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Chapter 6

Efficiency Ranking Method using SFA and

SDEA: Analysis and Discussion

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The proposed approach, in this chapter is based on the theme of integration of SFA and Super efficiency

model of DEA (SDEA). This model is called Efficiency Ranking Method by SFA and SDEA (ERM-SSD). It

acknowledges stochastic nature of data and recommends the best alternative whose average performance

is compared with the best alternative in the set of alternatives. The proposed model is illustrated using a

hypothetical data set with two inputs and two outputs and with three inputs and three outputs. The

proposed approach is validated using data obtained from PSU banks operating in India. The ranks

obtained by the proposed model are compared with the conventional models such as CRS-DEA and Super

efficiency DEA models using two different techniques namely: Spearman’s rank test and Mean Squared

Deviation (MSD). It is hoped that the proposed model is able to consider the case of multiple inputs and

multiple outputs in SFA framework and has good amount of discrimination power while ranking the

DMUs.

6.1 Introduction

Stochastic Frontier analysis (SFA) and Data Envelopment Analysis (DEA) are widely used

Multi-Criteria Decision Making (MCDM) tools for performance evaluation and benchmarking

(Bazrkar and Khalilpour, 2013; Odeck and Bråthen, 2012; Goncharuk, 2011; Wu et al., 2011;

Thoraneenitiyan and Avkiran, 2009; Reinhard et.al, 2000; Coelli and Perelman, 1999). SFA a

parametric technique and DEA a non-parametric technique are often used by academicians and

practicing managers in complex business situation. For measuring cost and revenue efficiency of

the property-casualty insurance companies, Park et al. (2009) proposed a model using SFA. The

study considered different types of insurance distribution systems in the U.S. to evaluate the

impact of the ownership pattern on the efficiency of these companies. Baltas (2005), applied

two-stage SFA model to find the consumer differences in food demand. In this study, SFA was

presented as an alternative methodology for analyzing and segmenting store clientele. Vencappa

and Thi (2007) applied SFA for decomposing the productivity growth of foreign banks in Czech

Republic, Hungary and Poland. Hiebert (2002) applied SFA for estimating cost and operational

efficiency of electric generating plants for the period 1988 to 1997.

Mohamad and Said (2012) used Super efficiency Data Envelopment Analysis (SDEA) model to

measure and assess the performance of selected largest listed companies in Malaysia. SDEA

model was able to provide distinct ranking to these companies as against those by usual DEA

model. It was concluded that that top-ranked companies on the basis of revenue generated are not

necessarily top-ranked performers. Chen et al. (2012) used SDEA and DEA to measure

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efficiency of financial and non-financial holding companies in Taiwan. DEA ranked multiple

companies as efficient whereas SDEA was able to further rank efficient companies and assign

distinct ranks. Yawe (2010) used standard DEA and SDEA models to measure efficiency of

hospitals operating in Uganda. Using SDEA model, ranking of the efficient units was possible.

Hospitals were further categorized into four groups: strongly super-efficient; super-efficient;

efficient and inefficient.

Few studies which integrated SFA and DEA to derive benefits from both of them can also be

cited. For example, Thoraneenitiyan and Avkiran (2009) used a three stage integrated model

using SFA and DEA to measure the impact of restructuring and country-specific factors on the

efficiency of post-crisis East Asian banking systems. In the first stage, a non-oriented Slack

based (SBM) DEA was used to assess technical efficiency of banks without considering

environmental effects. In the second stage, SFA was used for measuring the impact of the

environment on bank inefficiency. In stage three, SBM was repeated with adjusted data. The

results indicated that domestic mergers play a significant role in developing efficient banks, and

restructuring does not lead to more efficient banking systems. Banking system inefficiencies

were mostly attributed to country-specific conditions, particularly, high interest rates,

concentrated markets and economic development. Azadeh et al., (2009) presented an integrated

approach using DEA, Corrected ordinary least squares (COLS), SFA, Principal Component

Analysis (PCA) and Numerical Taxonomy (NT) for performance assessment, optimization and

policy making of electricity distribution units. This model accounted for both static and dynamic

aspects of information on environment due to involvement of SFA. The integrated approach gave

an improved ranking methodology and facilitates better optimization of electricity distribution

systems. To illustrate the usability and reliability of the proposed algorithm, thirty eight

electricity distribution units in Iran were considered, ranked and optimized by the proposed

algorithm.

Coelli and Perelman (1999) used SFA framework to investigate technical inefficiency in

European railways. The objective of the paper was to compare the results obtained from the three

alternative methods for estimating multi-output distance functions. Techniques applied were

Parametric frontier using Linear Programming (PLP); DEA and Corrected Ordinary Least

Squares (COLS). Input-orientated, output-orientated and constant returns to scale (CRS) distance

functions were estimated and compared. The results indicated a strong degree of correlation

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between the input and output oriented results for each of the three methods. Authors suggested a

method of combining the technical efficiency scores obtained from the three different methods

using geometric mean. Ghaderi, et al. (2006), integrated DEA, COLS and Principal Component

Analysis (PCA) in a two-stage model. In the first stage technical efficiency of the electricity

distribution units operating in Iran were obtained by DEAS and COLS separately. In the second

stage, the efficiency scores obtained in the first stage were considered as inputs in the PCA

model. The authors claim that the DEA-COLS-PCA model used in this paper provided better

ranking of units than would have done by DEA and COLS separately.

To assess the impact of regulatory and environmental factors and statistical noise on the

efficiency of public transit systems of Italian companies, Margari et al. (2007) used both SFA

and DEA approach. The proposed model decomposed DEA inefficiency measures into three

components: exogenous effects, managerial inefficiency and stochastic events. The developed

measure provided evidence on the determinants of input-specific efficiency differentials across

companies. The results also pointed out that managerial skills play a minor role, and emphasized

the relevance of regulatory policies aimed at replacing cost-plus subsidization with high-powered

incentive contracts as well as improving environmental conditions of public transit networks.

In another study by Ta et al. (2008), logistic center location problem was solved by applying

DEA and SFA separately and later the ranks obtained by both were compared using Spearman’s

rank correlation coefficient. Bazrkar and Khalilpour (2013) conducted a comparative study on

ranking banks using DEA and SFA approach. The differences between ranks assigned by these

two models were tested using Pearson’s correlation test. The results showed significant

difference between the ranks assigned by two approaches. Moreover, authors found SFA as the

superior method of measuring efficiency of the bank as compared to DEA. In order to establish

an evaluation system for key discipline scientific research level, Li and Wang (2011), used DEA

and SFA models. The objective of this study was to measure scientific research input and output

efficiency of fifty eight key disciplines in fifteen universities. The results showed significant

difference between input and output efficiencies in terms of numerical sequencing but good

consistency in terms of efficiency ranking.

