Introduction to Weibull Analysis Ver4

8/4/2019 Introduction to Weibull Analysis Ver4

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Product Quality Summit

January 24-28, 2005

INTRODUCTION TO

WEIBULL ANALYSISJanuary 2007


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Table Of Contents

Introduction To Weibull Distribution Weibull And Weibull Parameters

Weibull Probability Plots

Incomplete Data Time Methods And Data Dry Up

Weibull Estimation Methods

Bad Data and Bad Weibulls Weibull Process Flow

Determining A Significant Difference


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Learning Objectives

Be able to fit a Weibull distribution to a set of data. Predict failures in a population based on Weibull. Interpret what the Weibull parameter values tell you

about the data. Understand how to handle incomplete data, which time

method to use and when to use data dryup. Understand how to select and when to use each

Weibull estimation method.

Learn to identify Bad Weibulls, Bad Data, &

Uncertainties. Be able to determine if one population failure rate is

statistically different than another.


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Whats In It For You

Be able to predict failure rate with extremelysmall sample sizes.

Identify possible root causes very quickly.

Ability to identify bad data.

Detecting a difference between distributionswith a given confidence level.

Become more proficient with another statistical

distribution with wide applicability.


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Waloddi Weibull

1887- 1979

He invented the Weibull distribution

in 1937. He delivered a paper in 1951

in the United States on the distribution

and included 7 examples on its use.

These examples ranged from strength of

steel to height of adult males in the

British Isles.

The Weibull distribution is by far the

world's most popular statistical model

for life data. It is also used in many other

applications, such as weather

forecasting and fitting data of all kinds.It may be employed for engineering

analysis with smaller sample sizes than

any other statistical distribution.

Waloddi Weibull


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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 20 40 60

X value

ProbabilityDensityFunction

3 Ways to View a Statistical Distribution

1. Probability Density Function (PDF) Total Areaunder PDF curve

equals 1.0


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2. Cumulative Distribution Function (CDF)

CDF is the

Integral of thePDF

0%

20%

40%

60%

80%

100%

120%

0 20 40 60

X value

CumulativeDistrib

utionFuntion


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3. Failure Rate (or Hazard Function) Failure Rate is thePDF/(1-CDF)

0

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

0 20 40 60

X value

FailureR

ate

PDF, CDF, &

Hazard Function

are 3 ways to

view the same

thing


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0

0

1

0

0

o X-

X-t

X-

X-t

X-f(t) Exp

0

0

T

X X-

X-t-1dtf(t)F(t)

0

Exp

1

0

0

o X-

X-t

X-

F(t)-1

f(t)h(t)

0

0

T

XX-

X-tdth(t)H(t)

0

Cumulative

Distribution

Function

(like Weibull fit)

Probability

DensityFunction

Failure Rate or

Hazard Function

(like DRF fit)

Cumulative

Hazard

Function

Equations that Define the Failure Data


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Theta

Characteristic Life

Beta

Shape (slope) Parameter

63.2%

Theta and Beta


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Theta: In terms of y=mx+b, Theta is like b where theline crosses the y-axis, but Theta is the hours (or miles)where the best fit line crosses the 63.2 percentile.

Beta: Same as slope (rise over run) just like m iny=mx+b.

Theta and Beta


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Weibull Parameters Characteristic Life

Characteristic Parameter ( = Theta) is the life for 63.2% of the population (in terms of

number of hours, cycles, mileage or strength, etc.)

is the pivot point for the distribution and remains sofor any value or change in

It is analogous to the mean in a Normal distribution

0.6321-1t

-1CDF

ExpExp


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Weibull Parameters Shape (or Slope)

Shape Parameter ( = Beta) describes the shape of the distribution and in turn

indicates the type of problems inherent in the population

< 1 means there is a decreasing failure rate(declining DRF vs operating hours)

= 1 means there is a constant failure rate(constant DRF vs operating hours)

> 1 means there is an increasing failure rate(increasing DRF vs operating hours)


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Bathtub Curve

< 1Decreasing Failure Rate

= 1Constant Failure Rate

> 1Increasing Failure Rate

Useful Life


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0

5

10

15

20

25

30

35

NumberofFailures

inanInterval.

Useful Life

WearOut

0 20 201 1000

VEHR

Operating Hours

InfantMortality

21 200

mDRF = Avg Of [VEHR + DRF1 + DRF2]

DRF

1DRF

2

DRF

3 & 4 & 5

| mDRF Range |

Bathtub Curve

(Failures/x

hrs)


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Product VEHR DRF1 DRF2 DRF3 DRF4 DRF5

All Product Not Listed Below (Hours) 0-20 21-200 201-1000 1001-2000 2001-5000 5001-10000Medium Duty Truck Engines (Miles) 0-500 501-5000 5001-25000 25001-50000 50001-125000 125001-250000

Heavy Duty Truck Engines (Miles) 0-1000 1001-10000 10001-50000 100000 100001-250000 250001-500000

Standby Gensets (Hours) 0-10 11-100 101-300 301-600 601-1500 1501-3000

BCP (Hours) 0-10 11-100 101-500 501-1000 1001-2500 2501-5000

Marine Engines(Pleasure)(Hours) 0-20 21-100 101-500 501-1000 1001-2500 2501-5000

Agriclture Product (Hours) 0-20 21-100 101-500 501-1000 1001-2500 2501-5000

Utility Compactors(CB214-335)(Hours) 0-20 21-100 101-500 501-1000 1001-2500 2501-5000

BCP Work Tools (Hours) 0-20 21-200 201-1000 1001-2000 2001-5000 5001-10000

All Other Commercial Engines (Hours) 0-20 21-200 201-1000 1001-2000 2001-5000 5001-10000

