Uncertainty Propagation for the Structures with Fuzzy Variables and Uncertain-but-Bounded Variables

Various uncertain factors exist in the practical systems. Random variables, uncertain-but-bounded variables and fuzzy variables are commonly employed to measure these uncertain factors. Random variables are usually employed to define uncertain factors with sufficient samples to accurately estimate probability density functions (PDFs). Uncertain-but-bounded variables are usually employed to define uncertain factors with limited samples that cannot accurately estimate PDFs but can precisely decide variation ranges of uncertain factors. Fuzzy variables can commonly be employed to define uncertain factors with epistemic uncertainty relevant to human knowledge and expert experience. This paper focuses on the practical systems subjected to epistemic uncertainty measured by fuzzy variables and uncertainty with limited samples measured by uncertain-but-bounded variables. The uncertainty propagation of the systems with fuzzy variables described by a membership function and uncertain-but-bounded variables defined by a multi-ellipsoid convex set is investigated. The combination of the membership levels method for fuzzy variables and the non-probabilistic reliability index for uncertain-but-bounded variables is employed to solve the uncertainty propagation. Uncertainty propagation is sued to calculate the membership function of the non-probabilistic reliability index, which is defined by a nested optimization problem at each membership level when all fuzzy variables degenerate into intervals. Finally, three methods are employed to seek the membership function of the non-probabilistic reliability index. Various examples are utilized to demonstrate the applicability of the model and the efficiency of the proposed method.


Introduction
Based on classical probability theory, traditional probabilistic reliability analysis has been more and more perfect. The main purpose of probabilistic reliability analysis is to assess reliability or failure probability. Many practical methods, such as Monte Carlo simulation, the importance sampling method [1], the response surface method [2], the first-order reliability method (FORM) [3], the second-order reliability method (SORM) [4], the subset simulation [5], the directional method [6], the line sampling method [7] and the asymptotic method for SORM [8], have been proposed to achieve this aim and apply it to practical engineering problems.
However, the traditional probabilistic reliability model requires precise probability density functions of the random variables, which are difficult to obtain in many practical applications because the samples available in practical engineering problems are limited. Although the principle of maximum entropy has been employed as an efficient technique to model the concerned uncertainty with a probabilistic distribution [9], it has been pointed out that classical probability reliability may be extremely sensitive to the statistical distribution of the data and even small errors in the inputs may yield misleading results in where x c = (x c 1 , · · · , x c n ), which is presumed to lie in the reliable domain in this paper, is the nominal value vector of the uncertain-but-bounded variables; W ∈ R n×n is a symmetric positive definite matrix called the characteristic matrix of the ellipsoid convex set model, which describes the orientation and aspect ratio of the principal axes of the ellipsoid model; θ, which defines the size of the ellipsoid model (or the magnitude of the uncertain-butbounded variables variability) is a positive real number. In practical engineering problems, these parameters can be obtained from the available data, such as tolerance specifications provided by the producers. When only one uncertain-but-bounded variable is involved in Equation (1), the corresponding ellipsoid convex set model can be expressed as [20] x Then, Equation (2) can be further simplified into the following form This means that the hyper-box model with only one uncertain-but-bounded variable can be viewed as the specific instance of the single ellipsoid convex set model. In other words, the interval convex set with only one uncertain-but-bounded variable is a onedimensional single ellipsoid convex set model [20].

