Fractional Order Efficiency: A New DEA Paradigm Based on Memory Dependent Dynamics

Classical data envelopment analysis (DEA) is largely static: it benchmarks decision making units (DMUs) using momentary observations [۱,۱۴]. However, many manufacturing systems, especially energy systems, exhibit memory and path dependence, where current performance depends on historical paths [۱۸,۱۹]. In parallel, uncertainty in energy data is often epistemic and multi source, which is naturally modeled by fuzzy sets [۲۱,۲۵,۳۶]. 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 [۲۹-۳۱]. 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 [۱۱,۱۷]. 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 [۸,۱۵,۲۴].

کلیدواژه ها:

Fractional calculus ، data envelopment analysis ، fuzzy DEA ، memory effects ، energy efficiency