A Quick-Look Design-of-Experiments (DOE) Orientation

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A Quick-Look Design-of-Experiments (DOE) Orientation

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

Actual Function Hidden Within the Black Box

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. © 2008 SynGenics Corporation. All rights reserved.

“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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

2IV4-1 DOE Test Program in Comparison to Companion OFT
21 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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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. © 2008 SynGenics Corporation. All rights reserved.

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. © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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.

Two-Level Orthogonal Array of Exact Type Used in Halon-Replacement Test Program

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 © 2008 SynGenics Corporation. All rights reserved.

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∞ © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

Vane Cleaning Dial Model
Factors Response x1 x2 x3 x4 x5 Levels y1 A Full Factorial experiment would consists of 25 = 32 individual trials. © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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) © 2008 SynGenics Corporation. All rights reserved.

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. © 2008 SynGenics Corporation. All rights reserved.

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! © 2008 SynGenics Corporation. All rights reserved.

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! © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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 © 2008 SynGenics Corporation. All rights reserved.

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, "[http://www.fooledbyrandomness.com]"</ref> 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. © 2008 SynGenics Corporation. All rights reserved.

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