## 6 Discussion and Conclusions  
This paper introduces a new IM technique for structures incorporating input fuzzy variables from a probabilistic viewpoint, termed the fuzzy-failure-probability-based importance measure (FFP-IM). The FFP-IM index can discern the impact of input fuzzy variables on the failure probability of structural systems from a conventional probabilistic perspective, and it provides a ranking of the levels of influence. Designers can selectively focus on the variables with a large FFP-IM index according to the ranking, thus reducing the design difficulty. By transforming the unconditional and conditional FFP into the mathematical expectation of the index function of the failure domain, the proposed FFP-IM index becomes the variance-based IM index. Therefore, an efficient SDP-based solution algorithm is established. Compared to the conventional MCS method, the proposed SDP approach markedly decreases the number of calls to the LSF, while maintaining a high \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Method** & \(\tau_{d}\) & \(\tau_{c}\) & \(\tau_{P_{1}}\) & \(\tau_{P_{2}}\) & \(P_{f}\) & \(N^{c}\) & \(t\) \\ \hline SDP & \(6.935\times 10^{-6}\) & \(1.615\times 10^{-4}\) & \(1.032\times 10^{-5}\) & \(1.395\times 10^{-5}\) & \(0.0075\) & \(2800\) & \(1.815\times 10^{4}\) s \\ \hline \end{tabular}
\end{table} 
Table 3: Analysis results of the FFP-IM index, for example, 3.  
degree of accuracy that aligns closely with the outcomes derived from MCS. Therefore, the SDP-based method has more advantages for dealing with complex nonlinear structural systems. Numerical and practical engineering examples are used to verify the feasibility and practicality of the presented FFP-IM in practical engineering applications, as well as the correctness and efficiency of the established SDP solution algorithm. The established FFP-IM provides a novel and efficient tool to deal with fuzzy uncertainty IM analysis for complex structural systems, which is not only significant in theory but also shows a broader prospect in practical engineering applications. Traditional importance measures mainly focus on studying the impact of input random variables on input responses. For example, variance-based importance measures primarily reflect the influence of input uncertainty variables on the output response variance, while entropy-based importance measures reflect the degree to which input uncertainty variables affect the output response entropy. However, these important measures do not reflect the impact of input fuzzy variables on the failure probability of the structural system. The FFP-IM proposed in this paper studies the degree to which fuzzy input variables affect the failure probability of interest in engineering from a probabilistic perspective. Nevertheless, it should be noted that while the SDP algorithm exhibits improved computational efficiency, it may still encounter challenges when dealing with extremely high-dimensional systems. Future research could explore the extension of the proposed FFP-IM framework to address time-dependent reliability analysis and machine learning-based predictions. For time-dependent reliability, the framework could be adapted to account for the evolution of fuzzy uncertainties over time, enabling the assessment of structural systems under dynamic loading conditions. Additionally, introducing machine learning techniques, such as neural networks or Gaussian processes, could further enhance the computational efficiency and predictive accuracy of the FFP-IM, particularly for large-scale and complex systems. Conceptualization, G.L. and S.L. ; methodology, G.L. and M.Z. ; software, M.Z. ; validation, G.L., M.Z. and S.L. ; formal analysis, G.L. and M.Z. ; investigation, G.L. ; writing--original draft preparation, G.L. and M.Z. ; writing--review and editing, M.Z., G.L. and S.L. All authors have read and agreed to the published version of the manuscript.