Data Envelopment Analysis (DEA) is a data-oriented approach for relatively
evaluating the performance of a set of homogenous entities referred as Decision
Making Units (DMUs) by using a ratio of the weighted sum of outputs to the weighted
sum of inputs. DEA models first proposed by Charnes et
al. (1978) are formulated to measure the efficiency of DMUs for crisp
data. Although traditional DEA models such as CCR (Charnes-Cooper-Rhodes) and
BBC (Banker-Charnes-Cooper) are a powerful tool for efficiency measurement,
but are only developed for crisp data (Charnes et al.,
1978; Banker et al., 1984; Banker
and Morey, 1986; Zhou et al., 2007). Whereas,
in many real situations, inputs and outputs in DEA models are imprecise and
inaccurate. Therefore, how to evaluate the management or operation efficiency
of a set of DMUs in fuzzy environments is a worth-studying problem. Fuzzy set
theory has been proposed as a way to challenge imprecise data in DEA models
(Bellman and Zadeh, 1970).
Fuzzy DEA model is a fuzzy linear programming that provides a technique to
deal with the uncertainty in fuzzy objectives and/or fuzzy constraints (Buckley,
1988; Rommelfanger et al., 1989; Julien,
1994). We can meet various fuzzy DEA models to evaluate the efficiency of
DMUs in the DEA literature (Guo and Tanaka, 2001; Jahanshahloo
et al., 2004; León et al., 2003;
Saati et al., 2002; Guo, 2008;
Kao and Liu, 2000, 2003; Soleimani-Damaneh
et al., 2006). Most of the previous studies in the fuzzy DEA included
the computational complexity, hence, the decision maker cannot be clearly realized
the procedure of assessing DMUs. Therefore, we try to propose a simple approach
for imprecise environment to obtain a method that is usable and practicable
for real-world problems such as performance of bank branches.
The main subject in using fuzzy theory in DEA is the notion of ranking fuzzy
numbers. Several different approaches have been proposed for ranking fuzzy numbers.
These include methods based on the Coefficient of Variation (CV index), distance
between fuzzy sets, centroid point and original point and weighted mean value.
In this study, the main interest is in the approach proposed by Asady
and Zendehnam (2007), which uses a modification of a distance method called
sign distance. Asady and Zendehnams (2007) method
constructed a ranking for fuzzy numbers is very realistic and its method can
be used for ordering all types of the fuzzy numbers. Their method led to the
crisp point, which is the best related to a certain measure of distance between
the fuzzy number and a crisp point of support function. Some advantages of their
method are the following:
||In that approach used the nearest point of support function
for ranking fuzzy numbers and the calculation is very simple. It is
independent of family of the fuzzy numbers, horizon, upper and/or
||A distance is applied in space fuzzy number
||The distance minimization is used for ordering of fuzzy
In this study, by using ranking fuzzy numbers, we present a new defuzzification
approach to solve fuzzy CCR model. For the case of fuzzy data in the form
of trapezoidal fuzzy numbers, DEA models are linear programming and can
be solved by usual methods. For demonstrating the applicability of the
proposed method, three numerical examples including an application to
bank branches at Tehran city in Iran are represented. The banking industry
has grown into profitable competitive. Improving the efficiency of bank
branches is now considered to be important to the banking industry and
this case study aims to help clarify this subject by applying the proposed
PRELIMINARY DEFINITIONS OF FUZZY DATA
Some basic definitions of fuzzy sets are reviewed by Kauffman
and Gupta (1991), Zimmermann (1996) and Dubois
and Prade (1978).
Definition 1: Let U be a universe set. A fuzzy set
of U is defined by a membership function ,
indicates the degree of membership of
Definition 2: A real fuzzy number
= (a, b, c, d; w) is described as any fuzzy subset of the real line R
with membership function
which satisfies the following properties:
is a semi continuous mapping from R to the closed interval [0, w],
0 ≤ w ≤ 1
(x) = 0, for all xε[- ∞ , a]
is increasing on [a, b]
(x) = w for all xε[b, c], where w is a constant and 0<w ≤
is decreasing on [c, d]
(x) = 0, for all xε[d, ∞ ]
where, a, b, c and d are real numbers.
