# Multi-Objective Optimization of Square Corrugation Multilayer Nested Structures
## 3 The Methodology of Multi-Objective Optimization
### Update the parent population  
For \(t\) generation, the coalition of parent and offspring population is classified into two sets, the \(\delta\)-feasible set \(S_{1}=\{x\in U_{t}\mid g(x)\leq\delta(t)\}\) and \(\delta\)-infeasible set \(S_{2}=\{x\in U_{t}\mid g(x)>\delta_{j}(t),j\in\{1,\,\ldots,\,q\}\}\). This process is discussed in Algorithm 3. ```
Input: parent population \(P_{t}\), offspring population \(O_{t}\) Output: parent population \(P_{t\,+1}\)
1\(u_{t}=P_{t}\cup O_{t}\);
2 Classify into \(U_{t}\) the \(\delta\)-feasible set \(S_{1}=\{x\in U_{t}\mid g(x)\leq\delta^{(0)}\}\) and the \(\delta\)-infeasible set \(S_{2}=\{x\in U_{t}\mid g(x)>\delta^{(0)},j\in\{1,\,\ldots,\,q\}\}\);
3 if\(\mid S_{1}\mid\geq N\), then
4 Select \(N\) populations from \(S_{1}\) according to Algorithm 4;
5 Add \(N\) populations to \(P_{t\,+1}\);
6 else
7 Add population \(S_{1}\) to \(P_{t\,+1}\);
8 Rank elements of \(S_{2}\) in terms of non-constraint level
9 Add the first \(N-\left\lvert\,S_{1}\,\right\rvert\) solutions to \(P_{t\,+1}\);
10 end if
11 return \(P_{t\,+1}\). ```  
**Algorithm 3**Renew the parent population  
To screen for more feasible solutions, the following two scenarios were considered:Case 1. When \(|S_{1}|\geq N\), the non-dominated ranking and prime selection based on the reference point will proceed to choose \(N\) solutions from \(S_{1}\), and then, the \(N\) solutions will be appended to \(P_{t\ +1}\). Case 2. When \(|S_{1}|<N\), there are still blank elements in \(P_{t\ +1}\), after the solutions \(S_{1}\) are introduced. The \(N\)-\(S_{1}\) rest of the solutions are selected from \(\delta\)-infeasible set \(S_{2}\). Therefore, the solutions in \(S_{2}\) are sorted in terms of the non-constraint level, and we append the top \(N-|S_{1}|\) solutions inside the ranked \(S_{2}\) into \(P_{t\ +1}\). D. Non-dominated ranking based on reference point method. When the case \(|S_{1}|\geq N\) occurs, non-dominated ranking based on a reference point will proceed as the environmental selection. Figure 9 displays the environmental selection of DCNSGA-III, which contains two parts: (1) the non-dominated ranking aims to obtain non-dominated solutions inside the \(\delta\)-feasible set \(S_{1}\) to approach its PF; (2) the prime selection based on a reference point aims to retain the population multiplicity through providing a group of reference points with good spatial distribution characteristics. As shown in Figure 9, the first section is that the non-dominated ranking given \(m+1\) objectives: \((f_{1}(x),\ldots,f_{m}(x),cov(x))\) is employed to classify into different non-dominated levels (\(F_{1},F_{2},\ldots\)) and select prime solutions which have higher objective values and lower non-constraint levels. This approach helps to optimize goals and handle constraints simultaneously. In the second section, an interception approach is employed to intercept the optimal \(N\) solutions. The optimal \(k\) levels are determined, and \(k\) denotes the maximum which satisfies \(|F_{1}\cup F_{2}\cup\ldots\cup F_{k}|<N\). The \(k+1\) solutions \(|F_{1}\cup F_{2}\cup\ldots\cup F_{k}\cup F_{k\ +1}|-N\) are determined in terms of the admixture of solutions and the pre-defined reference points, which is depicted in Algorithm 4. In addition, the non-constraint objective will not be referred for the normalization of targets and the associated employment. The phenomenon is because infeasible solutions which have lower non-constraint levels are selected during the non-dominated ranking. The purpose of the selection based on a reference point is to pick solutions that are close to the basis edge so that all reference points maintain the great distributed feature of solutions. Otherwise, the solution will erratically distribute within the target space. E. Evaluate computational complicacy. It can be observed that the DCNSGA-III method does not requires a very large amount of calculation.