## 1 Introduction
In order to enhance the phenomenon of precocious search, MIGA has been devised. MIGA was significantly strengthened in maintaining the diversity of the population and had a better performance in finding the global optimal solution [9]. In order to evaluate the fire-resistance performance of the A60 BPP, transient heat transfer analysis based on the finite element method was performed by applying specimen specifications and heating temperature conditions specified in the FTP code. The transient heat transfer analysis results were verified via the actual fire test according to both the FTP code and international fire-resistance test procedure specified in International Organization for Standardization (ISO) [10]. Design of experiment (DOE) method was applied to generate the meta-models to be used for the approximate optimization, and the verified finite element model and transient heat transfer analysis results were integrated with the DOE. The length, diameter, and material type of A60 BPP, and the insulation density were included in the design variables, and the discretized specifications that can be actually manufactured were considered their variation level. The A60 BPP weight was applied as the objective function, and constraints were considered the fire-resistance temperature specified in the FTP code, productivity, and cost, respectively. The design matrix was generated by applying the full factor design method to consider the results of all the objective functions and the constraints that occur due to the level of variation of the design variables. From the DOE results, various meta-models such as Kriging, response surface method (RSM), and radial basis function-based neural network (RBFN) were generated. In the Kriging method, an interpolative meta-model is defined as the sum of the global model representing the global characteristics of an actual design space function to be approximated and the local model corresponding to the deviation between the actual function and the global model [11]. In RSM, a regressive meta-model is generated generally as a polynomial model with second-order terms using the least-squares method [12]. In RBFN, the interpolative meta-model is fitted using linear combinations of a radially symmetric function based on Euclidean distance, while the neural network is trained by a back-propagation algorithm [13; 14]. The accuracy of approximating the design space of A60 BPP for each meta-model was evaluated, and the convergence results were compared by applying it to the approximate optimization. The approximate optimum design problem based on the meta-model was formulated such that the discrete design variables were determined by minimizing the A60 BPP weight subject to the limit values of constraints. The results of the approximate optimum design explored for each meta-model were compared with the results of the actual constraints to evaluate the approximation accuracy. For the fire-resistance design of A60 BPP, it was found that the RBFN-based approximate optimization method can explore the optimum solution with the highest accuracy. This paper is comprised of the following sections: Section 2 addresses the fire-resistance design of A60 BPP and explains the FTP code regulation, the transient heat transfer analysis results, and the fire test verification; Section 3 addresses the DOE results and describes the meta-modeling and theoretical background; Section 4 addresses the results and discusses the characteristics of the approximate optimum solutions according to various meta-models; and Section 5 presents the concluding remarks of this study. Both the list of acronyms and the whole DOE data for meta-modeling are provided in Appendix A.