3. Factorial Experiments (Ch.5. Factorial Experiments)

Hae-Jin ChoiSchool of Mechanical Engineering,

Chung-Ang University

3. Factorial Experiments

(Ch.5. Factorial Experiments)

1DOE and Optimization

2

Introduction to Factorials

Most experiments for process and quality improvement involve several variables. Factorial experimental designs are used in such situations. Specially, by a factorial experiment we mean that in each complete trial or replicate of the experiment all possible combinations of the levels of the factors are investigated. Thus, if there are two factors A and B with a levels of factor A and blevels of factor B, then each replicate contains all abpossible combinations.

Some Basic Definitions

Definition of a factor effect: The change in the mean response when the

factor is changed from low to high

40 52 20 3021

2 2

30 52 20 4011

2 2

52 20 30 401

2 2

A A

B B

A y y

B y y

AB

Main effect of A

Main effect of B

Interaction effect

between A and B

3

The Case of Interaction:

50 12 20 401

2 2

40 12 20 509

2 2

12 20 40 5029

2 2

A A

B B

A y y

B y y

AB

Main effect of A

Main effect of B

Interaction effect

between A and B

4

The Battery Life Experiment

An engineer is designing a battery for use in a device that will be subjected to some extreme variations in temperature. The only design parameter that he can select at this point is the plate material for the battery, and he has three possible choices.

When the device is manufactured and is shipped to the field, the engineer has no control over the temperature extremes that the device will encounter. It is known from experience that temperature will probably affect the effective battery life. However, temperature can be controlled in the product development laboratory for the purposes of a test

The engineer decides to test all three plate materials at three temperature levels,15, 70, 125 oF, because these temperature levels are consistent with the product end-use environment.

DOE and Optimization 5


A = Material type; B = Temperature

1. What effects do material type & temperature have on life?

2. Is there a choice of material that would give long life regardless of

temperature (a robust product)?


General Two-Factor Factorial Experiment

a levels of factor A; b levels of factor B; n replicates

This is a completely randomized design


1,2,...,

( ) 1,2,...,

1,2,...,

ijk i j ij ijk

i a

y j b

k n

Statistical Model of Two-factor Factorial Design

The observations may be described by

where is the overall mean effect, is the effect of the ith level of

factor A, is the effect of the jth level of factor B, is the

effect of the interaction between A and B. is a NID (0, )

random error component.

i

j ( )ij

ijk 2


Hypotheses for Two-factor Analysis


Hypotheses of no significant factor A effect, no significant factor B

effect, and no significant AB interaction. That is,

H

at least one

at least one

for all i, j

H at least one ( )

o

i

1 ij

: ...

:

: ...

:

:( )

:

1 2

1

1 2

1

0

0

0

0

0

0

a

o b

j

o ij

H

H

H

H

Extension of the ANOVA to Factorials

2 2 2

... .. ... . . ...

1 1 1 1 1

2 2

. .. . . ... .

1 1 1 1 1

( ) ( ) ( )

( ) ( )

a b n a b

ijk i j

i j k i j

a b a b n

ij i j ijk ij

i j i j k

y y bn y y an y y

n y y y y y y

breakdown:

1 1 1 ( 1)( 1) ( 1)

T A B AB ESS SS SS SS SS

df

abn a b a b ab n



ANOVA Table – Fixed Effects Case



SS yy

abni

a

j

b

k

n

ijk 1 1 1

22...

SSy

bn

y

abnA

i

ai

1

2 2

.. ...

SSy

an

y

abnB

j

b j

1

2 2

. . ...



SSy

n

y

abnsubtotals

i

a

j

b ij

1 1

2 2

. ...

SS SS SS SSAB subtotals A B

SS SS SS SS SSE AB A B




ANOVA Table

Residual Analysis


Residual Analysis


Interaction Plot


DESIGN-EXPERT Plot

Life

X = B: Temperature

Y = A: Material

A1 A1

A2 A2

A3 A3

A: Material

Interaction Graph

Life

B: Temperature

15 70 125

20

62

104

146

188

2

2

22

2

2

MINITAB Practice


An engineer suspects that the surface finish of a metal part is

influenced by the feed rate and the depth of cut. She selects three

feed rates and four depths of cut. She then conducts a factorial

experiment and obtains the following data:

MINITAB Practice


Data input

MINITAB Practice


Two way ANOVA

Stat -> ANOVA -> Two way

Select response, row factor (control factor), and column factor (uncontrollable factor)

MINITAB Practice


Select Graph

Select residual plot (Four in one)

Select ‘feedrate’ and ‘depth of cut’ for residual

MINITAB Practice


AVOVA result

MINITAB Practice


Residual Analysis Result

MINITAB Practice


Residual Analysis results (vs factor level)

MINITAB Practice


Interaction plot

Stat -> ANOVA -> Interactions plot..

Select response and factors

3. Factorial Experiments (Ch.5. Factorial Experiments)

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