Off-Highway Tractors(768-776)(Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Small Off-Highway Trucks (769-775)(Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Wheel Loader(988-992) (Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Wheel Dozers(834-854)(Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Motor Grader 14H (Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Tractor Scrapers(631-657) (Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Large Track Tractor (D8-D9) (Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Large Excavators (345-385) (Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

Shovels (5080-5090) (Hours) 0-20 21-200 201-2000 2001-5000 5001-10000 10001-20000

| mDRF Range |

DRF Range Definitions


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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Repairsper100H

oursofUse

50 100 150 200 250 700 800 900 5000 6000 7000 8000 9000 10,000 12,000

Life

0.02 DRF

6,000 Hr Life

0.06 DRF

12,000 Hr Life

Reliability vs. Durability Bathtub Curves


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< 1.0Infant Mortality

Slope Examples(applicable for Weibayes Method)

Leaks, loose bolts, quality & assembly problems,inadequate burn-in

Chance failures (human & maintenance errors,foreign object damage, multi-part system or

multiple failure modes)

Design flaws, fatigue, pitting, spalling, corrosion,erosion, wear, excessive cycles

Material brittle/worn out, severe pitting/corrosion,design obsolescence, numerous critical partsfailing

~ 1.0Random

~ 1-4Wear out

4Old age


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Classic Mature Weibull Plot

Wearout ( Slope > 1)

Useful Life ( Slope = 1)

Infant Mortality( Slope < 1)

Three failure modes

potentially evidenton this part


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Weibull Distribution

It has the ability to fit different distributions, i.e.,Normal, Lognormal and others

= 1.0: identical to the exponential distribution

= 2.0: identical to the Rayleigh distribution

= 2.5: approximates the lognormal distribution

= 3.6: approximates the normal distribution

= 5.0: approximates the peaked normal distribution


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0

0.0002

0.0004

0.0006

0.0008

0.001

0 1000 2000 3000 4000


PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0 1000 2000 3000 4000FailureRate(orHazardFun

ction)

h(t)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 1000 2000 3000 4000

Cumulati

veHazardFunction

H(t)

Beta = 1 Theta = 1000 Xo = 0Baseline

When Beta = 1, the Failure Rate is Constant


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Beta = 2 Theta = 1000 Xo = 0

0

0.0002

0.0004

0.0006

0.0008

0.001

0 1000 2000 3000 4000


PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.0070

0.0080

0.0090


ction)

h(t)

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0 1000 2000 3000 4000

Cumulati

veHazardFunction

H(t)

When Beta > 1, the Failure Rate continually increases


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Beta = 3 Theta = 1000 Xo = 0

0

0.0002

0.0004

0.0006

0.0008

0.0010.0012

0.0014

0 1000 2000 3000 4000

ProbabilityDensityFun

ction

PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

CumulativeDistribution

Function

CDF

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

0 1000 2000 3000 4000Fa

ilureRate(orHazardFun

ction)

h(t)

0.0

10.020.0

30.0

40.0

50.0

60.0

70.0

0 1000 2000 3000 4000

Cumulat

iveHazardFunction

H(t)

The Failure Rate is only linear if Beta = 1 or Beta = 2


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Beta = 5 Theta = 1000 Xo = 0

0

0.0005

0.001

0.0015

0.002

0 1000 2000 3000 4000


PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.0000

0.2000

0.4000

0.6000

0.8000

1.00001.2000

1.4000


ction)

h(t)

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

0 1000 2000 3000 4000

Cumulati

veHazardFunction

H(t)

The larger Beta, the narrower the life variation

B 1 Th 1000 X 0


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0

0.0002

0.0004

0.0006

0.0008

0.001

0 1000 2000 3000 4000


PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012


ction)

h(t)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 1000 2000 3000 4000

CumulativeHazardFunction

H(t)



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Beta = 0.5 Theta = 1000 Xo = 0

0

0.0005

0.001

0.0015

0.002

0 1000 2000 3000 4000


PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025


ction)

h(t)

0.0

0.5

1.0

1.5

2.0

2.5

0 1000 2000 3000 4000


H(t)

When Beta < 1, the Failure Rate continually decreases.


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Beta = 0.2 Theta = 1000 Xo = 0

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0 1000 2000 3000 4000


PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025


ction)

h(t)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 1000 2000 3000 4000


H(t)

The smaller Beta, the wider the life variation.

B t 1 Th t 1000 X 0


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0

0.0002

0.0004

0.0006

0.0008

0.001

0 1000 2000 3000 4000


PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012


ction)

h(t)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 1000 2000 3000 4000


H(t)



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Beta = 1 Theta = 10,000 Xo = 0

0

0.00002

0.00004

0.00006

0.00008

0.0001

0.00012

0 1000 2000 3000 4000


ction

PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000

Cumula

tiveDistribution

Function

CDF

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012


ction)

h(t)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 1000 2000 3000 4000


H(t)

For the same Beta, increasing Theta increases the variation


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Weibull Parameters - Location

Location Parameter X0

Optional(for 3 Parameter Weibull onlyrarely used)

X0 is used only when the life of a product starts atsome designated number of hours of operation suchas with fatigue related data.

It is not used when the starting point is zero andgreatly simplifies the use of Weibull distribution.

