INTRODUCTION
Data Envelopment Analysis (DEA), as reported by Charnes
et al. (1978), is a data oriented nonparametric method which is
used to assess the efficiency of peer Decision Making Units (DMUs) with multiple
inputs and outputs. DEA has been widely applied in a variety of fields, including
banking (Seiford and Zhu, 1999; Amirteimoori,
2008), agriculture (Ghorbani et al., 2009;
Armagan, 2008; Laha and Kuri, 2011),
education (Rayeni et al., 2010; Rayeni
and Saljooghi, 2010), airlines (Zandieh et al.,
2009; Coli et al., 2011) and hospitals (Taher
and Malek, 2009) among many others.
Generally, DEA enhances the performance of inefficient DMUs by reducing the
inputs or increasing the outputs. However, sometimes DMUs involve both desirable
and undesirable factors. DEA literature has suggested several models which can
be used to measure the efficiency of DMUs with undesirable factors. For instance
models proposed by Fare et al. (1989), Seiford
and Zhu (2002), Vencheh et al. (2005) and
Amirteimoori et al. (2006) among others can be
observed in the literature. During the last few years, the focus of some studies
have been on twostage processes connected in series in which all outputs of
the first stage are intermediate products which constitute the inputs of the
second stage. For example, twostage DEA approaches have been developed to evaluate
the efficiency of the profitability and marketability of US commercial banks
(Seiford and Zhu, 1999), the best 500 companies (Zhu,
2000), the Major League Baseball (Sexton and Lewis,
2003). Chen and Zhu (2004) introduced a linear DEA
model, in which the efficiency of each stage is defined on its own production
possibility set. Later, Kao and Hwang (2008) developed
a new approach that the overall efficiency of the system is defined as the product
of the efficiencies of two individual stages. This approach can be only applied
under the assumption of Constant Returns to Scale (CRS). Chen
et al. (2009) modified this model to be applied in both CRS and Variable
Returns to Scale (VRS) assumptions, by using an additive form. Recently, Wang
and Chin (2010) generalized Chen et al. (2009)
model by considering the relative importance weight of two stages. None of these
studies have considered the undesirable factors that may be exist in twostage
processes. Based on Chen et al. (2009) twostage
DEA model, a DEA model for measuring the efficiency of twostage processes with
desirable and undesirable factors would be proposed in the present research.
According to the suggested approach, undesirable outputs are decreased and undesirable
inputs are increased to improve the performance of twostage processes. An empirical
example of 21 firms in the banking industry in Iran is used to illustrate the
model.
CHEN et al.’S TWOSTAGE DEA MODEL
Figure 1 shows a graphical representation of a twostage process. Consider n twostage processes, DMU_{j} (j = 1,…, n), with m inputs, p intermediate products and s outputs. Let x_{ij} (i = 1,…, m) be the inputs of the first stage and y_{rj} (r = 1,…, s) the outputs of the second stage of the DMU_{j}. Also, let z_{dj} (d = 1,…, p) be the intermediate products of DMU_{j} that are the outputs of first stage as well as the inputs of second stage. The efficiencies of DMU_{j} in the first and the second stage are defined as:
where, v_{i} (i = 1,..., m) and η_{d} (d = 1,..., p) are
the multipliers associated with inputs and outputs in the first stage and:
are the multipliers associated with inputs and outputs in the second stage.
Based on the efficiencies of DMU_{j} in the two individual stages,
Chen et al. (2009) proposed the overall efficiency
of DMU_{j} in the whole process as the weighted sum of the two individual
efficiencies, namely, E^{*} = w_{1}.E^{*}_{1}+w_{2}.E^{*}_{2},
where w_{1} and w_{2} are userspecified weights satisfying
w_{1}+w_{2} = 1.
These weights are proposed to reflect the relative importance or contribution of two stages, in measuring the overall efficiency of the whole process. Their model for measuring the overall efficiency of DMU_{o} under the assumption of Constant Returns to Scale (CRS) can be shown as follows:
Due to the series relationship between the two stages, they assumed that

