DEVELOPMENT OF DEA IN UNCERTAIN ENVIRONMENTS WITH EFFICIENCY MEASUREMENT OF INDIAN BANKING SECTOR

Abstract

Since last 30 years, Data Envelopment Analysis (DEA) (Charnes et al., 1978) has become a powerful analytical and quantitative tool for measuring the performance of wide range of similar organizations. It has been applied to a large number of different types of profit/non-profit units that are engaged in a wide variety of activities worldwide. It is a dataoriented approach for measuring the performance of a set of homogeneous peer entities called Decision-Making Units (DMUs), which utilize multiple inputs to produce multiple outputs. Numerous approaches for evaluating different efficiency measures have been developed in DEA (Cooper et al., 2007). The advantage of DEA over the other performance measuring techniques is that it does not require any prior assumptions concerning the shape of the production frontier. DEA mainly focuses on achieving a single measure of production efficiency for each DMU. However, in some real life applications, a DMU may perform different types of functions and thus a DMU can be separated into different independent or interdependent components or decision making sub-units (DMSUs). The study of a DMU with its components is known as multi-component DEA (MC-DEA) (Amirteimoori and Kordrostami, 2005b). In this approach, independent components constitute the parallel structure whereas interdependent components constitute the series structure of the DMSUs. Despite of several advantages, DEA and MC-DEA have some limitations. The biggest problem is the sensitivity of these approaches to input-output data. Since these approaches are based on frontiers, a very small change in input/output data can change the efficient frontiers significantly. Thus, for successful application of DEA and MC-DEA, there is a need of accurate measurements of the data values of inputs and outputs. However, in some real life performance assessment problems, inputs used and outputs produced in a production process are sometimes volatile and complex, and thus, involve difficulty while measuring the accurate data in precise form. Moreover, in many real life applications, the input-output data may not always be available in precise form. The data might be uncertain, i.e., data may include some degrees of uncertainty or imprecision which cannot be defined precisely. For better understanding of the above mentioned problem, take a real life situation of the banking sector in India. The input and output variables of banks/branches may possess uncertainties of interval or fuzzy or intuitionistic fuzzy forms as discussed below: i. Granting a loan is a risky output because of the risk for a loan to eventually become a bad (non-performing) loan. Bad loans directly affect the asset quality, credit creation, profitability and performance of a bank. It is difficult to exactly measure how many iv normal loans might become bad and how many bad loans might become normal. Thus, normal and non-performing loans and their respective prices possess fuzzy behaviour in real situations. ii. The income from investment of a bank varies daily on account of its market value which changes daily. Thus, income from investments also possesses fuzzy characteristics. iii. The demand (more or less) for labour mainly depends on the factors like work efficiency, work load, desire for work, skills and training etc., of existing labours. The other aspects are prevailing market conditions and area of operation (urban/semiurban/ rural/industrial). The above mentioned factors lead to the fuzziness in the labour data. iv. In real situations, fluctuations in operating expenses, in particular, uncontrollable expenses occur due to factors like temporary shutdown of branch’s operations due to some unavoidable reasons, large number of bad loans, inflation, managerial inefficiency etc. Thus, vagueness in operating expenses possess fuzzy or intuitionistic fuzzy behaviour. v. A higher bank management would be more interested in running a bank branch with less employees in order to reduce its labour cost, whereas a branch manager may be interested in having more employees at the disposal of the branch in order to accommodate more customers, augment branch’s profit and handle day-to-day increased workloads. That is, number of employees is likely to be an undesirable attribute for the bank management, however a desirable attribute for the branch manager. Thus, the difference of thought at management level and branch level may lead to the existence of hesitation in the demand and availability of employees at branch level. Hence, labour possesses intuitionistic fuzzy behaviour (Atanassov, 1986) at branch level. Certainly, crisp set theory is not appropriate to deal with such types of