# Multi-Scenario Robust Distributed Permutation Flow Shop Scheduling Based on DDQN
## 1 Introduction  
Since the early 21 st century, the manufacturing industry has entered a new era of intelligitization, informatization, and automation [1, 2]. With continuous breakthroughs in cutting-edge technologies such as artificial intelligence, the Internet of Things, robotics, and big data analytics, intelligent manufacturing has emerged as a key strategic approach for countries aiming to enhance their industrial competitiveness. As globalized production expands and manufacturing scales increase, enterprises face increasingly complex and diversified market demands. To maintain a competitive edge in the global market, large manufacturing enterprises are actively exploring ways to reduce production costs, improve efficiency, and enhance production flexibility. In this context, manufacturing paradigms are undergoing a fundamental shift from traditional centralized workshop-based production models to distributed production systems. The DPFSP refers to the simultaneous processing of multiple jobs across multiple factories or production lines, where jobs may be reassigned or permuted between different factories. The objective is typically to minimize the total completion time (makespan). Theproblem was first formally introduced by [PERSON] and [PERSON] [3], who proposed six mixed-integer linear programming formulations based on different modeling perspectives. Following this foundational study, contributions to the DPFSP literature have steadily increased. [PERSON] et al. [4] proposed a series of dispatching rules based on heuristics or experiences to optimize flow shop scheduling problems. These rules offer simplicity and practicality, enabling high-quality near-optimal solutions with relatively low computational costs. [PERSON] et al. [5] extended traditional routing-based dispatching rules using a simulation-driven approach and conducted a systematic evaluation of rule performance in dynamic environments, emphasizing the need for responsive and robust scheduling strategies in settings with complex constraints. [PERSON] et al. [6] introduced a novel competitive memetic algorithm. This algorithm effectively improves global optimization performance by dividing the search space and employing multiple subpopulations for localized search. [PERSON] et al. [7] proposed an improved hybrid tabu search algorithm, which enhances traditional tabu search by incorporating problem-specific features and optimization objectives to improve performance and computational efficiency. [PERSON] et al. [8] also introduced an iterative greedy algorithm. This method is particularly well-suited for solving scheduling challenges involving complex constraints and multiple objectives, such as those encountered in production scheduling and logistics optimization. [PERSON] [9] proposed an innovative approach in solving DPFSP to optimize the scheduling scheme utilizing constraint planning techniques. [PERSON] et al. [10] developed a self-adaptive population-based iterated greedy algorithm to address the DPFSP. With the advancement of intelligent technologies, emerging methods such as DQN and hybrid algorithms have provided powerful tools to address large-scale, dynamic, and complex DPFSP scenarios. By integrating uncertainty into the state and reward functions, [PERSON] et al. [11] explored the application of Q-learning to DPFSP with uncertain processing times, enabling agents to dynamically adjust decisions based on learned experiences. [PERSON] et al. [12] designed a K-means-based workload allocation rule to balance load among factories and applied Deep Q-Network to capture nonlinear relationships between states and actions. This enables the agent to generalize and solve problems of different scales after training. Due to the inherent complexity of DPFSP, no single algorithm has yet demonstrated superior performance across all scenarios [13]. Most existing studies focus on specific cases by simplifying problem models or assuming fewer real-world constraints to improve computational performance.