Can Fuzzy DEA Effectively Measure Bank Efficiency？
An Implication Evidence from Taiwan
Hsien-Chang Kuo1,a, Yang Li2, Gwo-Hshiung Tzeng3,and Ya-Hui Tsai1
1
Department of Banking and Finance, National Chi-Nan University
1, University Rd., Puli, Nantou 545, Taiwan
Tel: 886-49-2912698; Fax: 886-49-2914511; E-mail: hckuo@ncnu.edu.tw
a
Takming College
56, Sec.1, Huanshan Rd., Neihu, Taipei 114, Taiwan
Tel: 886-2-26585801; Fax: 886-2-26588448; E-mail: hckuo@takming.edu.tw
2
3
Institute of Economics and Management; National University of Kaohsiung
700, Kaohsiung University Rd., Nan Tzu Dist., Kaohsiung 811, Taiwan
Tel: 886-7-5919341; Fax: 886-7-5919342; E-mail: yangli@nuk.edu.tw
Institute
of Technology and Management; Institute of Management of Technology
National Chiao Tung University; 1001, Ta-Hsuch Rd., Hsinchu 300, Taiwan
Tel: 886-3-5712121 ext 57505; Fax: 886-2-23120519; E-mail: ghtzeng@cc.nctu.edu.tw
Department of Bussiness Administration, Kainan University
No.1, Kainan Road, Luchu, Taoyuan 338, Taiwan
Tel: 886-3-3412500 ext. 1101; Fax: 886-3-3412430; E-mail: ghtzeng@mail.knu.edu.tw
February 2006
Can Fuzzy DEA Effectively Measure Bank Efficiency？
An Implication Evidence from Taiwan
Abstract
Data Envelopment Analysis (DEA) is widely applied in evaluating the efficiency of banks
since it is a method capable of evaluating the efficiency of decision making units in utilizing
multiple inputs to produce multiple outputs. However, some outputs of banks, in fact,
possess fuzzy characteristics, while conventional DEA approach can only assess efficiency
with a crisp value and is unable to evaluate imprecise data. Theoretically, the fuzzy DEA
approach can evaluate banks’ efficiency more realistic and accurate since it can take the
fuzzy characteristics of inputs and/or outputs into consideration. This study adopts 48
Taiwanese commercial banks as an empirical example to demonstrate the feasibility and the
effectiveness in our proposed fuzzy DEA. The results show that the fuzzy DEA approach
could not only effectively characterize uncertainty, but also may have a higher capability to
discriminate banks’ efficiency than the conventional DEA approach.
Keywords:
Fuzzy DEA, bank efficiency, technical efficiency,  -cut, imprecise data.
1
I. Introduction
Data Envelopment Analysis (DEA) is a method capable of evaluating the efficiency of
decision making units (DMUs) in utilizing multiple inputs to produce multiple outputs;
hence, it is widely applied in evaluating the efficiency of banks (Sherman and Gold, 1985;
Aly et al., 1990; Yue, 1992; Miller and Noulas, 1996; Berger and DeYoung, 1997; Berger
and Humphrey, 1997).
However, the conventional DEA model can only assess efficiency
with a crisp value and is unable to evaluate imprecise data (Kao and Liu, 2000). Thus,
Cooper et al. (1999) proposed a unified approach to treating mixtures involving exact as
well as imprecise data, while some of the data may be know only within specified bounds,
the other data may be known only in terms of ordinal relations. Although the enhanced
study provided imprecise data using the DEA approach, it did not deal with data specified in
bounded forms.
The outputs of banks, including loan income and investment income, in fact, possess
fuzzy characteristics: for example, credit granting is a risky output because of the ex ante
risk for loans to eventually become non-performing loans (NPLs) (Li, 2003).
It is difficult
to accurately measure how many problem loans might become normal loans and how many
normal loans might become problem loans. As for investment, the investment income of a
bank, which is not a constant number, changes daily on account of the market value of the
investment target changes daily.
To evaluate banks’ efficiency more realistically and
accurately, this study employs the fuzzy DEA model with data specified in bounded forms
to measure the efficiency of banks.
Financial institutes in all countries are important parts of the chain in the financial
systems.