Multi-Ellipsoid Convex Set Model
In practical engineering applications, the considered uncertain-but-bounded variables may arise from different sources such as inaccuracies in geometry, variability in material properties, fluctuations in external loads and errors resulting from instrument measurements. Therefore, it is more reasonable to divide all the uncertain-but-bounded variables into several uncorrelated groups according to the uncertainty source and then the multiellipsoid convex model can be established based on these uncorrelated groups, where each group is defined by a sub-dimensional ellipsoid convex set model according to the corresponding uncertainty source. For example, the variability in material properties can be described by an ellipsoid convex set model and errors arising from the instrument measurements by another. Suppose that the uncertain-but-bounded variables can be classified into k uncorrelated groups, and the corresponding vector is defined by where x i ∈ R n i (i = 1, · · · , k) denotes the ith group of the uncertain-but-bounded variables, and n i is the total number of uncertainties belonging to the ith group, which satisfies the following relation ∑ k i=1 n i = n, where n represents the total number of the uncertain-butbounded variables.
The single ellipsoid convex set model in Equation (1) can be extended to the multiellipsoid convex set model. For the multi-ellipsoid convex set model, each group of uncertain-but-bounded variables can be defined by the following form with an individual ellipsoid convex set model [18][19][20]39,40].
where W i ∈ R n i ×n i is the characteristic matrix of the ith ellipsoid convex set model and θ i is a positive real number. W i and θ i possess an identical meaning to W and θ in Equation (1). When the number of the groups k is equal to 1, then the multi-ellipsoid convex set model reduces to a single one. Obviously, if each group consists of only one uncertain-but-bounded variable, similar to Equations (2) and (3), it will reduce to an interval expressed as Equation (6), and then the multi-ellipsoid convex set model will degenerate into a hyper-box model. This implies that the hyper-box convex set model is a specific instance of the multi-ellipsoid convex set model [20].
A comparison of the three cases of the convex model with three uncertain-but-bounded variables is given in Figure 1, where (a) denotes the hyper-box model with three intervals, (b) represents a single ellipsoid model with three correlated variables and (c) is a multiellipsoid model with two sub-dimensional ellipsoids: an interval and a single ellipsoid. Since the single ellipsoid convex set model and hyper-box convex set model are the simplified versions of the multi-ellipsoid model, and thus the non-probabilistic reliability index can first be explained according to the two simple models, and then be further extended to the complex model in the next section.
in Equation (1). When the number of the groups k is equal to 1, then the multi-ellipsoid convex set model reduces to a single one.
Obviously, if each group consists of only one uncertain-but-bounded variable, similar to Equations (2) and (3), it will reduce to an interval expressed as Equation (6), and then the multi-ellipsoid convex set model will degenerate into a hyper-box model. This implies that the hyper-box convex set model is a specific instance of the multi-ellipsoid convex set model [20].
A comparison of the three cases of the convex model with three uncertain-butbounded variables is given in Figure 1, where (a) denotes the hyper-box model with three intervals, (b) represents a single ellipsoid model with three correlated variables and (c) is a multi-ellipsoid model with two sub-dimensional ellipsoids: an interval and a single ellipsoid. Since the single ellipsoid convex set model and hyper-box convex set model are the simplified versions of the multi-ellipsoid model, and thus the non-probabilistic reliability index can first be explained according to the two simple models, and then be further extended to the complex model in the next section.

Normalization of Uncertain-But-Bounded Variables
The multi-ellipsoid convex set model in Equation (5) can be transformed into the normalized form expressed in Equation (9) by the standard transformation expressed in Equations (7) and (8) [20].
where i Q is an orthogonal matrix consisting of the normalized eigenvectors of The multi-ellipsoid convex set model in Equation (5) can be transformed into the normalized form expressed in Equation (9) by the standard transformation expressed in Equations (7) and (8) [20].
where Q i is an orthogonal matrix consisting of the normalized eigenvectors of W i , Λ i is a diagonal matrix comprising the eigenvalues of W i and I i is a unit matrix, q i is the normalized or standard vector of the ith group uncertain-but-bounded vector x i . Figure 2 shows the comparison of the three models in the normalized q-space [18].

