Completion of the paper ranking Efficient DMUs based on single virtual inefficient DMU in DEA

This paper builds upon the foundation laid by Shetty s research, aiming to enhance our understanding of decision-making unit (DMU) efficiency. In doing so, we introduce a novel approach that offers a more comprehensive method for ranking DMUs. Unlike traditional methods that assess DMUs individually, our proposed methodology centers on the creation of a virtual DMU. This virtual entity serves as a composite representation, synthesizing the inputs and outputs of all DMUs within the study. By aggregating this information, we establish a benchmark against which the efficiency of individual DMUs can be assessed. This approach not only simplifies the evaluation process but also provides a more holistic perspective, enabling researchers to discern patterns and trends across the entire dataset. As is widely acknowledged, the efficiency frontier is delineated by the efficient decision-making units (DMUs). The method proposed in the aforementioned paper proved to be efficacious particularly in scenarios where the number of efficient DMUs was limited, enabling the model to accurately rank them. However, challenges may arise as the population of efficient DMUs increases. This is due to the necessity of excluding these efficient DMUs from the efficiency frontier to obtain their efficiency ranks. Consequently, their ranking criterion would be determined by the efficiency scores of virtual DMUs generated by the revised efficiency frontier. In instances where the number of efficient DMUs expands, the process of excluding them from the efficiency frontier becomes more intricate. Furthermore, the reliance on virtual DMUs for ranking purposes introduces a layer of complexity, as the efficiency scores of these virtual entities are contingent upon the composition of the new efficiency frontier. Therefore, as the number of efficient DMUs escalates, the effectiveness of the proposed methodology in accurately ranking them may diminish. This underscores the importance of ongoing refinement and adaptation of methodologies to accommodate evolving datasets and analytical requirements in the assessment of DMU efficiency. If, following the removal of the efficient DMU, the efficiency score of a remaining DMU within the possibility production set is higher, it suggests that the respective DMU exhibits greater efficiency. Efficient Decision Making Units (DMUs) construct the defining hyperplane; therefore, the exclusion of these contributing efficient DMUs in an attempt to derive their ranking, amidst an increase in their numbers, will impede the acquisition of efficiency scores for virtual DMUs. Hence, achieving a comprehensive ranking of all DMUs is unattainable unless those positioned precisely on the defining hyperplane are included. In this complementary method, we delineate an anti-ideal virtual DMU encompassing all DMUs situated on the corresponding defining hyperplane, which may be oriented in various directions. Then, we use this method for ranking efficient DMUs. In this method, DMUs located on different intersecting defining hyperplanes may hold multiple ranks, from which the highest rank is deemed the most relevant. As the proposed method aligns with the aforementioned study, it incorporates all the advantages, including simplicity and stability, and notably eliminates the identified flaw.