Abstract
A centrally managed system (CMS) typically comprises several decision making units (DMUs) that operate under a central DMU. The central DMU allocates the total available resources under its control among different DMUs to optimize the performance of the whole system. This distinguishing feature is at the heart of centralized resource allocation (CRA) methods and should be taken into account when assessing individual efficiency of each DMU in CMS. We introduce a slacks-based model for measuring individual efficiency of each DMU in CMS. As we will discuss, there are different possible CRA plans leading different projection points of DMUs on the frontier of the production possibility set (PPS). We will however show that all DMUs are projected on the same supporting hyperplane of the PPS under all CRA plans. We therefore have a common reference base, a subset of the ordinary efficient frontier, using which individual efficiency of each DMU can be measured in CMS. Having measured the individual efficiency of each DMU, we can categorize the DMUs into CRA-efficient and CRA-inefficient. To distinguish between CRA-efficient DMUs, we further introduce an influence index that measures the maximum effect of a specific CRA-efficient DMU on the construction of the projection points of the DMUs in CMS. We then propose a linear model to measure the influence of each CRA-efficient DMU. We can therefore provide a complete ranking of the DMUs in CMS. The proposed approach is demonstrated using a real data set.
Similar content being viewed by others
References
Adhikari, A., Majumdar, A., Gupta, G., & Bisi, A. (2020). An innovative super-efficiency data envelopment analysis, semi-variance, and Shannon-entropy-based methodology for player selection: Evidence from cricket. Annals of Operations Research, 284(1), 1–32.
Afsharian, M., Ahn, H., & Thanassoulis, E. (2017). A DEA-based incentives system for centrally managed multi-unit organisations. European Journal of Operational Research, 259(2), 587–598.
Allen, R., Athanassopoulos, A., Dyson, R. G., & Thanassoulis, E. (1997). Weights restrictions and value judgements in data envelopment analysis: Evolution, development and future directions. Annals of Operations Research, 73, 13–34.
An, Q., Meng, F., & Xiong, B. (2018). Interval cross efficiency for fully ranking decision making units using DEA/AHP approach. Annals of Operations Research, 271(2), 297–317.
Andersen, P., & Petersen, N. C. (1993). A procedure for ranking efficient units in data envelopment analysis. Management Science, 39(10), 1261–1264.
Asmild, M., Paradi, J. C., & Pastor, J. T. (2009). Centralized resource allocation BCC models. Omega, 37(1), 40–49.
Banker, R. D., Chang, H., & Zheng, Z. (2017). On the use of super-efficiency procedures for ranking efficient units and identifying outliers. Annals of Operations Research, 250(1), 21–35.
Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30, 1078–1092.
Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2005). Linear Programming and Network Flows. New Jersey: John Wiley & Sons.
Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functionals. Naval Research Logistics Quarterly, 9(3–4), 181–186.
Charnes, A., Cooper, W. W., Golany, B., Seiford, L., & Stutz, J. (1985). Foundations of data envelopment analysis for Pareto–Koopmans efficient empirical production functions. Journal of Econometrics, 30(1–2), 91–107.
Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444.
Charnes, A., Cooper, W. W., & Thrall, R. M. (1986). Classifying and characterizing efficiencies and inefficiencies in data development analysis. Operations Research Letters, 5(3), 105–110.
Chen, Y., Li, Y., Liang, L., & Wu, H. (2019). An extension on super slacks-based measure DEA approach. Annals of Operations Research, 278(1), 101–121.
Chen, L., Wang, Y.-M., & Huang, Y. (2020). Cross-efficiency aggregation method based on prospect consensus process. Annals of Operations Research, 288(1), 115–135.
Davtalab-Olyaie, M. (2019). A secondary goal in DEA cross-efficiency evaluation: A “one home run is much better than two doubles’’ criterion. Journal of the Operational Research Society, 70(5), 807–816.
Davtalab-Olyaie, M., & Asgharian, M. (2021). On Pareto-optimality in the cross-efficiency evaluation. European Journal of Operational Research, 288(1), 247–257.
Davtalab-Olyaie, M., Ghandi, F., & Asgharian, M. (2021). On the spectrum of achievable targets in cross-efficiency evaluation and the associated secondary goal models. Expert Systems with Applications, 177, 114927, https://doi.org/10.1016/j.eswa.2021.114927.
Davtalab-Olyaie, M., Mahmudi-Baram, H., & Asgharian, M. (2021). Incentivizing units in centralized systems: A slacks-based approach. Journal of the Operational Research Society. https://doi.org/10.1080/01605682.2021.1932620.
Davtalab Olyaie, M., Roshdi, I., Jahanshahloo, G., & Asgharian, M. (2014). Characterizing and finding full dimensional efficient facets in DEA: A variable returns to scale specification. Journal of the Operational Research Society, 65(9), 1453–1464.
Davtalab-Olyaie, M., Roshdi, I., Partovi Nia, V., & Asgharian, M. (2015). On characterizing full dimensional weak facets in DEA with variable returns to scale technology. Optimization, 64(11), 2455–2476.
Dehnokhalaji, A., Ghiyasi, M., & Korhonen, P. (2017). Resource allocation based on cost efficiency. Journal of the Operational Research Society, 68(10), 1279–1289.