Non-Probabilistic Reliability Index Single Ellipsoid Convex Set Model
When all the uncertain-but-bounded variables can be described by a single ellipsoid convex set model, the normalized form of the single ellipsoid convex set model can be defined by a hyper-sphere with unit radius according to Equations (7)- (9).
where q = (q 1 , q 2 , · · · , q n ) and n is the total number of the uncertain-but-bounded variables. When all the uncertain-but-bounded variables can be described by a single ellipsoid convex set model, the normalized form of the single ellipsoid convex set model can be defined by a hyper-sphere with unit radius according to Equations (7)- (9).
where 12 ( , , , ) n q q q = q and n is the total number of the uncertain-but-bounded variables. Figure 3 gives a single ellipsoid convex set with two uncertain-but-bounded variables in the standard q -space [18]. The domains surrounded by the dashed-line circles are the expanded convex set. The region encircled by the solid-line circle with unit radius, which is centered at the coordinate origin, represents the convex set formed by all the possible values of the two uncertain-but-bounded variables. According to the basic principle of non-probabilistic reliability [10][11][12][13][14][15][16][17][18][19][20], when the circle enlarges proportionally in two directions, all the possible values of the two uncertain-but-bounded variables will locate in the reliable domain until the circle becomes tangential to the standard limit state curve. The maximum allowable variability, which can be employed to measure the reliability of the systems according to the concept of the non-probabilistic reliability index proposed first by Ben-Hain and Elishakoff [10][11][12][13][14][15], can be determined by the shortest distance from the coordinate origin to the standard limit state curve [18]: (11) where q is the normalized or standard vector of the uncertain-but-bounded vector x , and  min  is the minimum operation, and ( ) 0 g = q is the normalized failure boundary.  Figure 3 gives a single ellipsoid convex set with two uncertain-but-bounded variables in the standard q-space [18]. The domains surrounded by the dashed-line circles are the expanded convex set. The region encircled by the solid-line circle with unit radius, which is centered at the coordinate origin, represents the convex set formed by all the possible values of the two uncertain-but-bounded variables. According to the basic principle of nonprobabilistic reliability [10][11][12][13][14][15][16][17][18][19][20], when the circle enlarges proportionally in two directions, all the possible values of the two uncertain-but-bounded variables will locate in the reliable domain until the circle becomes tangential to the standard limit state curve. The maximum allowable variability, which can be employed to measure the reliability of the systems according to the concept of the non-probabilistic reliability index proposed first by Ben-Hain and Elishakoff [10][11][12][13][14][15], can be determined by the shortest distance from the coordinate origin to the standard limit state curve [18]: where q is the normalized or standard vector of the uncertain-but-bounded vector x, and min{·} is the minimum operation, and g(q) = 0 is the normalized failure boundary.

Multi-Ellipsoid Convex Set Model
When all the uncertain-but-bounded variables can be classified into k groups and each group consists of only one uncertainty, the multi-ellipsoid convex set model in Equation (9) reduces to a hyper-box model in the q -space, which can be written in the following form.
where the normalized uncertain-but-bounded variables c () Figure 4 gives three specific cases of the standard q -space for a structure with two

Multi-Ellipsoid Convex Set Model
When all the uncertain-but-bounded variables can be classified into k groups and each group consists of only one uncertainty, the multi-ellipsoid convex set model in Equation (9) reduces to a hyper-box model in the q-space, which can be written in the following form.
where the normalized uncertain-but-bounded variables q i = (x i − x c i )/∆x i . Figure 4 gives three specific cases of the standard q-space for a structure with two interval variables. Obviously, all the possible values of the two interval variables lie in the domain of the solid-line box, which is centered as the coordinate origin and has a sidelength of 2, namely {−1 ≤ q 1 ≤ 1, −1 ≤ q 2 ≤ 1}. Similar to the procedure of enlarging the convex set boundaries proportionally, the same conclusion can be drawn as follows. For case (a), the maximum variation that the system can tolerate is the value of the vertical coordinate of the critical point A, namely min q|g(q)=0 (max(q 1 , q 2 )) = q 2 ; For case (b), the allowable maximum variability is min q|g(q)=0 (max(q 1 , q 2 )) = q 1 ; For case (c), the maximum degree of variability that the structure allows is min q|g(q)=0 (max(q 1 , q 2 )) = q 1 = q 2 . Hence, according to the non-probabilistic reliability theory [10][11][12][13][14][15][16][17][18][19][20], the non-probabilistic reliability index for the case with two intervals can be expressed as [18] The result can be further extended to the hyper-box model with k interval variables and the non-probabilistic reliability index is [18] Obviously, the non-probabilistic reliability index, which is employed to measure the safety of the structure, is the infinity norm (L − ∞ or maximum norm) of the vector.
Since the multi-ellipsoid convex set model is the extension of the hyper-box interval convex set, and the form for the hyper-box model expressed as Equation (14) can be extended to the non-probabilistic reliability index of the multi-ellipsoid model, which can be defined by Obviously, the non-probabilistic reliability index, which is employed to measure the safety of the structure, is the infinity norm ( L − or maximum norm) of the vector. (a)

