Robust Optimization FDH Model for Data Envelopment Analysis in Education
محورهای موضوعی : policy making
1 - Ph.D. Student of Industrial Management, Production and Operations Management ,Fi.C., Islamic Azad University, Firoozkooh, Iran.
کلید واژه: Performance Evaluation, Robust Optimization, Free Disposal Hull (FDH), Data Envelopment Analysis (DEA), Uncertainty,
چکیده مقاله :
Traditional Data Envelopment Analysis (DEA) models generally rely on the assumption that all input and output data are deterministic and known with certainty. However, in many real-world scenarios, data uncertainty and measurement errors are unavoidable, which can significantly affect the reliability of efficiency evaluations. This study addresses this limitation by developing a robust version of the Free Disposal Hull (FDH) model, a non-convex DEA variant that does not require the convexity assumption. The proposed approach integrates robust optimization—a powerful methodology in operations research for managing uncertainty—into the FDH framework. Initially, an equivalent linear programming formulation of the FDH model is constructed to facilitate the application of robust optimization techniques. To improve the model's discrimination ability and to mitigate the occurrence of zero weights in the input and output variables, a method for determining appropriate lower bounds for weights is introduced. Subsequently, the robust counterpart of the FDH model is derived, incorporating uncertainty directly into the efficiency assessment. A practical case study is conducted to demonstrate the applicability and effectiveness of the proposed robust FDH model in real-world performance evaluation scenarios. The results confirm that the model maintains computational tractability while offering enhanced reliability in environments characterized by data ambiguity.
Traditional Data Envelopment Analysis (DEA) models generally rely on the assumption that all input and output data are deterministic and known with certainty. However, in many real-world scenarios, data uncertainty and measurement errors are unavoidable, which can significantly affect the reliability of efficiency evaluations. This study addresses this limitation by developing a robust version of the Free Disposal Hull (FDH) model, a non-convex DEA variant that does not require the convexity assumption. The proposed approach integrates robust optimization—a powerful methodology in operations research for managing uncertainty—into the FDH framework. Initially, an equivalent linear programming formulation of the FDH model is constructed to facilitate the application of robust optimization techniques. To improve the model's discrimination ability and to mitigate the occurrence of zero weights in the input and output variables, a method for determining appropriate lower bounds for weights is introduced. Subsequently, the robust counterpart of the FDH model is derived, incorporating uncertainty directly into the efficiency assessment. A practical case study is conducted to demonstrate the applicability and effectiveness of the proposed robust FDH model in real-world performance evaluation scenarios. The results confirm that the model maintains computational tractability while offering enhanced reliability in environments characterized by data ambiguity.
Agrell, P. J., & Tind, J. (2001). A Dual Approach to Nonconvex Frontier Models. Journal of Productivity Analysis, 16(2), 129–147.
Amin, G. R., & Toloo, M. (2004). A polynomial-time algorithm for finding ε in DEA models. Computers and Operations Research, 31(5), 803–805.
Arabmaldar, A., Jablonsky, J., & Saljooghi, F. H. (2017). A new robust DEA model and super-efficiency measure. Optimization, 66(5), 723–736.
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.
Bazaraa, M. S., Jarvis, J. J., & Sherali, H. D. (2009). Linear Programming and Network Flows (4th ed.). Wiley.
Ben-Tal A., El Ghaoui, L. and Nemirovski, A. (2009). Robust Optimization. Princeton University Press.
Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805.
Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25(1), 1–13.
Ben-Tal, Aharon, & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88(3), 411–424.
Bertsimas, D., Pachamanova, D., & Sim, M. (2004). Robust linear optimization under general norms. Operations Research Letters, 32(6), 510–516.
Bertsimas, D., & Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Programming, 98(1), 49–71.
Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53.
Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444.
Cook, W. D., Kress, M., & Seiford, L. M. (1996). Data Envelopment Analysis in the Presence of Both Quantitative and Qualitative Factors. The Journal of the Operational Research Society, 47(7), 945.
