# Research on Multi-Objective Evolutionary Algorithms Based on Large-Scale Decision Variable Analysis
## 2 Related Work
### Contribution Target Analysis of Decision Variables  
To measure the contribution degree of convergence decision variables to each objective, _od_ is defined as the optimization degree of each objective for a decision variable \(x_{i}\). For a population \(S=\{X_{1},X_{2},\ldots,X_{N}\}\), where \(1\leq n\leq N\), \(N\) is the number of individuals, and \(x_{i}\) is a variable, the optimization degree \(od_{im}\) of \(x_{i}\) for the \(m\)-th objective is given by Equation (4):  
\[od_{im}=|\lambda_{m}|, \tag{4}\]  
where \(1\leq m\leq M\), \(M\) is the number of objectives, \(1\leq i\leq D\), \(D\) is the number of decision variables, and \(\lambda_{m}=(z^{\text{n\[co_{i}=\arg\max_{1\leq m\leq M}od_{im}. \tag{5}\]  
In different detection spaces of the objective space, the contribution degree of decision variables to different objectives may vary. Suppose \(X_{1}\), \(X_{2}\), \(X_{3}\) are the three samples obtained in the three detection spaces, as shown in Figure 1. By analyzing the optimization degree of these samples for the variable \(x_{1}\), and performing similar calculations for other variables, we obtain the contribution targets of the variable \(x_{1}\) in different spaces: \(f_{1},f_{2},f_{3}\). Therefore, this method can be used to solve the contribution targets in each detection space, ensuring the comprehensiveness and accuracy of the optimization process. ```
0:\(P\) (Current population), \(nSel\) (Number of selected solutions), \(nPer\) (Number of perturbations on each solution), \(D\) (Number of decision variables), \(W\) (Weight vector)
0:\(DV\) (Diversity-related variables), \(CV\) (Convergence-related variables), \(RX\) (The property value of the variable)
1:for\(i=1\) to \(D\)do
2:\(S\leftarrow\) randomly select \(nSel\) solutions from \(P\)
3:for\(j=1\) to \(nSel\)do
4:\(SP\leftarrow\) Perturb the \(i\)-th decision variable for \(nPer\) times. 5:\(z^{\text{nadir}}\leftarrow(f_{1}^{\text{max}},f_{2}^{\text{max}}),z^{\text{ ideal}}\leftarrow(f_{1}^{\text{min}},f_{2}^{\text{min}})\) (Find the nadir point and ideal point in the \(SP\).) 6:\(Dist=\{d_{1},d_{2},\ldots,d_{nPer}\}\) (calculate the distance from the hyperplane.) 7: The \(j\)-th convergence correlation of \(x_{i}\): \(C_{x_{i}}^{j}=\left\|\frac{(x_{1}^{\text{max}}-x_{2}^{\text{max}})\cdot\mathbf{x} ^{\text{-}}\cdot\mathbf{w}}{\|\mathbf{x}^{\text{-}}\|\|\mathbf{w}\|}\right\|\)
8:\(S^{\prime}\leftarrow\) Calculates the projected points of each point on the hyperplane. 9:\(Dist\leftarrow\) calculate the distance between every two points. 10: The \(j\)-th diversity correlation of \(x_{i}\): \(D_{x_{i}}^{j}=\left\|(X_{1}^{S}-X_{2}^{S})-\frac{(X_{1}^{S}-X_{2}^{S})\cdot \mathbf{x}^{\text{-}}}{\|\mathbf{x}^{\text{-}}\|}\right\|\)
11: The \(j\)-th \(R_{x}\) of \(x_{i}\): \(R_{x_{i}}^{j}=C_{x_{i}}^{j}-D_{x_{i}}^{j}\)
12:endfor
13:\(R_{x}\leftarrow\) calculate the average of all \(R_{x}^{j}\)
14:endfor
15:for\(i=1\) to \(D\)do
16:if\(R_{x}<0\) or \(\|R_{x}\|<\theta\)then
17:\(DV\gets DV\cup x_{i}\)
18:else
19:\(CV\gets CV\cup x_{i}\)
20:endif
21:endfor
```  
**Algorithm 2** Decision Variable Quantitative Analysis (\(P\), nSel, nPer, D, W)  
Algorithm 3 provides the pseudocode for the contribution target analysis of decision variables. First, generate the weight vector, which represents different objective spaces. Select a sample solution from each space. For each sample solution, perturb it to produce \(n\) solutions (line 4); then, calculate the optimal value \(\lambda\) of each perturbed subpopulation (line 5). Next, calculate the optimization degree \(od_{im}\) of the decision variable \(x_{i}\) for each objective \(1\leq m\leq M\), and record the objective with the maximum \(od_{im}\) value as the contribution target \(co_{ij}\) of the decision variable \(x_{i}\) (lines 6-7). Finally, calculate the maximum contribution target \(co_{ij}\) (\(j\in[1,NS]\)).