Fig. 1: 
Two stage process 
By setting:
which, reflect the relative sizes of the two stages. By substituting (2) into
(1) and using (Charnes and Cooper, 1962) transformation,
model Eq. 1 becomes the following Linear Programming (LP)
program:
Note that from model (3), it is observed that the overall efficiency of DMU_{o} is measured by inputs x_{io} (i = 1,..., m), intermediate measures z_{do} (d = 1,..., p) and outputs y_{ro} (r = 1,..., s) where the intermediate products serve as both inputs and outputs of DMU_{o} at the same time.
Note that Chen et al. (2009) approach can be
applied under the Variable Returns to Scale (VRS). By following the method proposed
for CRS condition, the overall efficiency of DMU_{o} under VRS assumption
can be calculated using the following linear model:
A TWOSTAGE DEA MODEL WITH UNDESIRABLE FACTORS
Now, suppose a twostage process with undesirable factors. Let there are n DMUs with twostage structure, where each DMU_{j} (j = 1,…, n) in the first stage utilizes m_{1} desirable inputs x^{D}_{ij} (i = 1,…, m_{1}) and m_{2} undesirable inputs x^{U}_{ij} (i = 1,…, m_{2}) with m_{1}+m_{2} = m to produce p_{1} desirable intermediate products z^{D}_{dj} (d = 1,…, p_{1}) and p_{2} undesirable inputs z^{U}_{dj} (d = 1,…, p_{1}) with p_{1}+p_{2} = p. These intermediate products are used in the second stage to produce s_{1} desirable outputs y^{D}_{rj} (r = 1,…, s_{1}) and s_{2} undesirable outputs y^{U}_{rj} (r = 1,…, s_{1}) with s_{2}+s_{2} = s.
Improving the performance of twostage processes through reducing undesirable
outputs and increasing undesirable inputs is the aim. In order to achieve this
goal, the following twostage DEA model under VRS assumption would be proposed
on the basis of Chen et al. (2009) twostage
model:
where, v_{i}^{D} (i = 1,…, m_{1}), v_{i}^{U} (i = 1,…, m_{2}), u_{r}^{D} (r = 1,…, s_{1}), u_{r}^{U} (r = 1,…, s_{2}), η^{D}_{d} (d = 1,…, p_{1}) and η^{U}_{d} (d = 1,…, p_{2}) are the multipliers associated with desirable and undesirable inputs, outputs and intermediate products, respectively. Note that if u^{1} = u^{2} = 0 then model (5) becomes for CRS cases.
In order to evaluate the overall efficiency scores of the whole twostage processes, this model will be solved for n times, once for each DMU. On optimality, the efficiency scores of two stages of each DMU_{o} (o = 1,…,n), can be calculated as follows:
where,
are the optimal multipliers calculated from model (5). Using the model (5),
the overall efficiency of the DMU_{o} can be evaluated in such a way
that the operations of its two stages are taken into account. Also, recognizing
the inefficient subprocesses and making later improvements can be done through
Eq. 6.
NUMERICAL EXAMPLE
In this study, for a group of real data derived from 21 branches of Commercial Bank situated in 10 cities of one of the provinces in Iran; proposed model would be used. These data have been adopted from banking activities of mentioned branches during 2009. Production process in any of these banks has been divided into two stages: Absorbing resources and spending resources. Inputs of the first stage include the number of personnel (x_{1}), expenditures (x_{2}) and depreciation (x_{3}). Intermediate products (or output in the first stage) consist of total resources (z_{1}). Outputs of the second stage include income (y_{1}), usages (y_{2}) and receivables (y_{3}). Data set derived from 21 branches of the bank has been provided in Table 1. In this data set all factors are desirables and only output y_{3} is undesirable.
By applying the CRS version of model (5) and Eq. 6, the CRS
efficiency scores of the whole process and two stages of the 21 banks are calculated.
The results are shown in Table 2 under the heading CRS efficiency
scores.
Table 1: 
Data set of 21 bank branches in Iran 

Table 2: 
Efficiency scores 

It is worth mentioning that under the assumption of CRS only DMU 20 is efficient
in the first stage while DMUs 3, 10, 13, 15 and 16 are efficient in the second
stage. Because none of the banks perform efficiently in both stages, none of
them perform efficiently for the whole process.
Finally, the VRS efficiency scores of the whole process and two stages of the 21 banks, evaluated by model (5) and Eq. 6, are reported in Table 2 under the heading VRS efficiency scores. It can be seen that DMUs 8, 9 and 20 are efficient in the first stage and DMUs 3, 4, 8, 10, 13, 14, 15, 16 and 19 are efficient in the second stage. As a result, DMU 8 is known as the only VRS overall efficient DMU.
CONCLUSION
It has been realized that DMUs may have a twostage structure in some specific
applications, where the first stage utilizes inputs to generate outputs that
become the inputs of the second stage and the second stage the employs the first
stage outputs to produce its own outputs. The first stage outputs are called
intermediate products. Based on Chen et al. (2009)
model, the current paper provides a twostage DEA model in dealing with desirable
and undesirable factors in data envelopment analysis (DEA). An empirical example
has been examined using the proposed model.