Available at
http://pvamu.edu/pages/398.asp
ISSN: 1932-9466
Vol. 2, Issue 2 (December 2007), pp. 152 – 166
(Previously Vol. 2 No. 2)
Applications and Applied
Mathematics
(AAM):
An International Journal
Fuzzy Efficiency Measure with Fuzzy Production Possibility Set
T. Allahviranloo
Department of mathematics, Science and Research Branch
Islamic Azad University, Tehran-Iran
E-Mail: tofigh@allahviranloo.com
F. Hosseinzade Lotfi
Department of mathematics, Science and Research Branch
Islamic Azad University, Tehran-Iran
E-mail: hosseinzade_lotfi@yahoo.com
M. Adabitabar Firozja
Department of mathematics, Science and Research Branch
Islamic Azad University, Tehran-Iran
Department of mathematics
Islamic Azad University, Qaemshahr-Iran.
E-mail: mohamadsadega@yahoo.com
Received December 16, 2006; revised May 7, 2007; accepted May 10, 2007
Abstract
The existing data envelopment analysis (DEA) models for measuring the relative efficiencies of a
set of decision making units (DMUs) using various inputs to produce various outputs are limited
to crisp data. The notion of fuzziness has been introduced to deal with imprecise data. Fuzzy
DEA models are made more powerful for applications. This paper develops the measure of
efficiencies in input oriented of DMUs by envelopment form in fuzzy production possibility set
(FPPS) with constant return to scale.
Keywords: Data Envelopment Analysis; Fuzzy Number; α − cut.
MSC 2000; 41A30, 65L06
1. Introduction
Data Envelopment Analysis (DEA) was suggested with CCR model by Charnes et al. (1978) and
built on the idea of Farrell (1957) which is concerned with the estimation of technical efficiency
and efficient frontiers. Several models introduced
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153
for evaluating of efficiency such as Charnes et al. (1978) Banker et al. (1984) and Cooper et al.
(1999). Most of practical data in such situations economic evaluating, are imprecise. To deal
quantitatively with imprecision in decision progress, Bellman et al. (1970) introduce the notion of
fuzziness. Some researchers have proposed several fuzzy models to evaluate DMUs with fuzzy
data, without access to FPPS (see Chiang et al. (2000) Lertworasirikul et al. (2003) Wang et al.
(2005) Tanaka et al. (2001), Jahanshahloo et al. (2004) Leon et al. (2003) Kao et al. (2003)).
Unfortunately, some of the existing techniques only provide crisp solution. S. Lertworasirikul et
al. (2003) provided fuzzy efficiency measures in multiple model. In fuzzy efficiency measures
with fuzzy production possibility set (FPPS) in constant return to scale with input oriented such
that satisfy the initial concept by crisp data is proposed. Data envelopment analysis and fuzzy
system are considered in section (2) and (3), respectively. Subsequently, production possibility
set (PPS), to extended by fuzzy data and then the proposed fuzzy efficiency measures in CCR
model, is brought in section(4). A numerical example is presented in section (5). Finally, we saw
concludes the paper in section (6).
2. Data Envelopment Analysis (DEA)
DEA utilizes a technique of mathematical programming for evaluate of n DMUs . Suppose there
are n DMUs : DMU , DMU , ..., DMU . Let the input and output data for DMU j
1
2
n
X = (x , ..., x ) and Yj =(y1j ,..., ysj ) , respectively.
j
1j
mj
be
2.1 Production Possibility Set (PPS)
We will call a pair of input X ∈ R m and output Y ∈ R s an activity and express them by the
notation (X,Y) . The set of feasible activities is called the production possibility set (PPS) and is
denoted by P. We postulate the following properties of P:
1: The observed activities (X j Yj )∈ P; j = 1,..., n
2: If an activity (X, Y )∈ P , then the activity (tX , tY ) ∈ P for all t > 0 .
3: If an activity (X, Y )∈ P , then (X, Y )∈ P if X ≥ X and Y ≤ Y .
4: If activity (X, Y )∈ P and (X, Y )∈ P , then ( λX + (1-λ )X , λY + (1-λ )Y )∈ P for all¸ λ∈[0 , 1].
We show the set P as follow:
{
P= ( x , y )
}
x ≥∑n λ j x j , y ≤∑n λ j y j , λ j ≥0 ; j =1,...,n
j =1
j =1
2.2 The CCR Model
The CCR model proposed by Charnes et al. (1978) is as follow:
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minθ
s.tθx
≥ ∑ nλ= x j j
j 1
yλ0 y ≤ ∑ n= j j
j 1
λj ≥
0
0
(1)
j=
1,...,n
The constructions of CCR model require the activity (θ x0 , y0 )∈PPS , while the objective seeks
the min θ that reduces the input vector x0 radially to θ x0 while remaining in PPS. In CCR
model, we are looking for an activity in PPS that guarantees at least the output level y
of
0
DMU 0 in all components while reducing the input vector x0 proportionally (radially) to a value
as small as possible.
3. Fuzzy Systems
~
Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function
µ ~ : X →[ 0,1] and µ ~ ( x ) is interpreted as the degree of membership of element X in fuzzy set
A
A
~
~
A for each x∈X . A fuzzy set A is completely determined by the set of tuples
~
A ={( x , µ ~ ( x )) | x∈X } .
A
~α
~
Definition 1. An α − level set of a fuzzy A of X is a non-fuzzy set denoted by [ A ]
and is
lα uα
lα
~α
defined
by
where
and
[ A] = [ A , A ]
A
= min{ x ∈ X : µ ~ ( x ) ≥ α }
A
Auα = max{ x∈ X : µ ~ ( x ) ≥ α } .
A
~
Definition 2. A fuzzy number A is a fuzzy set of the real line with a normal, (fuzzy) convex and
continuous membership function of bounded support.
~
Definition 3. If [ Alα , Auα ] and [ Blα , Buα ] be α − levels of fuzzy numbers A and
respectively and λ ∈ R then:
lα uα
1- [ A , A ] + [ Blα , Buα ] = [ Alα + Blα , Auα + Buα ]
2- λ [ A
lα
uα
 lα
, A uα ] = [λAuα , λ A lα ]
[λA , λ A ]
if λ ≥0
if λ <0
3.1 Extension Principle
Let X be a Cartesian product of universes X = X × X × ... × X and
1
2
r
~
B
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~
~
A1,...,A r be
155
r
fuzzy sets in X , X ,..., X respectively. f is a mapping
1 2
r
from X to a universe Y , y = f (x ,..., x ) . Then the extension principle
1
r
allows us to define fuzzy set
~
B
in
Y
by:
{
~
B = ( y , µ B ( y )) y = f ( x1,..., xr ) , ( x1,..., xr )∈X1×...× X r
Where
}
{

