Importance analysis on failure credibility of the fuzzy structure

Abstract

To measure the effects of the fuzzy inputs on structural safety degree, this paper establishes the failure credibility-based global sensitivity by the fuzzy expected value of the absolute difference between the unconditional failure credibility and conditional one. To establish the failure credibility-based global sensitivity, the conditional failure credibility is firstly defined according to the original definition of conditional event and the relationship among the possibility, necessity and credibility, in which no extra assumption is introduced. After that, the equivalent expression of the failure credibility is deduced, on which the Bayesian transformation of the conditional failure credibility is obtained in this paper. Then, a single-loop method based on the sequential quadratic programming is applied to efficiently estimate the defined failure credibility-based global sensitivity. According to the result of the constructed failure credibility-based global sensitivity, designers can pay more attentions to the more important fuzzy inputs to have a better control of the structural safety degree. The presented examples demonstrate the feasibility of the constructed failure credibility-based global sensitivity and the efficiency of the proposed solution.

Keywords

Fuzzy input failure credibility global sensitivity fuzzy expected value conditional failure credibility sequential quadratic programming

1 Introduction

Sensitivity analysis is intended to study and analyze the susceptibility of the output response of the model to the input parameters and surrounding conditions [1]. As an important branch of sensitivity analysis, importance analysis (also known as global sensitivity analysis) is of great significance in engineering design and safety assessment. Importance analysis aims to determine the input variables that influence model output the most in their whole uncertainty ranges. With the ranking of the input variables obtained by the importance analysis, designers can pay more attentions or priorities to the input variables with high importance and neglect the input variables with low importance during the process of design and optimization [2]. At present, importance analysis approaches under the random uncertainties have been fully developed, such as the variance-based importance analysis and moment independent importance analysis. The variance-based importance analysis [3] is proposed to measure the variance contribution of the input variables to the variance of model output. The moment independent importance analysis [4] is proposed to measure the effect of the inputs over their entire distribution ranges on the probability density function of the model output. In structural reliability analysis and reliability-based design optimization, researchers are more interested in structural reliability or failure probability, thus, investigating the effects of input variables on failure probability over their whole variation domains and ranking the input variables according to their contributions to the failure probability are really significant. For this purpose, the failure probability-based importance analysis was proposed by Cui et al. [5], which measures the effect of the input variable over its whole variation domain on the failure probability by the expected absolute difference between the unconditional failure probability and conditional one on a fixed input, and the failure probability-based importance analysis can provide much useful information in safety design of the structure for target failure probability.

The importance analysis approaches mentioned above only deal with random uncertainties described by given probability distributions. Nevertheless, in actual engineering applications, the exact probability distributions of certain uncertainties involved usually cannot be obtained due to scarce or even absent statistical information, i.e., certain uncertainties should be described as epistemic uncertainties. Compared with the random uncertainty which describes the nature or physical variability associated with a quantity of interest, the epistemic uncertainty usually results from lack of knowledge of fundamental phenomena, incomplete information or insufficient data, which can be reduced with the increasing recognition or additional observational data. Thus, it is also referred to as subjective uncertainty, cognitive uncertainty or reducible uncertainty [6]. The information of epistemic uncertainty can be obtained by experts’ estimation and in most cases possesses an interval or fuzzy characteristic. Generally, it is unsuitable to describe the epistemic uncertainty by the probability distribution, because the information of the epistemic uncertainty is insufficient. In view of this, many non-probabilistic methods presently have been developed to describe the epistemic uncertainty, such as the interval analysis [7, 8], convex modeling [9], fuzzy set theory [10] and possibility theory [11]. The main goal of these methods is to derive computationally less expensive algorithms without a priori assumption about probability distribution of the input variable and give conservative approximation of structural safety degree. In this paper, the fuzzy set theory treating the epistemic uncertainty as fuzzy variable is applied.

Fuzziness, which generally results from the vague of boundary [12 –14] or the uncertainty of determining a proposition’s true or false in logic [15], is one of the important uncertainties in real world [16]. Fuzzy set theory was first introduced by Zadeh [10] in 1965 to transform linguistic variables into numerical variables [17], then, to measure a fuzzy event, Zadeh [11] proposed the concept of possibility measure in 1978, and the notion of necessity was also developed by duality [18]. Although the possibility and necessity measures have been widely used, they do not possess the self-duality property which is absolutely needed in both theory and practice [19]. In order to define a self-dual measure, Liu and Liu [20] defined the concept of credibility measure in 2002, after that, the credibility theory was founded by Liu [21] in 2004 as a branch of mathematics for studying the behavior of fuzzy phenomena. Recently, scholars have introduced the concept of fuzziness into the structural system safety analysis field. Different from the commonly acknowledged probabilistic safety measure (reliability or failure probability) of the structure in the presence of random inputs, there isn’t a perfect safety measure so far in general recognition for the structure under fuzzy inputs. In order to measure the safety level of the structure in the presence of fuzzy variables, three widely used fuzzy safety measures have been proposed based on the aforementioned possibility, necessity and credibility measures, i.e., failure possibility [22], failure necessity [23] and failure credibility [23 –25]. Failure possibility is defined as the lower bound of fuzzy safety confidence level, i.e., the structure must be safe if the membership degree is larger than the failure possibility; the failure necessity defined as 1 minus the safety possibility is also a popular fuzzy safety measure; and the failure credibility is the arithmetic average of the failure possibility and failure necessity which can get full information of membership function. Various strategies have been developed to estimate the failure possibility, failure necessity and failure credibility, for example, the interval approach [26], level cut optimization approach [27], fuzzy simulation method [19, 23] and so on. Among these methods, the fuzzy simulation approach has no limit on the form of the fuzzy variable’s membership function or the complexity of the limit state function, as long as the number of simulation samples of the fuzzy variables is large enough, the estimated failure credibility converges to its true value.

Although various researches have been done on the safety analysis of the structure in the presence of fuzzy inputs, the research on the importance analysis of the fuzzy structure is rare. Inspired by the failure probability-based global reliability sensitivity for the structure in the presence of random variables, this paper establishes the failure credibility-based global sensitivity to analyze the importance of the fuzzy inputs for the structure in the presence of fuzzy variables. The failure credibility-based global sensitivity is defined as the fuzzy expected value of the absolute difference between the unconditional failure credibility and conditional one on fixing the fuzzy input at a specific value. The key to construct the failure credibility-based global sensitivity includes two aspects, i.e., the definition of conditional failure credibility and the definition of fuzzy expectation operator. For the first aspect, this paper defines a conditional failure credibility by the conditional event’s original definition as well as the relationship among possibility, necessity and credibility. No extra assumption is introduced in the definition of conditional failure credibility. For the second aspect, there are many ways to define an expected value operator related to fuzzy variable and this paper adopts the most general one given by Liu and Liu [20].

In the constructed failure credibility-based global sensitivity, its equivalent expression is presented after the Bayesian expression of the conditional failure credibility is deduced, on which a single-loop method based on the sequential quadratic programming strategy is proposed to efficiently estimate the constructed failure credibility-based global sensitivity. The innovations of this paper include the following four aspects:

define the conditional failure credibility

construct the failure credibility-based global sensitivity

deduce the Bayesian transformation of the conditional failure credibility

propose a single-loop method to efficiently estimate the constructed failure credibility-based global sensitivity

The paper’s outline is as follows. Section 2 demonstrates a brief review of the failure credibility. Section 3 firstly defines the conditional failure credibility and deduces its Bayesian expression, then the failure credibility-based global sensitivity is established and its fuzzy simulation solution is provided. Section 4 presents the proposed single-loop method for estimating the constructed failure credibility-based global sensitivity. Three examples are given in Section 5 to illustrate the ranking results obtained by the constructed failure credibility-based global sensitivity, and the efficiency and accuracy of the proposed single-loop solution are also verified by the examples. Section 6 draws the conclusions.

2 Failure credibility

Suppose the n-dimensional fuzzy input vector of the structure is X = {X₁, X₂, ⋯ , X_n} ^T, which can be characterized by their joint membership function ρ_X ( x ). The membership function represents the degree of possibility that the fuzzy variable takes some prescribed value. For example, the membership degree ρ_X (x) =0 if x is an impossible point, and ρ_X ( x ) =1 if x is the most possible point that X takes. If the fuzzy input variables are independent with each other, the joint membership function of them can be expressed as $ρ_{X} (x) = min_{i = 1, 2, \dots, n} ρ_{X_{i}} (x_{i})$ [19], in which ρ_{X
_i} (x_i) is the membership function of fuzzy input variable X_i (i = 1, 2, ⋯ , n). Denote the limit state function of the structure as Y = g ( X ). The failure event occurs on X = x if g ( x ) ⩽0 and the structure is safe on X = x when g ( x ) ⩽0. Due to the input variables are fuzzy and described by their membership functions, the limit state function Y = g ( X ) is also a fuzzy variable which can be characterized by its membership function ρ_Y [g ( x )].

