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A Quick-Look Design-of-Experiments (DOE) Orientation Carol Ventresca John C. Sparks © 2008 SynGenics Corporation. All rights reserved.

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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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© 2008 SynGenics Corporation. All rights reserved. 3 Experiments with a Black Box Controllable inputs: X variables y1y1 Outputs: Y Variables Objective: Determine Y = F(X) In the presence of Z x1x1 x2x2 x3x3 x4x4 x5x5 z1z1 z2z2 z3z3 z4z4 z5z5 y2y2 y3y3 y4y4 Uncontrollable inputs: Z variables Standard DOE nomenclature for black box experimentation

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© 2008 SynGenics Corporation. All rights reserved. 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 2 6 = 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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© 2008 SynGenics Corporation. All rights reserved. 5 OFT Test Program Applied to a Specific Black Box Unknown Functional Mechanism: Y 1 = f(x 1,x 2,x 3,x 4 ) y1y1 Inputs Output x1x1 x2x2 x3x3 x4x4 Goals: 1) Use OFT test methods to maximize the output y 1 2) Discover the operating characteristics of the black box in terms of an algebraic equation relating cause to effect

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© 2008 SynGenics Corporation. All rights reserved. 6 Actual Function Hidden Within the Black Box y1y1 Inputs Output x1x1 x2x2 x3x3 x4x4 y 1 =45+12x 1 +8x 2 +10x 1 x 2 +5x 3 -2x 1 x 3 -6x 1 x 4 +x 4 Nature of which is TBD

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© 2008 SynGenics Corporation. All rights reserved. 7 OFT Factors and Factor Levels (Chosen for Illustration Purposes Only) FactorLoHi x1x1 1 x2x2 1 x3x3 1 x4x4 1 Note: A FF test program would consists of 2 4 = 16 individual tests. x1x1 x2x2 x3x3 x4x4 -1, 1, -1, , -1, -1, -1 1, 1, 1, 1

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© 2008 SynGenics Corporation. All rights reserved. 8 “Straight-to-the-Chase” with a Five-Run OFT Test Program Run: Commentx1x1 x2x2 x3x3 x4x4 Run Codey1y1 1: Baseline (1) 21 2: x 1 lockdown1 x1x1 51 3: x 2 lockdown11 x1x2x1x2 77 4: x 3 lockdown111 x1x2x3x1x2x3 83 5: x 4 off the optimum1111 x1x2x3x4x1x2x3x4 73 6: Final Lockdown111 x1x2x3x1x2x3 83

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© 2008 SynGenics Corporation. All rights reserved. 9 OFT Model Building with Five Data Points Start with an assumed fully determined linear model y 1 = a 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 + a 4 x 4 Where a 0 – a 1 – a 2 – a 3 – a 4 = 21 a 0 + a 1 – a 2 – a 3 – a 4 = 51 a 0 + a 1 + a 2 – a 3 – a 4 = 77 a 0 + a 1 + a 2 + a 3 – a 4 = 83 a 0 + a 1 + a 2 + a 3 + a 4 = 73 Solving for the five unknown coefficients y 1 = x x 2 + 3x 3 – 5x 4 Optimizing y 1 = (1) + 13(1) + 3(1) – 5(-1) =83

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© 2008 SynGenics Corporation. All rights reserved. 10 The OFT Model Fails to Predict For Many Combinations True: y 1 =45+12x 1 +8x 2 +10x 1 x 2 +5x 3 -2x 1 x 3 -6x 1 x 4 +x 4 OFT Model: y 1 = x x 2 + 3x 3 – 5x 4 CombinationOFTTrueCombinationOFTTrue -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, , 1, 1, , 1, 1, , 1, 1, , 1, 1, 173

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© 2008 SynGenics Corporation. All rights reserved. 11 OFT Advantages and Disadvantages Advantages 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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© 2008 SynGenics Corporation. All rights reserved. 12 An OFT Test Program that Fails to Identify the Maximum y 1 = 7 + 2x 1 – 3x 2 + x 3 + 2x 1 x 2 – 4x 2 x 3 Run: Commentx1x1 x2x2 x3x3 y1y1 1: Baseline 5 2: x 1 lockdown per economics1 5 3: x 2 lockdown1111 4: x 3 less than maximum1115 Final lockdown1111 True Maximum1115