problems; rather fuzzy set theory (Zadeh, 1965) and theory of interval numbers are more suitable. Thus, the uncertainty in the input-output data of banks can be well represented by fuzzy numbers (Zimmermann, 1996) or intuitionistic fuzzy numbers (Atanassov, 1986) or interval numbers. Therefore, in this study, we have represented the input-output variables of banks/branches in uncertain forms for analysing different efficiency measures in uncertain environments and to evaluate banks’ efficiency more realistically and accurately. The financial institutions like banks play a significant role in the economic development and growth of a country. Therefore, in recent times, the performance of banks has become a v major concern of planners and policy makers in India. The growth and financial stability of the country mainly depends on the financial soundness of its financial institutions. Since early 1990s, the Indian banking sector has noticed various changes in the policies and prudential norms to raise the banking standards in India. In this regard, major changes took place in the functioning of banks in India only after nationalization and liberalization. The problem of NPAs in Indian banking sector was ignored for many years but recently it has been given considerable attention by the bank experts and researchers after the post reform period of the banking sector in India. The maintenance, control and recovery of NPAs (bad loans) are very much needed for well functioning of the banks in India. Since credit is essential for economic growth in the country but NPAs beyond a certain level affect the smooth flow of credit and credit creation. Apart from this, NPAs also affect profitability of the banks because higher NPAs require higher provisioning, which means a large part of the profits is needed as a provision against NPAs. Therefore, gauging the problem of NPAs has become a major concern of the bank experts as well as policy makers who are engaged in the economic growth of the country. Therefore, in this study, we have treated NPA as an undesirable output and applied the proposed DEA methodologies in uncertain environments to the Indian banking sector. The banks that are considered in this study are public sector banks (PuSBs) and private sector banks (PrSBs), and the period of study ranges from 2008 to 2013. Finally, this study is motivated by the following observations: (1) a greater need to extend DEA and MC-DEA to fuzzy and fully fuzzy environments in order to accommodate real life applications more realistically, (2) a need to incorporate undesirable factors into the production processes of DEA and MC-DEA in crisp as well as uncertain environments, (3) a need to pay attention towards measuring the DEA and MC-DEA performance of Indian banking sector in the presence of undesirable outputs, and (4) a dearth of studies of banking sector in India for measuring different efficiency measures like technical, mix, multicomponent, cost, revenue, optimistic and pessimistic efficiency measures in uncertain environments. The chapter-wise summary of the thesis is as follows: Chapter 1 is introductory in nature. This chapter includes some basic definitions, notions and operations in fuzzy set theory (Zimmermann, 1996), intuitionistic fuzzy set theory (Atanassov, 1986) and uncertain/imprecise data (Cooper et al., 2001) forms used throughout the work. It also presents brief review of the work done in the areas of DEA, MC-DEA, FDEA and bank efficiency in India. It also presents the different approaches available in literature for vi the selection of input and output data variables for bank efficiency. It includes the input and output variables selected in the present study. Chapter 2 includes the comparative analysis of two important categories of banks in India, namely, PuSBs and PrSBs, and their respective categories. This chapter seeks to measure the overall technical, pure technical and scale efficiencies of PuSBs and PrSBs for the year 2009-2010 using conventional DEA methodology (Cooper et al., 2007). It determines the bank group-wise, category-wise and size-wise variations in the efficiency scores of all the banks. It also examines the returns-to-scale, input/output slacks and benchmarks for relatively inefficient banks. Further, sensitivity analysis is performed to ensure the validity and robustness of the efficiency results. The key findings and some concluding remarks and suggestions for the policy makers and bank experts are also provided to improve the performance of all the categories of PuSBs and PrSBs in India. Chapter 3 includes the multi-component performance analysis of PuSBs in India and their categories in terms of two components such that the first component deals with the productivity whereas the second component deals with the