Study on Multi-stage and Multi-Objective Hierarchical Virtual Optimization Algorithm for Nonlinear Workflow

Nonlinear workflow scheduling in intelligent manufacturing systems is often constrained by complex task–resource dependencies, multi-objective conflicts, and inconsistencies between production planning and real execution. To address these challenges, this study proposes a virtual multi-stage nonlinear workflow model (VGD) and develops a hierarchical multi-objective scheduling algorithm (VGDP). In the VGD model, process tasks and manufacturing resources are abstracted into virtual nodes, enabling nonlinear process chains to be transformed into linearized multi-stage structures by introducing detection nodes that capture intermediate quality and cost requirements. Building on this model, the VGDP algorithm performs reverse-stage optimization to compute local feasible solutions within each stage, followed by forward Pareto-guided integration to construct the global non-dominated scheduling path.A real-world sheet-metal workshop is used to validate the proposed method. Comparative experiments against two widely adopted multi-objective evolutionary algorithms, NSGA-II and MOEA/D, demonstrate that the VGDP algorithm achieves higher-quality schedules while maintaining stable computational performance. In particular, VGDP improves production quality by approximately 0.3% compared with NSGA-II and 11.7% compared with MOEA/D under the same cost and time constraints. Moreover, the multi-stage virtualization mechanism effectively reduces the complexity of nonlinear workflows and enhances optimization stability as system scale increases.Overall, the proposed VGD/VGDP framework provides an effective and scalable approach to nonlinear multi-objective scheduling, offering strong applicability to advanced manufacturing environments that require high customization and dynamic process adaptability.