Data Envelopment Analysis (DEA) has become one of the most influential nonparametric methods for measuring the relative efficiency of Decision-Making Units (DMUs) in economics, operations research, and energy systems [1], [2]. Since the CCR and BCC models [2], DEA has evolved into a comprehensive analytical framework including slack-based measures, network DEA, dynamic DEA, and window analysis, which allow for a more realistic representation of production and service systems [1], [2]. DEA-based approaches have been widely applied in energy efficiency, environmental performance, and sustainability assessment, demonstrating their methodological flexibility and strong theoretical foundations [3–5]. However, despite these advances, most DEA models remain essentially static or quasi-dynamic, relying on cross-sectional images or rolling windows that implicitly assume weak temporal dependence between successive observations [6], [7]. In practical energy and sustainability systems, this assumption is rarely valid. Power generation units, renewable energy infrastructures, and industrial production systems have strong path dependence resulting from capital rigidity, learning-by-doing effects, technology lock-in, regulatory inertia, and cumulative investment decisions [6], [8]. As a result, current efficiency levels are strongly influenced by historical input-output paths rather than by instantaneous realizations [8], [9].

Although dynamic DEA and window DEA partially account for intertemporal structures, they are usually formulated in integer-order Markovian frameworks that ignore long-term memory and heritability effects [6], [9]. In parallel, efficiency assessment in energy systems is inherently affected by epistemic uncertainty. Energy input and output data are often imprecise due to measurement errors, aggregation procedures, reporting delays, and reliance on expert judgment-based estimates [10]. To address this issue, fuzzy set theory has been widely integrated into DEA models, allowing for the representation of efficiency with intervals rather than point estimates [10], [11]. Recent studies have proposed fuzzy, robust, and fuzzy-stochastic DEA frameworks to improve reliability in sustainability assessment and energy planning under uncertainty [12–14]. However, existing fuzzy DEA approaches largely treat uncertainty and temporal dynamics as separate issues and do not explicitly consider memory effects in the efficiency generation mechanism. Fractional calculus provides a rigorous and physically interpretable mathematical framework for modeling systems with memory and heritable properties via nonlocal operators characterized by power-law kernels [15], [16]. In particular, fractional derivatives of the Caputo type allow the inclusion of long-term historical effects while preserving meaningful initial conditions, making them particularly suitable for dynamic economic and energy-related processes [17]. Recent advances in Fuzzy Fractional Differential Equations (FFDEs) allow the joint consideration of uncertainty and memory in a unified dynamical system framework [18]. Motivated by these theoretical and practical considerations, this paper introduces a new DEA paradigm based on fractional order dynamics under fuzzy uncertainty. Efficiency is interpreted as a path-dependent function governed by FFDEs in the sense of Caputo. The proposed framework integrates and generalizes classical DEA, dynamic DEA, and windowed DEA as special cases obtained for specific values of fractional-order structure or uncertainty [19], [20]. The main contributions of this study are threefold: 1) a formal definition of fractional-order efficiency that incorporates long term memory effects, 2) extraction of the best and worst case efficiency under fuzzy parameter uncertainty using α-cut representations [10], [11], and 3) a convergence theorem that proves that fractional order efficiency converges to a subset of windowed DEA efficiency scores as the fractional order approaches unity [19], [20]. A numerical study inspired by a regional power generation system demonstrates the practical relevance of the proposed framework for stability analysis and stability-based ranking under noisy multi-year data [5], [14].

Consider a set of n DMUs where each DMUj (j = 1, … , n) uses m inputs xij (i = 1, … , m) to produce s outputs yrj (r = 1, … , s). The classical CCR model assumes constant returns to scale and evaluates the relative efficiency of a target DMU0 through the fractional programming form [2]:

∑r=1 ur yr0

max , (1)

u,v ∑ vi xi0 s. t.

∑r=1 ur yrj

≤ 1, j = 1, … , n, ur ≥ 0, vi ≥ 0. (2)

∑ vi xij To linearize, we normalize the denominator of the target DMU [1], [2]:

∑ vi xi0 = 1. (3)

The dual (envelopment) form is given as [2]:

minθ, (4)

s. t.

∑ λj xij ≤ θxi0, i = 1, … , m, ∑ λj yrj ≥ yr0, r = 1, … , s, λj ≥ 0, j = 1, … , n. (5)

j=1 j=1

The scalar θ ∈ (0,1] denotes the technical efficiency score. A value θ = 1 indicates that DMU0 lies on the efficiency frontier [1], [2]. Although theoretically robust, the CCR model is essentially static, assessing efficiency at a single point in time and assuming independence between successive observations [6], [7]. This limitation motivates the development of a memory-dependent fractional efficiency framework, which is introduced next.

