## 6 Results of the Optimization and the Finite Element Modeling  
### Results of the Elaborated Optimization Method  
In contrast to single-objective optimization problems, the solution of a multi-objective optimization problem is not unique. Instead, a range of optimal solutions can be considered. In multi-objective optimization, the improvement of one objective may lead to the deterioration of the performance of the other objectives. Therefore, it is difficult to achieve multiple optimal values simultaneously, and the only way to do that is to make each objective as optimal as possible. Consistent with the formulation of the multi-objective optimization problem in Equations (2) and (4), the objectives of sandwich structure optimization were defined as reducing the overall weight (\(W_{t}\)) and cost (\(C_{t}\)) while ensuring the durability of the structure. In this study, different densities of honeycomb cores and hybridized face sheets (FML) with their associated parameters were used as design variables to achieve the final objectives. Since the contradiction between the total weight of the sandwich structure and the cost was obvious, we chose the Neighborhood Cultivation Genetic Algorithm to investigate the optimal solution using I sight software in conjunction with Excel software. The total number of feasible solutions for this study was approximately 16,600 design points. The changes in weight and cost in the optimized design are shown in Figure 7. The blue hollow circles represent the feasible solutions of the optimization problem, which satisfy all the constraints of the problem while achieving a certain level of performance with respect to multiple competing objectives. The green solid circles represent the set of non-dominated solutions, which achieve the best trade-off between multiple competing objectives, also known as the Pareto set, and represent the best solutions of the optimization problem. The NCGA algorithm acquired 13 data points, which were extracted in Figure 8 and represent the Pareto set. Pareto points tend to be convexly distributed due to the opposing objectives' behavior. Whenever the weight decreases, the cost of the sandwich structure increases and vice versa. Therefore, the knee point was considered the most satisfactory solution. The calculations showed that the knee point with \(W_{t}\) = 5.25 kg and \(C_{t}\) = 2.37 unit price should be appointed as the optimal solution, representing an ideal compromise between the opposite objectives. The corresponding Pareto  
Figure 6: Finite Element Model of the investigated train floor. The minimum weight, minimum cost, and knee design points are the most significant design points on the Pareto line. Therefore, the related data have been extracted in Table 6 for further consideration. The data include the main parameters related to the geometrical characteristics of sandwich structure components and configurations, and have been adopted in the FEM modeling. The optimization of weight and cost of sandwich structures is crucial, and the hybridization of face sheet materials has been shown to have a significant impact on achieving these objectives. One approach to achieving this is to create a hybrid structure that combines the strengths of various materials while complementing their weaknesses. The cost and weight of a hybrid composite sandwich structure is determined by the sum of the cost and weight of its constituents. Therefore, the final composite face sheet is a function of the composite ratio of each material. In this study, the percentage of the FRP materials in terms of weight and cost was considered, as depicted in Figure 9. The  
Figure 8: Pareto optimal points. Figure 7: Feasible design points. figure shows that a higher percentage of FRP results in a lighter and more expensive sandwich structure, whereas reducing the FRP materials tends to result in a heavier and less expensive structure.