1 3 A fuzzy DEA model with undesirable fuzzy outputs and its application 4 to the banking sector in India 5 6 7 Jolly Puri ⇑ Q1 , Shiv Prasad Yadav 8 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India 9 10 12 article info 13 14 Keywords: 15 Data envelopment analysis 16 Fuzzy data envelopment analysis 17 Undesirable outputs 18 Rank efficient units 19 Banking sector performance 20 21 abstract 22 Data envelopment analysis (DEA) is a widely used technique for measuring the relative efficiencies of 23 decision making units (DMUs) with multiple inputs and multiple outputs. However, in real life applica- 24 tions, undesirable outputs may be present in the production process which needs to be minimized. The 25 present study endeavors to propose a DEA model with undesirable outputs and further to extend it in 26 fuzzy environment in view of the fact that input/output data are not always available in exact form in 27 real life problems. We propose a fuzzy DEA model with undesirable fuzzy outputs which can be solved 28 as crisp linear program for each a in (0, 1] using a-cut approach. Further, cross-efficiency technique is 29 applied to increase the discrimination power of the proposed models and to rank the efficient DMUs 30 at every a in (0, 1]. Moreover, for better understanding of the proposed methodology, we present a 31 numerical illustration followed by an application to the banking sector in India. This is the first study 32 which attempts to measure the performance of public sector banks (PuSBs) in India using fuzzy input/ 33 output data for the period 2009–2011. The results obtained from the proposed methodology not only 34 depict the impact of undesirable output on the performance of PuSBs but also analyze efficiently the 35 influence of the presence of uncertainty in the data over the efficiency results. The findings show that 36 the efficiency results of many PuSBs vary with the variation in a during the selected period. 37 Ó 2014 Published by Elsevier Ltd. 38 39 40 41 1. Introduction 42 Data envelopment analysis (DEA) is a linear programming 43 based non-parametric technique to evaluate the relative efficiency 44 of a decision making unit (DMU) on the basis of multiple inputs 45 and multiple outputs. It constructs a non-parametric piecewise 46 frontier over the data. Using this frontier, DEA computes a maximal 47 performance measure for each DMU relative to that of all other 48 DMUs, with the restriction that each DMU lies on the efficient fron- 49 tier or is enveloped by the frontier. The DMUs which lie on the 50 frontier are the best practice (efficient) units and attain the effi- 51 ciency values equal to 1. On the other hand, the DMUs which attain 52 the efficiency values between 0 and 1 are the inefficient units and 53 can improve their efficiency values to reach the efficient frontier by 54 either increasing their current output levels or decreasing their 55 current input levels. The standard DEA models and some of their 56 extensions can be seen in Cooper, Seiford, and Tone (2007). These 57 DEA models were initially formulated only for desirable inputs and 58 outputs, and are usually based on the assumption that inputs have 59 to be minimized and outputs have to be maximized. However, in 60 real life problems, undesirable inputs and/or outputs may be pres- 61 ent in the production process which also needs to be minimized. 62 This can be understood by taking an example in the banking indus- 63 try. In banks, the risk of loans to become non-performing loans/ 64 assets (NPAs) always exists. NPAs directly affect the stability, asset 65 quality, credit creation and profitability of a bank. The augmenta- 66 tion in NPAs leads to instability, poor asset quality, less credit cre- 67 ation and most important reduction in profitability of a bank. Thus, 68 NPAs can be treated as undesirable outputs in the banking industry 69 of any country and these needs to be minimized in order to 70 increase the asset quality and profitability of a bank. By taking into 71 account the above mentioned problem, the present study is an 72 attempt to study the impact of NPAs on the performance of Indian 73 banking sector for the period 2009–2011. 74 In order to deal with the above problem, there exists wide range 75 of approaches in the existing literature to incorporate undesirable 76 outputs in DEA. Scheel (2001) classified them as direct and indirect 77 approaches. In indirect approaches, the values of the undesirable 78 outputs are transformed by a monotone decreasing function f so 79 that the undesirable outputs become desirable after the transfor- 80 mation. On the other hand, direct approaches avoid data transfor- 81 mation and incorporate the undesirable outputs directly into the http://dx.doi.org/10.1016/j.eswa.2014.04.013 0957-4174/Ó 2014 Published by Elsevier Ltd. ⇑ Corresponding author. Tel.: +91 75007 66529. E-mail addresses: [email protected](J. Puri), [email protected](S.P. Yadav). Expert Systems with Applications xxx (2014) xxx–xxx Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa ESWA 9282 No. of Pages 14, Model 5G 22 April 2014 Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model with undesirable fuzzy outputs and its application to the banking sector in India. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.04.013
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Expert Systems with Applications xxx (2014) xxx–xxx
ESWA 9282 No. of Pages 14, Model 5G
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Contents lists available at ScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier .com/locate /eswa
A fuzzy DEA model with undesirable fuzzy outputs and its applicationto the banking sector in India
http://dx.doi.org/10.1016/j.eswa.2014.04.0130957-4174/� 2014 Published by Elsevier Ltd.
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model with undesirable fuzzy outputs and its application to the banking seIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa.2014.04.013
Jolly Puri ⇑, Shiv Prasad YadavDepartment of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
Data envelopment analysis (DEA) is a widely used technique for measuring the relative efficiencies ofdecision making units (DMUs) with multiple inputs and multiple outputs. However, in real life applica-tions, undesirable outputs may be present in the production process which needs to be minimized. Thepresent study endeavors to propose a DEA model with undesirable outputs and further to extend it infuzzy environment in view of the fact that input/output data are not always available in exact form inreal life problems. We propose a fuzzy DEA model with undesirable fuzzy outputs which can be solvedas crisp linear program for each a in (0,1] using a-cut approach. Further, cross-efficiency technique isapplied to increase the discrimination power of the proposed models and to rank the efficient DMUsat every a in (0,1]. Moreover, for better understanding of the proposed methodology, we present anumerical illustration followed by an application to the banking sector in India. This is the first studywhich attempts to measure the performance of public sector banks (PuSBs) in India using fuzzy input/output data for the period 2009–2011. The results obtained from the proposed methodology not onlydepict the impact of undesirable output on the performance of PuSBs but also analyze efficiently theinfluence of the presence of uncertainty in the data over the efficiency results. The findings show thatthe efficiency results of many PuSBs vary with the variation in a during the selected period.
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1. Introduction
Data envelopment analysis (DEA) is a linear programmingbased non-parametric technique to evaluate the relative efficiencyof a decision making unit (DMU) on the basis of multiple inputsand multiple outputs. It constructs a non-parametric piecewisefrontier over the data. Using this frontier, DEA computes a maximalperformance measure for each DMU relative to that of all otherDMUs, with the restriction that each DMU lies on the efficient fron-tier or is enveloped by the frontier. The DMUs which lie on thefrontier are the best practice (efficient) units and attain the effi-ciency values equal to 1. On the other hand, the DMUs which attainthe efficiency values between 0 and 1 are the inefficient units andcan improve their efficiency values to reach the efficient frontier byeither increasing their current output levels or decreasing theircurrent input levels. The standard DEA models and some of theirextensions can be seen in Cooper, Seiford, and Tone (2007). TheseDEA models were initially formulated only for desirable inputs andoutputs, and are usually based on the assumption that inputs have
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to be minimized and outputs have to be maximized. However, inreal life problems, undesirable inputs and/or outputs may be pres-ent in the production process which also needs to be minimized.This can be understood by taking an example in the banking indus-try. In banks, the risk of loans to become non-performing loans/assets (NPAs) always exists. NPAs directly affect the stability, assetquality, credit creation and profitability of a bank. The augmenta-tion in NPAs leads to instability, poor asset quality, less credit cre-ation and most important reduction in profitability of a bank. Thus,NPAs can be treated as undesirable outputs in the banking industryof any country and these needs to be minimized in order toincrease the asset quality and profitability of a bank. By taking intoaccount the above mentioned problem, the present study is anattempt to study the impact of NPAs on the performance of Indianbanking sector for the period 2009–2011.
