Zia i, Shok ollah; Raissi, Sadigh
A icle
Ranking e icien DMUs using minimizing dis ance in DEA
Jou nal o Indus ial Enginee ing In e na ional
P o ided in Coope a ion wi h:
Islamic Azad Uni e si y (IAU), Teh an
Sugges ed Ci a ion: Zia i, Shok ollah; Raissi, Sadigh (2016) : Ranking e icien DMUs using minimizing
dis ance in DEA, Jou nal o Indus ial Enginee ing In e na ional, ISSN 2251-712X, Sp inge ,
Heidelbe g, Vol. 12, pp. 237-242,
h ps://doi.o g/10.1007/s40092-016-0141-2
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ORIGINAL RESEARCH
Ranking e icien DMUs using minimizing dis ance in DEA
Shok ollah Zia i
1
•Sadigh Raissi
2
Recei ed: 24 July 2015 / Accep ed: 4 Janua y 2016 / Published online: 22 Janua y 2016
ÓThe Au ho (s) 2016. This a icle is published wi h open access a Sp inge link.com
Abs ac In many applica ions, anking o decision
making uni s (DMUs) is a p oblema ic echnical ask p o-
cedu e o decision make s in da a en elopmen analysis
(DEA), especially when he e a e ex emely e icien
DMUs. In such cases, many DEA models may usually ge
he same e iciency sco e o di e en DMUs. Hence, he e
is a g owing in e es in anking echniques ye . The main
pu pose o his pape is o o e come he lack o in easi-
bili y and unboundedness in some DEA anking me hods.
The p oposed me hod is o anking ex eme e icien
DMUs in DEA based on exploi ing he lea e-one ou and
minimizing dis ance be ween DMU unde e alua ion and
i ual DMU.
Keywo ds Da a en elopmen analysis (DEA) Ranking 
E iciency Ex eme e icien
In oduc ion
Da a en elopmen analysis (DEA) was ini ia ed by Cha nes
e al. (1978) as a me hod o assess ela i e e iciency o
homogeneous decision making uni s wi h mul iple inpu s
and mul iple ou pu s. Then, Banke e al. (1984) ex ended
basic DEA models unde e u ns o scale. As ega ds, he
mos models o DEA a e in oduced he mo e han one
e icien DMU in e alua ing he ela i e e iciency DMUs,
hus he in es iga ing ank o e icien DMUs is an in e -
es ing esea ch opic. A DMU is called ex emely e icien
i i canno be ep esen ed as a linea combina ion (wi h
nonnega i e coe icien s) o he emaining DMUs (Coope
e al. 2007). In da a en elopmen analysis, he e a e se e al
me hods o anking o he ex eme e icien DMUs, e.g.
AP (Ande sen and Pe e sen 1993) me hod, MAJ (Meh a-
bian e al. 1999) me hod. Ande sen and Pe e sen p oposed
a new p ocedu e o ank e icien DMUs. The AP me hod
exhibi s he ank o a gi en DMU by emo ing i om he
e e ence se and by compu ing i s supe e iciency sco e.
Howe e , he AP model may be in easible in some cases. I
is p o ed ha supe e icien DEA models a e in easible
(see Th all 1996, Coope e al. 2007, Sei o d and Zhu 1999,
Cha nes e al. 1989). Meh abian e al. (Cha nes e al. 1978)
sugges ed as MAJ model o comple e anking e icien
DMUs, bu hei app oach lacks in easibili y in some cases,
oo. To o e come he d awbacks o he AP (Ande sen and
Pe e sen 1993) and MAJ (Meh abian e al. 1999) models,
Jahanshahloo e al. (2004a) p esen ed a me hod o ank he
ex emely e icien DMUs in DEA models wi h cons an
and a iable e u ns o scale using L
1
-no m. The p oposed
model is a nonlinea p og amming o m which has he
compu a ional complexi y in sol ing. A complex ea men
was applied in Jahanshahloo e al. (2004a) o con e he
nonlinea model in o a linea one which p o ides an
app oxima ely op imal solu ion. Wu and Yan (2010) ha e
also used an e ec i e ans o ma ion o con e he non-
linea model in Jahanshahloo e al. (2004a) in o a linea
model. Also Jahanshahloo e al. (2004b) ha e applied
g adien line o anking e icien uni s. Rezai Bal e al.