B t 1 Th t 1000 X 0


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0

0.0002

0.0004

0.0006

0.0008

0.001

0 1000 2000 3000 4000


ction

PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000


Function

CDF

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012


ction)

h(t)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 1000 2000 3000 4000


H(t)



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Beta = 1 Theta = 2000 Xo = 1000

0

0.0002

0.0004

0.0006

0.0008

0.001

0 1000 2000 3000 4000


ction

PDF

0%

20%

40%

60%

80%

100%

0 1000 2000 3000 4000


Function

CDF

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

0 1000 2000 3000 4000Fa

ilureRate(orHazardFun

ction)

h(t)

0.0

0.51.0

1.5

2.0

2.5

3.0

3.5

0 1000 2000 3000 4000


H(t)

Everything is zero for T < Xo


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Probability Plot = CDF

0%

20%

40%

60%

80%

100%

120%

0 20 40 60

X value

CumulativeDistributionFuntion

Normal Probability Plot

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

0 20 40 60

Stdevfromt

heMean

Linear Probability Plot

A nonlinear transformation is used to linearize the CDF


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Weibull Probability Plot

X = Horizontal Axis = LN(X)

Y = Vertical Axis = LN(LN(1/(1-F(x))))

This transformation makes data with a

Weibull Distribution plot as a straight line


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Weibull Plot

00 ,])([exp1)( xx

xxxF


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Weibull Plot with Parameters

X0

63.2

When 63.2% of allunits fail.


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Weibull Plot and B-Life

B1

Life=65hrs

1

0

1

50

B10

Life

=280hrs

B50

Life

=1000hrs

Product or componentdurability level is

commonly defined byits Bxx Life level.


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Weibull Predictive Stability

1 year after fix: 2 years after fix: 3 years after fix:

Weibull gives stable failure rate projection over time.

1.5% FR 1.7% FR 1.9% FR


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Incomplete Data Reliability analysis concentrates on describing the distribution of time-

to-failure for the entire population.

Collecting data on failed units alone is not adequate to judge reliability. Question: If failure data shows our population has 10 failures under 2000

hours, does this represent good or bad reliability?

Answer: We dont really know information about the rest of the

population (those that havent failed) is also needed. Consider the situation

of 10 failed units vs. 10,000 non-failed units. TAKE AWAY: Hours on failed units plus non-failed units defines

reliability. Reliability analysis needs to consider both.

Statistically we call a data set incomplete if it contains non-failedunits. For these units we have incomplete information on the hrs-to-

failure. We dont know their life only that it is longer than the currenthours.


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Analysis of Incomplete Data Sets

CPI projects usually have incomplete data, or datacontaining suspensions (accumulated hours on thenon-failed units).

Special methods exist for analyzing these data sets Weibull Regression fit methods (with Median or

Hazard ranking to handle suspensions)

Maximum Likelihood Estimation methods


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Serial Hours Serial Hours Serial Hours Serial Hours

Number Number Number Number

1 4GZ00093 22,726 44 4GZ00136 23,526 87 4GZ00179 29,973 130 4GZ00222 30,051

2 4GZ00094 37,023 45 4GZ00137 23,298 88 4GZ00180 30,209 131 4GZ00223 29,244

3 4GZ00095 35,586 46 4GZ00138 29,736 89 4GZ00181 30,538 132 4GZ00224 29,888

4 4GZ00096 32,039 47 4GZ00139 30,265 90 4GZ00182 30,253 133 4GZ00225 28,7685 4GZ00097 35,893 48 4GZ00140 32,802 91 4GZ00183 30,710 134 4GZ00226 31,796