profitability. This chapter aims to measure the aggregate performance and component-wise performance for every PuSBs and their categories for the period 2008-2013 using the conventional MC-DEA methodology (Amirteimoori and Kordrostami, 2005b), and to analyse the effect of productivity and profitability on aggregate performance of each PuSB. It also examines the average inefficiencies present in each component. Further, sensitivity analysis is performed to validate the use of MC-DEA instead of DEA and to ensure the robustness of the efficiency results. The findings are quite useful for planners and policy makers in order to identify efficiencies and weaknesses present in either productivity phase or profitability phase or both, and to suggest some directions for improvement in each component of PuSBs. In Chapter 4, input- and output-oriented CCR and SBM DEA models (Cooper et al., 2007) are extended to fuzzy environments in which input-output data are taken in the form of fuzzy numbers, in particular, triangular fuzzy numbers (TFNs) (Zimmermann, 1996). The proposed input- and output-oriented fuzzy CCR and fuzzy SBM models are then transformed into the family of crisp DEA models by using the α – cut approach (Kao and Liu, 2000a) for measuring the fuzzy CCR and fuzzy SBM input and output efficiencies in fuzzy DEA (FDEA), where α represents the satisfaction level of the decision maker. Further, the concept of mixefficiency is extended to fuzzy environment for measuring the fuzzy input and output mixefficiencies of the DMUs in FDEA. The ranking of DMUs on the basis of the different efficiency measures are obtained by using defuzzification method (Kataria, 2010). A new vii ranking method is developed to rank the DMUs on the basis of fuzzy input mix-efficiency. Numerical illustration is provided to ensure the validity of the proposed approach. An application of the proposed approach to the State Bank of Patiala in the Punjab State of India with districts as the DMUs for the period 2010-2011 is presented to ensure its acceptability in real situations. Chapter 5 endeavours to propose a DEA model in the presence of undesirable outputs and further to extend it in fuzzy environment in view of the fact that precise input/output data may not always be available in real life problems. In this chapter, a FDEA model with undesirable fuzzy outputs is developed which is further transformed into a crisp linear program by using an α – cut approach of Saati et al. (2002). To increase the discrimination power of the proposed models, ranking algorithms based on cross-efficiency technique (Adler et al., 2002) are presented. To ensure the validity and effectiveness of the proposed FDEA methodology, a numerical illustration and an application to PuSBs in India for the period 2009-2011 are presented. The practical implication of this chapter is that the results obtained from the proposed FDEA approach are quite robust and effective to recognize the impact of undesirable output (NPA) on the performance of PuSBs in India for different values of  (0,1] along with the effect of the uncertainty present in the input-output data over the efficiency results. The findings are enormously valuable for the bank experts and policy makers to identify the average inefficiencies in PuSBs at different α – levels, and to suggest directions for their improvements. In Chapter 6, we have incorporated the undesirable outputs into the production technology of DEA, and have represented all the input-output data in uncertain/imprecise forms like intervals or ordinal relations or fuzzy numbers to reflect inherent uncertainty of real applications. This chapter involves the development of the new improved DEA models in the presence of undesirable outputs and interval data to find interval efficiencies of the DMUs based on interval arithmetic and unified production frontier. For incorporation of uncertain data of ordinal and fuzzy forms in the new DEA models with interval data, we use their interval estimations (Aliakbarpoor and Izadikhah, 2012; Azizi, 2014; Wang et al., 2005). The new models measure the final efficiency of each DMU as (i) an interval bounded by the best lower and the best upper bound efficiencies for interval and ordinal data, and (ii) a fuzzy number for fuzzy data having α – cuts as the intervals. Moreover, comparison with the existing approaches of Farzipoor Saen (2010) and Aliakbarpoor and Izadikhah (2012) show that the new improved models are theoretically accurate, numerically efficient and measure less number of DMUs as efficient than the existing models. In addition, some numerical examples with different data viii sets and an application to PuSBs and PrSBs in India for the period 2012-2013 are presented to validate the acceptability of the improved models. In Chapter 7, owing