(2012) applied Tchebyche no m (L
1
-no m) in oduced in
(B iec 1998; Ta a es e al. 2001) o comple e anking
&Shok ollah Zia i
[email p o ec ed]
Sadigh Raissi
[email p o ec ed]
1
Depa men o Ma hema ics, Fi oozkooh B anch, Islamic
Azad Uni e si y, Fi oozkooh, I an
2
School o Indus ial Enginee ing, Islamic Azad Uni e si y,
Sou h Teh an B anch, Teh an, I an
123
J Ind Eng In (2016) 12:237–242
DOI 10.1007/s40092-016-0141-2
e icien uni s. Ami eimoo i e al. (2005) in oduced a
me hod o anking o ex eme e icien DMUs, based on
dis ance. Hashimo o (1999) p oposed a supe e iciency
DEA model wi h assu ance egion in o de o ank he
DMUs comple ely. To gesen e al. (1996) sugges ed a
me hod o anking e icien uni s, by hei impo ance as
benchma ks o he ine icien uni s. Sex on e al. (1986)
in es iga ed a anking me hod o DMUs based on a c oss-
e iciency a io ma ix. The c oss-e iciency anking
me hod compu es he e iciency sco e o each DMU ha
de e mines a se o op imal weigh s using linea p og ams
co esponding o each DMU. Then by aking he a e age o
sco es o gi en DMU is ob ained he ank o ha DMU.
Liu and Peng (2008) de e mined one common se o
weigh s o anking e icien DMUs, ha DMUs a e anked
acco ding o he e iciency sco e weigh ed by he common
se o weigh s. Bal e al. (2008) sugges ed a DEA model o
anking o DMUs based on de ining he coe icien o
a ia ion o inpu –ou pu weigh s. Khodabakhshi and
A ya ash (2012) p oposed a me hod o ank he e icien
DMUs. Acco ding o hei me hod, i s he minimum and
maximum e iciency alues o each DMU a e compu ed
unde he assump ion ha he sum o e iciency alues o
all DMUs is equal o uni y. Then, he ank o each DMU is
de e mined in p opo ion o a combina ion o i s minimum
and maximum e iciency alues. She y and Pakkala (2010)
sugges ed a me hod o anking e icien uni s, which is
c ea ed he a e age o he co esponding inpu s and ou pu s
o all DMUs. Ea ly, Jahanshahloo and Fi oozi Shahmi zadi
(2013) modi ied he model which was p oposed by Bal
e al. (2008). They in oduced wo new models o anking
e icien DMUs based on L
1
-no m and using mean o
inpu –ou pu weigh s. Fo ou new me hod i does no need
any addi ional cons ain s.
In his pape , we sugges a new me hod o anking
ex eme e icien DMUs. The es o he pape is o ganized
as ollows. In ‘‘DEA models and anking models e iew’’,
we e iew he concep o DEA amewo k. We e iew
some anking me hods in ‘‘The p oposed anking model o
e icien DMUs’’, ‘‘Ex ension o a iable e u ns o scale’’
p oposes he new model o anking e icien uni s. ‘‘Il-
lus a ed examples’’ includes some nume ical examples.
The las sec ion concludes he s udy.
DEA model and anking model e iew
DEA model e iew
DEA is a me hodology o assessing he ela i e e iciency
o decision making uni s (DMUs) whe e each DMU has
mul iple inpu s used o secu e mul iple ou pu s.
I is assumed in DEA ha he e a e nDMUs and o
each DMU
j
ðj¼1;...;nÞis conside ed a column ec o o
inpu s ðXjÞ o p oduce a column ec o o ou pu s ðYjÞ,
whe e Xj¼ðx1j;x2j;...;xmjÞTand Yj¼ðy1j;y2j;...;ysjÞT.
He e, he supe sc ip ðTÞindica es a ec o anspose. I is
also assumed ha Xj0;Yj0;Xj6¼ 0;and Yj6¼ 0 o
e e y j¼1;...;n.
The ollowing inpu -o ien ed CCR model [see (Coope
e al. 2007)] in he en elopmen o m wi h cons an
Re u ns o Scale measu es he le el o DEA e iciency ðhÞ
o he k h DMU ðXk;YkÞ:
h¼min h
s. . X
n
j¼1
kjxij hxik;i¼1;...;m
X
n
j¼1
kjy j y k; ¼1;...;s
kj0;j¼1;...;n
ð1Þ
He e, k¼ðk1;...;knÞTis a column ec o o unknown
a iables used o componen s o he inpu and ou pu
ec o s by a combina ion. h ep esen s he e iciency sco e
o DMU
k
in (1), whe e he supe sc ip (*) indica es
op imali y.
