Demystifying Weibull - Minitab · 1. Weibull Distribution Plotting a) The big picture of today [s presentation b) Scope c) Overview: analytical flow Minitab Insights 9/13/2016 This

Demystifying WeibullA Practical Means to Locate Performance in the Bathtub Curve

Joe Sipper

9/13/2016

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Copyright © 2014 Raytheon Company. All rights reserved.Customer Success is Our Mission is a registered trademark of Raytheon Company.

Scope of Presentation

• Intent is foundational, discuss a focused facet of Weibull • This is about plotting data and identifying phases in the

bathtub curve, it’s not about further Reliability analysis

• Why all these details if Minitab does all the work?

• Key discussion points include:1. Weibull distribution plotting

2. Data censoring and test type

3. Types of repair to the population

4. Creating proper data structure for Minitab

5. Weibull interpretations and the bathtub curve

6. Process for Weibull plots

7. Using Weibull plots to gain performance insight

Minitab Insights 9/13/2016


Slide 2

1. Weibull Distribution Plotting

a) The big picture of today’s presentation

b) Scope

c) Overview: analytical flow



Slide 3

1a1. The Big Picture: Today’s Presentation

• There are lots of great Weibull applications to predict system life, reliability, success rates, preventive maintenance intervals, warranty structures, repair strategies, etc

• The same source data can be used for everything we’ll discuss however the format needs to be in a format appropriate for each assessment

• Today’s outputs can feed into some Weibull applications but that’s for another day

• Today is about structuring failure data, analyzing it in Minitab, and making first-tier conclusions about where in the bathtub a performance falls



Slide 4

1a2. The Big Picture: Characterize Performances

• Understanding the problem is often the hardest part of the problem; getting and arranging good data is a much easier for a problem that is well understood than for one that is not

• Determine what distribution best models performance

• For the most part, the analytical process is similar or identical across many scenarios but the data arrangement differs

• Data drives a model, and is then replaced by its model

• The Weibull model, which is highly flexible and often applicable, gives insight into performance phases of the bathtub curve



Slide 5

1a3. The Big Picture: Use Weibull for Systems?

• A lot depends on the question, i.e., the needs of the assessment

• Some believe that Weibull can be used only for an individual failure mode, and multiple failure modes offset performance characteristics and muddy the analysis

• Others believe a system is simply a series of components

• System fails when one (random) component fails

• System time-to-failure = smallest of the component failure times

• This is equivalent to Weibull distribution

• Underlying premise to, but not explored in, today’s presentation

• Ideally run a unique Weibull assessment for each failure mode, however the method given here addresses needs of the assessment



Slide 6

1b1. Distribution Plotting for this Presentation

Ultimately make a set of usable, value-added data with appropriate groupings.

Non-Repairable (System or Element)

Repairable (System or Element) analyze patterns b/w failures

Repairable (Fleet)

Typical characterization

Single failure mode based on sample data

Individual system based on field data

Individual system based on field data

Fleet based on field data

Repair state New Like New Minimal Mixed

s.i.i.d. ? Yes Yes No No

Data analyzedElement fails vs. element usage

System failures vs. system usage

System failures vs. system usage

Fleet failures vs. fleet usage

Data timelineOrdered usage from T=0

Ordered usage from T=0

Superimposed Cumulative

s.i.i.d. = statistically independent and identically distributed

• Scope focuses on expectations of performance when the system or element is repaired, and the corresponding data structures



Slide 7

1b2. Repair Type and Data Structures

Replace “Like New” Repair “Minimal” Repair

• Brand new hardware restarts the clock

• Expectation is that the next failure is random (independent) because the history is reset, i.e., the repair returned a system that was as good as new

• Next failure depends on current usage, i.e., the system’s age prior to the repair

• System Reliability is the same after the repair as it was before the repair

XIT0 = 0 T1 T2 T3 T4 Tn

t1 t2 t3 t4 tnXX XX . . .

Note the differences in cumulative timelines:

XI XXX

XIt1

XIt2

It3 X

It4 X

XI XXXI

tn X

T0 = 0

XIt1

XIt2

It3 X

It4 X

XI XXXI

tn X

T0 = 0

XIT0 = 0 T1 T2 T3 T4 Tn

t1 t2 t3 t4 tnXX XX . . .

