15t Capability

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Process CapabilityAnalysis

Process Capability Indices: Pp, Ppk

Observed DPM, Expected DPM


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TopicsI. Concept of Process Capability

II. Commonly Used Process Capability Indices

Exclude Centering: Pp

Include Centering: Ppk

III. Interpreting Capability Indices

Identifying Mean Vs. Variation Concerns

IV. Process Capability and One-Sided Specifications

V. Process Capability Indices and DPM

VI. Sigma Level


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I. Concept of Process Capability measure how well the output of a process meets

specification limits.

Provide tool to help assess whether defects arerelated to mean or variation concerns.

LSL USLLSL USL

Mean = 20S= 1.82

DPM=6000

Mean = 23S= 0.8

DPM=6000


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Target, Tolerances &

Specifications Target (nominal) - desired value of a characteristic.

Tolerance - specifies an allowable deviation relative to a target valuewhere a characteristic is acceptable.

Example: Order Time Specification: 20 +/- 5 days Target___ Tolerance width ___ LSL___ USL ___

One Sided-Specification (LSL or USL) time < 25 days; fee > $700

Target

Upper Specification

Limit (USL)Lower Specification

Limit (LSL)

+ t- t


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Three Point Estimates

If you do not have specifications, you need to:

Estimate them by asking Process Owners or Customers (for theirexpectations).

Sample Questions for Process Owner: What is your estimate of the time the process will take?

Target

What are the minimum and the maximum times you expect?

LSL and USL

Note: it may take some analysis to finalize meaningful

specifications.


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Assessing Process Capability

(Low Dispersion,Off Target)

Ideal(On Target,

Low Dispersion)

(On Target,High Dispersion)

Compare:

Mean Target, and Variation - Tolerance Widths

Consider some possible conditions: Mean On or Off Target

Variation (Range)

Low or highrelative to tolerance width

Worst Condition(Mean Off Target,

High Dispersion)

LSL USL

LSL USL LSL USL LSL USL


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II. Process Capability Indices Common Indices Used in Process Capability

Pp, Cp

Ppk, Cpk

Key Issues:

Exclude or include process centering (mean).

Exclude: Pp, Cp

Include: Ppk, Cpk

Method of estimating variation

Pp, Ppk ~ use S- Sample Std Deviation Note: Cp, Cpk, ~ use inherent common cause or within subgroup

variation from Statistical Process Control (SPC) chart methods(covered in Black Belt Course)

Focus of this Course: Pp and Ppk


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Pp Index Pp measures the relationship between the tolerance width

and the predicted range of variation (estimated by 6S).

Pp does not consider the location of the mean andtherefore represents thepotentialof the process to meetspecifications ifthe mean were on Target.

6

LSL-USL=

sigma6

Tolerance WidthPp=

For Pp, estimate using sample std deviation, S

Sample Standard DeviationIn Excel: S=stdev(array) ( )

=

n

i

i

nxS x

1

2

1=


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Pp Examples

For the same Pp, the # of defects may vary significantly.

T

LSL USL

T

Case 1: Case 2:

defects defects

LSL USL

Case LSL T USL Mean StDev PpEstimated

DPM

1 15 20 25 20.5 1.2 1.4 91

2 15 20 25 22.0 1.2 1.4 6,210


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Ppk Index Ppk takes into account any difference between the

target and the process mean (X).

LSL USL

TargetX = Mean

PpkUPpkL

3=U

XUSLPpk

3=L

LSLXPpk

Ppk = min (PpkL, PpkU)

estimate s using sample std deviation, S


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Pp and Ppk Calculation Exercise Suppose Order Time is 20 +/- 5 days

Sample 50 Orders: Mean = 20.5; S= 1.2

Compute Pp and Ppk?


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Difference Between Pp & Ppk By including how close the mean is to target, Ppk

provides a better relationship to defects (DPM).

T

Case 1: Case 2:

defects

LSL USL

Case LSL T USL Mean StDev Pp PpkEstimated

DPM

1 15 20 25 20.5 1.2 1.4 1.3 912 15 20 25 22.0 1.2 1.4 0.8 6,210

LSL USL

T defects


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Pp and Ppk Criteria Desired Pp and Ppk values may vary by industry / application.