DMU
k
is ela i ely e icien i and only i on op imali y,
he objec i e o (1) equals o one and all he slacks a e ze o.
Simila ly, he ou pu -o ien ed CCR model, co espond-
ing o (1), is o mula ed as ollows:
/¼max /
s: :P
n
j¼1
kjxij xik;i¼1;...;m
P
n
j¼1
kjy j /y k; ¼1;...;s
kj0;j¼1;...;n
ð2Þ
He e, 1=/in ends he DEA e iciency sco e in he
ou pu -o ien ed model.
Also, he ollowing inpu -o ien ed BCC model [see
Banke e al. (1984)] in he en elopmen o m wi h a i-
able Re u ns o Scale measu es he le el o DEA e iciency
ðhÞo he k h DMU ðXk;YkÞ:
h¼min h
s. . P
n
j¼1
kjxij hxik;i¼1;...;m
P
n
j¼1
kjy j y k; ¼1;...;s
P
n
j¼1
kj¼1;
kj0;j¼1;...;n
ð3Þ
238 J Ind Eng In (2016) 12:237–242
123
DMU
k
is ela i e e icien i and only i on op imali y,
he objec i e o (3) equals o one and all he slacks a e ze o.
Simila ly, he ou pu -o ien ed BCC model, co espond-
ing o (3) which ob ains om (2) by adding cons ain ,
X
n
j¼1
kj¼1:
Mo eo e , he ollowing addi i e model is based on inpu
and ou pu slacks which accoun s he possible inpu
dec eases as well as ou pu inc eases simul aneously.
max X
n
j¼1
s
iþX
n
j¼1
sþ
s: :P
n
j¼1
kjxij þs
ixik;i¼1;...;m
P
n
j¼1
kjy j sþ
/y k; ¼1;...;s
kj;s
i;sþ
;0;
ð4Þ
DMU
k
is ela i e e icien i and only i on op imali y,
he objec i e o (4) equals o ze o.
Ranking models
In his subsec ion we e iew he some anking models in
da a en elopmen analysis. The i s anking model p o-
posed by Ande son and Pe e son (1993) which is he sup-
pe e iciency model. In he AP model DMU unde
e alua ion is excluded om e e ence se and by using
o he uni s, he ank o gi en DMU is ob ained.
The AP model using he CRS supe -e iciency model is
as ollows:
AP:min h
s: :P
n
j¼1j6¼k
kjxij hxik;i¼1;...;m
P
n
j¼1j6¼k
kjy j y k; ¼1;...;s
kj0;j¼1;...;n;j6¼ k
ð5Þ
The main d awbacks o his model a e in easibili y and
ins abili y o some DMUs. I is said ha a model is
s able i a DMU unde e alua ion is e icien , i is emains
e icien a e pe u ba ion on da a.
The second anking model unde in es iga ion p oposed
by Meh abian e al. (1999) o sol e in easibili y o AP
models in some cases. The ollowing model is MAJ model:
MAJ: min 1 þw
s. P
n
j¼1j6¼k
kjxij xik þw;i¼1;...;m
P
n
j¼1j6¼k
kjy j y k; ¼1;...;s
kj0;j¼1;...;n;j6¼ k
ð6Þ
The hi d anking model p oposed by Jahanshahloo e al.
(2004a), ha hei p oposed me hod o ank he ex emely
e icien DMUs in DEA models wi h cons an and a iable
Re u ns o Scale using he omi ed DMU unde e alua ion
om p oduc ion possibili y se and applying L
1
-no m. I is
shown ha he p oposed me hod is able o o e come he
exis ing di icul ies in he AP (Ande sen and Pe e sen
1993) and MAJ (Meh abian e al. 1999) models. On he
o he hand, he p oposed model is he o m o nonlinea
p og amming which is di icul o be sol ed. The model o
Jahanshahloo e al. (2004a) is p esen ed as ollows:
L1no m:min X
m
i¼1
xixik
jj
þX
s
¼1
y y k
jj
s: :P
n
j¼1j6¼k
kjxij xi;i¼1;...;m
P
n
j¼1j6¼k
kjy j y ; ¼1;...;s
xi0;y 0i¼1;...;m; ¼1;...;s
kj0;j¼1;...;n;j6¼ k
ð7Þ
The ou h anking model p oposed by Rezai Bal e al.