In order to deal with the above problem, there exists wide rangeof approaches in the existing literature to incorporate undesirableoutputs in DEA. Scheel (2001) classified them as direct and indirectapproaches. In indirect approaches, the values of the undesirableoutputs are transformed by a monotone decreasing function f sothat the undesirable outputs become desirable after the transfor-mation. On the other hand, direct approaches avoid data transfor-mation and incorporate the undesirable outputs directly into the
2 J. Puri, S.P. Yadav / Expert Systems with Applications xxx (2014) xxx–xxx
ESWA 9282 No. of Pages 14, Model 5G
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DEA model. The literature on the modeling of undesirable outputshave been studied earlier by Seiford and Zhu (2002), Färe andGrosskopf (2004), Korhonen and Luptacik (2004), Jahanshahloo,Lofti, Shoja, Tohidi, and Razavyan (2005), Yang and Pollitt (2009),Liang, Li, and Li (2009), Liu, Meng, Li, and Zhang (2010), You andYan (2011), Charles, Kumar, and Kavitha (2012), Jahanshahloo,Lofti, Maddahi, and Jafari (2012), Leleu (2013) and Ramli andMunisamy (2013). In the banking industry, direct approach is moresuitable to deal with NPAs because it is more convenient and evi-dent to incorporate undesirable outputs directly into the DEAmodel and analyze the impact of NPAs on the bank’s performance.One of the direct approach to deal with undesirable outputs isstudied by Korhonen and Luptacik (2004) which is to treat all thedesirable and undesirable outputs as the weighted sum, but usingnegative weights for the undesirable outputs. However, one short-coming of this approach is that there may exist such optimumweights which on the one hand maximizes the efficiency value ofthe targeted DMU but on the other hand may evaluate the effi-ciency value(s) of the other DMU(s) less than zero. This can beunderstood while using the cross-efficiency technique (Sexton,Silkman, & Hogan, 1986) in which each DMU is evaluated by usingthe optimal weights of the other DMUs. Sometimes, the modeldeveloped by Korhonen and Luptacik (2004) leads to the negativecross-efficiencies for some DMUs and this is verified by giving anillustration in the next section. However, the efficiencies andcross-efficiencies can never be negative for any DMU. Thus, inthe present study, to overcome the shortcoming of the existingmodel, we propose a new DEA model with undesirable outputs(DEA-UO).
Conventional DEA requires crisp input and output data, whichmay not always be available in real world applications. However,in real life problems, inputs and outputs are often imprecise. Todeal with imprecise data, the notion of fuzziness was introducedin DEA and the DEA was extended to fuzzy DEA (FDEA). The liter-ature on FDEA has been previously studied by Kao and Liu (2000),Saati, Memariani, and Jahanshahloo (2002), Wang, Luo, and Liang(2009), Hatami-Marbini, Emrouznejad, and Tavana (2011), Wangand Chin (2011), Angiz, Emrouznejad, and Mustafa (2012), Puriand Yadav (2013a), Shiraz, Charles, and Jalalzadeh (2014) andPuri and Yadav (2014). In the present study, we extend DEA-UOmodel to fuzzy environment in which all inputs, desirable outputsand undesirable outputs are taken as fuzzy numbers, in particulartriangular fuzzy numbers (TFNs) (Chen, 1994). Since the proposedmodel is extended to fuzzy environment, so it can be named asFDEA model with undesirable fuzzy outputs (FDEA-UFO). Severalapproaches have been developed to deal with fuzzy input and out-put data in FDEA. Hatami-Marbini et al. (2011) classified them as:(i) tolerance approach, (ii) a-cut approach, (iii) fuzzy rankingapproach and (iv) possibility approach. Among these approaches,a-cut approach has been widely used to solve FDEA models. There-fore, to solve FDEA-UFO model, we use the methodology suggestedby Saati et al. (2002) based on a-cut approach.
In the present competitive scenario, the risk factor behind loansand investments exists in the banking industry of almost everydeveloping nation. The risk of the conversion of a loan into a badloan comprises the degree of uncertainty present in the loans. Cer-tainly, crisp set theory is not appropriate to deal with such kind ofsituations; rather fuzzy set theory (Zimmermann, 1991) is moresuitable. Thus, in the present study, we represent the risk factorpresent in the input and output data of Indian banking industryas fuzzy numbers. However, a wide range of studies have been con-ducted by the researchers on the bank efficiency and issue of NPAsin the Indian banking industry based on crisp input–output data.The literature on the bank efficiency and issue of NPAs in theIndian context can be seen in the studies like Ranjan and Dhal(2003), Krishna (2004), Subramanyam and Reddy (2008), Rawlin,
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
Sharan, and Lakshmipathy (2012), Saravana, Reddy, andSubramanyam (2012), Puri and Yadav (2013b), Siraj and Pillai(2013), Kaur and Kaur (2013) and Bandyopadhyay (2013). All ofthe existing studies reveal that NPA adversely affects the profitabil-ity, asset quality and credit creation of Indian domestic banks inthe crisp environment. To the best of our knowledge, there ishardly any study related to the Indian bank efficiency in whichinput–output data are taken as fuzzy numbers. Thus, the presentstudy explores the impact of undesirable output NPAs on theperformance of banks in India along with the effect of uncertaintypresent in the input–output data over the efficiency results.Furthermore, the complete ranking of the DMUs in crisp aswell as in fuzzy environments is achieved by following thecross-efficiency technique developed by Sexton et al. (1986). Thefindings of the present study are found to be very interesting andnear to reality.
To sum up with all the above aspects, the aim of the presentstudy is sevenfold: (i) to propose new DEA-UO model which over-comes the shortcoming of the existing model and leads to the posi-tive cross-efficiencies of the DMUs, (ii) to extend DEA-UO model tofuzzy environments and propose FDEA-UFO model, (iii) to developalgorithms for finding cross-efficiencies in crisp as well as fuzzyenvironments and to achieve complete ranking of the efficientDMUs, (iv) to provide numerical illustration in order to validatethe proposed FDEA-UFO methodology, (v) to apply the proposedFDEA-UFO methodology on the banking sector in India and tomeasure the average efficiencies of public sector banks (PuSBs) inIndia and their categories, namely, Nationalized banks (NBs) andState Bank of India and its associates (SBI group) for the period2009–2011, (vi) to recognize the impact of undesirable outputson the bank efficiency results, and (vii) to analyze efficiently theeffect of the presence of uncertainty in the data over the efficiencyresults.
The practical significance of the present study is that it is extre-mely valuable for the bank experts and policy makers to identifythe average inefficiencies present in each PuSB of India at differentsatisfaction levels of a 2 (0,1], and to form policies in order toimprove the performance and growth of PuSBs and their respectivecategories.
This paper is organized as follows: Section 2 and Section 3presents the proposed DEA-UO model and FDEA-UFO modelrespectively. Section 3 also describes the methodology to solveFDEA-UFO model using a-cut approach. Section 4 presentsthe ranking algorithms using cross-efficiency technique to achievethe complete ranking of the DMUs in crisp and fuzzy environ-ments. Section 5 presents the numerical illustration. Section 6 pre-sents an application of the proposed approach to the bankingsector in India. The last Section 7 concludes the findings of ourstudy.
2. Proposed DEA-UO model
Assume that the performance of a homogeneous set of n DMUs(DMUj; j = 1, . . . ,n) is to be measured. The performance of a DMU ischaracterized by a production process of m inputs to yield s out-puts in which s1 outputs are desirable (good) and s2 outputs areundesirable (bad) such that s = s1 + s2. Let Y 2 Rn�s be the outputmatrix consisting of non-negative elements. Then the output
matrix Y can be decomposed as Y ¼ ½Yg Yb �T , where Yg 2 Rn�s1
and Yb 2 Rn�s2 are the matrices for desirable outputs and undesir-able outputs respectively. Let X 2 Rn�m be the input matrix consist-ing of non-negative elements. Further, let xik (i = 1, . . . ,m) be the minputs used by the kth DMU, and yg
rkðr ¼ 1; . . . ; s1Þ andyb
pkðp ¼ 1; . . . ; s2Þ be the s1 desirable and s2 undesirable outputs pro-duced by the kth DMU respectively.
ith undesirable fuzzy outputs and its application to the banking sector in.2014.04.013
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Korhonen and Luptacik (2004) suggested the following DEAmodel in which negative weights are taken for undesirableoutputs:
Model-1
max Ek ¼Ps1
r¼1ugrkyg
rk �Ps2
p¼1ubpkyb
pkPmi¼1v ikxik
subject to Ej ¼Ps1
r¼1ugrkyg
rj �Ps2
p¼1ubpkyb
pjPmi¼1v ikxij
6 1; 8 j ¼ 1;2; . . . ;n;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0;
where ugrk; u
bpk and vik are the weights for the rth desirable output,
pth undesirable output and ith input of the kth DMU respectively,and e is the non-Archimedean infinitesimal.
