Document Type : Research Article
Authors
Department of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.
10.30473/coam.2025.73893.1293
Abstract
Data Envelopment Analysis (DEA) is a well-established methodology for assessing the efficiency of decision-making units. In complex systems comprising multiple interconnected subsections, Network DEA provides a structured framework for efficiency evaluation. However, traditional DEA models rely on the assumption of deterministic data, which inadequately reflects the inherent uncertainty present in real-world scenarios. Traditional uncertainty-handling methods, such as fuzzy logic, stochastic models, and interval-based techniques, often fail when there is limited historical data and when expert opinions significantly influence the dataset. To address these limitations, this study introduces an uncertain network DEA model based on Liu’s uncertainty theory, facilitating a more accurate assessment of efficiency under conditions of data imprecision. The proposed model is designed for three interconnected subsections and is further extended into a generalized multi-stage framework, allowing it to adapt to increasingly complex systems. Its effectiveness and practical applicability are demonstrated through two numerical case studies in the banking industry, highlighting its capacity to support decision-making under uncertainty. The findings emphasize the model's potential to enhance efficiency evaluation methods, particularly in environments characterized by limited and uncertain data.
Highlights
- Introduces a novel network DEA framework integrating Liu’s uncertainty theory to effectively address data imprecision and inherent uncertainty.
- Extends the conventional three-stage network DEA model into a flexible, multi-stage (p-stage) framework capable of modeling increasingly complex, interconnected systems under uncertainty.
- Demonstrates the model’s robustness and practical utility through two comprehensive real-world case studies in the banking industry, emphasizing its applicability with limited and subjective data.
- Highlights the model’s superior ability to incorporate expert opinions and subjective judgments, improving decision-making in environments with scarce or imprecise data.
- Validates the framework’s capacity for accurate efficiency assessment and resource allocation despite data uncertainty, supporting strategic decision-making in diverse complex systems.
- Showcases the potential of the proposed approach to enhance traditional DEA methodologies by effectively capturing data variability and uncertainty in multi-agent and multi-stage environments.
Keywords
Main Subjects
[1] Ahmadi, O., Mortazavi, S.B., Mahabadi, H.A., Hosseinpouri, M. (2020). “Development of a dynamic quantitative risk assessment methodology using fuzzy DEMATEL-BN and leading indicators”, Process Safety and Environmental Protection, 142, 15-44, doi:https://doi.org/10.1016/j.psep.2020.04.038.
[2] Boloori, F., Afsharian, M., Pourmahmoud, J. (2016). “Equivalent multiplier and envelopment DEA models for measuring efficiency under general network structures”, Measurement, 80, 259-269, doi:https://doi.org/10.1016/j.measurement. 2015.11.012.
[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, doi:https://doi.org/10.1016/0377-2217(78)90138-8.
[4] Chen, Y., Liang, L., Zhu, J. (2009). “Equivalence in two-stage DEA approaches”, European Journal of Operational Research, 193(2), 600-604, doi:https://doi.org/10.1016/j.ejor.2007.11.040.
[5] 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, doi:https://doi.org/10.1287/mnsc.45.4.597.
[6] Dyson, R.G., Shale, E.A. (2010). “Data envelopment analysis, operational research and uncertainty”, Journal of the Operational Research Society, 61(1), 25-34, doi:https://doi.org/10.1057/jors.2009.145.
[7] Ebrahimnejad, A., Tavana, M., Lotfi, F.H., Shahverdi, R., Yousefpour, M. (2014). “A three-stage data envelopment analysis model with application to banking industry”, Measurement, 49, 308-319, doi:https://doi.org/10.1016/j.measurement. 2013.11.043.
[8] Emrouznejad, A., Tavana, M., Hatami-Marbini, A. (2014). “The state of the art in fuzzy data envelopment analysis”, Performance Measurement with Fuzzy Data Envelopment Analysis, 1-45, doi:https://doi.org/10.1007/978-3-642-41372-8_1.
[9] Färe, R., Grosskopf, S., Whittaker, G. (2007). “Network dea”, Modeling Data Irregularities and Structural Complexities in Data Envelopment analysis, 209-240, doi:https://doi.org/10.1007/978-0-387-71607-7_12.
[10] Fried, H.O., Lovell, C.K., Schmidt, S.S. (Eds.). (2008). “The measurement of productive efficiency and productivity growth”, Oxford University Press, doi:https://doi.org/10.1093/acprof:oso/9780195183528.001.0001.
[11] Ghaffari-Hadigheh, A., Lio, W. (2020). “Network data envelopment analysis in uncertain environment”, Computers & Industrial Engineering, 148, 106657, doi:https://doi.org/10.1016/j.cie.2020.106657.
[12] Guo, P., Tanaka, H. (2001). “Fuzzy DEA: A perceptual evaluation method”, Fuzzy Sets and Systems, 119(1), 149-160, doi:https://doi.org/10.1016/S0165-0114(99)00106-2.
