Advanced Six-Sigma Statistical Tools - RIT - Peoplejgvcqa/OtherFiles/RSQC03_Voelkel.pdfAdvanced Six-Sigma Statistical Tools ... Rev: 3/25/03 2 CQAS BB Six-Sigma Statistical Tools

1

Six Sigma: The Statistical Tool Box

Advanced Six-Sigma Statistical Tools

ASQ-RS Meeting, March 2003

Dr. Joseph G. VoelkelCQAS, [email protected]/~jgvcqa for material

Rev: 3/25/03 2CQAS

BB Six-Sigma Statistical Tools

ExamplesGage R&R StudiesControl ChartsExperimental DesignTaguchi/RobustAnalysis of VarianceRegression

Basic Level, mostly

Short/LongX-bar/R, p, c2k, 2k–p, simple RSMcontrol, noise, S/None-, two-wayone or two predictors

www.rit.edu/~jgvcqa for material

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Rev: 3/25/03 3CQAS

Why Are We Here?

Show example problems where basic Black Belt tools do not perform well.Second, show methods that might be used to solve such problems.Third, show how the methods can indeed solve such problems.

Rev: 3/25/03 4CQAS

People Solve Problems

Good methodsprovide insight, often visualquantify the extent of the problemmay predict results of improvements

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Rev: 3/25/03 5CQAS

Case Study 1: Reducing Dimensional Variation in Radiator Cores

Radiator CoreWant dimensional stability across the core (“positions”)

Not being achievedControl factors

CS, PSNoise factors

Position, Frame

Rev: 3/25/03 6CQAS

Radiator-Core Experiment

Fr

CS PS

CoreTwo cores tested, for each CS/PS/Frame combination

PosEach core measured at 3 positions

Each position measured twice

Meas

Seven frames used

Taguchi Approach?

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Rev: 3/25/03 7CQAS

Radiator-Core: Taguchi ApproachTaguchi approach

After averaging data over two measurements

Noise FactorsFrame

Control Factors 1 2 7Pos s

CS PS Rep 1 2 3 1 2 3 … 1 2 31 1 1 Y Y Y Y Y Y Y Y Y 0.436 0.0411 1 2 Y Y Y Y Y Y Y Y Y 0.432 0.0422 1 1 Y Y Y Y Y Y Y Y Y 0.434 0.0442 1 2 Y Y Y Y Y Y Y Y Y 0.430 0.0451 2 1 Y Y Y Y Y Y Y Y Y 0.429 0.0411 2 2 Y Y Y Y Y Y Y Y Y 0.436 0.0392 2 1 Y Y Y Y Y Y Y Y Y 0.429 0.0392 2 2 Y Y Y Y Y Y Y Y Y 0.426 0.042

Y

Rev: 3/25/03 8CQAS

Radiator-Core: Taguchi ApproachMain Effect Plots

PSCS

2121

0.4332

0.4324

0.4316

0.4308

0.4300

Yba

r

PSCS

2121

0.0429

0.0423

0.0417

0.0411

0.0405

sd

Y

s

5

Rev: 3/25/03 9CQAS

Radiator-Core: Taguchi ApproachANOVA/Regression and % contributions

s Term Coef T P %Contr Constant 0.041552 89.70 0.000 CS -0.001329 -2.87 0.046 50 PS 0.000818 1.77 0.152 19 CS*PS -0.000453 -0.98 0.383 6 Error 25

Y Term Coef T P %Contr Constant 0.431497 398.44 0.000 CS 0.001604 -1.48 0.213 23 PS 0.001807 -1.67 0.171 29 CS*PS 0.000884 -0.82 0.460 7 Error 41

Rev: 3/25/03 10CQAS

2 CS × 2 PS × 2 Reps × 7 Frames × 3 Pos = 168 data pointsSo, 168 d.f.

Radiator-Core: Critique of Taguchi Approach

Noise FactorsFrame

Control Factors 1 2 7Pos df s df

CS PS Rep 1 2 3 1 2 3 … 1 2 31 1 1 Y Y Y Y Y Y Y Y Y 0.436 1 0.041 201 1 2 Y Y Y Y Y Y Y Y Y 0.432 1 0.042 202 1 1 Y Y Y Y Y Y Y Y Y 0.434 1 0.044 202 1 2 Y Y Y Y Y Y Y Y Y 0.430 1 0.045 201 2 1 Y Y Y Y Y Y Y Y Y 0.429 1 0.041 201 2 2 Y Y Y Y Y Y Y Y Y 0.436 1 0.039 202 2 1 Y Y Y Y Y Y Y Y Y 0.429 1 0.039 202 2 2 Y Y Y Y Y Y Y Y Y 0.426 1 0.042 20