sup
min µ A ( x1),..., µ A ( xr )

1
r
µ ( y ) = ( x ,..., x )∈ f −1( y )
r
1
B

 0
where f −1 is the inverse of f .
}
f −1( y ) ≠φ
otherwise
4. Fuzzy Efficiency Measure
In this section we are going to obtain fuzzy efficiency measure in FPPS.
4.1 Fuzzy Production Possibility Set (FPPS)
~
~
Let the input and output data for DMUj (j = 1, ..., n) be X j =( ~
x1 j ,..., ~
xmj )t and Y j =( ~
y1 j ,.., ~
ysj )t ,
respectively, such as ~
yrj ( r =1,..., s ) for j = 1, ..., n be (m + s)n fuzzy
xij (i =1,...,m ) and ~
numbers. We call set of feasible activities with fuzzy data is fuzzy production possibility set
(FPPS)
and denote by
~
P.
(FPPS)
as follow:
=
P {(( X , Y ), µ  ( X , Y )) | X ∈ R m , Y ∈ R s }
P
where
=
X ( x ,..., x ) , =
( x , µ ( x )) ∈ x i 1,...m

We define the fuzzy production possibility set
1
m
i
xi
i
i
=
Y ( y ,..., y ) , (=
y , µ ( y )) ∈ y i 1,...s
1
s
i yi i
i
Then with extension principle
(2)
(3)
(4)
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µ  ( X , Y ) = max
P
min {µ
λ j >0
x1 j
( x )..., µ
( x ), µ
( y ),..., µ
( y )}
1j
xm j mj
y1 j 1 j
y s j sj
n
X≥ ∑ λjX j
j =1
n
Y ≤ ∑ λ jY j
j =1
λ j ≥0, j =
1,..., n
s.t
(5)
( X , Y ) ∈ supp( X , Y )
j j
j j
~ ~
such as ( X j ,Y j ) ( j =1, 2,...,n ) is observed activities.
Notation 1. Let S be set of vectors then X = min S if and only if ∀ X ∈ S ; X ≥ X .
Notation 2. We define FPPS β as follow:

