A generalized Equilibrium Network DEA Approach in Presence of Fixed-Sum Undesirable Output
Abstract
In this paper, we address the issue of dealing with fixed-sum undesirable outputs in a two-stage Data Envelopment Analysis (DEA) network structure. In this regard, we first introduce models for evaluating the performance of Decision-Making Units (DMUs) in the presence of fixed-sum undesirable outputs, and then we introduce this model for a two-stage network structure. We show that all DMUs are projected on the equilibrium efficiency frontier and will be efficient. We show that the erupted model will always be feasible and that the proposed model can be converted into a linear model. We then illustrate the model with a numerical example and proposed the final results.
Keywords:
Network system, Generalized equilibrium, Efficient frontier, Data envelopment analysis, Fixedsum, Undesirable outputsReferences
- [1] United Nations. (2015). The Paris agreement. https://unfccc.int/process-and-meetings/the-paris-agreement?utm_source
- [2] Yang, M., An, Q., Hu, D., & Liang, L. (2021). Performance evaluation of China’s industry: A generalized equilibrium data envelopment analysis approach with fixed-sum undesirable output. INFOR: information systems and operational research, 59(2), 290–308. https://doi.org/10.1080/03155986.2021.1881360
- [3] Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European journal of operational research, 2(6), 429–444. https://doi.org/10.1016/0377-2217(78)90138-8
- [4] Cook, W. D. (2001). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software. JSTOR. https://www.jstor.org/stable/25062691
- [5] 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. https://doi.org/10.1287/mnsc.30.9.1078
- [6] Tavana, M., Toloo, M., Aghayi, N., & Arabmaldar, A. (2021). A robust cross-efficiency data envelopment analysis model with undesirable outputs. Expert systems with applications, 167, 114117. https://doi.org/10.1016/j.eswa.2020.114117
- [7] Feng, C., Chu, F., Ding, J., Bi, G., & Liang, L. (2015). Carbon emissions abatement (CEA) allocation and compensation schemes based on DEA. Omega, 53, 78–89. https://doi.org/10.1016/j.omega.2014.12.005
- [8] Fang, L. (2016). A new approach for achievement of the equilibrium efficient frontier with fixed-sum outputs. Journal of the operational research society, 67(3), 412–420. https://doi.org/10.1057/jors.2015.65
- [9] Yang, F., Wu, D. D., Liang, L., & O’Neill, L. (2011). Competition strategy and efficiency evaluation for decision making units with fixed-sum outputs. European journal of operational research, 212(3), 560–569. https://doi.org/10.1016/j.ejor.2011.02.029
- [10] Amirteimoori, A., Masrouri, S., Yang, F., & Kordrostami, S. (2017). Context-based competition strategy and performance analysis with fixed-sum outputs: an application to banking sector. Journal of the operational research society, 68(11), 1461–1469. https://doi.org/10.1057/s41274-017-0180-0
- [11] Amirteimoori, H., Amirteimoori, A., & Karbasian, M. (2020). Performance measurement of gas companies with fixed-sum inputs: A DEA-based model. Journal of economic studies, 47(7), 1591–1603. https://doi.org/10.1108/JES-06-2019-0285
- [12] Arabmaldar, A., Hatami-Marbini, A., Loske, D., Hammerschmidt, M., & Klumpp, M. (2024). Robust data envelopment analysis with variable budgeted uncertainty. European journal of operational research, 315(2), 626–641. https://doi.org/10.1016/j.ejor.2023.11.043
- [13] Arabmaldar, A., Mohammadi, A., & Tavana, M. (2023). Robust DDF model for undesirable outputs under uncertainty. Journal of cleaner production, 412, 137264.
- [14] Arabmaldar, A., Sahoo, B. K., & Tavana, M. (2023). Robust DDF in DEA. European journal of operational research, 311(2), 617–632.