Timelines shown on this slide are in terms of each system studied

Data structure: ordered usage from T0 = 0

Data structure: ordered usage from T0 = 0

Data structure: superimposed



Slide 8

1b3a. Fleet vs System

• Fleet failures and fleet time can stay random over time because of the mix of old and new systems in the fleet

• Failures for each individual system are tracked and composed additively on a cumulative timeline

• Cumulative timeline combines failures from different systems onto a single timeline

• Individual systems each provide usage data, and clustering of events becomes evident at similar usages

• Failures are superimposed onto a timeline

• Both use Power law statistically, but treat data differently



Slide 9

1b3b. Fleet vs Repairable Systems

XIX X XXSystem 1

XI X XXXSystem 2

XIX X XXSystem 3

I X XXXSystem n

XIX X XXXX XXXXX X XXX XXXSuperimpose

XIX X XX System 1

XI X XXX System 2

XIX X XX System 3

I X XXX System n

XIX X XX XX XXX XX X XX X XXXCumulative Timeline

Repairable Systems –Systems age in patterns, characteristics are evident when viewing clustering of superimposed failures

NOTE: In repairable systems, failures for non-repairable elements are s.i.i.d. however failures for the system or element are dependent

Fleet – Failures become random when time modeled because the fleet is composed of systems both new and old, i.e., assumption of independent failures is not valid



Slide 10

1c. Overview: Analytical Flow

Censored

Data ?

Test

and

Censoring

Considerations

(Test) Type I: Time Censoring – test time

fixed

Right Censored – lower bound known,

upper bound not fully known for all data

(Test)

Type

(Data)

SuspensionsAnalyze as

Complete Data

Yes

Analyze as

Censored Data

(Test) Type II: Failure Censoring –

number failures fixed

Left Censored – only have the upper

bound, not sure of exact time of failure

No

Interval Censored – all fail, have ideas of

times but not exact measures

MTTF

or

MTBF

?

Expectations

of Repair

Replaceable = New =

Non-repairable

(MTTF)

Repairable

(MTBF)Analyze as

MTTF

Minimal

New or

Like New

Superimpose

each System

Ordered Usage

from T=0

Identical

conditions

Data

Structure

Repair

Structure

Cumulative

Timeline

Fleet



Slide 11

2. Data Censoring and Test Typea) Complete data

b) Left censored data also called Interval data

c) Right censored data also called Suspended data

d) Singly censored data

e) Multiply censored data

f) Interval, or grouped, data

g) Test Type I also called Type I Censoring

h) Test Type II also called Failure Censoring



Slide 12

2a. Complete data

• All units run to failure, have exact usage times for each

• Another way to say this is that each unit has run through its entire lifetime and is characterized exactly

• Examples include:Sample 1 of n

Sample 2 of n

Sample 3 of n

Sample 4 of n

Sample 5 of n

Sample n of n

Usage (Time, Cycles, Distance, etc.)

Complete Data

I

I

I

I

I

I

Fail

Fail

Fail

Fail

Fail

Fail

. . .

. . .• Highly structured

lab testing

• Fully accessible field data with high failure rates



Slide 13

2b1. Left Censored data

• Check performance at some time, t

• Observe the unit(s) have failed, do not know exactly when

• Examples include:

Sample 1 of n

Sample 2 of n

Sample 3 of n

Sample 4 of n

Sample 5 of n

Sample n of n


Left Censored Data

I

I

]

I

]

I

Fail

Fail, not sure of exact usage

Fail

Fail

Fail, not sure of exact usage

Fail

. . .

. . .

• Testing or field data where data is not fully accessible



Slide 14

2b2. Interval Censored data

• Different from left-censored in that all intervals do not start with zero usage

• Check performance at some time, t

• Observe the units have failed, do not know exactly when


Sample 2 of n

Sample 3 of n

Sample 4 of n

Sample 5 of n

Sample n of n


Interval Censored Data

]

]

]

]

]

]

Fail in interval

Fail in interval

. . .

. . .

[

[

[

[

[

[

Fail in interval

Fail in interval

Fail in interval

Fail in interval

• Testing or field data where data is not always accessible



Slide 15

2c. Right Censored (or Suspended) data

• Important to include usage of survivors in timelines

• Check performances at some time, t

• Observe some units have failed, while the remaining are still operational; record all usage


Sample 2 of n

Sample 3 of n

Sample 4 of n

Sample 5 of n

Sample n of n


Right Censored Data

I

I

I

Operational

Operational

Fail

Fail

Fail

Operational

. . .

. . .

