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<noinclude>{{Banner Weibull Examples}}
<noinclude>{{Banner Weibull Examples}}
''This example appears in the [[Warranty_Data_Analysis#Warranty_Analysis_.28Usage_Format.29|Life Data Analysis Reference book]]''.</noinclude>
''This example appears in the [https://help.reliasoft.com/reference/life_data_analysis Life data analysis reference]''.


'''Warranty Analysis Usage Format Example'''


Suppose that an automotive manufacturer collects the warranty returns and sales data given in the following tables. Convert this information to life data and analyze it with a suitable distribution.
</noinclude>'''Warranty Analysis Usage Format Example'''


Suppose that an automotive manufacturer collects the warranty returns and sales data given in the following tables. Convert this information to life data and analyze it using the lognormal distribution.


{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+ '''Quality In-Service Data'''
|+ '''Quality In-Service Data'''
|-
|-
Line 37: Line 37:
| 30||Nov-10
| 30||Nov-10
|-
|-
|
|}
|}




{| border="1" class="wikitable" style="margin: 1em auto 1em auto"
{| border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+ '''Quality Return Data'''
|+ '''Quality Return Data'''
|-
|-
| align="center" style="background:#f0f0f0;"|'''Quantity Returned'''
| align="center" style="background:#f0f0f0;"|'''Quantity Returned'''
| align="center" style="background:#f0f0f0;"|'''Usage at Return Date'''
| align="center" style="background:#f0f0f0;"|'''Usage at Return Date'''
| align="center" style="background:#f0f0f0;"|'''Return Date'''
| align="center" style="background:#f0f0f0;"|'''Date In-Service'''
|-
|-
| 1||9072||Dec-09
| 1||9072||Dec-09
Line 76: Line 75:
| 1||3250||Nov-10
| 1||3250||Nov-10
|-
|-
|
|}
|}


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'''Solution'''
'''Solution'''


Create a warranty analysis folio and select the usage format. Enter the data from the tables in the '''Sales''', '''Returns''' and '''Future Sales''' sheets. On the control panel, set the '''End of Observation Period''' 12/1/2010. Analyze the data using the 2P-Weibull distribution and RRX analysis method. The parameters are estimated to be beta = 1.315379 and eta = 102,381.486165.  
Create a warranty analysis folio and select the usage format. Enter the data from the tables in the '''Sales''', '''Returns''' and '''Future Sales''' sheets. The warranty data were collected until 12/1/2010; therefore, on the control panel, set the '''End of Observation Period''' to that date. Set the failure distribution to '''Lognormal''', as shown next.


[[Image:Usage In-Service Weibull Data.png|center|500px| ]]


[[Image:Usage In-Service Weibull Data.png|thumb|center|500px| ]]
In this example, the manufacturer has been documenting the mileage accumulation per year for this type of product across the customer base in comparable regions for many years. The yearly usage has been determined to follow a lognormal distribution with <math>{{\mu }_{T\prime }}=9.38\,\!</math>, <math>{{\sigma }_{T\prime }}=0.085\,\!</math>. The Interval Width is defined to be 1,000 miles. Enter the information about the usage distribution on the Suspensions page of the control panel, as shown next.


The return data (accumulated mileage at time of failure and the date the failed unit was put in-service ( <math>DIS</math> )) is shown in the next figure.
[[Image:Specify Usage Distribution.png|center|250px| ]]


[[Image:Usage Return Weibull Data.png|thumb|center|500px| ]]
Click '''Calculate''' to analyze the data set. The parameters are estimated to be:


The above figure also shows the calculation End Date ( <math>ED</math> ), which can be set on the Analysis page of the Control Panel. In this example, the warranty data were collected until 12/1/2010.
::<math>\begin{align}
  & {{\mu }_{T\prime }}= & 10.528098 \\
& {{\sigma }_{T\prime }}= & 1.135150 
\end{align}\,\!</math>


The user has the option to input either a constant value that describes an average of usage or a distribution with intervals to be considered in the analysis. In this example, the manufacturer has been documenting the mileage accumulation per year for this type of product across the customer base in comparable regions for many years. The yearly usage has been determined to follow a lognormal distribution with  <math>{{\mu }_{T\prime }}=9.38</math> ,  <math>{{\sigma }_{T\prime }}=0.085</math> . Graphically, the distribution looks as follows.
The reliability plot (with mileage being the random variable driving reliability), along with the 90% confidence bounds on reliability, is shown next.
 
[[Image:Usage lognormal pdf.png|thumb|center|500px| ]]
 
 
The usage distribution is specified on the Usage page of the Control Panel, as shown next.
 
[[Image:Specify Usage Distribution.png|thumb|center|250px| ]]
 
Note that the Interval Width was defined to be 1000 miles (as seen in the above figure). This interval facilitates the calculations of mileage usage for the suspensions in the data set.
 