The Membership Levels Method
The membership levels method is usually employed to calculate the fuzzy variables in [24], as shown in Figure 5. Suppose the limit state function of a structure is expressed is the fuzzy vector defined by the membership functions. At each membership level α, the fuzzy variable Y i (i = 1, · · · , m) degenerates into a lower and an upper bound (or interval variable) The bounds of the output response M can be calculated by optimization or any other technique. Once all the variables are defined as membership functions, the bounds of the output response M at various a-cuts can be obtained, and then the approximation of the membership functions of the outputs can be obtained. In other words, uncertainty propagation from the fuzzy input Y i (i = 1, · · · , m) to the output response M can be achieved by the membership levels method.

The Membership Levels Method
The membership levels method is usually employed to calculate the fuzzy variables in [24], as shown in Figure 5. Suppose the limit state function of a structure is expressed as . The bounds of the output response M can be calculated by optimization or any other technique. Once all the variables are defined as membership functions, the bounds of the output response M at various a-cuts can be obtained, and then the approximation of the membership functions of the outputs can be obtained. In other words, uncertainty propagation from the fuzzy input to the output response M can be achieved by the membership levels method.

The Membership Function of the Non-Probabilistic Reliability Index
The non-probabilistic reliability index is employed to define the quantified measure of the reliability of the structure with the uncertain-but-bounded variables [18][19][20]. The membership levels method is utilized to calculate the fuzzy variables [24], as shown in Figure 5. However, for many engineering problems with incomplete available information, all the uncertain variables may arise from many different sources such as internal parameters, external loads, etc. Some of these uncertain variables may be uncertain-butbounded variables, which can be described by the multi-ellipsoid convex set, and others may be fuzzy variables, which can be defined by the membership function. It is necessary to investigate uncertainty propagation within engineering problems with uncertain-butbounded variables and fuzzy variables. The limit state function of a system with uncertain-but-bounded variables and fuzzy variables is expressed as

The Membership Function of the Non-Probabilistic Reliability Index
The non-probabilistic reliability index is employed to define the quantified measure of the reliability of the structure with the uncertain-but-bounded variables [18][19][20]. The membership levels method is utilized to calculate the fuzzy variables [24], as shown in Figure 5. However, for many engineering problems with incomplete available information, all the uncertain variables may arise from many different sources such as internal parameters, external loads, etc. Some of these uncertain variables may be uncertain-but-bounded variables, which can be described by the multi-ellipsoid convex set, and others may be fuzzy variables, which can be defined by the membership function. It is necessary to investigate uncertainty propagation within engineering problems with uncertain-but-bounded variables and fuzzy variables. The limit state function of a system with uncertain-but-bounded variables and fuzzy variables is expressed as where x T = x T 1 , · · · , x T k represent the k groups of uncertain-but-bounded variables defined by the multi-ellipsoid convex set, and Y = [Y 1 , · · · , Y m ] are the m fuzzy variables described by the membership functions.
The k groups of uncertain-but-bounded variables x T = x T 1 , · · · , x T k can be transformed into the normalized ones q T = q T 1 , · · · , q T k by Equations (7)- (9). At the membership level α i , the m fuzzy variables , y 2α i , · · · , y nα i and upper bounds y α i = y 1α i , y 2α i , · · · , y nα i by the membership levels method stated in Section 3. Then, the original limit state function (16) is mapped into the standard one g(q, y α i ).
In order to state the principal ideas conveniently, the q-space of a problem, which consists of a single ellipsoid convex set with two uncertain-but-bounded variables and a fuzzy variable, is given in Figure 6. In the ellipsoid convex set model introduced in Section 2, the normalized limit state curve g(q) = 0 divides the q-space into two parts: the reliable domain and the failure domain, which can be seen in Figures 3 and 4. However, as revealed by Figure 6, the normalized g(q, y α i ) = 0(y α i ∈ y α i , y α i ) consists of a cluster of normalized limit state curves and each single limit state curve corresponds to a possible realization of the intervals y α i = y α i , y α i . In other words, all the possible values of the uncertain-but-bounded variables and the degraded fuzzy variables that satisfy g(q, y α i ) = 0 form a banded geometry in the standard q-space. Hence, the q-space is partitioned into three parts: the reliable domain, the critical domain and the failure domain, as shown in Figure 6. Figure 7 gives the case consisting of a hyper-box model with two intervals and a fuzzy variable, the basic idea of which is the same as Figure 6.