(( X ,Y ) , µ  ( X ,Y )) ∈ P 
β
β
P
FPPS
( X ,Y )
=
 [ P ]
µ  ( X ,Y )≥ β


P
(6)
~~ ~
~
The CCR model by fuzzy data require the activity ( θ X , Y0 ) to belong to P , while for any
0
~ ~
(( X 0 ,Y0 ),α )∈( X 0 ,Y0 ) and 0<α ≤1 that µ ~ ( X 0 ,Y0 )≥ β the objective seeks the minimum
P
~
~β
θ (θ ∈θ ) that reduces input vector X 0 radially to θX 0 while (θX , Y ) ∈ supp P . Hence CCR
0 0
model is proposed as follows:
min
s.t
θ
(7)
(θ X , Y ) ∈ P
0 0
But
~~ ~
~
(θ X , Y ) ∈ P if and only if
0 0
[(θ~X~0 ,Y~0 )]α ⊆[ P~ ]β
0<α ≤1
0< β ≤1

( X 0 ,Y0 )∈[ P~ ]β for any
~ ~
( X 0 ,Y0 )∈[( X 0 ,Y0 )]α
~~ ~
If [(θ X 0 ,Y0 )]α =[ E1, E2 ] then with (7) and notation 1 θ lα is the efficiency measure with
maximum input and minimum output, hence
~
~
~
E1= min{(θX 0 ,Y0 )|θ ∈[θ ]α , X 0∈[ X 0 ]α ,Y0∈[Y0 ]α }=(θ lα X 0uα ,Y0lα )
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Similarity, θ uα is the efficiency measure with minimum input and maximum output, hence
~
~
~
E2 = max{θX 0 ,Y0 )|θ ∈[θ ]α , X 0∈[ X 0 ]α ,Y0∈[Y0 ]α }=(θ uα X 0la ,Y0uα )
Hence
(θ lα X uα ,Y lα )∈[ P~ ]β
0
0

(θ uα X lα ,Y uα )∈[ P~ ]β
0 0

0<α ≤1
~~ ~
~
(θ X , Y ) ∈ P if and only if 
0 0
0< β ≤1

( X uα ,Y lα )∈[ P~ ]β
 0 0
 lα uα ~ β
( X 0 ,Y0 )∈[ P ]
[
]
~
Let, for any ( X , Y ) ∈ ( X~ 0,Y~0 ) α with µ ~ ( X , Y ) ≥ β , [θ ](α , β ) =[θ l (α , β ) ,θ u (α , β ) ]
0 0
0 0
P
such that µ ~ (θX 0 ,Y0 )≥ β for all θ ∈[θ l (α , β ) ,θ u (α , β ) ] .( α ∈[ 0 ,1] is a membership function of
P
fuzzy DMU and β ∈[ 0 ,1] is a membership function of production frontier)
Hence:
θ l (α , β ) = min
s.t
θ
uα
θ X0 ≥
n
∑ λjX j
j =1
lα
Y0 ≤
n
∑ λ jY j
j =1
uα
X0 ≥
n
∑ λjX j
j =1
0
λj ≥
j 1,..., n
=
β
( X/ , Y ) ∈ [( X , Y )]
j j
j j
(8)
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θ u (α , β ) = min
s.t
θ
lα
θ X0 ≥
n
∑ λjX j
j =1
uα
Y0 ≤
n
∑ λ jY j
j =1
lα
X0 ≥
n
∑ λjX j
j =1
0
λj ≥
(9)
β
( X , Y ) ∈ [( X , Y )]
j j
j j
The lowest and highest relative efficiency of DMU 0 is found by proposed models as follows:
θ 'l (α , β ) = min
θ
=
j 1,..., n
s.t
lβ
uβ
n
n
λj X j+ ∑ µ j X j
∑
0=j 1=j 1
uα
θX ≥
uβ
lβ
lα
n
n
Y0 ≤
Y
λ
µ
+
j
∑
∑
j
jY j
=j 1=j 1
(10)
lβ
uβ
uα
n
n
X0 ≥
∑ λ j X j+ ∑ µ j X j
=j 1=j 1
θ ' u (α , β ) = min
s.t
λj ≥0
j=
1,..., n
µj ≥ 0
j=
1,..., n
θ
lα
uβ
lβ
n
n
θ X0 ≥
∑ λj X j+ ∑ µ j X j
=j 1 =j 1
lβ
uβ
uα
n
n
Y 0 ≤
∑ λj Y j+ ∑ µ j Y j
=j 1 =j 1
uβ
lβ
lα
n
n
X0 ≥
∑ λj X j+ ∑ µj X j
=j 1 =j 1
j=
1,..., n
λ j ≥0
µj≥ 0
j=
1,..., n
(11)
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Theorem 1. Model (10) is feasible always if α ≥ β .
Theorem 2. Model (11) is feasible always if α ≥ β .
Theorem 3. θ l (α , β ) =θ 'l (α , β ) .
Proof: Let, P β (8) and P β (10) are production possibility sets of models (8) and (10),
respectively. We show that production possibility set of model (8) is equal to production
possibility set of model (10).
uβ
lβ n
n


