## 4 Case Study
### Inversion Problem Description  
As shown in Figure 6, hangers H1, H2, H22-H28, H48, and H49 are considered short hangers. These short hangers are primarily located in critical regions of the main cable, where their tension deviations can significantly affect the final shape of the bridge and its structural performance. Short hangers are more sensitive to tension measurement errors due to their relatively large bending stiffness, making it more difficult to guarantee precise control during the construction process compared to longer hangers. As a result, tensioning these short hangers requires careful monitoring and adjustment to ensure the correct final configuration. Additionally, since the SLB provides a flexible constraint to the main cable, its tension directly affects the lateral position of the main cable during configuration transformation. Consequently, the permissible tension range of the SLB also significantly impacts the constructability and stability of the bridge. Given these factors, the tension range of the short hangers and the SLB is treated as adjustable assembly parameters, with the midpoint and radius of their respective intervals serving as optimization variables. The formulation of the fault-tolerant interval inversion model is described in Equation (4). \[ \begin{array}{ll}find&D^{l}=[d_{1}^{l},\ d_{2}^{l}\ \ldots\,\ d_{n}^{l}]^{T},d_{i}^{l}=[d_{i}^{c}-d_{i}^{w},d_{i}^{c}+d_{i}^{w}],i= 1,2,\ \ldots\ n\\ obj.&\overset{\wedge}{f_{1}}=min\big{\{}\Delta Z^{c}(D^{l})\big{\}}\\ &f_{2}=min\big{\{}\sum_{k=1}^{n}(\big{|}d_{k}^{c}-d_{k}^{*}\big{|}+max(\big{|}d _{k}^{c}-d_{k}^{w}-d_{k}^{*}\big{|},\big{|}d_{k}^{c}+d_{k}^{w}-d_{k}^{*}\big{|} ))\big{\}}\\ &f_{3}=min\ \big{\{}-W\big{\}}\\ s.t.&\overset{\wedge}{g_{1}}=\overset{\wedge}{g^{L}}(G(X,D^{l}))\geq\beta^{T}\\ &\overset{\wedge}{g_{2}}=max|\theta(D^{l})|\leq\theta^{u}\\ &\overset{\wedge}{g_{3}}=max|v(D^{l})|\leq\upsilon^{u}\\ &d_{k}^{c}\in[95\%d_{k}^{*},105\%\omega_{k}^{d}];\ d_{k}^{w}\in[0,\ 15\%\ d_{k}^{*}]\end{array}  \tag{4}\]  
The inversion model includes three objective functions and three constraint functions which aim to balance the trade-offs between constructability, design optimality, and reliability. The objective functions are as follows:  
Figure 7: Detailed scheme of stretching hangers. 1. Objective function \(f_{1}\): Minimize the \(\Delta Z\) (namely the deviation between the main cable shape at the completion stage and the design target), thus achieving the optimality of the design target. 2. Objective function \(f_{2}\): Minimize the deviation between the tension of the hangers at the completion stage and the design tensions, thus ensuring tension uniformity. \(d^{*}\) is the tension in the hangers given in the original design scheme. 3. Objective function \(f_{3}\): Maximize the constructability of the configuration transformation process, considering the feasibility of achieving the design target with minimal rework. The optimization process is constrained by three key factors: 1. Constraint function \(g_{1}\): The asymmetric tensioning of the hangers during mid-span and the uncertainties arising from construction errors can lead to an unbalanced force in the main cable. If the unbalanced force exceeds a predefined limit, it may cause local damage to the bridge tower. This constraint is designed to ensure that the longitudinal displacement of the tower top remains within acceptable limits, with the corresponding limit state function described as \(G(X)=a_{\max}-a(X)\). And \(a_{\max}\) is the allowed maximal longitudinal displacement of the bridge tower. 2. Constraint function \(g_{2}\): During configuration transformation, both the upper and lower anchor points of the hangers experience lateral and longitudinal angular deviations. To prevent bending damage, the maximum angular deviations \(\theta(D^{l})\) of the already tensioned and untensioned hangers at each stage must be controlled within the specified limits \(\theta^{u}\). 3.