Jolly Puri

A concept of fuzzy input mix-efficiency in fuzzy DEA and its application in banking sector

Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiency of homogeneous decision making units (DMUs) on the basis of multiple inputs and multiple outputs. There exist radial and non-radial models in DEA. Radial models only deal with proportional changes of inputs/outputs and neglect the input/output slacks. On the other hand, non-radial models directly deal with the input/output slacks. The slack-based measure (SBM) model is a non-radial model in which the SBM efficiency can be decomposed into radial, scale and mix-efficiency. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure requires crisp data which may not always be available in real world applications. In real world problems, data may be imprecise or fuzzy. In this paper, we propose (i) a concept of fuzzy input mix-efficiency and evaluate the fuzzy input mix-efficiency using @a - cut approach, (ii) a fuzzy correlation coefficient method using expected value approach which calculates the expected intervals and expected values of fuzzy correlation coefficients between fuzzy inputs and fuzzy outputs, and (iii) a new method for ranking the DMUs on the basis of fuzzy input mix-efficiency. The proposed approaches are then applied to the State Bank of Patiala in the Punjab state of India with districts as the DMUs.

Intuitionistic fuzzy data envelopment analysis

Extend FDEA to intuitionistic FDEA (IFDEA) for handling intuitionistic fuzzy data.Develop IFDEA models and ranking methods for optimistic and pessimistic situations.Design a hybrid performance decision process in IFDEA to study overall performance.Present numerical examples and comparison with existing approach in DEA/IFDEA.Present application to the bank branches in India with intuitionistic fuzzy inputs. Intuitionistic fuzzy set (IFS) is an extension of fuzzy set and an approach to define a fuzzy set where available information is not sufficient to define an imprecise concept by means of a conventional fuzzy set. The existing fuzzy DEA (FDEA) models for measuring relative fuzzy efficiencies of decision making units (DMUs) are limited to fuzzy input/output data. However, in real life applications, some inputs and outputs of subjective, linguistic and vague forms may possess intuitionistic fuzzy essence instead of fuzziness. Therefore, in the present study, we extend FDEA to intuitionistic fuzzy DEA (IFDEA) in which the input/output data are represented by intuitionistic fuzzy numbers (IFNs), in particular triangular IFNs (TIFNs). This is the first study in analysing optimistic and pessimistic efficiencies with intuitionistic fuzzy input/output data in DEA. In this study, we develop models to measure optimistic and pessimistic efficiencies of each DMU in intuitionistic fuzzy environment (IFE). By using super-efficiency technique, we develop algorithms to obtain the complete ranking of the DMUs when optimistic and pessimistic situations are considered separately. Further, to rank the DMUs when both optimistic and pessimistic situations are taken simultaneously as hybrid approach, we propose two alternate ranking methods based on levels of inefficiencies and efficiencies respectively. To address the overall performance using optimistic and pessimistic situations together in IFEs, we propose a hybrid IFDEA performance decision model. To validate the proposed methodology and proposed ranking methods, we illustrate different numerical examples and then compare the results with an existing ranking approach based on geometric average efficiency index. Moreover, we present an application of the proposed approach to the banking sector in which two inputs, namely, labour and operating expenses possess intuitionistic fuzzy essence at branch level, and are represented as TIFNs.

A fully fuzzy approach to DEA and multi-component DEA for measuring fuzzy technical efficiencies in the presence of undesirable outputs

DEA is a non-parametric technique for measuring the relative technical efficiencies of similar decision making units (DMUs) with multiple inputs and multiple outputs. In smoe real life situations, DMUs may perform different types of functions and can be segregated into independent components such that each component has its own set of inputs and outputs. Owing the importance of the internal structure of the DMUs, DEA has been extended to multi-component DEA (MC-DEA) which is a technique for measuring the technical efficiencies of the DMUs and their components. The conventional DEA and MC-DEA approaches require crisp input and output data which may not always be available precisely. However, in real life problems, data might be imprecise or vague which can suitably be represented by fuzzy numbers. Many researchers have proposed methods to deal with fuzzy parameters in DEA and MC-DEA. However, in real situations, the decision variables can also take fuzzy forms. Therefore, the aim of the present paper is fourfold: (i) to extend DEA in the presence of undesirable outputs to fully fuzzy DEA (FFDEA) for measuring the fuzzy technical efficiencies of the DMUs in fully fuzzy environments where all decision variables and parameters are taken as fuzzy numbers, in particular triangular fuzzy numbers, (ii) to extend MC-DEA in the presence of undesirable outputs to multi-component fully fuzzy DEA (MC-FFDEA) for measuring the fuzzy technical efficiencies of DMUs and their components in fully fuzzy environments, (iii) to propose a new ranking function approach to transform both the FFDEA and MC-FFDEA models into crisp linear programming problems, and (iv) to present numerical examples in order to validate the effectiveness and advantages of the proposed approach over the existing ones.

