## 1 Introduction
This approach effectively enhanced the decoupling ratio while satisfying the imposed constraints. [PERSON] treated the uncertain parameters of a powertrain mounting system as interval variables and developed the Chebyshev vertex method, which offers an efficient technique for rapidly computing the lower and upper bounds of natural frequencies and decoupling ratios [22]. Other works based on interval analysis methods, such as those presented in [23; 24], have calculated the variation in inherent frequency and decoupling rate of powertrain mounting systems when mount stiffness fluctuates within an interval range. These studies have demonstrated more robust results compared to certainty optimization calculations. [PERSON] et al. [25] combined evidence theory and the regression method to propose a robust design approach for optimizing the reliability of powertrain mounting systems. Their method accounts for inaccurate parameter information and integrates robustness and reliability objectives. In some cases, certain parts of the system possess sufficient parameter information, while others have limited data availability. This scenario necessitates the use of hybrid models that incorporate both probabilistic and non-probabilistic elements. [PERSON] et al. [26] addressed the inherent characteristic response of a powertrain mounting system by employing an interval stochastic hybrid model. This model was utilized to establish optimization objectives and reliability constraints for robust system optimization design. However, irrespective of whether, based on probabilistic, non-probabilistic, or hybrid models, the optimization methods employed under these uncertainties all share a common objective. The general approach involves assessing the variation in the objective function when the range of suspension stiffness fluctuation is known, allowing for a comprehensive evaluation of the objective function's robustness to stiffness changes. This solution approach exemplifies the progression from the underlying parameters towards the objective function. In the context of engine mount design, it is crucial to account for the influence of diverse uncertain factors on stiffness during manufacturing and usage. Therefore, it becomes essential to establish a quantitative mapping relationship between the underlying design variables and the objective function, ensuring a comprehensive consideration of these factors. By adopting a reverse thinking perspective, the challenge lies in determining the appropriate range of stiffness values in each direction of the mounting system, satisfying the design requirements for force transfer rate and main mode decoupling rate with a certain level of probability. Furthermore, while meeting the target performance, it is desirable to have a broader range of suspension stiffness values. To address these challenges, we propose a novel approach, called the generalized inverse cascade method. This method offers a specific application process that involves pre-setting the objective function requirements and establishing a quantitative mapping relationship between the underlying design variables and the upper objective function. Efficient and intelligent search techniques are then employed to determine the optimal value interval for each underlying parameter. Importantly, any combination of parameters within this interval guarantees that the objective function meets the desired confidence requirements. The solving process of this method essentially entails a reverse inference from the objective function to the underlying parameters. The key advantage of the generalized inverse cascade method is its ability to provide designers with an optimal parameter interval for the underlying design variables, while also offering insights into the probability of the objective function, meeting the design requirements within this interval. This method proves to be highly suitable for both single-objective and multi-objective optimization problems encountered in multi-level systems.