≥
+
λ
µ
X
X
X
∑
∑
j
j : λ j ≥ 0, j =1,..., n 
j
j

j =1
j =1


β
P (10) = ( X ,Y ):



lβ
uβ n
n

Y ≤ ∑ λ j Y j + ∑ µ j Y j : µ j ≥ 0, j =1,..., n 
j =1
j =1


n

X ≥ ∑ λ jX j;

j =1

β
P (8) = ( X ,Y ):

n
Y ≤ ∑ λ jY j ;

j =1

λ j ≥0, j =1,...,n,
~ ~
( X j ,Y j )∈[( X j ,Y j )]β







P β (10)⊆ P β (8) because if ( X ,Y )∈P β (10) then
lβ
uβ
n
n

X ≥ ∑ λ j X j + ∑ µ j X j
j =1
 j =1

lβ
uβ
n
n
Y ≤ ∑
λj Y j+ ∑ µ j Y j
 j =1
j =1
If λ j + µ j = k j then
hence ( X , Y ) ∈ P
β
n

X≥ ∑ k jX j
j =1
λ j =a j .k j 
n


µ
b
k
≤
.
Y
k jY
=
⇒
∑
 j j j 
j =1
a j +b j =1 
~ ~ β

 ( X j ,Y j )∈[( X j ,Y j )] , j =1,...,n

(8).
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Let P β (8) ⊄P β (10) then ∃ ( X ,Y );( X ,Y )∈P β (8) and ( X ,Y )∉ P β (10) hence one of the three
case is occurred:
lβ n
uβ
n

X
X
X
<
+
λ
µ
∑
∑
j
j
j
j

j =1
 j =1
∀λ , µ 
j
j
uβ n
lβ
n
Y ≤ ∑
λj Yj + ∑ µ jYj
 j =1
j =1
or

uβ
 X ≥ n λ Xlβ + n µ X
∑
∑
 j =1 j j j =1 j j

uβ n
lβ
 n
Y > ∑ λ j Y j + ∑ µ j Y j
j =1
 j =1
or
lβ n
uβ
n

X
X
X
<
+
λ
µ
∑ j j ∑ j j

j =1
 j =1

uβ n
lβ
n
Y > ∑
λj Yj + ∑ µ jYj
 j =1
j =1
Let,
 X < n λ Xlβ + n µ Xuβ
 j∑=1 j j j∑=1 j j
 n uβ n l β
Y > ∑ λ j Y j + ∑ µ j Y j
 j =1
j =1
If λ j + µ j = k j then
n

