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Fractional Order Efficiency: A New DEA Paradigm Based on Memory Dependent Dynamics
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Abstract
Classical Data Envelopment Analysis (DEA) is largely static: It benchmarks Decision Making Units (DMUs) using momentary observations. However, many manufacturing systems, especially energy systems, exhibit memory and path dependence, where current performance depends on historical paths. In parallel, uncertainty in energy data is often epistemic and multi‑source, which is naturally modeled by fuzzy sets. This paper introduces fractional‑order efficiency by embedding DEA in a dynamic framework driven by Fuzzy Fractional Differential Equations (FFDE) in the sense of Caputo. We define efficiency as a functional input‑output path with a fractional memory kernel, derive best (worst) case efficiency bounds under fuzzy parameter uncertainty, and develop a convergence theorem that connects the proposed fractional-order efficiency to a subset of discrete window DEA scores. A repeatable numerical study inspired by a regional power source demonstrates: How fractional-order controls inefficiency inertia (memory), how fuzzy uncertainty yields efficiency intervals, and how the method produces sustainability rankings under noisy multi‑year data.
Keywords
Fractional calculus Data envelopment analysis Fuzzy data envelopment analysis Memory effects Energy efficiencyReferences (20)
- [1] Tone, K. (2017). Advances in DEA theory and applications: With extensions to forecasting models. Wiley. https://doi.org/10.1002/9781118946688
- [2] Mergoni, A., Emrouznejad, A., & De Witte, K. (2025). Fifty years of data envelopment analysis. European journal of operational research, 326(3), 389–412. https://doi.org/10.1016/j.ejor.2024.12.049
- [3] Wang, Z., & Feng, C. (2015). A performance evaluation of the energy, environmental, and economic efficiency and productivity in China: An application of global data envelopment analysis. Applied energy, 147, 617–626. https://doi.org/10.1016/j.apenergy.2015.01.108
- [4] You, J., & Zhang, W. (2022). How heterogeneous technological progress promotes industrial structure upgrading and industrial carbon efficiency? Evidence from China’s industries. Energy, 247, 123386. https://doi.org/10.1016/j.energy.2022.123386
- [5] Hanoum, S., Shubbak, M., Ardiantono, D. S., & Korpysa, J. (2026). Data envelopment analysis applications for energy efficiency in power generation: A comprehensive review with bibliometric and content analysis. Energy conversion and management: x, 30, 101590. https://doi.org/10.1016/j.ecmx.2026.101590
- [6] Kao, C. (2013). Dynamic data envelopment analysis: A relational analysis. European journal of operational research, 227(2), 325–330. https://doi.org/10.1016/j.ejor.2012.12.012
- [7] Hu, J. L., Huang, Y. S., & You, C. Y. (2024). Renewable energy generation efficiency of Asian economies: An application of dynamic data envelopment analysis. Energies, 17(18), 1–22. https://doi.org/10.3390/en17184682
- [8] Stergiou, E., & Kounetas, K. (2021). European industries’ energy efficiency under different technological regimes: The role of CO2 emissions, climate, path dependence and energy mix. The energy journal, 42(1), 93–128. https://doi.org/10.5547/01956574.42.1.este
- [9] Wang, K., Yu, S., & Zhang, W. (2013). China’s regional energy and environmental efficiency: A DEA window analysis based dynamic evaluation. Mathematical and computer modelling, 58(5), 1117–1127. https://doi.org/10.1016/j.mcm.2011.11.067
- [10] Hatami-Marbini, A., Tavana, M., Gholami, K., & Beigi, Z. G. (2015). A bounded data envelopment analysis model in a fuzzy environment with an application to safety in the semiconductor industry. Journal of optimization theory and applications, 164(2), 679–701. https://doi.org/10.1007/s10957-014-0559-x
- [11] Izadikhah, M., Roostaee, R., & Emrouznejad, A. (2021). Fuzzy data envelopment analysis with ordinal and interval data. International journal of uncertainty, fuzziness and knowledge-based systems, 29(03), 385–410. https://doi.org/10.1142/S0218488521500173
- [12] Nehi, H. M., Saljooghi, F. H., Rahimi, A., Rathour, L., Mishra, L. N., & Mishra, V. N. (2025). Data envelopment analysis with imprecise data: Fuzzy and interval modeling approaches. Results in control and optimization, 19, 100531. https://doi.org/10.1016/j.rico.2025.100531
- [13] Jahani Sayyad Noveiri, M., & Kordrostami, S. (2021). Sustainability assessment using a fuzzy DEA aggregation approach: A healthcare application. Soft computing, 25(16), 10829–10849. https://doi.org/10.1007/s00500-021-05992-y
- [14] Tavana, M., Khalili-Damghani, K., Santos Arteaga, F. J., & Hosseini, A. (2019). A fuzzy multi-objective multi-period network DEA model for efficiency measurement in oil refineries. Computers & industrial engineering, 135, 143–155. https://doi.org/10.1016/j.cie.2019.05.033
- [15] Li, X. (2018). Numerical methods for fractional differential equations [Thesis].
- [16] Monje, C. A., Chen, Y., Vinagre, B. M., Xue, D., & Feliu, V. (2010). Fractional-order systems and controls: Fundamentals and applications. London: Springer London. https://doi.org/10.1007/978-1-84996-335-0_16
- [17] Mendes, E. M. A. M., Salgado, G. H. O., & Aguirre, L. A. (2019). Numerical solution of Caputo fractional differential equations with infinity memory effect at initial condition. Communications in nonlinear science and numerical simulation, 69, 237–247. https://doi.org/10.1016/j.cnsns.2018.09.022
- [18] Dwivedi, A., Rani, G., & Gautam, G. R. (2024). Analysis on the solution of fractional fuzzy differential equations. Rendiconti del circolo matematico di palermo series 2, 73(5), 1763–1791. https://doi.org/10.1007/s12215-024-01006-6
- [19] Peykani, P., Gheidar-Kheljani, J., Farzipoor Saen, R., & Mohammadi, E. (2022). Generalized robust window data envelopment analysis approach for dynamic performance measurement under uncertain panel data. Operational research, 22(5), 5529–5567. https://doi.org/10.1007/s12351-022-00729-7
- [20] Wang, C. N., Dang, T. T., Tibo, H., & Duong, D. H. (2021). Assessing renewable energy production capabilities using DEA window and fuzzy TOPSIS model. Symmetry, 13(2), 1–20. https://doi.org/10.3390/sym13020334