Abstract
We consider a new class of Data Envelopment Analysis (DEA) modeling, which we call ‘sequential DEA’. This new approach is a relatively simple generalization of the standard and popular in practice DEA. It allows for analyzing efficiency of the decision making units that consist of a sequence of sub-DMUs (e.g., branches of banks, hospital holding company running a number of hospitals at different locations, hotel chains, etc.). The approach is embedded in the Hilbert sequence space (\(\ell ^{2}\)) and therefore it allows for potentially different numbers of the sub-DMUs as well as different numbers of inputs and outputs used by different decision making units. We hope this approach will open up a new stream of literature in the sense that many existing variations from the already rich literature on DEA can be adapted to this approach.
Notes
See Luenberger (1969) on this topic in the context of optimization theory.
Shephard and Färe (1978) describe their theory as having inputs and outputs in Banach spaces and their theoretical foundation can be potentially used (with some adaptation) to further generalize our approach, to represent DEA models in sequences of the Banach space.
Also see Sickles and Zelenyuk (2019) for related discussions.
References
Afriat, S. N. (1972). Efficiency estimation of production functions. International Economic Review, 13(3), 568–598.
Banker, R. D. (1993). Maximum likelihood, consistency and data envelopment analysis: A statistical foundation. Management Science, 39(10), 1265–1273.
Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30(9), 1078–1092.
Caves, D. W., Christensen, L. R., & Diewert, W. E. (1982). The economic theory of index numbers and the measurement of input, output, and productivity. Econometrica, 50(6), 1393–1414.
Charnes, A., Cooper, W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444.
Dakpo, K. H., Jeanneaux, P., & Latruffe, L. (2017). Modelling pollution-generating technologies in performance benchmarking: Recent developments, limits and future prospects in the nonparametric framework. European Journal of Operational Research, 250(2), 347–359.
Debreu, G. (1951). The coefficient of resource utilization. Econometrica, 19(3), 273–292.
Färe, R., & Grosskopf, S. (1996). Intertemporal Production Frontiers: With Dynamic DEA. Norwell: Kluwer Academic Publishers.
Färe, R., & Grosskopf, S. (2009). A comment on weak disposability in nonparametric production analysis. American Journal of Agricultural Economics, 91(2), 535–538.
Färe, R., He, X., Li, S. K., & Zelenyuk, V. (2019). A unifying framework for Farrell efficiency measurement. Operations Research, 67(1), 183–197.
Färe, R., & Svensson, L. (1980). Congestion of production factors. Econometrica Journal of the Econometric Society, 48(7), 1745–1753.
Farrell, M. J. (1957). The measurement of productive efficiency. Jornal of the Royal Statistical Society Series A (General), 120(3), 253–290.
Gijbels, I., Mammen, E., Park, B. U., & Simar, L. (1999). On estimation of monotone and concave frontier functions. Journal of the American Statistical Association, 94(445), 220–228.
Kao, C. (2014). Network data envelopment analysis: A review. European Journal of Operational Research, 239(1), 1–16.
Karlin, S. (1959). Mathematical Methods and Theory in Games, Programming, and Economics (Vol. 2). Reading: Addison Wesley.
Kneip, A., Simar, L., & Wilson, P. W. (2015). When bias kills the variance: Central limit theorems for DEA and FDH efficiency scores. Econometric Theory, 31(2), 394–422.
Koopmans, T. C. (1951). Analysis of production as an efficienct combination of activities. Activity Analysis of Production and Allocation, 13, 33–37.
Korostelev, A., Simar, L., & Tsybakov, A. B. (1995). Efficient estimation of monotone boundaries. Annals of Statistics, 23(2), 476–489.
Luenberger, D. G. (1969). Optimization by vector space methods. In D. G. Luenberger (Ed.), Series in Decision and Control. New York: Wiley.
Shephard, R., & Färe, R. (1980). Dynamic Theory of Production Correspondences. Mathematical Systems in Economics. Berlin: Verlag Anton Hain.
Shephard, R. W., & Färe, R. (1978). Dynamic theory of production correspondences. part i, ii, iii. Technical Report.
Sickles, R., & Zelenyuk, V. (2019). Measurement of Productivity and Efficiency: Theory and Practice. New York: Cambridge University Press.
Simar, L., & Zelenyuk, V. (2018). Central limit theorems for aggregate efficiency. Operations Research, 166(1), 139–149.
Acknowledgements
The authors also acknowledge the financial support from ARC Grant (FT170100401). We also thank Bao Hoang Nguyen, Zhichao Wang, and Evelyn Smart for their proofreading feedback.
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
Färe, R., Zelenyuk, V. Sequential data envelopment analysis. Ann Oper Res 300, 307–312 (2021). https://doi.org/10.1007/s10479-020-03924-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-020-03924-x