## 5 Instance Verification & Algorithm Comparison  
To perform verification: (1) Accept the impact of the acceptance factor value on the maximum makespan and heat treatment equipment utilization rate herein; (2) Through comparing our algorithm with other algorithms based on other decoding rules, prove that the decoding strategy proposed herein has improved the algorithm convergence in terms of handling the FJSP with the heat treatment process. 12 instances MK01-12 are extended from [PERSON] (1993) and the detailed information is shown in Figure 7, where n indicates the job quantity of each instance, oj indicates the quantity scope of each job, m indicates the quantity of machining equipment in this workshop, mh indicates the quantity of batch processing equipment in this workshop, copro indicates the scope of the process quantity contained in each job, meq indicates the scope of the quantity of optional processing equipment for each machining process, proc indicates the scope of the processing time for each machining process, del indicates the scope of the job delivery date, SingleV indicates the volume of the single piece in each job, proh indicates the processing time of the heat treatment process of each job, and HMaxVol indicates the maximum volume of the heat treatment equipment in this workshop. All the above instance verification algorithms run on a personal computer with the processor of i5-2400 CPU, dominant frequency of 3.1 GHz and memory of 4 GB. The improved NSGA proposed herein was used to solve the problem. The number of iteration times of the algorithm was set to 200, the initial population size is set to 100, the maximum crossover probability was set to 0.45, the minimum crossover probability was set to 0.25, the maximum mutation probability was set to 0.1, and the minimum mutation probability was set to 0.01. Figure 6: Mutation Operator Based on the Process Coding. Figure 7: Instance Data. To obtain the optimal value of the acceptance factor, in this section, we calculated different values of the acceptance factor, and then obtained the solution result of the FJSP with the heat treatment process with the improved genetic algorithm by following the decoding rule proposed herein. For Case MK10 proposed above, we solved this problem with the acceptance factor, provided that the value range is [0, 1] and the value increment is 0.001. Solved this problem with the acceptance factor at each value for 20 times respectively, and investigated the maximum makespan, heat treatment equipment utilization rate and running time of the best result, which are shown in Figure 8 below. Because this paper studied the multi-objective optimization problem with the adoption of the non-dominated genetic algorithm, multiple non-dominated solutions may be obtained during each algorithm calculation. Therefore, only the solutions with the shortest running time in the Pareto optimal solution set are compared in this section. As shown in Figure 7, the case solution was obtained within the effective time with the decoding rule and adaptive cross-mutation operator proposed herein, and more than 90% of the utilization rate of the heat treatment equipment volume was obtained. It can be seen from this figure that when the acceptance factor is between [0.860, 0.871], the minimum \(C_{max}\) and maximum equipment utilization rate are obtained. Therefore, when the average value \(\delta=\frac{0.860+0.871}{2}=0.865\) is taken, the obtained maximum makespan is the smallest, and the utilization rate of the heat treatment equipment volume is the largest. To further verify the contribution of the decoding strategy proposed herein to the improvement of the algorithm convergence, we performed algorithm comparison with 12 cases proposed in this paper, the number of iteration times of the algorithm was set to 200, and the population quantity was set to 30. Table 3 shows the running results of above 12 cases with different decoding methods for different heat treatment processes in combination with the non-dominated genetic algorithm.