Journal of Applied Mathematics and Physics, 2013, 1, 1-5
doi:10.4236/jamp.2013.11001 Published Online February 2013 (http://www.scirp.org/journal/jamp)
Copyright © 2013 SciRes. JAMP
A Ranking Method of Extreme Efficient DMUs Using
Super-Efficiency Model*
Dariush Akbarian
Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran.
Email: d_akbarian@yahoo.com, d-akbarian@iau-arak.ac.ir
Received 2013
Abstract
In this paper, we present a method for ranking e xtreme efficient dec ision making units (DMUs) in data envelopment
analysis (DEA) models based on measuring distance between them and new PPS (after omissio n extreme efficie nt
DMUs ) along the input -axis or outp ut axis.
Keywords: Data envelopment ana lysis; Effici ency; Rank i ng.
1. Introduction
Data envelopment analysis (DEA) is a non-parametric
method for measuring efficiency of a set of Decision
Making Units (DMUs) such as firms or a public sector
agencies, first introduced by Charnes, Cooper and
Rhod es (CCR) [1] and extended by Banke r, Cha rnes, and
Cooper (BCC) [2]. One important issue in DEA which
has been studied by many DEA researchers, is to discri-
minate between efficient DMUs. Several authors have
proposed methods for ranking the best performers
([3]-[10] among others). Ying-Ming Wang et al. [10]
pro pose d a ranki ng methodology for DMUs by imposing
an appro priate minimum weight restriction on all inp uts
and outputs, which is decided by a decision maker (DM)
or an assessor in terms of the solutions to a series of li-
near programming (LP) models that are specially con-
structed to determine a maximin weight for each DEA
efficient unit. Jahanshahloo et al. [4] proposed a ranking
system based on changing reference set. in the proposed
ranking system, the evaluation for efficient DMUs is
dependent of the efficiency changes of all inefficient
units due to its absence in the reference set while the
estimate for inefficient DMUs depends on the influence
of the e xclus ion of each efficient unit from the reference
set. Fo r a review of ra n ki ng me tho d s, re ad ers are referred
to Adler et al. [8]; in which the previous methods were
divided into six categories. One of the six areas is
well-known as the super-efficiency approach, which was
first proposed by Andersen and Petersen (AP) [9] to rank
extreme efficient DMU s. The main idea of this approach
is to evaluate a DMU after this performer itself is ex-
cluded from the reference set. However, in some cases,
especially under the condition of variable returns to scale
(VRS) , the met hod ma y fail due to the i nfeasibili ty prob-
lem associated with the super-efficiency models. In this
paper, we intend to introduce a new ranking system for
extreme efficien t DMUs under the co ndition of VRS and
CRS. For this aim, we use a variance of super-efficiency
models (see models (6) and (7)) and obtai n the most dis-
tance between them and new PPS (after omission ex-
treme efficient DMUs) along the input-axis or out-
put-axis. Also, our proposed method is able to rank ex-
treme efficient DMUs even in presence of infeasibility.
This paper is organized as follows. Section 2 presents
some basic DEA models. Section 3 introduces our pro-
posal and states and proves some facts related to proper-
ties and characteristics of it. A numerical example is
given in Section 4 and Section 5 comprehends our con-
clusions.
2. Background
Consider a set of n DMUs which is associated with m
inputs and s outputs. Particularl y,  ( ϵ= {1, ..., n})
consume s amount xij of input i and produces amount yrj
of output r. Let =(1,… , ) in which 0 &
0 and
=1,…, in which 0 & 0.
The production possibility set (PPS) of CCR model de-
fine as follows:
( )
1
1
{, |,,0,}
nn
cj
j
jjj j
j
TXY XXYYjJ
λ λλ
=
=
=≥≤ ≥∈
and similarly the productio n possibility set of BCC mod-
el define as follows:
D. Akbarian ET AL.
Copyright © 2013 SciRes. JAMP
2
( )
1 11
{, |,,1,0,}
n nn
vjjjj jj
j jj
TXY XX YXjJ
λλλλ
== =
=≥≤= ≥∈
∑ ∑∑
By omitting ,
fro m , the new pro ducti on pos-
sibility set is as follows:
{}
{}
{(,)|,,0,{ }}.
nn
cjjjJpj jj
jJ p
TXYXX YYjJp
λ λλ
∈−
∈−
= ≥∈−≥≤
∑∑
In fig ur e (1) the polyhedral

and

are
and
, respectively.
