## 6 Conclusions  
In this paper, the multi-factor flexible job shop scheduling problem under parallel multi-equipment is studied, including machine setup time, job handling time and machine processing time, for multi-equipment production systems with multi-variety and small-batch job processing task orders. A multi-objective optimization model based on the maximum completion time, total number of machine adjustments, total number of workpiece handlings and total machine load was constructed. An improved discrete particle swarm optimization algorithm based on Pareto selection and an adaptive nonlinear inertia weight change strategy was proposed. Firstly, a hybrid initialization method combining process coding and machine coding was used to generate the initial population in the initialization phase to maintain the diversity of particles and improve the quality of the initial population. Second, an improved cross-variance operator was used to update the positions of the particles and promote good information sharing among the particles. Then, algorithm simulation experiments of different scales were conducted on Kacem and Brandimarte standard datasets and compared with other algorithms to verify the effectiveness of the IDPSO algorithm results. Finally, a multi-factor FISP example including setup, handling and processing time is designed to verify the feasibility and effectiveness of the IDPSO algorithm in solving multi-objective FJSPs with good practical value through multiple simulations.  
However, this study has three limitations. First, this paper studies the static job shop scheduling problem for multi-variety and small-batch processing tasks only. Second, the algorithm in this paper improves the local search capability of the algorithm by changing the adaptive nonlinear inertia weights. The optimal solution can be obtained when the instance size is small, but it can only be close to the known optimal solution in large-scale instances. Finally, in the algorithm process of selecting individual \"_gbest_\" particles in multi-objective decision-making, the fourth optimization objective, namely minimizing the total load of the machine, is chosen as a secondary target. The key to this objective lies in effectively allocating machine loads, reasonably distributing tasks, managing operation times and optimizing equipment utilization. However, in some actual production workshops, it may not be necessary to consider minimizing the total load of the machine because the complexity of the objective function may not warrant this goal. In subsequent research, the IDPSO algorithm can be combined with other local search algorithms to form a hybrid algorithm to further improve the local search capability of the algorithm and apply it to more complex job shop scheduling problems (such as dynamic planning and variable-batch problems). In addition, using green scheduling and production cost targets as the optimization objectives of this algorithm to actively respond to government policies of green and smart manufacturing in China can further enrich the application scope of the algorithm model.  
Conceptualization, Z.W. and J.K.; methodology, Z.W. and J.K.; validation, Z.W. and J.K.; formal analysis, Z.W.; investigation, Z.W.; resources, J.K.; writing--original draft preparation, Z.W. and J.K.; writing--review and editing, Z.W. and J.K.; supervision, J.K.; project administration, J.K. and Z.W.; funding acquisition, J.K. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Humanities and Social Science Youth foundation of the Ministry of Education of China, grant number 20 YJC630054.  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors declare no conflicts of interest.