• Most field data, especially those checked somewhat regularly

• Time truncated = Type I, failure truncated = Type II



Slide 16

2d. Singly Censored data

• Only one set of samples throughout all observations

• Population started together, data collected before all fail

• Record all usage – impacts cumulative percent fail


• Typical Type I or Type II testing in 2g and 2h



Slide 17

2e. Multiply (muhl-tuh-plee) Censored data

• Different run times and different number of interventions for each system

• More than one censoring point in a set of observations


• Examples (CP = censor point) include:

• Field data for a mature repairable product

• Controlled testing

• Test for n = 100 units stopped at 20,000 cycles CP1

• Remove n = 30 units, re-observe after 5,000 more cycles CP2Minitab Insights 9/13/2016


Slide 18

2f. Interval or Grouped data

• 2 applications

• Known individual data is plotted and recoded and grouped into bins after the fact for analysis, e.g., Pareto groupings

• Observations are made periodically across a population, exact failure times are unknown but exact quantities are recorded as described in 2b2



Slide 19

2g. Test Type I (or Type I Censoring)

• Test n samples, terminate after some predetermined usage regardless of the number of failures

• Some samples may, and likely do, survive the test (test conditions may be too harsh if all fail) and are coded as suspensions

• Test time is fixed, number of failures is variable


• Examples include:• Reliability growth testing – accelerations impose 10 years of accelerated life

on samples in 600 hours, then terminate test

• Able to obtain usage for each element or system in the population, cross-reference with those systems or elements that have failed at the time of the data collection



Slide 20

2h. Test Type II (or Failure Censoring)

• Test n samples, terminate after some predetermined number of failures regardless of the usage

• Apply when time is of the essence, guaranteed enough data to carry out assessments with statistical significance

• Number failures is fixed, code survivors as suspensions


• Examples include:• Reliability growth testing – run test until value-added information is

available for, say, n = 30 failures

• Able to obtain usage for each system in the population, keep selecting random systems from field population, viewing pass/fail status, until say n = 30 failures are identified



Slide 21

2i. Thought on Censoring (also called Suspensions)

• Implementation also depends on the question asked

• Field data has practical limitations, often obtainable only upon failure; suspensions in the data must be recorded

• These types of performances often span a lifetime that includes multiple failures for each system, i.e., system will fail it just hasn’t at the time of this data collection (sometime have eternal survivors)

• Suspensions may have a different failure mode from that being studied or may not have failed at all

• Field data is usually a snapshot, survivors at a data collection point will eventually fail and be treated as appropriate in terms of a data timeline



Slide 22

3. Types of Repair

a) Non-repairable system or element (Replaceable)

i. Replaceable = return to “new” condition

b) Repairable system

i. Repair to “like-new” condition “reset” system history

ii. Minimal repair fix only what is broken, retain historical information

c) Repairable fleet

i. Treat individual systems as a single population



Slide 23

3a. Non-Repairable System or Element

• Must be replaced; impractical or unsafe to return to reusable condition

• Hazardous materials

• Disposables

• Inexpensive products

• Each system life, production or repaired, begins at T=0 upon being placed into operation

• Number of failures, and time between failures, is random

• Data analyzed: system or element failures versus system or element usage



Slide 24

3b1. Repairable System – Like New

• All elements’ usage resets upon repair

• Analyzing the data between failures is, in effect, the equivalent of pulling all usage back to a common chronological point, i.e., T=0


• Overhauls, e.g., jet engines, dirt bikes, race cars, etc.

• Preventive Maintenance

• Customer expectations, viable or not, based on repair costs or perceptions of failing quickly upon return

• Generally NOT true but sometimes useful to consider

• Data analyzed: system failures versus system usage



Slide 25

3b2. Repairable System – Minimal

• Minimal Repair – fix only what is broken, improve nothing else

• Each elements’ usage retains its history upon repair the next failure is not independent, components untouched in the repair have a cumulative life and the next failure is influenced by this life


• Most personal services, e.g., car repair, dentist, etc

• Data analyzed: system failures versus system usage



Slide 26

3c. Repairable Fleet

• Provide a service where the function is more important than the means of accomplishing the function

• Fleet failures and fleet usage can stay random over time because of the mix of old and new systems in the fleet

• Failures for each individual system are tracked and composed additively on a cumulative timeline


• “Big picture” questions relevant to fleet, not individual systems

• Rental cars

• Capable computers for a large workforce

• Data analyzed: fleet failures versus fleet usageMinitab Insights 9/13/2016


Slide 27

4. Creating the Proper Data Structure for Minitab

a) The purpose here is to produce shape and scaleparameters and then replace the data with a model; 3-parameter Weibull includes a threshold parameter

b) Prepare as much relevant data as practical – the necessity is for

• Appropriate operational descriptors of usage (e.g., hours, distance, cycles, etc) for each failure being repaired

• Constructing the appropriate timeline and groupings



Slide 28

4a1. Shape, Scale, and Threshold Parameters

• Shape parameter (used in 2- and 3-parameter Weibull)

• In simple words: an influential value, defined by the underlying data, that drives the look (shape) of the Weibull plot

• More technically: Measure the rate of change in the unreliability function over the usage, where both the unreliability and usage are linearized using logarithms.

