Document Type : Research Paper


1 Department of Applied Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran.

2 Department of Mathematics, Technical and Vocational University (TVU), Rasht Branch, Rasht, Iran.


Naturally, managers are interested in analyzing the relative efficiency of the production systems with parametric or non-parametric methods. One of the non-parametric methods utilized in recent three decades is a Data Envelopment Analysis (DEA) that can evaluate the relative efficiency of these structures and determine their strengths and weaknesses. In the real world, most production systems have a network structure. So, Network Data Envelopment Analysis (NDEA) is a suitable non-parametric method to evaluate the performance of production processes, which considers the internal processes. In addition to the importance of treating consumed waters and pollutants in environmental protection in today’s world, increasing the profitability of the production unit can be an important motivation to design the proper model to evaluate the performance of these structures in terms of profitability. In this sense, we first introduce a two-stage feedback structure including undesirable factors. Then, by applying the assumption of weak disposability for undesirable factors, a method for analyzing the relative performance of such network structures is given. The focus would be on profitability maximization. Moreover, to illustrate the proposed approach, a real case on the ecological system of 31 regions of China is used.


Main Subjects

  1. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision-making units. European journal of operational research2(6), 429-444.
  2. Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the royal statistical society: series A (general)120(3), 253-281.
  3. Färe, R., Grosskopf, S., & Lovell, C. K. (1985). The measurement of efficiency of production(Vol. 6). Springer Science & Business Media.
  4. Tone, K. (2002). A strange case of the cost and allocative efficiencies in DEA. Journal of the operational research society53(11), 1225-1231.
  5. Camanho, A. S., & Dyson, R. G. (2008). A generalisation of the Farrell cost efficiency measure applicable to non-fully competitive settings. Omega36(1), 147-162.
  6. Fukuyama, H., & Weber, W. L. (2008). Profit inefficiency of Japanese securities firms. Journal of applied economics11(2), 281-303.
  7. Sahoo, B. K., Mehdiloozad, M., & Tone, K. (2014). Cost, revenue and profit efficiency measurement in DEA: a directional distance function approach. European journal of operational research237(3), 921-931.
  8. Färe, R., & Grosskopf, S. (2000). Network DEA. Socio-economic planning sciences, 34(1), 35-49.
  9. Amirteimoori, A. (2013). A DEA two-stage decision processes with shared resources. Central European journal of operations research21(1), 141-151.
  10. Kao, C. (2015). Efficiency measurement for hierarchical network systems. Omega51, 121-127.
  11. Nematizadeh, M., Amirteimoori, A., Kordrostami, S., & Vaez-Ghasemi, M. (2020). Assessment of mixed network processes with shared inputs and undesirable factors. Operations research and decisions30(1), 97-118.
  12. Kao, C. (2014). Network data envelopment analysis: a review. European journal of operational research239(1), 1-16.
  13. Balfaqih, H., Nopiah, Z. M., Saibani, N., & Al-Nory, M. T. (2016). Review of supply chain performance measurement systems: 1998–2015. Computers in industry82, 135-150.
  14. Mohajer Tabrizi, M., Karimi, B., & Mirhassani, S. A. (2016). A novel two-stage stochastic model for supply chain network design under uncertainty. Scientia Iranica23(6), 3046-3062.
  15. Saeedi Mehrabad, M., Aazami, A., & Goli, A. (2017). A location-allocation model in the multi-level supply chain with multi-objective evolutionary approach. Journal of industrial and systems engineering10(3), 140-160.
  16. Akbari-Kasgari, M., Khademi-Zare, H., Fakhrzad, M. B., Hajiaghaei-Keshteli, M., & Honarvar, M. (2020). a closed-loop supply chain network design problem in copper industry. International journal of engineering33(10), 2008-2015.
  17. Amirteimoori, A., Khoshandam, L., & Kordrostami, S. (2013). Recyclable outputs in production process: a data envelopment analysis approach. International journal of operational research18(1), 62-70.
  18. Wu, J., Zhu, Q., Ji, X., Chu, J., & Liang, L. (2016). Two-stage network processes with shared resources and resources recovered from undesirable outputs. European journal of operational research251(1), 182-197.
  19. Wu, H., Lv, K., Liang, L., & Hu, H. (2017). Measuring performance of sustainable manufacturing with recyclable wastes: A case from China’s iron and steel industry. Omega66, 38-47.
  20. Li, W., Li, Z., Liang, L., & Cook, W. D. (2017). Evaluation of ecological systems and the recycling of undesirable outputs: An efficiency study of regions in China. Socio-economic planning sciences60, 77-86.
  21. Zhang, L., Guo, C., & Wei, F. (2019). Multistage network data envelopment analysis: semidefinite programming approach. Journal of the operational research society70(8), 1284-1295.
  22. Hu, Z., Yan, S., Li, X., Yao, L., & Luo, Z. (2019). Evaluating the oil production and wastewater treatment efficiency by an extended two-stage network structure model with feedback variables. Journal of environmental management251, 109578.
  23. Nematizadeh, M., Amirteimoori, A., & Kordrostami, S. (2019). Performance analysis of two-stage network processes with feedback flows and undesirable factors. Operations research and decisions29(3), 51-66.
  24. Lozano, S. (2011). Scale and cost efficiency analysis of networks of processes. Expert systems with applications38(6), 6612-6617.
  25. Banihashemi, S., & Tohidi, G. (2013). Allocation efficiency in network DEA. International journal of data envelopment analysis1(2), 85-96.
  26. Banihashem, S., Sanei, M., & Manesh, Z. M. (2013). Cost, revenue and profit efficiency in supply chain. African journal of business management7(41), 4280-4287.
  27. Fukuyama, H., & Matousek, R. (2017). Modelling bank performance: a network DEA approach. European journal of operational research259(2), 721-732.
  28. Jahani Sayyad Noveiri, M., Kordrostami, S., & Amirteimoori, A. R. (2017). Cost efficiency of closed–loop supply chain in the presence of dual-role and undesirable factors. Journal of new researches in mathematics3(9), 5-16.
  29. Fathi Ajirlu, S., Amirteimoori, A., & Kordrostami, S. (2020). Measuring the cost efficiency in NDEA. Journal of new researches in mathematics6(27), 141-154.
  30. Mousavizadeh, R., Navabakhsh, M., & Hafezalkotob, A. (2020). Cost-efficiency measurement for two-stage DEA network using game approach: an application to electrical network in Iran. Sādhanā45(1), 1-14.
  31. Fukuyama, H., Hashimoto, A., Matousek, R., & Tzeremes, N. G. (2021). Analyzing bank “black boxes”: a two-stage Nerlovian profit inefficiency model. Expert systems with applications, 183(30), 115405.
  32. Liang, L., Cook, W. D., & Zhu, J. (2008). DEA models for two‐stage processes: Game approach and efficiency decomposition. Naval research logistics (NRL)55(7), 643-653.
  33. Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functionals. Naval research logistics quarterly9(3‐4), 181-186.
  34. Shephard, R. W. (2015). Theory of cost and production functions. Princeton University Press.
  35. Kuosmanen, T. (2005). Weak disposability in nonparametric production analysis with undesirable outputs. American journal of agricultural economics87(4), 1077-1082.
  36. Maghbouli, M., Amirteimoori, A., & Kordrostami, S. (2014). Two-stage network structures with undesirable outputs: a DEA based approach. Measurement48, 109-118.