(2012) which applies o anking ex eme e icien uni s
using he lea e-one-ou idea and L1-no m. The p oposed
model is always easible and so, i is able o emo e he
exis ing di icul ies in some me hods, such as Ande sen and
Pe e sen (1993). The model o Rezai Bal e al. (2012)is
o mula ed as ollows:
L1no m :min mk
s: :mkP
n
j¼1j6¼k
kjxij xik;i¼1;...;m
mky k P
n
j¼1j6¼k
kjy j; ¼1;...;s
kj0;j¼1;...;n;j6¼ k
mk0
ð8Þ
J Ind Eng In (2016) 12:237–242 239
123
The p oposed anking model o e icien DMUs
In his sec ion, we suppose ha he DMUk is ex eme
e icien . By excluding he DMUk om he CCR p oduc-
ion possibly se , i is ob ained a new e iciency on ie . In
o de o gain he anking sco e o DMUk by exploi ing he
new e iciency on ie , we sugges a new model by by
using he lea e-one ou idea and minimizing dis ance
be ween DMU unde e alua ion and i ual DMU. The
p oposed model is as ollows:
min P
m
i¼1
aiþP
s
¼1
b
s: :P
n
j¼1j6¼k
kjxij xik ai;i¼1;...;m
P
n
j¼1j6¼k
kjy j y k þb ; ¼1;...;s
kj0;j¼1;...;n;j6¼ k;
ai0;b 0i¼1;...;m ¼1;...;s;
ð9Þ
whe e a¼ða1;...;amÞ,b¼ðb1;...;bsÞand k¼
ðk1;...;kk1;kkþ1;...;knÞa e he a iables o he model
(9).
Theo em 1 The model (9)is easible and bounded.
P oo Fo p6¼ kwe se kp¼1;kj¼0;j¼1;...;n;j6¼
k;p;ai¼min xik xipg;i¼1;...;m;b ¼min y p
y kg; ¼1;...;s:
Ob iously, i can be seen ha ðk;a;bÞacco ding o
abo e selec ion is a easible solu ion o he model (9).
Mo eo e , he objec i e unc ion o model (9) is bounded
below ze o, because he a iables o model a e nonnega-
i e. Also, he a ge unc ion is ze o when ai¼0 and b ¼
0 o all i; h.
Ex ension o a iable e u ns o scale
In his sec ion, he p oposed model in p e ious sec ion is
ex ended o a iable Re u ns o Scale model. Fo his
pu pose, he model (9) is e o mula ed by adjoining he
ollowing con exi y cons ain o he model:
X
j¼1
n
j6¼k
kj¼1;kj0:
So, in o de o ge he anking sco e unde a iable e u ns
o Scale assump ion is sol ed he ollowing model:
min P
m
i¼1
aiþP
s
¼1
b
s: :P
n
j¼1j6¼k
kjxij xik ai;i¼1;...;m
P
n
j¼1j6¼k
kjy j y k þb ; ¼1;...;s
P
n
j¼1j6¼k
kj¼1;
kj0;j¼1;...;n;j6¼ k;
ai0;b 0i¼1;...;m; ¼1;...;s;
ð10Þ
Theo em 2 The model (10)is easible and bounded.
P oo The p oo o his heo em is simila o he p oo o
Theo em 1.
Table 1 Inpu and ou pu da a o Example 1
DMU Inpu 1 Inpu 2 Ou pu 1 Ou pu 2
1 81 87.6 5191 205
2 85 12.8 3629 0
3 56.7 55.2 3302 0
4 91 78.8 3379 8
5 216 72 5368 639
6 58 25.6 1674 0
7 112.2 8.8 2350 0
8 293.2 52 6315 414
9 186.6 0 2865 0
10 143.4 105.2 7689 66
11 108.7 127 2165 266
12 105.7 134.4 3963 315
13 235 236.8 6643 236
14 146.3 124 4611 128
15 57 203 4869 540
16 118.7 48.2 3313 16
17 58 47.4 1853 230
18 14 650.8 4578 217
19 0 91.3 0 508
Table 2 Resul s o anking by di e en models
DMU 12591519
AP anking esul s 4 1 3 – 2 –
MAJ anking esul s 5 3 2 6 4 1
L1-no m anking esul s 4 3 2 6 5 1
L1-no m anking esul s 5 2 3 6 4 1
P oposed model anking
esul s
53426 1
240 J Ind Eng In (2016) 12:237–242
123

Illus a ed examples
In his sec ion, we employ he abo e DEA model (6) and
(7) on he wo da a se s which hey a e in oduced he e,
wi h he assump ion o cons an e u ns o scale.
