Abstract
The zero sum gains data envelopment analysis models (ZSG-DEA models) are non-linear. In this paper, we first show that the ZSG-DEA models can be transformed to linear or parametric linear models and discuss the feasible domains of the parameters. Second, we show that the linear formulations of ZSG-DEA models under the equal output reduction strategy and the proportional output reduction strategy in a single output case are equivalent to the output-oriented super-efficiency model under variable returns-to-scale (VRS) assumption. As a matter of course, the models may encounter infeasibility. Third, we propose the linear transformations of ZSG-DEA models under constant returns-to-scale (CRS) assumption and compare them with the VRS models. In the end, we evaluate the participant countries at the Olympic Games by the linear equivalent models with multiple outputs under different weight restrictions. Our results are compared with the efficiencies obtained from the original ZSG-DEA model with an aggregated output under both CRS and VRS assumptions. It is found that the original method with aggregated output tends to underestimate the efficiencies of DMUs.
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Notes
In this paper, we call two optimization problems equivalent if from a solution of one, a solution of the other is readily found, and vice versa. Models (4) and (5) are equivalent because the feasible sets of the two are identical. A point is optimal for one, if and only if it is optimal for the other.
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Acknowledgements
The authors are grateful for the comments and suggestions by two anonymous reviewers. This research work is supported by grants from the National Natural Science Funds of China (No. 71171181), National Natural Science Funds of China for Innovative Research Groups (No. 71121061), China Postdoctoral Science Foundation (No. 2012M521259) and the Fundamental Research Funds for the Central Universities.
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Bi, G., Feng, C., Ding, J. et al. The linear formulation of the ZSG-DEA models with different production technologies. J Oper Res Soc 65, 1202–1211 (2014). https://doi.org/10.1057/jors.2013.69
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DOI: https://doi.org/10.1057/jors.2013.69