Ding, T., Chen, Y., Wu, H., & Wei, Y. (2018). Centralized fixed cost and resource allocation considering technology heterogeneity: A DEA approach. Annals of Operations Research, 268(1), 497–511.
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.
Du, J., Wang, J., Chen, Y., Chou, S.-Y., & Zhu, J. (2014). Incorporating health outcomes in Pennsylvania Hospital efficiency: An additive super-efficiency DEA approach. Annals of Operations Research, 221(1), 161–172.
Fang, L. (2015). Centralized resource allocation based on efficiency analysis for step-by-step improvement paths. Omega, 51, 24–28.
Fang, L., & Zhang, C. (2008). Resource allocation based on the DEA model. Journal of the Operational Research Society, 59(8), 1136–1141.
Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society: Series A (General), 120(3), 253–281.
Gan, G.-Y., & Lee, H.-S. (2021). Resolving the infeasibility of the super-efficiency DEA based on DDF. Annals of Operations Research, 307(1), 139–152.
Hammami, H., Ngo, T., Tripe, D., & Vo, D.-T., (2020). Ranking with a Euclidean common set of weights in data envelopment analysis: With application to the Eurozone banking sector. Annals of Operations Research. https://doi.org/10.1007/s10479-020-03759-6.
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 Applications, 37(12), 7483–7488.
Jahanshahloo, G., Roshdi, I., & Davtalab-Olyaie, M. (2013). Characterizing and finding full dimensional efficient facets of PPS with constant returns to scale technology. International Journal of Industrial Mathematics, 5(2), 149–159.
Kao, C., & Hung, H.-T. (2005). Data envelopment analysis with common weights: The compromise solution approach. Journal of the Operational Research Society, 56(10), 1196–1203.
Li, F., Wu, H., Zhu, Q., Liang, L., & Kou, G. (2021). Data envelopment analysis cross efficiency evaluation with reciprocal behaviors. Annals of Operations Research, 302(1), 173–210.
Liu, S.-T. (2018). A DEA ranking method based on cross-efficiency intervals and signal-to-noise ratio. Annals of Operations Research, 261(1), 207–232.
Liu, W., Wang, Y.-M., & Lv, S. (2017). An aggressive game cross-efficiency evaluation in data envelopment analysis. Annals of Operations Research, 259(1), 241–258.
Lovell, C. A. K. (1993), Production Frontiers and Productive Efficiency. In H. Fried, C. A. Knox Lovell, and S. S. Schmidt (Eds.), The measurement of productive efficiency: Techniques and applications (pp. 3–67). Oxford: Oxford University Press, Inc.
Lozano, S., & Villa, G. (2004). Centralized resource allocation using data envelopment analysis. Journal of Productivity Analysis, 22(1–2), 143–161.
Lozano, S., & Villa, G. (2005). Centralized DEA models with the possibility of downsizing. Journal of the Operational Research Society, 56(4), 357–364.
Mar-Molinero, C., Prior, D., Segovia, M.-M., & Portillo, F. (2014). On centralized resource utilization and its reallocation by using DEA. Annals of Operations Research, 221(1), 273–283.
Podinovski, V. V. (2016). Optimal weights in DEA models with weight restrictions. European Journal of Operational Research, 254(3), 916–924.
Razipour-GhalehJough, S., Hosseinzadeh Lotfi, F., Jahanshahloo, G., Rostamy-Malkhalifeh, M., & Sharafi, H. (2020). Finding closest target for bank branches in the presence of weight restrictions using data envelopment analysis. Annals of Operations Research, 288(2), 755–787.
Tone, K. (2001). A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research, 130(3), 498–509.
Tone, K., Toloo, M., & Izadikhah, M. (2020). A modified slacks-based measure of efficiency in data envelopment analysis. European Journal of Operational Research, 287(2), 560–571.
Varmaz, A., Varwig, A., & Poddig, T. (2013). Centralized resource planning and yardstick competition. Omega, 41(1), 112–118.
Xie, Q., Zhang, L. L., Shang, H., Emrouznejad, A., & Li, Y. (2021). Evaluating performance of super-efficiency models in ranking efficient decision-making units based on Monte Carlo simulations. Annals of Operations Research, 305(1), 273–323.
Yu, S.-H., & Hsu, C.-W. (2020). A unified extension of super-efficiency in additive data envelopment analysis with integer-valued inputs and outputs: An application to a municipal bus system. Annals of Operations Research, 287(1), 515–535.
Yu, Y., Zhu, W., & Zhang, Q. (2019). DEA cross-efficiency evaluation and ranking method based on interval data. Annals of Operations Research, 278(1), 159–175.
Acknowledgements
The authors would like to express their sincere thanks to the Editor, the AE who handled our submission and two referees for their constructive comments. The research of the third author is supported by the Natural Science and Engineering Research Council (NSERC) of Canada [NSERC RGPIN-2018-05618].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Davtalab-Olyaie, M., Mahmudi-Baram, H. & Asgharian, M. Measuring individual efficiency and unit influence in centrally managed systems. Ann Oper Res 321, 139–164 (2023). https://doi.org/10.1007/s10479-022-04676-6
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-022-04676-6