Cooper, W. W., Park, K. S., & Yu, G. (1999). IDEA and AR-IDEA: models for dealing with imprecise data in DEA. Management Science, 45(4), 597–607.
Deprins, D., Simar, L., & Tulkens, H. (1984). Measuring labor-efficiency in post offices. In The performance of public enterprises: concepts and measurements (pp. 243–268). Amsterdam: North-Holland.
Gorissen, B. L., Yanikoğlu, I., & den Hertog, D. (2015). A practical guide to robust optimization. Omega, 53, 124–137.
Hatami-Marbini, A., Emrouznejad, A., & Tavana, M. (2011). A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making. European Journal of Operational Research, 214(3), 457–472.
Hougaard, J. L. (1999). Fuzzy scores of technical efficiency. European Journal of Operational Research, 115(3), 529–541.
Jahanshahloo, G. R., Matin, R. K., & Vencheh, A. H. (2004). On FDH efficiency analysis with interval data. Applied Mathematics and Computation, 159(1), 47–55.
Kerstens, K., & Eeckaut, P. Vanden. (1999). Estimating returns to scale using non-parametric deterministic technologies: A new method based on goodness-of-fit. European Journal of Operational Research, 113(1), 206–214.
Khanjani, R., Fukuyama, H., Tavana, M., & Di, D. (2016). An integrated data envelopment analysis and free disposal hull framework for cost-efficiency measurement using rough sets. Applied Soft Computing Journal, 46, 204–219.
Khanjani Shiraz, R., Tavana, M., & Paryab, K. (2014). Fuzzy free disposal hull models under possibility and credibility measures. International Journal of Data Analysis Techniques and Strategies, 6(3), 286–306.
Khodabakhshi, M., & Kheirollahi, H. (2013). Performance evaluation of Iran universities with Stochastic Data Envelopment Analysis (SDEA). International Journal of Data Envelopment Analysis, 1(1), 7–13.
Leleu, H. (2006). A linear programming framework for free disposal hull technologies and cost functions : Primal and dual models. European Journal of Operational Research, 168, 340–344.
Olesen, O. B., & Petersen, N. C. (2016). Stochastic data envelopment analysis - A review. European Journal of Operational Research, 251(1), 2–21.
Omrani, H. (2013). Common weights data envelopment analysis with uncertain data: A robust optimization approach. Computers & Industrial Engineering, 66(4), 1163–1170.
Peykani, P., Mohammadi, E., Saen, R. F., Sadjadi, S. J., & Rostamy-Malkhalifeh, M. (2020). Data envelopment analysis and robust optimization: A review. Expert Systems, (September 2019), e12534.
Sadjadi, S. J., & Omrani, H. (2008). Data envelopment analysis with uncertain data: An application for Iranian electricity distribution companies. Energy Policy, 36(11), 4247–4254.
Sadjadi, S. J., Omrani, H., Abdollahzadeh, S., Alinaghian, M., & Mohammadi, H. (2011). A robust super-efficiency data envelopment analysis model for ranking of provincial gas companies in Iran. Expert Systems with Applications, 38(9), 10875–10881.
Salahi, M., Toloo, M., & Torabi, N. (2020). A new robust optimization approach to common weights formulation in DEA A new robust optimization approach to common weights formulation. Journal of the Operational Research Society, 1–13.
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.
Soyster, A. L. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21(5), 1154–1157.
Toloo, M. (2014). An epsilon-free approach for finding the most efficient unit in DEA. Applied Mathematical Modelling, 38(13), 3182–3192.
Toloo, M., & Mensah, E. K. (2019). Robust optimization with nonnegative decision variables: A DEA approach. Computers & Industrial Engineering, 127, 313–325.
Tulkens, H. (1993). On FDH Efficiency Analysis : Some Methodological Issues and Applications to Retail Banking , Courts , and Urban Transit, 210, 183–210.
Zohrehbandian, M., Makui, A., & Alinezhad, A. (2010). A compromise solution approach for finding common weights in DEA: an improvement to Kao and Hung’s approach. Journal of the Operational Research Society, 61(4), 604–610.
