DEVELOPMENT OF DEA MODELS IN CERTAIN AND UNCERTAIN ENVIRONMENTS WITH APPLICATIONS TO MGNREGA

Abstract

In recent years, Data Envelopment Analysis (DEA) has been widely applied to evaluate the performance of a diverse range of entities engaged in various activities in multiple contexts in many countries. The popularity of DEA is due to its ability to handle cases that are challenging for other methods, often resulting from the complex and unknown relationship between multiple inputs and outputs, which are frequently reported in non-comparable units. Examples include maintenance operations of U.S. Air Force bases at various locations (Bowlin (1987)), comparison of banks and their branches (Puri & Yadav (2013a)), education institutes like universities and schools (A. P. Singh, Yadav, & Tyagi (2022)), health care centres in India and other countries (Arya & Yadav (2018)) and transport sectors (Agarwal et al. (2010)) etc.. The technique has successfully provided an alternative location for shifting the capital city of Japan (away from Tokyo) (Takamura & Tone (2003)). DEA employs linear mathematical programming techniques that can handle a substantial number of variables and constraints, which relaxes the limitations that are often encountered when only a limited number of inputs and outputs can be selected. The technique of DEA depends entirely on the data of inputs and outputs used. So, it becomes necessary for the decision-maker to use precise data as the results are very sensitive. However, in real world, it is impossible to collect precise data. Indeterminacy is always present to some degree in the data. The issue of indeterminacy was addressed by the introduction of fuzzy sets by Zadeh et al. (1996). In traditional crisp set theory, an element is either considered part of the set or its compliment, as determined by its characteristic function. In contrast, the fuzzy set (FS) theory allows the decision maker to decide the degree of belongingness to be assigned to an element, which ranges from 0 to 1. The degree of non-membership is difference of 1 and degree of membership. FSs also have the ability to accurately represent linguistic variables mathematically. Some linguistic variables may prove to be significant inputs or outputs for a company. The membership function can take on any form and is typically represented as a continuous function that is triangular or trapezoidal in shape. These are referred to as triangular or trapezoidal fuzzy sets. FSs can also be extended to intuitionistic fuzzy sets (IFS), which were introduced by Atanassov (K. Atanassov (1988)). These sets take into account the hesitation or uncertainty of the decision-maker in their representation. In IFSs, the sum of the degree of membership and non-membership of an element lies within the range of 0 to 1. IFSs have a wide variety of applications in medical diagnosis (Ghafari Someh et al. (2020)), career determination (P. A. Ejegwa (2019)), pattern recognition (Dengfeng & Chuntian (2002)), Clustering algorithm (Kaur et al. (2013)), multi-attribute in formation classification (P. Singh et al. (2020)), banking (Puri & Yadav (2013b)), risk assessment methodology (Chang & Cheng (2010)), health sector (Arya & Yadav (2018)), transportation problem (S. K. Singh & Yadav (2015)) and many more. The studies in this field have shown that the applications of FS and IFS in DEA can provide more accurate and reliable results compared to the traditional crisp DEA approach. Moreover, the integration of these methods has opened up new avenues for research in areas such as multi-objective optimization, environmental efficiency and sustainability analysis. Despite the progress made in this field, there is still much to be explored and developed. For example, further research is needed to address the limitations of the current fuzzy DEA (FDEA) models and to develop more robust and efficient algorithms for solving the optimization problems associated with these models. Additionally, there is a need to explore the applications of FDEA and intuitionistic FDEA (IFDEA) in new domains and to develop new models for dealing with more complex and challenging real-world problems. The use of MGNREGA as a case study for exploring the concepts and models of DEA in crisp and uncertain environment provides a practical and meaningful context for the research. The scheme has been widely implemented in India and has had a significant impact on the livelihoods of millions of people, making it an ideal choice for investigating its efficiency. The analysis of the MGNREGA scheme using DEA and IFDEA can provide valuable insights into the effectiveness of the program in achieving its goals and the areas where improvements can be made. For example, the analysis can assess the efficiency of different States and Union Territories in providing employment within the mandated 15 days. Additionally, the IFS can take into account the uncertainty and vagueness in the data, allowing for a more accurate evaluation of the efficiency of the program. This is