###### Abstract
[17] formulated a mixed-integer programming model for railway vehicle turnaround optimization, aiming to minimize empty train mileage while considering periodic maintenance plans. [PERSON] et al. [18] proposed a train operation optimization model under multiple constraints, utilizing an improved branch-and-price algorithm for solution derivation. [PERSON] et al. [19] developed an integrated multi-train scheduling and crew dispatching model based on spatiotemporal networks, transforming column generation subproblems into resource-constrained shortest path problems for effective problem-solving. [PERSON] et al. [20] applied spatiotemporal network graphs to construct a minimum-cost maximum-flow optimization model, simplifying train operation problems into linear formulations for improved computational efficiency. Vehicle maintenance and scheduling are interdependent components of urban rail transit operations. Maintenance plans impose scheduling constraints, as vehicles require periodic servicing to restore operational status, while scheduling decisions dictate maintenance timing, subsequently affecting the overall schedule. To enhance operational efficiency and reduce costs, a scientifically integrated optimization approach is necessary. Coordinating preventive maintenance and train scheduling can mitigate disruptions caused by unexpected failures, reduce frequent rescheduling and resource inefficiencies, and prevent issues such as service delays, overcrowding, and diminished passenger satisfaction. Increasingly, researchers have begun exploring joint optimization methodologies. [PERSON] et al. [21] investigated multi-vehicle scheduling in spatiotemporally discrete network scenarios, formulating a linear integer programming model to minimize operational costs, solved via Lagrangian relaxation. [PERSON] et al. [22] optimized maintenance scheduling by minimizing maintenance downtime, constraining the operational time between consecutive maintenance actions. [PERSON] et al. (2020) analyzed subway and railway vehicle cases, demonstrating that integrating maintenance considerations into scheduling reduces overall maintenance expenses. [PERSON] et al. (2020) developed a scheduling model constrained by daily operational demands and maintenance plans, optimizing train frequencies and ensuring timely maintenance to prevent over- or under-maintenance. [PERSON] et al. (2020) established a short-term scheduling model incorporating maintenance constraints, solved using column generation. In summary, while existing research has developed relatively comprehensive joint optimization models for urban rail maintenance and scheduling using spatiotemporal network methods, several issues remain unaddressed: (1) Most optimization models structure maintenance planning based on traditional periodic inspections (e.g., bi-weekly or quarterly), failing to refine maintenance intervals based on actual task durations and lacking precision in defining preventive maintenance cycle constraints. (2) Although spatiotemporal network theory has been integrated into optimization models, current constructions predominantly employ one-dimensional, single-layer network representations that focus solely on train scheduling states, neglecting the intricate spatiotemporal interdependencies between maintenance and scheduling plans. The objectives and limitations of existing studies are shown in Table 1. To address these gaps, this study proposes a joint optimization approach for preventive maintenance and train scheduling based on spatiotemporal network graphs. A multi-layer spatiotemporal network is constructed to represent subway vehicle scheduling and maintenance interactions, and a corresponding optimization model is formulated. An improved genetic algorithm is employed for model solution derivation. Ultimately, the proposed method aims to achieve the integrated optimization of subway vehicle preventive maintenance and scheduling.