To quantify the safety level of the structure in the presence of fuzzy inputs, the failure possibility π_F defined as the supremum of the membership degree of the failure event F ={ g ( x ) ⩽0 } has been proposed, i.e.,

$\begin{matrix} π_{F} & = Poss {g (x) ⩽ 0} = sup {ρ_{Y} [g (x)] | g (x) ⩽ 0} \\ = sup {ρ_{X} (x) | g (x) ⩽ 0} \end{matrix}$ (1) in which Poss {·} represents the possibility operator and sup {·} denotes the supremum operator. The illustration of failure possibility π_F is shown in Fig. 1(a).

Fig. 1

Illustrations of possibility and necessity.

Similarly, the safety possibility π_S defined as the supremum of the membership degree of the safety event S ={ g ( x ) >0 } is expressed as follows,

$\begin{matrix} π_{S} & = Poss {g (x) > 0} = sup {ρ_{Y} [g (x)] | g (x) > 0} \\ = sup {ρ_{X} (x) | g (x) > 0} \end{matrix}$ (2)

The demonstration of safety possibility is presented in Fig. 1(b).

The failure necessity κ_F (just as shown in Fig. 1(c)) is defined as 1 minus safety possibility π_S, i.e.,

$\begin{matrix} κ_{F} = Nece {g (x) ⩽ 0} = 1 - π_{S} = 1 - Poss {g (x) > 0} \\ = 1 - sup {ρ_{Y} [g (x)] | g (x) > 0} \\ = 1 - sup {ρ_{X} (x) | g (x) > 0} \end{matrix}$ (3)

in which Nece{·} represents the necessity operator. The safety necessity κ_S (just as shown in Fig. 1(d)) is defined as 1 minus failure possibility π_F, i.e.,

$\begin{matrix} κ_{S} = Nece {g (x) > 0} = 1 - π_{F} = 1 - Poss {g (x) ⩽ 0} \\ = 1 - sup {ρ_{Y} [g (x)] | g (x) ⩽ 0} \\ = 1 - sup {ρ_{X} (x) | g (x) ⩽ 0} \end{matrix}$ (4)

The failure credibility c_F is the arithmetic average of failure possibility π_F and failure necessity κ_F, i.e.,

$\begin{matrix} c_{F} = Cred {g (x) ⩽ 0} = \frac{1}{2} (π_{F} + κ_{F}) = \frac{1}{2} (π_{F} + 1 - π_{S}) \\ = \frac{1}{2} [sup {ρ_{Y} [g (x)] | g (x) ⩽ 0} \\ + 1 - sup {ρ_{Y} [g (x)] | g (x) > 0}] \\ = \frac{1}{2} [sup {ρ_{X} (x) | g (x) ⩽ 0} + 1 - sup {ρ_{X} (x) | g (x) > 0}] \end{matrix}$ (5)

in which Cred {·} is the credibility operator. The safety credibility c_S is the arithmetic average of safety possibility π_S and safety necessity κ_S, i.e.,

$\begin{matrix} c_{S} = Cred {g (x) > 0} = \frac{1}{2} (π_{S} + κ_{S}) = \frac{1}{2} (π_{S} + 1 - π_{F}) \\ = \frac{1}{2} [sup {ρ_{Y} [g (x)] | g (x) > 0} \\ + 1 - sup {ρ_{Y} [g (x)] | g (x) ⩽ 0}] \\ = \frac{1}{2} [sup {ρ_{X} (x) | g (x) > 0} + 1 - sup {ρ_{X} (x) | g (x) ⩽ 0}] \end{matrix}$ (6)

The superiority of failure credibility c_F over failure possibility π_F and failure necessity κ_F has been summarized in [23 –25], which can be concluded as follows: (1) failure credibility can accurately distinguish the safety level of different structures whereas failure possibility and failure necessity cannot in some cases, and (2) failure credibility possesses self-duality property whereas the others do not have.

3 Importance analysis on the failure credibility

This section firstly defines the conditional failure credibility, on which the failure credibility-based global sensitivity is constructed to measure the importance of the fuzzy inputs and rank the fuzzy inputs according to their contributions to the failure credibility. Finally, the fuzzy simulation for estimating the defined failure credibility-based global sensitivity is introduced.

3.1 Conditional failure credibility

In Section 2, a brief review of failure credibility c_F is presented. Now, consider the credibility of the failure event F ={ g ( x ) ⩽0 } after the event { X_i = x_i } has occurred. This new credibility of the failure event F is called the conditional credibility of the failure event F given {X_i = x_i} or the conditional failure credibility given {X_i = x_i}. To define the conditional failure credibility, this paper firstly defines the conditional failure possibility given {X_i = x_i} as follows, $\begin{matrix} π_{F | X_{i} = x_{i}} = Poss {g (x) ⩽ 0 | X_{i} = x_{i}} \\ = sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) ⩽ 0} \\ = sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) ⩽ 0} \end{matrix}$ (7) where π_{F|X_i=x_i} is the conditional failure possibility given {X_i = x_i}, ρ_{Y|X_i=x_i} [·] is the membership function of model output given {X_i = x_i}. _X - i = { X₁, X₂, ⋯ , X_i-1, X_i+1, ⋯ , X_n } ^T denotes the vector consisting of all fuzzy inputs except the ith one, ρ_{_X-i} (_x - i) is the joint membership function of fuzzy inputs _X - i. Similarly, the conditional safety possibility π_{S|X_i=x_i} given {X_i = x_i} is

$\begin{matrix} π_{S | X_{i} = x_{i}} = Poss {g (x) > 0 | X_{i} = x_{i}} \\ = sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) > 0} \\ = sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) > 0} \end{matrix}$ (8)

The conditional failure necessity κ_{F|X_i=x_i} given {X_i = x_i} is defined as 1 minus the conditional safety possibility π_{S|X_i=x_i}, i.e.,

$\begin{matrix} κ_{F | X_{i} = x_{i}} = 1 - π_{S | X_{i} = x_{i}} = 1 - Poss {g (x) > 0 | X_{i} = x_{i}} \\ = 1 - sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) > 0} \\ = 1 - sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) > 0} \end{matrix}$ (9)

Likewise, the conditional safety necessity κ_{S|X_i=x_i} given {X_i = x_i} is defined as 1 minus the conditional failure possibility π_{F|X_i=x_i}, i.e.,

$\begin{matrix} κ_{S | X_{i} = x_{i}} = 1 - π_{F | X_{i} = x_{i}} = 1 - Poss {g (x) ⩽ 0 | X_{i} = x_{i}} \\ = 1 - sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) ⩽ 0} \\ = 1 - sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) ⩽ 0} \end{matrix}$ (10)

Finally, the conditional failure credibility c_{F|X_i=x_i} and conditional safety credibility c_{S|X_i=x_i} given {X_i = x_i} are respectively defined as follows, $\begin{matrix} c_{F | X_{i} = x_{i}} = Cred {g (x) ⩽ 0 | X_{i} = x_{i}} \\ = \frac{1}{2} (π_{F | X_{i} = x_{i}} + κ_{F | X_{i} = x_{i}}) \\ = \frac{1}{2} (π_{F | X_{i} = x_{i}} + 1 - π_{S | X_{i} = x_{i}}) \\ = \frac{1}{2} [Poss {g (x) ⩽ 0 | X_{i} = x_{i}} \\ + 1 - Poss {g (x) > 0 | X_{i} = x_{i}}] \\ = \frac{1}{2} [sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) ⩽ 0} \\ + 1 - sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) > 0}] \\ = \frac{1}{2} [sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) ⩽ 0} \\ + 1 - sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) > 0}] \end{matrix}$ (11) $\begin{matrix} c_{S | X_{i} = x_{i}} = Cred {g (x) > 0 | X_{i} = x_{i}} \\ = \frac{1}{2} (π_{S | X_{i} = x_{i}} + κ_{S | X_{i} = x_{i}}) \\ = \frac{1}{2} (π_{S | X_{i} = x_{i}} + 1 - π_{F | X_{i} = x_{i}}) \\ = \frac{1}{2} [Poss {g (x) > 0 | X_{i} = x_{i}} \\ + 1 - Poss {g (x) ⩽ 0 | X_{i} = x_{i}}] \\ = \frac{1}{2} [sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) > 0} \\ + 1 - sup {ρ_{Y | X_{i} = x_{i}} [g (x_{- i}, x_{i})] | g (x_{- i}, x_{i}) ⩽ 0}] \\ = \frac{1}{2} [sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) > 0} \\ + 1 - sup {ρ_{X_{- i}} (x_{- i}) | g (x_{- i}, x_{i}) ⩽ 0}] \end{matrix}$ (12)

Now, this paper has solved the first problem of defining the failure credibility-based global sensitivity, i.e., defining the conditional failure credibility. From Equations (7) to (12), it can be seen that to define the conditional failure credibility, this paper does not make any assumptions. It is just based on (1) the original definition of conditional event; (2) the necessity is the dual index of possibility; (3) the credibility is the arithmetic average of possibility and necessity. Therefore, the conditional failure credibility defined in this paper is reasonable.