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© 2008 SynGenics Corporation. All rights reserved IV 4-1 DOE Test Program in Comparison to Companion OFT x3x3 x1x3x1x3 x1x4x1x4 x4x4 y1y1 FIIF x2x2 x1x2x1x2 x1x1 FIF GM Column Assign (1) x3x4x3x4 x2x4x2x4 x2x3x2x3 x1x4x1x4 x1x3x1x3 x1x2x1x x1x2x3x4x1x2x3x4 Run y 1 = c 0 + c 1 x 1 + c 2 x 2 +c 3 x 1 x 2 +c 4 x 3 +c 5 x 1 x 3 +c 6 x 1 x 4 +c 7 x 4

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© 2008 SynGenics Corporation. All rights reserved. 14 Calculating Coefficients via Matrix Orthogonality (1/2) c0c0 c1c1 c2c2 c3c3 c4c4 c5c5 c6c6 c7c7 ●● =● c4c c0c0 c1c1 c2c2 c3c3 c4c4 c5c5 c6c6 c7c7 ● =» » 8c 4 = 40 » c 4 =5

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© 2008 SynGenics Corporation. All rights reserved. 15 Calculating Coefficients via Matrix Orthogonality (2/2) c0c0 c1c1 c2c2 c3c3 c4c4 c5c5 c6c6 c7c7 ●● =● c3c3 Each coefficient is calculated in like fashion resulting in y 1 = 45+12x 1 +8x 2 +10x 1 x 2 +5x 3 -2x 1 x 3 -6x 1 x 4 +x 4 Unveiling the black-box functional relationship

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© 2008 SynGenics Corporation. All rights reserved. 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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© 2008 SynGenics Corporation. All rights reserved. 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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© 2008 SynGenics Corporation. All rights reserved. 18 When Can We Use DOE? Fact: Any physical phenomenon or process that can be thought of in terms of a stimulus- response model can be analyzed using DOE. Stimulus A Stimulus B Stimulus C Stimulus D Response Ouch!

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© 2008 SynGenics Corporation. All rights reserved. 19 The Classic Dial Problem 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?” Factors Response -1 1 A B C D E F G H I J K L M N Levels Objective: Minimize the quantity of fire suppressant needed to extinguish a fire.

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© 2008 SynGenics Corporation. All rights reserved ,384 Rows X 15 Columns The Previous Question Equates to the Classic Dial Problem In the previous Air Force example, a full factorial test program consists of 2 14 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?

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© 2008 SynGenics Corporation. All rights reserved. 21 DOE Solves the Dial Problem! 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. Factors Response -1 1 A B C D E F G H I J K L M N Levels

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© 2008 SynGenics Corporation. All rights reserved. 22 Two-Level Orthogonal Array of Exact Type Used in Halon-Replacement Test Program

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© 2008 SynGenics Corporation. All rights reserved. 23 Some Typical DOE Compression Ratios for Two-Level Experiments # Two-Level Factors FF ,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 or or

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© 2008 SynGenics Corporation. All rights reserved. 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∞V∞

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© 2008 SynGenics Corporation. All rights reserved. 25 Factors, Levels, and Output Using Standard XYZ Descriptors FactorSound-AlikeXYZ-1 Level1 Level Orifice SizeOx1x1 0.07in0.1in Standoff DistanceSx2x2 0.5in1.0in PressurePx3x3 20KSI35KSI Feed RateFx4x4 20ipm30ipm Pump RPMRx5x5 1500rpm2000rpm Output%y1y1

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© 2008 SynGenics Corporation. All rights reserved. 26 Vane Cleaning Dial Model Factors Response -1 1 x1x1 x2x2 x3x3 x4x4 x5x5 Levels y1y1 A Full Factorial experiment would consists of 2 5 = 32 individual trials.

