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This paper presents a new formulation and valid constraints for a periodic capacitated vehicle routing problem with multiple depots, heterogeneous fleet, and hard time-windows (MDHFPCVRP-TW). The problem raises from a real-world application in the vending machine industry in Medellín, Colombia. Our main contribution is a novel formulation that replaces binary depot-client assignment variables with continuous auxiliary variables and implements depot replication, achieving both model simplicity and computational efficiency. We introduce preprocessing techniques and valid constraints, particularly focusing on capacity-based constraints with client combinations, which significantly strengthen the formulation’s linear relaxation. Computational experiments demonstrate that our formulation consistently outperforms previous approaches across different instance sizes, achieving optimality for small instances and maintaining single-digit optimality gaps for medium-sized instances where earlier formulations showed gaps above 12%. The formulation shows particularly strong performance in solution time, often requiring less time to find feasible solutions. While limitations persist for very large instances, our results suggest promising directions for developing hybrid exact-heuristic methods for industrial-scale problems.本文针对多车场、异构车队、硬时间窗周期容量车辆路径问题（MDHFPCVRP-TW）提出了全新的建模形式与有效约束集。该问题源自哥伦比亚麦德林市自动售货机行业的实际应用场景。本文的核心贡献在于一种全新建模形式：以连续辅助变量替代二进制车场-客户分配变量，并引入车场复制机制，同时实现了模型简洁性与计算效率的双重提升。本文还提出了预处理技术与有效约束集，尤其聚焦于基于客户组合的容量约束，可显著强化建模形式的线性松弛效果。计算实验结果表明，相较于已有方法，本文提出的建模形式在各类实例规模下均表现更优：小型实例可实现最优解，中型实例可保持个位数最优性间隙，而此前的建模形式在中型实例上的间隙均高于12%。该建模形式在求解时长上表现尤为突出，通常仅需更短时间即可找到可行解。尽管针对超大型实例仍存在一定局限性，但本文研究结果为面向工业级问题的精确-启发式混合方法开发指明了颇具前景的方向。