Global Sensitivity Analysis of Fuzzy Distribution Parameter on Failure Probability and Its Single-Loop Estimation

Abstract

An extending Borgonovo’s global sensitivity analysis is proposed to measure the influence of fuzzy distribution parameters on fuzzy failure probability by averaging the shift between the membership functions (MFs) of unconditional and conditional failure probability. The presented global sensitivity indices can reasonably reflect the influence of fuzzy-valued distribution parameters on the character of the failure probability, whereas solving the MFs of unconditional and conditional failure probability is time-consuming due to the involved multiple-loop sampling and optimization operators. To overcome the large computational cost, a single-loop simulation (SLS) is introduced to estimate the global sensitivity indices. By establishing a sampling probability density, only a set of samples of input variables are essential to evaluate the MFs of unconditional and conditional failure probability in the presented SLS method. Significance of the global sensitivity indices can be verified and demonstrated through several numerical and engineering examples.

1. Introduction

Two different uncertainty sources are involved in reliability engineering: aleatory uncertainty and epistemic uncertainty [1–3]. Aleatory uncertainty describes the inherent variability associated with a structural system, which is referred to as irreducible, objective uncertainty, while epistemic uncertainty results from lack of knowledge of fundamental phenomena and is related to our ability to understand, measure, and describe the system under study. Employing the probability theory, the aleatory uncertainty can be modeled as random variable with probability density function (PDF). When dealing with the epistemic uncertainty, more than one theory has been developed, such as the interval analysis [4–6], the evidence theory [7, 8], and the fuzzy set theory [9].

Input variables can be modeled as random variables when pure aleatory uncertainty is present. Thus available sensitivity analysis (SA) techniques can be applied to study how uncertainty in the output of a model can be apportioned to different sources of uncertainty in the model input. SA is commonly divided into two groups: local SA (LSA) and global SA (GSA). LSA is defined as the partial derivatives of model output with respect to model input, which is generally computed at one given point. GSA, also known as importance analysis, focuses on determining which of the input variables affects output most in the whole uncertainty. With the ranking of the input variables resulting from importance analysis, one can pay more attention to variables with high importance and neglect variables with low importance during the process of design and optimization, thus providing useful information and guidance for engineers. At present, many GSA techniques and indices are available in the case of randomness, such as nonparametric techniques [10–12], variance-based importance measure indices [13–16], and moment-independent importance measures [17–19], among which moment independent sensitivity analysis is attracting growing attention among both academics and practitioners. The moment independent sensitivity index 𝛿_𝑖 proposed by Borgonovo is widely extended in GSA: Castaings et al. [20], Plischke et al. [21], and Luo et al. [22] developed methods for moment independent indices recently; the moment independent model was applied to environmental models [23] and climate model [24] et al.; Wei et al. [25] and Borgonovo et al. [26] brought up new moment independent indices.

When fuzzy set theory is employed, assigned membership function (MF) is used to model the epistemic uncertainty. Subsequently, methods for calculating the possibility distribution of output response are proposed [27–32]. Based on the fuzzy set theory which models the epistemic uncertainty as a fuzzy variable represented by MF, some methods have been developed for evaluating the importance of fuzzy variables [29–31, 33], among which Song et al. [33] proposed a generalized Borgonovo’s sensitivity indicator to analyze the effect of fuzzy-valued variables on the output response, and optimization techniques are employed to calculate the MF of output response [34]. When two kinds of uncertainties are concomitant in reliability problems, where aleatory uncertainty is described by PDF and epistemic uncertainty is modeled by fuzzy set theory, GSA for both fuzzy and random input variables is constructed by Zhangchun et al. in recent years [35–37].

However, obviously, the importance of basic variable is affected by distribution parameters of all involved basic random variables which need to be designed for the required output response. Therefore, it is essential to analyze the importance of distribution parameter on probabilistic response in engineering applications. Recently, GSA is constructed for systems with both input variables and distribution parameters described by PDF [38–40], whereas in this paper, the input variables are described by PDF and the distribution parameters are modeled as fuzzy parameters with assigned MF [41–43].

