A introduction to Design of Experiments

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D How to conduct a DOE

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DHow conduct a DOE
An DOE (Design Of Experiments) is a test setup
with multiple factors on various levels combined in
one experiment. Instead of testing each factor
individually in a DOE multiple factors are variated
at once to reduce the amount of test with the
possibility to analyze interactions between factors.
What is a Factor
A factor is a input for a experiment that can change
the output when variating. Its like a dimmer (a
factor) of a lamp when turning the knob the
brightness of the lamp changes.

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DWhy use a complex DOE instead of
the standard approach?
With the classical approach
only one input factor is
changed to determine the
influence of it on the output
with a DOE more than one
input is changed to see the
influence of multiple factors
on the output.
Classical DOE
Test Factor 1 Factor 2 Factor 3 Factor 4
1 High High High High High
2 High High High High Low
3 High High High Low High
4 High High High Low Low
5 High High Low High High
6 High High Low High Low
7 High High Low Low High
8 High High Low Low Low
9 Low Low High High High
10 Low Low High High Low
11 Low Low High Low High
12 Low Low High Low Low
13 Low Low Low High High
14 Low Low Low High Low
15 Low Low Low Low High
16 Low Low Low Low Low
So the answer is to reduce the amount of experiments.
In other words reduce cost and time.

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DWhat is important for a successful
DOE?
1. The output can be measured, preferable in continuous
scale!!
2. The influencing factors are known
3. Important Factors can be controlled (variated on a desired
level or fixed on a constant level)
4. Try to control Noise (uncontrollable factors) or record
them (Environment temperature, Air pressure, different
operators, ect)
5. Keep the DOE simple as possible
6. DO the Confirmation Run!

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DLevel of the factor
Factor Levels
Shape of lamp Ball, Cone, Candle
Power 1, 2, 9, 30, 40 60 100Watt
Setting on the
dimmer
0%…100%
Input current 0...230V
Color of glass Clear, White, Silver, Green, Red
Type of lamp Light bulb, LED, TL
Armature Silver reflector, White reflector, No
reflector
Settings of levels
●
Try to chose realistic values for the levels (not impractical high or low outside
machine settings)
●
Avoid impossible combinations of the levels with other factors in the experiment

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DThe test Arrays
Name Full factorial Response surface Orthogonal
Type Full Factorial Array Box-Behnken design Orthogonal Array
Central Composite
design
Plackett–Burman Array
Amount of
tests
High (all combinations) Medium Small
No interactions
Usage Simple More complex Simple no interactions
Complex with
interactions
Interactions All interactions First level interactions Yes possible
Response
Surface
design
No Yes No

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DPoints in the array
Tests
Factors
A B C
1 -1 -1 -1
2 1 -1 -1
3 -1 1 -1
4 1 1 -1
5 -1 -1 1
6 1 -1 1
7 -1 1 1
8 1 1 1
9 0 0 0
10 0 0 0
11 0 0 0
12 -1.682 0 0
13 1.682 0 0
14 0 -1.682 0
15 0 1.682 0
16 0 0 -1.682
17 0 0 1.682
18 0 0 0
19 0 0 0
20 0 0 0
• Corner points 8
extremes
• Center points

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DArray selection
2 Levels 3 Levels
Factors Full Factorial Orthogonal Full Factorial Orthogonal Box-Behnken
Test Runs Test Runs Test Runs Test Runs Test Runs
2 4 4 9 9
3 8 4 27 9 16
4 16 8 81 9 26
5 32 8 243 18 45
6 64 8 729 18 54
7 128 8 2187 18
8 256 12 6561 27
The choice between arrays for a DOE. Looking to the table below it is clear that quite
fast a Full Factorial array is not economical.

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DAmount of test with different arrays
Test Full Factorial Array Box-Behnken design BB3 Orthogonal Array L9
Factor 1 Factor 2 Factor 3 Factor 1 Factor 2 Factor 3 Factor 1 Factor 2 Factor 3 Factor 4
1 -1 -1 -1 0 0 0 -1 -1 -1 -1
2 0 -1 -1 -1 -1 0 -1 0 0 0
3 1 -1 -1 1 -1 0 -1 1 1 1
4 -1 0 -1 -1 1 0 0 -1 0 1
5 0 0 -1 1 1 0 0 0 1 -1
6 1 0 -1 0 -1 -1 0 1 -1 0
7 -1 1 -1 0 0 0 1 -1 1 0
8 0 1 -1 0 -1 1 1 0 -1 1
9 1 1 -1 0 1 -1 1 1 0 -1
10 -1 -1 0 0 1 1
11 0 -1 0 -1 0 -1
12 1 -1 0 -1 0 1
13 -1 0 0 1 0 -1
14 0 0 0 1 0 1
15 1 0 0 0 0 0
16 -1 1 0
17 0 1 0
18 1 1 0
19 -1 -1 1
20 0 -1 1
21 1 -1 1
22 -1 0 1
23 0 0 1
24 1 0 1
25 -1 1 1
26 0 1 1
27 1 1 1

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D2 or 3 level designs
2 Levels 3 Levels
Amount of
tests
Low Higher
Non linear
response
No Yes
There are arrays with 2 or 3 levels and even with different amount of levels in
one array.
When generating Full Factorial array for each factor a dedicated amount of
levels can be chosen.

