## 4 Result and Analysis--A Case Study
### Optimization Results and Analysis
For the potential optimal compromise solution of ZDM, the maximum dominant value is particle 13, whose dominant value is 0.0297. For the global potential optimal compromise solution, particle 33 has the largest dominant value (0.01355), which comes from the optimization results of the initial state of the SBM. It is worth noting that these two dominant values come from different value intervals and are not comparable. For the optimization objectives of MMM and MEM, the initial state is too unreasonable, so the optimization results are not satisfactory. It is caused by the sensitivity of the SOM to the initial population under the M-MOPSO algorithm kernel. It is shown that when the initial population is too unreasonable, the optimization does not yield reliable results. Alternatively, the closest alternative to the Pareto optimal compromise solution obtained from the initial state optimization of the SBM is the LFM. LFM methods occupy 23 of the 76 sets of potentially optimal compromise solutions. It shows that if the initial  
Figure 14: The dominant value of the potential optimal compromise solution of ZDM. Figure 15: The dominant value of global potential optimal compromise solution. value is within a reasonable range, the result does not depend on the initial value, which is consistent with the findings of [PERSON] et al. (2018). In a reasonable range, even if all the cable forces are given the same initial value, a more reliable and reasonable bridge state can be obtained. Experienced bridge designers can estimate the approximate initial range of cable force values based on their experience, and then optimize the cable force to obtain a reasonable bridge condition, which is undoubtedly efficient and exciting. In addition to unreasonable cable force values after optimization, the corresponding design variables of each method after optimization and their values before and after optimization are shown in Table 3.  
The optimal compromise solution of the global Pareto front solution comes from the initial state determined by the SBM method. The vertical displacement of the main beam, bending moment of the main beam, and axial stress of the cable before and after optimization are shown in Figures 16-18. In Figure 16, the peak vertical displacement of the optimized main beam is reduced from \(-11\) mm to \(-6\) mm, which is a 45% reduction. The smaller displacement variation means that the alignment of the main beam is smoother and the force is more reasonable under dead load. At the same time, the smaller displacement variation means that the alignment of the main beam is smoother and the force is more reasonable under constant load. In Figure 17, it can be seen that the optimized main beam is subjected to tension on the lower side, which is obviously more conducive to the efficient performance of the mechanical properties of the main beam with tensioned reinforcement configured in the lower part of the neutral axis. The overall stiffness of the combined cable-stayed-T-frame bridge is greater than that of the conventional cable-stayed bridge. Therefore, the huge  
Figure 16: Vertical displacement of the optimized main beam. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Number of Stay Cables** & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** & **S7** & **S8** & **S9** \\ \hline ZDM & 5431 & 5440 & 5379 & 6090 & 6293 & 6212 & 6834 & 7533 & 7969 \\ Optimized value & 6139 & 680  
negative moment at the top of the pier in the tower-girder combination area is the local force state that needs to be focused on. The peak negative bending moments at the top of the pier before and after optimization are \(-\)654,971.0 kN.m and \(-\)426,875.9 kN.m, respectively, which are reduced by 34.8%. The local force of the optimized main beam is more reasonable. In Figure 18, the optimized cable force values are all increased, which shows that the more rational alignment of the main beam comes at the cost of an increase in cable force.