An Extended Aggregated Ratio Analysis in DEA
Liang Liang*
School of Management
University of Science and Technology of China
He Fei, An Hui Province, P.R. China 230026
Email: lliang@ustc.edu.cn
Jie Wu
School of Management
University of Science and Technology of China
He Fei, An Hui Province, P.R. China 230026
Email: jacky012@mail.ustc.edu.cn
[005-0135]
Submitted to POMS International Conference - Shanghai 2006
May 2006
An Extended Aggregated Ratio Analysis in DEA
Liang Liang, Jie Wu
Abstract: We propose an extended aggregated ratio analysis model that is similarly extended from
CCR to C2GS2 model in DEA. This extended model also offers an insight into frontier analysis.
Whether a DMU is on the frontier or efficient frontier can be informed by proposed extended
model. Several results developed in the paper are coincident with that in the literature.
Key words: Data envelopment analysis (DEA); C2GS2 model; Ratio; Efficient frontier
1. Introduction
Ratio analysis is a popular technology to evaluate the performance in economy. The ratio is used
to evaluate the relation among variables. In order to realize home the performance of DMU from
multi-perspectives, such as yield rate, liquidity and quality of asset, we select these variables. The
number of ratio lies on the aims of analysis. Because they are easily understood, ratio analysis is
used widely in some fields, like financial investment, credit card and insurance. But there are
some disadvantages in using. First, only by observation of one aspect of behavior of DMU, or
obtaining one numeral by combination of multi-perspectives, the result isn’t satisfying；Second,
the ratio obtained by financial descriptive data is very large, may puzzle persons and make
contradictions among ratio numbers. So, ratio analysis factually uses some limited principles to
evaluate the efficiency of unit. Due to incomplete consideration of process with multi-inputs and
multi-outputs, and difficult to find the optimal in peer set, classical ratio analysis usually is
ineffective in application to evaluate the efficiency of unit.
To overcome above limitations of ratio analysis, D. Wu et. al.(2005）presented an aggregated
ratio analysis model, and proved that the model proposed is equivalent to CCR model, that is, to
any given DMU, the efficiency of aggregated ratio model is the same as the efficiency of CCR
model, and the best DMU in aggregated ratio model is on the DEA frontier.
In this paper, we propose an extended aggregated ratio analysis model that is similarly
extended from CCR to C2GS2 model in DEA. This extended model also offers an insight into
frontier analysis. Whether a DMU is on the frontier or efficient frontier can be informed by
proposed extended model. The rest of this paper unfolds as follows. Section 2 presents the
extended aggregated ratio analysis model. The frontier analysis is introduced in section3. and
finally concluding remarks are made in section 4.
2. Extended aggregated ratio analysis model
Let the observed input and output vectors of DMUj ( j  1,2,..., n) be
X j  ( x1 j , x2 j ,..., xmj )T  0, j  1,2,..., n
Y j  ( y1 j , y2 j ,..., ysj )T  0, j  1,2,..., n,
respectively.
Then the mathematical programming problem of the C2GS2 ”ratio-form” DEA model is stated as:
s
m
h*  Max h0 (u, v, u0 )  ( ur yr 0  u0 ) /  vi xi 0
r 1
s
m
r 1
i 1
i 1
( ur yrj  u0 ) /  vi xij  1, j  1, 2,..., n.
s.t.
(1)
ur   , vi   , i  1,..., m; r  1,..., s.
and the C2GS2 Envelopment Form is:
 *  Min
n
s.t.
 x
j 1
j ij
n
 y
j 1
j 1
 yr 0 , r  1, 2,..., s.
j
rj
j
1
n

  xi 0 , i  1, 2,..., m.
（2）
 j  0, j  1,..., n.
*
*
Definition 1: DMU0 is DEA efficient if and only if there exists an optimal solution (ur , vi ) of
*
*
（1）and an optimal solution ( ,  j ) of (2) such that   h  1 .
*
*
Now we introduce aggregated ratio analysis model which can also be used to evaluate DMUs.
For
any
given
DMUj,
the
output-input
ratio
vector
wj
is
defined
as
 yrj

w j  wlj : l  1,..., m  s   : i  1,..., m; r  1,..., s  , we introduce the following
 xij

aggregated ratio model for DMU0:
m
 *  Max    pl wl 0  
lL
s.t.
i 1
m
p0
i 1
ij
 p w  x
l
lL
lj
p0
xi 0
 1, j  1,..., n.
(3)
pl   , l  L.
Definition 2: DMU0 is ratio efficient if there exists an optimal solution
p
*
l
: l  L and p0*
*
with pl  0 such that   1 .
*
Theorem 1: DMU0 is ratio efficient if and only if it is DEA efficient.
Proof: We only need to show that the optimal objective value of (1) is equal to the optimal
objective value of (3), i.e. h   .
*
*
First, we would like to show   h .
*
*
Let
p
*
ir
, i  1,..., m; r  1,..., s and p0* be the optimal solution of (3) and  * be its optimal
objective value, therefore,
s
yr 0 m p0*

