Super Efficiency with 2- Stage DEA Model

Super Efficiency with 2- Stage DEA Model

Sheila Eka Putri Department of Mathematics, University of Sumatera Utara Medan, Indonesia

AbstractDEA model estimate a set of evaluated DMU and use to estimate the efficiency score by evaluating each DMU in a data set. This research determined the new scheme of 2-stage DEA model analysis in obtain of new efficiency score, called super efficiency DEA model. We then extended the DEA model that was formulated by considering input-ouput oriented for each used data. The model formulated by a linear program and gives three major solutions: (1) an alternative new scheme of the 2-stage of DEA model, (2) super efficiency scores for a given data and (3) DMU ranking based on each its super efficiency score.

KeywordsData Envelopment Analysis (DEA), linear program, super efficiency, ranking

1. INTRODUCTION

   Data Envelopment Analysis (DEA) model is a method that used to estimate a frontier to evaluate the performances or the efficiency of all of the entities that are to be evaluated. In some previous studies on 2-stage DEA model was developed into some of applications. Banker and Natarajan [1] developed 2-stage DEA model by using linear regression analysis, Monte-Carlo simulation, that obtained 2-stage DEA estimator for a certain variable context with definite constraint in input vector on model. Simar and Wilson [2] extended the maximum likelihood method in order to determine 2-stage regression into DEA model that produced a DEA estimator for 2-stage DEA model as a result.

   Andersen et al. [3] developed a radial super-efficiency measure called AP model. This model was comparing the DMU that evaluated with a linear combination of other DMUs, while excluding the observations of the DMU being evaluated. This model then was extended by Tone [4] by considering the input-output slacks of a non-radial super efficiency called SBM model. Castelli et al. [5] proposed a comprehensive categorized overview of methods and models for different multi-stage production architectures in DEA model. Seiford and Shu [6] was studied a production process in banking sector by treating the two stages, independently,

   in two-stage process under the assumption of series relationship. Their modeling approach facilitates the linearization of a non-linear mathematical program and based on the assumption that the weight of the two stages is extendable to Variable Returns to Scale (VRS) assumption.

   In this paper we present an alternative new scheme of the 2-stage under the assumptions of the series relationship between the two stages. We then formulated the model by considering input-output oriented for each given data. We select some inputs orientation that produce the efficiency score for the first stage, e1. Then, outputs orientation for the second stage that produce e2, the efficiency score for the second stage. Thus, the overall efficiency score can obtained by a simple division rule.

   This paper unfolds as follows. Section II we review some background information on the study area and outline that adopted the 2-stage analysis. Section III we propose an alternative new scheme of 2-stage DEA model. In Section IV, we presents the result in comparison of the new model with a given set data studied by Wang et al. [9]. Finally, conclusion and future research are provided in the last section. The result of super efficiency scores and DMU ranking also provided in the Appendix section.

2. DATA ENVELOPMENT ANALYSIS (DEA) DEA model can be shown in two general forms, linear

   program and linear regression form. DEA model using linear program method which the weighted score as decision variable and produce the efficiency score for each Decision Making Units (DMUs) as solutions of DEA model (see Seiford and Thrall [10]; Lovell [11]; Cooper et al. [12] and Thanassoulis [13]). Charnes et al. [14] showed that DEA is a multi-factor productivity analysis model that used to estimate the efficiency relative score (E) of homogeny set of DMU that formulated as

   without assuming any relationship between the two stages. A novel approach then developed by Kao and Hwang [7] that consider a series relationship of the two stages and provide a

   E sum of output weight 100%

   sum of input weight

   (2.1)

   model that estimates the overall efficiency of the production process. This approach is based on the reasonable assumption that the immediate measures of the value is same without consider whether as outputs or input in the first stage. Chen et al. [8] then introduced the additive efficiency decomposition

   Assume n evaluated DMUs, DMU ( = 1, , ) for each using m inputs, ( = 1, , ), that produce r outputs,

   ( = 1, ), respectively. The efficiency score for DMU

   l can be defined as sum of all outputs weight divided by sum

   of all inputs weight where single input, ( = 1, , ) and single output, ( = 1, , ). Mathematically, it can be

   for j 1,, n ; r 1,, s ; i 1,, m and uk , vi 0 . For (x,y) in P, any semi positive activity (x, y) with x x and y y is

   formulated as follows

   l

   r

   k kl

   u y k 1

   m

   i il

   v x i1

   (2.2)

   included in P. Thus, any activity with input of no less than x in any component and with output no greater than y in any component is feasible.

