## 2 Model Establishment and Analysis
### Multi-Objective Optimization  
Lightweight design plays a crucial role in vehicle manufacturing. By reducing the weight of the fifth wheel, the overall vehicle weight can be reduced, fuel efficiency can be improved, and vehicle performance and safety can be enhanced. The fifth wheel is a key component connecting the tractor and semi-trailer. The semi-trailer operates under intense and complex working conditions, leading to significant fatigue performance challenges. If a lightweight design of the fifth wheel is implemented, the fatigue performance of the fifth wheel can be improved without compromising its weight reduction, and the overall vehicle performance will also be enhanced. In multi-objective optimization problems, conflicting objective functions must be optimized simultaneously and cannot be simplified into a single objective function. In this study, multi-objective optimization adopted the method of a multi-objective genetic algorithm ([PERSON] et al. [35]) and solved the multi-objective optimization problem through fitness sharing, Pareto frontier sorting, and crowding distance. The basic idea of MOGA is to search for the optimal solution set of multiple objective functions by simulating the genetic mechanism of nature, obtaining a series of equilibrium solutions that achieve a good balance between multiple objective functions. Most researchers use constrained fatigue life to minimize mass. For the fifth wheel, both lightweight design and improved fatigue life are important. Therefore, this study aims to achieve lightweight design and improve fatigue life simultaneously, rather than focusing solely on a single maximum value calculation. The multi-objective optimization in this study targets the minimum mass and maximum number of minimum lifetimes of the fifth wheel. The maximum stress under the three conditions was constrained to be less than 600 MPa. The multi-objective optimization mathematical model of the fifth wheel was formulated using Equation (5). \[\left\{ \begin{array}{l}\text{Find }X=[x_{1},x_{2}\cdots x_{10}]\\ \text{Minimize }f_{1}(A)=M(X)\\ \text{Maximize }f_{2}(A)=\text{min}(\text{Ni}(X))\\ x_{1}\in(6\text{ mm},7.5\text{ mm})\\ [x_{2},x_{3}\cdots x_{10}]\in(6\text{ mm},20\text{ mm})\\ \text{Constraint: Max}(\text{Si}(X))\leq 600\text{ MPa}\end{array} \right. \tag{5}\]  
In the shell unit diagram of the fifth wheel, the force-transforming element is located at the center as a solid unit. As shown in Figure 11a, the thickness of the saddle body is in the range of 6 mm to 7.5 mm. The corresponding function is \(x_{1}\) in \(X\). As shown in Figure 11b-j, the thicknesses of the upper transverse plate, lower transverse plate, reinforcing ribs, lower reinforcing plate, swing rod support seat, guide plate, carriages, inner baffles, and outer baffles, respectively, range from 6 mm to 20 mm. The corresponding function is [\(x_{2}\),\(x_{3}\),\(x_{4}\cdots x_{10}\)] in X. Figure 11k shows the force-transforming element. M(X) represents the total mass of the fifth wheel, corresponding to the thickness of each component. Ni(X) represents the fatigue life of each unit point of the fifth wheel, corresponding to the thickness of each component under the three conditions. Si(X) represents the equivalent stress of the fifth wheel under corresponding conditions. Because the first condition is static, the stress is very small; therefore, it is not constrained. Because the maximum stresses in conditions 2 and 3 are located at the lower transverse plate, the objective function is the minimum mass \(f_{1}\) and the maximum life\(f_{2}\) simultaneously. It is only necessary to constrain the maximum stress of conditions 2 and 3 to less than 600 MPa of the yield strength of SD600. HyperStudy was utilized for multi-objective optimization. In the multi-objective optimization results, the fatigue life under condition 2 mostly exceeds 1.0 \(\times\) 10\({}^{7}\) cycles. Therefore, only the weight-life optimization results for condition 3 were analyzed.