Template:Example: Warranty Analysis Prediction Example

Warranty Analysis Prediction Example

Using the data from Example 1, predict the number of warranty returns for October for each of the three shipment periods.

Solution

We use the Weibull parameter estimates found in Example 2 to determine the conditional probability of failure for each shipment time period, and then multiply that probability with the number of units that are at risk for that period as follows. The equation for the conditional probability of failure is given by:

[math]\displaystyle{ Q(t|T)=1-R(t|T)=1-\frac{R(T+t)}{R(T)} }[/math]

For the June shipment, there are 89 units that have successfully operated until the end of September ( [math]\displaystyle{ T=3 }[/math] [math]\displaystyle{ months) }[/math] . The probability of one of these units failing in the next month ( [math]\displaystyle{ t=1 }[/math] [math]\displaystyle{ month) }[/math] is then given by:

[math]\displaystyle{ Q(1|3)=1-\frac{R(4)}{R(3)}=1-\frac{{{e}^{-{{\left( \tfrac{4}{6.70} \right)}^{2.49}}}}}{{{e}^{-{{\left( \tfrac{3}{6.70} \right)}^{2.49}}}}}=1-\frac{0.7582}{0.8735}=0.132 }[/math]

Once the probability of failure for an additional month of operation is determined, the expected number of failed units during the next month, from the June shipment, is the product of this probability and the number of units at risk ( [math]\displaystyle{ {{S}_{JUN,3}}=89) }[/math] or:

[math]\displaystyle{ {{\widehat{F}}_{JUN,4}}=89\cdot 0.132=11.748\text{, or 12 units} }[/math]

This is then repeated for the July shipment, where there were 134 units operating at the end of September, with an exposure time of two months. The probability of failure in the next month is:

[math]\displaystyle{ Q(1|2)=1-\frac{R(3)}{R(2)}=1-\frac{0.8735}{0.9519}=0.0824 }[/math]

This value is multiplied by [math]\displaystyle{ {{S}_{JUL,2}}=134 }[/math] to determine the number of failures, or:

[math]\displaystyle{ {{\widehat{F}}_{JUL,3}}=134\cdot 0.0824=11.035\text{, or 11 units} }[/math]

For the August shipment, there were 146 units operating at the end of September, with an exposure time of one month. The probability of failure in the next month is:

[math]\displaystyle{ Q(1|1)=1-\frac{R(2)}{R(1)}=1-\frac{0.9519}{0.9913}=0.0397 }[/math]

This value is multiplied by [math]\displaystyle{ {{S}_{AUG,1}}=146 }[/math] to determine the number of failures, or:

[math]\displaystyle{ {{\widehat{F}}_{AUG,2}}=146\cdot 0.0397=5.796\text{, or 6 units} }[/math]

Thus, the total expected returns from all shipments for the next month is the sum of the above, or 29 units. This method can be easily repeated for different future sales periods, and utilizing projected shipments. If the user lists the number of units that are expected be sold or shipped during future periods, then these units are added to the number of units at risk whenever they are introduced into the field. The warranty analysis tool in Weibull++ performs this type of analysis for you.

Note that Weibull++ can perform this step automatically by using the `Generate Forecast' functionality.