The quality of the financial institutes determines the level of development of a
country’s economy.
A healthy financial institution relies on its asset quality, capacity, asset
flow, capital adequate ratio, and etc.
has recently grown fierce.
However, competition between financial institutes
Gradual increasing NPLs indeed affect economy seriously.
2
In
the late 1980s, Japanese banks generally lacked an awareness of risks (Hoshi and Kashyap,
1999). They turned banks into over-loaning enterprises.
By the early 1990s, when the
“bubble economy” collapsed, a large scale of enterprises went bankrupt resulting in the
insurgence of NPLs in a troubled financial system.
In 1988, more than 200 American
commercial banks went bankrupt (see Figure 1), the increase of bad assets and worsening
finance exposed the banking industry to liquidity risks and serious financial crisis.
In
1997-98, the Asian financial crisis caused all the countries involved, South Korea, Thailand,
Indonesia, Malaysia and so forth, to suffer from drastic increases in overdue loans and a
worsening financial condition (Park, 2002).
The affected countries shared common
problems of excessive risk-taking, especially after the economical liberalization in
Southeast Asia, mainly because these countries lacked an appropriate management system
(Kao and Liu, 2004).
Therefore, a high NPL ratio has been a general problem of the
global financial industry and results in the predicament of banks’ operations.
The main
purpose of this study is to take the fuzzy phenomena of outputs into consideration in order
to effectively incorporate NPLs to measure banks’ efficiency.
250
209
200
206
186
Banks Failure Number
158
150
141
116
100
78
45
50
32
10
7
1980
1981
0
1982
1983
1984
1985
1986
1987
Year
Data source: FDIC, Commercial Bank Report, 2005, US.
Figure 1: US Banks’ Bankruptcy in 1980’s
3
1988
1989
1990
After publishing the fuzzy set that began in 1965, Zadeh applied a mathematical
method to express the fuzzy situation in a real environment, and then fuzzy set approach
was introduced into engineering science, creating a foundation for the fuzzy set theory.
The theory is widely applied from medicine to engineering to the commercial field.
Fuzzy
DEA, combined fuzzy set theory with DEA model, is more capable of handling drawbacks
of data than the traditional DEA approach and is able to find the interval for the fuzzy
number efficiency score even if the correct efficiency score of the fuzzy number is
unavailable.
Furthermore, a possible interval for the efficiency score can be derived given
a different
α-cut, and the greater the  value is, the more possible it is to see an
accurate efficiency score of the fuzzy number.
Kao and Liu (2000a) applied  -cut and the extension principle which Zadeh (1965)
proposed to transform a fuzzy DEA model into a conventional crisp DEA model. The
upper and lower bounds on the membership functions of the efficiency score is derived
under a specific  level.
Saati et al. (2002) employed a fuzzy CCR model with an
asymmetrical triangular fuzzy number.
Entani et al. (2002) obtained the interval efficiency
score from the optimistic and the pessimistic viewpoints and extended the idea into interval
data or DEA model with fuzzy data.
This study mainly discusses the influence on bank efficiency with fuzzy outputs and
adopts fuzzy set theory to solve problems.
Fuzzy DEA may, in addition to measuring
fuzzy data, be applied to establish the upper and lower bounds of the efficiency score and
obtain valuable information from the intervals between the upper and the lower bounds
under a specific level.
The application of fuzzy DEA model can more realistically
represent real-world problems than the conventional DEA models (Lertworasirikul et al.,
2003).
The remainder of this paper is organized as follows.
In Section II the Fuzzy DEA
model and the process of measuring this problem are discussed and proposed.
4
An
implication example is used to illustrate the proposed model for measuring bank efficiency
in Section III. Finally, discussions are presented in Section IV and conclusions are given in
Section V.
II. The Fuzzy DEA Model
Suppose that there are H DMUs producing M outputs by using N inputs.
In Fuzzy
DEA, inputs and outputs are characterized by uncertainty, and therefore we use ~x jn and
~
y jm to represent input n and output m of DMUj, respectively.
x jn
Assume that the inputs ~
y jm are approximately known and can be represented by membership
and the outputs ~
functions ~x jn and  ~y jm of the fuzzy set, respectively.