Some studies have been carried out to compare the results obtained from these tools. For

example, Reinhard, et al. (2000), developed an analytical framework to calculate environmental

efficiency in the presence of multiple environmentally detrimental inputs. The model so

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developed enables the aggregation of environmentally detrimental inputs, and it allows the

calculation of the environmental efficiency using these inputs. It also indicates which

environmentally detrimental input is used most inefficiently, both on individual farms and in the

aggregate. In most of the other applications, DEA and SFA were used separately to measure

technical efficiency of a DMU and the results obtained were compared using either descriptive

statistics like; average or mean efficiency score and standard deviation in the scores or using

Spearman’s rank test (Odek, 2007; Wadud, 2003; Sharma et al., 1997; Mulwa et al.,2008, etc.).

While going through different studies in the literature on the application of DEA and SFA, one

may realize that in most of the cases, DEA and SFA have been applied separately and the results

obtained through these techniques are compared in the end by various methods. In some of the

studies where DEA and SFA have been used in an integrated approach, mostly SFA is used to

study the impact of the exogenous variables on the efficiency obtained using DEA. Also, in these

studies, mostly one output or an aggregated output with multiple inputs is considered. To the best

of our knowledge, there is no study being made which uses an integrated approach using SDEA

and SFA. In this study, an attempt has been made to propose an integrated model using SFA and

SDEA. Instead of CRS-DEA, Super efficiency DEA (SDEA) is specifically used to improve the

discrimination power of the proposed method.

Also, the proposed model doesn’t require the separation of the exogenous variables in the later

stage. Most importantly, the proposed model can accommodate a case of multiple outputs and

multiple inputs.

This chapter is organized as follows: In the following section 6.2, the need for integrated

approach is explained. Section 6.3 presents Spearman’s rank test and the MSD approach. These

methods are used to compare the ranks obtained using proposed model with those obtained using

conventional CRS-DEA and SDEA models. Section 6.4, presents the proposed integrated

framework with illustrations using hypothetical data of various input and output combinations.

Model is further verified using data from PSU Banks in section 6.5. Analysis and discussion are

presented in Section 6.6. Section 6.7 gives summary and conclusions.

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6.2 Need for Integrated Approach

In this section, SFA, DEA and SDEA as integrating tools are explained in terms of their model

specifications, advantages and disadvantages. This discussion carves the niche for proposed

integrated approach.

6.2.1 Stochastic Frontier Analysis (SFA)

Stochastic frontier is a parametric tool for the measurement of technical efficiency of a firm.

Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977) almost at the same

time proposed this model. They proposed a stochastic production function for the cross-sectional

data given by

( ) , i = 1, …, N, (Expression 6.1)

Where

denote output (or the logarithm of the output) of the i-th firm;

is a kx1 vector of functions of actual input quantities used by the i-th firm;

β is a vector of parameters to be estimated; and

is the composite error term which is further divided into two components defined as:

, i = 1, …, N,

Here ’s are assumed to be independently and identically distributed (iid) random errors, which

have normal distributions with mean zero and unknown variance independent of the random

variables which are assumed to account for technical inefficiency in production and are often

assumed to be iid truncations (at zero) of the N( µ, ) distribution. In this model, account for

random variation of production outside the control of the individual unit or producer.

The error term , is composed of two parts: (a) the traditional random error that captures the

effect of measurement error, other statistical noise, and random shock due to exogenous

(variables which are out of producer’s control) variables (if any); and b) , one-sided component

(as it is iid truncations (at zero) of the N (µ, ) distribution) which captures the effect of

inefficiency.

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Expression 6.1 in log-linear Cobb-Douglas form becomes

∑ (Expression 6.2)

Estimation of Expression 6.2 by Ordinary Least Square (OLS) method provides consistent

estimates of the s., but not of , since E( )= -E( ) ≤ 0.

Moreover, OLS doesn’t provide estimates of producer specific technical efficiency. But OLS

does provide a simple test of technical efficiency (TE).

If , then = =TE

if , then = TE which then is negatively skewed, and there is evidence of

technical inefficiency in the data. In the present study, FRONTIER version 4.1 (developed by

Coelli, T.J, Centre for Efficiency and Productivity Analysis, University of New England,

Australia) to compute TE of a DMU using above model is used

Some of the advantages of SFA are

Ability to incorporate stochastic data (Bazrkar and Khalilpour, 2013; Odeck and Bråthen,

2012),

Ability to test the result using statistical hypothesis testing technique (Reinhard et al.,

2000; Coelli and Perelman, 1999),

Accounts for statistical noise (which may be beyond the control of the producer) and

separates out technical inefficiency (Thoraneenitiyan and Avkiran, 2009; Reinhard et al.,

2000; Coelli and Perelman, 1999; etc.) and

Compares the performance of each DMU against the average performance of all the units

in the group (Ta, et al., 2008).

On the other hand, the limitations of this approach are

Requirement of a priori specification of the distribution of the inefficiency terms and the

functional form of the frontier (Perera and Skully, 2012; Saad and El-Moussawi, 2009;

Chen T-Y, 2002),

No indication of potential improvement possible in case of inefficient DMU and

Inability to handle a case of multiple outputs without developing an aggregate measure of

all outputs (Kumbhakar and Lovell, 2000).

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6.2.2 Data Envelopment Analysis (DEA)

Data Envelopment Analysis (DEA) is a non-parametric benchmarking tool, based on linear

programming technique. It was originally developed by Farrell (1957) and further extended by

Charnes, Cooper and Rhodes (1978). Charnes, Cooper and Rhodes (CCR) model measures the

relative efficiency of a set of firms that use a variety of inputs to produce a range of outputs

under the assumption of constant return to scale (CRS). In economics, return to scale is the term

related to the firm’s production function. It describes the behavior of the firm in terms of its rate

of change in the output/production as a result of change in its input/s. Constant return to scale

signifies a production process of a firm where output/production changes are in proportion to the

changes in firm’s input quantities. As a result the manufacturer is able to scale the inputs and

outputs linearly without increasing or decreasing efficiency.

An individual unit in this set (of firms) is referred to as DMU. A DMU, for instance, can include

hospitals, power plants, universities, schools, banks, bank branches, etc. Performance of a DMU

is measured using the concept of efficiency or productivity, which is defined as the ratio of total

weighted outputs to total weighted inputs. While measuring the performance, this model captures

not only the productivity efficiency of a firm at its actual scale size, but also the inefficiency

(Banker, 1984). The best performing unit in the set of DMUs is assigned a score of 100 percent

or 1, and the remaining DMUs get a score ranging between 0 and 100 percent, or equivalently

between 0 and 1, relative to the score of best performing DMU. DEA forms a linear efficiency

frontier which passes through the best performing units within the group whereas all the

remaining less efficient units lie off the frontier. The term efficiency used in DEA is the relative

efficiency and not the absolute efficiency. Banker, Charnes and Cooper (BCC) in1984, extended

the earlier CCR model to BCC model in which, variable return-to-scale (VRS) was introduced.

This was achieved by introducing one more constraint in the CCR model which ensures that

firms operating at different scales are recognized as efficient. In this case, inefficient firms are

compared only with the efficient firms of similar scale.

