Analyze Wise, LLC

Forecasting Warranty ReturnsWeibull Analysis

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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

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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 50 100 150 200 250 300 3500.00%

0.05%

0.10%

0.15%

0.20%

0.25%

Probability of Failure at a given value of Time

Time

P(Fa

ilure

)0 50 100 150 200 250 300 350

0%10%20%30%40%50%60%70%80%90%

100%

Cummulative % of Failures over Time

Time

% F

aile

d

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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

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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 Ratef(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 tR(t) = Reliability Function. Probability of no failure before time t

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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

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Life Data Analysis – 2 easy steps1. Obtain Time-To-Failure Data2. 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

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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

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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….

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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

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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.

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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

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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 ReturnsForecast for Feb 2011

“We expect 2 returns during Feb-11 that were manufactured in May-10”

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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”

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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

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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

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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.