In the above model, the weighted sum of all the desirable andundesirable outputs is used, but with negative weights for undesir-able outputs. However, due to these negative weights it may hap-pen that for any optimal solution ug�
k ;ub�k ;v�k
� �for DMUk of Model-1,
there exists some DMUj for which the efficiency Ej becomes nega-tive, i.e., for some DMUj,
Ej ¼Ps1
r¼1ugrkyg
rj �Ps2
p¼1ubpkyb
pjPmi¼1v ikxij
< 0; j 2 f1;2; . . . ; ng; j–k:
This can happen while using the cross-efficiency technique inwhich each DMU is evaluated by using the optimal weights ofthe other DMUs. For better understanding, consider an efficiencyassessment problem of five DMUs from Guo and Wu (2013) interms of two inputs (I1 and I2), one desirable output (DO1) andone undesirable output (UDO1). The input–output data and theefficiency results using Model-1 are presented in Table 1. Thecross-efficiency matrix is shown in Table 2. It indicates thatcross-efficiencies for some DMUs in the cross-efficiency matrixare obtained to be negative.
Since, in DEA, the efficiencies and cross-efficiencies cannot benegative for any DMU. So, the cross-efficiency results obtained byModel-1 are sometimes turning out to be unrealistic and cannotbe used for complete ranking. Therefore, in order to make effi-ciency non-negative for every DMUj, we propose a new DEA modelin which we include additional constraints Ej P 0, j = 1, 2, 3, . . . ,n inModel-1 and hence, the Model-1 becomes:
Model-2
max Ek ¼Ps1
r¼1ugrkyg
rk �Ps2
p¼1ubpkyb
pkPmi¼1v ikxik
subject to 0 6 Ej ¼Ps1
r¼1ugrkyg
rj �Ps2
p¼1ubpkyb
pjPmi¼1v ikxij
6 1 8 j ¼ 1;2; . . . ;n;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0:
By using Charnes–Cooper transformation (Cooper et al., 2007),Model-2 can be transformed into the linear programming problem(LPP) given by
Model-3
max Ek ¼Xs1
r¼1
ugrkyg
rk �Xs2
p¼1
ubpkyb
pk
subject toXm
i¼1
v ikxik ¼ 1;
Xs1
r¼1
ugrkyg
rj �Xs2
p¼1
ubpkyb
pj �Xm
i¼1
v ikxij 6 0 8 j ¼ 1;2; . . . ;n;
Xs1
r¼1
ugrkyg
rj �Xs2
p¼1
ubpkyb
pj P 0 8 j ¼ 1;2; . . . ; n;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0:
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
The Model-3 is known as the DEA model with undesirable out-puts (DEA-UO).
The cross-efficiency scores of the kth DMU obtained by usingthe weights of Model-3 leads to positive numbers. Consider an effi-ciency assessment problem from Guo and Wu (2013) of five DMUswith input–output data given in Table 1. The efficiency resultsusing Model-3 are presented in Table 3. The cross-efficiency matrixis shown in Table 4. It indicates that efficiency values in thecross-efficiency matrix are all positive. Thus, the DEA-UO modelovercomes the shortcoming of the existing model and leads tothe positive cross-efficiencies of the DMUs. Further, the efficientDMUs in Model-3 can now be ranked using cross-efficiencytechnique
Definition 1. A DMUk is said to be efficient if E�k ¼ 1.In real life problems, inputs and outputs are often imprecise/
fuzzy and in the present study, to deal with such situations, weextend the DEA-UO model to the fuzzy DEA (FDEA) model withundesirable fuzzy outputs (FDEA-UFO). In the extended model,the data values of each input, desirable output and undesirableoutput are taken as positive fuzzy numbers (FNs), in particular,positive triangular fuzzy numbers (TFNs) (Chen, 1994).
Definition 2. A TFN eA, denoted by (aL,aM,aR), is defined by themembership function leA given by
leAðxÞ ¼x�aL
aM�aL ; aL < x 6 aM;
x�aR
aM�aR ; aM6 x < aR;
0; otherwise:
8><>:
ith undesirable fuzzy outputs and its application to the banking sector in.2014.04.013
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ESWA 9282 No. of Pages 14, Model 5G
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A TFN eA ¼ ðaL; aM; aRÞ is said to be positive TFN if aL > 0.
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3. Proposed FDEA-UFO model
Assume that the performance of a homogeneous set of n DMUs(DMUj; j = 1, . . . ,n) is to be measured. The performance of a DMU ischaracterized by a production process of m fuzzy inputs to yield sfuzzy outputs in which s1 fuzzy outputs are desirable (good) and s2
fuzzy outputs are undesirable (bad) such that s = s1 + s2. Let eY bethe fuzzy output matrix consisting of positive fuzzy elements. Then
the fuzzy output matrix eY can be decomposed as eY ¼ eY g eY bh iT
,
where eY g and eY b are the matrices for desirable fuzzy outputs
and undesirable fuzzy outputs respectively. Let eX be the fuzzyinput matrix consisting of positive fuzzy elements. Further, let ~xik
(i = 1, . . . ,m) be the m fuzzy inputs used by the kth DMU, and~yg
rkðr ¼ 1; . . . ; s1Þ and ~ybpk ðp ¼ 1; . . . ; s2Þ be the s1 desirable and s2
undesirable fuzzy outputs produced by the kth DMU respectively.The fuzzy efficiency of the kth DMU with the undesirable fuzzyoutputs can be evaluated from the following FDEA-UFO model:
Model-4
max eEk ¼Xs1
r¼1
ugrk
~ygrk �
Xs2
p¼1
ubpk
~ybpk
subject toXm
i¼1
v ik~xik ¼ ~1;
Xs1
r¼1
ugrk
~ygrj �
Xs2
p¼1
ubpk
~ybpj �
Xm
i¼1
v ik~xij 6~0 8 j ¼ 1;2; . . . ; n;
Xs1
r¼1
ugrk
~ygrj �
Xs2
p¼1
ubpk
~ybpj P ~0 8 j ¼ 1;2; . . . ; n;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0;
where ugrk;u
bpk and vik are the weights for the rth desirable fuzzy out-
put, pth undesirable fuzzy output and ith fuzzy input of the kthDMU respectively, and e is the non-Archimedean infinitesimal.
