Completion of the Paper “Ranking Efficient DMUs based on Single Virtual Inefficient DMU in DEA”

Subject Areas : Mathematical Optimization

AmirReza Hassani Tabatabaee ¹ , Amir Gholam Abri ^{2 *} , Shokrollah Ziari ³ , Ebrahim Niknaghsh ⁴

1 - Department of Industrial Management, Firoozkooh branch, Islamic Azad University, Firoozkooh, Iran.
2 - Department of Mathematics, Firoozkooh branch, Islamic Azad University, Firoozkooh, Iran.
3 - Department of Mathematics, South Tehran branch, Islamic Azad University, Tehran, Iran.
4 - Department of management, FiroozKooh Branch, Islamic Azad University, FiroozKooh, Iran

Received: 2024-08-05 Accepted : 2024-09-01 Published : 2024-09-07

Keywords: Data Envelopment Analysis (DEA), Decision Making Units (DMU), Ranking, Virtual DMU, Anti Ideal Point (AIP),

Abstract :

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. 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. 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. Therefore, as the number of efficient DMUs escalates, the effectiveness of the proposed methodology in accurately ranking them may diminish. 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. As the proposed method aligns with the aforementioned study, it incorporates all the advantages, including simplicity and stability, and notably eliminates the identified flaw.

References:

[1] Adler N., Friedman L., Sinuany-Stern Z.,: Review of ranking methods in the data envelopment analysis context. Eur J Oper Res, 140,249-265 (2002).
[2] P. Andersen, N.C. Petersen, A, Procedure for ranking efficient units in data envelopment analysis, Management science 39 (1993), 1261-1264.
[3] Banker R.D., Charnes A., Cooper W.W.: Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci, 30(9), (1984), 1078-1092.
[4] A. Carnes, W.W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2(4)(1978), 429-444.
[5] A. Gholam Abri, G.R. Jahanshahloo, F.Hosseinzadeh Lotfi, M. Fallah Jelodar, A new method for ranking non-extreme efficient units in data envelopment analysis, Optimization Letters, 7(2013),309-324
[6] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, N. Shoja, M. Fallah Jelodar, Amir Gholam Abri, Ranking extreme efficient decision making units in data envelopment analysis, Mathematical and computational application S,15(2)(2010), 299-308
[7] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, H. Zhiani Rezai, F. Rezai Balf, Finding strong defining hyperplanes of production possibility set, European journal of operational research, 177(2007) 42-54
[8] Ruiyue Lin, Zhiping Chen, A directional distance based supper-efficiency DEA model handling negative data, Journal of the operational research society (2017)53-64.
[9] V. Shetty, T.P.M. Pakalla, Ranking efficient DMUs based on a single virtual inefficient DMU in DEA, OPSEARCH, 47(1)(2010) 50-72.
[10] Mehdi Shoja, Farhad Hosseinzadeh Lotfi, Amir Gholam Abri, Alireza Rashidi Komijani, Efficiency of Green supply chain in the presence of non-discretionary and undesirable features, using Data Envelopment Analysis, Business informatics vol.15 No3-2021.