Unless elsewhere specified, it is assumed that
is convex and bounded; i.e., - ∞ <a, d< ∞ . If w = 1,
is a normal fuzzy number and if 0<w<1,
is a non normal fuzzy number.
The membership function
can be expressed as:
where, fL: [a, b] → [0, w] and fR: [c, d]
→ [0, w].
In addition, a fuzzy number
in parametric form is denoted by
of function ,
which satisfies the following requirements:
||a(r) is a bounded increasing left continuous
is a bounded decreasing right continuous function
where, 0 ≤ r ≤ 1
Definition 3: A trapezoidal fuzzy number is widely best used for
solving pracMarch 14, 2009March 14, 2009tical problems. Trapezoidal fuzzy
number is determined by quadruples
of crisp numbers with two defuzzifier
and left fuzziness σ>0 and right fuzziness β>0, which
membership function can be defined as follows:
Trapezoidal fuzzy number in its parametric form can be obtained as follows:
Note that the trapezoidal fuzzy number is a triangular fuzzy number if ,
denoted by a triple (m, σ, β).
Definition 4: In fuzzy linear programming, the min T-norm is usually
applied to assess a linear combination of fuzzy quantities. Therefore,
for a given set of trapezoidal fuzzy numbers ,
j = 1, 2,
, n, a linear combination is defined as follows:
denotes the combination .
Ranking fuzzy numbers is important in decision-making, data analysis, artificial
intelligence and fuzzy linear programming. Many researchers have investigated
various ranking methods (Asady and Zendehnam, 2007; Wang
et al., 2006; Huijun and Jianjun, 2006; Cheng,
1998; Wang and Kerre, 2001a, b).
Recently, Asady and Zendehnam (2007) propose a method for
minimizing distance to order the fuzzy numbers. Interval EI ()
of a fuzzy number
is introduced independently by Dubois and Prade (1987)
and Heilpern (1992) as follows:
The middle point of interval EI ()
can be defined as follows (Asady and Zendehnam, 2007):
is used to rank fuzzy numbers. Therefore, for any two fuzzy numbers
Finally, if ,
In particular, if
be a trapezoidal fuzzy number, then:
The properties of this ranking method have been expressed by Asady
and Zendehnam (2007).
THE CCR MODEL WITH FUZZY COEFFICIENTS
The most frequently used DEA model is the CCR model, called after Charnes
et al. (1978). Suppose that there are n DMUs, each consumes the same
type of inputs and produces the same type of outputs. Let m be the number of
inputs and r is the number of outputs.
The CCR model is formulated as the following linear program:
where, yrp (r = 1,
, s) and xip (i = 1,
m) are the output and input values of DMUp, p = 1,
the DMU under consideration, respectively, ur (r = 1,
s) and vi (i = 1,
, m) the weights associated with rth output
and ith input, respectively. All inputs and outputs are assumed to be
nonnegative, but at least one input and one output are positive.
In recent years, fuzzy set theory has been proposed as a way to quantify
imprecise and vague data in DEA models. The CCR model with fuzzy coefficients
is as follows:
Many researchers are used triangular and trapezoidal fuzzy numbers in their
studies (Guo and Tanaka, 2001; Jahanshahloo
et al., 2004; León et al., 2003;
Saati et al., 2002). In this study also, inputs and outputs are assumed
to be fuzzy numbers with trapezoidal membership function.
