Design of Experiments Review

Client, Enterprise & Competitive Intelligence for Product, Process & Systems Innovation

Dr. Rick L. Edgeman, University of Idaho

Six Sigma

IXCUSTOMER & COMPETITIVE INTELLIGENCE

FOR SYSTEMS INNOVATION & DESIGN

S IGMAS DEPARTMENT OF

STATISTICSDR. RICK EDGEMAN, PROFESSOR & CHAIR – SIX SIGMA BLACK BELT

[email protected] OFFICE: +1-208-885-4410



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IXS IGMAS DEPARTMENT OF

STATISTICS

Design of Experimentsand

2k Factorial Designs



Six Sigma

IXS IGMAS

DEPARTMENT OF

STATISTICS

a highly structured strategy for acquiring, assessing, and applying customer, competitor, and enterprise intelligence for the purposes of product, system or enterprise innovation and

design.



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The Scientific Context of Quality

Improvement

The Scientific Method & Informed Observation;Data Driven Decision Making;

Directed Experimentation;

2k Factorial Experiments, Interaction and Scree Plots;

2k-p Fractional Factorial Designs;Central Composite Designs & Response Surface Methods;

Process Optimization - Selecting Your Settings.



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The Scientific Method of Informed Observation

Design ofExperiments

Sin



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Soliciting, Hearing & Acting Upon theVoice of the Process

“In God we trust. ... all others must bring data.”

Thank you Dr. Freud.It is key to effective

DECISION-MAKING!

DATA!You are DRIVEN!

Data-Driven Decision Making



Six SigmaWhere and What Do We Measure



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Directed Experimentation

Regression data is commonly observational in nature, having arisen simply from observing the response variable, Y, and noting the values of the driver variables which led to the response.

In contrast, data in an experimental design setting usually arise from planning or setting the values of interest of the driver variables and then observing the response variable.



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Experimental DesignsThe scheme used to determine the settings of the

drivervariables and of collecting the data is referred to as anexperimental design. Three very useful classes ofexperimental designs are:

2k factorial designs,

2k-p fractional factorial designs, and

Central Composite Designs (CCD)



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2k Factorial Designs2k factorial designs are experimental designs for which there are k factors (or driver variables) and each of these factors will be investigated at 2 levels,“high” and “low” or, symbolically, “+” and “-”.

All possible combinations of factor levels are used inthe investigation. That is, if there are k= three driver variables, then the data that would be collected would be represented as follows:



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23 Factorial Design Data

X1 X2 X3 Y

- - - Y1

- - + Y2

- + - Y3

- + + Y4

+ - - Y5

+ - + Y6

+ + - Y7

+ + + Y8



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The 23 Full Factorial Model

The full factorial model when k = 3 is given by:

Y = 0 + 1X1 + 2X2 + 3X3 + 12X1X2 + 13X1X3 + 23X2X3 + 123X1X2X3 +

It is rare to investigate the “three-factor interaction” term. A replicated 2k design (r2k)would gather r observations under each of the2k (factorial) combinations previously listed.



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Modified 23 Full Factorial Data Set

X1 X2 X3 X1X2 X1X3 X2X3 X1X2X3 Y

- - - + + + - Y1

- - + + - - + Y2

- + - - + - + Y3

- + + - - + - Y4

+ - - - + + + Y5

+ - + - + - - Y6

+ + - + - - - Y7

+ + + + + + + Y8



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STATISTICS

Example:Optimization of a

Flexible Packaging Material Process



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A 24 Factorial Design Example:Seal Strength of Flexible Packaging Material

A key characteristic of a flexible packaging material is its seal strength, measured in grams / square inch.

This is the force required to separate the seal once it has been made.

A flexible packaging material manufacturer has identified four variables which are believed to influence the seal strength (Y) of a particular material and has specified operating ranges for these variables which, it is thought, are broad enough to identify the impact of the variable if, in fact, there is an impact. These variables follow.



Six SigmaFlexible Packaging Material

VariablesResponse Variable: Y = Seal Strength (gm/si)Driver Variables: High = +1 Low = -1

Temperature in Degrees 300 250Pressure psi 100 80Material Thickness (inch) .03 .02Dwell in Seconds .20 .10



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Flexible Packaging Material Data

Degrees Pressure Gage Dwell Strength

-1 -1 -1 -1 150 -1 -1 -1 1 158 -1 -1 1 -1 141 -1 -1 1 1 163 -1 1 -1 -1 160 -1 1 -1 1 164 -1 1 1 -1 147 -1 1 1 1 168 1 -1 -1 -1 153 1 -1 -1 1 159 1 -1 1 -1 149 1 -1 1 1 160 1 1 -1 -1 170 1 1 -1 1 163 1 1 1 -1 171 1 1 1 1 178



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Factor Main Effects for Seal Strength of aFlexible Packaging Material

