## 5 Experimental Results  
### Feasibility of the MILP Model  
To test the feasibility of the MILP model formulated in Section 4.2, an example is first presented for analysis. Consider an FFS with two processing stages (s = 2), where three and two parallel machines are installed. For the convenience of analysis, in this example, we assume that the parallel machines in each stage are identical. Five jobs (n = 5) are prepared to be processed by the machines and transported by two AGVs (r = 2). Detailed data are shown below in a matrix form from Equations (20)-(22). The MILP model is programmed and executed using CPLEX commercial software. The experimental outcome in the CPLEX software is presented in Figures 2 and 3.  
//solution (optimal) with objective 63  // Quality Incumbent solution:  // MILP objective  // MILP solution norm |x| (Total, Max)  // MILP solution error (Ax=b) (Total, Max)  // MILP x bound error (Total, Max)  // MILP x integrality error (Total, Max)  // MILP slack bound error (Total, Max)  //  
Z = 63;  
In detail, Figure 2 plots the optimal makespan of the FFSP-AGV. Obviously, it is 63 s. Figure 3 plots the running process of the MILP model in CPLEX. The optimal solution is found when the red solid line intersects with the green dashed line. Figures 4 and 5 show the values of binary decision variables and continuous decision variables in the MILP model, respectively. The Gantt chart of the optimal schedule obtained via CPLEX is displayed in Figure 6. It can be seen that with the help of binary/continuous decision variables in Figures 4 and 5, the job sequence, machine assignment and AGV assignment are reasonably and uniquely determined. Therefore, according to these experimental results, the feasibility of the MILP model is verified.  
\[TT_{k}^{k^{\prime}}=\begin{bmatrix}&\text{M}_{11}&\text{M}_{12}&\text{M}_{13} &\text{M}_{21}&\text{M}_{22}\\ \text{M}_{11}&0&1&2&4&6\\ \text{M}_{12}&1&0&3&2&2\\ \text{M}_{13}&2&3&0&5&6\\ \text{M}_{21}&4&2&5&0&4\\ \text{M}_{22}&6&2&6&4&0\end{bmatrix} \tag{20}\]  
\[TT_{LU}^{k}=\begin{bmatrix}\text{M}_{11}&\text{M}_{12}&\text{M}_{13}&\text{ M}_{21}&\text{M}_{22}\\ 5&3&4&2&1\end{bmatrix} \tag{21}\]  
Figure 3: Running process of MILP in the CPLEX interface.  
Figure 2: Optimal makespan of the FFSP-AGV in the CPLEX interface.  
In the actual production, there are often practical constraints, such as collision avoidance, real-time AGV scheduling and idle time analysis optimization. For collision avoidance, a priority-based strategy can be introduced to assign priority levels to AGVs, allowing those with higher priority to pass the critical paths first. Speed adjustment strategies can also be used to prevent collisions by adjusting AGV speeds when potential collisions are detected. For real-time AGV scheduling, wireless communication networks and positioning technologies can be leveraged to enable real-time communication between the AGVs and the scheduling center, promptly updating AGV statuses and task completion. For idle time analysis optimization, the idle time includes both equipment idle time and AGV idle time, as shown in the Gantt chart in Figure 6. Drawing on the equipment-driven comprehensive scheduling algorithm for adjusting equipment idle periods, analyze the time sequences of processing and transportation tasks to identify potential idle periods and take optimization measures. For example, schedule subsequent tasks or perform maintenance for equipment about to become idle, and promptly assign new transportation tasks or recharge idle AGVs. Additionally, optimize the production processes and transportation routes to reduce waiting times between tasks, lowering the likelihood of idle time.