8 160

Y

df are grouped—less information

In Taguchi approach:

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Rev: 3/25/03 11CQAS

Radiator-Core: ANOVA ApproachSplit apart d.f.—More InformationStep 1: ESD (Extended Structure Diagram) tool used

CSTF(2) PSTF(2) FrameTF(7)

SectionUA(1)

PosCF(3)CoreUA(2)

MeasCA(2)

CS PS

Rev: 3/25/03 12CQAS

Radiator-Core: ANOVA Approach

Step 2: translate this to an ANOVA model

CSTF(2) PSTF(2) FrameTF(7)

SectionUA(1)SectionUA(1)

PosCF(3)CoreUA(2)CoreUA(2)

MeasCA(2)MeasCA(2)

Y=CS | PS | FrCore (CS PS Fr)PosPos*CoreError [Meas(Core Pos)]

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Rev: 3/25/03 13CQAS

Radiator-Core: ANOVA ApproachStep 3a: Run the ANOVA model

Source DF SS MS F PCS 1 0.0008646 0.0008646 5.20 0.030PS 1 0.0010966 0.0010966 6.60 0.016Fr 6 0.2531569 0.0421928 253.80 0.000Pos 2 0.0367108 0.0183554 74.95 0.000CS*PS 1 0.0002625 0.0002625 1.58 0.219CS*Fr 6 0.0024635 0.0004106 2.47 0.048CS*Pos 2 0.0010048 0.0005024 2.05 0.138PS*Fr 6 0.0029055 0.0004842 2.91 0.025PS*Pos 2 0.0025371 0.0012685 5.18 0.009Fr*Pos 12 0.2212360 0.0184363 75.28 0.000CS*PS*Fr 6 0.0016036 0.0002673 1.61 0.182CS*PS*Pos 2 0.0011605 0.0005802 2.37 0.103CS*Fr*Pos 12 0.0065032 0.0005419 2.21 0.023PS*Fr*Pos 12 0.0045778 0.0003815 1.56 0.132CS*PS*Fr*Pos 12 0.0029688 0.0002474 1.01 0.452Core(CS PS Fr) 28 0.0046549 0.0001662 3.74 0.000Pos*Core(CS PS Fr) 56 0.0137153 0.0002449 5.51 0.000Error 168 0.0074635 0.0000444Total 335 0.5648860

Rev: 3/25/03 14CQAS

Radiator-Core: ANOVA ApproachStep 3b: FindComponentsof Variance

Full ModelSource MS 10000*VC %CS 0.000865 0.021 0%PS 0.001097 0.028 0%Fr 0.042193 7.505 44%Pos 0.018355 1.078 6%CS*PS 0.000263 0.003 0%CS*Fr 0.000411 0.044 0%CS*Pos 0.000502 0.015 0%PS*Fr 0.000484 0.057 0%PS*Pos 0.001269 0.061 0%Fr*Pos 0.018436 6.497 38%CS*PS*Fr 0.000267 0.018 0%CS*PS*Pos 0.000580 0.020 0%CS*Fr*Pos 0.000542 0.106 1%PS*Fr*Pos 0.000382 0.049 0%CS*PS*Fr*Pos 0.000247 0.001 0%Core(CS PS Fr) 0.000166 0.203 1%Pos*Core(CS PS Fr) 0.000245 1.003 6%Error 0.000044 0.444 3%

17.151 100%

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Rev: 3/25/03 15CQAS


Step 3b:FindComponentsof Variance

Full Model

Source MS10000

*VC %Fr 0.042193 7.5 44%Pos 0.018355 1.1 6%Fr*Pos 0.018436 6.5 38%Pos*Core 0.000245 1.0 6%Error 0.000044 0.4 3%Rest 0.0 2%

16.5 100%

Rev: 3/25/03 16CQAS


How come we didn’t see this using the Taguchi approach?Taguchi approach depends totally on control (CS, PS) & control x noiseeffects

Full Model

Source MS10000


16.5 100%

This is what the Taguchi approach analyzed !!!