DwellGagePressureDegrees

165.0

162.5

160.0

157.5

155.0

Str

engt

hMain Effects for Flexible Packaging Material



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DegreesLow = -1 High = 1

High = 1 Gage Low = -1

PressureLow = -1 High = 1

High = 1

Dwell

Low = -1

163 168 160 178

158 164 159 163

141 147 149 171

150 160 153 170

Graphic Representation:Flexible Packaging Material Example



Six Sigma DegreesLow = -1 High = 1



High = 1

Dwell

Low = -1

163 168 160 178

158 164 159 163

141 147 149 171

150 160 153 170

Main Effect for Degrees

Right Cube vs. Left Cube:(153 + 149 + … + 178)/8 - (150 + 141 + … + 168)/8

= 1303/8 - 1251/8 = 162.875 - 156.375 = 6.50






High = 1

Dwell

Low = -1

163 168 160 178

158 164 159 163

141 147 149 171

150 160 153 170

Right Faces vs. Left Faces [(160 + … + 168 + 170 + … + 178)/8] - [(150 + … + 163 + 153 + … + 160)/8]

= 165.125 - 154.125 = 11.0

Main Effect for Pressure






High = 1

Dwell

Low = -1

163 168 160 178

158 164 159 163

141 147 149 171

150 160 153 170

Main Effect for Gage

Back Faces vs. Front Faces (141 + 163 + ….. + 178)/8 - (150 + 158 + ….. + 163)/8 =

159.625 - 159.625 = 0.0






High = 1

Dwell

Low = -1

163 168 160 178

158 164 159 163

141 147 149 171

150 160 153 170

Main Effect for Dwell

Top Faces vs. Bottom Faces

(158 + 163 + …. + 163 + 178)/8 - (150 + 141 + … + 170 + 171)/8 =

164.125 - 155.125 = 9.0



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1

1

-1-1

1

1-1

-1

1 1

-1-1

1

1-1

-1

1

1

-1-1

1

1

-1

-1

Degrees

Pressure

Gage

Dwell

Two Factor Interactions for Strength Data

Interaction Effects for Seal Strength of a

Flexible Packaging Material






High = 1

Dwell

Low = -1

163 168 160 178

158 164 159 163

141 147 149 171

150 160 153 170

Interaction Effect for Dwell with Degrees

(150 + 141 + 160 + 147 + 159 + 160 + 163 + 178)/8 -(158 + 163 + 164 + 168 + 153 + 149 + 170 + 171)/8 =

157.25 - 162.00 = -4.75



Six Sigma Analysis of Variance for Seal Strength

Source DF SS MS Fcalc Pvalue

Degrees 1 169.00 169.00 12.95 0.016Pressure 1 484.00 484.00 37.09 0.002Gage 1 0.00 0.00 0.00 1.000Dwell 1 324.00 324.00 24.83 0.004

Degrees*Pressure 1 72.25 72.25 5.54 0.065Degrees*Gage 1 42.25 42.25 3.24 0.132Degrees*Dwell 1 90.25 90.25 6.92 0.047Pressure*Gage 1 12.25 12.25 0.94 0.377Pressure*Dwell 1 30.25 30.25 2.32 0.188Gage*Dwell 1 156.25 156.25 11.97 0.018

Error 5 65.25 13.05

Total 15 1445.75



Six SigmaWhich Terms are Important?An Application of Scree Plots

10987654321

500

400

300

200

100

0

Index

Sum

_Sqr

s

Sum of Squares Scree Plot for Flexible Packaging Material

Sums of Squares Above the LineMay be Associated with ActiveEffects and Interactions.

SS Below the LineMay be Associated with Inactive Effects & Interactions.



Six Sigma Regression AnalysisStrength = 160 + 3.25 Degrees + 5.50 Pressure + 0.000 Gage + 4.50 Dwell + 2.12 temp*pr + 1.62 temp*gag - 2.38 temp*dw + 0.875 pres*gag - 1.37 pres*dw + 3.13 gage*dw

Predictor Coefficient Effect Std.Dev. Tcalc

Pvalue

Constant 159.625 ----- 0.9031 176.75 0.000

Degrees 3.250 6.500 0.9031 3.60 0.016 Pressure 5.500 11.000 0.9031

6.09 0.002 Gage 0.000 0.000 0.9031 0.00 1.000 Dwell 4.500 9.000 0.9031 4.98 0.004 temp*pr 2.125 4.250 0.9031

2.35 0.065 temp*gag 1.625 3.250 0.9031

1.80 0.132 temp*dw -2.375 -4.750 0.9031 -2.63 0.047 pres*gag 0.875 1.750 0.9031

0.97 0.377 pres*dw -1.375 -2.750 0.9031

-1.52 0.188 gage*dw 3.125 6.250 0.9031

3.46 0.018

S = 3.612 R2 = 95.5% R2adj =

86.5%



Six SigmaAnalysis of Variance:Overall Model

Source DF SS MS Fcalc Pvalue

Regression 10 1380.50 138.05 10.58 0.009Error 5 65.25 13.05

Total 15 1445.75

Unusual ObservationsObs Degrees Strength Fit StDev Fit Residual St Resid 10 1.00 159.000 154.875 2.995 4.125 2.04 R

R denotes an observation with a large standardized residual



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STATISTICS

End of Session

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Design of Experiments Review