## 3 Multi-Parameter Optimization and Sensitivity Analysis Method
### Multi-Parameter Sensitivity Analysis Based on Optimized Central Gradient Method  
Next, a sensitivity analysis will be conducted on each parameter to determine how changes in each parameter affect the rate of change in the objective function. This provides a basis for fuzzy optimization design and stiffness adjustment, focusing on parameters with high sensitivity while applying fuzzy optimization to those with low sensitivity.  
For the multi-parameter optimization problem of the three-dimensional seismic isolation device, each parameter is interrelated, and their relationship with the objective function is nonlinear. If traditional single-factor analysis methods are used, calculating the sensitivity of parameter \(x_{i}\) generally yields different results when other parameters take different baseline values. Thus, evaluating the sensitivity of parameter \(x_{i}\) loses significance.  
Therefore, it is essential to use the optimal parameters (i.e., optimization center) as a baseline and apply single-factor analysis methods (i.e., gradient methods) to calculate the sensitivity of each parameter. The optimal parameters \(\mathbf{X}^{*}\) have been obtained through genetic algorithm optimization. It has been found that the internal hollow dimensions of the three-dimensional rubber seismic isolation device significantly affect its vertical stiffness. Thus, using the vertical stiffness \(K_{V}\) of the isolation device as a sensitivity evaluation function allows for calculating the sensitivity value \(f(x_{i})\) of each parameter \(x_{i}\) using:  
\[f(x_{i})=\frac{K_{V}(x_{1}^{*},\ldots,x_{i}^{*}+\Delta x_{i},\ldots,x_{n}^{*}) -K_{V}(x_{1}^{*},\ldots,x_{i}^{*},\ldots,x_{n}^{*})}{\Delta x_{i}} \tag{8}\]  
To adjust the stiffness of the three-dimensional seismic isolation device, it is sufficient to modify the values of design parameters with high sensitivity.  
The flowchart for the initial design, multi-parameter optimization, and sensitivity analysis of the three-dimensional isolation device is shown in Figure 5.  
Based on the parameter sensitivity design method, the following fuzzy optimization design theory is established:  
For design variables \(x_{i}\) with low sensitivity, let \(a\leq x_{i}\leq b\), while maintaining the other variables at their optimal center values. If the following condition is satisfied:  
\[\frac{\left|W(x_{i}=a)-W(x_{i}=b)\right|}{\min\{W(x_{i}=a),W(x_{i}=b)\}}<5\% \tag{9}\]  
where \(W(\mathbf{X})\) is the objective function, as shown in Equation (4), then the design value of variable \(x_{i}\) can be freely chosen. This allows the isolation device to exhibit good versatility for different isolated objects, facilitating industrial processing.  
The topological optimization model of the isolation bearing was obtained using the progressive structural optimization method, which was then transformed into a design model characterized by quantifiable parameters.  
The optimization objective was to minimize the wideband acceleration-transmission ratio, and the optimal design parameters for the isolation bearing were obtained through an improved genetic algorithm.  
The optimized central gradient method was employed to analyze the sensitivity of each parameter, providing a basis for fuzzy optimization design and stiffness adjustment.