particularly important in the context of MGNREGA, where the data may be subject to measurement errors or may not reflect the true conditions on the ground. Hence the use of MGNREGA as a case study for exploring the concepts and models in DEA and IFS theory provides a practical and relevant context for the research and has the potential to provide valuable insights into the efficiency of government programs in providing employment and improving livelihoods. The motivation behind the thesis is to explore and expand our understanding of the efficiency of decision making units (DMUs) in different environments, taking into account the presence of undesirable outputs, uncertainty and vagueness in the data. The first motivation is to examine the efficiency of DMUs in a crisp environment in the presence of undesirable outputs. This is important because traditional DEA models do not explicitly take into account the presence of undesirable outputs, which can lead to an overstated assessment of the efficiency of DMUs. By incor porating the presence of undesirable outputs into the efficiency analysis, the thesis aims to provide a more accurate and robust evaluation of the efficiency of DMUs. The second motivation is to develop an IFDEA efficiency model from fuzzy to the IF environment. This will allow for a more accurate representation of the uncertainty and vagueness in the data, providing a more robust evaluation of the efficiency of DMUs in real-world scenarios. The third motivation is to determine the importance and effects of hesitation in IFSs on the efficiency analysis. IFSs provide a way to model uncertainty and vagueness in the data, but they also allow mathematical representation of hesitation, which can have a significant impact on the efficiency analysis. The thesis aims to explore the effects of hesitation on the efficiency analysis and to provide insights into the importance of incorporating the effects of hesitation in the efficiency evaluation. The fourth motivation is to analyze the performance of the MGNREGA scheme in India. By using MGNREGA as a case study, the thesis aims to provide a practical and relevant context for the exploration of the concepts and models presented in the thesis and to assess the efficiency of the program in providing employment and improving livelihoods. In conclusion, the main aim behind the thesis is to explore and expand our understanding of efficiency analysis in different environments, to provide a more accurate and robust evaluation of the efficiency of DMUs and to assess the performance of the MGNREGA scheme in India. A summary of each chapter in the thesis is presented as follows: Chapter 1 is introductory in nature. It includes basic concepts, definitions of terms, basic operation of fuzzy and IF sets. It presents a brief description of DEA, FDEA and IFDEA, the approach for selection of inputs and outputs. It presents the basic definitions and models related to the research. It also gives a brief description of the scheme of MGNREGA. Chapter 2 is focused on the application of DEA technique to assess the efficiency of MGNREGA in a crisp environment. It evaluates the efficient and inefficient states. For the inefficient states, it also identifies the sources of inefficiency in the form of the inputs excess and output shortfalls. The chapter also determines the best and worst performing States under MGNREGA in India. Chapter 3 aims to determine the efficiency of DMUs in presence of undesirable output. The study presents a two-phase model to identify the sources of inefficiency in a DMU and applies this model to MGNREGA scheme to assess the efficiency of States in presence of undesirable output. Chapter 4 develops a model for determining efficiency intervals of DMUs with IF inputs and outputs using α, β −cuts. This model is based on the Interval DEA model and provides efficiency intervals for each DMU. The chapter also proposes a new ranking approach for ranking the DMUs based on obtained efficiency intervals. Chapter 5 digs deeper into the concept of hesitation in an IF variable and determines the minimum and maximum hesitation in an input, output variable and a DMU. It calculates the effect of this hesitation on the efficiency of the DMU using the IFDEA model and provides the efficiency of each DMU incorporated with hesitation. The study provides a ranking method for comparison of DMUs and also defines the projected inputs and outputs for a DMU to become efficient. Chapter 6 uses the basic concepts of hesitation discussed Chapter 5 and calculates the efficiency of a DMU with IF inputs and outputs and uses the traditional IF Slack based measure to find the efficiency incorporated with hesitation. The results are verified using the MGNREGA scheme. Chapter 7 concludes the thesis by summarizing the results obtained in each chapter, drawing conclusions about the efficiency of MGNREGA and highlighting the contributions made to the fields of DEA and IFDEA. It also discusses the limitations of the study and proposes directions for future research.