Obtaining a Unique Solution for the Cross Efficiency by Using the Lexicographic method

Subject Areas : Data Envelopment Analysis

گ.ر.  جهانشاه لو ¹ (epartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran)
ر.  فلاح نژاد ² (epartment of Mathematics, Khorram Abad Branch, Islamic Azad University, Khorram Abad, Iran)

Received: 2010-07-30 Accepted : 2010-08-30 Published : 2010-09-01

Keywords: Data envelopment analysis, Ranking, Cross efficiency, Lexicographic Method,

Abstract :

Cross efficiency is a method with the idea of peer evaluation instead of self-evaluation, and is used for evaluation and ranking Decision Making Units (DMUs) in Data Envelopment Analysis (DEA). Unlike most existing DEA ranking models which can only rank a subset of DMUs, for example non-efficient or extreme efficient DMUs, cross efficiency can rank all DMUs, even non-extreme ones. However, since DEA weights are generally not unique, cross-efficiency which uses optimal weights corresponding to evaluation of DMUs may not be unique either. This deficiency renders the cross efficiency method useless. However, the secondary goals proposed to deal with this deficiency of cross efficiency have such drawbacks themselves as well. In this paper we present a new secondary goal for cross efficiency method based on the lexicographic method. The main advantage of the proposed method is that with the possibility of existence of alternative optimal weights at the end of the secondary goal problem, the performance and the rank of DMUs will be constant, while the previous secondary goal methods don't offer any suggestions to deal with their alternative optimal weights.

References:

[1] Charnes A., Cooper W.W., Rhodes E.,  Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 29-444, 1978.

[2] Andersen P., Petersen N. C., A procedure for ranking efficient units in data envelopment analysis. Management Science, 39, 1261–1264, 1993.

[3] Mehrabian S., Alirezaei M.R., Jahanshahloo G. R., A complete efficiency ranking of decision making units: an application to the teacher training university. Computational optimization and applications, 14, 261–266, 1998.

[4] Saati M.S., Zarafat Angiz M., Jahanshahloo G.R., A model for ranking decision making units in data envelopment analysis. Recrca operative 31, 47-59, 1999.

[5] Jahanshahloo G.R., Hosseinzadeh Lotfi F., Rezai Balf F., Zhiani Rezai H., Akbarian D., Ranking efficient DMUs using tchebycheff norm. Working Paper (2004).

[6] Jahanshahloo G.R., Memariani A., Hosseinzadeh Lotfi F., Rezai H.Z, A note on some of DEA models and finding efficiency and complete ranking using common set of weights. Applied Mathematics and Computation, 166, 265–281, 2005.

[7] Jahanshahloo G.R., Pourkarimi L., Zarepisheh M., Modified MAJ model for ranking decision making units in data envelopment analysis. Applied Mathematics and Computation: 174, 1054–1059, 2006.

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

[9] Shanling L., Jahanshahloo G.R., Khodabakhshi M., super-efficiency model for ranking efficient units in data envelopment analysis, Applied Mathematics and Computation, 184, 638-648, 2007.

[10] Adler N., Friedman L., Sinuany-Stern Z., Review of ranking methods in data envelopment analysis context. European Journal of Operational Research, 140, 249–265, 2002.

[11] Sexton T.R., Silkman R.H., Hogan A.J., Data envelopment analysis: Critique and extensions. In: Silkman, R.H. (Ed.), Measuring Efficiency: An Assessment of Data Envelopment Analysis. (Jossey-Bass, San Francisco,1996).

[12] Shang J., Sueyoshi T., A unified framework for the selection of a flexible manufacturing system. European Journal of Operational Research, 85 297–315, 1995.

[13] Talluri

S., Yoon K. P., cone-ratio DEA approach for AMT justification. International Journal of Production Economics, 66, 119–129, 2000.

[14] Green R., Doyle J., Cook W. D., Preference voting and project ranking using DEA and cross-evaluation. European Journal of Operational Research, 90, 461–472, 1996.

[15] Appa G., Williams H. P., A new framework for the solution of DEA models, European Journal of Operational Research, 172, 604–615, 2006.

[16] Doyle J., Green R., Efficiency and cross-efficiency in DEA: derivation, meanings and uses. Journal of Operation Research Society, 45, 567–578, 1994.

[17] Liang L., Wu J., Cook W. D., Zhu J., Alternative secondary goals in DEA cross-efficiency evaluation. International Journal of Production Economics, 113, 1025–1030, 2008.

[18] Figueira J., Greco S., Ehrgott M., Multiple Criteria Decision Analysis: State of the art surveys. (Springer Science +Business Media, Inc, United States of America, 2005.

[19] Ehrgott M., Multicriteria Optimization. 2nd edition, (Springer Berlin Heidelberg, Germany, 2005.

[20] Hosseinzadeh Lotfi F., Jahanshahloo G. R., Memariani A., A method for finding common set of weight by multiple objective programming in data envelopment analysis, southwest journal of pure and applied mathematics, 1, 44-54, 2000.

[21] Cook W.W. D., Zhu J., Within-group common weights in DEA: An analysis of power plant efficiency. European Journal of Operational Research, 178, 207–216, 2007.

[22] Dinkelbach W., On nonlinear fractional programming. Management Science, 1, 492–498, 1967.

 [23] Hwa F., Liu F., Peng H. H., Ranking of units on the DEA frontier with common weights. Computers & Operations Research, 35, 1624-1637, 2008.

[24] Bardhan I., Bowlin W. F., Cooper W. W., Sueyoshi T., Models for efficiency dominance in data envelopment analysis. Part I: Additive models and MED measures. Journal of Operation Research Society of Japan, 39, 322–332, 1996.