## 4 MIGA Based Approximate Optimization with Discrete Design Variables
The approximate optimization formulation of the A60 BPP design was defined as follows:  
\[Min.f=f(x_{1},\ x_{2},\ x_{3},\ x_{4}) \tag{18}\]  
\[s.t.g_{1}(x_{1},\ x_{2},\ x_{3},\ x_{4})=\text{Fire resistance temperature}\leq 170\ ^{\circ}\text{C}\]  
\[g_{2}(x_{1},\ x_{2},\ x_{3},\ x_{4})=\text{Productivity}\geq 9.5\]  
\[g_{3}(x_{1},\ x_{2},\ x_{3},\ x_{4})=\text{Cost}\leq W30,000\]  
\[x_{1}=150;160;170;\ \cdots;230;240;250\]  
\[x_{2}=6;8;10;12\]  
\[x_{3}=S45C;SU316L\]  
\[x_{4}=96;128;160.\]  
Since the accuracy tolerance of the temperature measurement sensor used in the fire test is allowed up to \(\pm 5\) at 940 \({}^{\circ}\)C [26], in Equation (18), the upper limit of the temperature inequality constraint is set to 170 \({}^{\circ}\)C, which is 10 \({}^{\circ}\)C lower than the fire resistance temperature regulated in the FTP code. Referring to the results of previous studies [27] on the sensitivity analyses of both productivity and cost in A60 BPP, the lower and upper limits of the inequality constraints on the productivity and cost were applied at 80% of the maximum values evaluated from the full-factorial experiments, as represented in Table 4. The ranges of design variables in the body length (\(x_{1}\)) and diameter (\(x_{2}\)) of A60 BPP were defined discretely as 10 mm and \(\phi\)2 intervals, which can be actually manufactured. The range of discrete design variables in the material type (\(x_{3}\)) of A60 BPP was applied to S45C and SUS316L, which are steel materials that can be used for fire protection apparatus. The range of discrete design variables for the insulation density (\(x_{4}\)) was considered the specifications actually produced for the A60 class insulation. The numerical search parameters of MIGA applied to the approximate optimization of A60 BPP are summarized in \[Min.f=f(x_{1},\ x_{2},\ x_{3},\ x_{4}) \tag{18}\]0 \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Sub-population size & 10 \\ \# of islands & 10 \\ \# of generations & 10 \\ Crossover rate & 1.0 \\ Mutation rate & 0.01 \\ Migration rate & 0.01 \\ Migration interval & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters used in multi-island genetic algorithm (MIGA). The approximate optimal solution result was explored by applying the meta-models of Kriging, RSM, and RBFN to the discrete design ranges defined in the process of formulating the optimal design of A60 BPP. The accuracy of the approximate optimal design result was compared with the results of the actual constraints for each meta-model. The optimization results are summarized in Table 7 for the optimal design solution, the approximate and actual calculation results of the constraints, and the weight minimization results. The results of the optimization convergence of the objective function are shown in Figure 9.  
As shown in Table 7, the results of approximation optimization represented the weight minimized design of more than 23% compared to the initial design while satisfying all the limit values of constraints. The approximate optimization result using the RBFN meta-model represented the highest accuracy because the approximate results of the constraints were coincident with the actual calculation results. In the case of applying Kriging and RSM meta-models, the weight reduction ratio was achieved by 23.5%, compared to the initial design, but the approximation accuracy results of the fire test temperature constraint (\(g_{1}\)) were inferior to the results of the case where the RBFN meta-model was applied. As shown in Table 7 and Figure 9, it was found that the convergence results and the iteration