Estimate the Membership Function of the Non-Probabilistic Reliability Index
Based on the membership levels method, three techniques are introduced to calculate the membership function of the non-probabilistic reliability index. Before all the procedures are performed, the membership level i  is supposed to take a value of 1 ( 0,1, , )

Estimate the Membership Function of the Non-Probabilistic Reliability Index
Based on the membership levels method, three techniques are introduced to calculate the membership function of the non-probabilistic reliability index. Before all the procedures are performed, the membership level i  is supposed to take a value of 1 ( 0,1, , )  Obviously, the shortest distance from the coordinate origin to the normalized limit state curve varies from η α i and η α i as demonstrated in Figures 6 and 7. According to the mathematical definition of the non-probabilistic reliability index described in Section 2, the non-probabilistic reliability index η α i for the problems g(q, y α i ) = 0(y α i ∈ y  1], the membership function of the non-probabilistic reliability index can be estimated. The following section will give some approaches to estimating the membership function of the non-probabilistic reliability index.

Estimate the Membership Function of the Non-Probabilistic Reliability Index
Based on the membership levels method, three techniques are introduced to calculate the membership function of the non-probabilistic reliability index. Before all the procedures are performed, the membership level α i is supposed to take a value of α i = i × 1 N (i = 0, 1, · · · , N), where N is the total number of partitions. In order to reduce the computational cost, N takes a value of N = 5 in the numerical examples. The following three methods are employed for the normalized limit state curve g(q, y α i ), where q are normalized uncertainbut-bounded variables and y α i are upper and lower bounds (degenerated fuzzy variables) for y.

Double-Loop Optimization
Based on these properties of the model with fuzzy variables and uncertain-butbounded variables, the lower η α i and the upper η α i bounds of the non-probabilistic reliability index can be calculated from Equations (17) and (18), as shown in Figure 8: where α i = i × 1 N (i = 0, 1, · · · , N) is the value of the membership level, the non-probabilistic reliability index corresponding to y α i is given as the following form  (19) The symbols in Equation (19) are identical to the ones in Equation (15).

Single-Loop Optimization
Firstly, the minmax optimization problem expressed as Equation (19) can be transformed into an equivalent minimization problem by introducing a variable  [19]. The symbols in Equation (19) are identical to the ones in Equation (15).

Single-Loop Optimization
Firstly, the minmax optimization problem expressed as Equation (19) can be transformed into an equivalent minimization problem by introducing a variable δ [19].
Then, with the combination of the three sub-optimization problems expressed in Equations (17), (18) and (20), respectively, the lower bound η α i and the upper bound η α i can be equivalently transformed into the single-loop optimization problems expressed as Equations (21) and (22).
where the notations in Equations (21) and (22) are in accordance with the ones in Equations (17) and (18), respectively. The flowchart of computing η α i and η α i is shown in Figure 9.  (22) where the notations in Equations (21) and (22) are in accordance with the ones in Equations (17) and (18)

The Outer Optimization by Random Sampling Method
The only constraints of the outer loop in Equations (17) and (18)

