Multi-stage job scheduling in collaborative heterogeneous clouds poses significant challenges due to uncertain task durations, cost and performance trade-offs, and the complexity of coordinating resource allocation across distributed infrastructures. To address these challenges, we formulate a multi-objective binary integer programming model for fuzzy job scheduling under a master–worker cloud architecture. This problem is NP-hard, making exact solutions impractical. To obtain high-quality approximate solutions, we propose MFJSHC, a hybrid meta-heuristic framework that combines NSGA-II with Deep Reinforcement Learning (DRL). NSGA-II is augmented with Dirichlet-based population initialization to enhance both the quality and diversity of the resulting Pareto front. Meanwhile, DRL introduces experience-driven policies that guide local search and refine stage-to-VM assignments. Experiments on real-world datasets show that MFJSHC consistently outperforms three state-of-the-art baselines in terms of Hypervolume, demonstrating its effectiveness and scalability for cloud scheduling.