- [15] Wu, J., Xia, P., Zhu, Q., & Chu, J. (2019). Measuring environmental efficiency of thermoelectric power plants: a common equilibrium efficient frontier DEA approach with fixed-sum undesirable output. Annals of operations research, 275(2), 731–749. https://doi.org/10.1007/s10479-018-2958-0
- [16] Yang, M., Li, Y., Chen, Y., & Liang, L. (2014). An equilibrium efficiency frontier data envelopment analysis approach for evaluating decision-making units with fixed-sum outputs. European journal of operational research, 239(2), 479–489. https://doi.org/10.1016/j.ejor.2014.05.013
- [17] Yang, M., Li, Y. J., & Liang, L. (2015). A generalized equilibrium efficient frontier data envelopment analysis approach for evaluating DMUs with fixed-sum outputs. European journal of operational research, 246(1), 209–217. https://doi.org/10.1016/j.ejor.2015.04.023
- [18] Yang, M., Hou, Y., Ji, Q., & Zhang, D. (2020). Assessment and optimization of provincial CO2 emission reduction scheme in China: An improved ZSG-DEA approach. Energy economics, 91, 104931. https://doi.org/10.1016/j.eneco.2020.104931
- [19] Yu, S., Lei, M., & Deng, H. (2023). Evaluation to fixed-sum-outputs DMUs by non-oriented equilibrium efficient frontier DEA approach with Nash bargaining-based selection. Omega, 115, 102781. https://doi.org/10.1016/j.omega.2022.102781
- [20] Wang, W., Xu, F., Chu, J., Dong, Y., & Yuan, Z. (2025). Determining the equilibrium efficient frontier by proportional frontier shifting for data envelopment analysis with fixed-sum outputs. Omega, 130, 103174. https://doi.org/10.1016/j.omega.2024.103174
- [21] Ma, Y., Jin, X., & Chen, L. (2022). A two-stage DEA model for wastewater treatment systems with fixed-sum undesirable outputs. Journal of environmental management, 310, 114675.
- [22] Li, J., Wei, F., & Chu, J. (2024). Analysis of CO2 emission performance of China’s thermal power industry: a meta-frontier Malmquist-Luenberger approach with fixed-sum CO2 emissions. Journal of environmental planning and management, 67(8), 1746–1774. https://doi.org/10.1080/09640568.2023.2180349
- [23] Li, Y., Hou, W., Zhu, W., Li, F., & Liang, L. (2021). Provincial carbon emission performance analysis in China based on a Malmquist data envelopment analysis approach with fixed-sum undesirable outputs. Annals of operations research, 304(1), 233–261. https://doi.org/10.1007/s10479-021-04062-8
- [24] Chu, J., Dong, Y., & Yuan, Z. (2024). An improved equilibrium efficient frontier data envelopment analysis approach for evaluating decision-making units with fixed-sum outputs. European journal of operational research, 318(2), 592–604. https://doi.org/10.1016/j.ejor.2024.06.003
- [25] Chu, J., Shao, C., Emrouznejad, A., Wu, J., & Yuan, Z. (2021). Performance evaluation of organizations considering economic incentives for emission reduction: A carbon emission permit trading approach. Energy economics, 101, 105398. https://doi.org/10.1016/j.eneco.2021.105398
- [26] Li, J., Wu, H., Zhu, C., & Goh, M. (2024). Evaluating and analyzing renewable energy performance in OECD countries under uncertainty: A robust DEA approach with common weights. Applied energy, 375, 124115. https://doi.org/10.1016/j.apenergy.2024.124115
- [27] Moon, T. K. (1996). The expectation-maximization algorithm. IEEE signal processing magazine, 13(6), 47–60. https://doi.org/10.1109/79.543975
- [28] Zhang, X., Xia, Q., & Wei, F. (2023). Efficiency evaluation of two-stage parallel-series structures with fixed-sum outputs: An approach based on SMAA and DEA. Expert systems with applications, 227, 120264. https://doi.org/10.1016/j.eswa.2023.120264
- [29] Zhu, Q., Li, X., Li, F., Wu, J., & Sun, J. (2021). Analyzing the sustainability of China’s industrial sectors: A data-driven approach with total energy consumption constraint. Ecological indicators, 122, 107235. https://doi.org/10.1016/j.ecolind.2020.107235
- [30] Zhou, X., Xu, Z., Chai, J., Yao, L., Wang, S., & Lev, B. (2019). Efficiency evaluation for banking systems under uncertainty: A multi-period three-stage DEA model. Omega, 85, 68–82. https://doi.org/10.1016/j.omega.2018.05.012
- [31] Kao, C., & Hwang, S.N. (2008). Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan. European journal of operational research, 185(1), 418–429. https://doi.org/10.1016/j.ejor.2006.11.041