## 5 Instance Verification & Algorithm Comparison
Among them, A1 is the improved non-dominated genetic algorithm herein based on the decoding strategy proposed herein; A2 is the inverse scheduling strategy proposed in [18] in combination with the non-dominated genetic algorithm. Table 3 shows that the solutions of 12 cases were calculated with two algorithms for 20 times respectively to evaluate the scheduling objective and the average running time of the algorithm for the best results. Figure 9 shows the convergence diagram of two algorithms for MK12 (where \(C_{max}\) is the average maximum total makespan of workpieces, TP is the average total delay penalty, \(\eta_{h}\) is the average utilization rate of the heat treatment equipment, and CT is the average running time of the algorithm). Figure 8: Optimized Curve of the Acceptance Factor for Case MK10. From the perspective of the algorithm convergence speed: two algorithms have excellent convergence at the initial stage, but A1 has the faster convergence speed and better algorithm performance than A2 during the entire iteration process. Figure 10 is the Pareto frontier of Case MK3. To display the output of the scheduling result with the heat treatment process guided by the decoding rules, we generated the equipment Gantt chart of a non-dominated solution of the Pareto frontier. Figure 11 shows the equipment Gantt chart of the MK3 result obtained with the algorithm proposed herein, where Equipment 6 is the heat treatment equipment. It can be seen from the figure and Table 4 that the heat treatment process has been taken under consideration timely, other machining processes are tightly scheduled on the equipment, and the equipment utilization rate is high [19]. Among them, the batches on the heat treatment equipment corresponding to MK3 are shown in Figure 12 below. It can be seen from the data in this figure that the heat \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & \(\mathbf{C_{max}}\) & **TP** & \(\mathbf{\eta_{h}}\) & **CT** \\ \hline MK3 & 327 & 0 & 93.27 & 71.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: A non-dominated optimization target value and running time of MK3. Figure 10: Pareto frontier. treatment equipment has been almost always in the running status after it started running, and the equipment utilization rate of each heat treatment batch is high, which has directly proved the effectiveness of the coding-encoding method containing the heat treatment process proposed herein to solve the actual problems. In addition, it can be seen from the following figure that the most of orders are processed in batch for the heat treatment, but the heat treatment processing time of various batches is concentrated, which is more beneficial to minimizing \(C_{max}\) and ensuring the delivery time of workpieces. Meanwhile, it has also proved the effectiveness of the decoding rules for the machining process based on the equipment bottleneck and the heat treatment process based on the heat treatment equipment volume and job delivery data. Figure 13 shows the equipment Gantt chart of the MK7 result obtained with the algorithm proposed herein, where Equipment 6 is heat treatment equipment, and Table 5 shows the optimization target value and running time of this non-dominated solution. As can be seen from the figure below, the machining equipment has almost been running during the work, the heat treatment equipment has rarely waited during the processing, the utilization rate of the heat treatment equipment is 94.13%, and heat treatment batches are connected with each other at an almost zero gap. Such phenomena have proved the effectiveness of the decoding rules for the heat treatment process based on the heat treatment equipment volume and delivery date proposed herein. Figure 12: Heat Treatment Batches on the Heat Treatment Equipment corresponding to MK3. Figure 14 shows the equipment Gantt chart of the MK12 result obtained with the algorithm proposed herein, where equipment 8 is heat treatment equipment as shown in the figure.