RETURNS TO SCALE AND SCALE ELASTICITY IN TWO-STAGE DEA

AbstractData Envelopment Analysis (DEA) provides a method to evaluate the relative efficiency of peer Decision Making Units (DMUs) that have multiple inputs and outputs. Production process in two-stage DEA is performed in the two consecutive phases and DMUs have intermediate measures, in addition to their inputs and outputs. A unique feature of the intermediate measures is that the outputs in the first stage are being treated as inputs in the second stage. The aim of this paper is to determine the returns to scale (RTS) classification and scale elasticity (SE) in two-stage DEA. Therefore an approach is introduced for estimating the RTS situation of DMUs with two-stage structure based on the consideration of SE quantity in each of the individual stages. The utilization of the proposed approach is demonstrated with a real data set.


INTRODUCTION
Data envelopment analysis is a scientific method for the performance analysis of peer decision making units, in the presence of multiple inputs and outputs.In the recent years, a number of DEA studies have focused on measuring the relative efficiency of DMUs with a two-stage structure (e.g.see [1][2][3][4][5][6]).In the two-stage DEA, DMUs have a two-stage structure and intermediate measures exist between two consecutive stages.Namely, the first stage uses the inputs to generate intermediate measures and later on the second stage uses them to produce outputs.Consequently, the intermediate measures which were determined by the first stage are all of the second stage inputs.
Meanwhile returns to scale and scale elasticity are two important topics in the production theory and since the beginning of DEA research, RTS has been widely discussed as an important economic implication of DEA efficiencies.These two concepts can determine the optimal size of efficient DMUs under variable returns to scale technology.Most of the previous attempts to deal with the two-stage DEA have only addressed measuring the performance of such two-stage processes.Therefore, this research attempts to measure RTS in the analytical framework of two-stage DEA.To achieve this goal, the production space in two-stage DEA under variable returns to scale (VRS) technology is investigated and a method for measuring RTS quality in this field, regarding the SE quantity and RTS classification in each of the individual stages, is proposed.
The structure of this research unfolds as follows: In section 2, the two-stage DEA under variable returns to scale (VRS) technology is introduced and also the cost minimization model is applied for evaluating each DMU with two-stage structure.The proposed method for estimating RTS and SE in two-stage DEA is discussed in section 3. Section 4 includes an application of the proposed method to 26 branches of an Iranian commercial bank.Finally, the concluding remarks are provided in section 5.

TWO-STAGE DEA UNDER VARIABLE RETURNS TO SCALE
It is assumed that there are n DMUs which their production activities are performed in two phases.In the first stage, each , in order to produce D outputs, , in the second stage.Therefore, .Figure 1 visually describes this production process for j DMU .The Production Possibility Set (PPS) under variable returns to scale (VRS) technology for each stage can be defined by the following DEA formulations: . If are two row vectors of dual variables related to the first and second sets of constraints in model (1).Similarly, are those related to the third and fourth sets of constraints in model (1).Finally, the dual variables, is an optimal solution of model ( 2), then the following lemmas hold.

Lemma1. For each feasible solution
of (1), we have: is a feasible solution of model ( 1) therefore the following inequalities are conducted: Considering the third sets of constraints in (2) and It follows from ( 3) and ( 5) that Similarly, it is obtained from the fourth sets of constraints in (2), and the relation Mathematically, the complementary slackness conditions can be specified as follows: Now, it follows from ( 6), ( 7), (10) Also, from (8), ( 9) and (11)  Remark2.The cost minimization model is a non-radial DEA model.Therefore a problem of multiple projections can be found and this issue is an effective factor on RTS measurement.Sueyoshi et al. [9] have investigated how to solve this difficulty.In this study, it is assumed that the projection is unique.
Remark3.Scale elasticity (SE) is an important topic in performance analysis related to (RTS) .In fact, it represents the quantitative part of (RTS) which is the proportional change in outputs resulting from the equi-proportionate change in inputs.Considering the optimal solution of model ( 2), the scale elasticity of each stage is simply determined as follows [9,10] A problem associated with the RTS measurement is that sometimes a supporting hyper-plane of each stage could not be uniquely determined.In other words, it is necessary to consider an occurrence of multiple optimal solutions on * 1  and * 2  , in model (2).For dealing with this problem see [9].In this paper it is assumed that model (2) has unique optimal solution.

MESUREMENT OF RTS AND SE IN TWO-STAGE DEA
To introduce a new approach for RTS measurement in two-stage DEA, consider the following brief description on the relation among a supporting hyper-plane, RTS and SE in DEA [8,9,11].
The Case 2) If SE 1 =1 and SE 2 <1, then the overall RTS will depend on the existence or nonexistence of contraction possibility in stage 1.Consequently, we have proposed the following model to investigate this possibility.
Case 3) If SE 1 =1 and SE 2 >1, then the existence or non-existence of expansion possibility in stage 1 is the recognition criterion of RTS measurement.In this case, model (17) examines this possibility.

 
. 0 , 0 , 0 Similarly, the type of RTS at   The discussion similar above is held when SE 1 <1 and SE 2 >1.