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Rev: 3/25/03 17CQAS


Step 4. Graph the results of the analysisWhat graph(s) should we make?

Full Model

Source MS10000


16.5 100%

Rev: 3/25/03 18CQAS

Radiator-Core: Interaction PlotCan’t we do better?

1 2 3

7654321

0.50

0.45

0.40

0.35

Fr

Pos

Mea

n

Make graph connect more to physical reality

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Rev: 3/25/03 19CQAS

Radiator-Core: Interaction Plot

76543210

0.53

0.51

0.49

0.47

0.45

0.43

0.41

0.39

0.37

0.35

Fr/Pos

Y

Rev: 3/25/03 20CQAS

Radiator-Core: One More Plot

Full Model

Source MS10000


16.5 100%

76543210

0.53

0.51

0.49

0.47

0.45

0.43

0.41

0.39

0.37

0.35

Fr/Pos

Y

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Rev: 3/25/03 21CQAS

Radiator-Core: One More Plot

76543210

0.55

0.45

0.35

Frame1

Y

Rev: 3/25/03 22CQAS

Radiator-Core: Lessons Learned Push the Taguchi button?Run ANOVA that reflects your design !Pull apart those degrees of freedom!

Push the Interaction-Graph button?Make up problem-specific graphs !

76543210

0.53

0.51

0.49

0.47

0.45

0.43

0.41

0.39

0.37

0.35

Fr/Pos

Y

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Rev: 3/25/03 23CQAS

Case Study 2Unbalance: 2D Gauge R&R Studies

Rev: 3/25/03 24CQAS

Unbalance

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Rev: 3/25/03 25CQAS

Unbalance

Unbalance

0.750.75 1.501.50 2.252.25

3.00 oz, 0.5 in0.75 oz, 2.0 in

* 1.00 oz, 1.5 in

Unitsoz-in (1.5 oz-in)g-mm

CoordinatesPolar (r, θ)Cartesian (X,Y)

Rev: 3/25/03 26CQAS

UnbalanceAnother measure

Unbalance= 1.5 oz-inTake weight of impeller into account

Eccentricitye=U / munits: in or mm

Center of Axis of Rotation

Center of Mass of Rotating Body

Light impellerHeavy impeller

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Rev: 3/25/03 27CQAS

Unbalance: 2D Gauge R&R

0.750.75 1.501.50 2.252.25

20 repeat measurements of unbalance — 1 impeller, appraiser, machineHow would you summarize the variability in these measurements?

Rev: 3/25/03 28CQAS

Unbalance: 2D Gauge R&R

One approach that’s been usedMeasure r from origin for each pointUse regular gauge R&R measures

0.750.75 1.501.50 2.252.25

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Rev: 3/25/03 29CQAS

1D Analysis of 2D Data

Now estimate range where 99% of values would beFind s = sample standard deviation = 0.42Find 5.15s = 2.17 = EV

Rev: 3/25/03 30CQAS


But consider(X,Y)(X+1,Y+1)

90° rotation *All have same variation

0.750.75 1.501.50 2.252.25

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Rev: 3/25/03 31CQAS


0.0 0.5 1.0 1.5 2.0 2.5 3.0r

(X+1,Y+1). EV=1.40

90 deg rotation. EV=3.67

(X,Y). EV=2.17

0.750.75 1.501.50 2.252.25

What is a better way?

Rev: 3/25/03 32CQAS

Extending 1D Summaries to 2D

Fundamental point:engineering tolerance is a circleThe proposed two-dimensional method

repeatability (equip-ment variation) EVEV ≡ diameter of circle capturing p = 99% of such readings.

0.750.75 1.501.50 2.252.25

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Rev: 3/25/03 33CQAS

Comparison of 2D to 1D Summary

EV = 2.17, 1.40, 3.67?