 *
xi 0 i 1 xi 0
m
 pir*
r 1 i 1
s
m
 pir*
and
r 1 i 1
yrj
m

xij
i 1
(4)
p0*
1
xij
(5).
Equation (4) can be rewritten as :
s
 p1*r yr 0
r 1
x10
s

s
 p2*r yr 0
  
r 1
x20
p
r 1
*
mr
yr 0
xm 0
(
p0* p0*
p*

   0 )   *
x10 x20
xm 0
(6)
zi 0  x10 x20  x(i 1)0 x(i 1)0  xm 0 .
By multiplying
On both numerator and denominator of (6), we have:
s
m
m
r 1
i 1
m
i 1
 yr 0  pir* zi 0  p0  zi 0
 *
1
( zi 0 ) xi 0

i 1 m
Let ur 
m
 pir* zi 0 , vi 
i 1
(7)
m
1
zi 0 , u0  p0  zi 0 .
m
i 1
s
u y
r 1
Then Eq.(7) can be written as
r
r0
 u0
m
v x
i 1
u y
Similarly
r
r0
 u0
m
v x
i 1
（8）and (9) indicate (ur 
m
p
i 1
z , vi 
*
ir  i 0
(8)
i i0
s
r 1
 *
1
(9)
i i0
m
1
zi 0 , u0  p0  zi 0 ) is a feasible solution to the
m
i 1
 *  h* .
C2GS2 model in (1) and thus
Now let us prove that   h .
*
*
*
*
*
Suppose that (ur , vi , u0 ) is an optimal solution of the C2GS2 model in (1) and h
*
is its optimal
objective value.
s
s
u y
Then
*
r r0
r 1
u y
m
 u0*  h*  vi* xi 0
(10)
*
r rj
r 1
and
 u0*
m
v x
i 1
1
(11)
*
i ij
i 1
Multiplying (10) by z i 0 and summarize on i , after simple arrangement we then have:
m
s
m
m
m
m
m
i 1
r 1
i 1
q 1
i 1
i 1 f 1
 zi 0  ur* yr 0  h*  zi 0  vq* xq 0  u0*  zi 0  h* vi*
m
xi 0

 u0* 
xf 0
i 1 xi 0
(12)
Where   zi 0 xi 0  x10 x20  xi 0  xm 0 .
s
m
m
r 1
i 1
m
 yr 0  ur* zi 0  u0* 
Eq.(12) can be written as
i 1

xi 0
m
x
v i0

xf 0
i 1 f 1
 h* 
(13)
*
i
Let pir 
ur*
m
m
 vi*
i 1 f 1
xi 0
xf 0
, p0 
u0*
m
m
 v
i 1 f 1
*
i
xi 0
xf 0
s
m
m
r 1
i 1
i 1
 yr 0  pir zi 0  p0 
Then (13) can be written as

 h*
xi 0
(14)
Divide both sides of (14) by  , we have:
m
s
m
yr 0 m p0

 h*
xi 0 i 1 xi 0
 pir
i 1 r 1
( pir 
Therefore
(15)
s
 pir
i 1 r 1
ur*
m
similarly,
m
x
v i0

xf 0
i 1 f 1
, p0 
*
i
u0*
m
m
x
v i0

xf 0
i 1 f 1
yrj
xij
m

i 1
p0
1
xij
, i  1,..., m.r  1,..., s.) is
a
feasible
*
i
solution of model (3) and thus   h .
*
*
Then   h .
*
*
3. The frontier analysis
In this section, we will introduce the character of the frontier.
Theorem 2: If there exist weights combination of pir : r  1,..., s.i  1,..., m. pir  0 and p0
s
such that
m
 pir
r 1 i 1
m
y
yr 0 m p0 Max s m
p


( pir rj   0 ) , then DMU0 is on the frontier.
j r 1 i 1
xi 0 i 1 xi 0
xij i 1 xij
Furthermore, if pir  0 , then DMU0 is on efficient frontier.
Proof:
pir 
pir  0
For
pir

, p0 
p0

,
we
denote
m
 p
and
s
m
( pir
j
r 1 i 1
yrj
xij
m

i 1
p0
)
xij
.Let
. we have pir  0
s
s
Max

r 1 i 1
ir
yrj
xij
m

i 1
p0

xij
m
( pir
r 1 i 1
yrj
xij
m

i 1
p0
)
xij

 1, j  1,..., n
It means that { pir : r  1,..., s.i  1,..., m} and p0 satisfies the first constraint of ratio model (3)
s
s
Since
m
 pir
r 1 i 1
yr 0 m p0


xi 0 i 1 xi 0
m
( pir
r 1 i 1
yr 0 m p0
 )
xi 0 i 1 xi 0

Now if pir  0 ,we know that pir 
*
p ir

 1 , DMU0 is on the frontier.
s
m
 0 and  *   pir*
r 1 i 1
Then { pir : r  1,..., s.i  1,..., m} is an optimal solution of (3) with
*
yr 0
1.
xi 0
 *  1 . DMU0 is efficient.
The theorem above indicates that DMU with the highest aggregated ratio of output divided by
input is on the frontier, it also proves the possibility of finding DMU on the frontier using different
combinations of input/output ratio.
4. Conclusions
In this paper, we propose an extended aggregated ratio analysis model that is similarly extended
from CCR to C2GS2 model in DEA. This extended model also offers an insight into frontier
analysis. Whether a DMU is on the frontier or efficient frontier can be informed by proposed
extended model.
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