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DInteractions
An interaction is when the result is not the sum of two factors. With an
interaction it can occur that the result of the to factors is lower or
higher as the sum of the result.
●
An interaction is not a common thing
●
Adding interactions will increase the array size especially in a 3 level
array
●
If it is not logic that there is an interaction between factors do not
include it!
Interaction
No interaction
with the typical crossing line
1

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D Example

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DExample with interactions
Factor Level1 Level2
Size 10mm 20mm
Weight 1kg 10kg
Color White Black
Interaction1 Size Weight
Interaction2 Size Color
Interaction3 Weight Color
We want to create a DOE with the following factors

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DPicking array
There are 3 factors, in a
L4 array is space for 3
Factors on 2 levels.
But we also want to
include interactions so a
bigger array is needed.
In an L8 is space for 7
factors.
Open this array.

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DConstruct array
Now add the first interaction. Column C is
now used for this interaction.
The second interaction. Column E is also
used for an interaction.
The third interaction. Column F is now
also used for an interaction.
The name of the factors and levels are
added.

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DBuilding and testing the samples
Now build the samples according the array.
Fist sample 10mm, 1kg and white etc.
Some important points
• Always build and test the complete array.
• When adding repetitions it is preferred is not to test all the
replicates on a row (samples with the same levels) but first
finish the first replicate then the next. This is to randomize
the order to prevent drift over time in the result.
• When testing try not to test all factors on the first level
than on the second level but try to randomize.
• Do not add an extra test in the array except Center points
this will create an unbalanced array, and will lead to wrong
results.
8samples

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DAnalyzing the result
•Size 20mm gives a higher output this
factor is also significant according the
Anova
•Weight Small influence not significant
But it looks that it has a big influence on
the variation! This can be seen in the
Response graph Column D and E.
•Color Small not significant influence
•The interaction between Size and Weight
is significant. So if Weight is combined
with Size Weight is significant (see
Anova)! In the Response graph Column H
and I the typical crossing lines that is an
indication for an interaction.
• The interaction is the cause of the
bigger variation of Weight at 1
kg!

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DAnova DOE
To check on significance influence of each factor in the
DOE. If a factor is not significant it can bee pooled, and the
influence of the non pooled factors is getting bigger.
Weight and color are not
significant.
After pooling Color (the
least significant factor)
weight is still not
significant.

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DDo the Confirmation Run!!!
In the Anova Dialog the optimal setting can selected in the column
Level confirmation run. To confirm output (14.43) of the DOE build
samples according these settings!
When an interaction is significant only set the level of the interaction
do not set them individual. If you do both you set the influence of the
factor twice!

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D Software

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DDevelve Software
• Gaug R&R
(measurement system
analysis)
• Basic statistics
• DOE
• Weibull analysis
(Reliability)

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DSet the DOE mode

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DSet the DOE mode

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DHow the DOE mode works
A B C D E F G
All
factors
Levels small Big heavy Light Dark white
10 10 10 10
20 20 20 20
30 30 30 30
40 40 40 40
50 50 50 50
60 60 60 60
70 70 70 70
80 80 80 40
Mean 45 35 55 40 50 45 45
n 8 4 4 4 4 4 4
Median 45 35 55 40 50 45 45
STDEV 24.49 23.8 23.8 25.82 25.82 28.87 23.8
Size Weight Color

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D Exercise

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DPaper Helicopter

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D
Select factors
and interactions

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D
Select and
construct Array

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D Build and Test

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D Test and analyze

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D Confirmation Run

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DTerms and abbreviations
Anova: Analyze of variance to analyze the differences among group means
Confirmation Run: Build the result of the experiment to confirm the result
DOE: Design Of Experiments
Factor: A factor is a input for an experiment that can change the output
when variating
Full Factorial array: Experiment with all the combination of factors on all
levels
Fractional array (Orthogonal): Orthogonal Arrays constructed with a
fraction of a Full factorial array but the orthogonality (in-dependency)
between the factors is kept
Interactions: Interaction is a kind of effect that occurs as two or more
objects have an effect upon one another.
Levels: Setting of a factor (High, Low etc.)
Response: Output of a setting
Test array: Array with all the tested factors on different levels
Center points: Center points are half way between all the high and low
values in the experiment so they are in the center (the row with only 0 in the
array)