6 4GZ00098 35,087 49 4GZ00141 32,589 92 4GZ00184 31,206 135 4GZ00227 31,112

7 4GZ00099 29,146 50 4GZ00142 31,593 93 4GZ00185 16,594 136 4GZ00228 31,357

8 4GZ00100 32,162 51 4GZ00143 31,111 94 4GZ00186 31,419 137 4GZ00229 33,069

9 4GZ00101 29,237 52 4GZ00144 30,863 95 4GZ00187 31,318 138 4GZ00230 15,301

10 4GZ00102 25,255 53 4GZ00145 30,416 96 4GZ00188 27,561 139 4GZ00231 28,699

11 4GZ00103 26,222 54 4GZ00146 30,502 97 4GZ00189 31,328 140 4GZ00232 28,724

12 4GZ00104 24,768 55 4GZ00147 30,855 98 4GZ00190 28,187 141 4GZ00233 27,705

13 4GZ00105 23,708 56 4GZ00148 29,951 99 4GZ00191 27,465 142 4GZ00234 28,590

14 4GZ00106 24,145 57 4GZ00149 30,146 100 4GZ00192 28,668 143 4GZ00235 28,627

15 4GZ00107 27,494 58 4GZ00150 29,134 101 4GZ00193 30,705 144 4GZ00236 28,758

16 4GZ00108 27,921 59 4GZ00151 29,147 102 4GZ00194 30,713 145 4GZ00237 25,776

17 4GZ00109 23,556 60 4GZ00152 30,382 103 4GZ00195 22,123 146 4GZ00238 25,478

18 4GZ00110 24,705 61 4GZ00153 26,776 104 4GZ00196 21,667 147 4GZ00239 25,637


21 4GZ00113 26,344 64 4GZ00156 29,803 107 4GZ00199 21,987 150 4GZ00242 25,431

22 4GZ00114 18,745 65 4GZ00157 29,609 108 4GZ00200 25,756 151 4GZ00243 25,169

23 4GZ00115 23,830 66 4GZ00158 26,147 109 4GZ00201 26,093 152 4GZ00244 25,232

24 4GZ00116 24,383 67 4GZ00159 24,476 110 4GZ00202 26,032 153 4GZ00245 26,318

25 4GZ00117 19,065 68 4GZ00160 30,004 111 4GZ00203 26,452 154 4GZ00246 29,932

26 4GZ00118 24,683 69 4GZ00161 28,516 112 4GZ00204 27,706 155 4GZ00247 29,746

27 4GZ00119 23,794 70 4GZ00162 22,378 113 4GZ00205 25,451 156 4GZ00248 31,462

28 4GZ00120 21,816 71 4GZ00163 22,898 114 4GZ00206 26,948 157 4GZ00249 30,532

29 4GZ00121 28,216 72 4GZ00164 22,320 115 4GZ00207 27,192 158 4GZ00250 26,830

30 4GZ00122 28,524 73 4GZ00165 30,626 116 4GZ00208 28,200 159 4GZ00251 25,987

31 4GZ00123 35,746 74 4GZ00166 30,342 117 4GZ00209 26,730 160 4GZ00252 23,385

32 4GZ00124 34,497 75 4GZ00167 31,610 118 4GZ00210 26,192 161 4GZ00253 19,921

33 4GZ00125 35,935 76 4GZ00168 19,584 119 4GZ00211 25,854 162 4GZ00254 18,786


36 4GZ00128 34,869 79 4GZ00171 29,459 122 4GZ00214 29,787 165 4GZ00257 23,724

37 4GZ00129 33,365 80 4GZ00172 29,584 123 4GZ00215 28,293 166 4GZ00258 22,329

38 4GZ00130 29,087 81 4GZ00173 30,860 124 4GZ00216 29,290 167 4GZ00259 30,180

39 4GZ00131 29,938 82 4GZ00174 25,602 125 4GZ00217 27,317 168 4GZ00260 23,006

40 4GZ00132 27,807 83 4GZ00175 26,833 126 4GZ00218 30,327 169 4GZ00261 28,666

41 4GZ00133 26,474 84 4GZ00176 28,849 127 4GZ00219 30,280 170 4GZ00262 24,345

42 4GZ00134 18,712 85 4GZ00177 27,582 128 4GZ00220 30,106 171 4GZ00263 24,051

43 4GZ00135 32,629 86 4GZ00178 30,324 129 4GZ00221 30,001 172 4GZ00264 25,179

Population Data

172 Machines

These are

SUSPENSIONS


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Nomenclature

Failures

Report of a failure on the part(s) under investigation Not all repairs reported are failures (nor same root-cause?)

Suspensions (or Successes)

# of non-failed machines under investigation (as defined inProblem Definition)

Can have a major impact on the analysis results (early or late?)

Population

The total number of machines being analyzed

(# of Suspensions + # of Failures = Population)

M h d Add F il & S i


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Methods to Address Failures & Suspensions

Time-to-Failure (default) product hours continue on after failure

Assumes the repair is the same as a product which didnt fail Use if multiple same parts on a machine (planet gears, injectors).

Use if repair is not better or worse than before failure

Light Bulb once failed, the product is removed Use to characterize time to initial failure

Use where repairs are not the same as factory-built (e.g. weld repairs) Use when dealer repair is permanent (e.g. factory assembly errors)

Use as a conservative method in any situation

Clock Reset once failed, product hours start over at zero Use for Component Replacements (component Weibull perspective)

Typically used for fatigue and wearout type failures

Use when repeat failures need to be included

Use with caution if slope is less than 1

E l f Add i F il & S i


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Example for Addressing Failures & Suspensions

A

B

C

No Failures

1 Failure

2 Failures

Start=0 Now

x1

x2 x3

Light Bulb

Anow,SBx1,FCx2,F

Time-to-Failure

Anow,SBx1,FBnow,SCx2,FCx3,FCnow,S

Clock Reset

Anow,SBx1,F

(Bnow- Bx1), SCx2,F

(CX3- Cx2), F(Cnow- Cx3), S2 Failures

1 Suspension 3 Failures, 3 Suspensions

3

machines

D D


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Data Dryup

Data Dryup is typically the machine hours after which we stop

hearing about failures (often near end of warranty coverage) Data Dryup will clip Suspension Hours and exclude Failures that

occur after the Data Dryup hours.

Data Dryup is applied to the data before the Time Method.

Data Dryup is used to limit the maximum machine operating

hours for predicting the L1 Failures. It does not impact theprediction of the L3 Failures.

Use Data Dryup if SIMS reports dont come in at higher

hours, or if data beyond a certain hour point is suspect (forexample, failure reports on a component beyond typical

overhaul hours or warranty period). Use caution if Data Dryup hours specified are less than the

typical warranty for that component.

Example for Addressing Failures &


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a p e o dd ess g a u es &Suspensions

A

B

C

No Failures

1 Failure

2 Failures

Start=0 Now

x1

x2 x3

Light Bulb

AT1,SBx1,FCx2,F

Time-to-Failure

AT1,SBx1,FBT1,SCx2,FCT1,S

Clock Reset

AT1,SBx1,F

(BT1- Bx1), SCx2,F

(CT1- Cx2), S

2 Failures

1 Suspension

3

machines

T1

Data Dryup

2 Failures, 3 Different Suspensions


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Effect of Suspensions on Weibull Analysis

A unit that has not failed by the failure mode in question

is a suspension or censored unit. (e.g, a bolt that fails inthe bolt head would be a suspension in a pull test forthread failures.)

An early suspension is one that was suspendedbefore the first failure time. A late suspension is

suspended after the last failure. Suspensions betweenfailures are called random or progressive suspensions. Early suspensions have negligible effect on the Weibull

plot.

Late suspensions have more significant effect, and may

reduce the slope . This particularly true when using MLE. Random or progressive suspensions increase the

characteristic life , but have little effect on the slope .