to the importance of internal structure of the DMUs in practical applications, we have proposed MC-DEA approach with uncertain data in the presence of undesirable outputs and shared resources. The intervals and ordinal relations in the study represent the uncertain data forms. To solve the MC-DEA model with uncertain data, we have developed a new common set of weights methodology by using interval arithmetic and unified production frontier. The new approach evaluates a common set of weights to measure interval efficiencies of the DMUs as well as their components. The final aggregate efficiency interval for each DMU is bounded by the best lower and best upper bound efficiencies. Numerical example is presented to validate the effectiveness of the proposed approach. Compared with the existing approaches of Amirteimoori and Kordrostami (2005a) and Ashrafi and Jaafar (2011), the results by the proposed approach are quite robust and effective. Moreover, we have presented an application of the proposed approach to PuSBs in India for the period 2011-2013 in order to ensure its acceptability in real life applications. This is the first study in Indian context to investigate the performance of each PuSB in terms of two distinct components, namely, general business and bancassurance business. The findings of the present study are valuable for the bank experts to identify weaknesses associated with aggregate performance and components’ performance of each PuSB, and to provide feasible improvement measures for their growth. In Chapter 8, we have extended the classical cost efficiency (CE) and revenue efficiency (RE) models (Cooper et al., 2007) to fully fuzzy environments in order to accommodate the real life situations, where input-output data and their corresponding prices are not known precisely. Owing to the importance of the existence of undesirable outputs, we have also incorporated these into the production technologies of the proposed models. This chapter endeavours to propose fully fuzzy CE (FFCE) and fully fuzzy RE (FFRE) models, where inputoutput data and prices include uncertainty of fuzzy forms, in particular, of triangular membership forms. Further, the concepts of fully fuzzy linear programming problems (FFLPPs) (Nasseri et al., 2013) and linear ranking function (Maleki, 2002) are employed to transform FFCE and FFRE models into the crisp LPPs, and to assess fuzzy CE and fuzzy RE measures of every DMU as TFNs in fully fuzzy DEA (FFDEA). The proposed models in FFDEA are then exemplified with an application to PuSBs in India for the period 2011-2013 in order to present their acceptability and effectiveness in real world systems. As per the consideration of fully fuzzy situations in this chapter, the findings of the study provide ix additional information to the bank experts that will further assist them to deal with uncertainties of real life problems and to do healthy improvements with the objectives of cost minimization and revenue maximization in PuSBs of India. In Chapter 9, we have extended FDEA to intuitionistic fuzzy DEA (IFDEA) to accommodate inputs and outputs of subjective, linguistic and vague forms possessing intuitionistic fuzzy essence (Atanassov, 1986) instead of fuzziness in real life applications. This extension involves the proposal of IFDEA models to measure optimistic and pessimistic efficiencies in intuitionistic fuzzy environments with inputs and outputs represented by triangular intuitionistic fuzzy numbers (Mahapatra and Roy, 2009). We have developed algorithms based on super-efficiency technique (Andersen and Petersen, 1993) to obtain complete ranking of the DMUs on individual optimistic and pessimistic situations. Further, we have proposed two alternate ranking methods based on the levels of inefficiencies and efficiencies respectively to achieve complete ranking of the DMUs when both the optimistic and pessimistic situations are considered simultaneously. We have also designed a hybrid IFDEA performance decision model for true decision process. To ensure the validity of the proposed IFDEA methodology and ranking methods, we have presented some numerical examples with different input-output data sets. Further, we have compared our ranking results with an existing ranking approach based on geometric average efficiency index (Wang et al., 2007). To validate the acceptability of the proposed IFDEA approach, we have presented its application to the branches of State Bank of Patiala in Amritsar district of the Punjab State in India for the period 2010-2011. In last chapter 10, conclusions are drawn and future extensions of the research work are suggested. This chapter also includes a summary of findings, conclusions and recommendations for the policy considerations along with some suggestions for improvements.