3.2 Failure credibility-based global sensitivity

To measure the importance of the fuzzy inputs to the failure credibility, this section defines the failure credibility-based global sensitivity. The absolute difference $ς_{X_{i}} (x_{i})$ between the unconditional failure credibility c_F and the conditional failure credibility c_{F|X_i=x_i} given {X_i = x_i} is $ς_{X_{i}} (x_{i}) = | c_{F} - c_{F | X_{i} = x_{i}} |$ (13)

Since variable X_i is fuzzy and characterized by its membership function, as a function of fuzzy variable X_i, $ς_{X_{i}} (x_{i})$ is also a fuzzy variable. Therefore, $ς_{X_{i}} (x_{i})$ cannot be directly applied to compare the importance of different fuzzy inputs. To measure the effects of different fuzzy input variables on the failure credibility, the notion of expected value of fuzzy variable is applied in this paper. Taking advantage of the expected value of fuzzy variable [20], this paper defines the failure credibility-based global sensitivity $ξ_{X_{i}}^{c_{F}}$ as the fuzzy expected value of $ς_{X_{i}} (x_{i})$ , i.e., $\begin{matrix} ξ_{X_{i}}^{c_{F}} = E [ς_{X_{i}} (x_{i})] = \int_{0}^{+ \infty} Cred {ς_{X_{i}} (x_{i}) ⩾ r} d r \\ - \int_{- \infty}^{0} Cred {ς_{X_{i}} (x_{i}) ⩽ r} d r \\ = E [| c_{F} - c_{F | X_{i} = x_{i}} |] \\ = \int_{0}^{+ \infty} Cred {| c_{F} - c_{F | X_{i} = x_{i}} | ⩾ r} d r \\ - \int_{- \infty}^{0} Cred {| c_{F} - c_{F | X_{i} = x_{i}} | ⩽ r} d r \end{matrix}$ (14)

The higher $ξ_{X_{i}}^{c_{F}}$ implies that X_i is more important to failure credibility c_F, i.e., the fuzzy variable X_i possesses larger effect on the structural safety.

Since ς_{X
_i} (x_i) = |c_F - c_{F|X_i=x_i}|⩾0 always holds, $\int_{- \infty}^{0} Cred {ς_{X_{i}} (x_{i}) ⩽ r} d r = 0$ . Hence, Equation (14) can be simplified as $ξ_{X_{i}}^{c_{F}} = \int_{0}^{+ \infty} Cred {ς_{X_{i}} (x_{i}) ⩾ r} d r$ (15)

The constructed failure credibility-based global sensitivity satisfies $0 ⩽ ξ_{X_{i}}^{c_{F}} ⩽ 1$ , which can be simply proved by Proof 1.

Proof 1.

Due to ς_{X
_i} (x_i) = |c_F - c_{F|X_i=x_i}| ⩾ 0, therefore, $ξ_{X_{i}}^{c_{F}} = \int_{0}^{+ \infty} Cred {ς_{X_{i}} (x_{i}) ⩾ r} d r ⩾ 0$ . Since 0 ⩽ c_F ⩽ 1 and 0 ⩽ c_{F|X_i=x_i} ⩽ 1, thus, 0 ⩽ ς_{X
_i} (x_i) = |c_F - c_{F|X_i=x_i}| ⩽ 1. Then, $\begin{matrix} ξ_{X_{i}}^{c_{F}} = \int_{0}^{+ \infty} Cred {ς_{X_{i}} (x_{i}) ⩾ r} d r \\ = \int_{0}^{1} Cred {ς_{X_{i}} (x_{i}) ⩾ r} d r \\ + \int_{1}^{+ \infty} Cred {ς_{X_{i}} (x_{i}) ⩾ r} d r \\ = \int_{0}^{1} Cred {ς_{X_{i}} (x_{i}) ⩾ r} d r ⩽ \int_{0}^{1} 1 d r = 1 \end{matrix}$ (16)

Algorithm 1: Algorithm of fuzzy simulation for estimating failure credibility-based global sensitivity.

(Estimate the unconditional failure credibility)

1. Uniformly generate N samples

S = {x_{1}, x_{2}, \dots, x_{N}}^{T}

of the fuzzy inputs in [ X ^L, X ^U], where

X^{L} = {X_{1}^{L}, X_{2}^{L}, \dots, X_{n}^{L}}^{T}

denotes the lower bounds of fuzzy inputs and

X^{U} = {X_{1}^{U}, X_{2}^{U}, \dots, X_{n}^{U}}^{T}

represents the upper bounds of fuzzy inputs.

2. Calculate

{g (x_{1}), g (x_{2}), \dots, g (x_{N})}^{T}

3. Record the failure samples as

{x_{1}^{F}, x_{2}^{F}, \dots, x_{N_{F}}^{F}}^{T}

and record the safety samples as

{x_{1}^{S}, x_{2}^{S}, \dots, x_{N_{S}}^{S}}^{T}

where N_F + N_S = N.

4. Estimate the unconditional failure credibility by

{\hat{c}}_{F} = (max_{j = 1, 2, \dots, N_{F}} ρ_{X} (x_{j}^{F}) + 1 - max_{j = 1, 2, \dots, N_{S}} ρ_{X} (x_{j}^{S})) / 2

(Estimate the conditional failure credibility)

Fori = 1 : n

5. Uniformly generate N_i samples

{x_{1}^{(i)}, x_{2}^{(i)}, \dots, x_{N_{i}}^{(i)}}^{T}

of the fuzzy input X_i in

[X_{i}^{L}, X_{i}^{U}]

Fork = 1 : N_i

6. Construct the matrix

S_{i} = {x_{1}^{'}, x_{2}^{'}, \dots, x_{N}^{'}}^{T}

, in which the ith column is

x_{k}^{(i)}

and the others are the corresponding columns from S.

7. Calculate

{g (x_{1}^{'}), g (x_{2}^{'}), \dots, g (x_{N}^{'})}^{T}

, record the failure samples as

{x_{1}^{F}, x_{2}^{F}, \dots, x_{N_{F}^{'}}^{F}}^{T}

and record the safety samples as

{x_{1}^{S}, x_{2}^{S}, \dots, x_{N_{S}^{'}}^{S}}^{T}

, and

N_{F}^{'} + N_{F}^{'} = N

8. Estimate the conditional failure credibility by

{\hat{c}}_{F | X_{i} = x_{k}^{(i)}} = (max_{j = 1, 2, \dots, N_{F}^{'}} ρ_{X} (x_{j}^{F}) + 1 - max_{j = 1, 2, \dots, N_{S}^{'}} ρ_{X} (x_{j}^{S})) / 2

End for

9. Obtain the conditional failure credibility

{{\hat{c}}_{F | X_{i} = x_{k}^{(i)}}, k = 1, 2, \dots, N_{i}}^{T}

10. Estimate

{{\hat{ς}}_{X_{i}} (x_{i}) = | {\hat{c}}_{F} - {\hat{c}}_{F | X_{i} = x_{k}^{(i)}} |, k = 1, 2, \dots, N_{i}}^{T}

End for

(Estimate the defined failure credibility-based global sensitivity)

Fori = 1 : n

11. Set e = 0,

a = min_{k = 1, 2, \dots, N_{i}} {\hat{ς}}_{X_{i}} (x_{k}^{(i)})

and

b = max_{k = 1, 2, \dots, N_{i}} {\hat{ς}}_{X_{i}} (x_{k}^{(i)})

12. Randomly generate N_i points {r₁, r₂, ⋯ , r_{N
_i}} ^T from [a, b].

Fork = 1 : N_i

13. Record the samples satisfying

ς_{X_{i}} (x_{i}) ⩾ r_{k}

S_{1} = {x_{1}^{(i)}, x_{2}^{(i)}, \dots, x_{N_{1}}^{(i)}}^{T}

, the samples satisfying

ς_{X_{i}} (x_{i}) < r_{k}

S_{2} = {x_{1}^{(i)}, x_{2}^{(i)}, \dots, x_{N_{2}}^{(i)}}^{T}

, and N₁ + N₂ = N_i.

Ifr_k ⩾ 0

14.

e = e + Cred {ς_{X_{i}} (x_{i}) ⩾ r_{k}} = e + (max_{x_{i} \in S_{1}} ρ_{X_{i}} (x_{i}) + 1 - max_{x_{i} \in S_{2}} ρ_{X_{i}} (x_{i})) / 2

else

15.

e = e - Cred {ς_{X_{i}} (x_{i}) < r_{k}} = e - (max_{x_{i} \in S_{2}} ρ_{X_{i}} (x_{i}) + 1 - max_{x_{i} \in S_{1}} ρ_{X_{i}} (x_{i})) / 2

End if

End for

16. Estimate the failure credibility-based global sensitivity by

ξ_{X_{i}}^{c_{F}} = E [ς_{X_{i}} (x_{i})] = max (a, 0) + min (b, 0) + e (b - a) / N_{i}

End for

3.3 Fuzzy simulation for estimating failure credibility-based global sensitivity

From the definition of failure credibility-based global sensitivity in Equation (14), it is shown that to estimate $ξ_{X_{i}}^{c_{F}}$ , it has to estimate the unconditional/conditional failure credibility and the expected value of their absolute difference. A direct method for estimating the defined failure credibility-based global sensitivity is fuzzy simulation. For the sake of completeness, this paper briefly introduces the steps of the fuzzy simulation method for estimating the defined failure credibility-based global sensitivity. The whole algorithm of fuzzy simulation for estimating $ξ_{X_{i}}^{c_{F}}$ includes three parts, i.e., (1) estimate the unconditional failure credibility; (2) estimate the conditional failure credibility; (3) estimate the defined failure credibility-based global sensitivity. Just as shown in Algorithm 1.

