The New Neutral Secondary Goal based on Ideal DMU Evaluation in‏‎‎ Cross-Efficiency

Document Type: Research Paper


Department of Mathematics, University of Mazandaran, Babolsar, Iran


Cross-efficiency is a famous ranking method for data envelopment analysis (DEA) that deletes unrealistic weights pattern with no need to a priori information ‎related ‎to ‎weights ‎restrictions. ‎This‎ method analyzes each decision making unit (DMU) taking into account the best weights resulted from assessing other DMUs. In cross-efficiency evaluation, secondary goals such as aggressiveness, benevolence and neutrality are used because there are alternative optimal solutions. The neutral secondary goal makes the decision maker have no problem in selecting the aggressive and benevolent secondary goals. In the article at hand, a new secondary purpose is introduced which ‎selects the ‎ ‎optimal ‎weight ‎among the ‎multiple‎ optimal ‎weights based on the evaluation of ideal virtual DMUs corresponding ‎to ‎each ‎DMU. Since this kind of ‎weight selection does not lead to any increase or decrease in other DMUs’ cross-efficiency, the new secondary goal is neutral. The advantage of this method over other methods is that the selected ‎weights are the best ‎possible weights, because it maximizes the ideal virtual DMUs’ efficiciency score corresponding ‎to ‎each ‎DMU. ‎For this purpose‎, some examples are used to illustrate  its difference with other methods‎.


Main Subjects

Article Title [Persian]

هدف ثانویه بی طرف جدید بر مبنای ارزیابی واحد تصمیم گیرنده ایده آل در کارایی متقاطع

Authors [Persian]

  • سید هادی ناصری
  • حمید کیایی
دانشکده علوم ریاضی، دانشگاه مازندران، بابلسر، ایران
Abstract [Persian]

کارایی متقاطع روش رتبه بندی معروفی در تحلیل پوششی داده ها است که الگوی وزن های غیرواقعی را بدون نیاز به اطلاعات برتر مرتبط با محدودیت های وزنی حذف می کند. این روش، هر واحد را از دیدگاه بهترین وزن های حاصل از ارزیابی واحدهای دیگر، مورد سنجش قرار می دهد. در ارزیابی کارایی متقاطع به دلیل وجود جواب های بهینه دگرین از هدف های ثانویه ای چون خوش بینانه، بدبینانه و بی طرف استفاده شده است. هدف ثانویه بی طرف باعث می شود تا تصمیم گیرنده، مشکلی برای انتخاب هدف های ثانویه خوش بینانه و بدبینانه نداشته باشد. در مقاله حاضر هدف ثانویه جدیدی معرفی می گردد که بر مبنای ارزیابی واحدهای مجازی ایده آل نظیر هر واحد، وزن های بهینه ای از میان وزن های بهینه دگرین انتخاب می کند و از آن جایی که این نوع گزینش وزن ها، تاثیری در افزایش یا کاهش کارایی متقاطع دیگر واحد ها ندارد، هدف ثانویه جدید بی طرف است. حسن این روش در مقایسه با روش های دیگر آن است که وزن های انتخاب شده، بهترین وزن های ممکن است چرا که نمره کارایی واحدهای مجازی ایده آل متناظر هر واحد را ماکزیمم می کند.

Keywords [Persian]

  • تحلیل پوششی داده ها
  • کارایی متقاطع
  • هدف ثانویه بی طرف
  • واحدهای مجازی ایده آل
  • رتبه بندی
Abolghasem, S., Toloo, M., & Amézquita, S. (2019). Cross-efficiency evaluation in the presence of flexible measures with an application to healthcare systems. Health Care Management Science,, 1-22.

Aldamak, A., & Zolfaghari, S. (2017). Review of efficiency ranking methods in data envelopment analysis. Measurement, 106, 161–172.

Amirteimoori, A., Kordrostami, S., & Nasrollahian, P. (2017). A Method for solving super-efficiency infeasibility by adding virtual DMUs with mean values. Iranian Journal of Management Studies, 10(4), 905-916.

Carrillo, J., & Jorge, J. M. (2018). An alternative neutral approach for cross-efficiency evaluation. Computers & Industrial Engineering, 120, 137–145.

Charnes‎, ‎A.‎, & Cooper‎, ‎W‎. ‎W‎. ‎(1962)‎. ‎Programming with linear fractional functional‎. Naval Research Logistics Quarterly‎, 9, 181-185‎.

Charnes‎, ‎A.‎, ‎Cooper‎, ‎W‎. ‎W.‎, & ‎Rhodes‎, ‎E‎. ‎(1978)‎. ‎Measuring the efficiency of decision‎ making units‎. ‎European Journal of Operational Research‎, 2, ‎429-444‎.