Then, a global sensitivity analysis is proposed to estimate the effect of fuzzy-valued distribution parameters on failure probability. Estimation of the established global sensitivity indices involves calculating the MFs of unconditional and conditional failure probability. For the problem that the computational cost for direct Monte-Carlo simulation (MCS) is rather too expensive to be accepted, a single-loop simulation (SLS) based on the concept of importance sampling is introduced to calculate the failure probability function (FPF), where a sampling PDF is introduced to generate a group of samples of the input variables. Thus, only one set of function calls is necessary to evaluate the sensitivity indices, and the efficiency has been greatly improved comparing with the existed direct MCS.

The paper is organized as follows. In Section 2, sensitivity indices of fuzzy-valued distribution parameters on failure probability are proposed and their properties are established. Section 3 gives the detailed procedures for direct MCS. Section 4 presents the SLS which can greatly save the computation cost. Applicability of sensitivity indices and accuracy and efficiency of the SLS are demonstrated by the numerical and engineering examples in Section 5. The paper ends with conclusions in Section 6.

2. Global Sensitivity Indices for Fuzzy Distribution Parameter Based on Failure Probability

2.1. Definitions of Global Sensitivity Indices

Considering structural systems with both epistemic and aleatory uncertainties, the output response can be given as 𝑌=𝑔(𝐗), (1) where 𝑌 is the output response. Let 𝐗 denote the aleatory variables whose PDF is given by 𝑓_𝐗(𝐱∣𝜽^∗), and 𝜽=(𝜃₁,𝜃₂,…,𝜃_𝑝) refers to the fuzzy-valued distribution parameter vector, and its characteristic is represented by a family of MFs noted by 𝜇_𝜃𝑖(𝜃_𝑖)(𝑖=1,2,…,𝑝). The membership grade 𝜇_𝜃𝑖(𝜔) quantifies the degree of possibility that parameter 𝜃_𝑖 takes value 𝜔. It is possible to evaluate fuzzy parameters using 𝜆-cuts or membership levels. Generally, the cut level 𝜆 takes value from the interval [0,1]. At each 𝜆-cut level, the variation of the fuzzy variable is defined by a lower bound 𝜃_𝑖𝐿(𝜆) and upper bound 𝜃_𝑖𝑈(𝜆).

Once the bounds of distribution parameters at 𝜆-cut are obtained by the corresponding MF 𝜇_𝜃𝑖(𝜃_𝑖), the bounds [𝑃^𝐿_𝑓(𝜽(𝜆)),𝑃^𝑈_𝑓(𝜽(𝜆))] of failure probability at various 𝜆-cut can be calculated; that is, the MF 𝜇_𝑃𝑓 of unconditional failure probability is obtained.

While the 𝑖th distribution parameter 𝜃_𝑖 was fixed at a value 𝜃^∗_𝑖, that is, 𝜽^∗=(𝜃₁,…,𝜃_𝑖−1,𝜃^∗_𝑖,𝜃_𝑖+1,…,𝜃_𝑝), the MF 𝜇^𝜃𝑖_𝑃′𝑓 of conditional failure probability can also be evaluated. Obviously, the shift between 𝜇_𝑃𝑓 and 𝜇^𝜃𝑖_𝑃′𝑓 measured by the area 𝑠_𝜃𝑖, which is the shadow area shown in Figure 1, can reflect the effect of the fixed parameter 𝜃_𝑖 on failure probability: 𝑠_𝜃𝑖=¹₀𝜇_𝑃𝑓−𝜇^𝜃𝑖_𝑃′𝑓𝑑𝑃_𝑓. (2)

Figure 1 

Shift between 𝜇𝑃𝑓 and 𝜇𝜃𝑖𝑃′𝑓.

The larger the value of 𝑠_𝜃𝑖 is, the more important the fixed value 𝜃_𝑖 for failure probability is. In average, the global sensitivity index of 𝜃_𝑖 can be measured by 𝜏_𝜃𝑖=^+∞_−∞𝜇_𝜃𝑖𝑠_𝜃𝑖𝑑𝜃_𝑖. (3)

However, in the engineering, the fuzzy-valued distribution parameters may have different dimensions. In order to eliminate the infections of the units of distribution parameters, we normalize the distribution parameters by introducing the equivalent PDF 𝜇_𝜃^(𝑒)_𝑖=𝜇_𝜃𝑖/∫𝜇_𝜃𝑖𝑑𝜃_𝑖, which satisfies the property that ∫𝜇_𝜃^(𝑒)_𝑖𝑑𝜃_𝑖=1. The equivalent PDF can eliminate the effect of the dimensions, while it does not change the distribution type of the MFs of distribution parameters [44] Thus, we have Γ_𝜃𝑖=^+∞_−∞𝜇_𝜃𝑖∫𝜇_𝜃𝑖𝑑𝜃_𝑖𝑠_𝜃𝑖𝑑𝜃_𝑖=^+∞_−∞𝜇_𝜃𝑖∫𝜇_𝜃𝑖𝑑𝜃_𝑖¹⁰𝜇_𝑃𝑓−𝜇^𝜃𝑖_𝑃′𝑓𝑑𝑃_𝑓𝑑𝜃_𝑖. (4)