(Most industries require at least a Pp and Ppk > 1.33)

Some general goals are as follows:

Criteria FormulaModerate

Goals

Six Sigma

Goals

Pp (or Cp) Pp > 1.67 Pp > 2.0

Ppk (or Cpk) Ppk > 1.33 Ppk > 1.5

6

LSLUSL

)3

,3

min(

LSLXXUSL

*Assumes 2-Sided Specification


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III. Interpreting Pp and Ppk

By definition: Ppk


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Pp and Ppk Examples

Objective: Ppk > 1.67 and Pp > 1.67 For each of the following, assess if likely have

a mean and/or variation problem:

Pp = 0.4 and Ppk = 0.3 M or V

Pp = 0.8 and Ppk = -0.2 M or V




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Summary of Combinations Combinations:

1. Mean and Variation Problem

High Dispersion, Off Target ~Low Pp, Ppk


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IV. Capability Indices for One-Side

Specifications For some processes, we may only have one

specification limit. Order Time < 25 days (USL = 25 days)

Fee > $700 (LSL = 700)

Here, we only compute Ppk.

3=

XUSLPpk

If USL only:

If LSL only:3

=LSLX

Ppk

For Ppk, estimate using sample standard deviation, S


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Process Improvement For one-sided specification, we may improve Ppk (and

reduce defects) by either shifting mean away fromspecification limit and/or reduce standard deviation.

USL=25

defects

USL1) Shift Mean 18Same Sigma (S=1.2)

USL

2) Reduce S 0.52

Same Mean (22)

Current State

Ppk = 0.8Mean = 22; S= 1.2

Future State

Ppk = 1.9

Ppk = 1.9

-OR


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V. Process Capability and DPM We often relate process capability with Observed or Expected DPM.

Observed DPM based on number of samples actually

observed outside the specification limits divided by totalnumber of samples.

Expected DPM based on fitting sample data to a distributionand determining the probability of a defect.

To calculate Expected DPMs, we need Specification Limits, Mean,Standard Deviation, and an Assumed Distribution (Normal) to find theprobability of a defect.

DPM = Probability of a Defect * 1 Million

Understanding how to compute Probability is a Black Belt Skill. Still,Green Belts should be able to use software to estimate DPMs.

QE Tools Process Capability Summary

DPM Calculator, or Process Capability Summary


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DPM Calculator Normal

(use if you only have summary statistics)

QETools Process

Capability Summary DPM Calculator

Enter the mean, standard

deviation, andspecifications in theyellow boxes to get DPM.

Mean = 20.5 andStandard Deviation = 1.2

Specification: 20 +/- 5

DPM Calculator Assuming Normal Distribution

DPM given Average, Sigma, USL, LSLinsert values in white boxes.

Average 20.5Standard Deviation 1.2

USL 25 Upper Specification LimitTarget 20 Nominal Value (optional)

LSL 15 Lower Specification Limit

DPM Probability Defect DPMDPM > USL 0.000088 88.4DPM < LSL 0.000002 2.3

DPM Total 0.000091 90.7

Pp 1.39Ppk 1.25

Predicted DPM 90.7Quality Yield 99.99%


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DPM Example

Suppose mean

shifts from 20.5 to22 days with thesame S= 1.2.

Is this a meanand/or variationconcern?

Hint: compare Pp,Ppk, DPM with priorexample

Average 22Standard Deviation 1.2

USL 25 Upper Specification Limit

Target 20 Nominal Value (optional)LSL 15 Lower Specification Limit

DPM Probability Defect DPMDPM > USL 0.006210 6,209.7DPM < LSL 0.000000 0.0DPM Total 0.006210 6,209.7

Pp 1.39Ppk 0.83

Predicted DPM 6,209.7Quality Yield 99.38%

Average 20.5Standard Deviation 1.2

USL 25 Upper Specification LimitTarget 20 Nominal Value (optional)

LSL 15 Lower Specification Limit

DPM Probability Defect DPMDPM > USL 0.000088 88.4DPM < LSL 0.000002 2.3DPM Total 0.000091 90.7

Pp 1.39Ppk 1.25

Predicted DPM 90.7Quality Yield 99.99%


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Process Capability - QETools QE tools may be used to compute process capability (Pp/

Ppk), Observed DPM, and Expected DPM.

