1. A Quick-Look Design-of-Experiments (DOE) Orientation
Carol Ventresca
John C. Sparks
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2. Presentation Topics Introductory Experiments with a Black Box
Using One-Factor-at-a-Time Methodology
Using an Orthogonal Array via a “Designed Experiment”
What Exactly is DOE?
History
When Applicable
The Classic “Dial Problem”
Air Force Example: Vane Cleaning Experiment
Summary and Resources
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3. Experiments with a Black Box
Controllable inputs:
X variables y1
Outputs:
Y Variables
Objective:
Determine Y = F(X)
In the presence of Z x1 x2 x3 x4 x5 z1 z2 z3 z4 z5 y2 y3 y4
Uncontrollable inputs:
Z variables
Standard DOE nomenclature for black box experimentation
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4. Goal of Testing and Associated Test Program Options
Via experimentation, engineer must assess the response of a system as a function of several variables or factors
Each factor has at least two different operating levels
Any change to any one factor necessitates an additional test
Traditional Options
Full Factorial (FF): Solid option, but quickly discarded with the ballooning of factor/level combinations
Example: FF for six two-level factors necessitates 26 = 64 individual tests in order to capture all factor/level combinations
Engineering Judgment: Normally a poor option since this approach by nature allows random pursuit of rabbit trails
Leads to a situation known as the “random test matrix”
One-Factor-at-a-Time (OFT): Poor option, process attempts to optimize in serial fashion with no regard to synergistic or “interactive” combinations
Once an individual factor comes up for optimization and has its level fixed, all other levels of the same factor are disregarded for the remainder of experimentation
Hence, interaction effects between factor levels are never fully assessed
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5. OFT Test Program Applied to a Specific Black Box
Unknown Functional
Mechanism:
Y1 = f(x1,x2,x3,x4) y1
Inputs
Output x1 x2 x3 x4
Goals:
1) Use OFT test methods to maximize the output y1
2) Discover the operating characteristics of the black box in terms of an algebraic equation relating cause to effect
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6. Actual Function Hidden Within the Black Boxy1
Inputs
Output x1 x2 x3 x4
y1=45+12x1+8x2+10x1x2+5x3-2x1x3-6x1x4+x4
Nature of which is TBD
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7. OFT Factors and Factor Levels (Chosen for Illustration Purposes Only)x1 x2 x3 x4
-1, 1, -1, 1 1 -1
-1, -1, -1, -1
1, 1, 1, 1
Factor Lo Hi x1 -1 1 x2 x3 x4
Note: A FF test program would consists of 24 = 16 individual tests.
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8. “Straight-to-the-Chase” with a Five-Run OFT Test Program
Run: Comment x1 x2 x3 x4
Run Code y1
1: Baseline -1
(1) 21
2: x1 lockdown 1 51
3: x2 lockdown
x1x2 77
4: x3 lockdown
x1x2x3 83
5: x4 off the optimum
x1x2x3x4 73
6: Final Lockdown
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9. OFT Model Building with Five Data Points
Start with an assumed fully determined linear model
y1 = a0 + a1x1 + a2x2 + a3x3 + a4x4
Where
a0 – a1 – a2 – a3 – a4 = 21
a0 + a1 – a2 – a3 – a4 = 51
a0 + a1 + a2 – a3 – a4 = 77
a0 + a1 + a2 + a3 – a4 = 83
a0 + a1 + a2 + a3 + a4 = 73
Solving for the five unknown coefficients
y1 = x1 + 13x2 + 3x3 – 5x4
Optimizing
y1 = (1) + 13(1) + 3(1) – 5(-1) =83
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10. The OFT Model Fails to Predict For Many Combinations
True: y1=45+12x1+8x2+10x1x2+5x3-2x1x3-6x1x4+x4
OFT Model: y1 = x1 + 13x2 + 3x3 – 5x4
Combination
OFT
True
-1, -1, -1, -1 21
1, -1, -1, -1 51
-1, -1, -1, 1 11 35
1, -1, -1, 1 41 31
-1, -1, 1, -1 27
1, -1, 1, -1 53 47
-1, -1, 1, 1 17 49
1, -1, 1, 1 43 37
-1, 1, -1, -1
1, 1, -1, -1 77
-1, 1, -1, 1
1, 1, -1, 1 67
-1, 1, 1, -1
1, 1, 1, -1 83
-1, 1, 1, 1 45
1, 1, 1, 1 73
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11. OFT Advantages and Disadvantages
Search process locates the maximum value
Similar search process locates the minimum value
Does so in five runs
Disadvantages
Leads to wrong functional model
Factor main effects only; no interactions
Poor overall prediction capability
Example
OFT predicts six settings out of sixteen
Blue is fortuitous
Due to unaccounted-for interactions
