## 4 Model Lightweight Algorithm
### Optimization Result Certification  
To verify the accuracy of the optimization results, an additional multi-objective optimization program based on NSGA-II (non-dominated sorting genetic algorithm II) was written using MATLAB. This program was used to compare the optimization results with those obtained from WPSO. NSGA-II is a multi-objective optimization algorithm based on the genetic algorithm and simulates the process of natural selection and evolution to find optimal solutions. The flowchart of the NSGA-II algorithm is displayed in Figure 13.  
Figure 12: The optimization curves. The same optimization function and constraints were set for both algorithms. The primary goal was minimizing the engineering quantity of the upper structure, and the secondary goals were minimizing the mid-span vertical deflection and axial internal force of the empty tube of the web bar under compression at the pier top, while ensuring the safety of the bridge structure. Five sets of better optimization results were then selected from the outcomes and the best data were compared. The optimization results obtained using NSGA-II are presented in Table 4, while the optimization results obtained using WPSO are shown in Table 5.  
Figure 13: The flowchart of the NSGA-II algorithm. From Table 4, it can be seen that Solutions 2 and 4 are the two methods that achieved the minimum engineering quantity for the upper structure. Although Solution 2 yielded a slightly better optimization result for the mid-span vertical deflection, Solution 4 had a larger optimization magnitude for the upper structure quantity. Moreover, Solution 2 resulted in the negative optimization of the axial internal forces of the stiffened string and the empty tube of the web bar under compression at the pier top, which would affect the structural load-bearing capacity. Therefore, Solution 4 was selected as the final optimization result for NSGA-II. From Table 5, it can be observed that the optimization magnitudes of each solution of WPSO were quite similar, and there were no extreme cases, as in Solution 2 of NSGA-II. Among them, Solution 3 had the largest optimization magnitude for the engineering quantity of the upper structure, while other optimization indicators did not display significant differences compared to other solutions. Therefore, Solution 3 was selected as the final optimization result for WPSO. The comparison of the final optimization coefficient for both methods is shown in Figure 14.  
Figure 14 compares the optimization results of the final solutions of the two optimization algorithms. Compared to the NSGA-II optimization results, WPSO yielded a 6.29% smaller mid-span vertical deflection, a 1.42% larger axial internal force of the stiffened string, a 1.5% smaller longitudinal internal force of the bridge deck bottom plate, a 0.21% smaller axial internal force of the empty tube of the web bar under compression at the pier top, and a 0.45% larger  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Optimization**} & \multicolumn{6}{c}{**Optimization Objective**} & \multicolumn{6}{c}{**Optimization Coefficient (\%)**} \\ \cline{2-10}  & **A** & **B** & **C** & **D** & **R1\({}^{\prime}\)** & **R2\({}^{\prime}\)** & **R3\({}^{\prime}\)** & **R4\({}^{\prime}\)** & **R5\(  
In comprehensive consideration of the optimization targets, it is preferred to minimize the upper structure quantity while ensuring safety, followed by minimizing the mid-span vertical deflection and the axial internal force of the empty tube of the web bar under compression at the pier top. Although the WPSO optimization solution displayed a slight disadvantage in the optimization coefficients of the upper structure quantity and the axial internal force of the stiffened string compared to the NSGA-II solution, it achieved an advantage in the optimization coefficients of the mid-span vertical deflection, the longitudinal internal force of the bridge deck bottom plate, and the axial internal force of the empty tube of the web bar under compression at the pier top.