   B. Banker, Charnes and Cooper (BCC) Model

   DEA model estimate a set of evaluated DMU and use to estimate the efficiency score by evaluating each n DMU in a data set. It is done by estimating a frontier point which gives interval of efficiency score, 0 1 to each n evaluated DMU. This efficiency score was obtained by comparing DMUs performances to all of evaluated DMUs performances in a certain data set. The obtained efficiency score by using DEA model gives the highest efficiency relative to interval 0 1 for each DMU ( = 1, , ).

   As linear regression form, DEA model is a nonparametric tool in analyzing efficiency score with multiple inputs and

   outputs that consider both qualitative and quantitative in a data set. Also, DEA is a linear programming model that

   The extended of CCR model then studied by Banker et al. [15]. This model has its production frontiers spanned by convex hull of existing DMUs that leads to Variable Returns to Scale (VRS) assumption characterizations. Banker et al.

   [15] published the BCC model whose production possibility set which defined by

   P {(x, y) | x X , y Y,e 1, 0}

   where = ( ) × dan = ( ) × are a given data set, and e is a row vector with all elements equal to 1. The BCC model differs from the CCR model that

   n

   calculate multiple inputs and outputs and evaluate DMUs both qualitatively and quantitatively by a linear program

   developed by adding convexity constraint

   j 1.

   j 1

   form. Generally, there are two basic DEA model as follows.

   A. Charnes, Cooper and Rhodes (CCR) Model

   DEA model as originally proposed by Charnes et al. [14] namely Charnes, Cooper and Rhodes (CCR) model to produce

   Mathematically, BCC model can be formulated as follows

   s

   max ur yro w

   r 1

   m

   (2.6)

   the efficiency frontier based on concept of Pareto optimum. This model was built on the assumption of Constant Returns to Scale (CRS) for the production frontier in the single input and single ouput case. More generally, this model assumed

   1. t

      vi xio 1

      i1

      s m

      ur yrj vi xij w 0

      (2.7)

      (2.)

      that the production possibility set

      r 1

      i1

      P {(x, y) | x X , y Y, 0}

      with the pairs of positive inputs and outputs vectors,

      for

      j 1,, n ; r 1,, s ; i 1,, m and uk , vi 0 .

      (x j , y j )( j 1,, n) belongs to P of n DMUs. Thus, we can assume that such a pair of semi positive input-outputs, x Rm and y Rn . Charnes et al. [14] developed CCR

      basic model input oriented DEA which contains of objective function, maximizing DMU efficiency score with constraint that efficiency score for all DMU less than or equal to 1 as follows

      s

3. 2-STAGE DEA MODEL WITH INPUT-OUTPUT ORIENTED

This research based on 2-stage DEA model developed by Despotis et al. [16]. We then extended the model that was formulated by considering input-ouput oriented for each used data. The objective of DEA model with input oriented is to minimize input which produced at least or equal to total output of given data. Whereas the aim of output oriented

model is to maximize the output which obtained not greater

max ur yro

r 1

m

(2.3)

than total input of a given data.

Despotis et al. [16] developed an additive decomposition

s. t

vi xio 1

i1

s m

ur yrj vi xij 0

(2.4)

(2.5)

model into 2-stage DEA model to estimate the efficiency score. The model is under the CRS assumptions by considering input-output oriented for each evaluated data. Fig. 1 shows scheme of 2-sta(2g.e6D) EA model by Despotis et al. [16]

r 1

i1

as follows

Stage 2. Input oriented

s

max ur yrjk

(2.19)

r 1

q

Fig 1. 2-stage DEA model by Despotis et al. [16]

Assume there exists n DMU ( = 1, , ) where each uses m inputs, ( = 1, , ), that produces s outputs,

( = 1, , ). For each efficiency score of evaluated DMU can be obtained by 2-stage DEA model that produce

s.t wp z pjk 1

p1

q m

wp z pj vi xij 0

(2.20)

(2.21)

1 and 2 , respectively

p1

i1

vi , wp ,ur 0

Stage 1. Output oriented

m

1

e

1 min vi xijk

jk i1

q

s. t wp z pjk 1

(2.9)

(2.10)