Two Options:

Process Capability Summary Normal (use if data are normal)

Process Capability Summary Non-Normal (use if data areskewed right or follow an exponential distribution).

Note: if data do not follow these patterns, use ObservedDPM (e.g., if bi-modal).


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Process Capability Normal Suppose you have wait time with a USL = 40.

See wait-time-B in excel file: capability.xls

Note:One-Sided Specification


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Sample Results: Wait-Time QETools Process Capability Summary Normal

Process Capability Graphical SummaryWait-Time-B

Summary Statistics

USL 40.00LSL

N 70

Mean 38.21

Sigma (R-bar) 4.28

Sigma (Overall) 4.55Range 22.00

Process Capability

CpCpk 0.14

Pp

Ppk 0.13

xp. vera er ormanceDPM < LSL

DPM > USL 347,424.0

DPM Total 347,424.0

Observed PerformanceDPM < LSLDPM > USL 342,857.1

DPM Observed 342,857.1

Histogram

0

2

4

6

8

10

12

14

26.90 29.10 31.30 33.50 35.70 37.90 40.10 42.30 44.50 46.70 48.90 51.10

USLDATA are

Close toNormal347K DPM


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Non-Normal Distributions For non-normal processes (e.g., skewed right

distributions), you should either: Use non-normal distribution (e.g., weibull distribution in QETools)

Use the % Observed DPM method (i.e., # of observed defects / #samples).

Careful with interpreting % observed with small sample sizes(e.g., N < 30)

USL


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Histogram

0

5

10

15

20

25

11.20

12.80

14.40

16.00

17.60

19.20

20.80

22.40

24.00

25.60

27.20

28.80

30.40

32.00

33.60

35.20

36.80

38.40

USLLSL

Process Capability Non-normal

(Skew Right) Data on order fill time for shipping industrial PCs.

Goal: 20 +/- 5 days Is this a Mean Off Target or Variation Problem?

USL 25.00

LSL 15.00

N 155

Mean 20.61

St Dev 4.51

Results: QETools >> Process Capability Summary Non Normal

Assumed Distribution: Weibull

Process CapabilityPp 0.39

Ppk 0.24

Exp.Overall PerformanceDPM < LSL 102,638.7

DPM > USL 149,455.2

DPM Total 252,093.8

Observed PerformanceDPM < LSL 58,064.5

DPM > USL 154,838.7

DPM Observed 212,903.2


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Process Capability

Six Sigma Project Measure Phase assess current state: Pp, Ppk, DPM

Analyze Phase identified sources of variation: in-house delays related to incomplete order information and

components not in stock.

Improve Phase created new systems to enter new order

information and manage inventory. Re-Assess Capability After Improvements are made.

CurrentState

PostImprovements

Sample 155 55

Mean 20.6 20.4

Sigma 4.51 2.50

Pp 0.39 0.66Ppk 0.24 0.61

Actual DPM 212,903 48,270


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VI. Sigma Level Converters

Yield DPM (DPMO) Sigma Level

30.9% 690,000 1.069.2% 308,537 2.0

93.3% 66,807 3.0

99.4% 6,210 4.099.98% 230 5.0

99.9997% 3.4 6.0

DPMO Defects per million opportunity

Given DPM or DPMO estimate, we may equate to a sigma level

(this is an index scale). Some sample sigma levels are:


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Sigma Level QE Tools QE Tools will compute Sigma Level given:

Yield DPM or DPMO level

Sigma Level from Quality Yield (Yield = 1 - % defective)

Quality Yield % Non-Conforming DPM Z Sigma Level

0.9999966 0.0000034 3.4 4.50 6.00

Sigma Level for DPM for Non Centered Process - Assumes 1.5 sigma shift (Normal Distributio

DPM Yield Prob Z Sigma Level

6210 99.38% 0.0062100 2.4999809 4.00

*QETools Process Capability Summary Sigma Level


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Summary Process capability indices (and Sigma Level) are additional

performance measures used in Six Sigma projects. Pp and Ppk together help identify if defects are related to

Mean and/or Variation problems. For example, Mean off target/low variation (Pp high, Ppk low);

Mean on-target/high variation (Pp and Ppk low, Pp=Ppk); Mean off target and high variation (Pp/Ppk low, Ppk

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