In recent years, grey models based on fractional-order accumulation and/or derivatives have attracted considerable research interest because they offer better performance in handling limited samples with uncertainty than integer-order grey models; however, there remains room for improvement. This paper considers a more flexible and general structure for the fractional grey model by incorporating a generalized fractional-order derivative (GFOD) that complies by memory effects, resulting in the development of a generalized fractional grey model (denoted as GFGM(1,1)). Specifically, we comprehensively analyse the modelling mechanism of the proposed GFGM(1,1) model, involving model parameter estimation and time response function derivation, and discuss the link between the proposed approach and existing special cases. Then, to further improve the efficacy of the proposed approach, four mainstream metaheuristic algorithms are employed to ascertain the orders of fractional accumulation and derivatives. Finally, we carry out a series of simulation studies and a real-world application case to demonstrate the applicability and advantage of the our approach. The numerical results show that GFGM(1,1) outperforms other benchmarks, and some significant insights are obtained from the numerical experiments.

近年来，基于分数阶累积和/或导数的灰色模型引起了相当大的研究兴趣，因为它们在处理具有不确定性的有限样本方面比整数阶灰色模型具有更好的性能；但是，仍有改进的余地。本文通过结合符合记忆效应的广义分数阶导数（GFOD），为分数灰度模型考虑更灵活和通用的结构，从而开发出广义分数灰度模型（表示为 GFGM(1,1) ）。具体来说，我们全面分析了所提出的 GFGM(1,1) 模型的建模机制，涉及模型参数估计和时间响应函数推导，并讨论了所提出的方法与现有特殊情况之间的联系。然后，为了进一步提高所提出方法的有效性，采用四种主流元启发式算法来确定分数累积和导数的阶数。最后，我们进行了一系列的模拟研究和一个真实的应用案例来证明我们方法的适用性和优势。数值结果表明 GFGM(1,1) 优于其他基准，并且从数值实验中获得了一些重要的见解。