The inp ut-o ri e nte d BC C and i np ut -oriented CCR models,
corresponds to,  , is given by (1) and (2), re-
spectively:
1
min ()
ms
ir
ia r
ss
θ
+−
= =
−∈ +
∑∑
. .1,,
1,, (1)
1
j ijiik
jJ
j rjrrk
jJ
j
jJ
stxsx im
y syrs
λθ
λ
λ
+
+= =…
−= =…
=
0
j
λ
jJ
0
i
s
1, ,im= …
0
r
s
+
θ
free
11
min min()
m
ir
s
ir
tt
θ
= =
+−
−∈ +
∑∑
. .1,,
1,, (2)
j ijiik
jJ
j rjrrk
jJ
stxtx im
y tyrs
λθ
λ
+
+= =…
−= =…
0
j
λ
jJ
0
i
t
1, ,im= …
0
r
t
+
θ
free
where ϵ is non-Archimedean small and positive number
and
+,
,
+and
, =1, …, , =1, … ,are called
slack variables belong to 0. Note that
and
represent input excesses; also
+and
+ represent output
shortfalls. The models (1) and (2) are called envelopment
forms (with non-Archimedean number).  is said to b e strong efficient (CCR-efficie nt) if and
only if:
*
= 1 and t*+ = 0, t*- = 0. Where the superscript
(*) indicates optimalit y. In similar manner the BCC- effi-
cient DMUs can be defined.
The AP model is as follows [9 ]:
{}
{}
. .1,,
1,, (3)
j ijip
jJ p
j rjrp
jJ p
stxxim
y yrs
λθ
λ
∈−
∈−
≤=…
≥=…
0
j
λ
jJ
θ
free
The Jahanshahloo’s method corresponding inefficient
 is as foll ows : [4]
,, 1
}
1
{
min min()
. .1,,
ms
ab abi
ir
r
j ijiia
jJ b
ss
stxsx im
θ
λθ
+−
=
=
∂ ∂=−+
+= =…
∑∑
{}
1,, (4)
j rjrra
jJ b
y syrs
λ
+
∈−
−= =…
0
j
λ
{}jJ b∈−
0
i
s
1, ,im=…
0
r
s
+
The efficiency of strong efficiency 
will be de-
noted by Ω and will be given by:
,
Ω
n
ab
aJ
b
n
=
in which is the set of non-strong efficienc y DMUs and
 is the number of non-strong efficiency DMUs.
Jaha nsha hlo o et al. [1 4] used 1-nor m in or der to ra nk the
extremely efficient DMUs in DEA models with constant
and variable returns to scale, and the proposed method
can remove the difficulties arising from AP and MAJ
models. Their proposed model is as follows:
11
minΓ(,) ms
c
pi iprrp
ir
XYx xyy
= =
= −+−
∑∑
{}
. .1,,
j iji
jJ p
stxxim
λ
∈−
≤=…
{}
1,, (5)
j rjr
jJ p
y yrs
λ
∈−
≥=…
0
i
x
1, ,im=…
0
r
y
0
j
λ
{}jJ p∈−
In this paper we rank DMUs in CCR model; in a similar
way one can also rank DMUs in BCC model. The fol-
lowing super-efficiency models are used for ranking ex-
treme efficient DMUs [12]:
D. Akbarian ET AL.
Copyright © 2013 SciRes. JAMP
3
min
{ }
..
pp
j ljllk
jJ p
stx x
λθ
∈−
{}
1, (6)
p
j ijik
jJ p
xximi l
λ
∈−
≤ =…≠
{}
1, ,
p
j rjrk
jJ k
y yrs
λ
∈−
≥=…
0
p
j
λ
{}j Jp
min p
q
ϕ
{}
. .1,,
p
j ijip
jJ k
stxxim
µ
∈−
≤=…
{}
,1,, (7)
p
j rjrp
jJ p
yy rsrq
µ
∈−
≤ =…≠
{}
pp
j qjq qp
jJ p
yy
µϕ
∈−
0
p
j
µ
in whi c h =1, … , and =1, … ,. In ord er to rank DMUs in BCC model t he constraint
=
{}1 is is added to (6) and (7) (see [13]).
