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Int. J. Data Envelopment Analysis (ISSN 2345-458X)
Vol.7, No.4, Year 2019 Article ID IJDEA-00422, 12 pages
Research Article
A Data Envelopment Analysis Model with
Triangular Intuitionistic Fuzzy Numbers
S. A. Edalatpanah*
Department of Industrial Engineering, Ayandegan Institute of Higher Education, Iran
Received 03 April 2019, Accepted 20 Octpber 2019
Abstract
DEA (Data Envelopment Analysis) is a technique for evaluating the relative effectiveness of
decision-making units (DMU) with multiple inputs and outputs data based on non-
parametric modeling using mathematical programming (including linear programming,
multi-parameter programming, stochastic programming, etc.). The classical DEA methods
are developed to handle the information in the form of a crisp number but have no capability
in dealing with fuzzy information like triangular intuitionistic fuzzy numbers (TIFNs),
which is flexible in reflecting the uncertainty and hesitation associated with the decision-
makers’ opinion. In this paper, an extended model of DEA is proposed under the triangular
intuitionistic fuzzy environment where the inputs and outputs of DMUs are TIFNs. At first,
the definition and characteristics of a classical model of DEA and the comparative TIFNs
are introduced. In addition, a new ranking function considering the interaction between
membership and non-membership values of different intuitionistic fuzzy sets are defined.
Then, the triangular intuitionistic DEA model and a new strategy to solve it is proposed.
Finally, the new approach is illustrated with the help of a numerical example.
Keywords: Data envelopment analysis, Efficiency, Intuitionistic fuzzy numbers, Ranking.
*. Corresponding author: Email: saedalatpanah@gmail.com
International Journal of Data Envelopment Analysis Science and Research Branch (IAU)
S. A. Edalatpanah / IJDEA Vol.7, No.4, (2019), 47-58
48
1. Introduction
The efficiency evaluation of every system
is important to find its weakness so that
subsequent improvements can be made.
Data envelopment analysis (DEA) is a
mathematical technique to evaluate the
relative efficiency of a set of some
homogeneous units called decision-
making units (DMUs) that use multiple
inputs to produce multiple outputs. DMUs
are called homogeneous because they all
employ the same inputs to produce the
same outputs. DEA by constructing an
efficiency frontier measures the relative
efficiency of decision-making units
(DMUs). Charnes et al. [1] developed a
DEA model (CCR) based on the seminal
work of Farrell [2] under the assumption
of constant returns to scale (CRS). Banker
et al. [3] extended the pioneering work
Charnes et al. [1] and proposed a model
conventionally called BCC to measure the
relative efficiency under the assumption
of variable returns to scale (VRS). DEA
technique has just been effectively
connected in various cases such as
broadcasting companies [4], banking
institutions [5-8], R&D organizations
[9-10], health care services [11-12],
manufacturing [13-14], telecommuni-
cation [15], and supply chain
management [16-19]. However, the
classical DEA methods are limited to deal
with the decision information in the
format of the crisp number, and unable to
handle uncertainty and imprecision
information. Due to the estimation
inaccuracies, knowledge deficiency and
data unavailability in practical problems,
DMs’ preferences are usually presented in
fuzziness and may exist some hesitations.
Zadeh [20] first proposed the theory of
fuzzy sets (FSs) against certain logic
where the membership degree is a real
number between zero and one. After this
work, many researchers studied this topic;
details of some researches can be
observed in [21-30].
The use of this theory in DEA can be
traced to Sengupta [31]. According to
Hatami Marbini, Emrouznejad, and
Tavana [32], DEA approaches using
fuzzy theory can be classified into four
primary categories: (a) parametric
approaches that convert a fuzzy DEA
model into a parametric model depending
on a parameter α level [33-36]; (b)
possibility approaches that represent
fuzzy variables by probability
distributions [37-38]; (c) ranking
approaches, with the main objective of
designing a fuzzy DEA model able to
yield fuzzy efficiencies that can be ranked
using different methods [39]; and (d)
defuzzification approaches that try to first
convert fuzzy values of inputs and outputs
into crisp values then to solve the
resulting DEA crisp model [40]. Many
other approaches have also been
introduced to fuzzy DEA development
[41-50].