Consider that all fuzzy inputs, desirable fuzzy outputs andundesirable fuzzy outputs are TFNs. Then Model-4 becomes:
Model-5
max eEk ¼Xs1
r¼1
ugrk yg L
rk ; yg Mrk ; yg R
rk
� ��Xs2
p¼1
ubpk yb L
pk ; yb Mpk ; y
b Rpk
� �
subject toXm
i¼1
v ik xLik; x
Mik ; x
Rik
� �¼ ð1;1;1Þ;
Xs1
r¼1
ugrk yg L
rj ; yg Mrj ; yg R
rj
� ��Xs2
p¼1
ubpk yb L
pj ; yb Mpj ; y
b Rpj
� �
�Xm
i¼1
v ik xLij; x
Mij ; x
Rij
� �6 ð0; 0;0Þ 8 j ¼ 1;2; . . . ;n;
Xs1
r¼1
ugrk yg L
rj ; yg Mrj ; yg R
rj
� ��Xs2
p¼1
ubpk yb L
pj ; yb Mpj ; y
b Rpj
� �
P ð0; 0;0Þ 8 j ¼ 1;2; . . . ;n;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0:
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3.1. Methodology to solve Model-5
Saati et al. (2002) developed a method to solve FDEA model byusing the concept of a-cut and variable substitution. Here, we willapply this method to solve Model-5. Firstly, introduce a-cuts of the
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
objective function and constraints in Model-5. Then Model-5becomes:
Model-6
max Eka ¼
Xs1
r¼1
ugrk ayg M
rk þ ð1� aÞyg Lrk ;ayg M
rk þ ð1� aÞyg Rrk
h i
�Xs2
p¼1
ubpk ayb M
pk þ ð1� aÞyb Lpk ;ayb M
pk þ ð1� aÞyb Rpk
h i
subject toXm
i¼1
v ik axMik þ ð1� aÞxL
ik;axMik þ ð1� aÞxR
ik
� �¼ ½1;1�;
Xs1
r¼1
ugrk ayg M
rj þ ð1� aÞyg Lrj ;ayg M
rj þ ð1� aÞyg Rrj
h i
�Xs2
p¼1
ubpk ayb M
pj þ ð1� aÞyb Lpj ;ayb M
pj þ ð1� aÞyb Rpj
h i
�Xm
i¼1
v ik axMij þ ð1� aÞxL
ij;axMij þ ð1� aÞxR
ij
h i
6 ½0;0� 8 j ¼ 1;2; . . . ;n;
Xs1
r¼1
ugrk ayg M
rj þ ð1� aÞyg Lrj ;ayg M
rj þ ð1� aÞyg Rrj
h i
�Xs2
p¼1
ubpk ayb M
pj þ ð1� aÞyb Lpj ;ayb M
pj þ ð1� aÞyb Rpj
h i
P ½0;0�; 8 j ¼ 1;2; . . . ;n;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0:
It can be seen that all the coefficients in Model-6 are intervals whichimplies that it is an interval problem. The following steps areinvolved in the evaluation of Model-6:
Step 1. Consider the variables xij; ygrj and yb
pj such that
xij 2 axMij þ ð1� aÞxL
ij;axMij þ ð1� aÞxR
ij
h i8 i; j;
ygrj 2 ayg M
rj þ ð1� aÞyg Lrj ;ayg M
rj þ ð1� aÞyg Rrj
h i8 r; j;
ybpj 2 ayb M
pj þ ð1� aÞyb Lpj ;ayb M
pj þ ð1� aÞyb Rpj
h i8 p; j:
By substituting these variables, Model-6 becomesModel-7
max Eka ¼
Xs1
r¼1
ugrkyg
rk �Xs2
p¼1
ubpkyb
pk
subject toXm
i¼1
v ikxik ¼ 1;
Xs1
r¼1
ugrkyg
rj �Xs2
p¼1
ubpkyb
pj �Xm
i¼1
v ikxij 6 0 8 j ¼ 1;2; . . . ;n;
Xs1
r¼1
ugrkyg
rj �Xs2
p¼1
ubpkyb
pj P 0 8 j ¼ 1;2; . . . ;n;
axMij þ ð1� aÞxL
ij 6 xij 6 axMij þ ð1� aÞxR
ij 8 i; j;
aygMrj þ ð1� aÞygL
rj 6 ygrj 6 aygM
rj þ ð1� aÞygRrj 8 r; j;
aybMpj þ ð1� aÞybL
pj 6 ybpj 6 aybM
pj þ ð1� aÞybRpj 8 p; j;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0:
ith undesirable fuzzy outputs and its application to the banking sector in.2014.04.013
J. Puri, S.P. Yadav / Expert Systems with Applications xxx (2014) xxx–xxx 5
ESWA 9282 No. of Pages 14, Model 5G
22 April 2014
Model-7 is a nonlinear programming problem which can be trans-formed into LPP by using the variable substitutions given in Step 2.
Step 2. Let�xij ¼ v ikxij ð8 i; jÞ; �yg
rj ¼ ugrkyg
rjð8r; jÞ and �ybpj ¼ ub
pkybpj ð8 p; jÞ: ð1Þ
By substituting these variables, Model-7 reduces to the followingLPP:
Model-8
max Eka ¼
Xs1
r¼1
�ygrk �
Xs2
p¼1
�ybpk
subject toXm
i¼1
�xik ¼ 1;
Xs1
r¼1
�ygrj �
Xs2
p¼1
�ybpj �
Xm
i¼1
�xij 6 0 8 j ¼ 1;2; . . . ;n;
Xs1
r¼1
�ygrj �
Xs2
p¼1
�ybpj P 0 8 j ¼ 1;2; . . . ;n;
v ik axMij þ ð1� aÞxL
ij
� �6 �xij 6 v ik axM
ij þ ð1� aÞxRij
� �8 i; j;
ugrk aygM
rj þ ð1� aÞygL
rj
� �6 �yg
rj 6 ugrk aygM
rj þ ð1� aÞygR
rj
� �8 r; j;
ubpk aybM
pj þ ð1� aÞybL
pj
� �6 �yb
pj 6 ubpk aybM
pj þ ð1� aÞybR
pj
� �8 p; j;
ugrk P e 8r; ub
pk P e 8p; v ik P e 8i; e > 0:
Model-8 is a crisp linear parametric programming problem wherea 2 (0,1] is a parameter and we have an optimal solution for eachvalue of a.
Definition 3. A DMUk is said to be efficient at given a 2 (0,1] ifEk�
a ¼ 1.
388
389
390
Theorem 1. Ek�
a26 Ek�
a1for any a1, a2 2 (0,1] and a1 6 a2, where Ek�
a2
and Ek�
a1are the optimum objective function values of Model-8 at a1
and a2 respectively.
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
Proof. Let ug�
1k;ug�
2k; . . . ;ug�
s1k;ub�
1k;ub�
2k; . . . ;ub�
s2k;v�1k;v�2k; . . . ;v�mk;�yg�
11;�yg�
21;�
. . . ;�yg�
s11; . . .�yg�
1n;�yg�
2n; . . . ;�yg�s1n;�yb�
11;�yb�21; . . . ;�y
b�s21; . . .�y
b�1n;�y
b�2n; . . . ; �yb�
s2n;�x�11;
�x�21; . . . ;�x�m1; . . .�x
�1n;�x
�2n; . . . ;�x
�mnÞ be the optimal solution to Model-8
at a = a2.Therefore,
Xm
i¼1
�x�ik ¼ 1;
Xs1
r¼1
�yg�
rj �Xs2
p¼1
�yb�pj �
Xm
i¼1
�x�ij 6 08 j ¼ 1;2; . . . ;n;
Xs1
r¼1
�yg�
rj �Xs2
p¼1
�yb�
pj P 0 8 j ¼ 1;2; . . . ;n;
9>>>>>>>>>>>>=>>>>>>>>>>>>;
ð2Þ
v�ik a2xMij þð1�a2ÞxL
ij
� �6 �x�ij6v�ik a2xM
ij þð1�a2ÞxRij
� �8 i; j;
ug�
rk a2ygMrj þð1�a2ÞygL
rj
� �6 �yg�
rj 6ug�
rk a2ygMrj þð1�a2ÞygR
rj
� �8 r; j;
ub�
pk a2ybMpj þð1�a2ÞybL
pj
� �6 �yb�
pj 6ub�
pk a2ybMpj þð1�a2ÞybR
pj
� �8 p; j;
9>>>>>>=>>>>>>;ð3Þ
ug�
rk P e 8r; ub�
pk P e 8p; v�ik P e 8i; e > 0: ð4Þ
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
Since ð~xikÞa1¼ a1xM
ij þ ð1� a1ÞxLij
� �; a1xM
ij þ ð1� a1ÞxRij
� �h i8 i
and ð~xikÞa2¼ a2xM
ij þ ð1� a2ÞxLij
� �; a2xM
ij þ ð1� a2ÞxRij
� �h i8 i
so for a1 6 a2; we get a1xMij þ ð1� a1ÞxL
ij
� �
6 a2xMij þ ð1� a2ÞxL
ij
� �8 i and
a2xMij þ ð1� a2ÞxR
ij
� �6 a1xM
ij þ ð1� a1ÞxRij
� �8 i:
Similarly; a1ygM
rj þ ð1� a1ÞygL
rj
� �
6 a2ygM
rj þ ð1� a2ÞygL
rj
� �8r and
a2ygM
rj þ ð1� a2ÞygR
rj
� �6 a1ygM
rj þ ð1� a1ÞygR
rj
� �8 r:
In a similar way; a1ybM
pj þ ð1� a1ÞybL
pj
� �
6 a2ybM
pj þ ð1� a2ÞybL
pj
� �8p and
a2ybM
pj þ ð1� a2ÞybR
pj
� �6 a1ybM
pj þ ð1� a1ÞybR
pj
� �8 p:
Consequently,
v�ik a1xMij þð1�a1ÞxL
ij
� �6 �x�ij6v�ik a1xM
ij þð1�a1ÞxRij
� �8 i; j;
ug�
rk a1ygM
rj þð1�a1ÞygL
rj
� �6 �yg�
rj 6ug�
rk a1ygM
rj þð1�a1ÞygR
rj
� �8 r; j;
ub�
pk a1ybM
pj þð1�a1ÞybL
pj
� �6 �yb�
pj 6ub�
pk a1ybM
pj þð1�a1ÞybR
pj
� �8 p; j:
9>>>>=>>>>;ð5Þ
From (2), (4), and (5), we can find that ug�
1k;ug�
2k;�
. . . ;ug�
s1k; ub�
1k;ub�
2k;
. . . ; ub�
s2k; v�1k; v�2k; . . . ;v�mk; �yg�
11; �yg�
21; . . . ; �yg�
s11; . . . �yg�
1n; �yg�
2n; . . . ; �yg�s1n; �yb�
11; �yb�21;
. . . ; �yb�s21; . . . �yb�
1n; �yb�2n; . . . ; �yb�
s2n; �x�11; �x�21; . . . ; �x�m1; . . . �x�1n; �x
�2n; . . . ; �x�mnÞ is a
feasible solution of Model-8 at a = a1. h
Therefore Ek�
a2¼Ps1
r¼1�yg�
rk �Ps2
p¼1�yb�
pk 6 Ek�
a1; where Ek�
a2and Ek�
a1are
the optimum objective function values of Model-8 at a1 and a2
respectively.Hence, Ek�
a26 Ek�
a1for any a1, a2 2 (0,1] and a1 6 a2.
