# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 2 Related Work
### Dual-Population Co-Evolutionary Stage
#### 3.5.3 Environmental Selection  
The selection of convergence-related variables is based on the new generation obtained by merging the offspring. This selection strategy selects individuals with higher convergence by selecting individuals with higher values of the PBI indicator. The main steps of the offspring selection strategy in this paper are shown in Algorithm 5. Specifically, first, the convergence indicator values of the offspring are calculated, and then the individuals are ranked based on their convergence indicator values. The \(d_{1}\) value of the PBI indicator is calculated using Equation (10), where \(d_{1}\) is used to calculate the convergence indicator value, and the offspring of the CP population is selected for the next generation:  
\[d_{1}=\frac{\|F(x)\cdot w\|}{\|w\|}. \tag{10}\]  
```
0:\(\mathit{CP}\) (Convergent populations), \(\mathit{SCP}\) (Offspring of Convergent populations)
0:newCP (New offspring resulting from environmental selection)
1:\(\mathit{newCP}\leftarrow\emptyset\)
2:\(\mathit{SCP}=[\mathit{SCP}\cup\mathit{CP}]\)
3:for\(i=1\) to \(|\mathit{SCP}|\)do
4: Select the smallest individual \(x\) from the \(\mathit{SCP}\) based on the \(d_{1}\) index
5:\(\mathit{newCP}=\mathit{newCP}\cup\{x\}\)
6:endfor
7:return\(\mathit{newCP}\)
```  
**Algorithm 5** Convergence-oriented Environmental Selection (SCP, CP)  
The selection of diversity variables is based on the new generation to select individuals with better diversity. This selection strategy selects individuals with higher diversity by comparing the diversity index values of different individuals. This paper uses the \(d_{2}\) indicator of the PBI calculation for the selection of diversity variables. The main steps of the diversity variable selection strategy are shown in Algorithm 6. Specifically, first, the diversity index values of the offspring obtained by crossover and mutation are calculated,and then the individuals are ranked based on their diversity index values. The calculation method of \(d_{2}\) is shown in Equation (11), and the individuals with the highest \(N\) values are selected to form the next generation of the DP population:  
\[d_{2}=\left\|F(x)-d_{1}\cdot\frac{w}{\|w\|}\right\|. \tag{11}\]  
```
0:\(DP\) (Diversity populations), \(SDP\) (Offspring of Diversity populations)
0:\(newDP\) (New offspring resulting from environmental selection)
1:\(newDP\leftarrow\emptyset\)
2:\(SDP=[SDP\cup DP]\)
3:for\(i=1\) to \(|SDP|\)do
4: Select the smallest individual \(x\) from the \(SDP\) based on the \(d_{2}\) index
5:\(newDP=newDP\cup\{x\}\)
6:endfor
7:return\(newDP\)
```  
**Algorithm 6** Diversity-oriented Environmental Selection (SDP, DP)  
At the end of the algorithm, the new offspring \(newCP\) obtained from the convergence population and the new offspring \(newDP\) obtained from the diversity population are merged and selected using the PBI indicator \(d_{2}\) (Equation (A1)) as the evaluation criterion for environmental selection. The individuals with the highest \(N\) values are selected based on the index values of all individuals to form the next generation of the population.