# Digital Twin Model Quality Optimization and Control Methods Based on Workflow Management
## 5 Micro-Level Model Quality Optimization Method
### Solution Selection  
A mathematical model for the optimal selection of the solution is developed. Maximizing the validity of the solution and minimizing the variance of the validity for each index are objectives 1 and 2, and objective 1 is transformed into a minimization objective by taking the inverse.  
\[F_{1}=\min(1/f_{sd}(s)) \tag{11}\]  
\[F_{2}=\min\bigl{(}f_{sp}(s)\bigr{)} \tag{12}\]  
\(A_{ik}\) identifies whether abnormal indicator \(T_{i}\) is assigned to the solution; \(X_{ik}\) indicates that each abnormal indicator should be assigned to a solution. The constraint is as follows:  
\[\begin{array}{l}\sum\limits_{k}A_{ik}=\begin{cases}1,i=1,2,\ldots,n\\ 2,i=1,2,\ldots,n\\ \end{cases}\\ A_{ik}\leq X_{ik},1\leq i\leq n,1\leq k\leq m,\\ X_{ik}\in\{0,1\},1\leq i\leq n,1\leq k\leq m,\end{array} \tag{13}\]  
Employing the constructed mathematical model, an optimal solution selection method is proposed based on the NSGAII algorithm [26; 27]. To take advantage of multiple solutions while not affecting their implementation with too many solutions, we set at least one and at most two solutions for each abnormal index. The steps of the solution are as follows:  
Step 1: The population and algorithm parameters are initialized.  
Step 2: The operation is initialized, and the fitness is calculated [28].  
Step 3: Operator manipulation is conducted, and individual selection is performed according to the environment selection strategy.  
Step 4: it is determined whether the iteration is finished. If yes, the TOPSIS method [29; 30] is used to set the relative weight vector, and the process ends by outputting the optimal solution; otherwise, the process proceeds to Step 2.  
1. Coding method. The chromosome code consists of the sequence of the desired solution, and the candidate solution codes are numbered and binary coded, based on which the initialized populations are generated.
2. GA operator manipulation strategy. The fine-grained original fitness assignment method and the density estimation method were combined to calculate the fitness. The roulette wheel selection method was adopted for individual selection, and GA crossover and mutation manipulation operators were executed to generate new solutions. For the crossover operator, two chromosomes were randomly selected, and crossover points were randomly selected for gene fragment exchanges. The number of crossover points was set to two. For the mutation operator, one gene in the chromosome is randomly selected for efficient regeneration in the chunk.
3. Environment selection strategy. Environment selection is used to evaluate the quality of contemporary and elite solutions, and the better solution will be selected for the next iteration. The evaluation strategies adopted are the non-dominated solution level and congestion calculation [28]. The non-dominated solution level indicates the dominance relationship between individuals in the population, and a higher level (smaller value) indicates a better solution. A greater congestion value indicates better distribution performance of the solution.