Test Design Using the Expected Failure Times Plot
In last month’s issue of HotWire, we used a new tool
in Weibull++8, the Expected Failure Times plot, to
get a clear view of the range of the expected failure times for two different water pump designs. In this article, we will
focus on the mathematical theory behind the estimated failure
times and demonstrate how the plot may be used to design
reliability tests.
Background
Test duration is one of the key factors that should be considered in designing a test. If the expected test duration can be estimated prior to a test, resources can be better allocated. In this section, we will explain how to estimate the expected test time based on the sample size and the assumed underlying failure distribution.
The binomial equation used in nonparametric demonstration test design is the basis for predicting expected failure times. The equation is:
where:
 CL is the required confidence level.
 r is the number of failures.
 n is the total number of test units.
 R_{TEST} is the demonstrated reliability at the test time.
If CL, r and n are given, the R value can be solved from the above equation. When CL=0.5, the solved R (or Q, the probability of failure whose value is 1R) is the so called median rank for the corresponding failure.
For example, given n = 8, r = 3 and CL = 0.5, the calculated Q is 0.3205189. This means
that at the time when the third failure occurs, the estimated system probability of failure is 0.3205189. The median rank can be calculated in
Weibull++ using the Quick Statistical Reference tool, as shown next:
Similarly, if we set r = 4 for the above example, we can get the probability of failure at the time when the fourth failure occurs. Using the estimated median rank for each failure and the assumed underlying failure distribution, we can calculate the expected time for each failure. Assume that the failure distribution is Weibull, then we know that:
where:
 β is the shape parameter
 η is the scale parameter
Using the above equation for a given Q, we can get the corresponding time t. The calculation gives the median of each failure time for CL = 0.5. If we set CL at different values, the confidence bounds of each failure time can be obtained. For example, if we set CL=0.9, then from the calculated Q we can get the upper bound of the time for each failure. The calculated Q is given in the next figure:
If we set CL=0.1, then from the calculated Q we can get the lower bound of the time for each failure. The calculated Q is given in the figure below:
Example: Estimating Test Duration
Eight units are allocated for a reliability test. The test engineers want to know how long the test would last if all the units were tested to failure. Based on previous experiments, they assume that the underlying failure distribution is a Weibull distribution with β = 1.8 and η = 500
hours.
Solution
The following figure shows the expected failure times plot with 80% 2sided confidence bounds.
The following report shows the expected failure times.
From these results, we can see that the upper bound of the last failure is about 1,130 hours. Therefore, the test
will probably last for around 1,130 hours.
As we know, with eight samples, the median rank for the third failure is 0.3205189. Using this value and the assumed Weibull distribution, the median value of the
third failure time is calculated as:
This value could be verified using Weibull++’s SimuMatic utility. To obtain the result, use the following settings in SimuMatic:
 On the Main tab, choose the 2PWeibull distribution and enter the given parameters (i.e., β = 1.8 and η = 500
hours).
 On the Settings tab, set the number of points to 8.
 On the Analysis tab, choose the MLE analysis method
and set the confidence bounds to 80.
 On the Reliabilities & Time tab, enter a reliability value of 0.6794811 (because 10.3205189=0.6794811).
For all other settings, use the default values.
The following worksheet shows the generated results. It shows that T(0.6794811) = 309.21
hours, which is close to the calculated value. Please note that your results may vary due to the random values generated by simulation.
Conclusion
In this article, we introduced the underlying theory behind the Expected Failure Times plot
that is now available in Weibull++8, and demonstrated how reliability engineers could use the plot to design tests by estimating the testing time and expected failure times.