It is noted that $ς_{X_{i}} (x_{i}) ⩾ 0$ is always satisfied, thus, in Algorithm 1, b ⩾ a ⩾ 0 always holds. Then, r_k ⩾ 0 always holds. Hence, $e = e - Cred {ς_{X_{i}} (x_{i}) < r_{k}}$ will not be used when estimating the failure credibility-based global sensitivity. In the meanwhile, the final estimator can be simplified as $ξ_{X_{i}}^{c_{F}} = E [ς_{X_{i}} (x_{i})] = a + e (b - a) / N_{i}$ .

From the whole process of the fuzzy simulation for estimating the defined failure credibility-based global sensitivity, it can be seen that the fuzzy simulation is a double-loop strategy. To estimate the conditional failure credibility ${{\hat{c}}_{F | X_{i} = x_{k}^{(i)}}, k = 1, 2, \dots, N_{i}}^{T}$ , it firstly has to generate N_i samples of X_i, then, at every point in ${x_{k}^{(i)}, k = 1, 2, \dots, N_{i}}^{T}$ , it has to generate the samples of all the fuzzy inputs except X_i to estimate the conditional

failure credibility. The total number of limit state function evaluations in the fuzzy

simulation for estimating the defined failure credibility-based

global sensitivity is $N + N \sum_{i = 1}^{n} N_{i}$ , where

N denotes the number of limit state function evaluations produced in estimating the unconditional failure credibility. In this paper, the number of samples of the fuzzy inputs used to estimate the unconditional failure credibility is same as that used to estimate the conditional one, thus, $N \sum_{i = 1}^{n} N_{i}$ denotes the number of limit state function evaluations produced in estimating the conditional failure credibility.

4 Proposed method for estimating failure credibility-based global sensitivity

This section presents the proposed single-loop method based on the sequential quadratic programming for estimating the defined failure credibility-based global sensitivity, which is concretely demonstrated as follows.

4.1 Equivalent expression of the failure credibility

Figure 2(a) shows the membership function ρ_{X
_i} (X_i) of fuzzy variable X_i. It can be seen that at a specific membership degree λ ∈ [0, 1], the fuzzy input X_i can be seen as an interval variable bounded in $[X_{i}^{L} (λ), X_{i}^{U} (λ)]$ , where $X_{i}^{L} (λ)$ and $X_{i}^{U} (λ)$ respectively denote the lower and upper bounds of X_i at a given membership level λ. The corresponding membership function of the limit state function is shown in Fig. 2(b), where g^L ( X (λ)) and g^U ( X (λ)) respectively represent the lower and upper bounds of the limit state function at λ membership degree.

Fig. 2

Membership function of fuzzy variable.

It is seen from Fig. 1(a) and Fig. 1(b) that the failure possibility π_F only depends on the lower bound of limit state function, and it can deduce that the failure possibility can be equivalently expressed as $π_{F}^{E} = \int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ$ (17)

in which X (λ) = { X₁ (λ) , X₂ (λ) , ⋯ , X_n (λ) } ^T denotes the fuzzy inputs at λ membership degree, and $X_{i} (λ) \in [X_{i}^{L} (λ), X_{i}^{U} (λ)]$ (i = 1, 2, ⋯ , n). I_F [·] denotes the failure indicator function and $I_{F} [g^{L} (X (λ))] = {\begin{matrix} 1 g^{L} (X (λ)) ⩽ 0 \\ 0 g^{L} (X (λ)) > 0 \end{matrix}$ (18)

Since g^L (X (λ)) >0 holds when π_F < λ ⩽ 1 (see Fig. 1(a)), and g^L (X (λ)) ⩽0 holds when 0 ⩽ λ ⩽ π_F (see Fig. 1(a) and (b)), hence, the equivalent relationship $π_{F}^{E} = π_{F}$ can be obtained and the corresponding proof is given in proof 2.

Proof 2.

$\begin{matrix} π_{F}^{E} & = \int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ \\ = \int_{0}^{π_{F}} I_{F} [g^{L} (X (λ))] d λ + \int_{π_{F}}^{1} I_{F} [g^{L} (X (λ))] d λ \\ = \int_{0}^{π_{F}} 1 d λ + \int_{π_{F}}^{1} 0 d λ = π_{F} \end{matrix}$ (19)

Similarly, the safety possibility π_S only relies on the upper bound of limit state function and the equivalent expression of the safety possibility is $π_{S}^{E} = \int_{0}^{1} I_{S} [g^{U} (X (λ))] d λ = 1 - \int_{0}^{1} I_{F} [g^{U} (X (λ))] d λ$ (20) where I_S [·] =1 - I_F [·] denotes the safety indicator function and $I_{S} [g^{U} (X (λ))] = {\begin{matrix} 1 g^{U} (X (λ)) > 0 \\ 0 g^{U} (X (λ)) ⩽ 0 \end{matrix}$ (21)

Since g^U (X (λ)) >0 holds when 0 ⩽ λ < π_S (just as shown in Fig. 1(a) and (b)), and g^U (X (λ)) ⩽0 holds when π_S ⩽ λ ⩽ 1 (just as shown in Fig. 1(b)), thus, it has the equivalent relationship $π_{S}^{E} = π_{S}$ and the proof of it is given in proof 3.

Proof 3.

$\begin{matrix} π_{S}^{E} & = \int_{0}^{1} I_{S} [g^{U} (X (λ))] d λ \\ = \int_{0}^{π_{S}} I_{S} [g^{U} (X (λ))] d λ + \int_{π_{S}}^{1} I_{S} [g^{U} (X (λ))] d λ \\ = \int_{0}^{π_{S}} 1 d λ + \int_{π_{S}}^{1} 0 d λ = π_{S} \end{matrix}$ (22) or $\begin{matrix} π_{S}^{E} = \int_{0}^{1} {1 - I_{F} [g^{U} (X (λ))]} d λ \\ = \int_{0}^{π_{S}} {1 - I_{F} [g^{U} (X (λ))]} d λ \\ + \int_{π_{S}}^{1} {1 - I_{F} [g^{U} (X (λ))]} d λ \\ = \int_{0}^{π_{S}} (1 - 0) d λ + \int_{π_{S}}^{1} (1 - 1) d λ = π_{S} \end{matrix}$ (23)

Based on the equivalent expressions of failure possibility and safety possibility, the equivalent expressions of failure necessity and safety necessity are respectively shown as follows,

$\begin{matrix} κ_{F}^{E} & = 1 - π_{S}^{E} = 1 - \int_{0}^{1} I_{S} [g^{U} (X (λ))] d λ \\ = 1 - \int_{0}^{1} {1 - I_{F} [g^{U} (X (λ))]} d λ \\ = \int_{0}^{1} I_{F} [g^{U} (X (λ))] d λ \end{matrix}$ (24) $κ_{S}^{E} = 1 - π_{F}^{E} = 1 - \int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ$ (25)

The equivalent expressions of failure credibility and safety credibility are respectively shown in Equations (26) and (27),

$\begin{matrix} c_{F}^{E} = \frac{1}{2} (π_{F}^{E} + κ_{F}^{E}) = \frac{1}{2} \\ [\int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ + 1 - \int_{0}^{1} I_{S} [g^{U} (X (λ))] d λ] \\ = \frac{1}{2} [\int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ + \int_{0}^{1} I_{F} [g^{U} (X (λ))] d λ] \end{matrix}$ (26)

$\begin{matrix} c_{S}^{E} = \frac{1}{2} (π_{S}^{E} + κ_{S}^{E}) = \frac{1}{2} \\ [\int_{0}^{1} I_{S} [g^{U} (X (λ))] d λ + 1 - \int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ] \\ = \frac{1}{2} [1 - \int_{0}^{1} I_{F} [g^{U} (X (λ))] d λ + 1 - \int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ] \\ = 1 - \frac{1}{2} [\int_{0}^{1} I_{F} [g^{L} (X (λ))] d λ + \int_{0}^{1} I_{F} [g^{U} (X (λ))] d λ] \end{matrix}$ (27)

4.2 Bayesian transformation of the conditional failure credibility

Based on the equivalent expression of the failure credibility in Equation (1), it can obtain the equivalent expression of the conditional failure credibility $c_{F | X_{i} = x_{i}}$ , i.e.,