4. Complete ranking of the efficient DMUs using cross-efficiencytechnique
Most of the times, more than one DMU is evaluated as efficientDMU using DEA and these efficient DMUs cannot be discriminatedfurther. Thus, lack of discrimination power is one of the majordrawbacks of DEA. In literature, many researchers have proposeddifferent approaches to increase the discrimination power ofDEA. The literature on complete ranking of the DMUs can be seenin the studies like Adler, Friedman, and Sinuany-Stern (2002),Liang et al. (2009), Jahanshahloo, Lofti, Khanmohammadi,Kazemimanesh, and Rezaie (2010), Wu, Sun, Liang, and Zha(2011) and Guo and Wu (2013). Adler et al. (2002) concluded thatDEA ranking can be classified into six techniques. One among themis a cross-efficiency technique in which the DMUs are self- andpeer-evaluated. In the present study, we follow the cross-efficiencytechnique to achieve the complete ranking in crisp as well as fuzzyenvironments. A self-evaluation means that each DMU gets anopportunity to self-evaluate its efficiency relative to the otherDMUs. On the other hand, a peer-evaluation means that eachDMU is evaluated by using the optimal weights of the other DMUs.The average of these efficiencies is known as the average cross-effi-ciency (ACE). In traditional DEA approach, ACE is used to rank allthe DMUs (efficient as well as inefficient). In the present study,we follow cross-efficiency technique to rank only the efficientDMUs and to increase the discrimination power of our proposed
ith undesirable fuzzy outputs and its application to the banking sector in.2014.04.013
Table 6Cross-efficiency matrix at level a 2 (0,1].
DMU 1 2 3 4 � � � N
1 E11a E12
a E13a E14
a� � � E1n
a2 E21
a E22a E23
a E24a
� � � E2na
..
. ... ..
. ... ..
. ... ..
.
n En1a En2
a En3a En4
a� � � Enn
a
6 J. Puri, S.P. Yadav / Expert Systems with Applications xxx (2014) xxx–xxx
ESWA 9282 No. of Pages 14, Model 5G
22 April 2014
models (Model-3 and Model-8) which is as discussed in Sections4.1 and 4.2.
4.1. Complete ranking of the efficient DMUs in Model-3
The basic idea of cross-efficiency evaluation can be explained intwo stages. In the first stage, the traditional DEA analysis is per-formed and the optimal weights are calculated for each DMU. Inour case, Model-3 is executed in the first stage and the optimalweights for each DMU are evaluated in this stage. In particular,
let ug�
k ;ub�
k ;v�k� �
be the optimal solution (weights) of the kth
DMU obtained from Model-3. Then the cross efficiency of DMUj,using the weights of DMUk, can be evaluated as
Ekj ¼Ps1
r¼1ug�
rk ygrj �
Ps2p¼1ub�
pkybpjPm
i¼1v�ikxij; j ¼ 1;2; . . . ; n;
At k = j, we have Ekk ¼ E�k.In the traditional cross-efficiency approach, all the DMUs (effi-
cient as well as inefficient) are ranked according to the decreasingvalues of ACE. However, in our approach, to get the complete rank-ing of the efficient DMUs, the ACEs corresponding to only the effi-cient DMUs are evaluated and are used as a ranking criterion forthe ranking of these DMUs in DEA-UO model. The following algo-rithm summarizes the concept of ranking.
4.1.1. Ranking algorithm for complete ranking of DMUs in Model-3
Step 1. Find the efficiency Ek of each DMU, and the weights v ik;ugrk
and ubpk ð8 i; r; pÞ using Model-3.
Step 2. Let In = {1,2,3, . . . ,n} be an index set. LetJ1 ¼ fDMUkjE�k ¼ 1; k 2 Ing and J2 ¼ fDMUkjE�k < 1; k 2 Ingbe the sets consisting of the efficient and inefficient DMUsrespectively.
Step 3. Let h0 = |J1|. If h0 > 1, go to Step 4. Otherwise, go to Step 6.Step 4. The cross efficiency of each DMUj using the weights of
DMUk obtained in Step 1, can be evaluated as
489490
492492
Table 5Cross-e
DMU
12
..
.
N
PleaseIndia.
Ekj ¼Ps1
r¼1ug�
rk ygrj �
Ps2p¼1ub�
pkybpjPm
i¼1v�ikxij; j; k 2 In: ð6Þ
493
494
495
496
497
498
499
500
501
502
503
Once the cross-efficiencies have been found using (6), we can con-struct the matrix called cross-efficiency matrix, shown in Table 5.
The ACE of an efficient DMUj, denoted by �EjðDMUj 2 J1Þ; isdefined by
Ej ¼1n
Xn
k¼1
Ekj; DMUj 2 J1 ð7Þ
Step 5. Rank the DMUs belonging to J1 according to the decreasingEjðDMUj 2 J1Þ. In this step, h0 efficient DMUs are rankedfrom rank 1 to rank h0.
Step 6. Rank the remaining , n � h0 DMUs according to thedecreasing Ej(DMUj 2 J2) by starting with the rank h0 + 1.
fficiency matrix.
1 2 3 4 � � � N
E11 E12 E13 E14 � � � E1n
E21 E22 E23 E24 � � � E2n
..
. ... ..
. ... ..
. ...
En1 En2 En3 En4 � � � Enn
cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wExpert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
4.2. Complete ranking of the efficient DMUs in Model-8
In order to achieve the complete ranking of the efficient DMUsin Model-8 at every a 2 (0,1], we pursue the following algorithm.
4.2.1. Ranking algorithm for complete ranking of DMUs in Model-8
Step 1. Find the efficiency Eka of each DMU, and the variables
�xij; �ygrj; �y
bpjð8i; r; p; jÞ;v ik;u
grk and ub
pk using Model-8 at givena 2 (0,1].
Step 2. Find the values of xij; ygrj and yb
pj8i; r; p; j at given a by using(1).
Step 3. Let In = {1, 2, 3, . . ., n} be an index set. LetJ1a ¼ fDMUkjEk�
a ¼ 1; k 2 Ing and J2a ¼ fDMUkjEk�
a < 1; k 2 Ingbe the sets consisting of the efficient and inefficient DMUsrespectively at given a.
Step 4. Let h ¼ jJ1aj: If h > 1, go to Step 5. Otherwise, go to Step 7.