Although the traditional fuzzy theory
provides a powerful framework to
characterize vagueness and uncertainty, it
ignores the hesitation of DMs in the
decision-making process. Also, this
theory cannot deal with certain cases in
which it is difficult to define the
membership degree using one specific
value. To overcome this lack of
knowledge, Atanassov [51] extended the
traditional fuzzy set in 1986 to the
intuitionistic fuzzy set (IFS) which
simultaneously considers the degrees of
membership and non-membership with
hesitation index. Generally, the DEA
models described with intuitionistic fuzzy
numbers are more exquisite than those
with fuzzy numbers.
Although the theory of IFS has been used
extensively in decision-making problems,
there are not many studies that have
incorporated IFS to handle uncertainty or
vagueness in DEA.
Rouyendegh [52] was the first to use the
intuitionistic fuzzy TOPSIS method in a
two-stage process to fully rank the
DMUs. He used a unification of Fuzzy
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49
TOPSIS and DEA to select the units with
the most efficiency. First, the alternative
evaluation problem is formulated by DEA
and separately formulates each pair of
units. In the second stage, he used the
opinion of experts to be applied to a
model of group Decision-Making called
the Intuitionistic Fuzzy TOPSIS method.
Gandotra et al. [53] proposed an
algorithm to rank DMUs in the presence
of intuitionistic fuzzy weighted entropy.
Hajiagha et al. [54] developed a DEA
model when input/output data was
expressed in the form of IFS. They further
extended the model to the case of a
weighted aggregated operator for IFS.
Puri and Yadav [55] developed optimistic
and pessimistic DEA models under
intuitionistic fuzzy input data. They also
presented the application of their
proposed models through a case from the
banking sector in India where some of the
inputs were represented as triangular
intuitionistic fuzzy numbers in the form
of
, , ; , , .
l m u l m u
A a a a a a a
Although these approaches are
interesting, however, some limitations
exist. One limitation is that the proposed
approaches appear time-consuming,
especially when many input and output
sets are employed or when the number of
DMUs under evaluation is important.
So, in this paper, we design a new model
of DEA with triangular intuitionistic
fuzzy numbers with a different form of
[55-57] and establish a new strategy to
solve it. The proposed method is based on
the ranking function and has a simple
structure.
The remainder of the paper is organized
as follows: In Section 2, the basic
concepts of TIFNs and comparative
methods are explained. The classical
DEA method is reviewed in Section 3. In
Section 4, some new ranking functions
are established. In Section 5, an extended
CCR model is proposed to handle
efficiency problems under the TIFNs
environment. In Section 6, a numerical
study is given to illustrate the validity and
practicality of the proposed method.
Finally, some concluding remarks are
drawn in Section 7.
2. TINFs and the comparison method
Definition 1 [58-59]. A TIFN
, , ; ,
a a
a a a a
is a special IFS on a
real number set R, its membership
function is defined as follow:
, if
if
, if
0, if or
a
a
a
a
x a a x a
a a
x a
xa x a x a
a a
x a x a
，
(1)
And its non-membership function can be
defined as:
, if
if
, if
1, if or
a
a
a
a
a x x a a x a
a a
x a
x
x a a x a x a
a a
x a x a
，
(2)
Where
a
and
a
are the maximal
membership degree and the minimal non-
membership degree respectively, and they
satisfy the condition:
0 1
a
,
0 1
a
,
0 + 1
a a
. Also,
1
a a a
x x x
,
a
x
is called the degree of
indeterminacy of the element
x
to
a
. It
reflects the hesitancy degree of the
element
x
to
a
, the smaller
a
x
, the
clearer the fuzzy number is. The
membership function and non-
membership function are illustrated in
Fig. 1.
S. A. Edalatpanah / IJDEA Vol.7, No.4, (2019), 47-58
50
Fig. 1. A TIFN
, , ; ,
a a
a a a a
Definition 2 [60]. Let
, , ; ,
a a
a a b c
be
a TIFN, its score function can be defined
as follows:
2
4
a a
a b c
S a
(3)
and its accuracy function can be defined
as:
2
4
a a
a b c
H a
(4)
Definition 3 [61]. Let
, , ; ,
a a
a a a a
and
, , ; ,
b b
b b b b
be two TIFNs, then
If
S a S b
, then
a b
;
If
S a S b
, then
1. If
H a H b
, then
a b
;
2. If
H a H b
, then
a b
.