Model (2.9 – 2.11) was derived the following combined to be a bi-objective linear program with the aim is to maximize the overall efficiency score from 2-stage DEA model that formulated as follows

p1

s m

q m max F max ur yrj

• vi xij

wp z pj vi xij 0

(2.11)

r 1

k k

i1

p1

i1

q

p pj

(2s.1.2t ) w z 1

k

Stage 2. Input oriented

s

p1

q m

wp z pj vi xij 0

(2.22)

2

1 max

ur yrj

k

(2.12)

p1

i1

e jk r 1 s q

q ur yrj wp z pj 0

s.t wp z pj

(2.13)

r 1

p1

k

p1

s q

vi , wp ,ur 0

ur yrj wp z pj 0

r 1 p1

wp ,ur 0

(2.14)

And 2-stage DEA model for estimate the overall efficiency score are

min

Furthermore, 2-stage DEA model are obtained for each input- output oriented as follows:

Stage 1. Output oriented

m

n n

s.t j z pj j zpj zpjk 0

j 1 j 1

n

j yrj yrj

(2.23)

min vi xij

(2.15)

k

j 1

k

i1

j , j 0

q

s.t wp z pjk 1

p1

(2.16)

for i 1,, m ;

p 1,, q ; r 1,, s .

q m

wp z pj vi xij 0

(2.17)

1. 2-STAGE DEA MODEL WITH INPUT-OUTPUT ORIENTED

   p1

   s

   i1

   q

   This research shows the extended of multi-stage process of DEA model, so we obtained an alternative 2-stage DEA

   ur yrj wp z pj 0

   (2.18)

   model. Let us consider that there are n DMU ( = 1, , )

   r 1

   p1

   where each using m inputs,

   ( = 1, , ). It produce l

   vi , wp ,ur 0

   outputs, ( = 1, , ) for Stage 1. Then, consider that we have ( = 1, , ) as inputs of Stage 2 that produce final outputs, ( = 1, , ). Fig. 2 shows the new scheme of the extended of 2-stage DEA model that we used.

   q

   s.t wp z pjo 1

   p1

   q m

   wp z pj vi xij 0

   (2.32)

   (2.33)

   p1 i1

   Fig. 2 Extended of 2-stage DEA model

   Notice that Stage 1 produce outputs, , that been processed by inputs, . Since the definition of efficiency score in Equation (1), we obtain 1 as the efficiency score of Stage 1. Afterward for Stage 2, the efficiency score is the estimated of ratio or comparison between the final outputs ( ) and inputs of Stage 2, ( ). We then obtained 2 as the efficiency score of Stage 2. So that, here we obtained DEA model Stage 1 and Stage 2 based on input-output oriented as follows.

   Stage 1. Output oriented

   q

   wp z pjo

   tk ,ur , vi , wp 0

   By a simple division rule, we obtained the overall efficiency score by

   o e e

   1 2

   e (2.34)

   2

   that then referred as super efficiency score of 2-stage DEA model for each evaluated DMU.

2. COMPUTATIONAL RESULTS

   The extended 2-stage DEA model (2.24 2.33) obtained then we apply to a data set that studied by Wang et al. [8] that

   e1

   p1

   (2.24)

   given in Table 1 (see Appendix). We used three outputs:

   jo m

   Deposits (z ), Fixed assets (z ) and IT data (z ) that has been

   vi xij

   1 2 3

   o

   i1

   q

   s.t wp z pjo 1

   p1

   q m

   wp z pj vi xij 0

   (2.25)

   (2.26)

   processed by an input: Number of employees (x1) which produce e1 as the efficiency score of DEA model Stage 1. On Stage 1, we obtained weighted output and weighted input respectively that gives e1 scores. For DEA model Stage 2, we use an input: Profit (p) that produce an output: Loans recovered (y1) and gives e2 as the efficiency score of DEA model Stage 2. Therefore, the overall efficiency score

   p1

   vi , wp 0

   Stage 2. Input oriented

   s

   i1

   obtained by model (2.34) that referred as super efficiency score.

