## 3 Principle of I-NSGA-III-VLC Method
### Crossover Operator  
For different coding stages, the ectopic single point crossover method was adopted in the rotation stage, and the two-point crossover method was adopted in the task stage. The crossover operator steps are as follows.  
Step 1: Given the crossover rate \(p_{c}\), \(0\leq p_{c}\leq 1\), the new population size is \(M\), where \(new\_pop=[\;]\), and \(s:=0\).  
Step 2: Select two parent individuals and define \(count_{1}:=0\) and \(count_{2}:=0\). The paternal chromosomes are segmented according to the number \"prefix\", as shown in Figure 4.  
Step 3: Combine the split parent individuals according to the same \"prefix\". Because the first \(c*(c-1)\) combinations belong to the rotation stage, the length of coding segments may be different, so the heterotopic single point crossover method is adopted.  
For the code segment combination in the rotation stage:  
Step 4: Randomly select two points \(num_{1,j}\) and \(num_{2,j}\), where \(num_{i,j}\) is the point selected from the coding segment with \"prefix\" \(j\) of the parent \(i\). In addition, \(0\leq num_{1,j}\leq len_{1,j}\leq g\), \(0\leq num_{2,j}\leq len_{2,j}\leq g\), \(0\leq len_{1,j}-num_{1,j}+num_{2,j}\leq g\), and \(0\leq len_{2,j}-num_{2,j}+num_{1,j}\leq g\), where \(len_{i,j}\) is the length of the coding segment of the parent \(i\) with \"prefix\" \(j\).  
Step 5: Generate a random number \(r_{c}\) between \(0\) and \(1\). If \(r_{c}\leq p_{c}\) and \(count_{1}<10\), crossover the first \(num_{1,j}\) genes of the coding segment with \(j\) as the \"prefix\" in parent1 and the first \(num_{2,j}\) genes of the coding segment with \(j\) as the \"prefix\" in parent2. \(count_{1}:=count_{1}+1\), generate crossover results, as shown in Figure 5, and enter Step 6. If \(p_{c}<r_{c}\), do not crossover, return Step 4. If \(count_{1}=10\), the parent individuals will be preserved in the new population.  
Figure 4: Segmented chromosome \(C_{1}\) and \(C_{2}\).  
Figure 5: Heterotopic single-point crossover result.  
The specific steps of modifying operator 1 are as follows.  
Step 6: Calculate the repetitions \(g_{i}\) of each number after crossover in the format \"Number: Repetitions\". Iterate through the numbers in \(g_{i}\). If the repetitions of the number are 1 and the set of numbers is a subset of the number \(S^{\prime}_{i}\), acquire \(S^{\prime\prime}_{i}\) and enter Step 7. Otherwise, return to Step 10.  
For the code segment combination in the task stage:  
Step 7: Adopt the method of two-point crossing and randomly select two points, \(pot_{1}\) and \(pot_{2}\), where \(pot_{1}<pot_{2}\), \(pot_{1}\%v=0\) and \(pot_{2}\%v=0\).  
Step 8: Generate a random number \(r^{\prime}_{c}\) between 0 and 1. If \(r^{\prime}_{c}\leq p_{c}\) and \(count_{2}<10\), exchange genes between two points, and if \(count_{2}:=count_{2}+1\), generate crossover results, as shown in Figure 6, and enter Step 9. If \(p_{c}<r^{\prime}_{c}\) and \(count_{2}<10\) do not cross, return to Step 7. If \(count_{2}=10\), the parent individuals are added to the new population, with \(s:=s+2\).  
The specific steps of modifying operator 2 are as follows.  
Step 9: Calculate the number of repetitions \(b_{i,j}\) of each character in each unit after crossover, where \(i\) represents the individual number of the offspring after crossover, and \(j\) represents the unit number in the format \"Number: Quantity of Repetitions\". According to the different units, traverse the number in \(b_{i,j}\). If the number of repetitions is less than the number of repetitions of the corresponding number in \(S^{\prime\prime}_{i}\) of the corresponding unit and the number of \(b_{i,j}\) is a subset of \(S^{\prime\prime}_{i}\), enter Step 10. Otherwise, return to Step 7.  
Step 10: Merge the cross results according to the \"prefix\" and add the results to the new population \(new\_pop=[\ ]\), where \(s:=s+1\). If \(s<M\), return to Step 2. If \(s=M\), the crossover operator ends.