The DEA model developed by Charnes, Cooper and Rhodes with constant return to scale

referred to as CRS-DEA is given as under;

Let there be N DMUs each with ‘k’ inputs and ‘m’ outputs. For the pth

DMU under evaluation,

the technical efficiency measured by using the CRS-DEA model is given by

Maximize

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∑

Subject to

∑

∑( ) ∑( )

with (Expression 6.3)

where,

: Efficiency of the pth DMU

value for input criteria i for pth

DMU

weight of input i

value for output criteria j for pth

DMU

weight of output j

: value for input criteria i for nth

DMU

: value for output criteria j for nth DMU

an infinitesimal or non-Archimedean constant usually in the order of 10-5

or 10-6

where and here note that n includes p.

6.2.3 Super Efficiency DEA (SDEA)

While ranking the DMUs using DEA model, many a time’s several DMUs achieve an efficiency

score of one. In such cases, ranking of efficient units is a major challenge faced by the decision

maker (DM). In order to overcome this problem, Andersen and Petersen (1993) proposed Super

efficiency DEA model called SDEA which is an extension of DEA model. SDEA has been

defined either with CRS or VRS assumption. In this study, SDEA model with CRS assumption is

considered. This is also a non-parametric method of benchmarking like DEA. The model is given

as follows:

∑

Subject to

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∑

∑

∑

(Expression 6.4)

The difference between the standard DEA model and the SDEA model lies in the treatment of

the efficient units (Saen, 2008). This model is similar to Banker, Charnes and Cooper (BCC)

model, except that in the approach presented by Andersen and Petersen, the DMU under

evaluation is excluded from the set, thus allowing the efficient DMUs to increase their inputs

proportionally while preserving their efficiency status and achieve an efficiency score above 1.

DEA as a MCDM tool has several advantages which are as follows;

No need to specify a-priori weights on the input-output criteria (factors). The DEA

approach, allows each DMU to choose a set of weights (also called as multipliers) for

the input-output criteria that enables it to appear in the best light (George and

Rangaraj, 2008; Sufian, 2007; Avkiran, 1999; Al-Faraj et al., 1993; Mester, 1996;

Banker, 1984).

DEA uses the data to derive an efficiency frontier. This frontier sets the benchmark

for less performing units. It is with the reference to this frontier that each DMU is

evaluated (Soteriou and Stavrinides, 2002; Ramanathan, 2005; Koster et al, 2009).

It’s ability to indicate the potential improvement in the performance of an inefficient

Decision Making Unit (DMU) (Duffy et al., 2006; Banker and Morey, 1986;

Sherman, 1984).

DEA, though popular, has few limitations. One of the limitations of DEA is its less

discrimination power due to two reasons:

a) When the sum of the number of inputs and outputs is large as compared to the total

number of DMUs in the sample (Andersen and Petersen, 1993; Zhu, 2001; Saen, 2008).

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b) At times, DEA assigns high efficiency to a DMU due to its very low value of single input

or very high value of output, even though that input or output is seen as relatively

unimportant (Seiford and Zhu, 1999; Shang and Sueyoshi, 1995).

Therefore, the discussion that follows considers an extended version of Data Envelopment

Analysis called Super Efficiency Model of DEA (SDEA). SDEA is known for better

discriminating power than DEA (Andersen and Petersen, 1993; Balf et al., 2012; Lovell and

Rouse, 2003).

Some of the advantages of SDEA are

Its ability to indicate the potential improvement in the performance of an inefficient

Decision Making Unit (DMU) (Lovell and Rouse, 2003; Adler et al., 2002).

And most importantly, its ability to rank efficient DMUs. (Andersen and Petersen, 1993;

Lovell and Rouse, 2003; Chen et al., 2010).

Some of the disadvantages of SDEA are

Sometimes, it is likely that a specific set of DMUs are ranked too high (Balf et al., 2012).

This model is not unit invariant. This means that the model is deficient in its treatment of

the non-zero slacks as its treatment of the slack does not yield a measure that is “unit

invariant” (Cooper et al, 2007).

In some cases, DMUs which are rated efficient (efficiency score equal to one) using

conventional DEA model do not have feasible solution in SDEA model (Lovell and

Rouse, 2003).

The model does not take into account the existence of slacks in the inputs and the outputs

(Tone, 2002).

It can be realized that these models as a stand-alone tool are silent on following issues

SDEA and CRS-DEA cannot incorporate stochastic nature of the real data and

measure technical efficiency of a DMU after separating out inefficiency and random

shock due to exogenous variables (if any),

Application of SFA without aggregation of output in case of multiple outputs,

CRS-DEA at times cannot provide a tie-breaking procedure and

In some cases, DMUs which are rated efficient (efficiency score equal to one) using

conventional DEA model do not have feasible solution in SDEA model

Sometimes, it is likely that a specific set of DMUs are ranked too high using SDEA

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CRS-DEA and SDEA both being non-parametric in nature does not provide a

diagnostic tool to test the validity of the results.

Based on above discussion, it is proposed that these issues can be addressed by integrating SFA

and DEA. The proposed integrated model will help to

a) Incorporate stochastic nature of the real data and measure technical efficiency of a DMU

after separating out inefficiency and random shock due to exogenous variables (if any),

b) Apply SFA with multiple outputs,

c) Provide a tie-breaking procedure and

d) Recommend the best alternative whose average performance is evaluated against the best

DMU in the sample under study.

6.3 Rank Evaluation Tools

In this section, a brief description of two different methods namely Spearman’s rank test and

Mean Squared Deviation (MSD) is given in brief. These methods are used for comparing ranks

obtained by different approaches.

6.3.1 Spearman’s Rank Test

The Spearman’s correlation coefficient is a measure of the linear association between two

variables which are available in ordinal scale. That is, it measures the strength of association

between two ranked variables. It is the nonparametric version of the Pearson product-moment

correlate ion. Spearman’s rank test is used to test the strength of a relationship between ranks

assigned by different criteria to the same set of units. In other words, it tests whether there is

agreement or disagreement between the ranks obtained by two different techniques. A perfect

Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone

function of the other. The test statistic is given by

∑

(Expression 6.5)

Where

= Spearman’s rank correlation coefficient

n = the number of items or individuals being ranked

= the rank of item i with respect to one variable/criterion

= the rank of item i with respect to a second variable/criterion

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di =

6.3.2 Mean Squared Deviation (MSD)

The second method used to compare the ranks assigned by the proposed model and the

conventional model is Mean Squared Deviation (MSD). This is calculated by first finding the

mean efficiency score for each DMU using each of the models under study. In this pair wise

approach, average value of the square of difference of ranks obtained is computed this is called

as Mean Squared Deviation (MSD). MSD is computed for all the pairs so formed. MSD closer to

the value of zero indicate no (or little) difference between the ranks assigned by two different

methods in the pair.

6.4 Proposed Framework

In this subsection, the proposed framework is presented. The theme of this approach is as

follows:

6.4.1 Efficiency Ranking Method using SFA and SDEA (ERM-SSD)

Initially, various combinations using each of the outputs and all inputs are considered. Therefore

if there are ‘l’ outputs under study then there will be ‘l’ such combinations formed called

dimensions. For each of these dimensions, technical efficiency for each DMU is computed using

SFA. Next, SDEA model is applied with these technical efficiencies obtained from SFA analysis

as inputs and dummy output of 1, to get an efficiency score called as SSD-I. Similarly, by

considering the technical efficiencies obtained from SFA analysis as outputs and dummy input of

1, SDEA model is applied to get an efficiency score called SSD-O. Finally, efficiency scores

obtained through SSD-I and SSD-O are combined together using arithmetic mean to define a

value called SSD. Here arithmetic mean is used to combine SSD-I and SSD-O scores so as to

ensure equal weightage. The five step procedure is explained as follows:

Step 1:

Identify ‘n’ DMUs or alternatives to be evaluated with ‘k’ inputs and ‘l’ outputs.

