Study on multi-stage and multi-objective hierarchical virtual optimization algorithm for nonlinear workflow

Nonlinear workflow scheduling in intelligent manufacturing systems is often complicated by complex task-resource dependencies, conflicting objectives, and strict intermediate inspection constraints. To address these issues, 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 hierarchical multi-stage structures through the introduction of detection nodes that encode intermediate quality and cost requirements. Based on this representation, the VGDP algorithm performs reverse-stage local feasibility optimization and forward Pareto-guided integration, while incorporating dynamic virtual pruning to reduce unnecessary path expansion during the search process. Experimental results based on a real sheet-metal workshop case show that, compared with representative multi-objective evolutionary algorithms, the proposed VGDP framework achieves competitive Pareto-front quality and exhibits comparatively stable convergence behavior and computational efficiency under strict nonlinear constraints. Evaluated using Hypervolume (HV) and Inverted Generational Distance (IGD), the method demonstrates competitive solution quality and stable optimization performance. More importantly, the primary contribution of this study lies in the integration of hierarchical workflow modeling and multi-objective optimization under intermediate inspection constraints, rather than in claiming universal numerical superiority over existing algorithms. Overall, the proposed VGD/VGDP framework provides an effective structured approach for nonlinear multi-objective scheduling within the evaluated intelligent manufacturing scenario.