[13] Guo, P., Tanaka, H. (2008). “Decision making based on fuzzy data envelopment analysis”, Intelligent Decision and Policy Making Support Systems, 39-54, doi:https://doi.org/10.1007/978-3-540-78308-4_3.
[14] 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., doi:https://doi.org/10.1016/j.ejor.2011.02.001.
[15] Henriques, I.C., Sobreiro, V.A., Kimura, H., Mariano, E.B. (2020). “Two-stage DEA in banks: Terminological controversies and future directions”, Expert Systems with Applications, 161, 113632, doi:https://doi.org/10.1016/j.eswa.2020.113632.
[16] Heydari, C., Omrani, H., Taghizadeh, R. (2020). “A fully fuzzy network DEA-Range Adjusted Measure model for evaluating airlines efficiency: A case of Iran”, Journal of Air Transport Management, 89, 101923, doi:https://doi.org/10.1016/j.jairtraman.2020.101923.
[17] Hwang, S.N., Kao, T.L. (2008). “Using two-stage DEA to measure managerial efficiency change of non-life insurance companies in Taiwan”, International Journal of Management and Decision Making, 9(4), 377-401, doi:https://doi.org/10.1504/IJMDM.2008.019362.
[18] Jahanshahloo, G.R., Vencheh, A.H., Foroughi, A.A., Matin, R.K. (2004). “Inputs/outputs estimation in DEA when some factors are undesirable”, Applied Mathematics and Computation, 156(1), 19-32, doi:https://doi.org/10.1016/S0096-3003(03)00814-2.
[19] Jamshidi, M., Sanei, M., Mahmoodirad, A., Lotfi, F.H., Tohidi, G. (2021). “Uncertain SBM data envelopment analysis model: A case study in Iranian banks”, International Journal of Finance & Economics, 26(2), 2674-2689, doi:https://doi.org/10.1002/ijfe.1927.
[20] Jemric, I., Vujcic, B. (2002). “Efficiency of banks in Croatia: A DEA approach”, Comparative Economic Studies, 44, 169-193, doi:https://doi.org/10.1057/ces.2002.13.
[21] Kao, C. (2014). “Network data envelopment analysis: A review”, European Journal of Operational Research, 239(1), 1-16, doi:https://doi.org/10.1016/j.ejor.2014.02.039
[22] Korostelëv, A.P., Simar, L., Tsybakov, A.B. (1995). “Efficient estimation of monotone boundaries”, The Annals of Statistics, 476-489, doi:https://doi.org/10.1214/aos/1176324531.
[23] Lertworasirikul, S., Fang, S.C., Joines, J.A., Nuttle, H.L. (2003). “Fuzzy data envelopment analysis (DEA): A possibility approach”, Fuzzy Sets and Systems, 139(2), 379-394, doi:https://doi.org/10.1016/S0165-0114(02)00484-0.
[24] Lio, W., Liu, B. (2018). “Uncertain data envelopment analysis with imprecisely observed inputs and outputs”, Fuzzy Optimization and Decision Making, 17, 357-373, doi:https://doi.org/10.1007/s10700-017-9276-x.
[25] Liu, B. (2009). “Some research problems in uncertainty theory”, Journal of Uncertain Systems, 3(1), 3-10.
[26] Liu, B. (2010). “Uncertainty theory: A new paradigm for handling uncertain systems”, Journal of Uncertain Systems, 3-10, doi:https://doi.org/10.1109/TKDE.2012.179.
[27] Mergoni, A., Emrouznejad, A., De Witte, K. (2024). “Fifty years of data envelopment analysis.” European Journal of Operational Research, 326(3), 389-412, doi:https://doi.org/10.1016/j.ejor.2024.12.049.
[28] Mohmmad Nejad, Z., Ghaffari-Hadigheh, A. (2018). “A novel DEA model based on uncertainty theory”, Annals of Operations Research, 264, 367-389, doi:https://doi.org/10.1007/s10479-017-2652-7.
[29] Olesen, O.B., Petersen, N.C. (2016). “Stochastic data envelopment analysis—A review”, European Journal of Operational Research, 251(1), 2-21, doi:https://doi.org/10.1016/j.ejor.2015.07.058.
[30] Paul, C.J.M., Nehring, R. (2005). “Product diversification, production systems, and economic performance in US agricultural production”, Journal of Econometrics, 126(2), 525-548, doi:https://doi.org/10.1016/j.jeconom.2004.05.012.
[31] Pourbabagol, H., Amiri, M., Taghavifard, M.T., Hanafizadeh, P. (2023). “A new fuzzy DEA network based on possibility and necessity measures for agile supply chain performance evaluation: A case study”, Expert Systems with Applications, 220, 119552, doi:https://doi.org/10.1016/j.eswa.2023.119552.
[32] Pourmahmoud, J., Bafekr Sharak, N. (2018). “Measuring cost efficiency with new fuzzy DEA models”, International Journal of Fuzzy Systems, 20(1), 155-162, doi:https://doi.org/10.1007/s40815-017-0316-z.