Step 2:

Define various combinations/dimensions using each output and all inputs. Apply

stochastic production function with error component model of SFA to compute technical

efficiency of each DMU. The efficiencies thus computed will be for ‘l’ output

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dimensions. For instance, for two inputs and two outputs there will be two such

dimensions, one for each output.

Step 3:

Use efficiency scores obtained in step 2 as Input Criteria (IC) and output as dummy

variable with constant value (say 1) (Dummy Output Criterion, DOC). Apply Super

Efficiency DEA with input orientation (SDEA-I) model to find efficiency score for each

DMU/alternative. This efficiency score is designated as (SSD-I)i, Where i= 1, 2, 3…n.

SDEA with input orientation will help in minimizing inputs (resources, cost etc.) for the

given level of outputs.

Step 4:

Use efficiency scores obtained in step 2 as Output Criteria (OC) and dummy variable

with constant value (say 1) as input (Dummy input Criterion (DIC). Apply Super

Efficiency DEA with output orientation (SDEA-O) model to find efficiency score for

each DMU/alternative. This efficiency score is designated as (SSD-O)i, Where i= 1, 2,

3…,n. SDEA with output orientation will help in maximizing outputs (revenue, profit,

efficiency etc.) for the given level of inputs.

Step 5:

Combine SSD-O and SSD-I values obtained in step 3 and 4 using arithmetic mean to get

a combined efficiency score (for each DMU). This efficiency score is called SSD score.

Use these SSD efficiency scores to rank the DMUs/alternatives in ascending order of

magnitude. In this case, arithmetic mean is used to combine two score; SSD-O and SSD-I

so as to give equal weightage to both the scores.

6. 4. 2 Illustration

In this section, an application of the proposed model ERM-SSD is illustrated using hypothetical

data set with two different input-output combinations. First a hypothetical dataset using twelve

DMUs with two outputs (O1 and O2) and two inputs (I1 and I2) and then a hypothetical data set

using eighteen DMUs with three outputs and three inputs is considered for illustration purpose.

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6.4.2.1 A hypothetical case with two outputs and two inputs

Data for a case of twelve DMUs with two outputs (O1 and O2) and two inputs (I1 and I2)

presented in Table 6.1.

Step 1:

Identify DMUs or alternatives to be evaluated with 2 inputs and 2 outputs.

Table 6.1: Hypothetical data with 2 outputs and 2 inputs

DMU O1 O2 I1 I2

A 21 65 34 43

B 95 68 35 39

C 29 60 55 45

D 86 91 21 82

E 1 49 30 8

F 16 50 53 83

G 89 99 80 39

H 60 46 73 63

I 4 62 76 29

J 61 17 75 88

K 67 30 82 59

L 5 35 52 65

Step 2:

First, various combinations/dimensions using single output and multiple inputs are formed and

stochastic production function with error component model given by Expression 6.1 is used to

compute technical efficiency of each DMU. In this illustration, with two outputs and two inputs,

two dimensions are formed. These dimensions are (O1, I1, I2) and (O2, I1, I2).

Step 3 and 4:

The efficiency scores (Input Criteria and Output Criteria) obtained in Step 2 are used to compute

SSDI and SSDO scores using the procedure explained in step 3 and 4 of the proposed

framework. The computations are presented in Table 6.2.

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Table 6.2: Computations of SSD-I and SSD-O scores (2 outputs and 2 inputs)

DMU OC1 OC2 DIC SSD-I DOC IC1 IC2 SSD-O

A 0.440 0.750 1 0.844 1 0.440 0.750 0.644

B 0.728 0.763 1 1.019 1 0.728 0.763 0.544

C 0.490 0.775 1 0.873 1 0.490 0.775 0.611

D 0.544 0.820 1 0.923 1 0.544 0.820 0.569

E 0.320 0.546 1 0.615 1 0.320 0.546 0.884

F 0.188 0.741 1 0.834 1 0.188 0.741 0.766

G 0.715 0.888 1 1.122 1 0.715 0.888 0.495

H 0.535 0.736 1 0.828 1 0.535 0.736 0.618

I 0.211 0.797 1 0.897 1 0.211 0.797 0.707

J 0.429 0.395 1 0.590 1 0.429 0.395 1.384

K 0.575 0.586 1 0.790 1 0.575 0.586 0.700

L 0.101 0.593 1 0.668 1 0.101 0.593 1.868

Step 5:

Upon combining SSD-I and SSD-O scores using arithmetic mean, efficiency scores called SSD

scores are obtained. These SSD scores are ranked to get the final ranking of DMUs. Ranks are

also obtained for each DMU using CRS-DEA and SDEA models. The ranks of these DMU using

the proposed model ERM-SSD, CRS-DEA and SDEA are presented in Table 6.3.

Table 6.3: Ranks using ERM-SSD, CRS-DEA and SDEA (2 outputs and 2 inputs)

DMU ERM-SSD-I

Score

ERM-SSD-O

Score

ERM-SSD

Score

ERM-SSD

Rank

CRS-DEA

Rank

SDEA

Rank

A 0.844 0.644 0.744 10 5 5

B 1.019 0.544 0.781 6 1 3

C 0.873 0.611 0.742 11 6 6

D 0.923 0.569 0.746 8 1 2

E 0.615 0.884 0.750 7 1 1

F 0.834 0.766 0.800 5 10 10

G 1.122 0.495 0.809 3 1 4

H 0.828 0.618 0.723 12 9 9

I 0.897 0.707 0.802 4 7 7

J 0.590 1.384 0.987 2 11 11

K 0.790 0.700 0.745 9 8 8

L 0.668 1.868 1.268 1 12 12

From Table 6.3 it is observed that four DMUs are tied at rank 1, when computed using CRS-

DEA approach. However, this tie no more exists in ERM-SSD and SDEA approaches. The

proposed approach (ERM SSD) has recommended DMU ‘L’ as the best DMU while SDEA has

given the first rank to DMU ‘E’.

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Hypothesis Testing

In order to verify the similarity of the results obtained using ERM-SSD, CRS-DEA and SDEA

model, Spearman’s rank test was carried out. For this purpose, following hypotheses were

framed.

Hypothesis1 (referred to as Hypothesis 5 in Chapter 3)

H0: There is no association/correlation between the ranks of individual DMUs obtained by

ERM-SSD and CRS-DEA.

H1: There is association/correlation between the ranks of individual DMUs obtained by ERM-

SSD and CRS-DEA.

Hypothesis 2 (referred to as Hypothesis 6 in Chapter 3)

H0: There is no association/correlation between the ranks of individual DMUs obtained by

ERM-SSD and SDEA.

H1: There is association/correlation between the ranks of individual DMUs obtained by ERM-

SSD and SDEA.