• Use median ranks to estimate unreliability for each failure

• Median Rank =rank in the failure order − 0.3

number of failures+0.4 Bernard’s approximation

• xi = ln time to fail i yi = ln ln1

1−median rank

• In application: β < 1 decreasing failure rate, β = 1 constant failure rate, β > 1 increasing failure rate



Slide 29


• Scale parameter (used in 2- and 3-parameter Weibull)

• In simple words: measure of the spread of the Weibull, the usage from the first viable failure potentially occurring to 63.2% units failing

• More technically: the characteristic life, which is the level of usage at which 63.2% of the systems or elements will fail.

1 − R = 1 − e−λt = 1 − e−t

τ = 1 − 0.368 = 0.632 when t = τ

• In application: if scale is large the distribution is more spread out, if scale is small the distribution is tighter; a scale of 10 implies 63.2% of population will fail in the first 10 hours, cycles, etc. while a scale of 500 implies the failures will occur over a much longer time



Slide 30


• Threshold parameter (used in 3-parameter Weibull)

• In simple words: distribution’s shift from 0 usage; can be used if failures do not start at t=0 but instead at some time +t or -t; the parameter describes the threshold at which failure occurs (also called shift, failure-free, or location parameter)

• More technically: provides an estimate of the earliest viable time-to-fail; < 0 negative on x-axis, > 0 positive on x-axis

• In application: track ship date but it takes 30 days for customer to install and use – this is a failure-free usage period; negative values may imply failure occurred prior to collecting performance data such as a quality escape, in transit, etc. (Minitab defines threshold only for non-negative values)



Slide 31

4b0a. Prepare as much Relevant Data as Practical

• Key pieces of raw data include, but are not limited to:

• System identifiers such as part number, serial number, etc.

• Usage at the time of the repair (often given as cumulative)

• Other data useful for further characterizations (dates, etc.)

• Ensuring the data is relevant per specific assessment needs

• Rather than trending performances over measures of usage, as in Paretos, group repairs chronologically by each system identifier

• Build a timeline valid for the application, either pull failures to T=0, superimpose failures on a common timeline, or system performances as appropriate

• Note: data accuracy can lead to conservative results, high integrity data may mean discarding some information



Slide 32

4b0b1. Prepare as much Relevant Data as Practical

• Establish qualifying characteristics in raw data to decide whether data is relevant or not – examples include:

• A repair disposition that a field observation is indeed a failure, valid per the terms of the assessment

• The information is considered complete, the observation is closed

• Able to apply to identified element, failure mode, or date range as appropriate

• Rather than plotting overall quantities chronologically, use Excel pivot table to group data by each individual system and study each system chronologically

• Serial number is an obvious choice for identifiers

• Good to lay out the Pivot Table Design in Tabular layout (Pivot table tools > Design > Report Layout > Show in Tabular Form)

• Also consider (Report Layout > Repeat All Item Labels)



Slide 33


Sample of fictitious data for illustration only



Slide 34


Excel formulas to drive fields added to pivot shown on prior slide

• Good Record? (cell N8 and drag through column N)=IF(AND(A8<>"",A8<>"(blank)",B8<>"",B8<>"(blank)",C8<>"",C8<>"(blank)",E8="Yes",F8<>"",F8<> "(blank)",J8="Closed"),"yes","no")

• Use in Weibull? (cell P8 and drag through column P)?=IF(O8<=0,"no",IF(COUNTIF(O8,"*data*")=0,"yes",IF(COUNTIF(O8,"*data*")=1,"no","")))

• Failure Free Hours (per structure of the data as presented)

• Replaceable (cell S8 and drag through column S)=IF(O8<=0,"",IF(N8="yes",IF((A8<>A7),VALUE(F8),IF(AND(N7="yes",B8<>B7),VALUE(F8)-VALUE(F7),"")),""))

