## 1 Introduction  
Discrete manufacturing accounts for around 62.1% of China's manufacturing market and is widely employed in fields like machinery and equipment, automotive parts and semiconductors [1, 2]. The discrete manufacturing system is a complicated process and contains numerous elements; transportation is crucial for maintaining the production process. Traditional transportation relies heavily on manual methods, which are inefficient and costly and cannot meet modern manufacturing needs for fast and flexible material distribution. AGVs, as advanced automated transporters, enhance material transportation efficiency, reduce manual intervention and lower transportation costs. Their implementation in discrete manufacturing has witnessed exponential growth, particularly in flexible flow shop (FFS) environments, where the co-optimization of production scheduling and AGV coordination (denoted as FFSP-AGV) has emerged as a critical research frontier for industrial applications. FFS is a highly flexible and productive manufacturing system that can adapt to multi-variety and small-batch production. However, traditional FFS scheduling research mostly overlooks the impact of transportation on production and is mainly confined to traditional manual transportation. This may lead to resource competition conflicts, such as idle equipment and AGV congestion, which can prolong the production cycle and reduce the system's energy efficiency. In turn, this gives rise to deviations between production planning and actual execution. Therefore, studying the integrated scheduling problem of FFS and AGV is important for enhancing the overall performance of discrete manufacturing systems. In this paper, the coupling mechanism and integrated scheduling problem of FFS and AGV (FFSP-AGV) are investigated by applying system engineering ideas. This approach is expected to fully utilize the advantages of integrated scheduling, promote the efficient operation of the entire discrete manufacturing system and generate more economic and social benefits for enterprises. When addressing a fresh scheduling problem, especially for an integrated scheduling problem (FFSP-AGV), it is important to establish a feasible and effective mixed-integer linear programming (MILP) model for it [3]. This is because such a feasible MILP model can further examine the inherent information of scheduling problems and be employed as a reference to construct more approximate approaches. When formulating MILP models for addressing shop scheduling problems, four modeling ideas are usually adopted: sequence-based idea, position-based idea, time-based idea and adjacent sequence-based idea. The differences among them lie in their approach for determining the job sequence [4]. Of these four ideas, the adjacent sequence-based idea is more preferred and will be used to model the FFSP-AGV in this paper due to its resulting lower model complexity and higher solution efficiency. Compared to previous related studies, the paper offers the following four main contributions:  
1. An integrated scheduling problem of FFS and AGV, called FFSP-AGV, is investigated by considering the interaction between the shop production link and the AGV transportation link. 2. A MILP model that minimizes the production cycle, i.e., makespan, is developed based on the adjacent sequence-based modeling idea. 3. The feasibility and model complexity (size complexity and computation complexity) of the MILP model are verified and evaluated through a series of specially designed test instances. 4. Some key parameters in FFSP-AGV are further analyzed, and the marginal effect of AGV numbers on the production cycle is clearly revealed. The rest of this paper is organized as follows. Section 2 reviews the existing works from three aspects (FFSP, AGV scheduling and MILP modeling) and summarizes the research gap, respectively. Section 3 introduces the problem description of FFSP-AGV and several necessary assumptions. In Section 4, a MILP model is built using the concept of adjacent sequence modeling idea.