Department of Mathematics
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Item Performance Evaluation by SBM DEA Model Under Fuzzy Environments Using Expected Credits(Springer, 2023-03) Agarwal, Shivi; Mathur, TrilokPerformance evaluation is always the interest of the manager in every sector like banks, supply chain management, etc. Data envelopment analysis (DEA) is an operations research methodology based on linear programming, which is used to evaluate the performance of decision-making units (DMUs). The available data in real-life scenarios may be present in the vague form. Therefore, the crisp DEA model cannot be used for efficiency evaluation. In the present study, the slacks based model (SBM) DEA model is integrated with the fuzzy set theory to overcome the problem. The expected credits approach is used to solve the fuzzy SBM DEA model in the proposed study. The relative efficiency of the Indian oil refineries is calculated using this methodology, and the rank of the companies is computed from their relative efficiency.Item Ranking of Efficient DMUs Using Super-Efficiency Inverse DEA Model(Springer, 2023-03) Mathur, Trilok; Agarwal, ShiviThe overall potential for improving the relative efficiency of decision-making units (DMUs) is revealed by applying the data envelopment analysis (DEA) model. This study proposes a ranking system for ordering efficient DMUs with a super-efficiency inverse DEA (IDEA) model under a constant return to scale (CRS) assumption. IDEA is applied to evaluate the expected output or input variation level while keeping the efficiency value unchanged. For a numerical illustration of the proposed model in real-life problems, firstly, this study calculated the efficiency score of all 52 bus depots of Rajasthan State Road Transport Corporation (RSRTC) for the year 2018–19, applying the DEA model under the CRS assumption. The results revealed that 7 bus depots are efficient. Secondly, these 7 efficient depots have been ranked using the proposed super-efficiency IDEA model.Item Assessing the Radial Efficiency Performance of Bus Transport Sector Using Data Envelopment Analysis(CRC Press, 2021) Mathur, Trilok; Agarwal, ShiviThis study proposed an application of data envelopment analysis (DEA) to estimate the overall technical efficiency (OTE) for 52 bus depots of Rajasthan State Road Transport Corporation (RSRTC) for the financial year 2016–2017. The proposed study measures the efficiency and thoroughly explores the scope for optimal utilization of the input resources owned by depots of the RSRTC. The new slack model (NSM) of DEA is used as it calculates slacks for input and output variables. Also, this model satisfies the radial properties, unit invariance, and translation invariance. An input-oriented NSM model explicitly reduces input quantities (X) proportionally and renders no change in output level (Y). Four inputs have been considered, namely the number of buses, number of employees, total fuel consumption, and number of routes, and a single output, passenger kilometers occupied. The empirical results indicated that only 4 out of 52 depots are on the efficient frontier. All other depots possess the OTE value in the range of 0.7 to 1 relative to efficient depots. This study enables policy-makers to evaluate inputs for consistent output up to the optimum level and improve the performance of the inefficient depots.Item A Novel Fuzzy Non-radial Data Envelopment Analysis: An Application in Transportation(EDP Sciences, 2021-08) Mathur, Trilok; Agarwal, ShiviThe slack-based measure (SBM) DEA model is a non-radial model used to calculate the relative efficiency, input, and output targets of the different decision-making units (DMUs) based on their best peers or efficient frontier. The conventional SBM DEA model used crisp inputs and outputs. But, it can be observed in real-life problems that sometimes the available data is in linguistic forms such as “few”, “many”, “small”, or missing data. The DEA technique is frontier based, and therefore, imprecise data may lead to untenable results. Fuzzy theory, which is already established to handle uncertain data, can overcome this problem. Furthermore, the sensitivity and stability analysis have been checked the robustness of fuzzy DEA models. In this study, sensitivity and stability analysis of the fuzzy SBM DEA has been performed. The lower and upper sensitive bounds for inputs and outputs variables have been obtained for both the inefficient and efficient DMUs to calculate the input and output targets. Finally, a real-life transportation problem for the validity of the study is presented for its depiction.Item Fuzzy Slack Based Measure of Data Envelopment Analysis: A Possibility Approach(Springer, 2014-01) Agarwal, ShiviSlack based measure (SBM) model of Data Envelopment Analysis (DEA) is very effective to evaluate the relative efficiency of decision making units (DMUs). It deals with the directly input excess and output shortfall to assess the effect of slacks on efficiency with common crisp inputs and outputs. In some cases, input and output data of DMUs can’t be precisely measured, so, the uncertain theory has played an important role in DEA. In these cases, the data can be represented as linguistic variable characterized by fuzzy numbers. This paper attempts to extend the traditional DEA model to a fuzzy framework, thus proposing a fuzzy SBM DEA model based