Cui‎, ‎Q.‎, &‎Li‎, ‎Y‎. ‎(2015)‎. ‎Evaluating energy efficiency for airlines‎: ‎An application of‎ VFB-DEA‎. ‎Journal of Air Transport Management‎, 44-45, ‎34-41‎.

Doyle‎, ‎J.‎, & ‎Green‎, ‎R‎. ‎(1994)‎. ‎Efficiency and cross-efficiency in DEA‎: ‎Derivations‎, meanings and uses‎. ‎Journal of the Operational Research Society‎, ‎45(5)‎, ‎567-578‎.

Entani, T., & Tanaka, H. (2006). Improvement of efficiency intervals based on DEA by adjusting inputs and outputs. European Journal of Operational Research, 172(3), 1004-1017.

Farell‎, ‎M‎. ‎(1957)‎. ‎The measurement of productive efficiency‎. ‎Journal of the Royal‎ Statistical Society‎,  ‎120, ‎253-281‎.

Ghasemi, M. R., Ignatius, J., & Rezaee, B. (2019). Improving discriminating power in data envelopment models based on deviation variables framework. European Journal of Operational Research, 278(2), 442-447.

Hou, Q., Wang, M., & Zhou, X. (2018). Improved DEA cross efficiency evaluation method based on ideal and anti-ideal points. Hindawi,, 1-9.

Jahanshahloo, G. R., Lotfi, F. H., Khanmohammadi, M., Kazemimanesh, M., & Rezaie, V. (2010). Ranking of units by positive ideal DMU with common weights. Expert Systems with Applications37(12), 7483-7488.

Jahanshahloo, G. R., Sadeghi, J., & Khodabakhshi, M. (2016). Fair ranking of the decision making units using optimistic and pessimistic weights in data envelopment analysis. RAIRO-Operations Research51(1), 253-260.

Khodabakhshi, M., & Aryavash, K. (2017). The cross-efficiency in the optimistic–pessimistic framework. Operational Research - An International Journal, 17(2), 619-632.

Kiaei, H., Kazemi Matin, R., & Nasseri, S. H. (2019).  Production trade-offs and weight restrictions in two-stage network data envelopment analysis. Int. J. Applied Decision Sciences, (in press).

Kritikos, M. N. (2017). A full ranking methodology in data envelopment analysis based on a set of dummy decision making units. Expert Systems with Applications, 77, 211-225.

Liang, L., Wu, J., Cook, W. D., & Zhu, J. (2008a). The DEA game cross-efficiency model and its Nash equilibrium. Operations Research, 56(5), 1278–1288.

Liang‎, ‎L.‎, ‎Wu‎, ‎J.‎, ‎Cook‎, ‎W. D.‎, & Zhu‎, ‎J‎. ‎(2008b)‎. ‎Alternative secondary goals in DEA cross‎ efficiency evaluation‎. ‎International Journal of Production Economics‎,113(2)‎, ‎1025-1030‎.

Liu, X., Chu J., Yin, P., & Sun, J. (2017). DEA cross-efficiency evaluation considering undesirable output and ranking priority: A case study of eco-efficiency analysis of coal-fired power plants. Journal of Cleaner Production, 142, 877-885.

Nasseri, S. H., Gholami, O., & Ebrahimnejad, A. (2014). On ranking decision making units using relative similar units in data envelopment analysis, Int. J. Applied Decision Sciences, 7(4), 424–436.

Nasseri, S. H, & Kiaei, H. (2016). Cross-Efficiency evaluation by the use of ideal and anti-ideal virtual DMUs’ assessment in DEA. International Journal of Applied Operational Research, 6(3), 69-79.

Nasseri, S. H., & Kiaei, H. (2018). Allocation of weights using simultaneous optimization of inputs and outputs contribution in cross-efficiency evaluation of DEA. Yugoslav Journal of Operations Research, 28(4), 521-538.

Nasseri, S. H., & Kiaei, H. (2019). Ranking of efficient units on the basis of distance from virtual ideal and anti-ideal units.‎ Int. J. Applied Decision Sciences, 12(4), 361-374.

Oukil, A., & Amin, G.R. (2015). Maximum appreciative cross-efficiency in DEA: A new ranking method. Computers & Industrial Engineering, 81, 14-21.

Ramo´n‎, ‎N.‎, Ruiz‎, ‎J. L.‎, & Sirvent‎, ‎I‎. ‎(2010a)‎. ‎On the choice of weights profiles in cross efficiency‎ evaluations‎. ‎European Journal of Operational Research‎, 207(3), 1564-72‎.

