Document Type: Original Article


1 Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran.

2 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

3 Department of Mathematics, Shahr.e Qods Branch, Islamic Azad University, Tehran, Iran.


Data Envelopment Analysis (DEA) is one of the methods most widely used for measuring the relative efficiency of DMUs in the world today.  The efficiency evaluation of the network structure opens the “black box” and considers the internal structure of systems. In this paper, a three-stage network model is considered with additional inputs and undesirable outputs and obtains the efficiency of the network, as interval efficiency in presence of the imprecise datum. The proposed model of this paper simulates a factory in the factual world with a production area, three warehouses and two delivery points. This factory is taken into consideration as a dynamic network and a multiplicative DEA approach is utilized to measure efficiency. Given the non-linearity of the models, a heuristic method is used to linearize the models. Ultimately, this paper concentrates on the interval efficiency to rank the units. The results of this ranking demonstrated that the time periods namely, (24) and (1) were the best and the poorest periods, respectively, in context to the interval efficiency within 24 phases of time.


Aghayi, N., Agrell, P., and Hatami-Marbini, A., (2013). "Imprecise data envelopment analysis for the two-stage precess", International Association for Research and Teaching, pp. 1-20.

Amirteimoori, A., and Kordrostami, S., (2014). "Data envelopment analysis with discrete-valued inputs and outputs ", Expert Systems, Vol. 31, pp.335-342

An, Q., Yang, M., Chu, J., Wu, J., and Zhu, Q., (2017). "Efficiency evaluation of an interactive system by data envelopment analysis approach ", Computers and Industrial Engineering, Vol. 103, pp. 17-25.

Azizi, H., (2013). "A note on data envelopment analysis with missing values: an interval DEA approach ", The International Journal of Advanced Manufacturing Technology, Vol. 66, pp.1817-1823.

Badiezadeh, T., Saen, R. F., and Samavati, T., (2018). "Assessing sustainability of supply chains by double frontier network DEA: A big data approach ", Computers and Operations Research, Vol. 98, pp. 284-290. 

Banker, R.D., Charnes, A., and Cooper W.W., (1984). "Some models for estimating technical and scale inefficiencies in data envelopment analysis ", Management Science, Vol. 30, No. 9, pp. 1078-1092.

Ben-Tal, A., and Nemirovski, A., (2000). "Robust solutions of linear programming problems contaminated with uncertain data ", Mathematical programming, Vol. 88, pp. 411-421.

Charnes A., Cooper W.W., and Rhodes, E., (1978). "Measuring the efficiency of decision making units", European Journal of Operational Research, Vol. 2, No. 6, pp. 429-444.

Chen, C., and Yan, H., (2011). "Network DEA model for supply chain performance evaluation", European Journal of Operational Research, Vol. 213, No. 1, pp. 147–155.

Chen, Y., Cook, W.D., Li, N., and Zhu, J., (2009). "Additive efficiency decomposition in two-stage DEA", European Journal of Operational Research, Vol. 196, pp. 1170-1176 

Cooper, W.W., Park, K.S., and Yu, G., (1999). "IDEA and AR-IDEA: Models for dealing with imprecise data in DEA", Management Science, Vol. 45, No. 4, pp. 597-607. 

Cooper, W.W., Park. K.S., and Yu, G., (2001). "IDEA (imprecise data envelopment analysis) with CMDs (column maximum decision making units) ", J. Oper. Res. Soc., Vol. 52, pp.176–181.

Despotis, D.K., Maragos, E.K. and Smirlis, Y.G., (2006). "Data Envelopment Analysis with Missing Values: An Interval DEA Approach", European J. Oper. Res., Vol. 140, pp. 24–36.

Despotis, D.K., and Smirlis, Y.G., (2002). "Data envelopment analysis with imprecise data", European Journal of Operational Research, Vol. 140, No. 1, pp. 24-36. 

Du, J., Asia Pac, and Oper, J. (2015). DEA Models for Parallel Systems: Game-Theoretic Approaches Res. 32, 1550008.

Entani, T., Maeda, Y., and Tanaka, Y., (2002). "Dual Models of Interval DEA and its Extension to Interval Data", European J. Oper. Res., Vol. 136, pp. 32–45.

Fard, A. M. F., and Hajiaghaei-Keshteli, M., (2018). "A bi-objective partial interdiction problem considering different defensive systems with capacity expansion of facilities under imminent attacks", Applied Soft Computing, Vol. 68, pp. 343-359.

Fard, A. M. F., and Hajaghaei-Keshteli, M., (2018). "A tri-level location-allocation model for forward/reverse supply chain", Applied Soft Computing, Vol. 62, pp. 328-346.

Fare, R., and Grosskopf, S., (2000). "Network DEA", Socio Economics Planning Science, Vol. 4, No. 1, pp. 35–49.

Fare, R., Grosskopf, S., Lovell, K., and Pasurka, C., (1989). "Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach", Review of Economics and Statistics, Vol. 71, No. 1, pp. 90–98.

Farrell, M.J., (1957). "The Measurement of Productive Efficiency", Journal of the Royal Statistical Society Series A (General), Vol. 120, pp. 253-290.

Farzipoor Saen, R., (2009). "A mathematical model for selecting third-party reverse logistics providers", International Journal of Procurement Management, Vol. 2, No. 2, pp. 180–190. 

Fathollahi-Fard, A. M., Hajiaghaei-Keshteli, M., and Mirjalili, S., (2018). "Hybrid optimizers to solve a tri-level programming model for a tire closed-loop supply chain network design problem", Applied Soft Computing, Vol. 70, pp. 701-722.

