A Robust Directional Distance Function for Fixed-Sum Undesirable Outputs: A Petrochemical Industry Study

Authors

  • Seyed Morteza Mousavi Department of Industrial Engineering, Na.C., Islamic Azad University, Najafabad, Iran.
  • Javad Gerami * Department of Mathematics, Shi.C., Islamic Azad University, Shiraz, Iran. https://orcid.org/0000-0001-6829-1412
  • Mohammad Reza Mozaffari Department of Mathematics, Shi.C., Islamic Azad University, Shiraz, Iran.
  • Roya Mohammadalipour Ahari Department of Industrial Engineering, Na.C., Islamic Azad University, Najafabad, Iran. https://orcid.org/0000-0002-8952-8682
  • Mohammad Reza Feylizadeh Department of Industrial Engineering, Shi.C., Islamic Azad University, Shiraz, Iran. https://orcid.org/0000-0002-1382-7328

https://doi.org/10.48314/anowa.v1i4.61

Abstract

Petrochemical industries, as energy-intensive and pollution-oriented sectors, operate under stringent environmental regulations, particularly regarding caps on total pollutant emissions. Under such conditions, the fixed-sum property of undesirable outputs creates interdependence among Decision-Making Units (DMUs) and violates the independence assumption of classical Data Envelopment Analysis (DEA) models. Moreover, operational data in these industries are inherently subject to uncertainty due to fluctuations in operating conditions and environmental variability, which may lead to instability in the estimated efficiency frontier. This study proposes a robust equilibrium framework based on the Directional Distance Function (DDF) to evaluate the environmental-economic performance of petrochemical complexes. In the proposed model, the fixed-sum constraint on undesirable outputs is explicitly incorporated into the production technology, and data uncertainty is addressed through a robust optimization approach. The model was applied to data from 36 active petrochemical complexes in 2024. The results indicate that the number of efficient units decreases from 7 in the classical DEA model to 1 in the robust equilibrium model, demonstrating increased discrimination power and improved realism of the proposed framework. Sensitivity analysis with respect to different uncertainty levels confirms the stability of the ranking results. The findings suggest that neglecting the fixed-sum property of undesirable outputs and data uncertainty may lead to misleading and unstable efficiency identification. The proposed framework can serve as an effective decision-support tool for managerial planning and environmental policy-making in the petrochemical industry.

Keywords:

Data envelopment analysis, Equilibrium efficiency frontier, Petrochemical industry, Fixed-sum undesirable outputs, Directional distance function, Robust optimization

