Reliability HotWire

Reliability HotWire

Issue 136, June 2012

Reliability Basics

Bayesian Reliability Demonstration Test Design

For life tests, especially for systems like missiles, sample size is always limited. It is a challenge to get accurate reliability estimates with limited samples, and Bayesian methods have been used to solve this problem. Weibull++ 8 now offers non-parametric Bayesian reliability demonstration test (RDT) design. Unlike traditional non-parametric test design methods, this new feature allows engineers to integrate engineering knowledge or subsystem testing results into system reliability test design. In this article, we will explain the theory of Bayesian RDT and illustrate how to use Weibull++ to design an efficient reliability demonstration test.

Theory of Bayesian RDT

The following binomial equation is often used in reliability demonstration test design:

Equation (1)

where CL is the required confidence level, r is the number of failures, n is the total number of units on test and R is the reliability that needs to be demonstrated.

Eqn. (1) has four unknowns: CL, r, n and R. Given any three of these, the remaining one can be solved. For example, given CL, r and R, the required sample size can be determined.

Eqn. (1) shows that the reliability is a random number with a beta distribution. It can be rewritten as:

Equation (2)

As discussed in [1], Eqn. (2) also can be obtained using the Bayesian theory when the non-informative prior distribution 1/R is used. In general, when a beta distribution is used as the prior distribution for reliability R, the posterior distribution obtained from Eqn. (1) is also a beta distribution. For example, assuming the prior distribution is Beta(R, α0, β0), the posterior distribution for R is:  

Equation (3)

Therefore, Eqn. (3) can be used for Bayesian RDT. For a random variable x with beta distribution Beta(x, α, β), its mean and variance are:

Equation (4)

If the expected value and the variance are known, parameters α and β in the beta distribution can be solved by:

Equation (5)

Example 1: Bayesian Test Design with Prior Information from Expert Opinion

Suppose you wanted to know the reliability of a system and the following prior knowledge of the system is available:

  • Lowest possible reliability: a = 0.80
  • Most likely reliability: b = 0.85
  • Highest possible reliability: c = 0.97

Based on this information, the mean and variance of the prior system reliability can be estimated as:

Equation

Using the above two values and Eqn. (5), the prior distribution for R is a beta distribution Beta(R, α0, β0) with:

Equation (6)

Given the above prior information, if there is 1 failure out of 20 test samples, what is the demonstrated reliability at a confidence level of CL =  0.8? The result is given in Figure 1.

Bayesian DRT with expert opinion as prior
Figure 1 - Bayesian DRT with expert opinion as prior

Figure 1 shows the demonstrated reliability is 85.103991%.

Without the prior information for system reliability, the demonstrated reliability is 85.75745%, as given in Figure 2:

DRT without expert opinion as prior
Figure 2 - DRT without expert opinion as prior

The prior expert opinion can have a significant effect on the RDT result. From this example, we can see that the Bayesian method and the regular non-parametric binomial method produce similar results. However, if the reliability is very high based on prior expert opinion, the Bayesian RDT will give very different results than the regular one. Therefore, you must be very careful when you apply any Bayesian method.

For example, if we change the expert opinion in this example to:

  • Lowest possible reliability: a = 0.90
  • Most likely reliability: b = 0.95
  • Highest possible reliability: c = 0.97

and keep the rest settings unchanged, the Bayesian RDT result is:

DRT without expert opinion as prior
Figure 3 - DRT with modified expert opinion as prior

Example 2: Bayesian Test Design with Prior Information from Subsystem Tests

For each subsystem i in a system, the reliability can also be modeled using a beta distribution. If there are ri failures out of ni test samples, Ri is a beta distribution with the following cumulative distribution function:

Equation (7)

where si = ni - ri is the number of successes. Therefore, the expected value and the variance of Ri are:

Equation (8)

Assuming that all the subsystems are connected reliability-wise in a series configuration, the expected value and the variance of the system’s reliability R can then be calculated as:

Equation (9)

From Eqn. (9), we can get the α and β parameters for the prior distribution of R in Eqn. (5).

Assume a system of interest is composed of three subsystems: A, B and C. Prior information from tests of these subsystems is given in the table below.

Subsystem Number of Units (n) Number of Failures (r)
A 20 0
B 30 1
C 100 4

 

Given the above information, in order to demonstrate the system reliability of 0.9 at a confidence level of 0.8, how many samples are needed in the test? Assume the allowed number of failures is 1.

The result is given in the figure shown next, which shows that at least 49 test units are needed.

Bayesian DRT with subsystem tests as prior
Figure 3 - Bayesian DRT with subsystem tests as prior

Conclusion

In this article, we discussed the theory of Bayesian reliability demonstration test design. Weibull++ was used to solve two examples. Considering prior information about system reliability allows us to design a better test and estimate the system reliability more accurately. This article shows how both expert opinion and subsystem test information can be used to incorporate the prior distribution of the system reliability.

Reference

[1] H. Guo, T. Jin and A. Mettas, "Designing reliability demonstration tests for one-shot systems under zero component failure," IEEE Transactions on Reliability, vol. 60, no. 1, pp. 286-294, March 2011.