Hypotheses are tested at 5% level of significance. The results of Spearman’s rank test are shown

in Table 6.4.

Table 6.4: Spearman’s Rank test results for 2 output and 2 input data (for all three models)

Spearman's rho (ρ) ERM-SSD Rank CRS-DEA Rank SDEA Rank

ERM-SSD Rank Correlation Coefficient 1.000 -.278 -.322

Sig. (2-tailed) - .382 .308

N 12 12 12

CRS-DEA Rank Correlation Coefficient -.278 1.000 .982**

Sig. (2-tailed) .382 - .000

N 12 12 12

SDEA Rank Correlation Coefficient -.322 .982** 1.000

Sig. (2-tailed) .308 .000 -

N 12 12 12

From Table 6.4 it is seen that there is no significant association/correlation between the ranks

assigned by ERM-SSD and CRS-DEA (-0.278 with p-value of 0.382) and between ERM-SSD

and SDEA (-0.322 with p-value of 0.308). This means that there is significant difference

between the ranks assigned by these models. Thus, the null hypotheses H0 for hypotheses 1 and

2 are accepted and respective alternate hypotheses H1 are rejected.

Further, the difference between the ranks assigned by above three approaches is verified by MSD

method. The results are shown in Table 6.5.

223

Table 6.5: MSD results for 2 output and 2 input data (for all three models)

DMU ERM-SSD Rank

(1)

CRS-DEA

Rank (2)

SDEA

Rank (3)

(1-2)2 (1-3)2

A 10 5 5 25 25

B 6 1 3 25 9

C 11 6 6 25 25

D 8 1 2 49 36

E 7 1 1 36 36

F 5 10 10 25 25

G 3 1 4 4 1

H 12 9 9 9 9

I 4 7 7 9 9

J 2 11 11 81 81

K 9 8 8 1 1

L 1 12 12 121 121

MSD 34.167 31.5

MSD measures the difference between the ranks using average of squared deviation. In case of

complete association or agreement between the ranks assigned by any 2 models under study

ideally, value of MSD should be 0. From Table 6.5, it can be seen that none of the MSD values

are closer to zero. The MSD value ranges from 31.5 between ERM-SSD and SDEA to 34.17

between ERM-SSD and CRS-DEA. So, this suggests that there is no association between the

ranks assigned by ERM-SSD and CRS-DEA and between ERM-SSD and SDEA models.

6.4.2.2 A hypothetical case with three outputs and three inputs

In order to check the consistency of the proposed model, one more case of hypothetical data with

three inputs and three outputs using a set of eighteen DMUs is considered. Similar analysis is

carried out for this data set.

Step 1:

The data required for the analysis is as shown in Table 6.6.

224

Table 6.6: Hypothetical data with 3 outputs and 3 inputs

DMU O1 O1 O3 I1 I2 I3

A 13 14 42 61 35 49

B 11 4 3 18 10 30

C 38 98 13 96 85 26

D 82 27 64 31 7 33

E 46 14 12 10 37 48

F 80 49 11 54 14 32

G 36 15 23 48 40 96

H 48 29 81 66 81 40

I 46 72 31 31 38 8

J 40 49 81 97 62 1

K 45 89 20 16 32 78

L 99 43 94 53 75 40 M 6 2 6 9 64 71

N 46 88 44 18 37 49

O 62 25 12 2 87 88

P 33 54 84 19 22 99

Q 66 87 79 20 3 22

R 16 34 42 43 38 19

Step 2:

First, multiple inputs with single output combinations/dimensions are formed and stochastic

production function with error component model given by Expression 6.1 is used to compute

technical efficiency of each DMU. In this illustration, with three outputs and three inputs, three

combinations/dimensions are formed. These dimensions are (O1, I1, I2, I3), (O2, I1, I2, I3) and

(O3, I1, I2, I3).

Step 3 and 4:

The efficiency scores (Input Criteria and Output Criteria) obtained in Step 2 are used to compute

SSD-I and SSD-O scores using the procedure explained in step 3 and 4 of the proposed

framework. The computations are presented in Table 6.7.

225

Table 6.7: Computations of SSD-I and SSD-O scores (3 outputs and 3 inputs)

DMU OC1 OC2 OC3 DIC SSDI DOC IC1 IC2 IC3 SSDO

A 0.137 0.135 0.294 1 0.299 1 0.137 0.135 0.294 0.511

B 0.126 0.045 0.037 1 0.126 1 0.126 0.045 0.037 3.409

c 0.356 0.921 0.090 1 0.921 1 0.356 0.921 0.090 0.412

D 0.933 0.282 0.518 1 0.935 1 0.933 0.282 0.518 0.138

E 0.530 0.171 0.262 1 0.531 1 0.530 0.171 0.262 0.237

F 0.854 0.481 0.073 1 0.897 1 0.854 0.481 0.073 0.508

G 0.404 0.145 0.174 1 0.405 1 0.404 0.145 0.174 0.294

H 0.476 0.281 0.660 1 0.706 1 0.476 0.281 0.660 0.179

I 0.429 0.814 0.436 1 0.818 1 0.429 0.814 0.436 0.223

J 0.289 0.522 0.836 1 0.854 1 0.289 0.522 0.836 0.242

K 0.531 1.000 0.295 1 1.007 1 0.531 1.000 0.295 0.213

L 0.997 0.429 0.864 1 1.213 1 0.997 0.429 0.864 0.101

M 0.070 0.025 0.148 1 0.151 1 0.070 0.025 0.148 1.843

N 0.515 0.993 0.664 1 1.020 1 0.515 0.993 0.664 0.170

O 0.785 0.372 0.777 1 0.877 1 0.785 0.372 0.777 0.123

P 0.400 0.584 0.996 1 1.173 1 0.400 0.584 0.996 0.175

Q 0.772 0.967 0.748 1 1.168 1 0.772 0.967 0.748 0.126

R 0.158 0.356 0.425 1 0.477 1 0.158 0.356 0.425 0.443

Step 5:

Arithmetic mean of SSDI and SSDO scores called as SSD are obtained. These SSD scores are

ranked to get the final ranking of DMUs using the proposed method ERM-SSD. The ranks

obtained are shown in Table 6.8. Ranks obtained for DMUs using CRS-DEA and SDEA models

are also shown in Table 6.8.

Table 6.8: Ranks using ERM-SSD, CRS-DEA and SDEA (3 outputs and 3 inputs)

DMU ERM-SSD-I

Score

ERM-SSD-O

Score

ERM-SSD

Score

ERM-SSD

Rank

CRS-DEA

Rank

SDEA

Rank

A 0.299 0.511 0.405 16 15 15

B 0.126 3.409 1.767 1 17 17

C 0.921 0.412 0.667 5 12 12

D 0.935 0.138 0.537 11 9 9

E 0.531 0.237 0.384 17 8 8

F 0.897 0.508 0.703 3 10 10

G 0.405 0.294 0.349 18 16 16 H 0.706 0.179 0.442 15 13 13

I 0.818 0.223 0.520 12 1 4

J 0.854 0.242 0.548 10 1 1

K 1.007 0.213 0.610 8 1 5

L 1.213 0.101 0.657 6 11 11

M 0.151 1.843 0.997 2 18 18

N 1.020 0.170 0.595 9 1 7

O 0.877 0.123 0.500 13 1 3

P 1.173 0.175 0.674 4 1 6

Q 1.168 0.126 0.647 7 1 2

R 0.477 0.443 0.460 14 14 14

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From Table 6.8 one may observe that the proposed approach ERM-SSD assigns rank 1 to only

one DMU namely DMU ‘B’. Similarly, SDEA assigns rank 1 to only one DMU namely DMU

‘J’. On the other hand, CRS-DEA assigns rank 1 to 7 different DMUs. So, in this case also

proposed model is able to provide a tie-breaking procedure. This suggests a better discrimination

power of the proposed model.