From the above definitions, we know that Γ_𝜃𝑖 represents the effect of the distribution parameter 𝜃_𝑖 on failure probability of structural model.

Generally, MFs of failure probability cannot be expressed by explicit expression, which are determined by the lower and upper bounds at various 𝜆-cuts. So the integral of (4) needs numerical integration to be evaluated; that is, Γ_𝜃𝑖=^𝑁_𝑗=1Δ𝜃_𝑖⋅𝜇_𝜃𝑖𝜃_𝑖,𝑗∫𝜇_𝜃𝑖𝑑𝜃_𝑖×^𝑀_𝑘=1Δ𝑃_𝑓⋅𝜇_𝑃𝑓𝑃_𝑓^(𝑘)−𝜇^𝜃𝑖_𝑃′𝑓𝑃_𝑓^(𝑘)𝑁_𝑗=11, (5) where Δ𝜃_𝑖 and Δ𝑃_𝑓 are the spans of isometric sampling, 𝜃_𝑖,𝑗 is the 𝑗th fixed value of the distribution parameter 𝜃_𝑖, and 𝑃_𝑓^(𝑘) is the 𝑘th value of unconditional failure probability 𝑃_𝑓.

Global sensitivity index for an individual distribution parameter can be easily extended to a group of distribution parameters. Then the sensitivity index for a group of parameters can be given as: Γ_{𝜃𝑖,𝜃𝑗}=^+∞_−∞^+∞_−∞𝜇_𝜃𝑖∫𝜇_𝜃𝑖𝑑𝜃_𝑖𝜇_𝜃𝑗∫𝜇_𝜃𝑗𝑑𝜃_𝑗×¹₀𝜇_𝑃𝑓−𝜇^{𝜃𝑖,𝜃𝑗}_𝑃′𝑓𝑑𝑃_𝑓𝑑𝜃_𝑖𝑑𝜃_𝑗Γ_{𝜃𝑖∣𝜃𝑗}=^+∞_−∞^+∞_−∞𝜇_𝜃𝑖∫𝜇_𝜃𝑖𝑑𝜃_𝑖𝜇_𝜃𝑗∫𝜇_𝜃𝑗𝑑𝜃_𝑗×¹₀𝜇^𝜃𝑖_𝑃′𝑓−𝜇^{𝜃𝑖,𝜃𝑗}_𝑃′𝑓𝑑𝑃_𝑓𝑑𝜃_𝑖𝑑𝜃_𝑗, (6) where 𝜇^{𝜃𝑖,𝜃𝑗}_𝑃′𝑓 is the MF of conditional failure probability with distribution parameters 𝜃_𝑖 and 𝜃_𝑗 taken the fixed values 𝜃^∗_𝑖 and 𝜃^∗_𝑗, respectively.

Similarly, the numerical integrations for (6) are shown as follows: Γ_{𝜃𝑖,𝜃𝑗}=^𝑁𝑖𝑁_𝑙=1^𝑗_𝑚=1Δ𝜃_𝑖Δ𝜃_𝑗⋅𝜇_𝜃𝑖𝜃_𝑖,𝑙∫𝜇_𝜃𝑖𝑑𝜃_𝑖𝜇_𝜃𝑗𝜃_𝑗,𝑚∫𝜇_𝜃𝑗𝑑𝜃_𝑗×^𝑀_𝑞=1Δ𝑃_𝑓⋅𝜇_𝑃𝑓𝑃_𝑓^(𝑘)−𝜇^{𝜃𝑖,𝜃𝑗}_𝑃′𝑓𝑃_𝑓^(𝑘)Γ_{𝜃𝑖∣𝜃𝑗}=^𝑁𝑖𝑁_𝑙=1^𝑗_𝑚=1Δ𝜃_𝑖Δ𝜃_𝑗⋅𝜇_𝜃𝑖𝜃_𝑖,𝑙∫𝜇_𝜃𝑖𝑑𝜃_𝑖𝜇_𝜃𝑗𝜃_𝑗,𝑚∫𝜇_𝜃𝑗𝑑𝜃_𝑗×^𝑀_𝑞=1Δ𝑃_𝑓⋅𝜇^𝜃𝑖_𝑃′𝑓𝑃_𝑓^(𝑘)−𝜇^{𝜃𝑖,𝜃𝑗}_𝑃′𝑓𝑃_𝑓^(𝑘). (7)

Mathematical properties of the sensitivity indices are discussed in Section 2.2.