The Outer Optimization by Random Sampling Method
The only constraints of the outer loop in Equations (17) and (18)  , y α i uniformly, and then estimate the non-probabilistic reliability index η(y j α i ) of the normalized limit state curve corresponding to the jth realization y j α i by Equation (19) or (20). Finally, the minimum and maximum of the sequences η(y j α i )(j = 1, 2, · · · , M) can be employed to approximate the lower and upper non-probabilistic reliability index η α i and η α i , which can be expressed as Equations (23) and (24). The corresponding flowchart of estimating η α i and η α i is given in Figure 10.   Three methods have been given for estimating the membership function of the nonprobabilistic reliability index in Sections 4.2.1-4.2.3. Here, we will discuss the computational cost relevant to the three methods.
For the double-loop optimization method, N = 5 membership levels have been employed, i.e., α i = i × 1 N (i = 0, · · · , N − 1). Thus, the total computational cost is is the number of optimization iterations for in Equation (17) and N η α i is the number of optimization iterations for solving η α i in Equation (18). During every optimization iteration, we need to solve η(y α i ) that is a minimum-maximum nesting optimization in Equation (19), which is a time-consuming process.
For the single-loop optimization method, N = 5 membership levels have been employed, i.e., α i = i × 1 N (i = 0, · · · , N − 1). Thus, the total computational cost is is the number of optimization iterations for solving η α i in Equation (21), and N η α i is the number of optimization iterations for solving η α i in Equation (22). Obviously, there is no nesting optimization when solving η α i and η α i , as shown in Equations (21) and (22), and thus the computational cost relevant to each optimation iteration is low. For the random sampling method, N = 5 membership levels have been employed, i.e., . In order to solve η α i and η α i , we first generate M realizations y j α i (j = 1, 2, · · · , M) within the intervals y j α i (j = 1, 2, · · · , M) uniformly, and then we can estimate the non-probabilistic reliability index η(y j α i ) of the normalized limit state curve corresponding to the jth realization y j α i (j = 1, 2, · · · , M). Thus, the total computational cost is is the computational cost for estimating η(y α i ) given in Equation (19) or (20). In general, the number of realizations (i.e., y j α i (j = 1, 2, · · · , M)) is not small, and this paper M takes a value of 1000, i.e., M = 1000.
It is obvious that the double-loop optimization method is the most complex procedure, the random sampling method is the second most complex, and the single-loop optimization method is the least complex. The computational cost relevant to the single-loop optimization method can be afforded as the number of uncertainties increases, as shown in Section 5.

Numerical Examples
It is prohibitive to approximate the membership function of the non-probabilistic reliability index accurately due to the large computational cost. In order to reduce the computational consumption, the lower and upper bounds of the non-probabilistic reliability index corresponding to six membership levels are estimated first, and then the membership function of the non-probabilistic reliability index can be obtained by linking these six values, namely α i = i × 1 N (i = 0, 1, · · · , 5). The total number of realizations is 1000 for the random sampling method. The symbol NOFC represents the number of function calculations.

A Simple Linear Performance Function
Give a simple performance function g(x) = x 1 + x 2 − x 3 − 2, where x 1 and x 2 are two uncertain-but-bounded variables defined by the following single ellipsoid convex set x 2 1 + x 2 2 ≤ 1; x 3 is a fuzzy variable with the following membership function: where µ x 3 (x 3 ) is the membership level. Tables 1 and 2 summarize the estimation of the membership function of the non-probabilistic reliability index by single-loop optimization and double-loop optimization, respectively. Figure 11 shows a comparison of the results achieved using the two methods.   Figure 11 shows a comparison of the results achieved using the two methods.  This is a simple linear problem, and single-loop optimization and double-loop optimization have obtained an accurate estimator of the membership function of the nonprobabilistic reliability index. The results show that the single-loop optimization method and double-loop optimization method can give good results for the linear performance function. This is a simple linear problem, and single-loop optimization and double-loop optimization have obtained an accurate estimator of the membership function of the nonprobabilistic reliability index. The results show that the single-loop optimization method and double-loop optimization method can give good results for the linear performance function.