Step 5. The cross efficiency of each DMUj at given a, using theweights of DMUk obtained in Step 1, can be evaluated as
ith und.2014.0
Ekja ¼
Ps1r¼1ug�
rk ygrj �
Ps2p¼1ub�
pkybpjPm
i¼1v�ikxij; j; k 2 In; a 2 ð0;1�: ð8Þ
Once the cross-efficiencies at given a have been found by using (8),we can construct the matrix called cross-efficiency matrix at levela 2 (0,1], shown in Table 6.The ACE at a 2 (0,1], i.e., (ACEa) of an efficient DMUj, denoted by�Eja ðDMUj 2 J1
aÞ; is defined by
Eja ¼
1n
Xn
k¼1
Ekja ; DMUj 2 J1
a; a 2 ð0;1�: ð9Þ
504
505
506
507
508
509
510
511
512
513
514
515
Step 6. Rank the DMUs belonging to J1a according to the decreasing
�EjaðDMUj 2 J1
aÞ at given a. In this step, h efficient DMUs areranked from rank 1 to rank h.
Step 7. Rank the remaining n � h DMUs according to the decreas-ing Ej
aðDMUj 2 J2aÞ at given a by starting with the rank
h + 1.
5. Numerical illustration
In order to get the deep insight of the proposed methodology,consider a performance assessment problem of 12 DMUs in termsof two fuzzy inputs, two desirable fuzzy outputs and one undesir-able fuzzy output. The entire input and output data are in terms ofpositive TFNs and are shown in Table 7. Since, the available real lifeinput–output data are always uncertain, therefore, the proposedapproach is more suitable in such situations than the crispapproach.
The values of Ek�
a and �Ek�a at different values of a 2 (0,1] are eval-
uated using Model-8 and Eq. (9) respectively, and the results areshown in Tables 8 and 9. The non-Archimedean infinitesimal e istaken to be 10�4. The efficiency results obtained at a = 1 are samewhen these are evaluated using crisp input–output data. Thegraphical representations of the efficiency results using crisp andfuzzy input–output data are shown in Figs. 1a and 1b respectively.
esirable fuzzy outputs and its application to the banking sector in4.013
Fig. 1b. Efficiency results with fuzzy data at different a 2 (0,1].
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8 9 10 11 12
Eff
icie
ncy
DMUs
Crisp input-output data
Fig. 1a. Efficiency results with crisp data.
J. Puri, S.P. Yadav / Expert Systems with Applications xxx (2014) xxx–xxx 7
ESWA 9282 No. of Pages 14, Model 5G
22 April 2014
Carefully observing Figs. 1a and 1b, we can analyze the impact offuzzy data on the efficiency results of the DMUs. Fig. 1b revealsthat the number of efficient DMUs are 8 at a 2 {0.1,0.2,0.3}, andthe number reduces to 7, 6, 5, 4 and 3 at a = 0.4, a 2 {0.5,0.6},a 2 {0.7,0.8}, a = 0.9 and a = 1 respectively. This implies that withthe variation in the satisfaction level a, the efficiency results ofalmost every DMU varies. Thus, the efficiency results obtained
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
using fuzzy input–output data are more realistic and robust ascompared to the results obtained using crisp data. Table 8 revealsthat DMU5, DMU6 and DMU8 are inefficient at everya 2 {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1}. The complete ranking
ith undesirable fuzzy outputs and its application to the banking sector in.2014.04.013
8 J. Puri, S.P. Yadav / Expert Systems with Applications xxx (2014) xxx–xxx
ESWA 9282 No. of Pages 14, Model 5G
22 April 2014
of the efficient DMUs at a 2 (0,1], using the ranking algorithm dis-cussed in Section 4.2, is presented in Table 10. It shows that rank-ing is different at different levels of a 2 (0,1]. Hence, we canconclude that the change in the satisfaction level a not only affectsthe efficiency results but also affects the rankings of the DMUs.
6. Application to the banking sector
The performance of banks has become a major concern of plan-ners and policy makers in India. The growth and financial stabilityof the country depends on the financial soundness of its financialinstitutions. Since early 1990s, the Indian banking sector hasnoticed various changes in the policies and prudential norms toraise the banking standards in India. Major changes took place inthe functioning of banks in India only after nationalization and lib-eralization. The problem of non-performing assets (NPAs) in Indianbanking sector was ignored for many years but recently it has beengiven considerable attention after liberalization of the banking sec-tor in India. The maintenance and control of NPAs (bad loans) arevery much needed for well functioning of the banking industry.Since credit is essential for economic growth in the country butNPAs beyond a certain level affect the smooth flow of credit andcredit creation. Apart from this, NPAs also affect profitability ofthe banks because higher NPAs require higher provisioning, whichmeans a large part of the profits is needed as a provision againstbad loans. Therefore, gauging the problem of NPAs has become amajor concern of the lenders as well as policy makers who areengaged in the economic growth of the country. In the presentstudy, we treat NPA as an undesirable output and apply the pro-posed methodology to evaluate the performance of public sectorbanks in India for the period 2009–2011.
6.1. Present structure of the Indian banking sector
Presently, the banking industry in India is a mixture of public,private and foreign ownerships. The present structure (Puri &
Reserve Bank o
Schedule
Commercial Banks
Scheduled Urban Co-op
Private Sector BanksPublic Sector Banks
Nationalized Banks SBI and its Associate Banks Old
Fig. 2. Structure of Indi
Table 10Ranks of the DMUs at different levels of a 2 (0,1].
Note: The rankings in bold represent the rankings of the efficient DMUs.
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
Yadav, 2013b) of Indian banking sector is given in Fig. 2. TheReserve Bank of India (RBI) is the central bank of the country andthe supreme monetary authority. Scheduled Banks are those bankswhich are listed on the second schedule of the RBI Act, 1934. Publicsector banks (PuSBs) are the commercial banks which acceptdeposits from their customers and grant loans and advances totheir clients. These include: (a) Nationalized banks (NBs), and (b)State Bank of India (SBI) and its associate banks (SBI group). Thepresent study is mainly focused on the performance assessmentof PuSBs for the period 2009–2011. The PuSBs are listed in Table11. The total numbers of PuSBs were 26 in 2009 and 2010, andbecame 25 in 2011. In 2011, State Bank of Indore was amalgam-ated with the SBI.
6.2. Data and variables
The choice of inputs and outputs to measure the bank efficiencyin DEA is a matter of long standing debate among researchers. Inliterature, two approaches (Humphrey, 1985; Puri & Yadav,2013b) are mostly followed for the selection of input–output vari-ables for a bank: (1) production approach and (2) Intermediationapproach. In the production approach, banks are assumed to usephysical inputs to produce outputs like deposits and loans. Onthe other hand, the intermediation approach considers banks asfinancial intermediaries. In this approach, the bank accepts depos-its from customers and transforms them into loans to clients.Therefore, the inputs used are labor, capital and deposits, and theoutputs produced are loans, investments or any other income gen-erating activities.
6.2.1. Selection of input and output variablesThe present study follows the intermediation approach for the
selection of input and output variables. The DMUs in our studyare PuSBs. The performance of each PuSB is evaluated in terms oftwo inputs, one desirable output and one undesirable output. Thetwo inputs used in our study are as follows:
Fig. 3. Bank group-wise NPA to total advances ratio (in %) during 2009–2011.
0.0
0.4
0.8
1.2
2009 2010 2011
%ag
e Sh
are
Years
Sub-Standard assetsDoubtful assetsLoss assets
Fig. 4. Classification of NPAs of all PuSBs during 2009–2011.
Table 11PuSBs during the period 2009–2011.
