An ODE-augmented matheuristic for the permutation flow shop scheduling problem with thermally driven processing times

This work analyzes a dynamic extension of the permutation flow shop scheduling problem (PFSP) in which processing times depend on the evolution of machine states. The proposed matheuristic combines a constructive heuristic, a biased randomization strategy, and a system of ordinary differential equations (ODEs) that models the thermal behavior of machines. This integration allows scheduling decisions to interact directly with time-varying machine conditions, producing schedules that adapt to operational degradation effects. The proposed ODE-enhanced matheuristic is evaluated on classical benchmark instances ranging from $20$ to $100$ jobs and up to $20$ machines. Processing times are modified through a state-dependent thermal model that introduces performance degradation as machine load increases. The computational study compares static schedules, the dynamic evaluation of those schedules, and the solutions obtained by the proposed method. The results show that ignoring thermal effects leads to a substantial increase in makespan. In contrast, the proposed matheuristic consistently mitigates this degradation, achieving average improvements of $36.2\mathrm{\%}$, $26.4\mathrm{\%}$, and $25.9\mathrm{\%}$ with respect to the dynamically evaluated static solutions. Furthermore, the proposed approach reduces the dispersion of makespan values across instances, indicating improved robustness under thermal fluctuations.