0.750.75 1.501.50 2.252.250.750.75 1.501.50 2.252.25

EV = 3.73

Rev: 3/25/03 34CQAS

Example

3 appraisers measured each of 10 parts twice (3 × 10 × 2 = 60).For more validity, they did this at each of two time periods (60 × 2 = 120)Will sometimes act here as if 6 appraisers instead (6 × 10 × 2 = 120). This is (pretty much) OK.We will see how this technique can summarize variation, just as in the 1D case

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Rev: 3/25/03 35CQAS

Example. Step 1: Set up ESD

Appraiser(3) Parts(10)

Time(2)

Meas(2)

Rev: 3/25/03 36CQAS

Example: All 3 Appraisers

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

0.0008

-0.0008

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

0.0008

X

Y

6 “appraiser’s” × 10 impellers × 2 trials = 120

Adjust data (graphs) to meanof (0,0) for each impeller.

-8.00E-04

-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

-8.00E-04

-6.00E-04

-4.00E-04

-2.00E-04

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

X

Y

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Rev: 3/25/03 37CQAS

Example. Step 2:ANOVA-based Diameter Summaries

-0.002

-0.001

0

0.001

0.002

-0.002 -0.001 0 0.001 0.002

X

Y

%D.tol %TotalD.tol 100% 114%

R&R 58% 66%Reprocibility 56% 63% Oper/Time Oper/Time*ImpRepeatability 17% 20%Impeller 66% 75%Total 88% 100%

Data adjusted to a mean of (0,0) for each impeller.

6 “appraiser’s” × 10 impellers × 2 trials = 120

±0.0013"

Rev: 3/25/03 38CQAS

Step 3: Analysis. Matt Only

-0.002

-0.001

0

0.001

0.002

-0.002 -0.001 0 0.001 0.002

X

Y



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Rev: 3/25/03 39CQAS

Step 3: Analysis. Tom Only

-0.002

-0.001

0

0.001

0.002

-0.002 -0.001 0 0.001 0.002

X

Y



Rev: 3/25/03 40CQAS

Step 3: Analysis. Ben Only

-0.002

-0.001

0

0.001

0.002

-0.002 -0.001 0 0.001 0.002

X

Y



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Rev: 3/25/03 41CQAS

Step 4: Graphical Analysis. TomData = Averages (N=2) of each Part-Time combination

0.50.5 1.01.0

Time 1Time 2

EV

Tom

R&R = 40%AV = 32%EV = 24%

Rev: 3/25/03 42CQAS

Step 4: Graphical Analysis. BenData = Averages (N=2) of each Part-Time combination

0.50.5 1.01.0

Time 1Time 2

EV

Ben

R&R = 14%AV = 10%EV = 9%

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Rev: 3/25/03 43CQAS

Results

Standardization was done based on Ben’s techniques.Measurement variation greatly reduced



Rev: 3/25/03 44CQAS

Case Study 3:Multivariate Methods: Experimental Design

Many experiments: Multiple ResponsesMultiple Responses=Multivariate CaseExample: L18 Film Experiment

4 Factors24 Responses

Many are MD/TD typesEmphasis in this talk: Responses only

See how the responses are associated“How many different features exist among the responses?”

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Rev: 3/25/03 45CQAS

Multivariate Methods: Exp’l Design

The Responses Microsoft Excel Worksheet

How are the responses associated?How would you try to find this out?



Minitab Worksheet

1. Correlations2. Matrix Plot3. Add Col #’s4. CorrCode5. PC’s6. Plots

Minitab

Rev: 3/25/03 46CQAS

Case Study 4: SPC with a Twist

Consider this Process data40 subgroupsDimensional measurementSubgroup size 6. Six parts in a row sampled from a machine

How would you look at these data?

Minitab Worksheet

Minitab

Microsoft Excel Worksheet

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Rev: 3/25/03 47CQAS

Case Study 4: SPC with a TwistBut here is the real situation

40 subgroups, Subgroup size 6.Six parts sampled from a Six-Cavity machine

1 2 3

Now, how would you analyze these data?Minitab ProjectMinitab

Rev: 3/25/03 48CQAS

The Actual Mold



Documents

Advanced Six-Sigma Statistical Tools - RIT - Peoplejgvcqa/OtherFiles/RSQC03_Voelkel.pdfAdvanced Six-Sigma Statistical Tools ... Rev: 3/25/03 2 CQAS BB Six-Sigma Statistical Tools