## 4 MIGA Based Approximate Optimization with Discrete Design Variables  
Genetic algorithm (GA), which is one of the global search methods based on evolutionary computation, adopts the principle of survival of the fittest in organisms. GA repetitively performs a typical genetic process such as regeneration, crossover, and mutation, which are the basic principles of biological inheritance, using the probability distribution of a large number of design groups per generation. Through the GA process, the objective function is optimized by generating new design variables or searching for an improved design solution. The derivative-based optimization method searches for the closest improved solution from the initial design specification in the continuous design space. On the other hand, the GA is usefully applied to non-deterministic polynomial hard optimization problems such that the design problem characteristics are nonlinear, the design space is discontinuous, there are many design variables and constraints, and the design variable type is discrete or integer [22]. The discrete design optimization problem is generally difficult to solve due to the fact that the discrete design space is non-differentiable and nonconvex. Therefore, a general gradient-based programming method and optimality criteria cannot be applied directly to the discrete design space. The global optimal solution of the discrete design optimization problem can be obtained only by an exhaustive search method such as the GA [23]. In order to minimize the weight of the A60 BPP subject to various constraints, the approximate optimization with the discrete design variables was carried out using the MIGA \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Meta-Model**} & \multicolumn{5}{c}{**Output Responses**} \\ \cline{2-6}  & \(\mathbf{f}\) & \(\mathbf{g_{1}}\) & \(\mathbf{g_{2}}\) & \(\mathbf{g_{3}}\) \\ \hline \multirow{3}{*}{Kriging} & \(E_{\text{avg}}\) & 0.997 & 0.957 & 0.968 & 0.969 \\  & \(E_{\text{max}}\) & 0.958 & 0.903 & 0.933 & 0.938 \\  & \(R^{2}\) value & 0.978 & 0.965 & 0.957 & 0.976 \\ \hline \multirow{3}{*}{RSM} & \(E_{\text{avg}}\) & 0.998 & 0.977 & 0.983 & 0.986 \\  & \(E_{\text{max}}\) & 0.937 & 0.897 & 0.917 & 0.907 \\  & \(R^{2}\) value & 0.999 & 0.978 & 0.998 & 0.999 \\ \hline \multirow{3}{*}{RBFN} & \(E_{\text{avg}}\) & 0.999 & 0.991 & 0.994 & 0.997 \\  & \(E_{\text{max}}\) & 0.998 & 0.985 & 0.991 & 0.986 \\ \cline{1-1}  & \(R^{2}\) value & 0.999 & 0.999 & 0.999 & 0.999 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of meta-modeling results. technique and the meta-models. In the MIGA method, the population of the general GA is transformed into sub-populations called \"island,\" and genetic processes such as selection, crossover, and mutation are carried out for each island [24; 25]. Individuals selected on the island from the genetic process are periodically moved to another island via a roulette selection method. The main parameters controlling the individual migration in the MIGA are the migration interval and rate to the island. The migration interval determines the number of generations between each movement, and the migration rate determines the number of individuals that are migrated from each island. Three types of meta-models such as Kriging, RSM, and RBFN were applied to the approximate optimization with discrete design variables for the minimum weight design of the A60 BPP. The approximate optimization problem was formulated such that the discrete design variables were determined by minimizing the weight of A60 BPP subject to various constraints such as fire resistance temperature, productivity, and cost. This study explored the approximate optimum design results according to the characteristics of the meta-models and identified the most pertinent optimization approach to the minimum weight design of the A60 BPP, compared to the actual results of the constraints.