In this paper, we address a new integrated optimization problem, which is the lot-sizing and scheduling problem in distributed heterogeneous hybrid flow shops (LSP_DHHFS). This problem is modeled as an integration of the lot-sizing sub-problem (LSP) and the distributed heterogeneous hybrid flow shop scheduling sub-problem (DHHFSP), these two sub-problems are closely intertwined. Considering its NP-hard property, a novel collaborative double DQN hyper-heuristic evolutionary algorithm (Col_D_DQNHEA) is proposed to address the LSP_DHHFS. The proposed Col_D_DQNHEA consists of three collaborative agents, each of which is a specialized double DQNHEA (D_DQNHEA). Specifically, the joint agent explores the global solution space to identify high-potential regions, while the lot-sizing and scheduling agents perform intensive local searches within these regions using problem-specific operations. This collaborative framework enables the Col_D_DQNHEA to iteratively refine solutions, efficiently converging to high-quality results within limited computation time. Experimental results demonstrate that the proposed algorithm outperforms state-of-the-art methods on the LSP_DHHFS.