1. Analyze Wise, LLC
Forecasting Warranty Returns
Weibull Analysis

2. 2
Reasons for Warranty Analysis
 Actual warranty return data can be analyzed to forecast:
– The number of units that are expected to be returned at any given time
during the warranty period
 This forecast is useful to:
– Plan for repair center resources
– Manage customer communications/relationships
– Validate assumptions on Warranty Expenses/Reserves
– Facilitate decisions on currently deployed products
 This forecast is NOT useful to:
– Measure the “quality” of recent months of product shipments

3. 3
Question: How Many RMA Returns?
 Theory: Past return history can be used to
predict future returns (for a population or
failure mode(s))
– Methodology: Statistical Warranty Forecasting
using a failure time distribution
1. Regress time to failure data to find an model w/
good fit
2. Use the model to predict out future time periods
– Assumptions:
• Failure Rate is not constant over time
• Past customer behavior is representative of future
behavior
• Failed units are replaced with new units with similar
field quality
• Lag time to install & use is negligible
0.00%
0.05%
0.10%
0.15%
0.20%
0.25%
0 50 100 150 200 250 300 350
P(Failure)
Time
Probability of Failure at a given value of Time
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0 50 100 150 200 250 300 350
%Failed
Time
Cummulative % of Failures over Time

4. 4
Why use a forecasting model?
 Smooth-out warranty return time distributions for easy/accurate
comparison with a goal curve
 Results in an equation that will allow forecast of future warranty
costs
 The failure distribution, f(t), can be described with a few
parameters
– i.e.
• a normal distribution can be described with mean & standard deviation
• a exponential distribution can be described with a rate
• a Weibull distribution can be described with shape & scale

5. 5
Failure distribution & prediction terms
 Typically, “Return Rate” or “Failure Rate” is used as a
parameter to describe failure distributions
– Often these terms imply constant failure rate
– Most products do NOT have constant failure rates
 “Hazard Rate”, h(t) is the Function that describes the
“instantaneous failure rate over time”
– Represents the likelihood to fail in the next instant given that it hasn’t
failed yet
h(t) = Hazard Rate
f(t) = PDF or Failure Function. Likelihood of a failure at this point in time (t)
F(t) = Cumulative Failure Distribution. Probability of failure before time t
R(t) = Reliability Function. Probability of no failure before time t

6. 6
Typical Warranty Forecasting Models
 Regression Distribution options
– Constant Hazard Rate: F(t) = Exponential Distribution
– Linear Hazard Rate: F(t) = Rayleigh Distribution
– Variable Hazard Rate: F(t)= Weibull Distribution
• Weibull is a flexible life model that can be used to characterize failure
distributions in all three phases of the bathtub curve

7. 7
Life Data Analysis – 2 easy steps
1. Obtain Time-To-Failure Data
2. Perform regression to choose best fit model & estimate
parameters (Using a statistical software package of your choice)
Common Distributions in Reliability
– Weibull
– Exponential
– Gamma
– Loglogistic

8. 8
Step 1: Obtain Time-To-Failure Data
Historical data is formatted in a standard “Nevada” Chart
 “2435 units shipped in May-10; 1 returned in Jun-10, 1 in Jul-10, 0 in Aug-10...
 “1113 units shipped in Jun-10; 8 returned in Jul-10, 1 in Aug-10, 4 in Sep-10…”
Return Month

9. 9
Time-To-Failure Diagonals
 Lowest diagonal = Units That Failed after 1 month in field
– 1+8+1+1+33+0+0+0 = 44
 Next diagonal = Units That Failed after 2 months in field
– 1+1+1+1+51+1+3+0 = 59
 Etc….

10. 10
Censored Data
Assuming the most recent data includes up to Jan-11
 Units That Survived 8 Months
– 2435-1-1-0-0-0-1-0-0= 2432
 Units That Survived 7 months
– 1113-8-1-4-1-2-1-0= 1096
 Etc….
#
Shipped

11. 11
Step 2: Using a statistical package…
Input historical data for Time-To-Failure and total surviving (Censored)
for each time frame. Then find best fit distribution.

12. 12
Weibull Distribution Functions
 pdf = probability density function.
– Likelihood of a failure at this point in time (t)
 cdf= cumulative distribution function.
– Probability of failure before time t
– “Area Under the curve” of the pdf
 β = shape parameter
 ŋ = scale parameter

13. 13
Using the Weibull cdf & conditional
probability to forecast future returns
From Ship
Month May
2010
F(1/8) = 1 - R( 1+ 8)
R(8)
F(1/8) = 1 - R(9)
R(8)
= 1- e-(9/459)1.2
e-(8/459)1.2
2432*.001054= 2 Returns
Forecast for
Feb 2011
“We expect 2 returns during Feb-11 that were manufactured in May-10”

14. 14
Repeat for the next month of manufacture…
For Ship Month
Jun 2010
F(1/7) = 1 - R( 1+ 7)
R(7)
F(1/7) = 1 - R(8)
R(7)
= 1- e-(8/459)1.2
e-(7/459)1.2
1096*.001025 = 1 ReturnForecast for
Feb 2011
“We expect 1 return during Feb-11 that was manufactured in Jun-10”

15. 15
Repeat for each Ship Month & Return Month
Return Month
Ship Month Jan-11 Feb-11 Mar-11 Apr-11 May-11 Jun-11 Jul-11 Aug-11 Sep-11
May-10 2 3 3 3 3 3 3 3 3
Jun-10 1 1 1 1 1 1 1 1 1
Jul-10 5 5 5 5 5 5 5 5 6
Aug-10 13 13 14 14 15 15 15 16 16
Sep-10 14 15 15 16 16 17 17 17 18
Oct-10 9 10 11 11 11 12 12 12 13
Nov-10 7 8 8 9 9 9 10 10 10
Dec-10 10 12 13 13 14 15 15 16 16
62 66 69 72 74 76 78 80 82

16. 16
How good is the forecast?
 In this real-world case, within +/- 1%; enabling sound assessment of
warrant reserve and supporting the investment in corrective action*
*counts on vertical axis hidden per client request

17. 17
Q&A
 Weibull is one of the most popular distribution for reliability testing, but there are
others. Did we review analysis using other distributions?
– Yes – A two-parameter Weibull is the simplest distribution that fits this data, but Minitab checks a
dozen by default.
 For Weibull, how did we derive the parameters we are using.
– Distribution ID & regression using Minitab analysis for all return data history for this product.
 For analysis, what is confidence level around the results.
– Confidence Interval around each forecast point is provided in the Minitab analysis. R-square value
for the previous chart was .98 --- this is an unusually good fit. Your results may vary due to failure
mode(s), manufacturing variability and use characteristics of your product.
 What does this data mean?
– The return pattern is higher than the planned target of .x% per year failure goal.
 How can this be used?
– The equation will predict the number of returns across any given time period; so resource needs,
such as those for analysis & repair, can be forecast.
– Any proposed actions to address returns can be evaluated based on trustworthy forecast numbers.