References

  1. [1] 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
  2. [2] Aparicio, J., Pastor, J. T., & Zofio, J. L. (2013). On the inconsistency of the Malmquist–Luenberger index. European journal of operational research, 229(3), 738–742. https://doi.org/10.1016/j.ejor.2013.03.031
  3. [3] Zhu, Q., Song, M., & Wu, J. (2020). Extended secondary goal approach for common equilibrium efficient frontier selection in DEA with fixed-sum outputs. Computers & industrial engineering, 144, 106483. https://doi.org/10.1016/j.cie.2020.106483
  4. [4] Zhou, P., Ang, B. W., & Poh, K. L. (2008). Measuring environmental performance under different environmental DEA technologies. Energy economics, 30(1), 1–14. https://doi.org/10.1016/j.eneco.2006.05.001
  5. [5] Färe, R., Grosskopf, S., Lovell, C. A. K., & Pasurka, C. (1989). Multilateral productivity comparisons when some outputs are undesirable: a nonparametric approach. The review of economics and statistics, 90–98.
  6. [6] Zhu, Q., Wu, J., Song, M., & Liang, L. (2017). A unique equilibrium efficient frontier with fixed-sum outputs in data envelopment analysis. Journal of the operational research society, 68(12), 1483–1490. 10.1057/s41274-017-0181-z
  7. [7] Chambers, R. G., Chung, Y., & Färe, R. (1996). Benefit and distance functions. Journal of economic theory, 70(2), 407–419. https://doi.org/10.1006/jeth.1996.0096
  8. [8] Sueyoshi, T., & Goto, M. (2015). DEA environmental assessment in time horizon: Radial approach for Malmquist index measurement on petroleum companies. Energy economics, 51, 329–345. https://doi.org/10.1016/j.eneco.2015.07.010
  9. [9] Arabmaldar, A., Sahoo, B. K., & Ghiyasi, M. (2023). A generalized robust data envelopment analysis model based on directional distance function. European journal of operational research, 311(2), 617–632. https://doi.org/10.1016/j.ejor.2023.05.005
  10. [10] Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations research, 52(1), 35–53. https://doi.org/10.1287/opre.1030.0065
  11. [11] Shokouhi, A. H., Hatami-Marbini, A., Tavana, M., & Saati, S. (2010). A robust optimization approach for imprecise data envelopment analysis. Computers & industrial engineering, 59(3), 387–397. https://doi.org/10.1016/j.cie.2010.05.011
  12. [12] 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
  13. [13] Zhu, Q., Li, X., Li, F., Wu, J., & Zhou, D. (2020). Energy and environmental efficiency of China’s transportation sectors under the constraints of energy consumption and environmental pollutions. Energy economics, 89, 104817. https://doi.org/10.1016/j.eneco.2020.104817
  14. [14] Seiford, L. M., & Zhu, J. (2002). Modeling undesirable factors in efficiency evaluation. European journal of operational research, 142(1), 16–20. https://doi.org/10.1016/S0377-2217(01)00293-4
  15. [15] Tone, K. (2001). A slacks-based measure of efficiency in data envelopment analysis. European journal of operational research, 130(3), 498–509. https://doi.org/10.1016/S0377-2217(99)00407-5
  16. [16] Lins, M. P. E., Gomes, E. G., Soares de Mello, J. C. C. B., & Soares de Mello, A. J. R. (2003). Olympic ranking based on a zero sum gains DEA model. European journal of operational research, 148(2), 312–322. https://doi.org/10.1016/S0377-2217(02)00687-2
  17. [17] 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
  18. [18] Gomes, E. G., & Lins, M. P. E. (2008). Modelling undesirable outputs with zero sum gains data envelopment analysis models. Journal of the operational research society, 59(5), 616–623. 10.1057/palgrave.jors.2602384
  19. [19] Yang, F., Wu, D., Liang, L., Bi, G., & Wu, D. D. (2011). Supply chain DEA: Production possibility set and performance evaluation model. Annals of operations research, 185(1), 195–211. 10.1007/s10479-008-0511-2
  20. [20] Chung, Y. H., Färe, R., & Grosskopf, S. (1997). Productivity and undesirable outputs: a directional distance function approach. Journal of environmental management, 51(3), 229–240. https://doi.org/10.1006/jema.1997.0146
  21. [21] 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
  22. [22] Ben-Tal, A., & Nemirovski, A. (2002). Robust optimization--methodology and applications. Mathematical programming, 92(3), 453–480. 10.1007/s101070100286
  23. [23] Cooper, W. W., Seiford, L. M., Tone, K. (2007). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software (Vol. 2). Springer. https://doi.org/10.1007/978-0-387-45283-8

Downloads

Published

2025-12-11

Issue

Vol. 1 No. 4 (2025): Annals of Optimization With Applications (ANOWA)

Section

Articles

How to Cite

Mousavi, S. M. ., Gerami, J. ., Mozaffari, M. R. ., Mohammadalipour Ahari, R. ., & Feylizadeh, M. R. . (2025). A Robust Directional Distance Function for Fixed-Sum Undesirable Outputs: A Petrochemical Industry Study. Annals of Optimization With Applications, 1(4), 241-268. https://doi.org/10.48314/anowa.v1i4.61

Similar Articles

11-20 of 20

You may also start an advanced similarity search for this article.