• Minimal Repair (cell Y8 and drag through column Y)=IF(P8="yes",F8,"")

• Fleet (cell AE9 and drag through column AE)=IFERROR(AE8+S9,AE8) NOTE: also include in cell AE8 =S8



Slide 35


Example 4b1, System Repair State: New and Like New

• Usage for each element or system put into operation, whether new or repaired, starts at T=0

• Use columns S, U, V, W of Excel worksheet in Minitab

• Table implies complete data

Sample of fictitious data for illustration

only



Slide 36

4b1b. Prepare as much Relevant Data as Practical

Excel formulas for complete data for “New” or “Like New”

• Bin Grouped by 100_1 (cell Q8 and drag through column Q)=IF(S8<=0,"",IFERROR(IF(S8<>"",ROUNDDOWN(S8/100,0)+1,""),""))

• Order Bin Size 100_1 (cell T8 and drag through column T)=IF(ROW()-7<=MAX(Q:Q),ROW()-7,"")

• Grouped Cycles_1 (cell U8 and drag through column U)=IF(T8<>"",T8*100,"")

• Occurrences Bin Size 100_1 (cell V8 and drag through column V)=IF(T8<>"",COUNTIF(Q:Q,"="&T8),"")

• Cum Pct Bin Size 100_1 (cell W9 and drag through column W)Since this is cumulative, initialize U8 with a different formula

=IF(T8<>"",IFERROR(V8/SUM(V:V),""),"")=IF(T9<>"",IFERROR(V9/SUM(V:V)+W8,W8),"")

Data now structured for Replaceable analysis in Minitab:Each usage to fail, grouped (binned) occurrences and cumulative percentage



Slide 37


• Example 4b2, System Repair State: Minimal

• Each system’s usage starts at T=0, each failure and usage is recorded, and failures are superimposed

• Use columns Z, AB, AC, AD of Excel worksheet in Minitab



only



Slide 38


Excel formulas for “Minimal Repair”

• Bin Grouped by 100_2 (cell X8 and drag through column X)=IF(Z8<=0,"",IFERROR(IF(Z8<>"",ROUNDDOWN(Z8/100,0)+1,""),""))

• Order Bin Size 100_2 (cell AA8 and drag through column AA)=IF(ROW()-7<=MAX(X:X),ROW()-7,"")

• Grouped Cycles_2 (cell AB8 and drag through column AB)=IF(AA8<>"",AA8*100,"")

• Occurrences Bin Size 100_2 (cell AC8 and drag through column AC)=IF(AA8<>"",COUNTIF(X:X,"="&AA8),"")

• Cum Pct Bin Size 100_2 (cell AD9 and drag through column AD)Since this is cumulative, initialize U8 with a different formula

=IF(AA8<>"",IFERROR(AC8/SUM(AC:AC),""),"")=IF(AA9<>"",IFERROR(AC9/SUM(AC:AC)+AD8,AD8),"")

Data now structured for Repairable analysis in Minitab:Each usage to fail, grouped (binned) occurrences and cumulative percentage



Slide 39


• Example 4b3, Fleet Repair

• Each system’s usage starts at T=0, is tracked cumulatively, and each system is then stacked additively

• The bins are much larger because the failure free cycles are cumulative

• Use columns AG, AI, AJ, AK of Excel worksheet in Minitab



only



Slide 40


Excel formulas for “Fleet” (Bin sizes = 1000 in this example)

• Bin Grouped by 1000_3 (cell AE8 and drag through column AE)=IF(AG8<=0,"",IFERROR(IF(AG8<>"",ROUNDDOWN(AG8/1000,0)+1,""),""))

• Order Bin Size 1000_3 (cell AH8 and drag through column AH)=IF(ROW()-7<=MAX(AE:AE),ROW()-7,"")

• Grouped Cycles_3 (cell AI8 and drag through column AI)=IF(AH8<>"",AH8*1000,"")

• Occurrences Bin Size 1000_3 (cell AJ8 and drag through column AJ)=IF(AH8<>"",COUNTIF(AE:AE,"="&AH8),"")

• Cum Pct Bin Size 1000_3 (cell AK9 and drag through column AK)Since this is cumulative, initialize U8 with a different formula

=IF(AH8<>"",IFERROR(AJ8/SUM(AJ:AJ),""),"")=IF(AH9<>"",IFERROR(AJ9/SUM(AJ:AJ)+AK8,AK8),"")

Data now structured for Fleet analysis in Minitab:Each usage to fail, grouped (binned) occurrences and cumulative percentage