A Fully Fuzzy DEA Approach for Cost and Revenue Efficiency Measurements in the Presence of Undesirable Outputs and Its Application to the Banking Sector in India

This paper extends the conventional cost efficiency (CE) and revenue efficiency (RE) models to fully fuzzy environments to account for real situations where input–output data and their corresponding prices are not known precisely. Owing to the importance of the presence of undesirable outputs in the production process, these are also incorporated into the production technologies of the proposed models. This paper endeavours to propose fully fuzzy CE (FFCE) and fully fuzzy RE (FFRE) models where input–output data and prices include uncertainty of fuzzy forms, in particular, of triangular membership forms. Further, the concepts of fully fuzzy linear programming problems (FFLPPs) and linear ranking functions are used to transform FFCE and FFRE models into the crisp linear programming problems (LPPs), and to evaluate fuzzy CE (FCE) and fuzzy RE (FRE) measures of the decision-making units as triangular fuzzy numbers. Moreover, the proposed models are compared with some existing approaches and are also illustrated with an application to the banking sector in India for proving their acceptability and effectiveness in real-world systems.

A new multi-component DEA approach using common set of weights methodology and imprecise data: an application to public sector banks in India with undesirable and shared resources

Owing to the importance of internal structure of decision making units (DMUs) and data uncertainties in real situations, the present paper focuses on multi-component data envelopment analysis (MC-DEA) approach with imprecise data. The undesirable outputs and shared resources are also incorporated in the production process of multi-component DMUs to validate real problems. The interval efficiencies of DMUs and their components in MC-DEA are often challenging with imprecise data. In many practical situations, different set of weights may be resulted into valid efficiency intervals for DMUs but invalid interval efficiencies for their components. Therefore, the present study proposes a new common set of weights methodology, based on interval arithmetic and unified production frontier, to determine unique weights for measuring these interval efficiencies. It is a two-level mathematical programming approach that preserves linearity of DEA and exhibits stronger discrimination power among the DMUs as compared to some existing approaches. Theoretically, the aggregate efficiency interval of each DMU lies between the components’ interval efficiencies. Further, the proposed approach is also applied to banks in India for proving its acceptability in practical applications. The performance of each bank is investigated in terms of two components: general business and bancassurance business for the years 2011–2013. The present study emphasized expanding pattern of bancassurance business in current market scenario with more percentage increase as contrasted to general business.

Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application

Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiencies of a homogeneous set of decision making units (DMUs) which utilize multiple inputs to produce multiple outputs. It consists of two types of DEA models: radial models and non-radial models. A radial model deals only with proportional changes of inputs/outputs and neglects the input/output slacks whereas a non-radial model deals directly with the input/output slacks. The slack based measure (SBM) model is a non-radial model that results into the SBM efficiency which can be further decomposed into radial, scale and mix-efficiency. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure is limited to crisp input and output data which may not always be available in real life applications. However, in real life problems, data may be imprecise or fuzzy. In this chapter, we extend the idea of mix-efficiency to fuzzy environments and develop a concept of fuzzy mix-efficiency in fuzzy DEA. We provide both the input and output orientations of fuzzy mix-efficiency. The α-cut approach is used to evaluate the fuzzy input as well as fuzzy output mix-efficiencies of each DMU. Further, a new method is provided for ranking the DMUs on the basis of fuzzy input and output mix-efficiencies. Moreover, to ensure the validity of the proposed methodology, we illustrate a numerical example and applied the proposed methodology to the banking sector in India.

Performance evaluation of public and private sector banks in India using DEA approach

This paper seeks to measure the OTE, PTE and SE of PuSBs and PrSBs for the year 2009 to 2010 using DEA. The findings show that: 1) PuSBs outperformed PrSBs in all categories of efficiencies; 2) the contribution of scale inefficiency in overall technical inefficiency has been observed to be smaller than the contribution of pure technical inefficiency; 3) in PuSBs, State Bank of India (SBI) % its Associates outperformed nationalised banks and in PrSBs, new PrSBs outperformed old PrSBs; 4) the highest and lowest levels of average overall technical inefficiency have been seen for old PrSBs (48.8%) and SBI % its Associates (2.2%), respectively; 5) the results of the present study, by sensitivity analysis, are quite robust to discriminate between efficient and inefficient banks.

A Dual SBM Model with Fuzzy Weights in Fuzzy DEA

The dual part of a SBM model in data envelopment analysis (DEA) aims to calculate the optimal virtual costs and prices (also known as weights) of inputs and outputs for the concerned decision-making units (DMUs). In conventional dual SBM model, the weights are found as crisp quantities. However, in real-world problems, the weights of inputs and outputs in DEA may have fuzzy essence. In this paper, we propose a dual SBM model with fuzzy weights for input and output data. The proposed model is then reduced to a crisp linear programming problem by using ranking function of a fuzzy number (FN). This model gives the fuzzy efficiencies and the fuzzy weights of inputs and outputs of the concerned DMUs as triangular fuzzy numbers (TFNs). The proposed model is illustrated with a numerical example.