Mathematical analysis of a fractional model for a scheduling problem with precedence constraints and application to ant colony optimization

We proposed an original hybrid approach that combined a continuous modeling framework based on Caputo fractional differential equations with an Earliest Deadline First–Ant Colony Optimization (EDF–ACO) algorithm for solving the parallel machine scheduling problem with precedence constraints. Unlike most existing works, where the fractional order is usually fixed at its classical value, we investigated the influence of the fractional order $\alpha$ in the interval $(0,1]$ and analyzed its impact on the optimization process. The results showed that intermediate values of $\alpha$ allowed the effective incorporation of memory effects, leading to improved numerical stability, smoother convergence, and a reduction of oscillatory behavior. The fractional evaluation mechanism was coupled with the pheromone update strategy of the EDF–ACO algorithm, providing a more stable guidance for the search process. The existence and uniqueness of the solution to the fractional model were established using Banach's fixed point theorem, ensuring the consistency of the proposed continuous evaluation framework. Numerical experiments confirmed the effectiveness of the approach in terms of solution quality and convergence stability across different scheduling configurations.