Dual Frontiers in Non-convex Data Envelopment Analysis: Efficiency and In-efficiency Assessment and Stability Analysis
Subject Areas : StatisticsNasim Nasrabadi 1 , Sheyda Ayati 2
1 - Department of Applied Mathematics (Operations Research), Faculty of Mathematical Sciences, Birjand University, Iran
2 - Department of Applied Mathematics (Operations Research), Faculty of Mathematical Sciences, Birjand University, Iran
Keywords: شعاع پایداری, مدل پوسته دسترسی پذیری آزاد, مرز کاملا کارا, مرز کاملا ناکارا, داده های بازه ای,
Abstract :
AbstractBasic models of Data Envelopment Analysis intrinsically evaluate the decision making units with an optimistic point of view, in the sense that the efficiency status of each unit is evaluated by means of calculating its distance from the efficiency frontier. The efficiency frontier is in fact composed of all units indicating the best practice, in the sense that for each one there exist no other (virtual) unit with a better performance. A unit located on this frontier is called fully efficient and non-efficient, otherwise. In order to provide a more precise assessment, one can evaluate units with a pessimistic point of view, in the sense that a frontier consisting of the worst performance, called the in-efficient frontier is formed and then each unit is evaluated with respect to its distance from this frontier, in a way that the closer the unit to the in-efficient frontier, the more in-efficient it is. In this paper, assuming that the production technology is non-convex, we perform efficiency and in-efficiency evaluation and then, based on the optimal value of the corresponding (in-) efficiency model, we partition all units in two subsets called (in-)efficient and non- (in-)efficient units. Then we investigate the concept of stability of the obtained partitions, by means of presenting related multi objective programs. In the next step, assuming that the input and output data of all units are real intervals, we deal with the efficiency and in-efficiency analysis of units and partition them into three subsets, in each case.
[1] Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society. Series A (General), 120(3), 253-290.
[2] Charnes, A., Cooper, W.W., Rhodes, E. (1978), Measuring the efficiency of decision making units, European Journal of Operational Research 2 (6), 429-444.
[3] Cooper, William W., Seiford, Lawrence M., Tone, Kaoru, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software (2007), Springer.
[4] Yamada, Y., Matsui, T., & Sugiyama, M. (1994). An efficiency measurement method for management-systems. Journal of the Operations Research Society of Japan, 37, 158–168.
[5] Entani, T., Maeda, Y., & Tanaka, H. (2002). Dual models of interval DEA and its extension to interval data. European Journal of Operational Research, 136(1), 32-45.
[6] Aldamak, A., Hatami-Marbini, A., & Zolfaghari, S. (2016). Dual frontiers without convexity. Computers & Industrial Engineering, 101, 466-478.
[7] Deprins, D., Simar, L., & Tulkens, H. (2006). Measuring labor-efficiency in post offices. In Public goods, environmental externalities and fiscal competition (pp. 285-309). Springer, Boston, MA.
[8] Tulkens, H. (2006). On FDH efficiency analysis: some methodological issues and applications to retail banking, courts and urban transit. In Public goods, environmental externalities and fiscal competition (pp. 311-342). Springer, Boston, MA.
[9] Kerstens, K., & Van De Woestyne, I. (2014). Solution methods for nonconvex free disposal hull models: A review and some critical comments. Asia-Pacific Journal of Operational Research, 31(01), 1450010.
[10] Charnes, A., Cooper, W. W., Lewin, A. Y., Morey, R. C., & Rousseau, J. (1984). Sensitivity and stability analysis in DEA. Annals of Operations Research, 2(1), 139-156.
[11] Charnes, A., & Neralić, L. (1990). Sensitivity analysis of the additive model in data envelopment analysis. European Journal of Operational Research, 48(3), 332-341.
[12] Zhu, J. (1996). Robustness of the efficient DMUs in data envelopment analysis. European Journal of operational research, 90(3), 451-460.
[13] Cooper, W. W., Li, S., Seiford, L. M., Tone, K., Thrall, R. M., & Zhu, J. (2001). Sensitivity and stability analysis in DEA: some recent developments. Journal of productivity analysis, 15(3), 217-246.
[14] Neralić, L., & Wendell, R. E. (2019). Enlarging the radius of stability and stability regions in Data Envelopment Analysis. European Journal of Operational Research, 278(2), 430-441.