Hypothesis Testing

In order to verify the similarity of the results obtained using ERM-SSD, CRS-DEA and SDEA

model, Spearman’s rank test was carried out. For this purpose, same hypotheses as stated above

were tested at 5% level of significance. The results are presented in Table 6.9.

Table 6.9: Spearman’s Rank test results for 3 output and 3 input data (for all three models)

Spearman's rho (ρ) ERM-SSD Rank CRS-DEA Rank SDEA Rank

ERM-SSD Rank Correlation Coefficient 1.000 -.020 -.049

Sig. (2-tailed) - .937 .848

N 18 18 18

CRS-DEA Rank Correlation Coefficient -.020 1.000 .971**

Sig. (2-tailed) .937 - .000 N 18 18 18

SDEA Rank Correlation Coefficient -.049 .971** 1.000

Sig. (2-tailed) .848 .000 -

N 18 18 18

From Table 6.9 it is seen that there is no significant association/correlation between the ranks

assigned by ERM-SSD and CRS-DEA (-0.020 with p-value of 0.937) and between ERM-SSD

and SDEA (-0.049 with p-value of 0.848). This means that there is significant difference

between the ranks assigned by these models. Thus, the null hypotheses H0 for hypotheses 1 and

2 are accepted and respective alternate hypotheses H1 are rejected.

Further, the difference between the ranks assigned by above three approaches was checked by

MSD method. The results are shown in Table 6.10.

227

Table 6.10: MSD results for 3 output and 3 input data (for all three models)

DMU ERM-SSD

Rank (1)

CRS-DEA

Rank (2)

SDEA

Rank (3)

(1-2)2 (1-3)2

A 16 15 15 1 1

B 1 17 17 256 256

C 5 12 12 49 49

D 11 9 9 4 4

E 17 8 8 81 81

F 3 10 10 49 49

G 18 16 16 4 4

H 15 13 13 4 4

I 12 1 4 121 64

J 10 1 1 81 81

K 8 1 5 49 9

L 6 11 11 25 25

M 2 18 18 256 256

N 9 1 7 64 4

O 13 1 3 144 100

P 4 1 6 9 4

Q 7 1 2 36 25

R 14 14 14 0 0

MSD 68.5 56.444

As seen from Table 6.10, in this case as well it is seen that none of the MSD values is zero.

The MSD value ranges from 56.44 between ERM-SSD and SDEA to 68.5 between ERM-

SSD and CRS-DEA. So, this suggests that there is no association between the ranks assigned

by ERM-SSD and CRS-DEA and between ERM-SSD and SDEA models.

6.5 An Application of the Proposed Approach: PSU Banks

In this section, an application of the proposed framework using data on Public Sector Unit (PSU)

Banks operating in India is presented. For the purpose of analysis, all 26 PSU Banks are selected.

These twenty-six PSUs control more than ninety percent of all deposits, assets and credits of the

Indian banking sector (www.rbi.org.in). The parameters on which data are collected are as

follows:

Net Profit (output)

Total Income (output)

Operating Expenses (input) and

Total Assets (input)

228

Data are obtained for the financial year 2012-13 from the official website of Indian Bank

Association (www.iba.org.in)

Step 1:

Select ‘26’ PSU Banks operating in India in the sample. Data are presented in Table 6.11.

Table 6.11: PSU Bank data with 2 outputs and 2 inputs

(Fig. are in ₹ Crores)

Sr.

No.

Banks Net

Profit

Total

Income

Operating

expenses

Total

Assets

1 Allahabad Bank 1185.21 18912.6 2958.1 204373.2

2 Andhra Bank 1289.13 13957.11 2037.21 146298.9

3 Bank of Baroda 4480.72 38827.28 5872.11 547135.4

4 Bank of India 2749.35 35674.96 5331.54 452602.7

5 Bank of Maharashtra 759.52 10525.43 1796.64 116952.8

6 Canara Bank 2872.1 37230.94 5141.99 412342.6

7 Central Bank of India 1014.96 23527.98 4232.33 268129.5

8 Corporation Bank 1434.675 16942.02 1996.7812 193442.3

9 Dena Bank 810.38 9554.85 1299.7 113440.4

10 Indian Bank 1581.136 15179.42 2750.8608 162822.6

11 Indian Overseas Bank 567.23 22649.63 3407.84 244656

12 Oriental Bank of Commerce 1327.95 19359.49 2665.18 200697.2

13 Punjab & Sind Bank 339.22 7757.28 1119.33 80477.9

14 Punjab National Bank 4747.67 46109.25 8165.06 478877

15 Syndicate Bank 2004.42 18295.04 3178.83 215122

16 UCO Bank 618.19 17703.88 2176.62 198651.4

17 Union Bank of India 2157.93 27676.73 4512.17 311860.8

18 United Bank of India 391.9 10318.06 1503.92 114615.1

19 Vijaya Bank 585.61 9658.88 1362.97 110981.8

20 State Bank of India (SBI) 14104.98 135691.9 29284.42 1566261

21 State Bank of Bikaner & Jaipur 730.24 8224.47 1579.22 86016.83

22 State Bank of Hyderabad 1250.22 13423.41 2105.07 136078.1

23 State Bank of Mysore 416.1 6513.74 1104.76 67232.76

24 State Bank of Patiala 666.76 10323.08 1590.32 108550.6

25 State Bank of Travancore 615.0424 9287.862 1430.2167 101579.3

26 IDBI Ltd. 1882.08 28283.81 3134.36 322768.5

Source: www.iba.org.in

Step 2:

Multiple inputs with single output combinations/dimensions are formed and stochastic

production function with error component model given by Expression 6.1 is used to compute

technical efficiency of each DMU. Here, with two outputs and two inputs, two dimensions are

formed. These dimensions are (O1, I1, I2) and (O2, I1, I2).

229

Step 3 and 4:

The efficiency scores (Input Criteria and Output Criteria) obtained in Step 2 are used to compute

SSD-I and SSD-O scores using the procedure explained in step 3 and 4 of the proposed

framework. The computations are presented in Table 6.12.