Example 1 As can be seen om Table 1, he da a se
consis s o 19 DMUs wi h 2 inpu s and 2 ou pu s. The da a
o iginally a e used by Rezai Bal e al. (2012). Table 2
epo s he esul s o anking o 6 ex emely e icien
DMUs ðD1;D2;D5;D9;D15;D19Þin model (7) wi h con-
s an Re u ns o Scale and he p oposed me hod is com-
pa ed wi h Ap, MAJ, L1and L1. The esul s imply ha he
model p oposed in his pape p o ides a easy ool o
anking ex emely e icien DMUs. The alue o inpu s and
ou pu s.
Example 2 (Empi ical example). We employ DEA model
(10) on he empi ical example used in Zhu (1998), wi h he
assump ion o a iable Re u ns o Scale. The da a se in
Table 3p o ides 13 open coas al Chinese ci ies and i e
Chinese special economic zones in 1989. Two inpu s and
h ee ou pu s we e chosen o cha ac e ize he echnology o
hose ci ies/zones. Two inpu s include In es men in ixed
asse s by s a e-owned en e p ises, Fo eign unds ac ually
used. Th ee ou pu s include To al indus ial ou pu alue,
To al alue o e ail sales and Handling capaci y o coas al
po s. Table 4 epo s he esul s o anking o 10 ex e-
mely e icien DMUs ðD1;D2;D5;D6;D7;D9;D10;D11;
D13;D16Þin model (10) wi h a iable e u ns o scale and
he p oposed me hod a e compa ed wi h o he me hods.
Conclusion
Many DEA esea ches a e p oposed on anking o e icien
decision making uni s, bu hey ha e a p oblem, e.g. he AP
model may be in easible in some cases. In he p esen
pape , we p oposed a model o anking ex eme e icien
DMUs in DEA by exploi ing he lea e-one ou and mini-
mizing dis ance be ween DMU unde e alua ion and i -
ual DMU. The p oposed model is linea o m and always
easible and bounded. The e o e, i is able o ank all
ex eme e icien DMUs in he DEA me hods wi h con-
s ain and a iable Re u ns o Scale and so, elimina e he
Table 3 The alue o inpu s
and ou pu s DMU# ci ies/zones Inpu 1 Inpu 2 Ou pu 1 Ou pu 2 Ou pu 3
Dalian 2874.8 16,738 160.89 80,800 5092
Qinhuangdao 946.3 691 21.14 18,172 6563
Tianjin 6854.0 43,024 375.25 44,530 2437
Qingdao 2305.1 10,815 176.68 70,318 3145
Yan ai 1010.3 2099 102.12 55,419 1225
Weihai 282.3 757 59.17 27,422 246
Shanghai 17,478.6 116,900 1029.09 351,390 14,604
Lianyungang 661.8 2024 30.07 23,550 1126
Ningbo 1544.2 3218 160.58 59,406 2230
Wenzhou 428.4 574 53.69 47,504 430
Guangzhou 6228.1 29,842 258.09 151,356 4649
Zhanjiang 697.7 3394 38.02 45,336 1555
Beihai 106.4 367 7.07 8236 121
Shenzhen 4539.3 45,809 116.46 56,135 956
Zhuhai 957.8 16,947 29.20 17,554 231
Shan ou 1209.2 15,741 65.36 62,341 618
Xiamen 972.4 23,822 54.52 25,203 513
Hainan 2192.0 10,943 25.24 40,267 895
Table 4 Resul s o se e al models anking
DMU 12567910111316
AP 9184632 7 5 10
MAJ 183497106 2 5
L1-no m 483691107 2 5
L1-no m 384691107 2 5
P oposed
model
173289105 6 4
J Ind Eng In (2016) 12:237–242 241
123
exis ing di icul ies in some me hods. In addi ion, i can be
easily used when he numbe o inpu s and ou pu s is much
la ge han he numbe o DMUs. Illus a i e examples a e
included o show good anking esul s by he p oposed
me hod.
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made.
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