Predicting Warranty Returns in Weibull++ 7
Performing warranty return predictions can be a very useful analysis tool
when trying to budget for warranty costs or to prepare for a required warranty
pool. Usually when dealing with warranty data, one encounters data in the form
of sales per time period (which can be in weeks, months, quarters, etc.) and
subsequent returns per time period.
Weibull++ 7 can be used to
transform this type of data into traditional time-to-failure data so that a
failure distribution can be fitted and then predictions can be performed. In
this article we present an example of how Weibull++ 7 can be used to
perform those predictions and also how to report the results of the analysis
through the use of different plots.
Example
Suppose that we want to predict future warranty returns of our product over
the next year. The warranty period that we offer is 12 months and the data that
we have already collected is presented in the next table.
Furthermore, with the help of our marketing department we are able to make
the assumption that we will have sales of 3,000 units per month over the next
year. To enter this data in Weibull++ 7 we need to add a Warranty
folio (Project > Add Specialized Folio > Add Warranty) and choose
a Nevada format since the data is in the form of sales per period and associated
returns at each period. Figure 1 shows the setup for the Warranty folio given that
we have sales data for 10 months starting in March 2010, return data for 9 months
starting in April 2010, and we also want to include future sales for the
next 12 months.
Figure 1: Warranty Folio Setup
Now we can simply input the sales and return information into the data entry
sheets that the software creates. Figure 2 shows the sales data. Note that
the cells that are colored yellow indicate future sales.
Figure 2: Sales data in Weibull++ 7
Figure 3 shows the returns data. Note that the returns data corresponds
only to actual sales periods and not to future sales.
Figure 3: Returns data in Weibull++ 7
Weibull++ 7 can automatically transform this
data to time-to-failure data and fit a distribution. (For more details on
this process please refer to [1].) As it can be seen in
Figures 2 and 3, we have chosen to fit a 2-parameter Weibull distribution
using MLE as the parameter estimation method due to the large number of
suspended data points in our data set. Once we have obtained a failure
distribution, we can use the concept of conditional reliability in order
to calculate the probability of failure for the remaining units after each
sales period then make predictions of future returns. The equation
of the conditional probability of failure is:
Where Q(t|T) is the unreliability of a unit for
the next t months given that it has already
operated successfully for T months.
For example, if we look at the units that were shipped on March 2010, given
that those units have been out in the field for 9 months, the probability that
one of those units will fail in January 2011 is:
Out of the 1,623 units that were sold in March 2010, 1,543 units are still
out in the field as of the end of December 2010. That is calculated by
subtracting the total number of returns of the March shipment from the
number of sales for that month. Given that, the total number of returns that
we expect to see in January 2011 is:
Where NMar is the number of units out
of the March shipment that are still out in the field.
We could follow the exact same process to calculate the expected returns for
the other time periods. Furthermore, we can add some confidence bounds to
the expected number of failures. In order to do that we would use Eqn. (1)
again. However, instead of using the median estimate of the conditional probability
of failure, we would use the upper or lower confidence bound at a given
confidence level.
Now let us see how Weibull++ 7 can perform those calculations for
us. In order to make a return prediction we’ll need to generate a
forecast. Figure 4 shows the Forecast Setup window (Data > Generate
Forecast).
Figure 4: Weibull++ Forecast Setup window
As we can see, the forecast would start at January of 2011 and would be
for 12 months since this is our requirement. We also have the option of
calculating the upper or lower bounds of expected failures at a given
confidence level. Using the options shown in Figure 4, we can obtain a
warranty forecast as shown in Figure 5.
Figure 5: Generated warranty forecast
As we can see in Figure 5, the expected number of failures in January 2011
out of the March 2010 shipment is 26, which is the same number that we
calculated using Eqn. 1. Also notice that we have specified that our
warranty length is 12 months (in the control panel on the right side of
the window) so
the software won’t generate a forecast for time periods that are out of
warranty and are therefore of no interest to us.
Weibull++ 7 also gives us the option to generate a number of plots
in order to report the results of this analysis. Figure 6 shows the expected
number of failures for each month of 2011 along with the upper and
lower 1-sided 90% confidence bounds.
Figure 6: Expected number of failures
Figure 7 shows the cumulative expected number of failures for each month
of 2011 along with the upper and lower 1-sided 90% confidence bounds.
Figure 7:
Cumulative expected number of failures
Finally, Figure 8 shows the expected failures for each month of 2011 as
a percentage of the total number of units that are still out in the field
along with the upper and lower 1-sided 90% confidence bounds. For example,
for January 2011 this percentage represents the total number of returns that
were predicted for that month divided by the total number of units that are
still in the field and corresponds to sale periods that had returns in
January 2011 (in this case, March through December of 2010).
Figure 8: Expected failures as
a percentage
Conclusions
In this article we saw how we can perform warranty return predictions in
Weibull++ 7 based on warranty data that is in the form of number of
sales and number of returns for each time period. By creating a
Warranty folio with the Nevada format in Weibull++ 7 we were able to
fit a failure distribution to our data and predict future returns using the
concept of the conditional probability of failure. Finally we saw different
plots that can be used to report the results of the analysis.
References
[1] ReliaSoft Corporation, Life Data Analysis
Reference, Tucson, AZ: ReliaSoft Publishing, 2008.
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