Slide 41

4b4. Prepare as much Relevant Data as Practical

• Data structuring stops here due to time limitations

• Censoring can be considered by simply expanding the Excel table to include a “censoring” column and also importing this information into Minitab

• Consideration that this data set is complete

• Many data collection processes gather only failures

• For the purposes of the plotting here, each system will eventually fail (either permanently or will be repaired) and that information will be added to future assessments



Slide 42

5a. Weibull Interpretations and the Bathtub Curve



Slide 43

5b. Weibull Interpretations and the Bathtub Curve

• The bathtub curve is a compilation of 3 distributions, it is not continuous

• Weibull can indeed be made to fit all 3 distributions of the bathtub curve as three separate plots

• Shape parameter is the rate of change in failure rate over time

• Weibull characterization arranges data, plots a representation of each failure, and determines the rate of failure rate over usage along with the spread of failures via characteristic life



Slide 44

6. Process for Weibull Plots

a) Start with the appropriate data

b) Produce frequency plots for failures

i. Occurrences

ii. Cumulative failure percentage

c) Use Minitab to select the best distribution for the data

d) Use Weibull parameters to create a model for the data

e) Determine times to failure based on the model

f) Final validity check of distribution selected



Slide 45

6a1. Start with the Appropriate Data

• The analysis starts to become more systemic once the appropriate data is established

Sample of fictitious data for illustration only – note the differences

in occurrences and cumulative percentages



Slide 46

6a2. Start with the Appropriate Data

• Aside from censoring considerations, which may or may not be influential but are not discussed at this time, the analysis now becomes a repeatable process

• Censoring impacts cumulative percentage • Minitab – Stat > Reliability/Survival > Distribution Analysis

(Right Censoring) > Distribution ID Plot > Censor

• Work through the process in the rest of section 6 with one set of data

• Since the data has been structured per the needs of the analysis, the specifics of the data are not further discussed

• Section 7 shows analytics that reflect each phase of the bathtub curve



Slide 47

6b1. Produce Frequency Plots for Failure Times

Occurrences 49541 255 152 82 47 38 23 22

Percent 4.144.7 21.1 12.6 6.8 3.9 3.1 1.9 1.8

Cum % 100.044.7 65.8 78.4 85.2 89.1 92.2 94.1 95.9

Grouped_Cycles Other4000350030002500200015001000500

1200

1000

800

600

400

200

0

100

80

60

40

20

0

Occu

rre

nce

s

Pe

rce

nt

Pareto Chart of Grouped_Cycles

Use Minitab Assistant

> Graphical Analysis > Pareto Chart

“New” or “Like New”

14000120001000080006000400020000

160

140

120

100

80

60

40

20

0

100.0%

80.0%

60.0%

40.0%

20.0%

0.0%

Grouped Cycles_1 - "New" or "Like New"

Occu

rre

nce

s B

in S

ize

10

0_

1

Cu

m P

ct

Bin

Siz

e 1

00

_1

Occurrences Bin Size 100_1

Cum Pct Bin Size 100_1

Variable

Scatterplot of Occurrences Bin , Cum Pct Bin Size vs Grouped Cycles_1

Graph > Scatterplot > Simple

Y variables X variablesOccurrences Grouped CyclesCumulative Pct Grouped Cycles

Multiple Graphs > Multiple Variables > Overlaid on the same graph

Plot results with both variables on primary axis> Double click Y-axis

> Secondary > Assign secondary scale for Cum_Pct



Slide 48

The scatterplot and Pareto chart here look similar to the Weibull distribution plot in 7a of this presentation but they are different.

These are essentially the same plot. The first 5 points in the scatterplot make the first bar in the Pareto.

6b2. Produce Frequency Plots for Failure Times

• Copy Failure Free Hours to a new column in Excel, detached from table derived throughout this presentation

• Sort from smallest to largest

• Create a pair for each data point by adding an adjacent column filled with zeros (value = x is now = x, 0)

• Create scatterplot in Excel

• Compress entire output to produce format shown below

01

0 2000 4000 6000 8000 10000 12000 14000 16000

Cumulative Timeline for Previous Plot

Compress Occurrences into a Cumulative Timeline (Excel)

Clearly see a bias toward failures earlier in the

cumulative timeline

14000120001000080006000400020000

160

140

120

100

80

60

40

20

0

100.0%

80.0%

60.0%

40.0%

20.0%

0.0%

Grouped Cycles_1 - "New" or "Like New"

