Ranking Decision Making units in Data Envelopment Analysis based on cooperative games

Subject Areas : Statistics

sanaz asadi rahmati ¹ (Department of Mathematics, Khorramabad branch, Islamic Azad University, Khorramabad, Iran)
Reza Fallahnejad ² (Department of Mathematics, Khorramabad branch, Islamic Azad University, Khorramabad, Iran)

Received: 2020-02-26 Accepted : 2021-05-15 Published : 2021-09-23

Keywords: مقدار شپلی, بازی همکارانه, تحلیل پوششی داده‌ها, رتبه‌بندی واحدهای کارا,

Abstract :

Evaluation of decision-making units is very importantin economic and management systems. Data envelopment analysis is one of the scientific and practical techniques for evaluating decision making units. In the conventional models of data envelopment analysis, the units are divided into efficient and inefficient categories, where the efficiency of each efficient unit is one and there is no distinction between efficient units.This study intends to propose a new way of ranking efficient units based on the concepts of cooperative game. The proposed method is that efficient units are considered as players of a cooperative game. A subset of these players is defined as the coalitionS.Then the sum of the efficiency of the inefficient units according to the production possibility set that is created by the inefficient units and the members of the coalitionS is defined as the characteristic function of S, which is used to determine the marginal effect of the efficient units in the various coalitions. Finally, the Shapley value is used to determine the cooperative game solution and rank the efficient units.In the same way that it is able to rank non-extreme units, inefficient units are also effective in ranking, alsois feasiblein any circumstances of return to scale.

References:

[1] A. Charnes, W.W. Cooper and E. Rhodes. Measuring the efficiency of decision making units. European journal of operational research 2(6): 429-444 (1978).

 

[2] R.D. Banker, A. Charnes and W.W. Cooper. Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management science 30(9): 1078-1092 (1984).

 

[3] K. Tone. A slacks-based measure of efficiency in data envelopment analysis. European journal of operational research. 130(3) 498-509 (2001).

 

[4] F. Hosseinzadeh Lotfi, G.R. Jahanshahloo, M. Khodabakhshi, M. Rostamy-Malkhlifeh, Z. Moghaddas, and M. Vaez-Ghasemi. A review of ranking models in data envelopment analysis. Journal of Applied Mathematics. 2013 (2013).

 

[5] A. Aldamak, S. Zolfaghari. Review of efficiency ranking methods in data envelopment analysis. Measurement 106: 161-172 (2017).

 

[6] T.R. Sexton, R.H. Silkman and A.J. Hogan. Data envelopment analysis: Critique and extensions. New Directions for Program Evaluation (32): 73-105 (1986).

 

[7] Z. Sinuany-Stern, A. Mehrez and A. Barboy. Academic departments efficiency via DEA. Computers & Operations Research. 21(5): 543-556 (1994).

 

[8] G.R. Jahanshahloo, H.V. Junior, F.H. Lotfi and D. Akbarian. A new DEA ranking system based on changing the reference set. European Journal of Operational Research. 181(1): 331-337 (2007).

[9] M. Rezaeiani, A. Foroughi. Ranking efficient decision making units in data envelopment analysis based on reference frontier share. European Journal of Operational Research. 264(2): 665-674 (2018).

 

[10] P. Andersen, N.C. Petersen. A procedure for ranking efficient units in data envelopment analysis. Management science. 39(10): 1261-1264 (1993).

 

[11] K. Nakabayashi, K. Tone. Egoist’s dilemma: A DEA game. Omega. 34, 135–148 (2006).

 

[12] S. Lozano. DEA production games. Eur. J. Oper. Res. 231, 405–413(2013).

 

[13] C.Y. Lee. Mixed-Strategy Nash Equilibrium in Data Envelopment Analysis. European Journal of Operational Research. 266(3): 1013-24(2018).

 

[14] H. Omrani, K. Shafaat and A. Alizadeh. Integrated data envelopment analysis and cooperative game for evaluating energy efficiency of transportation sector: a case of Iran. Annals of Operations Research. 274(1-2): 471-499 (2019).

 

[15] H. Omrani, M. Amini, M. Babaei and K. Shafaat. Use Shapley value for increasing power distinguish of data envelopment analysis model: An application for estimating environmental efficiency of industrial producers in Iran. Energy & Environment. 31(4):656-675 (2020).

 

[16] J. Doyle, R. Green. Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the operational research society 45(5): 567-578 (1994).

 

[17] W. Jie, L. Liang and Y.c. ZHA. Determination of the weights of ultimate cross efficiency based on the solution of nucleolus in cooperative game. Systems Engineering-Theory & Practice 28(5): 92-97 (2008).

 

[18] J. Wu, L. Liang and Y.C. Zha. Determination of the weights of ultimate cross efficiency based on the solution of nucleolus. Xitong Gongcheng Lilun yu Shijian/System Eng. Theory Pract. 28: 92–97 (2008)

 

[19] J. Wu, L. Liang, F. Yang and H Yan. Bargaining game model in the evaluation of decision making units. Expert Systems with Applications 36(3): 4357-4362 (2009).

 

[20] Y. Li, J. Xie, M. Wang and L. Liang. Super efficiency evaluation using a common platform on a cooperative game. European Journal of Operational Research 255(3):  884-892 (2016).

 

[21] T.S.H. Driessen. Cooperative games, solutions and applications. Vol. 3: Springer Science & Business Media (2013).

 

[22] A. Amirteimoori and S. Kordrostami. Efficient surfaces and an efficiency index in DEA: a constant returns to scale. Applied Mathematics and Computation 163(2): 683-691 (2005).

 

[23] A. Barzegarinegad, G. R.Jahanshahloo and M. Rostamy-Malkhalifeh, A full ranking for decision making units using ideal and anti-ideal points in DEA. The Scientific World Journal,(2014).

 

[24] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, V. Rezaie and M. Khanmohammadi, Ranking DMUs by ideal points with interval data in DEA, Applied Mathematical Modelling, 35(1), 218–229, (2011).