Truncation of the Integral Used in the Comparison Wizard
in Weibull++
In the August 2010 issue of Hotwire, the
Reliability Basics article discussed the integral used
in the
Weibull++
Comparison Wizard to calculate the probability that Design A will outlast
Design B.
This comparison is performed over the entire time domain,
therefore the limits of integration are time = 0 and time
= infinity. This article uses simulations
in
RENO to explain how changing the limits of integration
affects the output of the Comparison Wizard in Weibull++.
One day when Joe the reliability engineer was using the
Comparison Wizard in Weibull++ to help him decide
between two designs for his upcoming project, he noticed
that there is a place in the wizard to change the limits
of the integration. "Hmm," Joe thought. "I wonder if changing
the upper limit could help me to compare two designs whose
lives have a maximum, say, in the case of a warranty."
Joe used his data sets for Design A and Design B from
his previous work, as shown in Table 1.
Table 1  Failure times for Designs
A and B
Design A 
Design
B 
1059 
4869 
2245 
5348 
2781 
5951 
3130 
6257 
3535 
6561 
4512 
6763 
5632 
6796 
6218 
7774 
6819 
7992 
6891 
8133 
He recalled that the probability that Design B would
outlast Design A was 80.7452% using the default limits of
integration from 0 to infinity. Then he reviewed the MultiPlot
of the reliability functions of the designs and noted that
most of the components from both Design A and Design B have
failed by 10,000 hours, as shown in Figure 1. (Click
here to access
Joe’s Weibull++ file.)
Figure 1  Input data and calculated
parameters for Design B
Based on this information, Joe decided to
create a table of outputs of the Comparison Wizard using
values of 1,000 to 10,000 for the warranty period.
Next, Joe opened the Comparison Wizard in Weibull++.
On the Setup tab, he selected the Override
autocalculated limits check box to allow him to
enter new limits of integration. He decided to set the number
of quadratures (i.e., the number of intervals that the integral
is divided into) to 1,000. Then he set the lower limit to
0 and the upper limit (i.e., warranty time) to 1,000. Finally,
he chose to compare Design A to Design B (the data for these
are in data sheets named "A" and "B" of the same folio). These settings are displayed
next.
Figure 2  Comparison Wizard settings
Joe returned to the Common Probability tab of the Comparison
Wizard and clicked Compare to view the result, as
shown in Figure 3. He copied the probability that A would
outlast B (0.0002%) and pasted it into a table he created
in a general spreadsheet.
Figure 3  Comparison of Designs
A and B from 0 to 1,000 hours
Next, he calculated the probability that Design B will
outlast Design A. Then he
entered the calculated probability (4.6180%) into the general
spreadsheet. He repeated this procedure for multiples of
1,000 hours up to 10,000 hours. The results are shown in
Table 2.
Table 2  Comparison Wizard results
for warranty times from 1,000 to 10,000 hours
Joe was surprised to see that, for a given warranty time,
the probability that Design A outlasts Design B and the
probability that Design B outlasts Design A did not add
up to 100%. "For a warranty time of 1,000 hours, what happens
in the remaining 95% of the time that is not represented in
the table?" he wondered. He did notice that as the warranty
time gets closer to infinity, the sum of the probabilities
in each row approaches 100%. In addition,
the probability that Design B outlasts Design A for a warranty
time of 10,000 hours was identical to the value he calculated
using the default Comparison Wizard limits of integration.
Joe looked at the equation that is used by the Comparison
Wizard to compute the probability that Design B outlasts
Design A, as shown below, and decided to come up with an
hypothesis.
For a given value of time, t, the expression inside the
integral represents the probability that Design A lasts
exactly t and Design B lasts longer than t. The integral
is the sum of those probabilities for all possible outcome
values of life. When a finite upper limit is used, outcomes
in which the life of Design A is greater than the upper
limit are excluded from the calculation. Similarly, when
computing the probability that Design A outlasts Design
B, the outcomes in which the life of Design B is greater
than the upper limit are excluded from the calculation.
This observation prompted Joe to create a table of the possible
combinations of the outcomes of the lives of Designs A and
B, as shown in Table 3. (Note that there is also a very
small probability that the lives of both designs will
be less than the upper limit and equal to each other. This
case is excluded from Table 3.) Based on his analysis of
the above equation, Joe realizes that only the first column
of Table 3 can contribute to the Comparison Wizard result
for the probability that Design B outlasts Design A, and
only the first row of the table can contribute to the Comparison
Wizard result for the probability that Design A outlasts
Design B. So the result of comparing Design A to Design
B using the Comparison Wizard in Weibull++ is the
sum of the italicized values in Table 3, and the result
of comparing Design B to Design A using the Comparison Wizard
is the sum of the underlined values. He hypothesized that
the remaining outcomes—i.e., those in the bottom right cell
of Table 3—account for the fact that the rows of Table 2
do not add up to 100%. He decided to test this hypothesis
by building a simulation in RENO to examine the
effect of an upper limit. (Click
here to access
his RENO file.)
Table 3  Combinations of lives
of Designs A and B using an upper limit
The flowchart that Joe created for his RENO
simulation follows these steps, as illustrated in Figure 4:
 Choose one random failure time using the Weibull
parameters for each design. For any failure time that
is greater than the warranty time, set that failure
time equal to the warranty time.
 Subtract the failure time for Design B from Design
A. Use this difference to determine which design lasts
longer.
 If the difference is positive, Design A outlasts Design B. Add 1 to the value of the
"A Outlasts B" storage block.