Fourth-Order Polynomial Performance Function
Give a nonlinear performance function g(x) = 1 40 x 4 1 + 2x 2 2 + x 3 + 3, where x 1 is a fuzzy variable described by the following membership function and x 2 and x 3 are two uncertain-but-bounded variables defined by the following single ellipsoid convex set x 2 2 + x 2 3 ≤ 1, Tables 3-5 give the summarization of the estimation of the membership function of the non-probabilistic reliability index by the single-loop optimization, double-loop optimization and random sampling method, respectively. The comparison of the estimated results is shown in Figure 12. x are two uncertain-but-bounded variables defined by the following single ellipsoid convex set 22 23 1 xx + , Tables 3-5 give the summarization of the estimation of the membership function of the non-probabilistic reliability index by the single-loop optimization, double-loop optimization and random sampling method, respectively. The comparison of the estimated results is shown in Figure 12.   For this example, it can be seen that the results by single-loop optimization are the best; the ones by the random sampling method and double-loop optimization are almost the same. The computational cost of the single-loop optimization is the least while that of the random sampling method is very large.

A High Nonlinear Performance Function
Suppose a high nonlinear performance function g(x) = 1.016 − 400, where x 1 and x 2 are two fuzzy variables with the membership functions depicted as follows and x 3 and x 4 are two correlated uncertain-but-bounded variables with the single ellipsoid convex set expressed as Tables 6-8 give the approximation of the membership function of the non-probabilistic reliability index by the single-loop optimization, double-loop optimization and the random sampling method, respectively. Figure 13 shows a comparison of the estimated results. As can be seen, the random sampling method has achieved the best estimated results but with expensive computational costs, while the single-loop optimization method has given suboptimal results with cheap computation costs. In addition, the single-loop optimization method and random sampling method have achieved good estimated results for the lower bound of the membership function of the non-probabilistic reliability index.

A Cantilever Beam
A cantilever beam subjected to a concentrated force P is shown in Figure 14. The beam has a length of L , a width of b and a height of h . Young's modulus of the material is E. The structure becomes unsafe when the tip displacement is greater than 0.15 in. Thus, the limit-state function is defined as Figure 13. The estimated membership function of the non-probabilistic reliability index.

A Cantilever Beam
A cantilever beam subjected to a concentrated force P is shown in Figure 14. The beam has a length of L, a width of b and a height of h. Young's modulus of the material is E. The structure becomes unsafe when the tip displacement is greater than 0.15 in. Thus, the limit-state function is defined as Materials 2023, 16, x FOR PEER REVIEW 20 of 25 Figure 14. A cantilever beam. Tables 10-12 summarize the results of the membership function by the three methods. A comparison of the estimated results of the non-probabilistic reliability index is shown in Figure 15. As shown in Tables 10-12 and Figure 15, the random sampling method has achieved the best estimated result for the membership function of the nonprobabilistic reliability index but with heavy computational costs, while the single-loop optimization method has obtained suboptimal results but with low computation costs. In addition, all three methods have achieved good estimated results for the upper bound of the membership function of the non-probabilistic reliability index. In this example, E(psi.), P(lb.) and L(in.) are described by fuzzy variables and the membership functions are expressed as the following three relationships. In addition, b and h are considered to be uncertain-but-bounded variables, and their uncertainty information is summarized in Table 9.  Table 9. Information of uncertain-but-bounded variables for a cantilever beam.

Uncertain Variable Nominal Value Convex Model Description
Tables 10-12 summarize the results of the membership function by the three methods. A comparison of the estimated results of the non-probabilistic reliability index is shown in Figure 15. As shown in Tables 10-12 and Figure 15, the random sampling method has achieved the best estimated result for the membership function of the non-probabilistic reliability index but with heavy computational costs, while the single-loop optimization method has obtained suboptimal results but with low computation costs. In addition, all three methods have achieved good estimated results for the upper bound of the membership function of the non-probabilistic reliability index.