Definition 4 [61]. Let
, , ; ,
a a
a a a a
and
, , ; ,
b b
b b b b
be two TIFNs, and
is a real number, some new arithmetic
operations on TIFNs considering
interactions are defined as follows:
(i)
+ , , ;min{ , },max{ , } ,
a a
b b
a b a b a b a b
(ii)
- , , ;min{ , },max{ , } ,
a a
b b
a b a b a b a b
(iii)
, , ;min{ , }, max{ , } , 0, 0
,
a a
b b
ab ab ab ab if a b
(iv)
/ , / , / ;min{ , },max{ , } , 0, 0
,
a a
b b
ab a b a b a b if a b
(v)
, , ; , if
0
, , ; , if 0
.
a a
a a
aaa
a
a a a
Definition 5 [29]. Suppose
A
and
B
be
two TIFNs, then
(i)
A B
iff
( ) ( )
R A R
B
,
(ii)
A B
iff
( ) ( )
R A R
B
.
3. The classical Data Envelopment
Analysis
The efficiency of a DMU is established as
the ratio of sum weighted output to sum
weighted input, subjected to happen
between one and zero. Let a set of
n
DMUs, with each DMUj ( 1, 2
,..., )
j n
by
using m inputs
ij
x
( 1, 2
,..., )
i m
and
producing
s
outputs
rj
y
( 1, 2
,..., )
r s
. If
DMUp is under consideration, the CCR
model for the relative efficiency is the
following model [1]:
*1
1
1
1
m a x
. .
1,
, 0
,
s
r rp
r
pm
i ip
i
s
r rj
r
m
i ij
i
r i
u y
v x
s t
u y
j
v x
u v r i
(5)
Where ( 1, 2
,...., )
r
u r s
and ( 1,2
,...., )
i
v i m
are the weights of the
i
th input and
r
th
output. This fractional program is
calculated for each DMU to find out its
best input and output weights. To
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51
simplify the computation, the nonlinear
program shown as (6) can be converted to
a linear programming (LP) and the model
was called the CCR model:
*
1
1
1 1
m a x
. .
1
0
,
, 0
,
s
p r rp
r
m
i ip
i
s m
r r j i i j
r i
r i
u y
s t
v x
u y v x
u v r i
(6)
We run Model (6)
n
-times to work out
the efficiency of
n
DMUs. The DMUp is
efficient if *
1
, otherwise, it is
inefficient.
4. New Ranking Functions
Here, we propose a ranking function.
Definition 6. One can compare any two
TIFNs based on the ranking functions.
Let
, , ; ,
a a
a a a a
be a TIFN; then:
(1
) .
6a a
a a a
R a
Example 1. Let
1, 2, 3 ;
0.5, 0.3
a
then
1.2
.
R a
Also. For
1 1,1,1 ;
1, 0
,
0 0, 0, 0 ;
1,1
we have:
1 1
R
and
0 0
.
R
Also, since
,
R a b R a R b
we define
an aggregation ranking function as
follows:
Definition 7. Let
, , ; ,
i i i i i i
a a a a
be n
TIFNs. Then the aggregation ranking
function is as follows:
1 1
1
( )
(1 min max ) 1
x
(1 min max ) ( ).
6
n n
i
i i i
i i
i i
n
i i
i i i
i
R a
R a
a a a
Example 2. Let
1, 2, 3 ;
0.5, 0.3
a
and
4,8,10 ;
0.2, 0.6
b
then:
1.2, 2.2, 5,10,13 ; 0.2, 0.6
.
R a R b a b
But:
2.8 1.2 2.2
.
R a b R a R b
So,
(1 0.2 0.6) (6 22) 2.8
.
6
R a b R a b
5. Triangular Intuitionistic Fuzzy Data
Envelopment Analysis
In this section, we establish DEA under
triangular intuitionistic fuzzy
environment. Consider the input and
output for the
j
th DMU as
, , ; ,
ij ij
l m u
ij ij ij ij x x
x x x x
and
, , ; ,
rj rj
l m u
rj rj rj rj y y
y y y y
which are the
triangular intuitionistic fuzzy numbers
(TIFNs). Then the triangular intuitionistic
fuzzy CCR model that called TIFN-CCR
is defined as follows:
*
1
1
1 1
m a x
. .
1
0
,
, 0
,
s
p r rp
r
m
i ip
i
s m
r r j i ij
r i
r i
u y
s t
v x
u y v x
u v r i
(7)
Next, to solve the Model (7), we propose
the following algorithm:
Algorithm 1.
Step 1. Consider the DEA model (7) that
the inputs and outputs of each DMU are
TIFNs.
Step 2. Using Definition 4, the model of
Step 1 can be transformed into the
following model:
*
1
max , , ; ,
rp rp
s
l m u
p r rp rp rp y y
r
u y y y
S. A. Edalatpanah / IJDEA Vol.7, No.4, (2019), 47-58
52
1
1
1
. .
