## 5 Multi-Objective Optimization
### Optimization Results  
When there are multiple objectives, a solution that is best for one objective may be worse for others due to the presence of conflicting and incomparable objectives. These solutions, which necessarily weaken at least one other objective function while improving another, are called Pareto solutions. The set of optimal solutions for a set of objective functions is called a Pareto front. In a multi-objective optimization problem, there is more than one objective function to be maximized or minimized, so the ultimate goal is not to seek a single optimal solution but to seek a Pareto front. This Pareto front needs to be selected from the set of Pareto optimal solutions according to the specifics of the problem. In order to obtain an ideal shrink tube energy-absorbing structure, it is required that the peak force \(F_{\text{PC}}\) is the minimum and the specific energy absorption \(E_{\text{SA}}\) is the maximum. In this paper, multi-objective optimal design was carried out using the GRSM algorithm with cone angle \(\alpha\) and pipe wall thickness \(T\) as design variables, and the obtained Pareto front solution set is shown in Figure 9. Figure 9 shows that the peak force \(F_{\text{PC}}\) of the shrinkage tube and the specific absorption energy \(E_{\text{SA}}\) have a positive correlation, and as the specific absorption energy \(E_{\text{SA}}\) increases, the peak force \(F_{\text{PC}}\) of the shrinkage tube also increases. Therefore, the two objectives of the minimum peak force \(F_{\text{PC}}\) and the maximum specific absorption energy \(E_{\text{SA}}\) have mutual exclusivity, which makes it difficult to select the optimal structural parameters.  
Figure 9: Pareto front: (**a**) Pareto front of specific energy absorption versus the peak force; (**b**) Pareto front of mean force target value and peak force.  
Although the Pareto solution set has more advantages in solving optimal design solutions, the optimal solution must be selected to quantitatively analyze the optimization results. Combined with the technical requirements of the actual engineering problems on the mean force, a set of solutions with the mean force \(F_{\text{MC}}\) closest to 600 kN was selected as the optimal solution when the optimization objective three took the minimum value. The optimized parameters were a cone angle \(\alpha\) value of 20.2\({}^{\circ}\), a pipe wall thickness \(T\) value of 7.1 mm, a peak force \(F_{\text{FC}}\) of 658.16 kN, a specific energy absorption \(E_{\text{SA}}\) of 29.73 kJ/kg and a mean force \(F_{\text{MC}}\) of 613.23 kN. The error between the mean force and the design objective was 2.2%, which meets the requirement of a mean force error of less than or equal to 7.5%. In order to verify the reliability of the optimization prediction results, finite element simulation calculations were performed on the structural parameters obtained from the optimization results. The comparison between the optimization prediction results and the finite element simulation calculation results is shown in Table 4. It could be seen that the finite element calculation results were close to the optimization prediction results. The peak force error was 1.46%, the specific energy absorption error was 1.33%, and the mean force error was 1.31%. All errors were less than 1.5%, indicating that the optimization results have high accuracy and reliability.