X< ∑ k jX j
j =1
λ j =a j .k j 
n


Y > ∑ k jY j
µ j =b j .k j ⇒
j =1
a j +b j =1 
~
~

 ( X j ,Y j )∈[( X j ,Y j )]β , j =1,...,n

hence ( X ,Y )∉ P β (10) which regarding to assumption
completed.
u (α , β )
'u (α , β )
Theorem 4. θ
=θ
.
this isn’t correct. Hence proof is
Proof: Similar to proof of Theorem 1.
Lemma 1. If θ 'l (α , β ) and θ 'u (α , β ) be optimal solution of models (10) and (11), respectively,
then θ 'l (α , β ) ≤θ 'u (α , β ) . The equality is held when all component of DMU 0 generate from
fuzzy data to exact data.
Proof: Regarding to theorem 1 and theorem 2 and the models (8) and (9) proof is evident
( θ 'u (α , β ) is a feasible solution of model (8)).
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Lemma 2. If α1≤α 2 then θ
both models (10) and (11).
'l (α1, β )
≤θ
'l (α 2 , β )
and θ
'u (α 2 , β )
161
≤θ
'u (α1, β )
for all β ∈( 0,1] in
uα1 uα 2
lα1 lα 2
Proof: If β is constant and α1 ≤ α 2 then Y0 ≤ Y0 and X 0 ≥ X 0 hence regarding to model
lα1 lα 2
'l (α1, β ) 'l (α 2 , β )
(10) θ
. Similarly, if β is constant and α1 ≤ α 2 then X 0 ≤ X 0 and
≤θ
uα1 uα 2
'u (α 2 , β ) 'u (α1, β )
.
Y0 ≥ Y0 hence regarding with model (11) θ
≤θ
Lemma 3. If β1 ≤ β 2 then θ
in both models (10) and (11).
'l (α , β1 )
≤θ
'l (α , β 2 )
[
and θ
'u (α , β1 )
≤θ
] [
'u (α , β 2 )
for all α ∈( 0,1]
]
~ ~ β
~ ~ β
Proof: If α is constant and β1 ≤ β 2 then ( X j ,Y j ) 2 ⊆ ( X j ,Y j ) 1 hence regarding to
'l (α , β1 ) 'l (α , β 2 )
. Similarly, if α is constant and β1≤ β 2 then
theorem 1 and model (8) θ
≤θ
~ ~ β
~ ~ β
[( X j ,Y j )] 2 ⊆[( X j ,Y j )] 1 hence
regarding to theorem 2 and model (9)
'u (α , β1 ) 'u (α , β 2 )
.
θ
≤θ
~
Proposition 1. Model (10) is non-feasible iff ( X 0uα ,Y0lα )∉P β .
~
Proposition 2. Model (11) is non-feasible iff ( X 0lα ,Y0uα )∉P β .
~ ~
~
Proposition 3. If both models (10) and (11) are feasible ([( X ,Y )]α ⊆ P β )
then θ 'l (α , β ) ≤1 and θ 'u (α , β ) ≤1 .
5. Numerical Example
A simple numerical example with fuzzy single-input and single-output was introduced by C. Kao
and S.T. Liu (2000). We will consider this example with its data listed in table 1. These DMUs
(A,B,C and D) are evaluated by proposed models in (10 ) and (11).
Table 1:
DMU s
A
B
C
D
Input
α −cut
Output
α −cut
(11,12,14)
(30,30,30)
(40,40,40)
(45,47,52,55)
[11+ α ,14-2 α ]
[30,30]
[40,40]
[45+2 α ,55-3 α ]
(10,10,10)
(12,13,14,16)
(11,11,11)
(12,15,19,22)
[10,10]
[12+ α ,16-2 α ]
[11,11]
[12+3 α ,22-3 α ]
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Allahviranloo et al.
~
The lower and upper bounds of ( α , β )-cut of θ A is calculated as follows:
l (α , β ) = min
θA
s.t
θ
(12)
(14− 2α )θ ≥(11+ β )λ1+30λ2 + 40λ3 + (45+ 2β )λ4 + (14− 2β ) µ1 +30µ2
+40µ3 + (55−3β ) µ4
10≤10λ1+ (16− 2β )λ2 +11λ3 + (22−3β )λ4 +10µ1+ (12+ β ) µ2 +11µ3
+(12+3β )µ4
