Dynamics of 4D supply chain system with fractal fractional derivatives insight of stability analysis and ANN prediction.

This paper investigated 4D supply chain finance system utilizing fractal fractional derivatives with the generalized Mittag-Leffler kernel. The model incorporates key interactions between customer demand, distributor inventory, and retailer orders, creating a dynamic system governed by fractional-order differential equations. We analyze the positiveness and boundedness of the systems solutions. Stability analysis is performed using local asymptotic and global stability criteria, and we further develop linear control strategies employing linear feedback. Additionally, a neural network (ANN) model is employed to predict the system's future states, including two scenarios of predictions, the first and third future values. Performance is evaluated using RMSE, MSE, and MAE metrics. The ANN model demonstrates excellent accuracy, achieving RMSE as low as 0.00011 for the first future value and 0.00029 for the third future value, confirming its robustness in capturing the system's across varying prediction scenarios. Numerical simulations illustrate the system's complex dynamics, and the proposed control and prediction methods show significant potential in managing supply chain systems. Understanding the financial system's long-term behavior, as well as learning about its resilience and crisis potential, are two benefits of fractional order financial model stability analysis. The study shows that Artificial Neural Networks (ANNs) can accurately forecast future states of fractional systems.