NBs NBs
S. No. BC Bank names S. No. BC Bank names
1 AlB Allahabad Bank 15 SB Syndicate Bank2 AnB Andhra Bank 16 UCOB UCO Bank3 BoB Bank of Baroda 17 UBoI Union Bank of India4 BoI Bank of India 18 UnBoI United Bank of India5 BoM Bank of Maharashtra 19 VB Vijaya Bank6 CaB Canara Bank SBI group7 CnBoI Central Bank of India S. No. BC Bank names8 CoB Corporation Bank 20 SBI State Bank of India9 DB Dena Bank 21 SBoBJ State Bank of Bikaner and Jaipur
10 IB Indian Bank 22 SBoH State Bank of Hyderabad11 IOB Indian Overseas Bank 23 SBoM State Bank of Mysore12 OBoC Oriental Bank of Commerce 24 SBoP State Bank of Patiala13 PSB Punjab and Sind Bank 25 SBoT State Bank of Travancore14 PNB Punjab National Bank 26 SBoI State Bank of Indore
Note: BC stands for Bank Code; In 2009–2010, PuSBs were from 1 to 26, and in 2011, these were from 1 to 25.
J. Puri, S.P. Yadav / Expert Systems with Applications xxx (2014) xxx–xxx 9
ESWA 9282 No. of Pages 14, Model 5G
22 April 2014
� Labor – it is the number of employees working in each PuSB.� Total deposits (TDs) – this is the sum total of demand deposits,
saving banks deposits and term deposits.
The one desirable output used in our study is as given below:
� Performing loans/assets (PAs) – these are the performingassets which are calculated by subtracting non-performingloans/assets from total advances.
The one undesirable output taken in the present study is thenon-performing loans/assets (NPAs).
� Non-performing loans/assets (NPAs) – these are the assets thatcease to generate income for banks. According to RBI, NPA is aloan/asset where interest and/or instalment of principal remainoverdue for a period of more than 90 days in respect of the termloan. According to RBI (2011b), banks should classify theirassets into the following broad groups, viz.
1. Standard assets – these assets are not considered as NPAs,but involve business risks. These require a minimum of0.25% provision on global portfolio and not on domesticportfolio.
2. Sub-standard assets – with effect from 31st March 2005, asub-standard asset would be one, which has remained NPAfor a period less than or equal to 12 months. In such cases,the current net worth of the borrower/guarantor or the cur-rent market value of the security charged is not enough toensure recovery of the dues to the banks in full. A generalprovision of 10% on total outstanding should be made with-out making any allowance for guarantee cover and securi-ties available.
3. Doubtful assets – with effect from 31st March 2005, anasset would be classified as doubtful if it has remained inthe substandard category for a period of 12 months. Inregard to the secured portion, provision may be made atthe rates ranging from 20% to 100% depending upon theperiod for which the asset has remained doubtful.
4. Loss assets – a loss asset is one where loss has been iden-tified by the bank or internal or external auditors or the RBIinspection but the amount has not been written off wholly.The provisions required are 100% of the outstanding bal-ance of the loan assets.
Standard assets are considered as PAs and the remaining cate-gories of assets such as sub-standard, doubtful and loss assets
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are regarded as NPAs. Fig. 3 presents the bank group-wise percent-age of NPA ratio which is calculated as a ratio (in percentage) ofNPA to total advances during the period 2009–2011. It reveals thatthere was an increase in NPA ratio of PuSBs from 1.74% in 2009 to2.08% in 2011. It also shows that the NPA ratio is higher in SBIgroup as compared to NBs during the selected period. The percent-age share of sub-standard assets, doubtful assets and loss assets inNPAs of PuSBs are shown in Fig. 4 for the selected period. It showsthat the percentage share of sub-standard assets increased from0.9% in 2009 to 1.1% in 2011, and the percentage share of doubtfulassets also increased from 0.98% in 2009 to 1.04% in 2011. On theother hand, the percentage share of loss assets remained 0.18%(approximately) from 2009 to 2011. It concludes that the majorrisk lies in sub-standard assets. So, more efforts should be madeby the PuSBs to restrain and recover the sub-standard assets.
6.2.2. Fuzzification of input and output dataIn the present study, the data are taken from RBI (2009) and RBI
(2011a). This input–output data of each bank are provisional andare available in crisp form. However, there always exist somedegree of uncertainty in the data which can be represented by
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FNs. In banks, uncertainty occurs due to the difference between theactual data and the available data. For example, let actual amountof TDs for a bank be 2345.65 crores and the possible data amountsof TDs available at different places be 2345 or 2345.6 or 2346crores. Then the difference of 0.65 or 0.05 or 0.35 crores results intothe occurrence of uncertainty in the data which further may affectthe efficiency results of the banks. Also there always exists a riskfactor behind loans. The current economy forces bank to pursuehigh risk loans in order to survive in the market and to competewith the other banks. The risk of the conversion of a loan into abad loan always persists in the market and in the minds of a bankmanager which directly affects the profit, goodwill and perfor-mance of that bank. Therefore, it is evident that there exist somedegree of uncertainty in PAs (loans) and NPAs (bad loans). Cer-tainly, crisp set theory is not appropriate to deal with such typeof problems/situations; rather fuzzy set theory (Zimmermann,1991) is more suitable. Therefore, in the present study, we fuzzifythe data as TFNs. One input (TDs), all desirable and undesirableoutputs are taken as TFNs. The collected crisp data from RBI is rep-resented by aM and the corresponding TFN is represented by (aL,aM,aR). Further, for input TDs, aL is calculated by taking the integerpart of aM and aR is calculated by adding 1 to aL; while for desirableoutput PAs and undesirable output NPAs, aL and aR are calculatedby subtracting 1% of aM from aM and by adding 0.001% of aM toaM respectively. Here aL and aR in PAs and NPAs are obtained aftera thorough discussion with the experts of the banks. The mainobjective of our study is to measure the efficiencies Ek�
a and �Ek�a of
all PuSBs at different levels of a 2 (0,1] for the period 2009–2011and to recognize the impact of undesirable output (NPAs) on theperformance of PuSBs along with the effect of uncertainty in thedata over the efficiency results.
6.3. Empirical results and discussion
In this section, the efficiencies Ek�
a for every DMU are obtainedby executing a MATLAB program of Model-8 at different levels of
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a for the period 2009–2011. The efficiency results ata 2 {0.1,0.5,0.7,1} are shown in Table 12 and at eacha 2 {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1} are shown graphically infigures from Figs. 5–7. Table 12 reveals that the two banks withbank codes AnB and CoB are overall efficient from 2009 to 2011and at each a 2 {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1}. It meansthat the undesirable output NPA does not show any bad impacton the performance of these banks. Also the percentage of NPAratio for these banks is not changing immensely with the year inthe selected period. So, these banks act as outliers/benchmarksfor the other PuSBs. Figs. 5–7 reveals that SBoT is efficient in2009 and 2011 for every a but inefficient for some a in 2010. Itimplies that even a small degree of uncertainty can make a bankeither efficient or inefficient. Overall we can depict from Figs. 5–7 that our proposed model has shown valuable results to analyzethe impact of undesirable output NPA on PuSBs for the selectedperiod and the efficiency results at different levels of a show theeffect of the presence of uncertainty into the data over the effi-ciency results. The present study depicts that the efficiency resultsof many PuSBs change with the change in the satisfaction levela 2 (0,1] during the selected period. Hence, the study producesvaluable results regarding the performance of PuSBs which furtherassists the bank experts to take appropriate decisions in order toenhance efficiencies of the banks.
6.3.1. Descriptive statistics for PuSBs, NBs and SBI groupThe descriptive statistics of the average efficiencies and average
inefficiencies at different satisfaction levels a 2 (0,1] for PuSBs as awhole and its categories (NBs and SBI group) during the period2009–2011 are shown in Table 13 and Fig. 8 respectively. Table13 reveals that at a = 0.1, the SBI group on average is 4.64%,7.16% and 6.1% more efficient in 2009, 2010 and 2011, respectivelythan the NBs. And at a = 1, the SBI group on average is 4.62%, 7.24%and 6.13% more efficient in 2009, 2010 and 2012, respectively thanthe NBs. Thus the average efficiency of SBI group is higher than NBsat each a in 2009–2011. Fig. 8 reveals that the average inefficiency
a of PuSBs at different levels of a during 2010–2011.
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for PuSBs ranges between 5.02% and 5.61% in 2009, 7.98–8.62% in2010, 8.65–9.38% in 2011 at different levels of a. The average inef-ficiency for NBs is high for every a as compared to SBI group in theselected period.
