Computational Method and Simulation of Reliability for Series, Parallel, and k-Out-of-n Systems with Interval Parameters

For series, parallel, and k-out-of-n voting system reliability calculation methods, the six σ principles have been proposed in this study to derive the interchange relationship between interval parameters and random parameters. )e interval reliability index can be expressed in the function of the random reliability index. )e interval reliability index can then be transformed into a random reliability index. )e computational method of the reliability for series, parallel, and k-out-of-n voting systems with interval parameters is established. Finally, it has been shown that the proposed method is rational, practical, and applicable with two engineering practical simulations.


Introduction
In real-world engineering practice, it is very difficult to collect enough and accurate whole service life data for engineering structures, machinery and systems, etc., and the random reliability methods often fail to meet the actual needs. e literature [1][2][3] points out that the random reliability methods are often used in reliability assessment, but in practice, there should be other more effective approaches to supplement the deficiencies of random reliability methods. When the random reliability methods are unable to fit the models, the reliability problems should be solved using the interval, fuzzy, or convex set models to describe the uncertainty parameters [4][5][6][7]. Structural nonprobabilistic reliability models based on the interval theory have been proposed, and the structural nonprobabilistic reliability indices have been analyzed under interval loads [8].
In most real-life engineering problems, structures and machinery are not isolated but exist in the form of a system. e main common systems are series, parallel, and k/n system (SPKS). In the paper by Jiang et al. [9], the SPKS are introduced, and the reliability computational methods are presented, respectively. e random reliability method and interval reliability method are compared [10]. It is believed that the two reliability methods have their advantages and can complement each other in reliability calculation. e conversion of interval and random reliability methods has been studied [11,12]. Different measure spaces are transformed into a unified random space. e structural reliability is computed by using the random reliability theory, and the accuracy of the calculation can be improved [13].
In the paper by Fang et al. [14,15], the dynamic reliability of the repairable k/n vote is studied, and its reliability computational model is proposed. e reliability and maintenance of SPKS are also studied, and the optimal maintenance plan with repair time is proposed [16]. Load and strength with interval parameters will normally be encountered during the service period of engineering systems. e random reliability calculation method has also not been applied to SPKS reasonably. At present, there are limited studies on this topic.
Based on the above literature review, the computational method for interval reliability of SPKS will be studied in this paper. e interval parameters will first be transformed into random parameters; then, the random reliability of each link within the SPKS will be obtained, and finally, the random reliability of the whole SPKS will be computed. Two numerical examples are used to illustrate the proposed method for practicability and effectiveness.

Series, Parallel, and k/n Systems with Random Parameters
A system is denoted as S s . ere are n links within the system, and the ith link (i from 1 to n) is subjected to the load of which the mean is denoted as μ Li and the standard deviation is σ Li . e mean of the strength for the ith link (i from 1 to n) is denoted as μ qi , and the standard deviation is σ qi . e reliability index of the ith link is denoted as β i , and the reliability (probability of failure) is denoted as P i . Based on the configuration of the system, the reliability of the series system (P ss ), parallel system (P ps ), and k/n system (P ks ) can be calculated as follows:

Basic eory.
A system with interval parameters is denoted as S I . ere are m links within the system, and the ith link (i is from 1 to m) is subjected to the load which contains interval parameters too. e interval number is given as x Li is the lower bound of the interval and x Li is the upper bound of the interval.
Its interval mean and deviation of the load can be calculated as follows: Similarly, the strength of the -th link (i is from 1 to m) with interval parameters is given as , where x qi , x qi ∈ R, and x qi ≤ x qi . Its interval mean and deviation of the strength can be calculated as follows: x r qi � Based on the load and strength of the ith link with interval parameters, its reliability computational equation is shown as follows: If η i > 1, the ith link in the system is reliable. However, if η i < − 1, the ith link in the system is not reliable. If −1 ≤ η i ≤ 1, the higher the value of η, the more reliable the link will achieve.

Proposed Reliability Computational
Approach. For SPKS with random and interval parameters, the following relationship for the ith link subjected to the load can be established using the 6 σ rule: Based on equations (4) and (5), equations (9) and (10) can be rearranged as follows: e relationship between interval deviation of the load and standard deviation of THE random load parameter can be obtained from equation (12): Similarly, the same relationship can be established using the 6 σ rule for the strength of the ith link with interval and random parameters: x r qi � 6σ qi .
en, based on the derived equations (13) and (15), the relationship between the interval reliability index and random reliability index of the ith link within the SPKS can be established by using the first-order second moment (FOSM) method as follows: (20) For the sake of conservatism, the interval reliability index can be reduced to Equation (21) shows that the interval reliability index corresponds to the random reliability index of the ith link.
is index is conservative which is useful in engineering practice. In addition, equation (19) can explain why the system is safe when η i > 1, why the system is a failure when η i < − 1, why the system is more reliable when η is larger, and why the system may be safe or fail when −1 ≤ η i ≤ 1. It can also be used to determine the degree to which the system may be safe. e reliability (probability of failure) P Ii of the ith link is calculated using the reliability index as follows: where Φ is the standard normal distribution. Based on equations (1)-(3), the reliability (probability of failure) of SPKS with interval parameters can be obtained once the reliability P Ii of the ith link has been calculated from equation (22). Based on the configuration of the system, the reliability of the series system (P sI ), parallel system (P pI ), and k/n system (P kI ) with interval parameters can be calculated as follows:  Table 1. After applying the proposed approach, the reliability of each link is shown in Table 2. e reliability of the mechanical system, when the system has been designed as series, parallel, and 3/4 systems, is shown in Table 3.
It can be seen from Table 3 that the reliability of the series system is the lowest with 0.999999, and the reliability of the parallel system is the highest with the value of 1. is result is consistent with the system reliability obtained based on the random parameters. Figure 1 [11]. Its parameters are shown in Table 4. S is the allowable yield strength, U is the allowable strength, θ is the allowable torsional angle, τ is the allowable shear stress, and w is the allowable deflection of the hollow cylindrical cantilever beam. Parameters 1 and 2, as shown in Table 4, are the positional and scale parameters for the Gumbel distribution, respectively. For the normal distribution, parameters 1 and 2 are the mean and standard deviation, respectively. For the uniform distribution and interval distribution, parameters 1 and 2 are the lower and upper bounds, respectively.

Example 2. A hollow cylindrical cantilever beam is shown in
Five criteria are considered in this example, which are strength, stiffness, shear force, torsional angle, and deflection. e reliability of the cantilever beam based on these criteria in this paper is shown in Table 5. If only considering one of the five failure criteria, it is a parallel system. However, by considering all these five criteria, it is a series system. If three of them do not fail, then the cantilever beam does not fail. Obviously, the cantilever beam is a 3/5 voting system. e reliabilities of these systems are shown in Table 6. Each criterion has been calculated by the Monte Carlo (MC) method with 10 6 samples, respectively, and the reliability results are shown in Table 6. e reliability of the hollow cylindrical cantilever beam based on yield strength has been calculated using the proposed approach and the value is 0.99519. is example has also been calculated in [14] and the result is identical. is validates the proposed approach in reliability estimation. In

Conclusion
e reliability calculation method for interval parameters within SPKS has been proposed. e interchange relationship of the interval parameters and random parameters has been derived by using the 6σ principle. e relationship between the interval reliability index and the random reliability index has been established. e interval reliability index of individual link within SPKS can be transformed into the random reliability index. Once the reliability of the individual link within the system has been obtained, the reliability of the whole SPKS can be calculated. It has been verified that the proposed method is reasonable, practical, and applicable with two numerical examples.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.