## 4 Elaboration of the Optimization Method  
The main aim of composite sandwich structure optimization is to minimize the structural weight and cost while achieving the structural integrity of the optimized structure. This paper considers weight and cost objective functions to optimize a high-speed train's floor by using a lightweight composite sandwich structure. It is worth mentioning that the investigated objective functions are in conflict with each other, so the multi-objective optimization problem is formulated with the associated design variables and design constraints. It is essential that the components of the sandwich structure have sufficient strength to withstand the applied loads; therefore, strength has been defined as one of the design constraints. The investigated sandwich structure consisted of hybrid laminates (FML) for the face sheets and an aluminum honeycomb core.  
The objective functions applied in this paper involve the minimization of both the weight and cost of the composite sandwich structure. In the context of the floor structure under consideration, the aim was to identify the Pareto solution of sandwich constructions that are optimal for achieving low weight and low cost simultaneously.  
During the optimization, the optimal material constituents and geometrical parameters of the components (face sheets and core) were sought for a given practical application in case of the given loading conditions.  
The optimization procedure was managed using the Neighborhood Cultivation Genetic Algorithm (NCGA) optimization algorithm, which was implemented through the I-sight software environment integrated with Excel software. The main steps of the optimization and FEM validation processes are illustrated in the flowchart shown in Figure 5.  
Figure 5: Flowchart of the elaborated multi-objective optimization method for the investigated composite sandwich structure.  
During the optimization, the Neighborhood Cultivation Genetic Algorithm (NCGA) was applied to obtain an optimal sandwich construction. In this technique, each objective function is treated separately and a Pareto front is constructed by selecting feasible non-dominated designs. NCGA is particularly suitable for discrete multi-objective problems and has been shown to provide a robust solution. The computational process of NCGA consists of several steps as it illustrated in Table 3[30]:  
It is common for multi-objective optimizations not to find a single optimal solution, but to have multiple solutions, each containing an element of optimality, which is referred to as Pareto-optimal solutions [31]. The Pareto solution refers to a set of optimal solutions that fulfill the specified objectives. However, further selection is required to determine the best solution [32]. Therefore, in this study, the Improved Minimum Distance Selection Method (IMDSM) [33] was used to determine the most satisfactory point, referred to as the knee point. The knee point can be expressed as follows:  
\[D_{min}=\sqrt{\left(\frac{f_{W_{t}}(x)}{min(f_{W_{t}}(x))}-1\right)^{2}+\left( \frac{f_{C_{t}}(x)}{min(f_{C_{t}}(x))}-1\right)^{2}}, \tag{1}\]  
where \(D_{min}\) represents the minimum distance from the ideal point (minimum weight, minimum cost) to any point on the Pareto line, \(f_{Wt}(x)\) refers to the value of the weight objective function on the Pareto line, and \(f_{Ct}(x)\) represents the value of the cost objective function on the Pareto line. \(D_{min}\) was calculated for 13 values, which corresponds to the number of Pareto points. However, the minimum distance value was identified to be the optimal solution.