   Table 2 (see Appendix) reports the efficiency scores obtained by applying model (2.24 2.33) on the data of

   e

   ur yrjo

   2 r 1

   l

   jo

   tk hkjo

   k 1

   q

   s.t wp z pjo 1

   p1

   q l

   ur yrj tk hkj 0

   (2.27)

   (2.28)

   (2.29)

   Table 1. Table 2 shows the comparison results of the efficiency score between model that developed by Despotis et al. [16] and model (2.24 2.33). From the results, we obtained input and output weight of DEA model Stage 1 and Stage 2, respectively. From each super efficiency score of each evaluated DMU, eo, we determined DMU ranking that given in Table 3 (see Appendix) that summarizes the results obtained from model (2.24 2.33).

   p1 l

   k 1 q

3. CONCLUSIONS AND FUTURE RESEARCH

tk hkj wp z pj 0

(2.30)

This research introduced a new alternative scheme of 2-

k 1

p1

stage DEA model to obtain super efficiency score of

tk ,ur , vi , wp 0

Then, under the CRS assumptions, we can formulated the 2- stage DEA model as

s m

evaluated DMU in a data set. 2-stage DEA model then was formulated into linear program for an based on a new scheme that showed in Fig. 2. This model was extended of CCR model by considering input-output oriented in a data set. The basic idea of this model based on input-output

oriented on Stage 1 and output oriented on Stage 2, so that

max ur yrjo vi xijo

(2.31)

super efficincy score obtained from model (2.34). Testing

r 1

i1

our models with data sets taken from previous studies [9], shows that results obtained are comparable to those reported in literature as given in Table 2. In future

research, we will extend a new alternative scheme of 2- stage DEA model by considering input and output interval in a data set.

REFERENCES

1. Banker, R.D. and Natarajan, R. Evaluating contextual variables affecting productivity using Data Envelopment Analysis, Operations Research, Vol. 56 (2008), pp. 48-58.

2. Simar, L. and Wilson, P.M. Two-stage DEA: Caveat emptor,

   Journal of Productivity Analysis, Vol. 35 (2011), pp. 205-218.

3. Andersen, P. and Petersen, N.C. A procedure for ranking efficient units in Data Envelopment Analysis, Management Science, Vol. 39 (1993), pp. 1261-1294.

4. Tone, K. A slack-based measure of super-efficiency in Data Envelopment Analysis. European Journal of Operational Research, Vol. 143 (2002), pp. 32-41.

5. Castelli, L., Presenti, R., and Ukovich, W. A classification of DEA models when the internal structure of the Decision Making Units is considere, Ann. Operational Research, Vol. 173 (2010), pp. 207-235.

6. Seiford, L.M. and Zhu, J. Profitability and marketability of the top 55 US commercial banks. Manage Science, Vol. 45 (1999), pp. 1270-1288.

7. Kao, C. and Hwang, C.N. Decomposition of technical and scale efficiencies in two-stage production systems. European Journal of Operational Research, Vol. 211 (2001), pp. 515-519.

8. Chen, Y., Cook, W.D., and Zhu, J. Deriving the DEA frontier for two-stage DEA processes. European Journal of Operational Research, Vol. 202 (2010), pp. 138-142.

9. Wang, C.H., Gopal, R. and Zionts, S. Use of Data Envelopment Analysis in assessing information technology impact on firm performance, Ann. Operational Research, Vol. 73 (1997), pp. 191-213.

10. Seiford, L.M.., and Thrall, R.M. Recent developments in DEA, Journal of Econometrics, Vol. 46 (1990), pp. 7-38.

11. Lovell, C.A.K. Linear programming approaches to the measurement and analysis of productive efficiency, Journal of Spanish Society of Statistics and Operations Research, Vol. 2 (1994), pp.175-248.

12. Cooper, W.W., Seiford, L.M. and Tone, K. Data Envelopment Analysis, Kluwer Academic Publishers (2000), pp. 301-313.

13. Thanassoulis, E. The use of Data Envelopment Analysis in the regulation of UK water utilities, European Journal of Operational Research, Vol. 126 (2001), pp. 436-453.

14. Charnes, A., Cooper, W.W., and Rhodes, E. Measuring the efficiency of decision making units, European Journal of Operational Research, Vol. 2 (1978), pp.429-444.

15. Banker, R.D., Charnes, A. and Cooper, W.W. Some model for estimating, technical and scale efficiencies in Data Envelopment Analysis, Management Science, Vol. 30 (1984), pp. 1078-1092.

16. Despotis, D.K., Koronakos, G. and Sotiros, D. Additive decomposition in two-stage DEA: an alternative approach, ICOCBA (2012).

APPENDIX

Table 1. IT data (Source: Wang et al. [9])

Table 2. Results of IT data

Despotis et al. [16] Model (2.24 2.33)

DMUj 1 2 o

Weighted output

Weighted

input e1

Weighted output

Weighted input

e2 eo

Table 3. DMU ranking based on its super efficiency score

0.570