$\begin{matrix} c_{F | X_{i} = x_{i}}^{E} = & \frac{1}{2} [\int_{0}^{1} I_{F} [g^{L} (X_{- i} (λ), x_{i})] d λ \\ + \int_{0}^{1} I_{F} [g^{U} (X_{- i} (λ), x_{i})] d λ] \end{matrix}$ (28) where _X - i (λ) = { X₁ (λ) , X₂ (λ) , ⋯ , X_i-1 (λ) , X_i+1 linebreak (λ) , ⋯ , X_n (λ) } ^T. In Equation (3) λ can be seen as the random variable uniformly distributed in [0,1], thus, Equation (3) can be transformed into the following probabilistic density function integral form, i.e., $\begin{matrix} c_{F | X_{i} = x_{i}}^{E} = \frac{1}{2} [\int_{0}^{1} I_{F} [g^{L} (X_{- i} (λ), x_{i})] f_{λ} (λ) d λ \\ + \int_{0}^{1} I_{F} [g^{U} (X_{- i} (λ), x_{i})] f_{λ} (λ) d λ] \\ = \frac{1}{2} [Prob {g^{L} (X_{- i} (λ), X_{i}) ⩽ 0 | X_{i} = x_{i}} \\ + Prob {g^{U} (X_{- i} (λ), X_{i}) ⩽ 0 | X_{i} = x_{i}}] \end{matrix}$ (29) where f_λ (λ) =1 is the probability density function of uniform variable λ and Prob {·} denotes the probability operator. According to the Bayesian theory, it can obtain

$\begin{matrix} Prob {g^{L} (X_{- i} (λ), X_{i}) ⩽ 0 | X_{i} = x_{i}} = \frac{Prob {g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} \cdot Prob {X_{i} = x_{i} | g^{L} (X_{- i} (λ), X_{i}) ⩽ 0}}{Prob {X_{i} = x_{i}}} \end{matrix}$ (30) where Prob{ g^L ( X _-i (λ) , X_i) ⩽0 } denotes the probability of the failure event {g^L ( X _-i (λ) , X_i)⩽0 }, which can be obtained by simultaneously treating λ and X_i as the random variables. Prob{ X_i = x_i } denotes the probability of the event {X_i = x_i}. Prob{ X_i = x_i|g^L ( X _-i (λ) , X_i) ⩽0 } represents the conditional probability of the event {X_i = x_i} on {g^L ( X _-i (λ) , X_i)⩽0 }.

Likewise, based on the Bayesian theory it can obtain

$\begin{matrix} Prob {g^{U} (X_{- i} (λ), X_{i}) ⩽ 0 | X_{i} = x_{i}} = \frac{Prob {g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} \cdot Prob {X_{i} = x_{i} | g^{U} (X_{- i} (λ), X_{i}) ⩽ 0}}{Prob {X_{i} = x_{i}}} \end{matrix}$ (31) where Prob{ g^U ( X _-i (λ) , X_i) ⩽0 } denotes the probability of failure event {g^U ( X _-i (λ) , linebreak X_i)⩽0 }, which can be obtained by simultaneously treating λ and X_i as random variables. Prob{ X_i = x_i|g^U ( X _-i (λ) , X_i) ⩽0 } represents the conditional probability of event {X_i = x_i} on {g^U (_X - i (λ) , X_i)⩽0 }.

Remark: Prob{ g^L ( X _-i (λ) , X_i) ⩽0 } and Problinebreak{ g^U ( X _-i (λ) , X_i) ⩽0 } are the failure probabilities which simultaneously take λ and X_i as random variables. As aforementioned, λ can be seen as a uniformly distributed random variable in [0,1]. The fuzzy input X_i now can be seen as a random variable uniformly distributed in $[X_{i}^{L}, X_{i}^{U}]$ .

Taking Equations (30) and (31) into Equation (29), the defined conditional failure credibility can be equivalently transformed into

$c_{F | X_{i} = x_{i}}^{E} = \frac{1}{2} [\begin{matrix} \frac{Prob {g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} \cdot Prob {X_{i} = x_{i} | g^{L} (X_{- i} (λ), X_{i}) ⩽ 0}}{Prob {X_{i} = x_{i}}} \\ + \frac{Prob {g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} \cdot Prob {X_{i} = x_{i} | g^{U} (X_{- i} (λ), X_{i}) ⩽ 0}}{Prob {X_{i} = x_{i}}} \end{matrix}]$ (32)

4.3 The proposed method for estimating failure credibility-based global sensitivity

This section illustrates the algorithm of the proposed method for estimating the defined failure credibility-based global sensitivity. As mentioned above, to estimate the defined failure credibility-based global sensitivity, it has to estimate the unconditional/conditional failure credibility and the fuzzy mean of their absolute difference. Thus, this paper will concretely demonstrate the proposed method according to these key points.

4.3.1 Estimation of the unconditional failure credibility

To estimate the defined failure credibility-based global sensitivity, the first task is to estimate the unconditional failure credibility. Based on Equation (26), the unconditional failure credibility can be estimated by

${\hat{c}}_{F} = \frac{1}{2} [\frac{1}{N} \sum_{j = 1}^{N} I_{F} [g^{L} (X (λ_{j}))] + \frac{1}{N} \sum_{j = 1}^{N} I_{F} [g^{U} (X (λ_{j}))]]$ (33) where {λ_j, j = 1, 2, ⋯ , N } ^T denote the samples of λ uniformly generated in [0,1]. The detailed steps of the proposed method for estimating the unconditional failure credibility are demonstrated in Algorithm 2.

Algorithm 2: Algorithm of proposed method for estimating unconditional failure credibility.

1. Uniformly generate N samples {λ₁, λ₂, ⋯ , λ_N } ^T of λ within [0,1].

Forj = 1 : N

2. Obtain the lower and upper bounds of fuzzy inputs at λ_j membership degree, i.e.,

[X_{L} (λ_{j}), X_{U} (λ_{j})]

, in which

X_{L} (λ_{j}) = {X_{i}^{L} (λ_{j}), X_{2}^{L} (λ_{j}), \dots, X_{n}^{L} (λ_{j})}^{T}

and

X_{U} (λ_{j}) = {X_{i}^{U} (λ_{j}), X_{2}^{U} (λ_{j}), \dots, X_{n}^{U} (λ_{j})}^{T}

3. Estimate the lower and upper bounds of limit state function (i.e., g^L ( X (λ_j)) and g^U ( X (λ_j))) by the sequential quadratic programming under the constraints

X (λ_{j}) \in [X_{L} (λ_{j}), X_{U} (λ_{j})]

4. According to the signs of g^L ( X (λ_j)) and g^U ( X (λ_j)), obtain the failure indicator function values I_F [g^L ( X (λ_j))] and I_F [g^U ( X (λ_j))].

End for

5. Estimate the unconditional failure credibility by Eq.(33).

From the steps of the proposed method for estimating the unconditional failure credibility, it can obtain that the number of limit state function evaluations is $\sum_{j = 1}^{N} N_{j}$ , where N_j denotes the computational cost produced in estimating the lower and upper bounds of the limit state function at λ_j membership degree by the sequential quadratic programming, N denotes the number of samples of λ.

4.3.2 Estimation of the conditional failure credibility

As seen in Equation (32), to estimate the conditional failure credibility, it has to estimate the components of Prob{ g^L ( X _-i (λ) , X_i) ⩽0 }, Prob{ g^U ( X _-i (λ) , X_i) ⩽0 }, Prob{ X _i = x_i }, Prob{ X_i = x_i|g^L (_X - i (λ) , X_i) ⩽0 } and Prob{ X_i = x_i|g^U (_X - i (λ) , X_i) ⩽0 }. The following parts provide the estimations of these components.

(1) Estimate Prob{ g^L ( X _-i (λ) , X_i) ⩽0 } and Prob{ g^U (_X - i (λ) , X_i) ⩽0 } Prob{ g^L (_X - i (λ) , X_i) linebreak ⩽ 0 } and Prob{ g^U (_X - i (λ) , X_i) ⩽0 } can be equivalently expressed as

${\begin{matrix} Prob {g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} = \int_{X_{i}} \int_{λ} I_{F} [g^{L} (X_{- i} (λ), x_{i})] f_{λ} (λ) f_{X_{i}} (x_{i}) d λ d x_{i} \\ Prob {g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} = \int_{X_{i}} \int_{λ} I_{F} [g^{U} (X_{- i} (λ), x_{i})] f_{λ} (λ) f_{X_{i}} (x_{i}) d λ d x_{i} \end{matrix}$ (34)

where f_{X
_i} (x_i) is the probability density function of X_i. As aforementioned, the fuzzy input X_i now can be seen as a random variable uniformly distributed in $[X_{i}^{L}, X_{i}^{U}]$ . Then, it can obtain

${\begin{matrix} Prob {g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} \approx \frac{1}{N} \sum_{j = 1}^{N} I_{F} [g^{L} (X_{- i} (λ_{j}), x_{i}^{(j)})] \\ Prob {g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} \approx \frac{1}{N} \sum_{j = 1}^{N} I_{F} [g^{U} (X_{- i} (λ_{j}), x_{i}^{(j)})] \end{matrix}$ (35) where $S_{i} = {x_{i}^{(j)}, j = 1, 2, \dots, N}^{T}$ are the samples of fuzzy input X_i uniformly generated in $[X_{i}^{L}, X_{i}^{U}]$ .