A Performance Function with 12 Variables
The function is given by 1 2 3 4 5 6 7 8 9 10 11 12 i xi = are uncertain-but-bounded variables defined by a multi-ellipsoid convex set with two single ellipsoid convex set models, respectively.
Tables 13-15 give the corresponding estimated results of the membership functions. The estimated membership functions of the non-probabilistic reliability index are given in Figure 16. The results show that the three methods have achieved good estimated results, and the computational consumption cost of the single-loop optimization is the least. The random sampling method is not practical for many engineering applications owing to the prohibitive computational cost. The results show that three methods can achieve good results for the nonlinear performance function with multiplication operation, addition operation and subtraction operation.

A Performance Function with 12 Variables
The function is given by where x i (i = 1, · · · , 4) and x i (i = 5, · · · , 8) are uncertain-but-bounded variables defined by a multi-ellipsoid convex set with two single ellipsoid convex set models, respectively.
and x i (i = 9, · · · , 12) are fuzzy variables expressed as Tables 13-15 give the corresponding estimated results of the membership functions. The estimated membership functions of the non-probabilistic reliability index are given in Figure 16. The results show that the three methods have achieved good estimated results, and the computational consumption cost of the single-loop optimization is the least. The random sampling method is not practical for many engineering applications owing to the prohibitive computational cost. The results show that three methods can achieve good results for the nonlinear performance function with multiplication operation, addition operation and subtraction operation.

Conclusions
This paper investigates the uncertainty propagation for systems with fuzzy variables and uncertain-but-bounded variables, and the membership function of the non-probabilistic reliability index is employed to define the uncertainty propagation. The proposed methods can be applied for uncertain analysis of any systems (such as structures and machines) that are subjected to uncertain factors with limited samples and uncertainty relevant to human knowledge and expert experience. Three algorithms, namely, single-loop optimization, double-loop optimization and random sampling are proposed to solve the membership function of the non-probabilistic reliability index. Five examples with linear and nonlinear problems are employed to demonstrate the applicability of the proposed methods. The results show that single-loop optimization is more efficient and stable than double-loop optimization. Although the results using the random sampling method are better than those using single-loop optimization for most cases, the former approach is not suitable for many engineering applications due to its huge computation cost. In addition, the results also show that the single loop method fits to the linear performance functions and nonlinear performance functions with multiplication operation, addition operation and subtraction operation. Meanwhile, the computational cost relevant to the singleloop optimization method can be afforded as the number of uncertainties increases. The main contribution of the proposed method is to propose a model to deal with the uncertainty analysis for systems with fuzzy variables and uncertain-but-bounded variables and to give three methods for solving this issue. Future research can focus on the more adaptable approaches that can find the global optimal solutions for linear performance functions and nonlinear performance functions.
Author Contributions: Algorithm analysis, Y.X. and Z.T.; manuscript writing, Y.X., L.D. and P.L.; methods research, Y.X. and Z.T. All authors have read and agreed to the published version of the manuscript.

Conclusions
This paper investigates the uncertainty propagation for systems with fuzzy variables and uncertain-but-bounded variables, and the membership function of the non-probabilistic reliability index is employed to define the uncertainty propagation. The proposed methods can be applied for uncertain analysis of any systems (such as structures and machines) that are subjected to uncertain factors with limited samples and uncertainty relevant to human knowledge and expert experience. Three algorithms, namely, single-loop optimization, double-loop optimization and random sampling are proposed to solve the membership function of the non-probabilistic reliability index. Five examples with linear and nonlinear problems are employed to demonstrate the applicability of the proposed methods. The results show that single-loop optimization is more efficient and stable than double-loop optimization. Although the results using the random sampling method are better than those using single-loop optimization for most cases, the former approach is not suitable for many engineering applications due to its huge computation cost. In addition, the results also show that the single loop method fits to the linear performance functions and nonlinear performance functions with multiplication operation, addition operation and subtraction operation. Meanwhile, the computational cost relevant to the single-loop optimization method can be afforded as the number of uncertainties increases. The main contribution of the proposed method is to propose a model to deal with the uncertainty analysis for systems with fuzzy variables and uncertain-but-bounded variables and to give three methods for solving this issue. Future research can focus on the more adaptable approaches that can find the global optimal solutions for linear performance functions and nonlinear performance functions.