2.2. Mathematical Properties of the Proposed Sensitivity Indices

Similar to the sensitivity indices of the random input variables proposed by Song et al. [33], the mathematical properties of the proposed sensitivity indices for fuzzy distribution parameters can be explained as follows.

Property 1. Γ_𝜃𝑖≥0.

Property 2. If failure probability 𝑃_𝑓 is independent of 𝜃_𝑖, then Γ_𝜃𝑖=0.

Property 3. If 𝑃_𝑓 is dependent of 𝜃_𝑖 but independent of 𝜃_𝑗, then Γ_{𝜃𝑖𝜃𝑗}=Γ_𝜃𝑖, Γ_{𝜃𝑖∣𝜃𝑗}=0, and Γ_{𝜃𝑗∣𝜃𝑖}=Γ_𝜃𝑖.

Proofs for the above properties are similar with those in [19].

Note that since the sensitivity index 𝛿_𝑖 proposed by Borgonovo et al. has the property of transformation invariance [45, 46], thus the algorithms of computing the delta index can converge even faster if a monotonic transformation is applied when dealing with large distribution region of the model output. From the definitions in (2) and (3), we can see that the inner statistic of Γ_𝜃𝑖 and inner statistic of 𝛿_𝑖 both are 𝐿¹-norm on densities; thus; the moment independent sensitivity index for fuzzy-valued parameters is also monotonic transformation invariant. For knowing about the detailed property of moment-independent sensitivity index, one could refer to [45, 46].

3. MCS for Evaluating MFs of Failure Probability

Failure probability 𝑃_𝑓 is a function of the basic fuzzy-valued distribution parameter 𝜽=(𝜃₁,𝜃₂,…,𝜃_𝑝), given by 𝑃_𝑓=𝜓(𝜃₁,𝜃₂,…,𝜃_𝑝). Details of MCS for evaluating the MF of failure probability are as follows.(1)At 𝜆_𝑘-cut (𝑘=1,2,…,𝑁_𝜆), the lower bounds and upper bounds of the parameters 𝜽=(𝜃₁,𝜃₂,…,𝜃_𝑝) can be obtained as [𝜃_𝑖𝐿(𝜆_𝑘),𝜃_𝑖𝑈(𝜆_𝑘)](𝑖=1,2,…,𝑝) according to the MF 𝜇_𝜃𝑖(𝜃_𝑖) of basic fuzzy-valued parameter 𝜃_𝑖. Generate 𝑁 samples 𝜽_𝑡=(𝜃_1𝑡,𝜃_2𝑡,…,𝜃_𝑝𝑡)(𝑡=1,2,…,𝑁) according to the intervals of distribution parameters at the given 𝜆_𝑘-cut.(2)For each sample 𝜽_𝑡, generate 𝑁 samples 𝐱_𝑠=(𝑥_1𝑠,𝑥_2𝑠,…,𝑥_𝑛𝑠)(𝑠=1,2,…,𝑁) of aleatory variables 𝐗, and then the corresponding 𝑁 samples 𝑃^𝑠_𝑓(𝑠=1,2,…,𝑁) of failure probability 𝑃_𝑓 can be evaluated. Thus, the lower bound 𝑃^𝐿_𝑓(𝜆_𝑘) and upper bound 𝑃^𝑈_𝑓(𝜆_𝑘) of unconditional failure probability at given 𝜆_𝑘-cut can be obtained.(3)Iterate (1) and (2) for 𝑁_𝜆 times; then, the MF of unconditional failure probability 𝜇_𝑃𝑓 is evaluated.(4)Change parameter matrix 𝜽_𝑡=(𝜃_1𝑡,𝜃_2𝑡,…,𝜃_𝑝𝑡) to 𝜽^∗_𝑡=(𝜃_1𝑡,𝜃_2𝑡,…,𝜃^∗_𝑖𝑡,…,𝜃_𝑝𝑡) by fixing the 𝑖th distribution parameter 𝜃_𝑖 at its fixed value 𝜃_𝑖𝑡(𝑖=1,2,…,𝑝,𝑡=1,2,…,𝑁).(5)For each sample 𝜽^∗_𝑡, generate 𝑁 samples 𝐱^∗_𝑠=(𝑥_1𝑠,𝑥_2𝑠,…,𝑥_𝑛𝑠)(𝑠=1,2,…,𝑁) of aleatory variables 𝐗; then, the corresponding 𝑁 samples 𝑃^𝑠∗_𝑓(𝑠=1,2,…,𝑀) of failure probability 𝑃^∗_𝑓 can be evaluated. Thus, the lower bound 𝑃^𝐿∗_𝑓(𝜆_𝑘) and upper bound 𝑃^𝑈∗_𝑓(𝜆_𝑘) of conditional failure probability at given 𝜆_𝑘-cut can be obtained.(6)Iterate (4) and (5) for 𝑁 times; then, the 𝑁 MFs 𝜇_{𝑃𝑓∣𝜃𝑖=𝜃𝑖𝑡}(𝑡=1,2,…,𝑁) of conditional failure probability can be obtained.(7)Iterate (4), (5), and (6) for 𝑝 times; all the MFs of conditional failure probability of distribution parameters 𝜇_{𝑃𝑓|𝜃𝑗=𝜃𝑗𝑡}(𝑗=1,2,…,𝑝,𝑡=1,2,…,𝑁) can be evaluated.