Hajiaghaei-Keshteli, M., and Fathollahi-Fard, A. M., (2018). "A set of efficient heuristics and metaheuristics to solve a two-stage stochastic bi-level decision-making model for the distribution network problem", Computers and Industrial Engineering, Vol. 123, pp. 378-395. 

Hajijabbari, A., and Sarabadani, M., (2008). "Practical guide to performance assessment of the executive organization", Tehran: Industrial Research and Training Centre of Iran.

Hwang, S. N., Chen, C., Chen, Y., Lee, H. S., and Shen, P. D., (2013). "Sustainable design performance evaluation with applications in the automobile industry: Focusing on inefficiency by undesirable factors", Omega, Vol. 41, No. 3, pp. 553-558. 

Jafarian Moghaddam, A.R., and Ghoseiri, K., (2011). "Fuzzy dynamic multi-objective Data Envelopment Analysis model", Expert Systems with Applications, Vol. 38, pp. 850–855.

Kao, C., (2006). "Interval efficiency measures in data envelopment analysis with imprecise data", European Journal of Operational Research, Vol. 174, pp. 1087–1099.

Kao, C. (2009). "Efficiency decomposition in network data envelopment analysis: A relational model", European Journal of Operational Research, Vol. 192, No. 3, pp. 949-962. 

Kao C., and 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, Vol. 180, No. 1, pp. 418-429. 

Kawaguchi, H., Tone, K., and Tsutsui, M., (2014). "Estimation of the efficiency of Japanese hospitals using a dynamic and network data envelopment analysis model", Health Care Management Science, Vol. 17, pp. 101-112. 

Khalili-Damghani, K., Tavana, M., and Haji-Saami, E., (2015). "A data envelopment analysis model with interval data and undesirable output for combined cycle power plant performance assessment", Expert Systems with Applications, Vol. 42, pp. 760-773. 

Korhonen, P.J., and Luptacik, M., (2004). "Eco-efficiency analysis of power plants: an extension of data envelopment analysis", European Journal of Operational Research, Vol. 154, Vol. 2, pp. 437-446.

Kou, M., Chen, K., Wang, S., and Shao, Y., (2016). "Measuring efficiencies of multi-period and multi-division systems associated with DEA: An application to OECD countries, national innovation systems", Expert Systems with Applications, Vol. 46, pp. 494-510. 

Li, Y., Chen, Y., Liang, L., and Xie, J., (2012). "DEA models for extended two-stage network structures", Omega, Vol. 40, No. 5, pp. 611-618.

Liang, L., Cook, W. D., and Zhu, J., (2008). "DEA models for two‐stage processes: Game approach and efficiency decomposition", Naval Research Logistics (NRL), Vol. 55, No. 7, pp. 643-653.

Liu, J., Lu, L., and Lu, W.M., (2016). "Research fronts in data envelopment analysis", Omega, Vol. 58, pp. 33-45. 

Lu, W.M. and Lo, S.F., (2007). "A closer look at the economic– environmental disparities for regional development in China", European Journal of Operational Research, Vol. 183, No. 2, pp. 882–894. 

Sengupta, J.K., (1995). "Dynamic of Data Envelopment Analysis: Theory of Systems Efficiency", Springer Science and Business Media, Netherlands. 

Shabanpour, H., Yousefi, S., and Farzipoor Saen, R., (2017). "Future planning for benchmarking and ranking sustainable suppliers using goal programming and robust double frontiers DEA", Transportation Research Part D: Transport and Environment, Vol. 50, pp.129–143.

Shahriari, S., (2013). "A Network Data Envelopment Analysis (NDEA) Model to evaluate firm’s Strategic Entrepreneurship. (Unpublished doctoral dissertation) ", Tehran University, Tehran, Iran. (In Persian).

Tone, K., and Tsutsui, M., (2009). "Network DEA: a slacks-based measure approach", European Journal of Operational Research, Vol. 197, No. 1, pp. 243-252. 

Wang, K., Yu, S., and Zhang, W., (2013). "China’s regional energy and environmental efficiency: A DEA Window Analysis Based Dynamic Evaluation", Mathematical and Computer Modelling, Vol. 58, pp. 1117–1127. 

Wang, M., Luo, Y., and Liang, L., (2009). "Fuzzy data envelopment analysis based upon fuzzy arithmetic with an application to performance assessment of manufacturing enterprises", Expert Systems with Applications, Vol. 36, pp. 5205–5211. 

Wang, W., Lu, W., and Liu, P., (2014). "A fuzzy multi-objective two-stage DEA model for evaluating the performance of US bank holding companies", Expert Systems with Applications, Vol. 41, pp. 4290-4297 

Wang, Y.M., Greatbanks, R., and Yang, J.B., (2005). "Interval efficiency assessment using data envelopment analysis", Fuzzy Sets and Systems, Vol. 153, No. 3, pp. 347–370. 

Wu, J., Lv, L., Sun, J., and Ji, X., (2015). "A comprehensive analysis of China's regional energy saving and emission reduction efficiency: from production and treatment perspectives", Energy Policy, Vol. 84, pp. 166-176. 

Wu, J., Zhu, Q., Ji, X., Chu, J., and Liang, L., (2016). "Two-stage network processes with shared resources and resources recovered from undesirable outputs", European Journal of Operational Research, Vol. 251, No. 1 , pp. 182-197.

Yousefi, S., Soltani, R., Saen, R. F., and Pishvaee, M. S., (2017). "A robust fuzzy possibilistic programming for a new network GP-DEA model to evaluate sustainable supply chains", Journal of Cleaner Production, Vol. 166, pp. 537-549. 

Zhou, X., Luo, R., Tu, Y., Lev, B., and Pedrycz, W. (2018). "Data envelopment analysis for bi-level systems with multiple followers", Omega, Vol. 77, pp. 180-188.