## 5 Results and Discussion
### Weight Vector Analysis  
In multi-objective optimization, weighting method combines different objectives into a single objective function by giving them corresponding weights. Three weight vectors (2,8), (5,5) and (8,2) are set and their effects on two objective functions (\(F_{obj1}\), \(F_{obj2}\)) are analyzed  
Figure 8: Load cases.  
Figure 9: Influence of control points on the deck vertical displacement.  
in order to find a reasonable allocation of weights. With the number of 5 control points, the calculation results are shown in Figure 10.  
From Figure 10, in the initial stage of the iteration process, the value of the first objective function (\(F_{obj1}\)) is higher than that of the second objective function (\(F_{obj2}\)), but with the progress of the iteration, the gap between the two objective functions gradually narrows. In the whole iteration process, the numerical change of the second objective function is small. This indicates that in the early stages of optimization, the first objective function has a more significant impact on the results, which helps to quickly reduce the displacement of the structure under dead load. As the iteration progresses, the sizes of the two objective functions gradually become uniform. This weighting strategy not only ensures the stability of the algorithm, but also adjusts the relative importance of the two objective functions by changing the weights.  
Figure 11 shows the influence of different weight vectors on the displacement of the deck. The results show that the displacement of the deck corresponding to the three weight vectors is similar under dead load case, and the maximum vertical displacement within the main deck is less than 0.030 m. The configuration of weight vectors has a relatively small impact on the displacement of the deck under dead load. Under three weight vectors, the maximum deflection of the deck caused by live load is 1.575 m, 1.581 m, and 1.589 m, respectively, with weight vectors (2,8) corresponding to the smallest deflection of the deck. Increasing the weight of the second objective function can improve the structural stiffness. Figure 12 shows the bending moment of the deck under dead load with different weight vectors. It is noted that the distribution of bending moments under different weight vectors is similar. This indicates that the configuration of weight vectors has little effect on the stress of the deck under dead load.  
Figure 11: Influence of weight vector on deck vertical displacement.  
Figure 10: Evolution of objective function under three weight vectors.  
Figure 13 shows the cable pre-tension forces under different weight vectors. The results show that the cable pre-tension force is evenly distributed along the deck. The distribution of cable pre-tension forces under each weight vector is similar, showing that the pre-tension forces near the tower is smaller, and the pre-tension forces away from the tower is gradually increased. In order to analyze the relationship between the allocation ratio of the deck' dead load between the crossing cables and its anchoring position, the ratio of the vertical components of the pre-tension force of the crossing cables under different weight vectors is summarized in Table 4.  
From Table 4, it can be seen that under different weight vectors, the allocation of the vertical component force of the crossing cable is similar. The vertical component of the pre-tension force of the crossing cable between the side tower and the mid tower at the mid span is basically equal. As the anchoring position moves towards the side tower, the vertical component of the crossing cable from the mid tower gradually decreases. The results show that increasing the weight of the second objective function is beneficial for increasing the stiffness of the structure, while having a relatively small effect on the dead load permanent state.  
Figure 12: Bending moments of deck under different weight vectors.  
Figure 13: Cable pre-tension forces with different weight vectors.