We can see from the procedures in Figure 2 that using MCS combing optimization techniques 𝑁×𝑁×𝑁_𝜆 and 𝑝×𝑁×𝑁×𝑁×𝑁_𝜆 function calls are needed to calculate the MFs of unconditional and conditional failure probability, which is unacceptable especially in engineering problems. Thus, a single-loop method is proposed to evaluate the FPF of the distribution parameters in Section 4. (𝑁 is the sample number of distribution parameters and 𝑁_𝜆 is the cut number of the membership functions.)

(a) MCS for evaluating MF of unconditional failure probability

(b) MCS for evaluating MFs of conditional failure probability

Figure 2 

Flow chart of MCS for evaluating the MFs of unconditional and conditional failure probability.

4. SLS for Evaluating MFs of Failure Probability

By definition, FPF is expressed as 𝑃_𝑓(𝜽)=_{𝑔(𝐱)≤0}𝐼_𝐹(𝐱)𝑓(𝐱𝜽)𝑑𝐱. (8)

Suppose now we have a joint PDF ℎ_𝐱(𝐱) of model input vector with constant distribution parameter vector 𝜽^∗=(𝜃^∗₁,𝜃^∗₂,…,𝜃^∗_𝑚). Then the FPF can be derived as 𝑃_𝑓(𝜽)𝐼=^𝐹(𝐱)𝑓_𝐱(𝐱∣𝜽)ℎ_𝐱(𝐱)ℎ_𝐱(𝐱)𝑑𝐱=𝐸_ℎ𝐼_𝐹(𝐱)𝑓_𝐱(𝐱∣𝜽)ℎ_𝐱(𝐱), (9) where ℎ_𝐱(𝐱) is an arbitrary selected joint PDF of 𝐗 with constant distribution parameters 𝜽^∗. Equation (9) indicates that 𝑃_𝑓(𝜽) can be expressed as the expectation of the function 𝐼_𝐹(𝐱)𝑓_𝐱(𝐱∣𝜽)/ℎ_𝐱(𝐱). Given a set of model input vector samples 𝐱^(𝑘)=(𝑥₁^(𝑘),𝑥₂^(𝑘),…,𝑥_𝑛^(𝑘))(𝑘=1,2,…,𝑁) generated by the joint PDF ℎ_𝐱(𝐱), then 𝑃_𝑓(𝜽) can be estimated by 𝑃_𝑓(𝜽)=1𝑁^𝑁_𝑘=1𝐼_𝐹𝐱^(𝑘)𝑓_𝐱𝐱^(𝑘)𝜽ℎ_𝐱𝐱^(𝑘). (10)