(2) Estimate Prob{ X_i = x_i }

Let Δx_i be a increment of fuzzy variable X_i and $x_{i}^{I} = [x_{i} - Δ x_{i} / 2, x_{i} + Δ x_{i} / 2]$ a small interval around x_i. Then, Equation (36) can be obtained to estimate Prob{ X_i = x_i }. $\begin{matrix} Prob {X_{i} = x_{i}} \\ \approx lim_{Δ x_{i} \to 0^{+}} Prob {X_{i} \in [x_{i} - Δ x_{i} / 2, x_{i} + Δ x_{i} / 2]} \\ = lim_{Δ x_{i} \to 0^{+}} Prob {X_{i} \in x_{i}^{I}} \\ \approx \frac{the number of samples from S_{i} falling in x_{i}^{I}}{the number of samples in S_{i}} \end{matrix}$ (36)

Algorithm 3: Algorithm of proposed method for estimating conditional failure credibility.

Fori = 1 : n

1. Uniformly generate N samples

S_{i} = {x_{i}^{(1)}, x_{i}^{(2)}, \dots, x_{i}^{(N)}}^{T}

of X_i within

[X_{i}^{L}, X_{i}^{U}]

Forj = 1 : N

2. Obtain the lower and upper bounds of fuzzy inputs _X - i at λ_j membership degree, i.e.,

[X_{- i}^{L} (λ_{j}), X_{- i}^{U} (λ_{j})]

3. Estimate the lower and upper bounds of limit state function (i.e.,

g^{L} (X_{- i} (λ_{j}), x_{i}^{(j)})

and

g^{U} (X_{- i} (λ_{j}), x_{i}^{(j)})

) by the sequential quadratic programming under the constraints

X_{- i} (λ_{j}) \in [X_{- i}^{L} (λ_{j}), X_{- i}^{U} (λ_{j})]

4. According to the signs of

g^{L} (X_{- i} (λ_{j}), x_{i}^{(j)})

and

g^{U} (X_{- i} (λ_{j}), x_{i}^{(j)})

, obtain the failure indicator function values

I_{F} [g^{L} (X_{- i} (λ_{j}), x_{i}^{(j)})]

and

I_{F} [g^{U} (X_{- i} (λ_{j}), x_{i}^{(j)})]

End for

5. Estimate Prob{ g^L ( X _-i (λ) , X_i) ⩽0 } and Prob{ g^U (_X - i (λ) , X_i) ⩽0 } by Eq.(35).

6. Record the samples of X _i satisfying g^L ( X _-i (λ) , X _i) ⩽0 and g^U ( X _-i (λ) , X_i) ⩽0 respectively as

X_{i}^{(F^{L})}

and

X_{i}^{(F^{U})}

7. Set Δx_i a small real number.

8. Randomly generate M samples

{x_{i}^{(1) *}, x_{i}^{(2) *}, \dots, x_{i}^{(M) *}}^{T}

of X_i in

[X_{i}^{L}, X_{i}^{U}]

Form = 1 : M

9. Record the number of samples from S _i falling in

[x_{i}^{(m) *} - Δ x_{i} / 2, x_{i}^{(m) *} + Δ x_{i} / 2]

, estimate Prob{ X_i = x_i } by Equation (36).

10. Record the number of samples from

X_{i}^{(F^{L})}

falling in

[x_{i}^{(m) *} - Δ x_{i} / 2, x_{i}^{(m) *} + Δ x_{i} / 2]

, estimate Prob{ X_i = x_i|g^L (_X - i (λ) , X_i) ⩽0 } by Equation (37).

11. Record the number of samples from

X_{i}^{(F^{U})}

falling in

[x_{i}^{(m) *} - Δ x_{i} / 2, x_{i}^{(m) *} + Δ x_{i} / 2]

, estimate Prob{ X_i = x_i|g^U (_X - i (λ) , X_i) ⩽0 } by Equation (38).

12. Estimate the conditional failure credibility

c_{F | X_{i} = x_{i}^{(m) *}}

given

{X_{i} = x_{i}^{(m) *}}

by Equation (32).

End for

13. Estimate the absolute difference between the unconditional failure credibility and conditional failure credibility, i.e.,

{{\hat{ς}}_{X_{i}} (x_{i}^{(m) *}), m = 1, 2, \dots, M}^{T}

End for

(3) Estimate Prob{ X_i = x_i|g^L (_X - i (λ) , X_i) ⩽0 } and Prob{ X_i = x_i|g^U (_X - i (λ) , X_i) ⩽0 } Similar to Equation (36), it can obtain

$\begin{matrix} Prob {X_{i} = x_{i} | g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} \approx lim_{Δ x_{i} \to 0^{+}} Prob {X_{i} \in [x_{i} - Δ x_{i} / 2, x_{i} + Δ x_{i} / 2] | g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} \\ \approx lim_{Δ x_{i} \to 0^{+}} Prob {X_{i} \in x_{i}^{I} | g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} \\ \approx \frac{the number of samples from S_{i} satisfying g^{L} (X_{- i} (λ), X_{i}) ⩽ 0 and in x_{i}^{I}}{the number of samples from S_{i} satisfying g^{L} (X_{- i} (λ), X_{i}) ⩽ 0} \end{matrix}$ (37)

$\begin{matrix} Prob {X_{i} = x_{i} | g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} \approx lim_{Δ x_{i} \to 0^{+}} Prob {X_{i} \in [x_{i} - Δ x_{i} / 2, x_{i} + Δ x_{i} / 2] | g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} \\ \approx lim_{Δ x_{i} \to 0^{+}} Prob {X_{i} \in x_{i}^{I} | g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} \\ \approx \frac{the number of samples from S_{i} satisfying g^{U} (X_{- i} (λ), X_{i}) ⩽ 0 and in x_{i}^{I}}{the number of samples from S_{i} satisfying g^{U} (X_{- i} (λ), X_{i}) ⩽ 0} \end{matrix}$ (38)

(4) Estimate the conditional failure credibility

Taking Equations (35)∼(38) into Equation (32), the conditional failure credibility estimate can be obtained. The detailed steps of the proposed method for estimating conditional failure credibility proceed as Algorithm 3. The steps of the proposed method for estimating conditional failure credibility in Algorithm 3 show that the number of limit state function evaluations is $\sum_{i = 1}^{n} \sum_{j = 1}^{N} N_{j}^{(i)}$ , where $N_{j}^{(i)}$ denotes the number of limit state function evaluations produced in estimating the lower and upper bounds of limit state function at λ_j membership degree by the sequential quadratic programming when fuzzy input X_j is fixed on $x_{i}^{(j)}$ .

4.3.3 Estimation of the failure credibility-based global sensitivity

Algorithm 4: Algorithm of proposed method for estimating failure credibility-based global sensitivity.

Fori = 1 : n

1. Set e = 0,

a = min_{j = 1, 2, \dots, M} {\hat{ς}}_{X_{i}} (x_{i}^{(j) *})

and

b = max_{j = 1, 2, \dots, M} {\hat{ς}}_{X_{i}} (x_{i}^{(j) *})

2. Randomly generate M points {r₁, r₂, ⋯ , r_M } ^T in [a, b].

Fork = 1 : M

e = e + \frac{1}{2} [max_{j = 1, 2, \dots, M} {ρ_{X_{i}} (x_{i}^{(j) *}) | {\hat{ς}}_{X_{i}} (x_{i}^{(j) *}) ⩾ r_{k}} + 1 - max_{j = 1, 2, \dots, M} {ρ_{X_{i}} (x_{i}^{(j) *}) | {\hat{ς}}_{X_{i}} (x_{i}^{(j) *}) < r_{k}}]

End for

4. Estimate the failure credibility-based global sensitivity

ξ_{X_{i}}^{c_{F}}

of fuzzy input X_i by

ξ_{X_{i}}^{c_{F}} = a + e (b - a) / M

End for

After obtaining the unconditional/conditional failure credibility and their absolute difference, to estimate the failure credibility-based global sensitivity, the last task is to estimate the fuzzy mean of ς_{X
_i} (x_i) = |c_F - c_{F|X_i=x_i}|, i.e., $ξ_{X_{i}}^{c_{F}} = E [ς_{X_{i}} (x_{i})]$ . This paper adopts the fuzzy simulation to estimate $ξ_{X_{i}}^{c_{F}} = E [ς_{X_{i}} (x_{i})]$ . Based on the algorithm of fuzzy simulation for estimating $ξ_{X_{i}}^{c_{F}}$ in section 3.3, the defined failure credibility-based global sensitivity can be estimated by the steps demonstrated in Algorithm 4.