Note that the crisp value can be
represented by degenerated membership functions in which there is only one value in their
domain. The fuzzy DEA model can be written as follows:
E j  max
s.t.


M

u 
um y mj  u0

n 1 n nj
u y mh

n 1 n nh
M
m 1
m1 m
0
N
N
u1 ,..., uM , v1 ,..., vN    0, u0
is
where  is a small non-Archimedean quantity.
vx
v x  1,
h  1,..., H
free
(1)
Kao and Liu (2000) proposed a way to
transform the fuzzy DEA model to the traditional crisp DEA model by applying  -cut
approach.
x jn and ~
y jm are defined as follows:
The  -cut of ~
5
( x jn )  {x jn  S ( ~
x jn ) |  ~x jn ( x jn )   }  [( x jn )L , ( x jn )U ]
 [min {x  S ( ~
x ) |  ~ ( x )   },
x jn
jn
jn
x jn
(2a)
jn
max {x jn  S ( ~
x jn ) |  ~x ( x jn )   }]
x jn
jn
( y jm )  { y jm  S ( y jm ) |  y jm ( y jm )   }  [( y jm )L , ( y jm )U ]
 [min{ y jm  S ( y jm ) |  y jm ( y jm )   },
(2b)
y jm
max{ y jm  S ( y jm ) |  y jm ( y jm )   }]
y jm
y jm ) are the support of ~
x jn ) and S ( ~
x jn and ~
y jm , respectively.
where   [0, 1] , and S ( ~
The intervals indicate the corresponding inputs and outputs ranges at each possibility level
.
According to Zadeh’s extension principle (1978), the membership function of
efficiency evaluation for DMUj may be defined as follows:
  x  ,   y  , j, m, n z  E  x, y 
E ( z )  sup min x
j
x, y
jn
jn
y jm
jm
j
(3)
where E j ( x, y ) is the efficiency score calculated by the conventional BCC model under a
set of x and y.
~
In order to find the bounds of the intervals for E j ( x, y) at each possibility
level α, Kao and Liu (2000) suggest a pair of mathematical programming, which is
function of α, as follows:
E 
L
j 
 max
s.t



M
M
u ( y jm
m1 m
M

) u  
) u  
N
u ( y jm )L  u0
m1 m
u ( yhm
m1 m
N
L

0
U

v ( x jn )U
n 1 n
0
v ( x jn )U  1
n 1 n
N
v ( xhn )L  1, h  1,..., H , h  j
n 1 n
u1 ,..., uM , v1 ,..., vN    0, u0
6
is
free.
(4a)
E 
U
j 
 max
s.t



M
M
u ( y jm
m1 m
M

) u  
) u  
u ( y jm )U  u0
m1 m
u ( yhm
m1 m
N
U

v ( x jn )L
v ( x jn )L  1
0
n 1 n
0
n 1 n
L

N
n 1 n
N
v ( xhn )U  1, h  1,..., H , h  j
u1 ,..., uM , v1 ,..., vN    0, u0
is
(4b)
free.
Model (4a) means that in order to find the minimal relative efficiency of the DMUj
compared with others, the data should be applied from the lowest output value of the DMUj
and the lowest input value of other DMUs as well as the highest input value of the DMUj
and the highest output value of other DMUs.
Similarly, model (4b) indicates that for the
maximal relative efficiency of the DMUj compared with others, the data should be applied
from the highest output value of the DMUj and the highest input value of other DMUs as
well as the lowest input value of the DMUj and the lowest output value of other DMUs.
Furthermore, model (4a) and (4b) are conventional DEA models capable of translating into
linear programming to obtain the optimal weights. The -cut set of fuzzy efficiency score
~
E j may be established as ( E j )  [( E j )L , ( E j )U ] . A different  value means a different
range/interval and a level of uncertainty of the efficiency score. The greater the  value is,
the smaller the range/interval of upper and lower bounds is and the lower the level of
uncertainty is.
The value   0 means the widest range that the efficiency score will
emerge, and   1 means the efficiency score that is most likely to be achieved.