Ramo´n‎, ‎N.‎, ‎Ruiz‎, ‎J. L.‎, & ‎Sirvent‎, ‎I‎. ‎(2010b)‎. ‎Reducing differences between profiles of weights‎: ‎A‎ ‎"peer-restricted"‎ ‎cross-efficiency evaluation‎. ‎Omega‎, 39(6), 634-641.

Rezaie, V., & Khanmohammady, M. (2010). Ranking DMUs by ideal PPS in Data Envelopment Analysis. International Scholarly and Scientific Research & Innovation, 4(7), 565-570.

Russell, R. R. (1985). Measure of technical efficiency. Journal of Economic Theory, 35(1), 109-126.

Sexton‎, ‎T‎. ‎R.‎, ‎Silkman‎, ‎R‎. ‎H.‎, & ‎Hogan‎, ‎A‎. ‎(1986)‎. ‎Data envelopment analysis‎: ‎Critique‎ and extensions‎. ‎In R. H‎. ‎Silkman‎ ‎(Ed.)‎, ‎Measuring efficiency‎: ‎An assessment of‎ data envelopment analysis‎ (‎vol‎. ‎32, pp‎. ‎73-105). ‎ San Francisco‎, ‎CA: Jossey-Bass ‎.

Song, M., Zhu, Q., Peng, J., & Santibanez, E. D. R. (2017). Improving the evaluation of cross efficiencies: A method based on Shannon entropy weight. Computers & Industrial Engineering, 112, 99-106.

Sun, J., Wu, J., & Guo, D. (2013). Performance ranking of units considering ideal and anti-ideal DMU with common weights. Applied Mathematical Modelling, 37(9), 6301-6310.

Tan, Y., Zhang, Y., & Khodaverdi, R. (2017). Service performance evaluation using data envelopment analysis and balance scorecard approach: An application to automotive industry. Annals of Operations Research, 248(1-2), 449-470.

Tofallis‎, ‎C‎. ‎(1997)‎. ‎Input efficiency profiling‎: ‎An application to airlines‎. ‎Computers & Operations Research‎,24(3)‎, ‎253-258‎.

Wang‎, ‎Y. M.‎, & ‎ Chin‎, ‎K. S‎. ‎(2010a)‎. ‎Some alternative models for DEA cross-efficiency evaluation‎. International Journal of Production Economics‎, ‎128(1), ‎332-338‎.

Wang‎, ‎Y‎. ‎M.‎, &‎ Chin‎, ‎K‎. ‎S‎. ‎(2010b)‎. ‎A neutral DEA model for cross-efficiency‎ evaluation and its extension‎. ‎Expert Systems with Applications‎, ‎37(5)‎, ‎3666-3675‎.

Wang‎, ‎Y. M.‎, ‎Chin‎, ‎K. S.‎, & ‎Luo‎, ‎Y‎. ‎(2011)‎. ‎Cross-efficiency evaluation based on ideal and anti-ideal decision making units‎. ‎Expert Systems with Applications‎,38(8)‎, ‎10312-10319‎.

Wang‎, ‎Y. M.‎, ‎Chin‎, ‎K. S.‎, & ‎Jiang‎, ‎P‎. ‎(2011)‎. ‎Weight determination in the cross-efficiency evaluation‎. ‎Computers & Industrial Engineering‎, 61(3), 497-502.‎‎

Wang, Y. M., & Luo, Y. (2006). DEA efficiency assessment using ideal and anti-ideal decision making units. Applied Mathematics and Computation, 173(2), 902-915.

Wong‎, ‎Y‎. ‎H‎. ‎B.‎, &‎ Beasley‎, ‎J‎. ‎E‎. ‎(1990)‎. ‎Restricting weight flexibility in data‎ envelopment analysis‎. ‎Journal of the Operational Research Society‎,41(9)‎, 829-835‎.

Wu‎, ‎J.‎, ‎Chu‎, ‎J.‎, ‎Sun‎, ‎J.‎, &‎ Zhu‎, ‎Q‎. ‎(2016)‎. ‎DEA cross-efficiency evaluation based on Pareto‎ improvement‎. ‎European Journal of Operational Research‎, 248(2), 571-579.

Wu, ‎J.‎, ‎Chu‎, ‎J.‎, ‎Sun‎, ‎J.‎, ‎Zhu‎, ‎Q.‎, & ‎Liang‎, ‎L. (2016). ‎Extended secondary goal models for weights‎ selection in DEA cross-efficiency evaluation. Computers & Industrial Engineering‎, 93, 143-151.