###### Abstract  
To address the multi-objective flexible job shop scheduling problem in rolling production mode (FJSP-RFM), this study proposes a Multi Objective Improved of Salp Swarm Algorithm (MISSA) that simultaneously optimizes equipment utilization and total tardiness. The MISSA generates initial population through various heuristic strategies to improve the initial population quality. The exploitation capability of the algorithm is enhanced through the global crossover strategy and variety of local search strategies. In terms of improvement strategies, the MISSA (using all three strategies) outperforms other incomplete variant algorithms (using only two strategies) in three metrics: Generational Distance (GD), Inverted Generational Distance (IGD), and diversity metric, achieving superior results in 9 test cases, 8 test cases, and 4 test cases respectively. When compared with NSGA2, NSGA3, and SPEA2 algorithms, the MISSA demonstrates advantages in 8 test cases for GD, 8 test cases for IGD, and 7 test cases for the diversity metric. Additionally, the distribution of the obtained solution sets is significantly better than that of the comparative algorithms, which validats the effectiveness of the MISSA in solving FJSP-RPM. flexible job shop; salp swarm algorithm; multi-objective optimization; date of delivery 20 May 202522 May 20252  
workshops was verified through experiments. [PERSON] et al. [3] studied the integrated scheduling problem of FJSP and Automated Guided Vehicle (AGV) with the objective function of minimizing the maximum completion time. A new mixed integer linear programming model and dual population collaborative genetic algorithm was proposed. [PERSON] et al. [4] proposed a hybrid algorithm combining quantum particle swarm optimization and variable neighborhood search to efficiently solve FJSP with the goal of maximum completion time. The stability and applicability of the algorithm were verified through testing on standard test cases and an industrial case. [PERSON] et al. [5] used memetic algorithm to solve the problem of green type-2 fuzzy flexible workshop. [PERSON] et al. [6] proposed a simulation optimization framework for solving stochastic flexible job shop scheduling problems in the semiconductor manufacturing industry. [PERSON] et al. [7] proposed a new solution algorithm based on quantum annealing, which combines the power of quantum annealing with classical techniques for calculating FJSP. [PERSON] et al. [8] proposed an FJSP model with flexible discrete operation order, considering that some job operations in the real job process do not have any order constraints. They verified the effectiveness of the model through CPLEX (a mathematical programming solver developed by IBM). In summary, most current FJSP models aim to minimize the maximum completion time as their core objective. Metaheuristic optimization algorithms have demonstrated significant advantages in solving the FJSP. However, traditional FJSP has two limitations. First, it is too idealistic. Current research is mostly based on the assumption of static scheduling, generating the global optimal solution at once. This deviates from the actual production of the enterprises. In actual machining operations, the scheduling workload is generally greater than the equipment capacity. New parts will be rolled into production at regular production cycles. The scheduling plan should be a periodic rolling adjustment. Therefore, this article describes a rolling production mode (defined in Section 3). The second limitation of traditional FJSP is that although FJSP has multiple scheduling objectives, current research mostly focuses on minimizing the maximum completion time as the core objective (without considering the delivery time of each workpiece). In the process of actual machining operations, each workpiece has an expected delivery date. On the one hand, enterprises need to avoid delays, and on the other hand, they hope to reduce equipment idle rates.