Fuzzy DEA may result in a fuzzy efficiency score.
Picking the best DMU from
numerous fuzzy efficiency scores cannot be solely determined by the fuzzy efficiency score.
Therefore, ranking fuzzy efficiency score becomes the key to finding the best DMU.
7
There are many ranking methods for fuzzy numbers (Chen, 1985, Tseng and Klein, 1989,
Chen and Klein, 1997, Saati et al., 2002; Kao and Liu, 2003).
However, most of the
ranking approaches require known membership functions, which are difficult to acquire in
the real world.
This study adopts the approach, proposed by Chen and Klein (1997), to
rank the fuzzy numbers only based on  -cut.
Let k be the maximum height of  E~ j ,
j  1,..., H .
Suppose that k is equally divided into T intervals such that  i  ik T ,
i  0, 1,..., T .
Chen and Klein (1997) defined the following index to rank the fuzzy
efficiency scores:
I j  i 0 (( E j )Ui  c)
T

T
i 0

(( E j )Ui  c)  i 0 (( E j )Li  d ) ,
T
where c  min i , j {( E j )Li } and d  max i , j {( E j )Ui } .
T 
(5)
The DMU with the favored fuzzy
efficiency score will have the larger index number.
The fuzzy DEA model of this study may be divided into four steps, as described below:
Step 1. Determine the type of fuzzy membership function
We may know, based on the fuzzy set theory, that the membership function sits
between 0 and 1.
Generally, the membership function can be classified into a
trigonometric function, trapezoidal function, Z-function, and sigmoid function; the
trigonometric function and trapezoidal function are more often applied.
Step 2. Calculate the upper and lower interval values of theα-cut
In accordance with the fuzzy situation of the multi-value, to induce fuzzy space to
calculate the upper and lower interval values of the α-cut.
Step 3. Calculate the upper and lower bounds of the efficiency score
Calculate the upper and lower bounds of the efficiency score of each DMU, and then
8
repeat to find the upper and lower bounds of the efficiency scores of all DMUs.
Step 4. Rank the fuzzy efficiency
According to the index proposed by Chen and Klein (1997), we can rank the fuzzy
efficiency scores of all DMUs to find the better performing DMU only based on α-cut.
Then, we use the model to measure the bank efficiency. An empirical example is illustrated
to show the proposed model in next Section.
III. An Empirical Example for Measuring Bank Efficiency
This study takes Taiwanese commercial banks as an example, calculates the data
collected on the basis of the trapezoidal function to obtain the upper and lower interval
values of each  value, and finds the upper and lower bounds of the efficiency score using
model (4).
We then rank the calculated upper and lower bounds of the efficiency score to
find the best performing banks.
The sample is the annual data of 48 Taiwanese commercial banks in 2002.
The
sources of the data are Taiwan Economic Journal Data Bank, “Conditions and Performance
of Domestic Banks Quarterly,” “Financial Institutions Major Business Statistics” published
by the Economic Research Department of the Central Bank of Taiwan, and statistics data of
foreign bank branches in Taiwan provided by the Bureau of Monetary Affairs, Ministry of
Finance in Taiwan.
The choice of inputs and outputs is perhaps the most important task in employing DEA
to measure the relative efficiency of the DMUs.
Two approaches are widely used to
identify a bank’s inputs and outputs, the production approach (Sherman and Gold, 1985),
and the intermediation approach (Aly et al., 1990; Yue, 1992; Miller and Noulas, 1996;
Favero and Pepi, 1995).
Under the production approach, banks are treated as a firm to
produce loans, deposits, and other assets by using labor and capital.
9
However, banks are
considered as financial intermediaries to transform deposits, purchase funds and labors into
loans and other assets under the intermediation approach.
More specifically, deposits are
treated as an input under the production approach and an output under the intermediation
approach.
This study views banks as intermediation institutions and adopts the concept of the
intermediation approach to recognize outputs and inputs.
Hence, the outputs include loan,
investment, and other income, while the inputs consist of the number of employees, total
fixed asset, and deposits.
We treat loan and investment as fuzzy items and analyze the
efficiency of the banking industry with a trapezoidal function from the fuzzy set theory.
