A New Fractional Grey Cobb–Douglas Framework for Reliability Modelling under Uncertainty: Mathematical Development Properties

Abstract

Background:

Reliability approaches using uncertainty-demand models that describe degradation progression, memory properties, and time-dependent changes in a system's response. To achieve this target, a fractional-order Grey Cobb–Douglas (GCD) system is proposed in this paper that combines a fractional-order grey model with time-variant elasticities. The derivation permits the conventional GCD paradigm to be extended to account for uncertainty and dynamic system response.

Materials and Methods:

In addition, the proposed model is theoretically supported through rigorous analysis establishing the existence, uniqueness, stability, and identifiability of the induced reliability functions. The classical and fractional-order models are applied to a real dataset from the AI4I 2020 predictive maintenance source, extracted for this study, and analytical expressions for the failure rate, hazard function, and survival function under both models are derived.

Results:

The comparison results show that the DE model produces smoother degradation patterns and better reflects localized system dynamics. Although numerical differences between the two models are occasionally small, our fractional version leads to easier interpretation and better capture of the data’s temporal properties. These results indicate that the inclusion of fractional grey accumulation provides a convenient and flexible methodology for reliability modeling when information is scarce or imprecise.

Conclusion:

The structural flexibility and the universality of the proposed model were demonstrated through analytical derivations, simulations, and comparisons. Beyond its empirical validity, the proposed framework makes a new theoretical contribution, strongly buttressed by mathematical proofs establishing the existence, uniqueness, stability, and identifiability of the resulting reliability formulation. The fractional model may not always be supported by performance measures, but its simplicity and parsimony do not diminish its academic and practical importance. Hybrid calibration methods and adaptation to more demanding degradation environments might be considered in future studies.