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12. An OFT Test Program that Fails to Identify the Maximum
y1 = 7 + 2x1 – 3x2 + x3 + 2x1x2 – 4x2x3
Run: Comment x1 x2 x3 y1
1: Baseline -1 5
2: x1 lockdown per economics 1
3: x2 lockdown 11
4: x3 less than maximum
Final lockdown
True Maximum 15
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13. 2IV4-1 DOE Test Program in Comparison to Companion OFT21 49 31 47 77 x3
x1x3
x1x4 x4 y1 F I x2
x1x2 x1 GM
Column
Assign
(1)
x3x4
x2x4
x2x3 73
x1x2x3x4
Run
y1 = c0 + c1x1 + c2x2 +c3x1x2 +c4x3 +c5x1x3+c6x1x4 +c7x4
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14. Calculating Coefficients via Matrix Orthogonality (1/2)-1 c0 c1 c2 c3 c4 c5 c6 c7 ● 21 49 31 47 77 73 =
» 8c4 = 40 » c4=5
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15. Calculating Coefficients via Matrix Orthogonality (2/2)1 -1 c0 c1 c2 c3 c4 c5 c6 c7 ● 21 49 31 47 77 73 =
Each coefficient is calculated in like fashion resulting in
y1 = 45+12x1+8x2+10x1x2+5x3-2x1x3-6x1x4+x4
Unveiling the black-box functional relationship
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16. DOE Advantages and Disadvantages
Allows for the inclusion of interactions into mathematical models and higher order terms when needed
Allows efficient evaluation of the coefficients associated with the mathematical model via the use of orthogonal arrays
Allows for multiple use and examination of test data per a variety of statistically sound techniques
Allows needed data to be generated using a minimum number of individual tests—time and cost savings!
Requires more up-front planning than traditional testing in that several pre-test issues must be addressed in asystematic fashion
Requires that the full DOE test program be executed in order to properly interpret data and results
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17. What Exactly is DOE? DOE is one of the core “Six-Sigma” methodologies
Statistically selects “axiomatic points” in the design space
Selection enables maximum information return on investment made
Used to systematically analyze the nature and cause of variation by means of controlled testing (as opposed to examining available data)
Cause is linked to effect by establishing through experimentation the coefficients for pre-determined “best-fit” models
Linear models: two-level experiments
Piece-wise linear models: multi-level “orthogonal type” experiments
Non-linear (general second-order quadratic model): response surface methods
Test programs built upon sound DOE principles are
Significantly compressed and extremely efficient
Produce high-quality and reusable data
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18. When Can We Use DOE? Response Ouch!
Stimulus A
Stimulus B
Stimulus C
Stimulus D
Response
Ouch!
Fact: Any physical phenomenon or process that can be thought of in terms of a stimulus-response model can be analyzed using DOE.
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19. The Classic Dial Problem
Factors
Response A B C D E F G H I J K L M N
Levels
In the early 1990s, the Air Force conducted a Halon Replacement test program that examined the effects of 14 two-level factors upon a single response variable: “pounds of fire suppressant needed to extinguish a fire”. The question was asked, “What are the best settings for our 14 dials in order to minimize the response variable?”
Objective: Minimize the quantity of fire suppressant needed to extinguish a fire.
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20. The Previous Question Equates to the Classic Dial Problem
In the previous Air Force example, a full factorial test program consists of 214 or 16,384 production runs
This many runs is definitely out of the question!
Can you image the size of the associated matrix!
We are limited by time and money and can typically make only fifty production runs or so.
But how do we pick the right fifty?
By experience?
By guessing?
By convenience?
16,384 Rows X 15 Columns
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21. DOE Solves the Dial Problem!
Factors
Response A B C D E F G H I J K L M N
Levels
In the Halon Replacement test program, a special-purpose “orthogonal array” having just 32 rows (one row per run) was used. This array not only solved the dial problem but also produced high-quality experimental results extremely useful in identifying a minimum.
© 2008 SynGenics Corporation. All rights reserved.