Table 6.12: Computations of SSD-I and SSD-O scores (PSU Banks)

Sr. No. PSU Bank OC1 OC2 DIC SSD-I DOC IC1 IC2 SSD-O

1 Allahabad Bank 0.627 0.957 1 0.963 1 0.627 0.957 0.909

2 Andhra Bank 0.982 0.981 1 0.995 1 0.982 0.981 0.840

3 Bank of Baroda 0.947 0.803 1 0.947 1 0.947 0.803 1.084

4 Bank of India 0.684 0.870 1 0.876 1 0.684 0.870 0.980

5 Bank of Maharashtra 0.704 0.906 1 0.912 1 0.704 0.906 0.942

6 Canara Bank 0.771 0.979 1 0.986 1 0.771 0.979 0.870

7 Central Bank of India 0.391 0.902 1 0.908 1 0.391 0.902 1.030

8 Corporation Bank 0.915 0.962 1 0.972 1 0.915 0.962 0.864

9 Dena Bank 0.868 0.894 1 0.904 1 0.868 0.894 0.927

10 Indian Bank 1.000 0.931 1 1.007 1 1.000 0.931 0.876

11 Indian Overseas Bank 0.252 0.969 1 0.975 1 0.252 0.969 1.486

12 Oriental Bank of Commerce 0.740 0.994 1 1.006 1 0.740 0.994 0.863

13 Punjab & Sind Bank 0.483 0.974 1 0.980 1 0.483 0.974 0.916

14 Punjab National Bank 0.967 0.987 1 0.998 1 0.967 0.987 0.838

15 Syndicate Bank 0.997 0.879 1 0.997 1 0.997 0.879 0.919

16 UCO Bank 0.375 0.969 1 0.975 1 0.375 0.969 0.949

17 Union Bank of India 0.733 0.931 1 0.938 1 0.733 0.931 0.915

18 United Bank of India 0.394 0.931 1 0.937 1 0.394 0.931 0.972

19 Vijaya Bank 0.625 0.911 1 0.917 1 0.625 0.911 0.950

20 State Bank of India (SBI) 0.801 0.912 1 0.921 1 0.801 0.912 0.921

21 State Bank of Bikaner & Jaipur 0.873 0.922 1 0.932 1 0.873 0.922 0.902

22 State Bank of Hyderabad 0.987 0.988 1 1.007 1 0.987 0.988 0.835

23 State Bank of Mysore 0.672 0.945 1 0.952 1 0.672 0.945 0.912

24 State Bank of Patiala 0.681 0.962 1 0.968 1 0.681 0.962 0.897

25 State Bank of Travancore 0.684 0.930 1 0.936 1 0.684 0.930 0.923

26 IDBI Ltd. 0.718 0.985 1 0.991 1 0.718 0.985 0.873

Step 5:

Upon combining SSDI and SSDO scores using arithmetic mean, efficiency scores called SSD

scores are obtained. These SSD scores are ranked to get the final ranking of DMUs. The ranks

obtained are shown in Table 6.13. Table 6.13 also shows ranks obtained for DMUs using CRS-

DEA and SDEA models.

230

Table 6.13: Ranks using ERM-SSD, CRS-DEA and SDEA (PSU Banks)

Sr.

No.

PSU Bank SSD-I SSD-O SSD ERM-SSD

Rank

CRS-DEA

Rank

SDEA

Rank

1 Allahabad Bank 0.96 0.91 0.94 9 17 17

2 Andhra Bank 0.99 0.84 0.92 24 1 7

3 Bank of Baroda 0.95 1.08 1.02 2 1 1

4 Bank of India 0.88 0.98 0.93 17 26 26

5 Bank of Maharashtra 0.91 0.94 0.93 18 22 22

6 Canara Bank 0.99 0.87 0.93 16 15 15

7 Central Bank of India 0.91 1.03 0.97 3 25 25

8 Corporation Bank 0.97 0.86 0.92 23 1 2

9 Dena Bank 0.90 0.93 0.92 26 20 20

10 Indian Bank 1.01 0.88 0.94 8 10 10

11 Indian Overseas Bank 0.98 1.49 1.23 1 16 16

12 Oriental Bank of Commerce 1.01 0.86 0.93 10 1 6

13 Punjab & Sind Bank 0.98 0.92 0.95 7 9 9

14 Punjab National Bank 1.00 0.84 0.92 22 1 4

15 Syndicate Bank 1.00 0.92 0.96 5 1 8

16 UCO Bank 0.97 0.95 0.96 4 11 11

17 Union Bank of India 0.94 0.92 0.93 19 23 23

18 United Bank of India 0.94 0.97 0.95 6 19 19

19 Vijaya Bank 0.92 0.95 0.93 11 21 21

20 State Bank of India (SBI) 0.92 0.92 0.92 20 24 24

21 State Bank of Bikaner & Jaipur 0.93 0.90 0.92 25 14 14

22 State Bank of Hyderabad 1.01 0.83 0.92 21 1 5

23 State Bank of Mysore 0.95 0.91 0.93 14 11 12

24 State Bank of Patiala 0.97 0.90 0.93 12 13 13

25 State Bank of Travancore 0.94 0.92 0.93 15 18 18

26 IDBI Ltd. 0.99 0.87 0.93 13 1 3

As seen from Table 6.13, Indian Overseas Bank emerges as the best bank according to ERM-

SSD, whereas Bank of Baroda is the top performing bank according to SDEA. However, in this

case, CRS-DEA has assigned first rank to eight different banks showing poor discrimination

power. So, in this case as well, the discrimination power of the proposed model ERM-DT is as

good as that of SDEA.

Hypothesis Testing

In order to verify the similarity of the results obtained using ERM-SSD, CRS-DEA and SDEA

model, Spearman’s rank test was carried out. The results are given in Table 6.14.

231

Table 6.14: Spearman’s Rank test results for PSU Bank data (for all three models)

Spearman's rho (ρ) ERM-SSD Rank CRS-DEA Rank SDEA Rank

ERM-SSD Rank Correlation Coefficient 1.000 .022 .030

Sig. (2-tailed) .916 .883 N 26 26 26

CRS-DEA Rank Correlation Coefficient .022 1.000 .985**

Sig. (2-tailed) .916 .000

N 26 26 26

SDEA Rank Correlation Coefficient .030 .985** 1.000

Sig. (2-tailed) .883 .000

N 26 26 26

From Table 6.14 it is seen that there is no significant correlation between the ranks assigned by

ERM-SSD and CRS-DEA (0.022 with p-value of 0.916) and between ERM-SSD and SDEA

(0.030 with p-value of 0.883). This means that there is significant difference between the ranks

assigned by these models. Thus, the null hypotheses H0 for hypotheses 1 and 2 are accepted and

respective alternate hypotheses H1 are rejected.

Further, the agreement between the ranks assigned by above four approaches is checked by MSD

method. The results are shown in Table 6.15.