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© 2008 SynGenics Corporation. All rights reserved III 5-2 Designed Experiment Showing the “Alias Structure” x4x4 x3x3 x2x2 x1x x1x3x1x3 x5x5 y1y1 x2x4x2x4 x2x5x2x5 x2x3x1x5x2x3x1x5 x1x4x1x4 Alias Structure x1x2x1x2 GM x3x4x3x4 x3x5x3x5 x4x5x4x5 Assumed Model Form: y 1 = c 0 +c 1 x 1 +c 2 x 2 +c 3 x 1 x 2 +c 4 x 3 +c 5 x 1 x 3 +c 6 x 4 +c 7 x 5

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© 2008 SynGenics Corporation. All rights reserved. 28 Coefficient Pareto Chart and “Scree Line” for Half Effects x3x3 x1x3x1x3 x2x2 x4x4 x1x1 x5x5 x1x2x1x2 Scree is the rubble at the bottom of a cliff Red columns: deemed insignificant and will be rolled into error Blue columns: significant or part of significant two-factor interaction

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© 2008 SynGenics Corporation. All rights reserved. 29 Cube Plot for the Three Retained Factors x 1, x 2, and x x2x2 x3x3 x1x1 -1,-1,-1: , 1, 1: ,-1, 1: , 1, -1: 7.8 1,-1,-1: , 1,-1: 9.2 1, 1, 1: ,-1, 1: X 3 effect = 3.45 c 4 = X 2 effect = c 2 = X 1 effect = 0.20 c 1 = 0.1

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© 2008 SynGenics Corporation. All rights reserved. 30 The x 1 x 3 Interaction Plot x3x3 x1x1 -1, -1: , 1: , -1: , 1: X 1 X 3 effect = -1.5 c 5 = x1x1 x 1 trends upward when x 3 = 1 x 1 trends downward when x 3 =

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© 2008 SynGenics Corporation. All rights reserved. 31 Linear Model and Optimal Settings y 1 = x x 1 x x x 1 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 y 1 Set x 3 = -1 Set x 1 x 3 = 1 which implies x 1 = 1 Set x 2 = 1 Minimum: y 1 = Implies theoretical best y 1 = Must be verified through a series of confirmation experiments

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© 2008 SynGenics Corporation. All rights reserved. 32 Vane Cleaning ANOVA Table FactorCiCi SSComment GM Shows a process! x1x Part of sig. 2FI x2x2 0.0 x1x2x1x2 Include with error x3x Big driver x1x3x1x x4x Include with error x5x Include with error Totals949.86Also, we have Σ c i 2 = SourceVSSF RatioSignificanceComment GM ,051.86>>99% x3x % x1x3x1x % x2x % x1x Must includePart of sig. 2FI Error30.043divisor

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© 2008 SynGenics Corporation. All rights reserved. 33 Overall 90% Confidence Interval Calculating the overall 90% CI where F 90% (1, 3) = 5.54 S GE = 8/(1+4) = 1.6 I HL = (5.54x0.043 / 1.6) 0.5 = 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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© 2008 SynGenics Corporation. All rights reserved. 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 A A B B C C D 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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© 2008 SynGenics Corporation. All rights reserved. 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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© 2008 SynGenics Corporation. All rights reserved. 36 A Small Central Composite Design For Three Factors x 1, x 2, and x 3 Run x1x1 x2x2 x3x Run x1x1 x2x2 x3x Run x1x1 x2x2 x3x Factorial Points Center Points Axial Points 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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© 2008 SynGenics Corporation. All rights reserved. 37 Run Diagram Showing Factorial, Center, and Axial Points x2x2 x3x3 x1x1 -1,-1,-1 -1, 1, 1 1,-1, 1 1, 1, -1 1,-1,-1 -1, 1,-1 1, 1, 1 -1,-1, , 0, 0 0, 0, 0 0, 0,1.68 0, 1.68, , 0, 0 0, -1.68, 0 0, 0,-1.68

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© 2008 SynGenics Corporation. All rights reserved. 38 SynGenics Two-Day DOE Course Description and Objectives Course Description 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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© 2008 SynGenics Corporation. All rights reserved. 39

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