10 0 0 0 0 0 0 0 0 0 0
D11 0 0 0 0 0 0 0 0 0 0
D12 0 0 1 2 2 8 0 0 0 0
D13 0 0 0 0 0 0 0 0 0 3
D14 0 0 0 0 0 0 0 0 0 0
D15 11 0 2 0 0 0 0 1 0 0
D16 0 0 0 0 0 0 0 0 0 0
D17 0 0 4 4 1 2 1 0 1 0
D18 0 0 0 0 0 0 0 0 0 0
D19 0 0 0 0 0 0 0 0 0 0
D20 0 0 1 6 6 1 0 0 0 0
Table8
Thenumberofeahrankassignedto eahDMU.
DMU P11 P12 P13 P14 P15 P16 P17 P18 P19 P20
D1 0 0 2 0 0 0 0 0 0 0
D2 1 2 2 1 0 0 0 0 0 0
D3 0 0 0 0 0 0 0 0 0 0
D4 0 0 0 0 0 0 0 0 0 0
D5 3 0 0 0 0 0 0 0 0 0
D6 0 1 1 9 0 1 0 0 0 0
D7 0 0 0 0 0 0 0 0 0 0
D8 1 7 1 0 0 2 0 0 0 0
D9 1 2 7 0 0 0 0 0 0 0
D10 0 0 0 0 0 0 0 0 2 12
D11 0 0 0 1 2 9 2 0 0 0
D12 0 0 0 0 0 0 0 0 0 0
D13 8 2 1 0 0 0 0 0 0 0
D14 0 0 0 0 0 0 1 10 1 2
D15 0 0 0 0 0 0 0 0 0 0
D16 0 0 0 1 12 0 0 1 0 0
D17 0 0 0 1 0 0 0 0 0 0
D18 0 0 0 1 0 2 9 2 0
D19 0 0 0 0 0 0 1 2 11 0
Table9
Proposed rankforDMUs.
DMU Resultsofmodel1 Proposedmodel
D1 0.16044733 7
D2 0.10973526 10
D3 0.15981827 8
D4 0.77260492 2
D5 0.12913588 9
D6 0.09965829 14
D7 0.43713193 3
D8 0.10257234 12
D9 0.10143072 13
D10 0.0598103 20
D11 0.07504528 16
D12 0.2199945 6
D13 0.10777632 11
D14 0.06504684 18
D15 1 1
D16 0.07861349 15
D17 0.27070503 5
D18 0.07097821 17
D19 0.06350943 19
D20 0.27427886 4
As an beseenin Table9,rstposition isassigned toD15. It an beseenin Table8that
this unit getsrank 1by11 models, rank3by 2models andrank8 by onlyone model. Beause
ithasbeenassignedrankone by11votes,outofatotalof 14votes, thereforehoosing thisunit
asthehighest-rankingunit islogial. A similarpointholdsforunits 4and7,whihgetpositions
2 and 3. Butit may seem that this method will assign the highest rankto theunit whih has
reeivedthemostvotes. Thismeansthat ifvotersgivethemostvotesforthet'thpositiontothe
l'thunit,thentheproposedmethoddoesso,aswell. Also,itmaybesuspetedthattheseletion
isbasedon lexiographimaximum. Butthis isnotthease, beauseaordingto lexiographi
maximumriteriaunit 17mustberankedhigherthanunit 20, whiletheproposed model assigns
position4to unit20andposition5tounit17.
5 Conlusion
IthappensoftenthatinrealproblemsinDataEnvelopmentAnalysis,wewouldliketorankthe
DeisionMakingUnits. ThereexistmanydierentDEAmodels. Hene,seletingthebestmodel
for ranking DMUs is a main questionin DEA. These models do not often haveany theoretial
problems dealing with most data. Beause eah of these models onsiders a ertain theory for
ranking,theymaygivedierentranks. So,ofteninpratie,hoosingarankingmodeltheresults
ofwhihtheDeisionMaker(DM)wouldbeabletotrustis animportantissue. Inthisartilea
method hasbeenproposedbywhih rankingwillbedonebyusingtheranksofdierentranking
models, eah ofwhih is importantand signiant. This method is basedonthe voting system.
Invotingsystems,one andidatemayreeivedierentvotesin dierentrankingplaes. Thetotal
soreof eah andidateis the weightedsum ofthe votesthat the andidate reeives in dierent
plaes. The andidate that has the biggest total sore has the highest rank. In this paper we
haveonsideredvarious rankingmodelsas voterswhihan rankDMUsfrom thetoptothe end
and DMUs as andidates, and tried to obtain a full rank from their votes. One of the most
one viewpoint,so itan besaidthat eahof therankingmodelsshowsonlysomeperent ofthe
realityand,therefore,usingonlyonemodel,intheasethatwewouldbeabletohooseit,should
beuntrustworthy. Therefore,theproposed methodprovidesthepossibilityofusingtheresultsof
allexistingrankingmodels. So,itsresultswillbemorereliablefortheDM.
Another strong point of the proposed method is that, as mentioned before, eah of the
ranking models is onstruted based on aertain theory or viewpoint, but it may happen that
theDMisinlinedtorankingbaseuponviewpointsthathavenotdenedanyrankingmodelsfor
them. Theproposedmodelhastheexibilitytodealwithsuhases. Inthisartile,theproposed
methodhasbeenusedforranking20Iranianbankbranhestodemonstrateitsvalidity.
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