(14− 2α )≥(11+ β )λ1+30λ2 + 40λ3 + (45+ 2β )λ4 + (14− 2β ) µ1+30µ2
+40µ3 + (55−3β ) µ4
λ j ≥=
0 , j 1, ... ,4 ,
θ
µ j ≥=
0 , j 1, ... ,4
u (α , β )
θ
= min
A
s.t (11+α )θ ≥ (14 − 2β )λ1 +30λ2 + 40λ3 + (55−3β )λ4 + (11+ β ) µ1
+30µ2 + 40µ3 + (45+ 2β ) µ
4
10≤10λ1 + (12 + β )λ2 +11λ3 + (12 +3β )λ4 +10 µ1 + (16 − 2 β ) µ2
+11µ3 + (22 −3β ) µ4
(13)
(11+α )≥(14 − 2 β )λ1 +30λ2 + 40λ3 + (55−3β )λ4 + (11+ β ) µ1 + 30 µ2
+40µ3 + (45+ 2 β ) µ4
λ =
≥ 0 , j 1, ... , 4 ,
j
µ =
≥ 0 , j 1, ... , 4
j
The lower and upper bounds of (α , β ) − cuts of
~
~
θ B , θC and θ~D can be solved similarly. The
results are showed in table 2 and table 3 for α =0.0, .1, .2, ...,1 and β =.3 and .7 . In DEA, we
consider that ( x0 , y0 )∈PPS and then we want to find the min θ where (θ x0 , y0 )∈PPS . In
l α uα
~ .3
) ∉ P hence the model is infeasible (inf) and also in table
table 2 for α = 0.0, .1, .2 . ( X , Y
A A
~.7
uα )∉P
3 for α =0.0, .1, .2, ..., .6 ( X lAα ,YA
. Hence the model is infeasible (inf) (see Proposition
1,2).
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Table 2: (α , .3) - cuts of efficiency measures
α
l ,θ r ]
l ,θ r ]
l
r
[θ A
[θ B
[θ , θ ]
A
B
C C
0.0
.2
.3
.4
.5
.6
.7
.8
.9
1.0
*[.81,inf]
*[.83,inf]
[.84,1.0]
[.86,.99]
[.87,.98]
[.88,.97]
[.90,.97]
[.91,.96]
[.93,.95]
[.94,.94]
~
~
[.45,.60]
[.46,.59]
[.46,.58]
[.47,.57]
[.47,.56]
[.47,.56]
[.48,.55]
[.48,.54]
[.49,.53]
[.49,.53]
~
[.31,.31]
[.31,.31]
[.31,.31]
[.31,.31]
[.31,.31]
[.31,.31]
[.31,.31]
[.31,.31]
[.31,.31]
[.31,.31]
~
Figure 1: ( θ A ∗ ∗∗) , ( θ B × ××) , ( θC ∆∆∆) and ( θ D  )
163
l ,θ r ]
[θ D
D
[.25,.55]
[.26,.53]
[.27,.52]
[.28,.51]
[.29,.49]
[.29,.50]
[.30,.48]
[.31,.48]
[.33,.46]
[.32,.47]
164
Allahviranloo et al.
Table 3: (α , .7 ) - cuts of efficiency measures
α
[θ Al , θ Ar ] [θ Bl , θ Br ] [θ Cl , θ Cr ]
0.0
*[.84,inf]
[.47,.62]
[.32,.32]
.1
*[.85,inf]
[.47,.62]
[.32,.32]
.2
*[.86,inf]
[.48,.61]
[.32,.32]
.3
*[.87,inf]
[.48,.60]
[.32,.32]
.4
*[.89,inf]
[.48,.59]
[.32,.32]
.5
*[.90,inf]
[.49,.58]
[.32,.32]
.6
*[.91,inf]
[.49,.58]
[.32,.32]
.7
[.93,1.0]
[.50,.57]
[.32,.32]
.8
[.94,.99]
[.50,.56]
[.32,.32]
.9
[.96,.98]
[.50,.55]
[.32,.32]
1.0
[.97,.97]
[.51,.55]
[.32,.32]
~
~
~
~
Figure 2: ( θ A ∗ ∗∗) , ( θ B × ××) , ( θC ∆∆∆) and ( θ D  )
l
r
[θ D
,θ ]
D
[.26,.57]
[.26,.56]
[.27,.55]
[.28,.54]
[.29,.53]
[.30,.52]
[.30,.51]
[.31,.50]
[.32,.49]
[.33,.48]
[.34,.47]
AAM: Intern. J., Vol. 2, Issue 2 (December 2007) [Previously Vol. 2 No. 2]
165
6. Conclusions
In the real world there are many problems which have fuzzy parameters. In this paper, we
proposed input oriented CCR model, for evaluation of relative efficiency of DMUs in fuzzy
production possibility set by considering initial concepts of DEA. This model can be extended to
the other topics and models of DEA.
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