Ranking units in DEA by using the voting system

Full text

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Int. J.IndustrialMathematisVol. 3,No. 3(2011)181-191

Ranking Units in DEA by Using the Voting

System

Iran.

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

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

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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

[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

The fth researh area ranks ineÆient units through proportional measures of

ineÆ-ieny. The last approah requires the olletion of additional, preferential information

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

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

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[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

ignore. So,itseemslogialto trydierentmodelsandombinetheresultsofthedierent

models and viewpoints. In this paper, we onsider the various ranking modelsas voters

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

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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

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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.

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

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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

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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

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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

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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

asthehighest-rankingunit islogial. A similarpointholdsforunits 4and7,whihgetpositions

2 and 3. Butit may seem that this method will assign the highest rankto theunit whih has

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.

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

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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.

Referenes

[1℄ N. Adler, L. Friedman, Z. Sinuany-Stern, Review of ranking methods in data envelopment

analysisontext,Eur.J.Oper.Res.140(2002)249-265.

[2℄ A.R. Amirteimoori, G.R. Jahanshahloo, S. Kordrostami, Ranking of deision makingunits

in data envelopment analysis adistane-based approah, Appl. Math. Comput. 171(2005)

122-135.

[3℄ A.R. Amirteimoori, S. Kordrostami, EÆient surfaes and an eÆieny index in DEA: A

onstantreturnsto sale,Appl.Math.Comput.163(2005)683-691.

[4℄ P. Andersen, N. C. Petersen, A proedure for ranking eÆient units in data envelopment

analysis,Mngt.Si.39(1993)1261-1264.

[5℄ A.Charnes,W.W.Cooper,E.Rhodes,MeasuringtheeÆienyofdeisionmakingunits.Eur.

J.Oper.Res.2(1978)429-444.

Si.36(1990)1302-1310.

[7℄ F.H.FranklinLiu, H.H.Peng,Rankingofunits onthe DEAfrontier withommonweights,

Comput.Oper.Res.35(2008)1624-1637.

[8℄ L. Friedman, Z. Sinuany-Stern, Saling units via theanonial orrelation analysis and the

dataenvelopmentanalysis.Eur.J.Oper.Res.100(3)(1997),629-637.

[9℄ R.H. Green,J.R. Doyle, W.D.Cook,Preferene voting andprojetrankingusingDEA and

ross-evaluation,Eur.J.Oper.Res.90(1996)461-472.

[10℄ A. Hashimoto, A ranked voting system using aDEA/AR exlusion model: a note, Eur. J.

Oper.Res.97(1997)600-60.

ofweightbymultipleobjetiveprogrammingindataenvelopmentanalysis,Southwestj.pure

appl.Math.1(2000)44-54.

[12℄ G.R. Jahanshahloo, F. Hosseinzadeh Lot, F. Rezai Balf, H. Zhiani Rezai, D. Akbarian,

RankingeÆientDMUsusing thebyhenorm.(2004)WorkingPaper.

L1-normin dataenvelopmentanalysis,Appl.Math.Comput.153(2004)215-224.

[14℄ G.R.Jahanshahloo,H.V. Junior, F.Hosseinzadeh Lot, D. Akbarian, A new DEAranking

systembasedonhangingtherefereneset, Eur.J.Oper.Res.181(2007)331-337.

[15℄ G.R. Jahanshahloo, A. Memariani, F. Hosseinzadeh Lot, H.Z. Rezai, A note on some of

(11)

[16℄ G.R.Jahanshahloo,L.Pourkarimi,M.Zarepisheh,ModiedMAJmodelforrankingdeision

makingunits indataenvelopmentanalysis,Appl.Math.Comput.174(2006)1054-1059.

[17℄ M.Khodabakhshi,A super-eÆienymodel basedonimprovedoutputsindataenvelopment

analysis,Appl.Math.Comput. 184(2007)695-703.

[18℄ S.Mehrabian,M.R. Alirezaei,G.R. Jahanshahloo,A ompleteeÆieny rankingofdeision

makingunits: anappliationto theteahertraininguniversity, Comp.Opt.Appl.14(1998)

261-266.

[19℄ H.Noguhi,M.Ogawa,H.Ishii,TheappropriatetotalrankingmethodusingDEAformultiple

ategorizedpurposes,J.Comput.Appl.Math.146(2002)155-166.

[20℄ K. Tone, A slaks-based measure of super-eÆieny in data envelopment analysis, Eur. J.

Oper.Res.143(2002)32-41.

[21℄ T.Obata,H.Ishii,AmethodfordisriminatingeÆientandidateswithrankedvotingdata,

Eur.J.Oper.Res.151(2003)233-237.

[22℄ M.S.Saati,M.ZarafatAngiz,G.R.Jahanshahloo,Amodelforrankingdeisionmakingunits

indataenvelopmentanalysis.Reraoperative.31(1999)47-59.

[23℄ T.R.Sexton,R.H.Silkman,A.J.Hogan,Dataenvelopmentanalysis:Critiqueandextensions.

In:Silkman,R.H.(Ed.),MeasuringEÆieny: AnAssessmentofDataEnvelopmentAnalysis.

Jossey-Bass,SanFraniso,CA,pp.73-105,1986.

[24℄ L.Shanling,G.R.Jahanshahloo,M.Khodabakhshi,super-eÆienymodelforrankingeÆient

unitsindataenvelopmentanalysis,Appl.Math. Comput.184(2007)638-648.

[25℄ Z. Sinuany-Stern,L. Friedman. Data envelopmentanalysis andthe disriminant analysis of

ratiosforrankingunits.Eur.J.Oper.Res.111(1998)470-478.

envelop-mentanalysis.Comput.Oper.Res.21(5) (1994)543-556.

andanappliationto Japaneseagriultureooperatives.Omega.27(1999)315-326.

[28℄ Y.M. Wang, K.S. Chin, Disriminating DEA eÆient andidates by onsidering their least

relativetotalsores,J.Comput.Appl.Math.206(2007)209-215.

[29℄ N.S.Wang,R.H.Yi, D.Liu,AsolutionmethodtotheproblemproposedbyWanginvoting

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