22. Two-Level Orthogonal Array of Exact Type Used in Halon-Replacement Test Program
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23. Some Typical DOE Compression Ratios for Two-Level Experiments
Factors 2 3 4 5 6 7 8 9 10 11 12 13 14 FF 16 32 64
128
256
512
1,024
2,048
4,096
8,192
16,384
DOE
Ratio 1
1/2
1/4
1/8
1/16
1/32
1/64
1/128
1/256
1/512
Unique Individual
Tests Required
Standard DOE
Nomenclature
22-0 or 22
23-0 or 23
24-1
25-1
26-2
27-3
28-4
29-4
210-5
211-6
212-7
213-8
214-9
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24. An Actual Air Force Gas Turbine Engine Vane Cleaning Experiment
A gas-turbine engine vane becomes corroded during service and requires periodic cleaning. Very high pressure water is delivered through a tiny nozzle orifice in order to cleanse the vanes. The response variable (Quality Characteristic) is percent contamination remaining after the cleansing procedure. A designed experiment is conducted in order to find the factor-level combination that minimizes the quality characteristic. (Lower is better.) V∞
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25. Factors, Levels, and Output Using Standard XYZ Descriptors
Sound-Alike
XYZ
-1 Level
1 Level
Orifice Size O x1
0.07in
0.1in
Standoff Distance S x2
0.5in
1.0in
Pressure P x3
20KSI
35KSI
Feed Rate F x4
20ipm
30ipm
Pump RPM R x5
1500rpm
2000rpm
Output % y1
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26. Vane Cleaning Dial Model
Factors
Response x1 x2 x3 x4 x5
Levels y1
A Full Factorial experiment would consists of 25 = 32 individual trials.
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27. 2III5-2 Designed Experiment Showing the “Alias Structure”-1
10.1 1
11.9
9.2
11.3
8.9
13.5
7.8
13.1
x1x3 x5 y1
x2x4
x2x5
x2x3
x1x5
x1x4
Alias
Structure
x1x2 GM
x3x4
x3x5
x4x5
Assumed Model Form: y1 = c0+c1x1+c2x2+c3x1x2+c4x3+c5x1x3+c6x4 +c7x5
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28. Coefficient Pareto Chart and “Scree Line” for Half Effects
0.2
0.4
0.6
0.8 1
1.2
1.4
1.6
1.8 x3
x1x3 x2 x4 x1 x5
x1x2
Scree is the rubble at the bottom of a cliff
Blue columns: significant or part of significant two-factor interaction
Red columns: deemed insignificant and will be rolled into error
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29. Cube Plot for the Three Retained Factors x1, x2, and x3
10.625 x2 x3 x1
-1,-1,-1: 10.1
-1, 1, 1: 11.3
1,-1, 1: 13.5
1, 1, -1: 7.8
1,-1,-1: 8.9
-1, 1,-1: 9.2
1, 1, 1: 13.1
-1,-1, 1: 11.9
9.0
12.45
X3 effect = 3.45
c4 = 1.725
10.725
10.875
10.35
11.1
X2 effect = -0.75
c2 =
X1 effect = 0.20
c1 = 0.1
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30. The x1x3 Interaction Plot
-1, -1: 9.65
1, 1: 13.3
1, -1: 8.35
-1, 1: 11.6
9.0
12.45
10.625
10.725
10.875
X1X3 effect = -1.5
c5 = -0.75
x1 trends upward when x3 = 1
x1 trends downward when x3 = -1
11.475
9.975
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31. Linear Model and Optimal Settings
y1 = x x1x x x1
Only factors deemed “significant” by themselves or part of a significant “two-factor” interaction are included
The others are part of the error
Methodology for minimizing y1
Set x3 = -1
Set x1x3 = 1 which implies x1 = 1
Set x2 = 1
Minimum: y1 =
Implies theoretical best y1 = 7.975
Must be verified through a series of confirmation experiments
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32. Vane Cleaning ANOVA Table
Factor Ci SS
Comment GM
10.725
Shows a process! x1
0.1
0.08
Part of sig. 2FI x2
0.0
x1x2
Include with error x3
1.725
23.805
Big driver
x1x3
-0.75
4.5 x4
-0.125
0.125 x5
0.05
0.02
Totals
949.86
Also, we have Σ ci2 =
Source V
F Ratio
Significance 1
19,051.86
>>99%
492.85
99%
93.16
1.125
23.29
95%
1.86
Must include
Error 3
0.043
divisor
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33. Overall 90% Confidence Interval
Calculating the overall 90% CI
where F90% (1, 3) = 5.54
SGE = 8/(1+4) = 1.6
IHL = (5.54x0.043 / 1.6)0.5 = 0.385
CI is (7.975 – 0.385, ) = (7.59, 8.36)
95% CI is (7.453, 8.496)
99% CI is (7.018, 8.932)
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34. General Applicability of the DOE Process as Presented
Even with the introduction of advanced techniques and models, the general DOE procedural protocol as presented in this orientation is still applicable.