[33] Pourmahmoud, J., Bagheri, N. (2021). “Providing an uncertain model for evaluating the performance of a basic two-stage system”, Soft Computing, 25, 4739-4748, doi:https://doi.org/10.1007/s00500-020-05481-8.
[34] Pourmahmoud, J., Bagheri, N. (2023). “Uncertain malmquist productivity index: An application to evaluate healthcare systems during COVID-19 pandemic”, Socio-Economic Planning Sciences, 87, 101522, doi:https://doi.org/10.1016/j.seps. 2023.101522.
[35] Pourmahmoud, J., Eini, M., Darvishi, D., Mehrabian, S. (2025). “A new approach in DEA with three-parameter interval grey numbers: investigating the effect of the drug-TPA on stroke patients”, Grey Systems: Theory and Application, 15(3), 547-573, doi:https://doi.org/10.1108/GS-08-2024-0097.
[36] Puri, J., Yadav, S.P. (2014). “A fuzzy DEA model with undesirable fuzzy outputs and its application to the banking sector in India”, Expert Systems with Applications, 41(14), 6419-6432, doi:https://doi.org/10.1016/j.eswa.2014.04.013.
[37] Ratner, S.V., Shaposhnikov, A.M., Lychev, A.V. (2023). “Network DEA and its applications (2017–2022): A systematic literature review”, Mathematics, 11(9), 2141, doi:https://doi.org/10.3390/math11092141.
[38] Rostamy-Malkhalifeh, M., Poudineh, E., Payan, A. (2018). “A fully fuzzy method of network data envelopment analysis for assessing revenue efficiency based on ranking functions”, Control and Optimization in Applied Mathematics, 3(2), 77-96, doi:https://doi.org/10.30473/coam.2019.46226.1115.
[39] Shakouri, R., Salahi, M., Kordrostami, S. (2019). “Stochastic p-robust approach to two-stage network DEA model”, Quantitative Finance and Economics, 3(2), 315-346, doi:https://doi.org/10.3934/QFE.2019.2.315.
[40] Simar, L. (1996). “Aspects of statistical analysis in DEA-type frontier models”, Journal of Productivity Analysis, 7, 177-185, doi:https://doi.org/10.1007/BF00157040.
[41] Simar, L., Wilson, P.W. (2000). “A general methodology for bootstrapping in non-parametric frontier models”, Journal of Applied Statistics, 27(6), 779-802, doi:https://doi.org/10.1080/02664760050081951.
[42] Simar, L., Wilson, P.W. (2011). “Two-stage DEA: Caveat emptor”, Journal of Productivity Analysis, 36, 205-218, doi:https://doi.org/10.1007/s11123-011-0230-6.
[43] Soleimani-Damaneh, M., Jahanshahloo, G.R., Abbasbandy, S. (2006). “Computational and theoretical pitfalls in some current performance measurement techniques; and a new approach”, Applied Mathematics and Computation, 181(2), 1199-1207, doi:https://doi.org/10.1016/j.amc.2006.01.085.
[44] Thanassoulis, E. (1999). “Data envelopment analysis and its use in banking”, Interfaces, 29(3), 1-13, doi:https://doi.org/10.1287/inte.29.3.1.
[45] Wang, C.N., Nguyen, T.L., Dang, T.T. (2021). “Analyzing operational efficiency in real estate companies: An application of GM (1, 1) and dea malmquist model”, Mathematics, 9(3), 202, doi:https://doi.org/10.3390/math 9030202.
[46] Wen, M., Guo, L., Kang, R., Yang, Y. (2014). “Data envelopment analysis with uncertain inputs and outputs”, Journal of Applied Mathematics, 2014(1), 307108, doi:https://doi.org/10.1155/2014/307108.
[47] Yeh, C.C., Chi, D.J., Hsu, M.F. (2010). “A hybrid approach of DEA, rough set and support vector machines for business failure prediction”, Expert Systems with Applications, 37(2), 1535-1541, doi:https://doi.org/10.1016/j.eswa.2009.06.088.
[48] Zadeh, L. (1965). “Fuzzy sets”, Inform Control, 8, 338-353, doi:https://doi.org/10.1016/S0019-9958(65)90241-X.
[49] Zadeh, L.A., Klir, G.J., Yuan, B. (1996). “Fuzzy sets, fuzzy logic, and fuzzy systems: Selected papers”, (Vol. 6). World Scientific, doi:https://doi:10.1142/2895doi:10.1142/2895.
[50] Zhou, Z., Lin, L., Xiao, H., Ma, C., Wu, S. (2017). “Stochastic network DEA models for two-stage systems under the centralized control organization mechanism”, Computers & Industrial Engineering, 110, 404-412, doi:https://doi.org/10.1016/j.cie.2017.06.005.
[51] 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, doi:https://doi.org/10.1016/j.omega..2018.05.012.
)