Occu

rre

nce

s B

in S

ize

10

0_

1

Cu

m P

ct

Bin

Siz

e 1

00

_1

Occurrences Bin Size 100_1

Cum Pct Bin Size 100_1

Variable

Scatterplot of Occurrences Bin , Cum Pct Bin Size vs Grouped Cycles_1



Slide 49

6c. Use Minitab to Select Best Distribution for Data

Weibull is a valid model for this particular dataStat > Quality Tools > Individual Distribution Identification

Data are arranged as a Single Column‘Failure Free Hours …’

1000050000-5000

99.99

99

90

50

10

1

0.01

Failure Free Hours at each retu

Pe

rce

nt

1000010001001010.1

99.99

90

50

10

1

0.01


Pe

rce

nt

1000010001001010.10.01

99.99

90

50

10

1

0.01


Pe

rce

nt

1000010001001010.10.010.001

99.999990

50

10

1

0.01

Failure Free Hours at each retuP

erc

en

t

Weibull

AD = 1.514

P-Value < 0.010

Gamma

AD = 3.374

P-Value < 0.005

Goodness of Fit Test

Normal

AD = 98.059

P-Value < 0.005

Exponential

AD = 15.622

P-Value < 0.003

Probability Plot for Failure Free Cycles_1 - "New" or "Like New"

Normal - 95% CI Exponential - 95% CI

Weibull - 95% CI Gamma - 95% CI



Slide 50

120001000080006000400020000

900

800

700

600

500

400

300

200

100

0

Failure Free Cycles_1 - "New" or "Like New"

Fre

qu

en

cy

Shape 0.8540

Scale 974.3

N 1215

Histogram of Failure Free Cycles_1Weibull

6d. Use Parameters to Create a Model for the Data

Graph > Histogram > Simple

Graph variables‘Failure Free Cycles_1’

Multiple Graphs > Multiple Variables > Overlaid on the same graph > OK

Data View > Distribution > check ‘Fit distribution’(choose Weibull, leave ‘Historical Parameters’ blank)

OK to make plot, then adjust the bins. Right-click bars on histogramEdit Bars > Binning > Cutpoint

These parameters drive the Weibull

model that replaces the data



Slide 51

6e. Determine Times-to-Failure based on the Model

Explanation:

Per the selected model,

• 1% of the failures occur

between 5 and 8-9 cycles

• 10% of the failures occur

between 73 hours and 96

cycles

• The average failure is

between 922 and 1038

cycles

STAT > Reliability/Survival > Distribution Analysis (Right Censoring) > Distribution ID PlotVariables: ‘Failure Free Cycles_1’ Specify Weibull

There are options available to draw further distinctions on censoring that are beyond today’s scope.




Slide 52

1000010001001010.10.01

99.99

95

80

50

20

5

2

1

0.01

Failure Free Cycles_1 - "New" or "Like New"

Perc

en

t

Weibull

0.995

Correlation Coefficient

Probability Plot for Failure Free Cycles_1LSXY Estimates-Complete Data

Weibull

6f. Final Validity Check of Distribution Selected

Explanation:

For this particular

set of data, the

Weibull model is

quite solid between

50-5,000 cycles

however is a little

shaky outside of

this range.

We need to pay

close attention to

processes

impacting early life

failures, e.g.,

• Testing

• Burn-in

• Shipping

STAT> Reliability/Survival > Distribution Analysis (Right Censoring) > Distribution ID Plot. Minitab Insights 9/13/2016


Slide 53

0.0025

0.0020

0.0015

0.0010

0.0005

0.0000

Cycles to Failure_1 - "New" or "Like New"

Pro

ba

bil

ity D

en

sit

y

634.3

0.5

0

Distribution PlotWeibull, Shape=0.854, Scale=974.3, Thresh=0

7a. Using Weibull Plots to Gain Performance Insight

Per the selected model, e.g., 50% fail at ≈ 634.3 cycles

Graph > Probability Distribution Plots > View Probability

Distribution: Weibull‘Shape’ and ‘Scale’ parameters from “Distribution ID Plot”

Shaded Area:Right Tail with Probability = 0.5

634.3 cycles creates

an area under the

curve of 50%

Shape parameter < 1 implies performance is in Infant Mortality

Same source data, manipulated as

appropriate, used for 7a – 7d




Slide 54

0.0025

0.0020

0.0015

0.0010

0.0005

0.0000

Cycles to Failure_1 - "New" or "Like New"