In addition to 14 failures, the data set contains 212 suspensions spread according to the defined usage distribution. This data set contains 12 groups of suspensions that have the same age. To illustrate the analysis procedure explained above, let us use one suspension group, the first group. The first group contains 9 units that went in service on a given date; 1 unit failed from that batch. Therefore, 8 suspensions have survived from December 2009 until the beginning of December 2010, a total of 12 months. The calculations are summarized as follows.
 
[[Image:Usage Suspension Allocation.PNG|center|500px| ]]
 
The two columns on the right constitute the calculated suspension data (number of suspensions and their usage) for the first group. The calculation in the above table is then repeated for each of the remaining 11 suspension groups in the data set. We then add the resulting data and the failure data and proceed with estimating a failure distribution model.
 
The data set that is used to perform the estimation of the failure distribution model can be extracted and displayed by selecting Transfer Life Data to New Folio from the Data menu. The failures and suspensions data set, as presented in the Standard Folio, is partially shown next. (Note that the Folio contains 66 rows.)
 
[[Image:Usage Example Weibull Std Folio Data.png|thumb|center|500px| ]]
 
The entered sales and return data set, the usage distribution and the interval width in the Warranty Folio can be used to derive a failure distribution. The manufacturer assumes that the observed failures are typically modeled by a lognormal distribution. The parameters of the distribution are estimated (by clicking on the Calculate button) to be:


::<math>\begin{align}
[[Image:Usage Example Reliability Plot.png|center|600px| ]]
  & {{\mu }_{T\prime }}= & 10.4747 \\
& {{\sigma }_{T\prime }}= & 1.1159 
\end{align}</math>


The reliability plot (with mileage being the random variable driving reliability), along with the 90% confidence bounds on reliability, is shown next.
In this example, the life data set contains 14 failures and 212 suspensions spread according to the defined usage distribution. You can display this data in a standard folio by choosing '''Warranty > Transfer Life Data > Transfer Life Data to New Folio'''. The failures and suspensions data set, as presented in the standard folio, is shown next (showing only the first 30 rows of data).


[[Image:Usage Example Reliability Plot.png|thumb|center|500px| ]]
[[Image:Usage Example Weibull Std Folio Data.png|center|550px| ]]

Latest revision as of 18:55, 18 September 2023

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This example appears in the Life data analysis reference.


Warranty Analysis Usage Format Example

Suppose that an automotive manufacturer collects the warranty returns and sales data given in the following tables. Convert this information to life data and analyze it using the lognormal distribution.

Quality In-Service Data
Quantity In-Service Date In-Service
9 Dec-09
13 Jan-10
15 Feb-10
20 Mar-10
15 Apr-10
25 May-10
19 Jun-10
16 Jul-10
20 Aug-10
19 Sep-10
25 Oct-10
30 Nov-10


Quality Return Data
Quantity Returned Usage at Return Date Date In-Service
1 9072 Dec-09
1 9743 Jan-10
1 6857 Feb-10
1 7651 Mar-10
1 5083 May-10
1 5990 May-10
1 7432 May-10
1 8739 May-10
1 3158 Jun-10
1 1136 Jul-10
1 4646 Aug-10
1 3965 Sep-10
1 3117 Oct-10
1 3250 Nov-10


Solution

Create a warranty analysis folio and select the usage format. Enter the data from the tables in the Sales, Returns and Future Sales sheets. The warranty data were collected until 12/1/2010; therefore, on the control panel, set the End of Observation Period to that date. Set the failure distribution to Lognormal, as shown next.

Usage In-Service Weibull Data.png

In this example, the manufacturer has been documenting the mileage accumulation per year for this type of product across the customer base in comparable regions for many years. The yearly usage has been determined to follow a lognormal distribution with [math]\displaystyle{ {{\mu }_{T\prime }}=9.38\,\! }[/math], [math]\displaystyle{ {{\sigma }_{T\prime }}=0.085\,\! }[/math]. The Interval Width is defined to be 1,000 miles. Enter the information about the usage distribution on the Suspensions page of the control panel, as shown next.

Specify Usage Distribution.png

Click Calculate to analyze the data set. The parameters are estimated to be:

[math]\displaystyle{ \begin{align} & {{\mu }_{T\prime }}= & 10.528098 \\ & {{\sigma }_{T\prime }}= & 1.135150 \end{align}\,\! }[/math]

The reliability plot (with mileage being the random variable driving reliability), along with the 90% confidence bounds on reliability, is shown next.

Usage Example Reliability Plot.png

In this example, the life data set contains 14 failures and 212 suspensions spread according to the defined usage distribution. You can display this data in a standard folio by choosing Warranty > Transfer Life Data > Transfer Life Data to New Folio. The failures and suspensions data set, as presented in the standard folio, is shown next (showing only the first 30 rows of data).

Usage Example Weibull Std Folio Data.png
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