The advantage of the trapezoidal function is that imprecise outputs still retain the fuzzy
characteristics (interval values) when α=1. The trigonometric function, another widely
used function in fuzzy set theory, regresses into a single value when α=1 without the fuzzy
phenomena.
The leftmost value and the rightmost value of loan and investment, characterized by
fuzzy phenomena, in the trapezoidal function ( a1 ,a5 in Figure 2) must be determined first.
In terms of the loan variable, this study places the loan variable in the middle ( a3 ).
Adding and subtracting NPLs (loans × NPLs ratio) from a3 results in a1 and a5 .
Adding and subtracting one half of NPLs from a3 results in a2 and a4 .
In terms of the
investment variable, considering that investment involves profits and losses from two items,
including buying and selling stocks and bills and long-term investment of equity shares, this
study places the investment variable in the middle ( a3 ). Adding and subtracting two times
the profit or loss from buying and selling stocks and bills and long-term investment of
equity shares from a3 results in a 1 and a 5 .
Adding and subtracting the profit or loss
from buying and selling stocks and bills and long-term investment of equity shares from a3
results in a2 and a4 .
The fuzzy output of -cut, to any   [0, 1] , is expressed as
[( a2  a1 )  a1 , (a4  a5 )  a5 ] by the interval method.
10
Insert a1 ~ a5 respectively, and
match them with 11  values, i.e.,   0, 0.1, 0.2,..., 1 , to find the upper and lower
interval values of loan and investment of all the sample banks so as to calculate the fuzzy
efficiency score of each bank in this empirical example.
α
1.0
0.0
α1
α2 α3 α4
α5
Input/Output
Figure 2 Trapezoidal Membership Function
After the upper and lower bounds of loan and investment of all sample banks are
derived in step 2, we can make use of the concept from models (4a) and (4b) to find the
upper and the lower bounds of the relative efficiency score of each bank.
Table 1 shows
the possible upper and lower bounds of efficiency scores of each sample bank under each
 -cut. The value   0 means the range that the efficiency score must fall within, and
  1 means what the efficiency score is most likely to be. The greater the  value is,
the smaller the interval will be between the upper and lower bounds of the efficiency score.
For example, the efficiency score of Bank 5 (the name of banks and their corresponding
series numbers of sample banks list in Appendix) is not less than 0.916 and not greater then
0.969 (   0 ), while its efficiency score is most likely between 0.929 and 0.956 (   1 ).
Table 1 also shows that the minimal value is 0.479 and the maximum value is 1.
There are
14 banks (Bank 1, 3, 4, 7, 11, 16, 20, 30, 31, 35, 36, 43, 45, and 47) with the upper and
lower bounds of an efficiency score of 1.
Hence, these 14 banks have efficiency score 1
11
with a crisp value.
Table 1: Fuzzy Efficiency Scores of Taiwanese Commercial Banks under  Value
Sample
Banks
  0.0
  0.2
  0.4
  0.6
  0.8
  1.0
1
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
2
(0.588, 0.803) (0.596, 0.791) (0.605, 0.780) (0.613, 0.769) (0.622, 0.757) (0.632, 0.746)
3
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
4
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
5
(0.916, 0.969) (0.918, 0.966) (0.921, 0.964) (0.924, 0.961) (0.926, 0.958) (0.929, 0.956)
6
(0.932, 1.000) (0.946, 1.000) (0.960, 1.000) (0.974, 1.000) (0.988, 1.000) (1.000, 1.000)
7
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000,1.000) (1.000,1.000)
8
(0.870, 1.000) (0.880, 1.000) (0.890, 1.000) (0.900, 1.000) (0.911, 1.000) (0.921, 1.000)
9