Pro

ba

bil

ity D

en

sit

y

100

0.1333

0


7b. Using Weibull Plots to Gain Performance Insight

Per the selected model, e.g., ≈ 13.3% fail 100 cycles

Graph > Probability Distribution Plots > View Probability

Distribution: Weibull‘Shape’ and ‘Scale’ parameters from “Distribution ID Plot”

Shaded Area:X Value, Left Tail, Value:100

100 cycles creates an

area under the curve

of 13.38%

Shape parameter < 1 implies performance is in Infant Mortality






Slide 55

0.00025

0.00020

0.00015

0.00010

0.00005

0.00000

Cycles to Failure_2 - Minimal Repair

Pro

ba

bil

ity D

en

sit

y

2294

0.5

0

Distribution PlotWeibull, Shape=1.164, Scale=3143, Thresh=0

7c. Using Weibull Plots to Gain Performance Insight

Per the selected model, e.g., 50% fail ≈ 2300 cycles

2294 cycles

creates an area

under the curve of 50%

Shape parameter ≈ 1 implies performance is in Steady-State

(Data from “New” or “Like New” repairs shows early-life tendencies however the failures upon “Like New” repairs have higher failure rates than those systems minimally repaired)



Minimal Repair



Slide 56

0.000012

0.000010

0.000008

0.000006

0.000004

0.000002

0.000000

Cycles to Failure_3 - Fleet

De

nsit

y

51366

0.5

0


7d. Using Weibull Plots to Gain Performance Insight

Per the selected model, e.g., 50% fail ≈ 2300 cycles

51,366 cycles

creates an area

under the curve of 50%

Shape parameter ≈ 1 implies performance is in Steady-State however fleet data has considerations when doing Weibull



Fleet



Slide 57

0.00030

0.00025

0.00020

0.00015

0.00010

0.00005

0.00000

Steady-State Cycles to Failure

Pro

ba

bil

ity D

en

sit

y

100

0.02217

0


7e. Using Weibull Plots to Gain Performance Insight

Model: Weibull

• Shape 1.13633

• Scale 2827.53698

Fail 95% CI (hours)

1% 24.7 105.9

5% 126.6 351.7

10% 259.6 599.8

50% 1640.1 2514.0

<0.1% fail within 100 hrs

Note early shape of data (spread out) and model (skewed but thick in the beginning).

Results for illustration of Steady State:

Data not reviewed in this presentation

120001000080006000400020000

5

4

3

2

1

100.0%

80.0%

60.0%

40.0%

20.0%

0.0%

Example of Steady-State

Occu

rre

nce

s_

Bin

_S

ize

_1

00

_S

S

Cu

m P

ct_

Bin

_S

ize

_1

00

_S

S

Occurrences_Bin_Size_100_SS

Cum Pct_Bin_Size_100_SS

Variable

Scatterplot of Occurrences_Bin_, Cum Pct_Bin_Size vs Grouped Cycles_B

Steady-State



Slide 58

0.00014

0.00012

0.00010

0.00008

0.00006

0.00004

0.00002

0.00000

Wearout Cycles to Failure

Pro

ba

bil

ity D

en

sit

y

10000

0.6113

0

Weibull, Shape=4.33205, Scale=11777.6, Thresh=0

Example of Wearout Distribution Plot

7f. Using Weibull Plots to Gain Performance Insight

Results for illustration of Wearout:

Data not reviewed in this presentation

Fail 95% CI (hours)

1% 822.9 5323.3

5% 2182.7 7053.5

10% 3354.0 7996.7

50% 9797.8 11701.7

Model: Weibull

• Shape 4.33205

• Scale 11777.56298

>61% fail after 10,000 hrs

• Note how shape is close to normal. This is by virtue of the shape value.

• Shape parameter =3.4-3 .6 closest to normal distribution

14000120001000080006000400020000

4

3

2

1

0

100.0%

80.0%

60.0%

40.0%

20.0%

0.0%

Example of Wearout

Occu

rre

nce

s_

Bin

_S

ize

_1

00

_W

O

Cu

m P

ct_

Bin

_S

ize

_1

00

_W

O

Occurrences_Bin_Size_100_WO

Cum Pct_Bin_Size_100_WO

Variable

Scatterplot of Occurrences_Bin_, Cum Pct_Bin_Size vs Grouped Cycles_B

Wearout



Slide 59

Documents

Demystifying Weibull - Minitab · 1. Weibull Distribution Plotting a) The big picture of today [s presentation b) Scope c) Overview: analytical flow Minitab Insights 9/13/2016 This