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Weibull Estimation Methods

There are 3 main Weibull analysis estimation methodsused at CAT Maximum Likelihood Estimation Method w or w/o RBA Median Rank Regression Method Weibayes Method (Weibull when is known)

There are several tools available at CAT & from thirdparties that perform these estimation methods. Notevery tool performs every method. Most Used: CPI, Web Weibull, Minitab, RWA Weibull,

Others: WATk, WidGet, Weibull ++, Excel


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Maximum Likelihood Estimation Method

The maximum likelihood Weibull analysis method consists of finding

the values for Beta & Theta which maximize the likelihood, ofobtaining & , given the observed data. The likelihood is expressedin Weibull probability density form. It is a function of the data and theparameters & .

This method works on most data sets & handles late

suspensions much better than median rank regression. Maximum Likelihood methods can also establish Weibull curves

for data sets with only 1 failure.

Caution: for small numbers of failures a Reduced Bias Adjustment(RBA) may need to be added to reduce the Beta.

Maximum Likelihood also contains a fully developed hypothesis testfor comparing baseline and after-fix populations.

Tools: CPI, Web Weibull, Minitab, RWA Weibull


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Median Rank Regression Method

The median rank regression method uses a best-fitstraight line, through the data plotted on Weibull paper,to estimate the Weibull parameters Beta & Theta. The best-fit line is found using the method ofleast squares.

This method works best on data sets with 10 or morefailures & is considered best practice by most.

This method also provides a graphical plot of the data.

Caution: This method doesnt handle large numbers of

late suspensions as well as the maximum likelihoodmethod.

Weibayes Method


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Weibayes Methodor Weibull when is Known

Weibayes is defined as Weibull analysis withan assumed Beta parameter.

Method was developed to solve problemswhen traditional Weibull analysis has largeuncertainties or cannot be used becausethere are no or few failures.

Weibayes offers significant improvements in

accuracy compared to small sample Weibullsif Beta is known.

Weibayes Method


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Weibayes Methodor Weibull when is Known (Cont.)

Weibayes method allows CPI Projects to calculatefailures, benefits, & CPI score at the earliest indicationsof a problem without having to fail a few more.

When using this method, typical questions to be raisedare How valid is the assumed slope?

How sensitive are the resulting failure intervals to differentpossible values of ?

With a redesign, what is the probability that a new failure modeis present?

Is the confidence in the improvement high enough for 6 Sigmawarranty benefits to be L1 or should they be L3?

B d W ib ll


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Bad Weibulls

Bi-slope Weibull (inaccurate fits) Dry-up (warranty report vs failure mode dryup) Batch Issues (non-uniform failure vs time basis) Wrong Population (mixed products/applications)

Wrong Population (varied part arrangements?) Missing Part Numbers (DTF search) Discussion of data problems and assumptions

Types of Bad Weibulls,


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Types of Bad Weibulls,

Bad Data, & Uncertainties

After the Weibull Model has been created it is veryimportant to make sure that it is reasonable.

Listed below are several warning signs to look for in themodel to make sure it is reasonable to use Suspect Outliers

Curved Weibulls

Type I

Type II

Data Dry-up

Steep Slopes Incomplete DataSuspensions

Predicted failures dont match Actual

S t O tli


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Suspect Outliers

Suspect Outliers are sometimes the1st or last data points in the data thatare suspected as invalid points.

The points may or may not beimportant to the life data analysis.

Review authenticity of SN claimdetails & hours and possible otherfailure mode differences.

Data is precious & should not berejected without sufficientengineering evidence.

In some cases statistics may beable to help in the investigation.

Time-to-Failure

ProbabilityCDF(%)

Time-to-Failure

ProbabilityCDF(%)

C d W ib ll T I


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Curved Weibulls: Type I

Multiple failure modes

(Most Likely Cause) Identify different modes

and separate into distinctWeibull analyses

Batch Problems Use data segmentation

techniques to handlebatch problems

Negative Start Time,

parts degrade beforebeing used (this is rare) Use a T0 adjustment

Negative t0or

classic Bi-Weibull

Time-to-Failure

ProbabilityCDF(%)

Negative t0or

classic Bi-Weibull

Time-to-Failure

ProbabilityCDF(%)

C d W ib ll T II


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Curved Weibulls: Type II

Batch Problems Use data segmentation

techniques to handle batchproblems

Warranty Report Dry-up Use Data Dry-up (data absence

after warranty period) Multiple failure modes

Identify different modes andseparate into distinct Weibullanalyses

Positive Start Time, Partscant fail until used later Use a T0 adjustment

Batch problemsor Positive t0

or Log Normal

Time-to-Failure

ProbabilityCDF(%)

Batch problemsor Positive t0

or Log Normal

Time-to-Failure

ProbabilityCDF(%)

C d W ib ll T II D t D U


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Curved Weibulls: Type II Data Dry-Up

Review Parts Sales

Does parts sales indicate thatfailures are still occurring, butnot being reported?

Identify population affectedand adjust suspension timesback in time to match end ofwarranty period

Rerun Weibull analysis withboth Median Rank &Maximum Likelihoodestimation methods

Compare results to partssales to see if Weibullanalysis is reasonable

Time-to-Failure

ProbabilityCDF(%)

Data dries up after a certaintime period such as

warranty period

Time-to-Failure

ProbabilityCDF(%)

Data dries up after a certaintime period such as

warranty period

St Sl


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Steep Slopes

Caution is suggested for

steep slopes ( > 3).

The steep plot often hides badWeibull data.

All the messages from datasuch as curves, outliers,doglegs tend to disappear.

Apparently good Weibullsmay have poor fits.

At first glance the plots appearto be good fits, but there iscurvature and perhaps anoutlier.