Remark 1: In (6) and (7), for each and ,
=
=
1 if and only i f st rong e ffi cient  lies on the str ong
defining hyperplane of PPS. In fact it is an interio r point
of strong defini ng h yper p lane .
Figure 1: The value distance of 
from new PP S (v)
a long t he x-a xis and y-axis
Remark 2: In (6) (or (7)), if for some (or ),
>1
(or
<1) or if for some l(or q), model (6) (or model
(7)) is infeasible then, stro ng efficient 
lies on the
extreme ray (edge) of PPS and vice versa. (For more
details on models (6) and (7) see [12].)
We call all efficie nt DMUs lying on extreme ray (edge)
of PPS of CCR model as extreme efficient DMUs, he-
reafter.
Remark 3: In multiple output case, if for some
model
(7) is infeasible then, virtual DMU
( )
'11
,,, ,,,,
kkmk kqksk
DMUxx yyy
γ
=……−…
in
which > 0, is on the weak defining hyperplane of PPS
vertical to hyperp lane =0.
Remark 4: In multiple inputs case, if for a t le a st one
,
model (6) is infeasible then v irtual DMU

=(1,…,+,…,,1,…,) in whic h
> 0, is on the weak defining hyperplane which passes
through th axis of input .1
3. A proposed method for ranking by su-
per-efficiency model
We state the following theorem without proof.
Theorem 1: If there exist at least two DEA-efficient
DMUs then, there is at least one
(or
) so that model (6)
(or model (7)) is feasible.
First, we evaluate each , ( ϵ ), by models (2).
Suppose that DMUs are strong efficient. Without lose
of generality we can assume that these efficient DMUs
are 1 ,…,  . Consider the set = {1 , ..., }.
Then, corr espondi ng to each 
, ( ϵ ), we solve the
models (6) and (7). In view of remarks 1,2 we can iden-
tify all extreme efficient DMUs.
Corresponding each extreme efficient 
we ob-
tain
 and
, = 1,…, , = 1,..., .
Note that for so me and the models (6) and (7) may be
infeasible. But by theorem 1 for 
there exist at
least for one or so that the models (6) or (7) have fi-
nite optimal solution.
We have:
 
 1=The value distance of 
from new
PPS (Tc) along the th axis of input.
 1 
=The value distance of 
from new
PPS (Tc) along the th axis of output.
Where it is understood that the above value distances are
taken ove r existing
and
(see Fig. 1).
Let
=max
,{ 
 1, 1
}
In order to judge which DMU has better rank in compar-
ison with other DMUs, the following definition is given:
Definition.  has a better rank in comparison with
 if
>
.
The following theorem shows that our proposed method
has more influence to the PPS of DEA models than
1-method has .
1For mor e det ail see [12 ].
D. Akbarian ET AL.
Copyright © 2013 SciRes. JAMP
4
Theorem 2:
(,) 
.
Proof. The proof is straightforward.
4. Numerical Example
We evaluated with our method the data of 20 branch
banks of Iran. This data was previously analyzed by
Amirteimoori and Kordrostami [11] and is listed in Ta ble
1. The use of our method generated the analysis shown in
Table 2, in which, the statement “infs” means “infeasi-
ble”. Table 3 shows a comparison of our proposal and
some other ranking approaches. All these approaches are
implemented in input-oriented version under the condi-
tion of CRS. As reported in Table 3, 15 is the most
efficient one in our method and other methods. Accord-
ing to t he results , the ran kings o f DMUs by the four me-
thods are almost similar; in particular, the results of our
method are more similar to the method [4].
5. Conclusion
In this paper we propose a method for ranking extreme
efficient DMUs based on measuring distance between
extreme efficient DMUs and new PPS (after omission
extreme efficient DMUs) along the input-axis or out-
put-axis, using supper efficiency models (6) and (7). It
seems that our approach is more robust than other me-
thod [14]; because, as it was shown in theorem 2, our
proposed method has more influence in new PPS than
the proposed method by Jahanshahloo et al. [14]. Initial
studies had shown that our approach also can be applied
with BCC model. We suggest as future works a deeper
analysis in t hi s subject.
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