, , ; ,
1
, , ; ,
, , ; , ,
, 0 ,
ip ip
rj rj
ij ij
m
l m u
i ip ip i p x x
i
s
l m u
r rj rj rj y y
r
m
l m u
i ij ij ij x x
i
r i
s t
v x x x
u y y y
v x x x
u v r i
(8)
Step 3. Transform Model (8) into the
following model:
*
1
1
1
1
( ) max ( , , ; , )
. .
( , , ; , ) (1)
( , , ; , )
( , , ; , ),
, 0 ,
rp rp
ip ip
rj rj
ij ij
s
l m u
p r rp rp rp y y
r
m
l m u
i ip ip ip x x
i
s
l m u
r rj rj rj y y
r
m
l m u
i ij ij ij x x
i
r i
R R u y y y
s t
R v x x x R
R u y y y
R v x x x
u v r i
(9)
Step 4. Based on Definition 7, convert the
Model (9) into the following crisp model:
*11
1
11
1
11
1
1
(1 min max )
( ) max ( )
6
. .
(1 min max )
( ) 1,
6
(1 min max )
( )
6
(1 min
rp rp
ip ip
rj rj
ij
s
y y l m u
r s r s
p r rp rp rp
r
m
x x l m u
i m i m
i ip ip ip
i
s
y y l m u
r s r s
r rj rj rj
r
x
i m
R u y y y
s t
v x x x
u y y y
1
1
max )
( ),
6
, 0 ,
ij
m
xl m u
i m
i ij ij ij
i
r i
v x x x
u v r i
Step 5. Run the crisp model of Step 4 and
obtain the optimal solution.
Theorem 1. The models (6) and (7) are
equivalent.
Proof. By considerate the aggregation
ranking function and Algorithm.1, it is to
see that every optimal feasible solution of
Model (7) is an optimal feasible solution
of Model (6), on the other hand, every
optimal feasible solution of Model (6) is
an optimal feasible solution of Model (7).
Theorem 1 also shows that the sets of all
feasible solutions of Model (7) and Model
(6) are the same. Furthermore, if
, ,
r i
ˆ ˆ
,
r i
u v
is an optimal solution for Model
(7), then
ˆ ˆ
,
r i
u v
is an optimal solution for
the Model (6). Moreover, if Model (6)
does not have an optimal solution, then
Model (7) does not have an optimal
solution either.
6. Numerical Experiment
For the purpose of interpreting the
practicability and the feasibility of the
new method proposed in this paper, a
numerical example is employed. There
are five DMUs that consume two inputs
to produce two outputs. These inputs and
outputs are given by triangular
intuitionistic numbers and do not have
obligatory symmetrical triangular truth
and falsity membership functions. Table 1
provides the data for this example.
Now, we use Algorithm.1 to solve the
performance assessment problem. For
example, Algorithm.1 for DMU1 can be
used as follows:
Table 1. Data for the five DMUs used in the numerical example
DMU
DMU
1
DMU
2
DMU
3
DMU
4
DMU
5
Input 1
<(
3.5,4,4.5
);
0.7,0.3>
<(
2.9,2.9,2.9
);
0.6,0.2>
<(
4.4,4.9,5.4
);
0.6,0.1>
<(
3.4,4.1,4.8
);
0.4,0.2>
<(
5.9,6.5,7.1
);
0.7,0.3>
Input 2
<(
1.9
,
2.1
,
2.3); 0.4,0.5>
<(
1.4
,
1.5
,
1.6
);
0.8,0.1>
<(
2.2
,
2.6
,
3.0
);
0.7,0.2>
<(
2.2
,
2.3
,
2.4
);
1.0, 0.0>
<(
3.6
,
4.1
,
4.6
);
0.9,0.1>
Output 1
<(
2.4
,
2.6
,
2.8); 0.9,0.1>
<(
2.2
,
2.2
,
2.2
);
0.9,0.0>
<(
2.7
,
3.2
,
3.7
);
0.7,0.2>
<(
2.5
,
2.9
,
3.3
);
0.7,0.1>
<(
4.4
,
5.1
,
5.8
);
0.8,0.2>
Output 2
<(
3.8
,
4.1
,
4.4); 0.8,0.1>
<(
3.3
,
3.5
,
3.7
);
1.0, 0.0>
<(
4.3
,
5.1
,
5.9
);
0.7,0.1>
<(
5.5,5.7
,
5.9
);
0.4,0.1>
<(
6.5
,
7.4
,
8.3
);
0.5,0.2>
IJDEA Vol.4, No.2, (2016).737-749
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53
First, we construct a DEA model with
mentioned TIFNs:
*
1 1
2
1
2
1
2
1
ma x( 2 .4, 2 .6, 2 .8; 0 .9, 0 .1
3.8 , 4.1, 4.4 ;0 .8, 0.1 )
. .
