## 3 Multi-Parameter Optimization and Sensitivity Analysis Method
### Multi-Parameter Identification Based on Improved Genetic Algorithm  
Genetic algorithms are intelligent optimization techniques known for their strong global search capabilities, particularly in optimizing objective functions that cannot be addressed by traditional gradient-based methods, such as discontinuous, non-differentiable, or complex objective functions. Genetic algorithms offer unique advantages in these scenarios. However, directly applying the basic genetic algorithm to solve multimodal optimization problems can lead to convergence at local optima (a phenomenon known as \"premature convergence\"), making it difficult to achieve global optima [28]. Since the optimization of design parameters for the three-dimensional seismic isolation device may be a multimodal optimization problem, it is necessary to improve the basic genetic algorithm to address its shortcomings. The operational process of the basic genetic algorithm includes the following steps:  
Step 1: Variable Encoding: The commonly used binary encoding method represents variable \(x_{i}\) as \(b_{l}b_{l-1}\ldots b_{2}b_{1}\) (where \(b_{i}\) = 0 or 1), with its corresponding value given by:  
\[x=x_{\text{imin}}+(\sum_{i=1}^{l}b_{i}\cdot 2^{i-1})\cdot\frac{x_{\text{ imax}}-x_{\text{imin}}}{2^{l}-1} \tag{5}\]  
Step 2: Population Initialization: The population of the \(t\)-th generation is denoted as \(P(t)\). All individuals in the initial population \(P(0)\) are randomly generated. Step 3: Fitness Calculation: The fitness of each individual in the population \(P(t)\) is calculated to determine the probability of its genetic contribution to the next generation. Since this problem is a minimization problem, the relationship between the fitness function \(F(\mathbf{X})\) and the objective function \(W(\mathbf{X})\) is established as:  
\[F(\mathbf{X})=\frac{1}{W(\mathbf{X})} \tag{6}\]  
Step 4: Selection and Genetic Operations: Selection is performed based on the proportion of individual fitness within the overall fitness (using a roulette wheel method), followed by crossover and mutation operations according to preset probabilities to obtain the next generation population \(P(t+1)\). Step 5: Steps 3 and 4 are repeated iteratively until \(t\) exceeds the preset maximum number of generations \(T\). To improve optimization efficiency and overcome the \"premature convergence\" issue of the basic genetic algorithm, the following enhancements are proposed:  
(1) Establishment of a small habitat algorithm with a fusion mechanism:  
After each generation of evolution, implement a \"last position replacement\" mechanism within each small habitat, eliminating the worst individual of that generation while ensuring that the best individual from the previous generation automatically enters this generation. This ensures that the optimal individual in the next generation is not worse than that of the previous generation, allowing the genetic algorithm to continuously evolve towards optimality without regressing. Implement an \"inter-habitat exchange\" mechanism, where the second-worst individual in the \(i\)-th small habitat is eliminated, and the best individual from the \((i-1)\)-th small habitat automatically enters the \(i\)-th small habitat. This increases population diversity and reduces the probability of the genetic algorithm converging to local optima. (2) Establishment of a method for identifying global optima in multimodal problems:  
Let \(A_{i}\) be the peak point to which the \(i\)-th small habitat converges after evolution (with an objective function value of \(F_{i}\) and corresponding variables \(\mathbf{X}_{i}\)). When there are sufficiently many small habitats, the point with the maximum objective function value is likely to be the global optimum, denoted as \(A^{*}\), with an objective function value of \(F^{*}\) and corresponding variables \(\mathbf{X}^{*}\).