Run 1 2 3 4 5 6 7 8 9 A B C D R
___ L9
To the right is an L9, which can be used as a full factorial design for two three-level piecewise linear factors or as a fully-saturated design for four three-level piecewise linear factors. All general DOE process topics still apply even though previously discussed computational methods will need to modified to accommodate the additional levels.
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35. A Short Laundry List of More Advanced DOE Topics
Use of “non-geometric”, fully-saturated screening designs such as L12, L20, and L28
Use of piecewise-linear, multi-level designs such as the L9 just shown
Use of center points in a design to check for quadrature
Design resolution, aliasing, and use of “fold-over” designs
Use of blocking and blocking factors
Use of and limitations of response-surface methodologies
e.g. Central-composite and Box-Bhenken
When DOE might not work past screening phase
Highly interactive and non-linear phenomena such as turbulence
Use of DOE as a preprocessor to major computer codes
Any analysis code can be looked upon as a numerical “test facility”
DOE can be used to pre-screen input parameters, cutting down on number of runs and subsequent total runtime
GE and Pratt-Whitney notable examples
Use of DOE to analyze “available data”
Requires systemic data mining and elucidation of patterns
Can be very tough to perform!
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36. A Small Central Composite Design For Three Factors x1, x2, and x3
Factorial Points
Center Points
Axial Points
Run x1 x2 x3 1 -1 2 3 4 5 6 7 8
Run x1 x2 x3 9 10 11 -0 12
Run x1 x2 x3 13
-1.68 14
1.68 15 16 17 18 + +
Center points are used to check curvature. If curvature is significant, then axial points are added to build a quadratic model. Axial points are not usually added for insignificant main effects. Continuous factors are a must!
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37. Run Diagram Showing Factorial, Center, and Axial Points
-1,-1,-1
-1, 1, 1
1,-1, 1
1, 1, -1
1,-1,-1
-1, 1,-1
1, 1, 1
-1,-1, 1
1.68, 0, 0
0, 0, 0
0, 0,1.68
0, 1.68, 0
-1.68, 0, 0
0, -1.68, 0
0, 0,-1.68
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38. SynGenics Two-Day DOE Course Description and Objectives
Basic introduction to “two-level” DOE that includes
The importance of experimental design
How to plan and design an experiment
The role and use of “orthogonal” arrays
How to conduct a statistically designed experiment
How to analyze results from a statistically designed experiment
Take-away tool box in this course is limited to two-level designs and associated analysis techniques
Course Objectives
Be able to plan, execute, and analyze a simple two-level designed experiment
Be able to understand and assess more complex two-level designed experiments as presented by Air Force contractors
Be cognizant of advanced DOE methodologies that go beyond the basic two-level designs
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39. The theory was described by [[Nassim Nicholas Taleb]] in his book ''[[The Black Swan]]''. Taleb regards many scientific discoveries as black swans—"undirected" and unpredicted. He gives the rise of the Internet, the personal [[computer]], the [[first world war]], as well as the [[September 11, 2001 attacks]] as examples of Black Swan events.<ref>Nassim Nicholas Taleb, "[
The term ''black swan'' comes from the ancient [[Western culture|Western]] conception that 'All [[swan|swans]] are [[white]]'. In that context, a [[black swan]] was a [[metaphor]] for something that could not exist. The [[17th Century]] [[Black_Swan_emblems_and_popular_culture#European_myth_and_metaphor|discovery of black swans]] in [[Australia]] metamorphosed the term to connote that the perceived impossibility actually came to pass. Taleb notes that [[John Stuart Mill]] first used the black swan narrative to discuss falsification.
There are theories that the existencce of the black swan could be predicted from the statistical analysis. With some thought, this could be extended to the development methodology.
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