3.5 , 4.0, 4 .5; 0.7 , 0.3
1. 9, 2.1, 2.3; 0 .4, 0.5 1,
2. 4, 2 .6, 2 .8; 0. 9, 0. 1
3.8 , 4.1, 4.4 ;0 .8, 0.1
3.5 , 4.0, 4 .5; 0.7 , 0.3
1. 9, 2.1, 2.3; 0 .4,
u
u
s t
v
v
u
u
v
2
1
2
1
2
0.5 ,
2. 2, 2 .2, 2 .2; 0 .9, 0 .0
3.3 , 3.5, 3 .7;1 .0, 0 .0
2. 9, 2 .9, 2 .9; 0 .6, 0 .2
1. 4,1. 5,1. 6; 0.8 , 0.1 ,
v
u
u
v
v
1
2
1
2
1
2
1
2
2 .7, 3 .2, 3 .7; 0 .7 , 0.2
4 .3, 5 .1, 5. 9; 0 .7 , 0 .1
4 .4, 4 .9 , 5. 4; 0 .6 , 0. 1
2 .2, 2 .6, 3 .0; 0 .7 , 0.2
,
2 .5, 2 .9 , 3. 3; 0. 7, 0 .1
5. 5, 5 .7 , 5. 9; 0 .4, 0 .1
3. 4, 4.1, 4. 8;0.4 , 0.2
2.2, 2.3, 2 .4;1.0 ,0.0
,
4 .4, 5 .1
u
u
v
v
u
u
v
v
1
2
1
2
, 5. 8; 0. 8, 0. 2
6 .5, 7 .4, 8. 3; 0. 5, 0. 2
5. 9, 6 .5, 7 .1; 0.7 , 0. 3
3. 6, 4.1, 4. 6; 0.9 ,0.1
,
, 0 , 1, 2
.
r i
u
u
v
v
u v r i
Finally based on Step 4 of Algorithm.1,
we convert the above model to the
following model:
*
1 1 2
1
m a x ((1 .7 ) ( 7 .8 1 2 .3
))
6
. .
u u
s t
1 2
1 2 1 2
1 2 1 2
1 2 1 2
1 2 1 2
1(0.9)(12 6.3 ) 1,
6
((1.7)(7.8 12.3 )) ((0.9)(12 6.3 )) 0
,
((1.9)(6.6 10.5 )) ((1.4)(8.7 4.5 )) 0
,
((1.5)(9.6 15.3 )) ((1.4)(14.7 7.8 )) 0
,
((1.3)(8.7 17.1 )) ((1.2)(12.3 6.9 ))
v v
u u v v
u u v v
u u v v
u u v v
1 2 1 2
0
,
((1.3)(15.3 22.2 )) ((1.4)(19.5 12.3 )) 0
,
, 0 , 1,2.
r i
u u v v
u v r i
After computations with Matlab, we
obtain *
1
1.000
for DMU1 . Similarly,
for the other DMUs, we report the results
in Table2.
7. Conclusion
In this paper, an extended model of DEA
is proposed to handle performance
evaluation problems under the TFINs
environment. Because the existing
arithmetic operations of TIFNs are
deficient and cannot take into account the
interaction between non-membership
function and membership function of
different TIFNs, a new ranking function
is proposed in this paper to address the
existing problem. In addition, an
aggregation measured method is put
forward to handle the summation of
TIFNs effectively. A novel algorithm is
developed to use these ranking functions
to calculate the weight of each evaluation
value in DMUs, which can effectively
avoid unreasonable evaluation values.
Finally, we use an example to illustrate
the practicality and validity of the
proposed method. In comparison with the
classical and fuzzy DEA methods, the
significant characteristic of the extended
DEA method is that it can handle the
triangular intuitionistic fuzzy information
simply and effectively.
Table 2. The efficiencies of the DMUs
DMUs
1
2
3
4
5
Efficiency
1.0000
0.8587
0.5760
0.7779
0.5934
Ranking
1
2
5
3
4
S. A. Edalatpanah / IJDEA Vol.7, No.4, (2019), 47-58
54
References
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