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Further, carefully observing Fig. 8, we can analyze that at a = 1,the average inefficiency increase in NBs is 3.71% in 2009–2010 andonly 0.28% in 2010–2011. However, the average inefficiencyincrease in SBI group is 1.09% in 2009–2010 and 1.39% in
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2010–2011 at a = 1. This implies that in 2011, there is moreincrease in the average inefficiency of SBI group banks than theNBs. It can also be verified from Fig. 3 which provides the percent-age increase or decrease of NPA ratio in the PuSBs, NBs and SBIgroup during the selected period of the study. It can be seen fromFig. 3 that the NPA ratio of SBI group is less than NBs in 2009 and isapproximately same in 2010. But the NPA ratio of SBI groupincreases immensely and exceeds NBs in 2011. This implies thatin 2011, the impact of NPAs at every a should be more on SBI groupas compared to NBs. Therefore, the proposed methodology showsthe robustness of the results. However, due to less average ineffi-ciency value in each year from 2009 to 2011, SBI group outperformNBs in the selected period.
6.3.2. Description of the efficiency results using NPA ratio ofPuSBs at a = 1
The bank-wise description of NPA ratio (NPA to Total Advancesratio in percentage) and efficiency Ek�
a at a = 1 of each PuSB during2009–2011 are shown in Fig. 9.
The dotted lines in Fig. 9 reveal that the pattern of increase ordecrease in efficiency of almost every PuSB is same as the patternof increase or decrease in NPA ratio during the selected period. Forexample, the NPA ratios of CnBoI are 2.67%, 2.32% and 1.82% in2009, 2010 and 2011 respectively. And, the efficiency values ofCnBoI at a = 1 are obtained to be 0.8336, 0.8451 and 0.9202 respec-tively. Similarly, the efficiency of IOB is 0.9625 in 2009 whichdecreases to 0.8175 in 2010 because the NPA ratio increases from
1
2
3
4
5
AlB
AnB Bo
BBo
IBo
MC
aBC
nBoI
CoB DB IB
IOB
OBo
C
0.7
0.8
0.9
1.0
NPA
rat
io (i
n %
age
)E
ffic
ienc
y
Pu
Fig. 9. Bank-wise NPA ratio (in%) and efficienc
Please cite this article in press as: Puri, J., & Yadav, S. P. A fuzzy DEA model wIndia. Expert Systems with Applications (2014), http://dx.doi.org/10.1016/j.eswa
2.54% in 2009 to 4.71% in 2010 and further efficiency increases to0.8847 in 2011 as the NPA ratio decreases to 2.71% in 2011. In asimilar way, the efficiency of UCOB also decreases from 0.8928 in2009 to 0.8015 in 2011 as the NPA ratio increases from 2.21% in2009 to 3.31% in 2011. Therefore, we can conclude from Fig. 9 thatthe results obtained from the proposed model are vigorous andvaluable to analyze the impact of undesirable output NPA on PuSBsfor the selected period.
6.3.3. Descriptive statistics of efficient and inefficient banks in PuSBs,NBs and SBI group
In the study, the number of efficient and inefficient bankschanges with the change in satisfaction level a 2 (0,1]. Table 14provides the number of PuSBs, NBs and SBI group banks whichare efficient and inefficient for each a 2 (0,1]. It reveals that in2009 only 30.77% PuSBs, 26.32% NBs and 42.85% SBI group banksare efficient at each a. Since, the NPA ratio increases from 1.74%,
PSB
PNB SB
UC
OB
UBo
IU
nBoI VB
SBI
SBoB
JSB
oHSB
oMSB
oPSB
oTSB
oI
200920102011
SBs
y Ek�
a at a = 1 of PuSBs during 2009–2011.
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Note: The rankings in bold represent the rankings of the efficient banks in the years 2009–2011.
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1.77% and 1.64% for PuSBs, Nb, and SBI group respectively in 2009to 2.08%, 1.99% and 2.37% for PuSBs, NBs and SBI group respectivelyin 2011. Therefore, the percentage of efficient banks decreases to12%, 10.53% and 16.7% for PuSBs, NBs and SBI group respectivelyin 2011. It implies that increase in NPA ratio results into thedecrease in the number of efficient banks.
810
811
812
813
814
815
816
817
818
6.3.4. Ranking of PuSBs at different levels of aThe values of ACEa, i.e., Ek�
a for each efficient DMU ata 2 {0.1,0.5,0.7,1} are evaluated as discussed in Section 4.2 andare listed in Table 15. Further, these ACEa scores are used to obtaincomplete ranking of PuSBs during the selected period which isshown in Table 16. It reveals that the ranking of the banks differsin different levels of a.
819
820
821
822
823
824
825
826
827
828
829
830
831832
833
834
835
7. Conclusions
In this paper, we have firstly explored a shortcoming in theexisting DEA model in which undesirable outputs are incorporateddirectly into the model but with negative weights. This modelleads to the negative cross-efficiencies for some DMUs. However,efficiencies as well as cross-efficiencies can never be negative inreal applications. Thus, to overcome this shortcoming in the pres-ent study, we have developed a new DEA model known as DEA-UOmodel which results into the positive cross-efficiencies of all theDMUs. Further, DEA-UO model is extended to fuzzy environmentwhich is named as FDEA-UFO model. We have used the methodol-ogy suggested by Saati et al. (2002) for transferring the FDEA-UFOmodel to Model-8 which is a crisp model. The optimum objectivefunction value of Model-8 at any a 2 (0,1] is the final efficiencyscore of the kth DMU at that a. Moreover, ranking algorithms based
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on cross-efficiency technique are presented to increase thediscrimination power of the proposed DEA-UO and FDEA-UFOmodels. This FDEA-UFO methodology is further utilized to analyzethe performance of banking sector in India for the period2009–2011. This is the first study to analyze the performance ofPuSBs in India and their categories, namely, NBs and SBI groupwith fuzzy input–output data. The present study has also exploredthe impact of fuzzy input–output data over the efficiency results ofthe PuSBs in India, and the effect of undesirable output NPAs on theperformance of each PuSB.
The practical implication of the present study is that the resultsobtained from the proposed methodology are quite robust andeffective to recognize the impact of NPA on the performance ofPuSBs in India at different levels of a 2 (0,1] along with the effectof the uncertainty present in the input–output data over the effi-ciency results. Therefore, the findings of the current study areenormously valuable for the bank experts and policy makers toidentify the average inefficiencies present in each bank group,and to form appropriate policies and take right decisions in orderto improve the performance and growth of PuSBs and their respec-tive categories. Finally, we present some efficiency results whichwill be highly useful for policy makers to improve the performanceof PuSBs and their categories. The results of our study indicate that(i) two banks AnB and CoB are overall efficient from 2009 to 2011and at each a 2 (0,1] which implies that these banks act as outli-ers/benchmarks for the other PuSBs, (ii) the efficiency results ofmany PuSBs changes with the change in the satisfaction levela 2 (0,1] during the selected period, (iii) the average efficiency ofSBI group is higher than the average efficiency of NBs at each ain 2009–2011. However, in 2011, the average inefficiency increaseis more in SBI group than in NBs. This implies that in 2011, theimpact of NPAs at every a is more on SBI group as compared to
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NBs. But due to less average inefficiency value of SBI group banks,SBI group outperform NBs in the selected period, (iv) the pattern ofincrease (decrease) in efficiency of almost every PuSB is same asthe pattern of increase (decrease) in NPA ratio for the selectedperiod. This validates that the results obtained from the proposedmodel are efficient to analyze the impact of undesirable outputNPA on PuSBs, (v) the number of efficient and inefficient bankschanges with change in satisfaction level a 2 (0,1], (vi) increasein NPA ratio results into the decrease in the number of efficientbanks during 2009–2011, and (vii) the complete ranking of PuSBsreveals that the ranking of PuSBs also varies with the variation in a.
The future scope of the present study is very wide. The futureresearch can be extended in various directions. We plan to extendour methodology to interval and ordinal data. The study can alsobe extended to the comparative study of public, private and foreignbanks in India. Furthermore, different input–output variables inthe banking industry which possess some degree of uncertaintiescan also be analyzed.
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Acknowledgements
The authors are thankful to the reviewers for their fruitfulcomments and suggestions. The first author is also thankful tothe University Grants Commission (UGC), Government of India,New Delhi for financial assistance.
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