Importance Measure for Fuzzy Structural Systems from the Probabilistic Perspective and Its Solving Algorithms  
[PERSON]  
1 School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian 116024, China; [EMAIL_ADDRESS] (G.L. ); 22303204 [EMAIL_ADDRESS] (M.Z. )1 School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian 116024, China; [EMAIL_ADDRESS] (G.L. ); 22303204 [EMAIL_ADDRESS] (M.Z. )1 assess the degree of safety of structural systems, and the fuzzy safety factor was defined to quantify the safety level for fuzzy uncertainty structures. [PERSON] [10] introduced the fundamental concepts and theoretical framework of fuzzy reliability analysis based on the possibilistic theory and further developed the possibility degree reliability analysis approach for general systems [11]. [PERSON] and [PERSON] [12; 13] proposed a fuzzy security level index based on the confidence interval, which provides both the upper and lower bounds for the security level of a structural system. [PERSON] and [PERSON] [14] suggested employing the minimal Chebyshev distance from the coordinate origin to the failure domain within the standard space as a metric for a fuzzy safety index. Although the index possesses a clear geometric interpretation and has shown practical value in specific applications, its implementation is strictly limited to situations where the membership function of the input variable conforms to a Gaussian distribution. Therefore, [PERSON] et al. [15; 16] modified the distance index presented by [PERSON] based on interval analysis theory, and the modified index can be widely applied to cases where membership functions of input variables are arbitrary, but it is still the fuzzy possibility measure. On this basis, [PERSON] et al. [16; 17] established a fuzzy structural failure probability analysis model based on a probability perspective by introducing random variables that obey the uniform distribution of [0, 1] into the membership level of the fuzzy variable. This model not only gives the fuzzy structure safety evaluation based on familiar probability theory but also considers the case of different fuzzy variable membership levels to different values. The model exhibits a broad spectrum of application prospects. In the structural reliability analysis, besides evaluating the reliability of uncertain structural systems, another important task is to effectively identify the impact of input uncertain parameters on the output response of the structures [17], which can guide the engineers to carry out the design more targeted, and this technique is called as sensitivity analysis (SA), which mainly includes local SA (LSA) and global SA (GSA), with GSA also referred to as an importance measure (IM). LSA is characterized by the partial derivatives of failure probability or reliability in relation to the distribution parameters of random variables, assessed at their nominal values. However, LSA fails to consider the impact of the entire range of random variables on the structural response, resulting in a lack of global perspective and computational stability [18]. Therefore, scholars have proposed IM techniques, among which the most widely used are variance-based IM [19; 20] and moment-independent sensitivity [21; 22; 23]. Variance-based IM is defined by the variance of the conditional mean of the output response given the input variables. This measure directly captures the impact of individual input variables and their interactions on the variance of the output response from an average perspective. Since its introduction in the mid-1990s, this technique has remained the dominant method in IM. However, [PERSON] and [PERSON] [24] argued that the IM index ought to encompass the full output distribution rather than merely one of its moments, advocating for moment-independent measures. [PERSON] [21], [PERSON] [22], and [PERSON] [23] independently moment-independent IM that assesses the influence of input variables across the entire output probability distribution.