Time-to-Failure

ProbabilityCDF(%)

Case 1 Case 2

Time-to-Failure

ProbabilityCDF(%)

Case 1 Case 2

Weibull with Multiple Failure Modes


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Bearing-Sleeve

After sorting out non-

relevant failures

Including all the failures



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Hose As.

After sorting out non-relevant failures

Including all the failures

Weibull Needing X Adjustment


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Weibull Needing X0 Adjustment

Without X0

With X0

Weibull with Batch Problem


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Weibull with Batch Problem

Excluding thebatch problems

Including the batch problem

Curved Weibull


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Curved Weibull

Boom Cylinder Rod

Containing twodifferent

failure modes

Curved Weibull


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Curved Weibull

Boom Cylinder Rod

After splittingthe two failure

modes

Flow Chart For Selecting Life Data Analysis


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g y

Standard Data( You know

failure andsuspension

times)

Zero failures?

Weibayes

Yes

One Failure

LateSuspensions

?

Beta Known

No

Yes

Yes

Weibayes

Yes

Weibayes

MLE

No

No

Less Than21 failures?

Weibayes

Beta known ?Median Rank

Regression (MRR)

MLE w/RBA

DoModels Seem

Reasonable?

MRR & MLEAgree?

Use More

Conservative ofMRR & MLE w/

RBA

Check for batchproblems

No

Yes

Yes

No

yes

Yes No

A1

No

No

Contact MBB

See The New Weibull Handbook For Reference

Flow Chart For Selecting Life Data Analysis


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g y( Continued)

A1 Median Rank

Regression (MRR)

Acceptable

Fit?

Use Median Rank

Regression Result

DistributionAnalysis using

Rank Regressionin Minitab

(Contact MBB)

AcceptableFit for Any

Distributions

Cross CheckW/MLE ?

Contact MBB Contact MBB

Distribution

Analysis usingMLE

RR & MLEAgree?

More than100 Failures?

Use RR result

Use MoreConservative of

RR & MLE Results

Use MLE Result

Yes

No

No

Yes

No

YesYes

No

No

Yes

See The New Weibull Handbook For Reference

Determining a Significant Difference


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Determining a Significant Difference

Confidence intervals at B5 and/or B10 etc. Maximum Likelihood Ratio Hypothesis Test

Confidence Intervals


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Confidence IntervalsDuo-Cone Seals

Hours

Percent

100001000

30

20

10

5

3

2

1

Table of Statistics

226 2490

1.17559 61433.6 1398.974 60 1107

0.99904 33396.4 11663.027

Shape

711 2958

Scale AD* F C

1.00676 59591.8 4922.555

Model

777

773

775

Probability Plot for Hours

Censoring Column in Status - ML Estimates

Weibull - 90% CI

When the confidence intervals do not

overlap we can say the populationsare statistically different at thatconfidence level.

In this case the 777 the B5 life isstatistically different than the 773 B5life and the 775 B5 life.

Slopes are very similar indicating a

common failure mode but Theta isdifferent.



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Co de ce te a sDuo-Cone Seals

Table of Percentiles 777

Standard 90.0% Normal CI

Percent Percentile Error Lower Upper

5 1152.27 88.4247 1015.63 1307.30

6 1451.26 102.160 1292.58 1629.41

7 1765.67 115.279 1585.88 1965.84

8 2094.52 128.003 1894.21 2316.02

9 2437.11 140.542 2216.57 2679.60

10 2792.91 153.092 2552.11 3056.42

Table of Percentiles 773

Standard 90.0% Normal CI

Percent Percentile Error Lower Upper

5 2005.95 217.792 1677.88 2398.17

6 2643.77 266.273 2240.15 3120.12

7 3343.09 320.364 2855.57 3913.85

8 4101.30 381.960 3518.79 4780.25

9 4916.53 452.648 4225.63 5720.40

10 5787.43 533.686 4972.93 6735.34



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72

Hours

Per

cent

100000.0

10000.0

1000.0

100.0

10.01.00.1

403020

10

5

32

1

0.01

Correlation 0.977

Shape 0.684089

Scale 1071873

Mean 1387617

StDev 2083837

Median 627282

IQR 1554401

Failure 33

Censor 1884

AD* 832.872

Table of Statistics


Censoring Column in Status - LSXY Estimates

Weibull - 90% CI



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Comparing old and new machines

with the same part failing. The

new machines have emission

compliant engines. Failure

differences are attributed to

increased vibration due to higherinjection pressures.

Hours

Percent

1000

00

.00

100

00

.00

1000

.00

100

.00

10

.00

1.00

0.10

0.01

99.99

9580

50

20

5

2

1

0.01

0.57763 3285729 0.981 28 1301

2.96661 5395 0.994 5 583

Shape Scale Corr F C

Table of Statistics

Tier 2

Tier 3

Population


Censoring Column in Status - LSXY Estimates

Weibull - 90% CI

Maximum Likelihood Ratio Hypothesis Test


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Statistical methodology to establish 90% confidenceNull & Alternative Hypothesis (prove the null hypothesis false with data)

Ho(null): The baseline reliability is same as after-fix reliability.

Ha(alternative): The after-fix reliability is different than baseline

Use Max Likelihood Ratio test to compare baseline to after-fix. If

the confidence exceeds 90% that these are different, you canbegin Financial Control. Test works even if after-fix data haszero failures (given sufficient maturity).

A Weibull curve can be fit to the after-fix data (even if zero failures).These Weibull parameters are then used to establish size of

improvement.



