Six Sigma: The Statistical Tool Box Advanced Six-Sigma Statistical Tools ASQ-RS Meeting, March 2003 Dr. Joseph G. Voelkel CQAS, RIT [email protected]www.rit.edu/~jgvcqa for material Rev: 3/25/03 2 CQAS BB Six-Sigma Statistical Tools Examples Gage R&R Studies Control Charts Experimental Design Taguchi/Robust Analysis of Variance Regression Basic Level, mostly Short/Long X-bar/R, p, c 2 k , 2 k–p , simple RSM control, noise, S/N one-, two-way one or two predictors www.rit.edu/~jgvcqa for material
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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
3
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?
4
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
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
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.