15t capability
TRANSCRIPT
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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
Pp = 2.0 and Ppk = 0.2 M or V
Pp = 1.5 and Ppk = 0.5 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