Performance Measurement of Two-Stage Production Systems with Undesirable Factors by Data Envelopment Analysis

INTRODUCTION

Data Envelopment Analysis (DEA), as reported by Charnes et al. (1978), is a data oriented non-parametric 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 two-stage 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, two-stage 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 two-stage processes. Based on Chen et al. (2009) two-stage DEA model, a DEA model for measuring the efficiency of two-stage 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 two-stage processes. An empirical example of 21 firms in the banking industry in Iran is used to illustrate the model.

CHEN et al.’S TWO-STAGE DEA MODEL

Figure 1 shows a graphical representation of a two-stage process. Consider n two-stage 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^* = w₁.E^*₁+w₂.E^*₂, where w₁ and w₂ are user-specified weights satisfying w₁+w₂ = 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:

(1)

Due to the series relationship between the two stages, they assumed that

Fig. 1:	Two stage process

By setting:

(2)

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:

(3)

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:

(4)

A TWO-STAGE DEA MODEL WITH UNDESIRABLE FACTORS

Now, suppose a two-stage process with undesirable factors. Let there are n DMUs with two-stage structure, where each DMU_j (j = 1,…, n) in the first stage utilizes m₁ desirable inputs x^D_ij (i = 1,…, m₁) and m₂ undesirable inputs x^U_ij (i = 1,…, m₂) with m₁+m₂ = m to produce p₁ desirable intermediate products z^D_dj (d = 1,…, p₁) and p₂ undesirable inputs z^U_dj (d = 1,…, p₁) with p₁+p₂ = p. These intermediate products are used in the second stage to produce s₁ desirable outputs y^D_rj (r = 1,…, s₁) and s₂ undesirable outputs y^U_rj (r = 1,…, s₁) with s₂+s₂ = s.

Improving the performance of two-stage processes through reducing undesirable outputs and increasing undesirable inputs is the aim. In order to achieve this goal, the following two-stage DEA model under VRS assumption would be proposed on the basis of Chen et al. (2009) two-stage model:

(5)

where, v_i^D (i = 1,…, m₁), v_i^U (i = 1,…, m₂), u_r^D (r = 1,…, s₁), u_r^U (r = 1,…, s₂), η^D_d (d = 1,…, p₁) and η^U_d (d = 1,…, p₂) are the multipliers associated with desirable and undesirable inputs, outputs and intermediate products, respectively. Note that if u¹ = u² = 0 then model (5) becomes for CRS cases.

In order to evaluate the overall efficiency scores of the whole two-stage 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:

(6)

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 sub-processes 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 (x₁), expenditures (x₂) and depreciation (x₃). Intermediate products (or output in the first stage) consist of total resources (z₁). Outputs of the second stage include income (y₁), usages (y₂) and receivables (y₃). 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 y₃ 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.