Global Sensitivity Analysis of Fuzzy Distribution Parameter on Failure Probability and Its Single-Loop Estimation
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.
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 .
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 proposed by Borgonovo is widely extended in GSA: Castaings et al. , Plischke et al. , and Luo et al.  developed methods for moment independent indices recently; the moment independent model was applied to environmental models  and climate model  et al.; Wei et al.  and Borgonovo et al.  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.  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 . 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 where is the output response. Let denote the aleatory variables whose PDF is given by , and refers to the fuzzy-valued distribution parameter vector, and its characteristic is represented by a family of MFs noted by . The membership grade quantifies the degree of possibility that parameter takes value . It is possible to evaluate fuzzy parameters using -cuts or membership levels. Generally, the cut level takes value from the interval . At each -cut level, the variation of the fuzzy variable is defined by a lower bound and upper bound .
Once the bounds of distribution parameters at -cut are obtained by the corresponding MF , the bounds of failure probability at various -cut can be calculated; that is, the MF of unconditional failure probability is obtained.
While the th distribution parameter was fixed at a value , that is, , the MF of conditional failure probability can also be evaluated. Obviously, the shift between and measured by the area , which is the shadow area shown in Figure 1, can reflect the effect of the fixed parameter on failure probability:
The larger the value of is, the more important the fixed value for failure probability is. In average, the global sensitivity index of can be measured by
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 , which satisfies the property that . The equivalent PDF can eliminate the effect of the dimensions, while it does not change the distribution type of the MFs of distribution parameters  Thus, we have
From the above definitions, we know that represents the effect of the distribution parameter 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 -cuts. So the integral of (4) needs numerical integration to be evaluated; that is, where and are the spans of isometric sampling, is the th fixed value of the distribution parameter , and is the th value of unconditional failure probability .
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: where is the MF of conditional failure probability with distribution parameters and taken the fixed values and , respectively.
Similarly, the numerical integrations for (6) are shown as follows:
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. , the mathematical properties of the proposed sensitivity indices for fuzzy distribution parameters can be explained as follows.
Property 1. .
Property 2. If failure probability is independent of , then .
Property 3. If is dependent of but independent of , then , , and .
Proofs for the above properties are similar with those in .
Note that since the sensitivity index 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 and inner statistic of both are -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 is a function of the basic fuzzy-valued distribution parameter , given by . Details of MCS for evaluating the MF of failure probability are as follows.(1)At -cut , the lower bounds and upper bounds of the parameters can be obtained as according to the MF of basic fuzzy-valued parameter . Generate samples according to the intervals of distribution parameters at the given -cut.(2)For each sample , generate samples of aleatory variables , and then the corresponding samples of failure probability can be evaluated. Thus, the lower bound and upper bound of unconditional failure probability at given -cut can be obtained.(3)Iterate (1) and (2) for times; then, the MF of unconditional failure probability is evaluated.(4)Change parameter matrix to by fixing the th distribution parameter at its fixed value .(5)For each sample , generate samples of aleatory variables ; then, the corresponding samples of failure probability can be evaluated. Thus, the lower bound and upper bound of conditional failure probability at given -cut can be obtained.(6)Iterate (4) and (5) for times; then, the MFs of conditional failure probability can be obtained.(7)Iterate (4), (5), and (6) for times; all the MFs of conditional failure probability of distribution parameters can be evaluated.
We can see from the procedures in Figure 2 that using MCS combing optimization techniques and 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. ( is the sample number of distribution parameters and 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
4. SLS for Evaluating MFs of Failure Probability
By definition, FPF is expressed as
Suppose now we have a joint PDF of model input vector with constant distribution parameter vector . Then the FPF can be derived as where is an arbitrary selected joint PDF of with constant distribution parameters . Equation (9) indicates that can be expressed as the expectation of the function . Given a set of model input vector samples generated by the joint PDF , then can be estimated by
Detailed procedures for SLS are as follows.(1)Generate samples according to the joint PDF . For each , calculate the values of selected joint PDF and failure indicator as and .(2)At -cut , the lower and upper bounds of the parameters can be obtained as according to the MF of basic fuzzy-valued parameter. Generate samples according to the intervals of distribution parameters at the given -cut.(3)For each , calculate the values of the original joint PDF as . Then (10) can be calculated as ; that is, is obtained.(4)Iterate (3) for times; then, is calculated. Thus, the lower bound and upper bound of unconditional failure probability at given -cut are obtained.(5)Iterate (3) and (4) for times, and the MF of unconditional failure probability can be evaluated.(6)Change parameter matrix to by fixing the th distribution parameter at its fixed value .(7)For each , calculate the values of the original joint PDF as . Then (10) can be calculated as ; that is, is obtained.(8)Iterate (7) for times; then, is calculated. Thus, the lower bound and upper bound of conditional failure probability at given -cut are obtained.(9)Iterate (7) and (8) for times, and the MF of conditional failure probability can be evaluated.(10)Iterate (7), (8), and (9) for times, and all the MFs of conditional failure probability are evaluated.
The procedure in Figure 3 shows that given any -cut , the interval of distribution parameters 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 at given interval of distribution parameters. Only one set of samples of the aleatory variables is used in the procedure at different -cut and only 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
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 ; 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 -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.
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 has two variables. Case 1: , are fuzzy-valued distribution parameters. Case 2: , are fuzzy-valued parameters. The MFs are listed as follows:
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.
Case 1. See Figure 4 and Table 1.
Case 2. See Figure 5 and Table 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 . Thus we take as shown in Table 1.
Example 2. The response function includes three variables, where . And the MFs of distribution parameters are given as
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.
Results of sensitivity indices in Table 3 indicate that the fuzzy-valued parameter 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 is applied on the roof truss, and the uniformly distributed load can be transformed into the nodal load . Taking the safety and applicability into account, the perpendicular deflection of the peak of structure node not exceeding 2.8 cm is taken as the constraint condition, and the performance response function can be constructed by , where is the function of the basic random variables and , and , , , , and , 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 . And the coefficients are listed in Table 5.
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.
Results of sensitivity indices in Table 6 show that the rank of fuzzy valued distribution parameter is , among which has little effect on the failure probability of the fuzzy structural model, and the effect of with other parameters is the smallest among all the . 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.
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.
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).
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