Real-world input and output data are seldom precise. To capture epistemic uncertainty, we model them as triangular fuzzy numbers [10], [11]

L M U L M U

x ̃ij = (xij, xij , xij ), y ̃rj = (yrj, yrj , yrj). (19)

L M U

The membership function for a triangular fuzzy number a ̃ = (a , a , a ) is [10]:

a − a , a ≤ a ≤ a , a − a

μ (a) = U (20)

a ̃ a − a , a ≤ a ≤ a , a − a { 0, otherwise.

Using the α-cut representation (α ∈ [0,1]), we obtain intervals [11]:

α L M L U U M

x ̃ij = [xij + α(xij − xij), xij − α(xij − xij )], (21)

α L M L U U M

y ̃rj = [yrj + α(yrj − yrj), yrj − α(yrj − yrj )]. (22)

min max

For each α-cut level, the fractional efficiency Eq. (8) becomes an interval [Eα (t), Eα (t)] that can be obtained by solving two optimisation problems: minimising and maximising the ratio under the fuzzy constraints [10], [11]. In practice, we compute the lower and upper efficiency bounds using the approach of [10], [11]. The fuzzy fractional model thus integrates both memory (via α) and uncertainty (via α-cuts) in a unified DEA framework [12], [13].

We illustrate the proposed fractional-order fuzzy DEA model on a realistic dataset of five thermal power plants (DMUs) over three consecutive years (2022–2024). Each DMU uses two inputs: fuel consumption and operational cost (thousand USD), and produces one output: electricity generated (MWh). The data, including fuzzy uncertainty (±8% around the nominal value), are given in Table 1 [5], [14]. The fuzzy numbers are triangular: e.g. fuel = (95,100,105), cost = (47,50,53), output = (118,125,132). Table 1 shows Input-output data (triangular fuzzy numbers) for five power plants (midpoints shown; lower/upper bounds are ±8%). The table below includes the full set of fuzzy data used in the numerical example. Each entry is a triangular

L M U L U M

fuzzy number (a , a , a ) where a and a are ±8% of the nominal midpoint a . For brevity, only midpoints are shown in the main text; the complete fuzzy intervals are presented here.

This paper introduced a novel paradigm for efficiency measurement that overcomes the limitations of classical static DEA by embedding a fractional-order dynamic framework with fuzzy uncertainty. The proposed fractional-order efficiency explicitly models long-term memory effects through Caputo fractional derivatives and a power-law kernel, while fuzzy α-cuts capture epistemic imprecision in input-output data. The framework unifies and generalizes several existing DEA models: classical static DEA, dynamic DEA, and window DEA are recovered as special cases (e.g., when the fractional order α → 1 or when uncertainty is absent). The theoretical contributions include a formal definition of memory-dependent efficiency, a method for computing best- and worst-case efficiency intervals under fuzzy uncertainty, and a convergence theorem linking the fractional efficiency to window DEA scores. The numerical study, based on a realistic regional power system with five thermal plants over three years, demonstrated three key practical insights:

I. Memory control: The fractional order α directly governs the inertia of inefficiency. Smaller α (stronger memory) yields smoother and more stable efficiency trajectories, which is desirable for capital-intensive industries where past investments dominate current performance. Larger α (weaker memory) makes efficiency more responsive to recent changes, suitable for rapidly evolving technologies.

II. Uncertainty quantification: Fuzzy inputs produce efficiency intervals that provide decision-makers with realistic confidence bounds. These intervals narrow under fractional averaging, offering more precise assessments compared to static CCR models.

III. Convergence and comparability: As α → 1 , the fractional efficiency converges monotonically to window DEA scores, ensuring backward compatibility with established methods. Moreover, fractional DEA produces rankings that are more robust to data noise and better reflect gradual performance improvements than static or window approaches.

The MATLAB implementation provided generates two essential figures: the efficiency trajectory of a representative DMU under different memory strengths, and the convergence plot of all DMUs as varies. These tools enable practitioners to choose an appropriate fractional order based on the desired memory behaviour and to assess the sensitivity of efficiency rankings to uncertainty. Future work should extend the model to include undesirable outputs (e.g., CO₂ emissions), develop data-driven algorithms for optimal selection per DMU, and apply the framework to large-scale energy transition datasets. The proposed fractional order DEA paradigm opens a new avenue for dynamic performance evaluation in systems where history matters, and data are imprecise, a common situation in sustainability assessment, public policy, and industrial management. The integration of fractional calculus with DEA not only enriches the theoretical toolkit of efficiency analysis but also provides a practical, flexible, and interpretable method for real-world decision-making under memory and uncertainty.

No funds, grants, or other support were received.

The data are available from the corresponding author upon reasonable request.

The authors declare no conflicts of interest concerning the research, authorship, or publication of this article.

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