## 4 Result and Analysis--A Case Study
### Optimization Results and Analysis  
The initial states obtained by the ZDM, MEM, SBM, MMM, and LFM methods are optimized separately to obtain the respective sets of individual Pareto front solutions. As an example, the optimization results of the ZDM method are shown in Figure 12.  
For the Pareto front solution set, the fitness value of the principal objective function D along the horizontal axis increases, but the performance decreases. Similarly, along the vertical axis, the cable force range per unit length R increases, which means that the design target performance decreases. Therefore, in a Pareto front solution set diagram, there are potential optimal compromise solutions in the region that can simultaneously balance the performance of the two design objectives, as shown in the dashed box in Figure 12. There are 34 groups of potential optimal compromise solutions in the individual Pareto front solution set of the ZDM. In Figure 13, a summary comparison of the Pareto front solution sets optimized by the five initial states is shown. A total of 76 sets of solutions to be selected are boxed by dashed lines, in which there are potentially optimal compromise solutions. Comparing the Pareto front solutions of all methods, the main objective performance of ZDM does not perform  
Figure 11: Influence of each cable on the vertical displacement of the main beam. Figure 12: The individual Pareto front solution set of ZDM. well, even for the individual optimal Pareto front solutions of ZDM. The globally optimal Pareto front compromise solution of all methods is the initial optimization result obtained by the SBM calculation. Fuzzy Set Theory  
The above Pareto front solution set contains a series of nondominated solutions, from which the empirically selected solutions cannot reach the global optimum. [PERSON] [26] proposed a fuzzy-set-theory-based approach to extract one of the Pareto optimal solutions as a sound compromising solution and applied it effectively to environmental/economic power dispatch problems. Define membership functions \(S_{i}^{j}\):  
\[S_{i}^{j}=\left\{ \begin{array}{l}1,f_{i}^{j}\leq f_{i}^{\min}\\ \frac{f_{i}^{\max}-f_{i}^{\min}}{f_{i}^{\max}-f_{i}^{\min}},f_{i}^{\min}\leq f_ {i}^{j}\leq f_{i}^{\max}\\ 0,f_{i}^{j}\leq f_{i}^{\max}\end{array} \right. \tag{17}\]  
where \(f_{i}^{\max}\) and \(f_{i}^{\min}\) are, respectively, the maximum value and minimum value of the \(i\)th optimization objective in the Pareto front solution set; \(f_{i}^{j}\) and \(S_{i}^{j}\) are the current value and the affiliation value of the \(i\)th optimization objective of the \(j\)th solution, respectively. Define the domination function \(\varphi_{k}\),  
\[\varphi_{k}=\frac{\sum\limits_{i}^{2}S_{i}^{k}}{\sum\limits_{j=1}^{l}\sum \limits_{i=1}^{2}S_{i}^{j}} \tag{18}\]  
where \(l\) is the number of solutions in the Pareto front solution set. The dominant value of each noninferior solution in the Pareto front solution set can be obtained by calculating \(\varphi_{k}\). The dominant value reflects the comprehensive performance of the solution. The solution with larger dominance value is selected as the optimal solution. The dominance value of the potential optimal compromise solution in the individual Pareto front solution set of ZDM is calculated by the dominance value calculation formula, as shown in Figure 14. The dominance values of each solution for the global potential optimal compromise solution composed of all methods are shown in Figure 15.  
Figure 13: The global Pareto front solution sets. All orange points in Figure 14 are the dominant values of all Pareto front solutions in the dashed box in Figure 12. The maximum dominance value of the orange solid point pointed by the arrow is the best compromise solution of ZDM. All blue points in Figure 15 are the dominance values of all Pareto front solutions in the dashed box in Figure 13. The maximum dominance value of the blue solid point indicated by the arrow is the global optimum compromise solution.