(0.928, 1.000) (0.938, 1.000) (0.948, 1.000) (0.958, 1.000) (0.968, 1.000) (0.978, 1.000)
10
(0.762, 1.000) (0.775, 1.000) (0.788, 1.000) (0.801, 1.000) (0.815, 1.000) (0.829, 0.985)
11
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
12
(0.786, 0.855) (0.791, 0.854) (0.797, 0.853) (0.802, 0.852) (0.808, 0.851) (0.814, 0.850)
13
(0.635, 0.768) (0.642, 0.762) (0.649, 0.756) (0.655, 0.750) (0.662, 0.744) (0.669, 0.738)
14
(0.804, 0.861) (0.807, 0.858) (0.810, 0.855) (0.812, 0.852) (0.815, 0.849) (0.818, 0.846)
15
(0.676, 0.843) (0.683, 0.834) (0.691, 0.826) (0.700, 0.818) (0.710, 0.810) (0.719, 0.802)
16
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
17
(0.912, 1.000) (0.928, 1.000) (0.945, 1.000) (0.961, 1.000) (0.978, 1.000) (0.995, 1.000)
18
(0.880, 1.000) (0.886, 0.998) (0.892, 0.992) (0.898, 0.985) (0.904, 0.979) (0.910, 0.973)
19
(0.982, 1.000) (0.988, 1.000) (0.994, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
20
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
21
(0.745, 0.807) (0.749, 0.805) (0.753, 0.803) (0.757, 0.801) (0.761, 0.798) (0.765, 0.796)
22
(0.632, 0.744) (0.637, 0.738) (0.643, 0.732) (0.649, 0.727) (0.654, 0.721) (0.660, 0.716)
23
(0.616, 0.835) (0.622, 0.821) (0.627, 0.809) (0.634, 0.796) (0.642, 0.784) (0.650, 0.772)
24
(0.479, 1.000) (0.483, 1.000) (0.491, 1.000) (0.512, 1.000) (0.550, 1.000) (0.593, 1.000)
25
(0.886, 0.965) (0.890, 0.961) (0.894, 0.958) (0.898, 0.954) (0.903, 0.950) (0.907, 0.946)
26
(0.952, 1.000) (0.954, 1.000) (0.956, 1.000) (0.958, 1.000) (0.960, 1.000) (0.962, 1.000)
27
(0.786, 0.921) (0.793, 0.914) (0.800, 0.908) (0.807, 0.901) (0.814, 0.895) (0.820, 0.888)
28
(0.652, 0.734) (0.656, 0.730) (0.660, 0.725) (0.664, 0.721) (0.668, 0.716) (0.672, 0.712)
29
(0.814, 0.958) (0.821, 0.951) (0.828, 0.944) (0.836, 0.937) (0.844, 0.930) (0.851, 0.923)
30
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
12
Table 1: (continued)
31
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
32
(0.600, 0.736) (0.605, 0.729) (0.611, 0.722) (0.616, 0.714) (0.622, 0.707) (0.628, 0.700)
33
(0.629, 0.752) (0.636, 0.747) (0.642, 0.741) (0.649, 0.736) (0.656, 0.730) (0.663, 0.725)
34
(0.721, 0.882) (0.728, 0.873) (0.735, 0.865) (0.742, 0.856) (0.750, 0.848) (0.757, 0.839)
35
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
36
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
37
(0.759, 1.000) (0.777, 1.000) (0.796, 1.000) (0.815, 1.000) (0.834, 1.000) (0.854, 1.000)
38
(0.849, 0.962) (0.855, 0.956) (0.860, 0.951) (0.866, 0.946) (0.872, 0.940) (0.878, 0.935)
39
(0.569, 0.672) (0.575, 0.667) (0.580, 0.662) (0.586, 0.657) (0.591, 0.652) (0.596, 0.647)
40
(0.545, 0.702) (0.552, 0.694) (0.560, 0.686) (0.567, 0.678) (0.575, 0.670) (0.583, 0.662)
41
(0.617, 0.824) (0.627, 0.814) (0.639, 0.804) (0.653, 0.794) (0.668, 0.783) (0.678, 0.773)
42
(0.663, 0.988) (0.663, 0.961) (0.663, 0.933) (0.663, 0.906) (0.663, 0.883) (0.663, 0.860)
43
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
44
(0.988, 1.000) (0.988, 1.000) (0.988, 1.000) (0.988, 1.000) (0.988, 1.000) (0.988, 1.000)
45
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
46
(0.918, 0.968) (0.920, 0.965) (0.922, 0.963) (0.924, 0.960) (0.927, 0.957) (0.929, 0.955)
47
(1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000) (1.000, 1.000)
48
(0.917, 1.000) (0.917, 1.000) (0.917, 1.000) (0.919, 1.000) (0.921, 1.000) (0.923, 1.000)
Note: (1) the number of left side in parentheses is the lower bound of fuzzy efficiency score, and the
number of right side in parentheses is the upper bound of fuzzy efficiency score;
(2) this table lists only 6  values, e.g.  = 0, 0.2, 0.4, 0.6, 0.8, 1;
(3) see Appendix for the serial numbers of sample banks.