on possibility approach to deal with the efficiency measuring problem with the given fuzzy input and output data. Finally, numerical examples are presented to illustrate the proposed fuzzy SBM model. By extending to fuzzy environment, the DEA approach is made more powerful for applicationItem Sensitivity analysis in data envelopment analysis(IDEAS is a RePEc, 2014) Agarwal, ShiviData envelopment analysis (DEA) is a non-parametric technique and therefore hypothesis testing is very difficult. So, to determine the robustness of the efficiency scores obtained by DEA, sensitivity analysis is applied. Sensitivity analysis is used to know how sensitive the solution values and efficiency scores of the DMUs are to the numerical observations. In this paper, we propose a new model of sensitivity analysis in data envelopment analysis (DEA). The proposed new model examines the robustness of DEA efficiency scores by changing the reference set of the decision making units (DMUs). The model is also used for ranking the efficient DMUs and to identify the outliers on the frontier. Super efficiency is also estimated by applying the model as omitting the DMU itself from its reference set. Applying the proposed sensitivity model, this article empirically examines the robustness of the efficiency scores of 15 regions of Uttar Pradesh State Road Transport Corporation (UPSRTC) in India obtained by new slack model of DEA. The results of empirical illustration of sensitivity analysis reveal that the efficiency scores of the regions are robust, i.e., they are not sensitive to the efficient regions.Item Efficiency Measure by Fuzzy Data Envelopment Analysis Model(Taylor & Francis, 2018-11) Agarwal, ShiviData envelopment analysis (DEA) is a non-parametric technique to measure the relative efficiencies of a set of decision making units (DMUs) with common crisp inputs and outputs. Input and output data of DMUs often fluctuate. These fluctuating data can be represented as linguistic variable characterized by fuzzy numbers. This paper attempts to extend the traditional DEA model to a fuzzy framework, thus proposing a fuzzy DEA model based on α-cut approach to deal with the efficiency measuring and ranking problem with the given fuzzy input and output data. Finally, a numerical example is presented to illustrate the fuzzy DEA model. Since the efficiency measures are expressed by membership functions rather than by crisp values, more information is provided for management. By extending to fuzzy environment, the DEA approach is made more powerful for application.Item A DEA Model for Systematic Screening of Sustainable Suppliers with Uncertainty(Solid State Technology, 2020) Agarwal, Shivi; Mathur, Trilok; Routroy, SrikantaSuppliers play a vital role to create a sustainable and environment friendly supply chain of manufacturers. The manufacturer must have a systematic screening of available sustainable suppliers along various dimensions including their environmental sustainability dimension so that they can make the right outsourcing decision. It was observed that there are quantitative, qualitative factors and vagueness of experts' opinions involved in the screening, so a modified integrated model of Data Envelopment Analysis (DEA) integrated with the concepts of fuzzy logic is proposed which can handle the above factors for studying, analyzing, comparing and screening the sustainable suppliers. To describe the salient features of the proposed model a numerical illustration as case study is presented.Item A Benchmarking Approach with Missing Values Using Data Envelopment Analysis for Non-Symmetrical Fuzzy Data(Solid State Technology, 2020) Agarwal, Shivi; Mathur, TrilokThis study proposes a benchmarking approach where the missing data point denoted by fuzzy numbers and aims to outstretch the crisp Data Envelopment Analysis (DEA) model in a fuzzy system. The fuzzy DEA integrated model is capable for benchmarking of homogeneous decision making units (DMUs) with the missing data as non-symmetrical fuzzy data in terms of fuzzy efficiency. A numerical illustration is presented as a case study to explain the integrated fuzzy DEA model.Item A novel fuzzy non-radial data envelopment analysis: An application in transportation(RAIRO-Oper. Res, 2021-07) Agarwal, Shivi; Mathur, TrilokThe slack-based measure (SBM) DEA model is a non-radial model used to calculate the relative efficiency, input, and output targets of the different decision-making units (DMUs) based on their best peers or efficient frontier. The conventional SBM DEA model used crisp inputs and outputs. But, it can be observed in real-life problems that sometimes the available data is in linguistic forms such as “few”, “many”, “small”, or missing data. The DEA technique is frontier based, and therefore, imprecise data may lead to untenable results. Fuzzy theory, which is already established to handle uncertain data, can overcome this problem. Furthermore, the sensitivity and stability analysis have been checked the robustness of fuzzy DEA models. In this study, sensitivity and stability analysis of the fuzzy SBM DEA has been performed. The lower and upper sensitive bounds for inputs and outputs variables have been obtained for both the inefficient and efficient DMUs to calculate the input and output targets. Finally, a real-life transportation problem for the validity of the study is presented for its depiction.