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Statistical methodology to establish 90% confidence

Calculate the Statistic for Likelihood Ratio Test

Test_value = -2 x (LogLik_before + LogLik_after LogLik_Combined)

Use the Test_value in a Chi-Square test with 1 degree of freedom todetermine establish statistical confidence that after-fix populationis different than baseline

Detailed step by step instructions in the appendix.

Summary and Take-Aways


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Summary and Take-Aways

Weibull distribution useful to predict life and failures. Weibull is good analytical tool to help to identify the

failure mode based on beta. Understand how to handle incomplete data, which time

method to use, and when to use data dryup. Understand how to select and when to use each

Weibull estimation method.

Learn to identify Bad Weibulls, Bad Data, &Uncertainties.

Shortcomings of the failure data and suspension data.

Be able to determine if one population failure rate isstatistically different than another.

Who to Call for Help


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Who to Call for Help

There are several experts that can help createa reasonable Weibull analysis if you are havingtrouble.

Please start in your local organization with yourMaster Black Belts (MBB)

If they need help they can call on the 6 Sigma Coreteam and / or Corporate Quality & Reliability

Helpful Weibull References


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Helpful Weibull References

The New Weibull Handbook, Dr. Robert B. Abernethy http://www.barringer1.com/tnwhb.htm

Weibull Analysis, Dodson, B., ASQ Quality Press, Milwaukee, WI, 1994. Statistical Design and Analysis of Engineering Experimentsby Charles

Lipson and Narendra J.Sheth QRWB Weibull Analysis

http://cti.corp.cat.com/qrwb/prmbrowser/weibull/weibull.html CAT Weibull Software

http://tsd.cat.com/e-tsd/docs/index.cfm?H=2&tech_id=5221 http://ris.moss.cat.com/index.html Weibull Analysis Mathematics

http://gold.pic.cat.com/weibulltools/weibullmath/ CAT Weibull User KN Community

https://kn.cat.com/message.cfm?id=529&parent=26790&type=Broadcast

Using Excel for Weibull http://www.qualitydigest.com/jan99/html/weibull.html

Supersmith Weibull software http://www.barringer1.com/supers.htm

Appendix 1


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January 24-28, 2005

Reliability Data Analysis

SIMS Data System


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Provides reports past warranty period

Provides timely information

Ten years of field data

Used for over 25 years

SIMS Data System(Service Information Management System)

SIMS Data System


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Field Repair

Dealer

CAT ComputerData System

SIMSReport

WarrantyClaim(Overlays

SIMSReport)

SIMS Data SystemService Information Management System

Fi ld D t t d t SIMS


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Engine Serial Number

Failure Hours/Mileage

Failure Mode Code

Failed Part Number

Brief Comment by Mechanic

Repairing Dealer

Field Data reported to SIMS

Sales Data reported to SIMS


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Arrangement Number

Build Date

Sold Date

Work Code

Selling Dealer / OEM

Budget Code

Engine Code Model Name

Sales Data reported to SIMS

Data Provided by Cross Referencing Failures Serial Number;

SIMS Data System


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Field FailureData

SalesData

Dealer Repair Frequency Weibull Analysis

Top Contributor Reports Predict Plots Other Reports

(Service Information Management System)

SIMS Data System


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What Are ??

- DTF Codes- PD Codes- odd part numbers


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DTF Codes Are:

F = Part Number = Last 3 Digitsassigned from: part number causing failure

DT = First 3 Digits = Group Number

assigned from:group number causing failure

Six digit code numbers: such as 212-111

H DTF U d?


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How are DTF Used?

- Grouping multiple Part Numbers

- Reliability Targets

- DRF Apportionment- Defining Profit Center Warranty


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DT Code


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90

F Code


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PD Code

Oth T p f P t N b


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Other Types of Part Numbers

SPXXXX - Substitute Part Number

XXXXXX - Substitute Group Number

PSXXXX - Product Support Program

PIXXXX - Product Improvement Program

PSP / PIP Definitions


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PSP /PIP Definitions

PSP - Product Support Program Incidents Both before failure (assigned PD code = 56) or after

failure (assigned PD code = 96) event Get a regular DT code in failure file and are

counted in Weibull or DRF, if after failure (PD=96).

PIP - Product Improvement Program Incidents Before failure event Get a 900 DT code and are not counted in Weibull or

DRF.

SIMS Data System


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SIMS Entry/Warranty

Claim

ComputerData Base

ReliabilityAnalysis

DealerFailure

SIMS Data System

What are the top three things


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What are the top three thingsour Customers value?

1. Reliability

2. Reliability

3. Reliability

Reliability


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Reliability

Probability that equipment willsatisfactorily perform intended functionfor a period of time under specifiedconditions.

Measured in percent Unacceptable

Two Reliability Measurements


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Mean DRF :

Weibull :

Two Reliability Measurements

Systems

Components

(components for special cases)

CDIM Project Launch


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How to get started;

1. Verify complete model/SN listing

2. Verify correct part numberssss, or DT and F code

3. Establish when problem started (date or SN range)

4. Quick check of warranty/failure rate severity

SIMS Failures or Warranty

Product Sales PopulationFailure Rate or Warranty $/unit ~

CDIM Project Launch


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Some typical questions; What is the primary failure mode concern?

Which product models are involved?

Which applications - machine, truck, industrial, marine, EPG?

Does problem involve selected arrangements or attachments?

Are other related parts claimed for this failure mode?

Is a day one problem, quality hiccup, or latest plateau period

evident?

Verify CPI Project Scope Definition (RWA)


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1. List top SN prefixes (models/applications) for failing part(s) named.

2. List top related parts reported (DTF search) for these SN prefixes.

3. Merge appropriate top system parts AND top machine/engineapplications (related SN prefixes) into project. Add related non-failing (immature) SN prefixes also, as appropriate.