This study employs the index proposed by Chen and Klein (1997) to rank the fuzzy
efficiency scores of all banks.
A bank with the larger index number is associated with a
more preferred fuzzy efficiency score.
efficiency (Farrell, 1957).
scores.
Note that the ranking index is not the Farrell
The indices only serve to rank DMUs with the fuzzy efficiency
Theoretically, this approach requires the number of -cuts, T, to be large enough.
However, Chen and Klein (1997) argued that T = 3 or 4 is sufficient to discriminate DMUs.
Table 2 shows that there are 14 efficient banks (with a ranking index of 1), which are
corresponding to those banks with the upper and lower bounds of an efficiency score of 1 in
13
Table 1. Other banks are inefficient (with a ranking index to be less than 1), which are
those banks with at least one lower bound of less than 1; for example, the ranking index is
0.989 for Bank 19, which has the upper and lower bounds of an efficiency score of 1 when
  0.6 while its lower bounds are less than 1 when   0.4 (upper bounds are still 1).
Table 2 Ranking of Fuzzy Efficiency Score of Taiwan’s Commercial Banks
Ranking
Sample Bank
Ranking Index
Ranking
Sample Bank
Ranking Index
1
1
1.000
25
18
0.829
1
3
1.000
26
25
0.821
1
4
1.000
27
38
0.774
1
7
1.000
28
29
0.733
1
11
1.000
29
37
0.728
1
16
1.000
30
10
0.717
1
20
1.000
31
27
0.684
1
30
1.000
32
14
0.665
1
31
1.000
33
12
0.651
1
35
1.000
34
34
0.594
1
36
1.000
35
21
0.568
1
43
1.000
35
42
0.568
1
45
1.000
37
15
0.531
1
47
1.000
38
24
0.518
15
19
0.989
39
41
0.475
16
44
0.977
40
23
0.468
17
6
0.940
41
13
0.441
18
26
0.924
42
2
0.431
19
17
0.918
43
33
0.423
20
9
0.917
44
28
0.419
21
48
0.865
45
22
0.414
22
46
0.863
46
32
0.382
23
5
0.862
47
40
0.318
24
8
0.833
48
39
0.302
14
IV. Discussions
Empirical results show that there are 14 banks (Bank 1, 3, 4, 7, 11, 16, 20, 30, 31, 35,
36, 43, 45, and 47) with the upper and lower bounds of an efficiency score of 1; in other
words, they are efficient banks surely.
conventional crisp DEA model.
These banks, of course, are also efficient under the
However, there are another 5 banks, Bank 6, 9, 17, 19,
and 44, that are efficient (with efficiency score 1) under non-fuzzy condition, while their
ranking indices are less than 1 with rankings between 15 and 20 (i.e., at least one lower
bound is less than 1) under fuzzy condition.
This may indicate that the fuzzy DEA
approach might have a higher ability to discriminate between efficient and inefficient banks
than the conventional DEA approach.
The average NPLs ratio of Taiwanese commercial
banks is 9.08% in 2002, while in our study it is 20.07% for Bank 44.
This average ratio
from 2002 implies that its quality of loans is far below average in Taiwan’s banking industry.
The traditional crisp DEA model without adjusting the quality of loans indicates that
Taiwan’s banking industry is efficient, while the fuzzy DEA model with adjusting for the
quality of loans identifies the banking industry to be inefficient.
Hence, a better approach
to measuring banks’ efficiency is to adjust effectively the loan and/or investment quality.