APPLICATION
In this section the proposed method is applied to 26 branches of an Iranian commercial bank that each branch has a two-stage structure.The two inputs to the first stage are personnel privilege and interest on deposit.The two intermediate measures (or the outputs from the first stage) are the total sum of four deposits and bank commission.The three outputs from the second stage are facilities, bank interest and other resources.Table 1 reports the data set and the last row of it indicates the unit costs associated with the inputs and intermediate measures.
Note that the first input i.e. the personnel privilege is composed of some effective factors on quality of the personnel including the record, university degree, educational major, skills, salary, and so on.This input is obtained after normalizing these factors and therefore it is a non-dimensional quantity.Moreover, dollar has been considered as the unit of the other inputs and outputs.The interest on deposit is an amount that each branch pays to the clients for long-term and short-term deposit accounts.The first intermediate measure contains the sum of four deposits which are opened by the clients in each branch.These deposits are long-term and short-term deposit accounts and current and savings accounts.Since each branch allocates 20 percent of each deposit as   for determining the RTS classification of the units which they have the same RTS situation in their two stages.the interest to the depositor, therefore $0.2 is considered as the unit cost associated with the first intermediate measure.The bank commission is an amount that each branch receives from the clients for providing different services.The facilities are the loans and other credit facilities that each branch pays to the clients.The second output i.e., bank interest is the amount of interest that each branch receives from the customers for providing facilities.The last output i.e., other resources are the revenues that each branch makes by investing on different projects.
Table 2 reports the results from models (1), ( 2), ( 15) and (17).The optimal inputs and intermediate measures from model (1) for each branch are reported under columns 2, 3, 4 and 5.The cost efficiency corresponding to each branch appears in 6 th column.Using model (2), the columns 7 and 8 of table 2 represent the SE quantity for stages 1 and 2, respectively.The 8 th column reports the value of *  for RTS measurement of some branches based upon models (15) or (17).Note that each branch needs the value of *  for improving their size according to relations (16) and (18).The last column of table 2 reports the state of overall RTS for cost efficient projection of each branch.

CONCLUDING REMARKS
The current paper discusses the problems of SE and RTS measurement in twostage DEA.In a two-stage process, the first stage outputs which are the intermediate measures, will serve as the inputs of the second stage.In fact, this research has extended the RTS concept from classical DEA to the two-stage DEA.The proposed method determines the SE quantity and the type of overall RTS for the two-stage process considering the SE quantity in each stage.
Congestion indicates an economic state where inputs are overly invested.In other words, congestion is identified when an increase in one or some inputs causes the worsening of one or more outputs.From an economic theory, the issues of RTS and congestion are closely interrelated.Therefore investigating the congestion concept in two-stage DEA can be the future research issue.
jDMU is characterized by the two consecutive production stages, the first stage from data set related to inputs, intermediate measures and outputs, respectively.Semi-positive vectors of weights, connect inputs and intermediate measures of n DMUs in stage 1 and intermediate measures and outputs of n DMUs in stage 2, respectively.e is a row vector with all elements equal to 1. 0 is a zero vector, whose dimension depends upon its corresponding vector comparison.

Figure 1 -
Figure 1-The production process of two column vectors of inputs cost in stage 1 and intermediate measures cost in stage 2, respectively.Based on an optimal solution this model, the cost efficiency can be defined by o as the cost efficient projection of o DMU .For introducing a common * z between stage 1 and stage 2, the values of slack variables have been ignored in the cost efficient projection.The dual model of (1) is expressed by

1  and 2 
, are due to the fifth and sixth constraints of model (1), respectively.If   The proof is not difficult regarding the constraints of model (1) and the complementary slackness conditions.Remark1.Note that   of the points on the efficiency frontier of stages 1 and 2, respectively, and we want to measure the RTS at two-stage DEA.
Therefore for each real number  the following equations hold:

The relation among SE 1, * 1 
and the type of RTS in stage 1 at   * * ,z x is as follows:

2 Case 1 )
similar relation exists among SE 2 , * and the type of RTS in stage 2 at   consideration of SE quantity in stages 1 and 2, 5 different cases must be perused.If SE 1 =1 and SE 2 =1, then the two-stage overall RTS will be considered as CRS and O SE =1; ( O SE denotes the overall SE of two-stage).
will be no contraction possibility in stage 1 and CRS with O SE =1 will be considered as the two-stage overall RTS.

)
will not be any expansion possibility in stage 1, so CRS with O SE =1 will be considered as the two-stage overall RTS.(ii) If 0 *   then the two-stage overall RTS will be IRS with O SE = SE 1 × SE 2 at

Case 4 )Case 5 )
If SE 1 <1 and SE 2 <1 (SE 1 >1 and SE 2 >1) then DRS (IRS) with O SE =SE 1 × SE 2 will be considered as the two-stage overall RTS and for resizing the ) (model(17)) must be solved.If SE 1 >1 and SE 2 <1, three possibility for overall RTS can occur.In this case, first the value of k is calculated according to the following equation: k = SE 1 × SE 2 (19) Therefore, three options may happen: (a) If k=1 then CRS with O SE =1 will be considered as an overall RTS.(b) If k<1 then the type of RTS will be determined by solving model (15), like case2.(c) If k>1 then the type of RTS will be determined by solving model (17), similar case3.

Table 1 .
Data Set