Detailed procedures for SLS are as follows.(1)Generate 𝑁 samples 𝐱_𝑠=(𝑥_1𝑠,𝑥_2𝑠,…,𝑥_𝑛𝑠)(𝑠=1,2,…,𝑁) according to the joint PDF ℎ_𝐱(𝐱). For each 𝐱_𝑠(𝑠=1,2,…,𝑁), calculate the values of selected joint PDF ℎ_𝐱(𝐱_𝑠) and failure indicator 𝐼_𝐹(𝐱_𝑠) as ℎ^𝑠 and 𝐼^𝑠_𝐹.(2)At 𝜆_𝑘-cut (𝑘=1,2,…,𝑁_𝜆), the lower and upper bounds of the parameters 𝜽=(𝜃₁,𝜃₂,…,𝜃_𝑝) can be obtained as [𝜃_𝑖𝐿(𝜆_𝑘),𝜃_𝑖𝑈(𝜆_𝑘)](𝑖=1,2,…,𝑝) according to the MF 𝜇_𝜃𝑖(𝜃_𝑖) of basic fuzzy-valued parameter. Generate 𝑁 samples 𝜽_𝑡=(𝜃_1𝑡,𝜃_2𝑡,…,𝜃_𝑝𝑡)(𝑡=1,2,…,𝑁) according to the intervals of distribution parameters at the given 𝜆_𝑘-cut.(3)For each 𝜽_𝑡(𝑡=1,2,…,𝑁), calculate the values of the original joint PDF 𝑓_𝐱(𝐱_𝑠∣𝜽) as 𝑓^𝑠. Then (10) can be calculated as (1/𝑁)∑^𝑁_𝑠=1𝐼^𝑠_𝐹𝑓^𝑠/ℎ^𝑠; that is, 𝑃_𝑓(𝜽_𝑡) is obtained.(4)Iterate (3) for 𝑁 times; then, 𝑃_𝑓(𝜽) is calculated. Thus, the lower bound 𝑃^𝐿_𝑓 and upper bound 𝑃^𝑈_𝑓 of unconditional failure probability at given 𝜆_𝑘-cut are obtained.(5)Iterate (3) and (4) for 𝑁_𝜆 times, and the MF 𝜇_𝑃𝑓 of unconditional failure probability can be evaluated.(6)Change parameter matrix 𝜽_𝑡=(𝜃_1𝑡,𝜃_2𝑡,…,𝜃_𝑝𝑡) to 𝜽^∗_𝑡=(𝜃_1𝑡,𝜃_2𝑡,…,𝜃^∗_𝑖𝑡,…,𝜃_𝑝𝑡) by fixing the 𝑖th distribution parameter 𝜃_𝑖 at its fixed value 𝜃_𝑖𝑡(𝑖=1,2,…,𝑝,𝑡=1,2,…,𝑁).(7)For each 𝜽^∗_𝑡(𝑡=1,2,…,𝑁), calculate the values of the original joint PDF 𝑓_𝐱(𝐱_𝑠∣𝜽) as 𝑓^𝑠∗. Then (10) can be calculated as (1/𝑁)∑^𝑁_𝑘=1𝐼^𝑠_𝐹𝑓^𝑠∗/ℎ^𝑠; that is, 𝑃_𝑓(𝜽^∗_𝑡) is obtained.(8)Iterate (7) for 𝑁 times; then, 𝑃_𝑓(𝜽^∗) is calculated. Thus, the lower bound 𝑃^𝐿∗_𝑓 and upper bound 𝑃^𝑈∗_𝑓 of conditional failure probability at given 𝜆_𝑘-cut are obtained.(9)Iterate (7) and (8) for 𝑁_𝜆 times, and the MF 𝜇_{𝑃𝑓∣𝜃𝑖=𝜃𝑖𝑡} of conditional failure probability can be evaluated.(10)Iterate (7), (8), and (9) for 𝑁×𝑝 times, and all the MFs 𝜇_{𝑃𝑓∣𝜃𝑖=𝜃𝑖𝑡}(𝑖=1,2,…,𝑝,𝑡=1,2,…,𝑁) of conditional failure probability are evaluated.