 If the difference is negative, Design B outlasts Design A. Add 1 to the value of the
"B Outlasts A" storage block.
 If the difference is zero, one cannot tell the
difference in the lives of Designs A and B.
 Subtract the original failure time for Design A (i.e., the failure
time before adjusting for warranty time) from the warranty time. Use this difference to determine
if both designs fail before or after the warranty time.
 If the difference is positive, Designs A and
B fail at the same time (less than the warranty
time). Add 1 to the value of the storage block labeled
"A Equal to B and Less Than Warranty Time."
 If the difference is negative or zero, both
Designs A and B last as long as or longer than the
warranty time. Add 1 to the value of the storage
block labeled "A and B Greater Than or Equal to
Warranty Time."
Figure 4  Comparison Wizard upper
limit simulation flowchart
At the end of each simulation, the results in the storage
blocks are divided by the number of simulations to obtain
the probability of each outcome occurring. Joe used 100,000
simulations per run, and he used one run for each warranty
time in Table 2. The results of his simulations are shown
in Table 4, along with the results of Table 2 for comparison.
Note that in all simulations for all warranty times considered,
the life of Design A was never both equal to the life of
Design B and less than the warranty time.
Table 4  Simulation results for
warranty times from 1,000 to 10,000 hours (shown with Comparison
Wizard results)
Joe was pleased to confirm that his hunch was correct,
as seen by comparing the simulation and Comparison
Wizard results
in Table 4. In addition, he realized that the probability
that both Designs A and B outlast the warranty time is the
product of the probabilities that each design lasts longer
than the warranty time. Since the probability that an item
lasts longer than a specific time is the reliability, R,
of that item at the specified time, the probability, P,
that both A and B outlast the warranty time is given by:
Joe computed these values for each warranty time using
the Function Wizard in the general spreadsheet that he had
created in Weibull++ and added them to the information
above as shown in Table 5. He confirmed that the sum of
the results of the two Comparison Wizard calculations and
the probability that both designs outlast the warranty time
computed from the above equation is 100% for each row of
the table.
Table 5  Simulation and Comparison
Wizard results for warranty times from 1,000 to 10,000 hours,
including probabilities that both designs last longer than
the warranty time
Joe was excited to see his results confirmed his intuition
about the use of the upper limit in the Comparison Wizard.
He decided to repeat the process for the lower limit. Joe
imagined that one might use the lower limit to examine the
effect of burnin on the lives of two designs.
First, Joe made a table of possible combinations of the
outcomes of the lives of Designs A and B when using a lower
limit, as shown in Table 6. Again, he thought about the
equation that the Comparison Wizard employs. Using a lower
limit excludes the outcomes in which the life of Design
A is less than the lower limit from the calculation of the
probability that Design B outlasts Design A, and it excludes
the outcomes in which the life of Design B is less than
the lower limit from the calculation of the probability
that Design A outlasts Design B. So the first column of
the table is excluded from the calculation of the probability
that Design B outlasts Design A, and the first row of the
table is excluded from the calculation of the probability
that Design A outlasts Design B. Therefore, only the outcomes
in the lower right cell of Table 6 are used to compute the
probability that Design A outlasts Design B (italicized
value) or that Design B outlasts Design A (underlined value).
Any outcomes in which at least one design fails before the
burnin time are excluded from the results of the Comparison
Wizard calculations.
Joe knew that the probability that at least one of the
designs would fail before the burnin time is the complement
of the probability that both designs last at least as long
as the burnin time. This probability is given by:
Joe computed these values for each burnin time using
the Function Wizard in the general spreadsheet that he had
created in Weibull++.
Table 6  Combinations of the lives
of Designs A and B using a lower limit
Next, Joe created his simulation in RENO. The
flowchart he created to examine the effect of a lower limit
follows these steps, as illustrated in Figure 5:
 Choose one random failure time using the Weibull
parameters for each design. For each design, compare
the failure time to the burnin time and, if either
failure time is less than the burnin time, add 1 to
the value of the "A and/or B Does Not Survive Burn
In" storage block.
 Subtract the failure time for Design B from Design
A. Use this difference to determine which design lasts
longer.
 If the difference is positive, Design A outlasts
Design B. Add 1 to the value of the "A Outlasts
B" storage block.
 If the difference is negative, Design B outlasts
Design A. Add 1 to the value of the "B Outlasts
A" storage block.
 If the difference is zero, Designs A and B fail
at the same time. Add 1 to the value of the storage
block labeled "A Equal to B and Greater Than Burn
In Time."
Figure 5  Comparison Wizard lower
limit simulation flowchart
Finally, Joe constructed a table of results from the
simulation and the Comparison Wizard, as shown in Table
7. The agreement between the results using the two methods
gives Joe confidence that now he has a firm understanding
of the use of the upper and lower limits in the Comparison
Wizard tool in Weibull++.
Table 7  Simulation and Comparison
Wizard results for burnin times from 1,000 to 10,000 hours,
including probabilities that at least one design fails before
the burn in time
Conclusion
In this article, we discussed the use of the upper and
lower limits in the Comparison Wizard in Weibull++
and illustrated the use of the limits by performing simulations
in RENO. We saw that using limits causes the Comparison
Wizard to exclude from the results some of the possible
outcomes for whether Design A will outlast Design B or Design
B will outlast Design A. Specifically, when using an upper
limit, the percentage of time where both Design A and Design
B are greater than the upper limit is excluded from the
calculations. When using a lower limit, the percentage of
time that either Design A or Design B is less than the lower
limit is excluded from the calculations.