From the whole process of the proposed method for estimating the defined failure credibility-based global sensitivity, it can be seen that to estimate the conditional failure credibility, the proposed method only needs to generate a group of samples of fuzzy input X_i and λ, thus, the proposed method can be seen as a single-loop strategy. The total number of limit state function evaluations of the proposed single-loop strategy for estimating the failure credibility-based global sensitivity is $\sum_{j = 1}^{N} N_{j} + \sum_{i = 1}^{n} \sum_{j = 1}^{N} N_{j}^{(i)}$ , where $\sum_{j = 1}^{N} N_{j}$ denotes the number of limit state function evaluations produced in estimating the unconditional failure credibility by the sequential quadratic programming, $\sum_{i = 1}^{n} \sum_{j = 1}^{N} N_{j}^{(i)}$ represents the number of limit state function evaluations produced in estimating the conditional failure credibility by the sequential quadratic programming.

5 Examples

This section presents three examples to demonstrate the feasibility of the defined failure credibility-based global sensitivity as well as the efficiency and accuracy of the proposed method for estimating the defined failure credibility-based global sensitivity.

5.1 A cantilever beam structure

A cantilever beam structure with the loads p₁ and p₂ is shown in Fig. 3. The critical ultimate bending moment of this beam is M_cr, and the actual bending moment of the beam is p₁d₁ + p₂d₂. Obviously, if p₁d₁ + p₂d₂ ⩾ M_cr, the failure event of this structure will occur. Thus, the limit state function of this structure can be constructed as g ( X ) = M_cr - p₁d₁ - p₂d₂, in which the input variables X = { d₁, d₂, p₁, p₂, M_cr } ^T are fuzzy and independent with each other. The membership functions of these fuzzy inputs are shown in Table 1.

Fig. 3

A cantilever beam structure.

Table 1

The membership functions of fuzzy variables of example 1

Variable	Membership function	Variable	Membership function
d ₁	$ρ_{d_{1}} (d_{1}) = {\begin{matrix} (d_{1} - 1.8) / 0.2 1.8 ⩽ d_{1} < 2 \\ (2.2 - d_{1}) / 0.2 2 ⩽ d_{1} ⩽ 2.2 \end{matrix}$	p ₂	$ρ_{p_{2}} (p_{2}) = {\begin{matrix} (p_{2} - 1.7) / 0.3 1.7 ⩽ p_{2} < 2 \\ (2.3 - p_{2}) / 0.3 2 ⩽ p_{2} ⩽ 2.3 \end{matrix}$
d ₂	$ρ_{d_{2}} (d_{2}) = {\begin{matrix} (d_{2} - 4.5) / 0.5 4.5 ⩽ d_{2} < 5 \\ (5.5 - d_{2}) / 0.5 5 ⩽ d_{2} ⩽ 5.5 \end{matrix}$	M _cr	$ρ_{M_{cr}} (M_{cr}) = {\begin{matrix} (M_{cr} - 22) / 422 ⩽ M_{cr} < 26 \\ (30 - M_{cr}) / 426 ⩽ M_{cr} ⩽ 30 \end{matrix}$
p ₁	$ρ_{p_{1}} (p_{1}) = {\begin{matrix} (p_{1} - 4.4) / 0.6 4.4 ⩽ p_{1} < 5 \\ (5.6 - p_{1}) / 0.6 5 ⩽ p_{1} ⩽ 5.6 \end{matrix}$

The unconditional failure credibility estimate ${\hat{c}}_{F}$ , the total number of limit state function evaluations (No. of calls) and the computational time (Runtime) of estimating the failure credibility-based global sensitivity are also listed in Table 2. It is seen that the proposed method can reduce the computational cost dramatically compared with the fuzzy simulation method (FS).

Table 2

The result comparison of example 1

Method	${\hat{c}}_{F}$	No. of calls	Runtime(s)
FS	0.164	10⁸	6337.0
Proposed	0.162	2.2×10⁵	101.4

The membership function of conditional failure credibility obtained by the fuzzy simulation method and the proposed method is shown in Fig. 4, from which we can see that the membership function of the conditional failure credibility obtained by the proposed method is in agreement with that obtained by the fuzzy simulation method.

Fig. 4

The membership function of conditional failure credibility.

The estimated failure credibility-based global sensitivity is illustrated in Fig. 5. It can be concluded that the proposed method can accurately estimate the failure credibility-based global sensitivity in the light of the result obtained by the fuzzy simulation method. It also can be acquired that the importance ranking of the fuzzy inputs is M_cr > p₂ > p₁ > d₂ > d₁. The critical ultimate bending moment M_cr has the largest effect on the failure credibility and d₁ possesses the smallest effect on structural failure credibility. The designers can pay more attentions to M_cr to have a better control of the structural safety.

Fig. 5

The failure credibility-based global sensitivity estimate of example 1.

5.2 A composite beam

A composite beam is shown in Fig. 6. The beam is A (mm) in width, B (mm) in height, and L (mm) in length long with the Young’s modulus E_w (GPa). The bottom of this beam is an aluminum plate with C (mm) in width and D (mm) in height along with the Young’s modulus E_a (GPa). Along the beam, six additional vertical forces, i.e., P₁ (kN), P₂ (kN), P₃ (kN), P₄ (kN), P₅ (kN) and P₆ (kN) act on six different application points, i.e., L₁ (mm), L₂ (mm), L₃ (mm), L₄ (mm), L₅ (mm), and L₆ (mm). The membership functions of these 19 fuzzy input variables are given in Table 3.

Fig. 6

A composite beam.

Table 3

The membership functions of fuzzy variables of example 2

Variable	Membership function	Variable	Membership function
A	$ρ_{A} (A) = {\begin{matrix} A - 9999 ⩽ A < 100 \\ 101 - A 100 ⩽ A ⩽ 101 \end{matrix}$	L	$ρ_{L} (L) = {\begin{matrix} (L - 1400) / 141386 ⩽ L < 1400 \\ (1414 - L) / 141400 ⩽ L ⩽ 1414 \end{matrix}$
B	$ρ_{B} (B) = {\begin{matrix} (B - 175) / 2173 ⩽ B < 175 \\ (177 - B) / 2175 ⩽ B ⩽ 177 \end{matrix}$	E _a	$ρ_{E_{a}} (E_{a}) = {\begin{matrix} (E_{a} - 66) / 266 ⩽ E_{a} < 68 \\ 168 ⩽ E_{a} < 70 \\ (72 - E_{a}) / 270 ⩽ E_{a} ⩽ 72 \end{matrix}$
C	$ρ_{C} (C) = {\begin{matrix} (C - 80) / 0.8 79.2 ⩽ C < 80 \\ (80.8 - C) / 0.8 80 ⩽ C ⩽ 80.8 \end{matrix}$	E _w	$ρ_{E_{w}} (E_{w}) = {\begin{matrix} (E_{w} - 8.15) / 0.2 8.15 ⩽ E_{w} < 8.35 \\ 1 8.55 ⩽ E_{w} < 8.75 \\ (8.95 - E_{w}) / 0.2 8.75 ⩽ E_{w} ⩽ 8.95 \end{matrix}$
D	$ρ_{D} (D) = {\begin{matrix} (D - 20) / 0.2 19.8 ⩽ D < 20 \\ (20.2 - D) / 0.2 20 ⩽ D ⩽ 20.2 \end{matrix}$	P ₁	$ρ_{P_{1}} (P_{1}) = 2 {[1 + exp (\frac{π \| P_{1} - 15 \|}{\sqrt{3} \times 0.01})]}^{- 1}$
L ₁	$ρ_{L_{1}} (L_{1}) = {\begin{matrix} (L_{1} - 200) / 2198 ⩽ L_{1} < 200 \\ (202 - L_{1}) / 2200 ⩽ L_{1} ⩽ 202 \end{matrix}$	P ₂	$ρ_{P_{2}} (P_{2}) = 2 {[1 + exp (\frac{π \| P_{2} - 15 \|}{\sqrt{3} \times 0.01})]}^{- 1}$
L ₂	$ρ_{L_{2}} (L_{2}) = {\begin{matrix} (L_{2} - 400) / 4396 ⩽ L_{2} < 400 \\ (404 - L_{2}) / 4400 ⩽ L_{2} ⩽ 404 \end{matrix}$	P ₃	$ρ_{P_{3}} (P_{3}) = 2 {[1 + exp (\frac{π \| P_{3} - 15 \|}{\sqrt{3} \times 0.01})]}^{- 1}$
L ₃	$ρ_{L_{3}} (L_{3}) = {\begin{matrix} (L_{3} - 600) / 6594 ⩽ L_{3} < 600 \\ (606 - L_{3}) / 6600 ⩽ L_{3} ⩽ 606 \end{matrix}$	P ₄	$ρ_{P_{4}} (P_{4}) = 2 {[1 + exp (\frac{π \| P_{4} - 15.5 \|}{\sqrt{3} \times 0.01})]}^{- 1}$
L ₄	$ρ_{L_{4}} (L_{4}) = {\begin{matrix} (L_{4} - 800) / 8792 ⩽ L_{4} < 800 \\ (808 - L_{4}) / 8800 ⩽ L_{4} ⩽ 808 \end{matrix}$	P ₅	$ρ_{P_{5}} (P_{5}) = 2 {[1 + exp (\frac{π \| P_{5} - 15.5 \|}{\sqrt{3} \times 0.01})]}^{- 1}$
L ₅	$ρ_{L_{5}} (L_{5}) = {\begin{matrix} 0.1 L_{5} - 99990 ⩽ L_{5} < 1000 \\ 101 - 0.1 L_{5} 1000 ⩽ L_{5} ⩽ 1010 \end{matrix}$	P ₆	$ρ_{P_{6}} (P_{6}) = 2 {[1 + exp (\frac{π \| P_{6} - 15.5 \|}{\sqrt{3} \times 0.01})]}^{- 1}$
L ₆	$ρ_{L_{6}} (L_{6}) = {\begin{matrix} (L_{6} - 1200) / 121188 ⩽ L_{6} < 1200 \\ (1212 - L_{6}) / 121200 ⩽ L_{6} ⩽ 1212 \end{matrix}$