The procedure in Figure 3 shows that given any 𝜆_𝑘-cut (𝑘=1,2,…,𝑁_𝜆), the interval of distribution parameters [𝜽_𝐿(𝜆_𝑘),𝜽_𝑈(𝜆_𝑘)] can be obtained; thus, the failure probability can be estimated using (10); that is, (10) can be treated as the imitate explicit function of the distribution parameters. Then any arbitrary optimization method can be used to get the upper and lower bounds of failure probability [𝑃^𝐿_𝑓(𝜆_𝑘),𝑃^𝑈_𝑓(𝜆_𝑘)] at given interval [𝜽_𝐿(𝜆_𝑘),𝜽_𝑈(𝜆_𝑘)] of distribution parameters. Only one set of samples of the aleatory variables is used in the procedure at different 𝜆_𝑘-cut and only 𝑁 function calls are needed to calculate the MF of unconditional failure probability. The procedures for calculating the MFs of conditional failure probability are similar, and the only difference is that the one or a group of distribution parameters are fixed at certain value.

(a) SLS for evaluating MF of unconditional failure probability

(b) SLS for evaluating MFs of conditional failure probability

Figure 3 

Flow chart of SLS for evaluating the MFs of unconditional and conditional failure probability.

Hence the above procedures for estimating the FPF need only one set of samples for implementing it; that is, the function calls for SLS is 𝑁; thus, it is computationally efficient.

Considering that the computational cost is rather expensive, optimization techniques can be employed to calculate the minimum and maximum values of the failure probability within the specified bounds of the cuts of the fuzzy-valued parameters corresponding to 𝜆-cut. Thus, combing MCS and SLS with local/global optimization techniques can be efficient. Accuracy and efficiency of the SLS can be established in Section 5.

5. Examples

A number of examples are used to demonstrate the availability and feasibility of the presented sensitivity indicator. These examples involve some cases met in engineering commonly, for example, normal, triangular, or trapezoidal types MFs, and MFs of failure probability are calculated by employing the optimization techniques.

Example 1. The response function 𝑧=4−𝑥₁−𝑥₂ has two variables. Case 1: 𝑥_𝑖∼𝑁(𝜇_𝑖,2), 𝜇_𝑖(𝑖=1,2) are fuzzy-valued distribution parameters. Case 2: 𝑥_𝑖∼𝑁(1,𝜎_𝑖), 𝜎_𝑖(𝑖=1,2) are fuzzy-valued parameters. The MFs are listed as follows: 𝜇_𝜇𝑖𝜇_𝑖=4−𝜇_𝑖3≤𝜇_𝑖𝜇≤4_𝑖−22≤𝜇_𝑖𝜇≤3,_𝜎𝑖𝜎_𝑖=4−𝜎_𝑖3≤𝜎_𝑖𝜎≤4_𝑖−22≤𝜎_𝑖≤3. (11)

The MFs of failure probability are drawn in Figures 4 and 5, and the results of global sensitivity indices calculated by MCS and SLS combing with optimization techniques are listed in Table 1.

Table 1 

Results of global sensitivity indices for Example 1, Case 1.

Figure 4 

MFs of failure probability for Example 1, Case 1.

Figure 5 

MFs of failure probability for Example 1, Case 2.

Case 1. See Figure 4 and Table 1.

Case 2. See Figure 5 and Table 2.

Table 2 

Results of global sensitivity indices for Example 1, Case 2.

Figures 4 and 5 show that the MFs of failure probability obtained by the SLS are in good accordance with those obtained by MCS, which testifies the accuracy of SLS. And Tables 1 and 2 indicate that the sensitivity indices of the two distribution parameters are equal, which is reasonable in this example. In addition, the accuracy of SLS is proved again by the results estimated by the two methods in Tables 1 and 2. In addition, the function calls of the two methods are also listed in Table 1, proving that SLS is greatly efficient in evaluating the proposed sensitivity indices.

To make the numerical results more robust, we carry out the calculations of sensitivity indices at increasing sample sizes to demonstrate convergence. Figures 6 and 7 show the convergence plot for Case 1 using MCS and SLS, respectively. One can easily find that the results begin to converge at the level of 2×10³. Thus we take 𝑁=2×10³ as shown in Table 1.

Figure 6 

Convergence plot of MCS for Example 1, Case 1.

Figure 7 

Convergence plot of SLS for Example 1, Case 1.