When input-output data are fuzzy numbers, model (8) can be expressed
as the following fuzzy LP problem:
Model (9) is a fuzzy linear programming. There are several fuzzy approaches
to solve it in the literature which mentioned in introduction part. In this
study the proposed ranking fuzzy numbers method by Asady
and Zendehnam (2007) would be used to solve this fuzzy linear programming
problem. Therefore, based on definition 4, model (9) can be transformed as follows:
It is obvious that model (10) is a linear programming model. Therefore,
the conventional LP method can be applied to solve it.
It should be noted that by crisp data, model (10) will be converted to
standard CCR model.
To avoid complex operation in ranking of DMUs, the linear scale transformation
is used to transform the different efficiency values into comparable scale.
The normalized efficiencies can be represented as:
Therefore, DMUp, p = 1,
, n is an efficient DMU if Wp
= 1, otherwise it is an inefficient one. It is evident that there exists
at least one efficient DMU.
Here, we examine three numerical examples using the proposed fuzzy DEA model.
The first two examples are taken from Soleimani-Damaneh
et al. (2006) and León et al. (2003)
for the purpose of comparison and the last example is an application of the
proposed model to sixteen banks branches at Capital city of Iran.
Example 1: Consider the example provided by Soleimani-Damaneh
et al. (2006) for 6 DMUs with single fuzzy input and single fuzzy
output. First two rows of Table 1 provide the data of this
example. The efficiency scores obtained by Soleimani-Damaneh
et al. (2006) and by the proposed approach in this study are recorded
in the last two columns of Table 1.
It can be seen that A and F are became efficient units in Soleimani-Damaneh
et al. (2006) method. Indeed when using the proposed approach, A
is inefficient and fell in second rank. Therefore, the proposed approach can
be used for assessing and ranking of efficient DMU in Soleimani-Damaneh
et al. (2006) approach.
Example 2: Consider the example investigated by León
et al. (2003), in which eight DMUs are evaluated against single fuzzy
input and single fuzzy output. The data are presented in the columns 1 and 2
of Table 2, respectively. According to (2003), A, B, C and
G are efficient, while the proposed approach only expressed C as efficient and
other DMUs can finally be ranked as A, B, H, D, D, E, G and F.
Example 3: Consider the collected data of sixteen branches at
Tehran city in Iran (2004) which each branch consumes three crisp and
two fuzzy inputs to produce two crisp outputs. The data for each DMU is
depicted in Fig. 1.
There are used linguistic variables shown in Fig. 2
to determine fuzzy inputs. Data of sixteen bank branches are shown in
Table 3 and 4.
|| Data and obtained efficiency scores by León
et al. (2003) and proposed approach in example 2
|| Conceptual model
|| Data of crisp inputs and outputs in example 3
|| Data of fuzzy inputs in example 3
|| Obtained results by proposed model
|| Linguistic variable
The results of the proposed method for ranking the branches are presented
in Table 5. It can be shown that branch 8 is ranked
at the first place by suggested method.
Since, Data Envelopment Analysis (DEA) was proposed in 1978, it has been got
comprehensive attention both in theory and application. Most of the studies
assume that all inputs and outputs are precisely known. But in more general
cases, uncertainty is an attribute of data. Hence, it may be more suitable to
interpret the experts understanding of the data as fuzzy numerical data
which can be represented by means of fuzzy numbers. In this study, an application
of ranking fuzzy numbers is introduced and CCR model with fuzzy inputs and outputs
is extended to propose a new fuzzy DEA (FDEA) model for evaluating the efficiencies
of DMUs. This approach transforms the fuzzy model into crisp LP model by ranking
fuzzy numbers. Also, the obtained efficiencies from the proposed model reflect
the inherent fuzziness in assessment problems. We compare the results of proposed
model with Soleimani-Damaneh et al. (2006) results
and León et al. (2003) by representing
a numerical example introduced by them. Furthermore, a case study for ranking
of banks branches in Iran is discussed. This application demonstrates
that fuzzy DEA models are quite powerful in evaluating real problems under uncertainty.
Because uncertainty always exists in human thinking and judgment, fuzzy DEA
models can play an important role in the real-world problems.