4. Plot DRF trends (at right time ranges) by build month for qualitytrends (by application subgroup and by SN prefix).

5. Restrict project to top machine/engine applications (SN prefixes andbaseline build date range), based on known design & process

change history, Weibulls, and part system warranty/reliability targets.6. Restrict by claim story/comment or failure codes to INCLUDE

only failures of interest for problem description failure criteria.

RESTRICT

EXPAND

Sum Weighting of DRF Metric


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VEHR + I1xDRF1 + I2xDRF2 + I3xDRF3 + I4xDRF4

Total Hour IntervalDRFsum =

VEHR + 180xDRF1 + 800xDRF2

1000DRFmean =

2. Poor-Mans Weibull* (good for trend analysis);

1. Mean DRF Metric (good for product monitoring);

*Note: More accurate when failure Beta slope ~ 1.0.

Which DRF Range Should I Use?


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Hours

FailureR

ate%

1 10 100 1000 10000

Hours

FailureR

ate%

1 10 100 1000 10000

Early Hour Issue < 1.0)

Midlife Issue ~ 1-4)

mDRF metric(1000 hour)

works to trackquality trendsfor this issue

Sum DRF metric(longer hour)

works to trackquality trendsfor this issue

Problem Root Cause Failure Paretos


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DRF trends by build month for quality trends.

DRF trends by repairing dealer for regional service trends. DRF trends by selling dealer(OEM) for application trends.

DRF trends by customer-code for application trends.

DRF trends by SN prefix, product type, sales model,

build mfg facility, arrangement, PD code, work code,power application code, or HP rating for other design,

build, or application trends.

Pareto of warranty comment (or story) top keywords

found. Pareto of warranty parts used in repair (contingent

damage?).

Failures trends by customer state/country location.

Where, when, and what is the sore spot?

9KS Marine Pan Gasket DRF Trend


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0.00

0.02

0.04

0.06

0.08

0.10

0.12

1998-

12

1999-

02

1999-

04

1999-

06

1999-

08

1999-

10

1999-

12

2000-

02

2000-

04

2000-

06

2000-

08

2000-

11

2001-

02

2001-

04

2001-

08

Build Date

0-1000HrDR

F,repairs/100h

rs

0.015 DRF

MLS Hsg Gskt

0.910 DRF

PerfCore Hsg Gskt

Appendix 2


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January 24-28, 2005

Maximum Likelihood Ratio TestDetailed Instructions


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Maximum Likelihood Ratio Test (pg2)


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3. Use Minitab to perform Weibull on baseline and after-fix data

combined into one data set.Save log-likelihood from session window as LogLik_Combined

4. Calculate the Statistic for Likelihood Ratio Test

Test_value = -2 x (LogLik_before + LogLik_after LogLik_Combined)

5. Use the Test_value in a Chi-Square test with 1 degree of freedom todetermine establish statistical confidence that after-fix population isdifferent than baseline

Maximum Likelihood Ratio Test step 1


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1. Tabulate data accumulated hours for each member(failures and non-failures) of before and after-fixpopulation. Create 3 columnsLife (accumulated hours), Type (failure or suspension), When (Before or

After Fix)

Maximum Likelihood Ratio Test Step 2 (zero failure case)


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2. Use Minitab to perform Weibull on baseline and after-fix data sets

Case A: After-fix data has zero failures.

a) Create distinct Hrs and Type columns for before & after

b) do max-likelihood Weibull on baseline. Save log-likelihood from session window asLogLik_Before into excel spreadsheet

c) do max-likelihood Weibull on after-fix (Bayes Analysis) with Weibull slope specified to be equal tothe before fix Weibull slope. Minitab does not report a log-likelihood enter LN(0.5) asLogLik_After in excel spreadsheet

Maximum Likelihood Ratio Test Step 2 (zero failure case)


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Case A: After-fix data has zero failures.

a) Create distinct Hrs and Type columns for before & after

b) do max-likelihood Weibull on baseline. Save log-likelihood from session window as LogLik_Beforeinto excel spreadsheet

c) do max-likelihood Weibull on after-fix (Bayes Analysis) with Weibull slope specified to beequal to the before fix Weibull slope. Minitab does not report a log-likelihood enterLN(0.5) as LogLik_After in excel spreadsheet

Maximum Likelihood Ratio Test - Step 2 (1 or morefailures)


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Case B: After-fix data has 1 or more failures.

do max-likelihood Weibull on both before and after in single step using Typecolumn to stratify data into 2 groups. Set Minitab option for common Weibullslope for the two populations

Save log-likelihoods from session window as LogLik_Before and LogLik_after

Maximum Likelihood Ratio Test Step3


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3. Use Minitab to perform Weibull on baseline and after-fix data

combined into one data set.Save log-likelihood from session window as LogLik_Combined

Maximum Likelihood Ratio Test Step 4 & 5


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4. Calculate the Statistic for Likelihood Ratio Test

Test_value = 2 x (LogLik_before + LogLik_after LogLik_Combined)

5. Use the Test_value in a Chi-Square test with 1 degree of freedom todetermine establish statistical confidence that after-fix population isdifferent than baseline

90% Confidence Test for Fix Calculations

Log-

Likelihood

Combined

Log-

Likelihood

Before Fix

Log-

Likelihood

After Fix Bef+Aft Test_value

# of Degrees

of Freedom P-Value Confidence

-223.462 -211.204 -11.919 -223.123 0.678 1 0.41027627 59%

In example 90%Confidence not

meet. Wait and

recalculate in 1

or 2 months

Documents

Introduction to Weibull Analysis Ver4