Example 2. The response function 𝑔=sin𝑥₁+7sin²𝑥₂+0.1𝑥⁴₃sin𝑥₁ includes three variables, where 𝑥_𝑖∼𝑁(0,𝜎_𝑖). And the MFs of distribution parameters 𝜎_𝑖 are given as 𝜇_𝜎𝑖𝜎_𝑖=0𝜎_𝑖5≤0.3𝜋𝜋𝜎_𝑖−1.50.3𝜋≤𝜎_𝑖≤0.5𝜋10.5𝜋≤𝜎_𝑖≤0.7𝜋8−10𝜎_𝑖𝜋0.7𝜋≤𝜎_𝑖≤0.8𝜋0𝜎_𝑖≥0.8𝜋. (12)

The MFs of failure probability are drawn in Figure 8, and the results of sensitivity indices calculated by MCS and SLS combing with optimization techniques are listed in Table 3.

Table 3 

Results of global sensitivity indices for Example 2.

Figure 8 

MFs of failure probability for Example 2.

Results of sensitivity indices in Table 3 indicate that the fuzzy-valued parameter 𝜎₃ makes more contribution to failure probability of the structural model than the other parameters. And the results evaluated by SLS are in good agreement with those evaluated by MCS.

Example 3. A roof truss is shown in Figure 9, and the top boom and the compression bars are reinforced by concrete, and the bottom boom and the tension bars are steel. Assume the uniformly distributed load 𝑞 is applied on the roof truss, and the uniformly distributed load can be transformed into the nodal load 𝑃=𝑞𝑙/4. Taking the safety and applicability into account, the perpendicular deflection Δ_𝐶 of the peak of structure node 𝐶 not exceeding 2.8 cm is taken as the constraint condition, and the performance response function can be constructed by 𝑔(𝑥)=0.028−Δ_𝐶, where Δ_𝐶 is the function of the basic random variables and Δ_𝐶=(𝑞𝑙²/2)(3.81/𝐴_𝐶𝐸_𝐶+1.13/𝐴_𝑆𝐸_𝑆), and 𝐴_𝐶, 𝐴_𝑆, 𝐸_𝐶, 𝐸_𝑆, and 𝑙, respectively, are sectional area, elastic modulus, length of the concrete, and steel bars; the distribution parameters of these independent normal basic random variables are listed in Table 4. It is assumed that the standard deviation of the variables is fuzzy parameters with membership function 𝜇_𝜎𝑖(𝜎_𝑖)=exp(−(𝑥−𝑎)²/2𝑏²). And the coefficients are listed in Table 5.

Table 4 

Distribution parameters of the input variables of roof truss.

Table 5 

Coefficients for membership function of 𝜎_𝑥.

(a)

(b)

Figure 9 

Schematic diagram of a roof truss.

The MFs of failure probability are drawn in Figure 10, and the results of sensitivity indices calculated by MCS and SLS combing with optimization techniques are listed in Table 6.

Table 6 

Results of global sensitivity indices for Example 3.

Figure 10 

MFs of failure probability for Example 3.

Results of sensitivity indices in Table 6 show that the rank of fuzzy valued distribution parameter is 𝜎_𝐴𝑠>𝜎_𝐸𝑠>𝜎_𝐴𝑐>𝜎_𝐸𝑐>𝜎_𝑙>𝜎_𝑞, among which 𝜎_𝑞 has little effect on the failure probability of the fuzzy structural model, and the effect of 𝜎_𝑞 with other parameters is the smallest among all the Γ_{𝜃𝑖𝜃𝑗}. In addition, the sensitivity indices can be estimated by the SLS guaranteeing acceptable accuracy with only one set of samples, which can greatly improve the computational efficiency. As testified by the results of Examples 1–3, properties of the sensitivity indices are obviously satisfied.

6. Conclusion

A global sensitivity index of fuzzy-valued distribution parameter on the fuzzy failure probability is defined in this work. The key to calculate the sensitivity index is to evaluate the MFs of failure probability. For the problem that the computational cost of MCS is rather expensive, a SLS is introduced to estimate the proposed sensitivity indices, which can greatly improve the computational efficiency considerably with acceptable accuracy. Applications of the sensitivity indices proposed in this paper to the numerical and engineering examples show that fuzzy sensitivity indices can provide useful information about the sequence of the distribution parameters on the fuzzy uncertainty of failure probability, and the proposed sensitivity indices can identify the relative important fuzzy distribution parameters of the structural model, thus providing useful information for engineers in the process of reliability design and optimization.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. NSFC 51175425), the Aviation Foundation (Grant nos. 2011ZA53015), and the Special Research Fund for the Doctoral Program of Higher Education of China (Grant nos. 20116102110003).