There are 12 banks, Bank 6, 8, 9, 10, 17, 18, 19, 24, 26, 37, 44, and 48, with upper
value of 1when  = 1 (the widest interval), indicating that the best efficiency score of these
banks might be 1 (i.e., each of them is a possible efficient bank).
Nevertheless, their lower
bounds, in which the efficiency scores will never fall below, show great variation from
0.479 to 0.988 with rankings between 15 and 38.
The other 22 banks are unlikely to be
efficient since their upper bounds are less than one for all  values.
Furthermore, the
banks with lower quality of loans and/or investment will have a wider interval, suggesting
that the fuzzy DEA model can characterize uncertainty effectively.
15
Operational performance is especially more important under the fierce competition and
rapid changing environment of the banking industry. Compared with previous studies of
evaluating bank efficiency with the DEA approach, the greatest feature of Fuzzy DEA is
that it is able to resolve the uncertainty in the production process (Cooper et al., 1999; Kao
and Liu, 2000; Guo and Tanaka, 2001; Despotis and Smirlis, 2002), which the conventional
DEA could not handle.
V. Conclusions
DEA is widely applied to measure the performance of the banking system since it is
capable of evaluating the efficiency of DMUs with multi-output and multi-input.
Previous
studies used the conventional DEA model to measure banks’ efficiency by assuming that all
inputs and outputs are crisp.
However, loan and investment, the outputs of banks, do
posses fuzzy phenomena because of the imprecise NPLs and the sizeable volatility of the
stock market.
The fuzzy DEA approach can evaluate the efficiency of banks more
accurately and realistically since it can take into account the fuzzy characteristics of inputs
and/or outputs.
This study takes the fuzzy phenomena of loan income and investment income into
consideration and employs the fuzzy DEA approach to evaluate Taiwanese commercial
banks.
The empirical results show that the fuzzy DEA approach could not only
successfully characterize the uncertainty of efficiencies, but also may have the higher ability
to discriminate banks’ efficiency than the traditional DEA model.
as follows:
The other findings are
(1) There are 14 banks with precise efficiency score of 1 even though their
loans and investments characterized by fuzzy phenomena; (2) Twelve inefficient banks with
at least one upper bound to be one still has the possibility to be efficient; (3) There are 22
banks that are unlikely to be efficient since all of their upper and lower bounds are less than
1.
16
There are several possible extensions for this study.
We can apply this approach to analyze
other financial institutions, such as foreign-owned banks and credit unions. Furthermore,
this method can help us to investigate financial institutions with missing and/or incomplete
data.
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19
Appendix: Serial numbers of sample banks:
No
Bank
No
Bank
1
Chiao Tung Bank
25
Taishin International Bank
2
The Farmers Bank of China
26
Fubon Bank
3
Bank of Taiwan
27
Ta Chong Bank
4
Taipei Bank
28
Jih Sun International Bank
5
Bank of Kaoshiung
29
EnTie Bank
6
Land Bank of Taiwan
30
Chinatrust Commercial Bank
7
Taiwan Cooperative Bank
31
Chinfon Bank
8
First Commercial Bank
32
Macoto Bnk
9
Hua Nan Commercial Bank
33
Sunny Bank
10
Chang Hwa Commercial Bank
34
Bank of Panshin
11
The International Commercial Bank of China
35
Lucky Bank
12
United World Chinese Commercial Bank
36
Kao Shin Commercial Bank
13
Hua Chiao Bank
37
Taiwan Business Bank
14
The Shanghai Commercial & Savings Bank Ltd.
38
International Bank of Taipei
15
Grand Commercial Bank
39
Hsinchu International Bank
16
Union Bank of Taiwan
40
Taichung Commercial Bank
17
The Chinese Bank
41
Tainan Business Bank
18
Far Eastern International Bank
42
Kaohsiung Business Bank
19
Fuhwa Bank
43
Enterprise Bank of Hualien
20
Bank Sinopac
44
Taitung Business Bank
21
E. Sun Bank
45
Cathay United Bank
22
Cosmos Bank, Taiwan
46
Hwatai Bank
23
Pan Asia Commercial Bank
47
COTA Commercial Bank
24
Chung Shing Bank
48
United-Credit Commercial Bank
20