## 4 Conclusions  
In this paper, we present an integrated task- and path-planning approach for mobile robots in a smart factory. The basic idea is that in the stage of task assignment, the real paths for mobile robots are identified and the time and energy consumed by mobile robots and machining centers are calculated. Then, a greedy strategy in conjunction with the Looking-backward Search Strategy or Regret-based Search Strategy is used to obtain task assignments in time series that satisfy the proposed objectives, which enables a joint efficient solution to be established for both task assignment and path planning. The real time consumed on the planned paths is used as the basis to adjust and improve the selection of mobile robots and task assignments, and the precedence constraints between sequential manufacturing tasks are considered simultaneously.  
The performance of our proposed methods is compared with NBS-ISPS, a popular method for PC-TAPF problems in a simulated factory environment with small-scale problems. The results show that the success rate of the proposed methods outperforms that of the NBS-ISPS method, and the average time consumption of the proposed methods is close to that of the NBS-ISPS method. Moreover, our proposed methods run much faster than NBS-ISPS, which allows smoother deployment of these methods in real-world scenarios.  
We also analyze the feasibility of our proposed methods with large-scale problems. The results show that the proposed LSS method and RSS method are better than the traditional method (Baseline) when the number of tasks or robots increases, especially when the  
Figure 11: A simulation environment developed in MATLAB 2023a to verify the feasibility of generated paths for mobile robots [28]. The numbers in blue circles and green circles are the indices of robots and start positions of transportation tasks, respectively.  
Figure 10: Computational efficiency of proposed methods with 5 robots for large-scale problems: (**a**) relationship between number of search iterations and number of tasks (\(n\)); (**b**) relationship between runtime and number of tasks (\(n\)). The Baseline method is not included in (**a**) because it does not utilize any searching strategy for assigning tasks, i.e., no search iterations are involved.  
number of tasks is large. In addition, when the weight of time consumption increases, the advantage of our approach becomes more noticeable. The proposed approach generally requires a short computation time, and it can help reduce the idle time of machining centers and make full use of these resources to improve the overall operational efficiency of smart factories.  
One limitation of this study is that the size of the studied factory is relatively small. We will examine the reliability and computational efficiency of the proposed approach in a large-scale factory environment where the planning and scheduling of transportation tasks for mobile robots can be more difficult. The complex relationship between manufacturing time and consumed energy will also be considered in future work.  
Conceptualization, S.L., Y.B., B.F. and D.Y.; methodology, S.L.; formal analysis, S.L., Y.B., B.F. and D.Y.; data curation, S.L.; writing--original draft preparation, S.L. and B.F.; writing--review and editing, Y.B.; funding acquisition, Y.B. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key R&D Program of China (2022 YFB4702400).  
Not applicable.  
The data supporting the conclusions of this article will be made available by the authors upon request.  
The authors declare no conflicts of interest.