The failure event of this composite beam will occur if the maximum binding normal stress σ_max of the beam exceeds the allowable stress S = 0.02 (GPa), and the corresponding limit state function can be expressed as g = S - σ_max, where σ_max can be computed by $σ_{max} = max_{p = 1, 2, \dots, 6} σ_{p}$ , in which σ_p (p = 1, 2, ⋯ , 6) represents the binding normal stress of the cross-section where P_p (p = 1, 2, ⋯ , 6) is applied. σ_p (p = 1, 2, ⋯ , 6) can be analytically solved by the following equations, $σ_{1} = \frac{[\frac{L_{1}}{L} \sum_{q = 1}^{6} P_{q} (L - L_{q})] Y_{max}}{W}$ (39)

$\begin{matrix} σ_{p} = & \frac{[\frac{L_{p}}{L} \sum_{q = 1}^{6} P_{q} (L - L_{q}) - \sum_{q = 1}^{p - 1} P_{q} (L_{p} - L_{q})] Y_{max}}{W} \\ (p = 2, 3, \dots, 6) \end{matrix}$ (40)

$Y_{max} = \frac{0.5 {AB}^{2} + DC (B + D) E_{a} / - E_{w}}{AB + DC E_{a} / - E_{w}}$ (41)

$\begin{matrix} W = & \frac{{AB}^{3}}{12} + AB {(Y_{max} - \frac{B}{2})}^{2} \\ + \frac{{CD}^{3} E_{a}}{12 E_{w}} + \frac{{CDE}_{a}}{E_{w}} {(\frac{D}{2} + B - Y_{max})}^{2} \end{matrix}$ (42)

The unconditional failure credibility estimate obtained by the proposed method and the fuzzy simulation method is listed in Table 4. The computational costs of these two methods for estimating the failure credibility-based global sensitivity, including the number of limit state function evaluations and the computational time, are listed in Table 4. It is seen that to estimate the failure credibility-based global sensitivity, the fuzzy simulation method needs to evaluate the limit state function 4 × 10⁸ times, whereas the proposed method only needs to evaluate the limit state function 2 × 10⁵ times. As for the computational time, the fuzzy simulation method needs about 3.67 × 10⁵ seconds, whereas the proposed method only needs approximate 2.38 × 10⁴ seconds. Hence, the proposed method significantly improves the computational efficiency of estimating the failure credibility-based global sensitivity.

Table 4

The result comparison of example 2

Method	${\hat{c}}_{F}$	No. of calls	Runtime(s)
FS	0.254	4×10⁸	3.67×10⁵
Proposed	0.248	2×10⁵	2.38×10⁴

The membership function of conditional failure credibility of this example estimated by the fuzzy simulation method and the proposed method is depicted in Fig. 7. It is seen that the membership function of the conditional failure credibility estimated by the proposed method is consistent with that obtained by the fuzzy simulation method.

Fig. 7

The membership function of conditional failure credibility of example 2.

The comparison of the failure credibility-based global sensitivity estimates obtained by the proposed method and the fuzzy simulation method is illustrated in Fig. 8. It can be seen that the fuzzy inputs A, B, L₆, L, E_a and E_w have large effect on the failure credibility of the structure, and the importance ranking of these important fuzzy inputs is B > L > L₆ > E_w > A > E_a. The effect of other fuzzy inputs on structural failure credibility can be neglected compared with that of the important fuzzy inputs. The failure credibility-based global sensitivity estimated by the proposed method is similar as that estimated by the fuzzy simulation method.

Fig. 8

The failure credibility-based global sensitivity estimate of example 2.

5.3 The lifting mechanism of a hatch

The main components of a lifting mechanism of a hatch are shown in Fig. 9. In the lifting mechanism, there are many hinge joints and telescopic cylinders which produce friction force, and the friction force seriously affects the performance of the lifting mechanism. Just as shown in Fig. 9, the fuzzy inputs of this example include the friction factors of the hinge joints (X₁, X₂, ⋯ , X₁₅) and the friction factors of the telescopic cylinders (X₁₆, X₁₇), the membership functions of these fuzzy inputs {X₁, X₂, ⋯ , X₁₇ } ^T are

$\begin{matrix} ρ_{X_{i}} (x_{i}) & = & {\begin{matrix} \frac{x_{i} - 0.01}{0.005}, 0.01 ⩽ x_{i} ⩽ 0.015 \\ \frac{x_{i} - 0.02}{- 0.005}, 0.015 ⩽ x_{i} ⩽ 0.02 \end{matrix} \\ (i = 1, 2, \dots, 17) \end{matrix}$ (43)

Fig. 9

The lifting mechanism of a hatch.

When opening the hatch, a certain amount of force or torque is needed to open the door smoothly. In this paper, the design force is 155(N), hence, it can define the following limit state function of this example, $g (X) = 155 - F (X_{1}, X_{2}, \dots, X_{17})$ (44) where X = { X₁, X₂, ⋯ , X₁₇ } ^T are the fuzzy inputs, F (X₁, X₂, ⋯ , X₁₇) denotes the actual force required to open the hatch. g ( x ) >0 implies the lifting mechanism of the hatch works normally, whereas g (x) ⩽0 represents the lifting mechanism of the hatch disabled. The actual required force is a function of the fuzzy inputs, which can be obtained by the simulation in the software “Automatic Dynamic Analysis of Mechanical Systems (ADAMS)”.

In this example, the unconditional failure credibility estimate obtained by the fuzzy simulation method and the proposed method is shown in Table 5. It is seen that the proposed method can obtain an accurate failure credibility estimate in the light of the result obtained by the fuzzy simulation method. The computational costs of the proposed method and the fuzzy simulation method for estimating the defined failure credibility-based global sensitivity are also listed in Table 5, which demonstrate that the proposed method reduces the computational cost dramatically.

Table 5

The result comparison of example 3

Method	${\hat{c}}_{F}$	No. of calls	Runtime(s)
FS	0.291	3.6×10⁸	1.80×10⁵
Proposed	0.295	1.2×10⁷	1.46×10⁵

The membership function of the conditional failure credibility estimated by the fuzzy simulation method and the proposed method is presented in Fig. 10, from which we can see that the membership function of conditional failure credibility obtained by the proposed method is in agreement with that obtained by the fuzzy simulation method.

Fig. 10

The membership function of conditional failure credibility of example 3.

The comparison of the failure credibility-based global sensitivity of this lifting mechanism is demonstrated in Fig. 11, which illustrates that the failure credibility-based global sensitivity obtained by the proposed method and the fuzzy simulation method is consistent. It can be concluded that the fuzzy inputs X₁, X₂, X₄ and X₁₆ have large effect on structural failure credibility, and the importance ranking of these important fuzzy inputs is X₁ > X₂ > X₁₆ > X₄. To have a better control of the structural safety level, designers can pay more attentions to these important fuzzy inputs.

Fig. 11

The failure credibility-based global sensitivity estimate of example 3.

6 Conclusions

For structural systems involving fuzzy input variables, an importance analysis model is firstly established by extending the traditional importance analysis idea of the random input variables. The importance measure is defined for the fuzzy inputs by taking account of their effects on structural safety degree (measured by failure credibility), which is the fuzzy expected absolute difference between the unconditional failure credibility and conditional one. Since the direct fuzzy simulation for estimating the defined failure credibility-based global sensitivity is a nested double-loop strategy which is inefficient, this paper deduces the Bayesian expression of the defined failure credibility-based global sensitivity, on which a single-loop method based on the sequential quadratic programming is proposed to efficiently estimate the defined failure credibility-based global sensitivity. The analysis results of the examples illustrate that the established failure credibility-based global sensitivity can reasonably measure the importance of the fuzzy inputs, and reflect their effects on the structural safety degree (failure credibility). Therefore, it can provide important guidance for the design optimization of the structural systems with fuzzy inputs. The proposed method can improve the computational efficiency of calculating the constructed failure credibility-based global sensitivity with acceptable accuracy, which provides a highly efficient alternative for the computation of the importance analysis model. The proposed method is a combination of numerical simulation approach and optimization approach, thus, it can integrate with more efficient numerical simulation approach, optimization strategy or surrogate model for further reducing the computational cost.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. NSFC 52075442) and National Science and Technology Major Project (Grant no. 2017-IV-0009-0046).

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