232

Table 6.15: MSD results for PSU Bank data (for all three models)

Sr. No. Banks ERM-SSD

Rank (1)

CRS-DEA

Rank (2)

SDEA

Rank (3)

(1-2)2 (1-3)2

1 Allahabad Bank 9 17 17 64 64

2 Andhra Bank 24 1 7 529 289

3 Bank of Baroda 2 1 1 1 1

4 Bank of India 17 26 26 81 81

5 Bank of Maharashtra 18 22 22 16 16

6 Canara Bank 16 15 15 1 1

7 Central Bank of India 3 25 25 484 484

8 Corporation Bank 23 1 2 484 441

9 Dena Bank 26 20 20 36 36

10 Indian Bank 8 10 10 4 4

11 Indian Overseas Bank 1 16 16 225 225

12 Oriental Bank of Commerce 10 1 6 81 16

13 Punjab & Sind Bank 7 9 9 4 4

14 Punjab National Bank 22 1 4 441 324

15 Syndicate Bank 5 1 8 16 9

16 UCO Bank 4 11 11 49 49

17 Union Bank of India 19 23 23 16 16

18 United Bank of India 6 19 19 169 169

19 Vijaya Bank 11 21 21 100 100

20 State Bank of India (SBI) 20 24 24 16 16

21 State Bank of Bikaner & Jaipur 25 14 14 121 121

22 State Bank of Hyderabad 21 1 5 400 256

23 State Bank of Mysore 14 11 12 9 4

24 State Bank of Patiala 12 13 13 1 1

25 State Bank of Travancore 15 18 18 9 9

26 IDBI Ltd. 13 1 3 144 100

MSD 134.65 109.08

From Table 6.15 it is seen that MSD values for all pairs are different from zero. The MSD value

for ERM-SSD and CRS-DEA is 134.65 whereas that for the pair ERM-SSD and SDEA is

109.08. This means that the ranks assigned by different methods are different. So, this suggests

that there is no association between the ranks assigned by ERM-SSD and CRS-DEA and

between ERM-SSD and SDEA models.

233

6.6 Analysis and Discussion

From the analysis done for three different cases above, one may acknowledge few common

points which are important to notice.

For all three cases analyzed above one may notice that the proposed model ERM-SSD

has given consistent results and has provided a tie-breaking procedure. It can also be

noted that in the proposed approach, the best DMU has emerged out after comparing its

average performance with the performance of the best DMU in the sample.

Although data under study consisted of multiple outputs, the proposed approach is able to

compute efficiency scores in SFA framework without aggregating the outputs.

The proposed model acknowledge the stochastic nature of the data by decomposing the

error term into two parts: (a) the traditional random error that captures the effect of

measurement error, other statistical noise, and random error; and b) , one-sided

component (as it is iid truncations (at zero) of the N (µ, ) distribution) which captures

the effect of inefficiency. Thus, the technical efficiency obtained for each DMU in Step 2

is more real in the sense that this efficiency is obtained after separating out the

inefficiency and the effect of random shock due to exogenous variables (if any).

Also, for various input-output combinations and bank data analyzed, it can be generalized

that there is no association between the ranks obtained by the proposed model ERM-SSD,

CRS-DEA model and SDEA model.

6.7 Conclusions

SFA is a parametric tool for measuring technical efficiency of a DMU. It benchmarks the

performance of a DMU against the average performance of the group of DMUs under study.

SDEA is a non-parametric tool which is an extension of conventional DEA model. It is been

developed to overcome some of the limitations of DEA such as poor discrimination power of

conventional DEA model in ranking efficient DMUs. In this chapter, an integrated approach

using SFA and SDEA is proposed which overcomes some of the shortcomings of the individual

integrating tools (in this case SFA and SDEA) and tries to make best use of their individual

strengths.

The proposed model ERM-SSD has number of advantages which are discussed below.

234

1. In a case where there are multiple outputs and multiple inputs, computation of technical

efficiency for each DMU using SFA is not very easy. In this case, quite often revenue

(adjusted for price differences) is considered as an output measure. For example, in a case

of evaluation of a performance of a bank, revenue will be a function of multiple sources

such as interest charged on loans, processing fee charged, interest earned on investments,

other income etc. So, unless income from these multiple sources is combined or

aggregated in the form of revenue, it is not possible to apply SFA. In other cases, an

aggregate measure for multiple outputs is defined to study production technology. In a

case where aggregation of outputs is not possible or information on price is not available,

one may have to resort to compute technical efficiency by considering multiple inputs

with single output at a time. An efficiency score thus obtained for each DMU is

incomplete in a sense that this efficiency score so obtained is based only on single output

at a time and not based on all outputs together. The proposed model ERM-SSD is able to

overcome this problem.

2. The proposed model has desired amount of discrimination power while ranking the

DMUs.

3. The proposed model ERM-SSD acknowledges the stochastic nature of the data and

measure the technical efficiency of each DMU after separating out efficiency from

inefficiency and effect of random shocks due to exogenous variables (if any).

4. ERM-SSD recommends the best DMU after comparing its average performance with the

performance of the best DMU in the sample of DMUs.

5. There are several ways of combining the scores obtained using various techniques. In one

of the studies done by Coelli and Perelman (1999), geometric mean is used to combine

the scores obtained by Parametric frontier using Linear Programming, Corrected

Ordinary Least Squares and DEA. Geometric mean is generally used when several

quantities are multiplied together to produce a product. On the other hand, arithmetic

mean is best suited in a situation where several quantities are to be added together to get a

total. Hence, the proposed model advocates the use of arithmetic mean as both the scores

SSD-I and SSD-O are to be added with equal weightage to form the final score using

ERM-SSD.

235

6. The score obtained by the proposed model will always have bound on both the sides as

the score will always lie between 0 and 1. One of the limitations of SDEA of not having

the upper bound is thus removed by this method.

In all the cases discussed above, it is seen that the ranks assigned by all three models namely;

ERM-SSD, CRS-DEA and SDEA are significantly different. This may be due to the fact that the

proposed model recommends the best alternative/ DMU after comparing its average performance

with the best alternative in the sample of alternatives/DMUs. The results are quite encouraging

as findings are similar by both Spearmen’s rank test and by MSD model. The algorithm so

developed is able to accommodate a case of multiple outputs with multiple inputs together.

In a business scenario, at times performance of a business unit may get affected by various

exogenous variables. For example, performance of a bank-branch may get affected by number of

factors (which are out of control of the DM) such as the geographical location it operates in,

population in the vicinity of the branch, number of years of its existence, competitors presence in

that location. The proposed model ERM-SSD will be most suited in such scenario. The

developed ERM- SSD model can acknowledge the effect of such external factors.

Also, the proposed approach can be used for continuous monitoring of the business. For

example, in case of a chain of a restaurant, the performance of the individual restaurant is likely

to vary over a period of time. Moreover, the performance of an individual unit is likely to get

affected by external factors like, the prevalent economic situation, competition by peers etc.

Depending upon the current business performance, management may make a decision of closing

some of the units, or increasing its resources to boost its performance or even merging some of

the less performing units with efficient units.

While benchmarking the relative performance of a business unit, generally, the best performance

of an individual unit is considered. But, in a highly dynamic and complex business environment,

a need may arise where instead of selecting the best decision with less chance of occurrence

(with more risk), the DM may have to settle for a less attractive decision which has more chance

to occur (with less risk). The proposed tool in this study facilitates benchmarking of the average

performance which is most likely to occur vis-à-vis the best in the group. It will certainly provide

a relative performance benchmarking which will be more realistic and more likely to occur as the

best performance can be considered as one of its kind and may not have high chance of

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occurrence. It is hoped that the proposed approach will help the researchers and practicing

managers alike.