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Statistics and DOE ME 470 Spring 2012

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We will use statistics to make good design decisions! We will categorize populations by the mean, standard deviation, and use control charts to determine if a process is in control. We may be forced to run experiments to characterize our system. We will use valid statistical tools such as Linear Regression, DOE, and Robust Design methods to help us make those characterizations.

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What is DOE Sometimes we would like to model or optimize a design or process, but the “first principles” analytic approach (Conservation laws, etc.) is too challenging. Empirical (experimental) models are the best next choice. The organized approach using DOE provides a good modeling tool for optimization of designs and processes

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Example Suppose you are trying to get faster cycle time on an injection molder and are looking for the best settings for several temperatures and pressures. The transient heat transfer and transient fluid mechanics (with changing viscosity) problem is challenging. Instead we change the inputs in an organized way, measure the outputs, and develop an experimental model that can be optimized.

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Here is an email that made my day! I'm working on a project that is nearing data collection. The study is focused on Thumb-Tip force resulting from muscle/tendon force. We're working with cadaveric specimens so this is awesome lab work. Some of the relationships are expected to be nonlinear so we're looking 10 levels of loading for each tendon. We also wish to document first order and possible second order interactions between tendons if they are significant. Last year with the human powered vehicle team we used minitab to create a test procedure for testing power output resulting from chain ring shape, crank length, and rider. There were 3 chain ring shapes, 3 different crank lengths. If possible, in this current study I would like to run a preliminary factorial experiments to determine which interactions are significant before exhaustively testing every combination at every level of loading. If such a method is appropriate it could save us a lot of time. Could you suggest a reference that I might be able to find at the library or on amazon?

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DOE Objectives At the end of this module, the participant will be able to : Understand the basic concepts and advantages of designed experiments Understand key terminology used in experimental design Use different techniques to deal with noise in an experiment Make good design decisions!!!

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Statistical Design of Experiments Chapter 13 of your text. Your text only looks at DOE in the context of Robust Design. This is one of several important uses of DOE: Determining Significant Factors Constructing a Model of System Performance Design and Analysis of Experiments, Montgomery and Hines is an excellent reference. Look at the Frisbee Thrower from Catapult

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What Experiments Can Do Characterize a Process/Product determines which X’s most affect the Y’s includes controllable and uncontrollable X’s identifies critical X’s and noise variables identifies those variables that need to be carefully controlled provides direction for controlling X’s rather than control charting the Y’s Optimize a Process/Product determines where the critical X’s should be set determines “real” specification limits provides direction for “robust” designs

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Definition of Terms Factor - A controllable experimental variable thought to influence response (example air flow rate, or in the case of the Frisbee thrower: angle, tire speed, tire pressure) Response - The outcome or result; what you are measuring (cycle time to produce one bottle, distance Frisbee goes) Levels - Specific value of the factor (fast flow vs. slow flow, 15 degrees vs. 30 degrees) Interaction - Factors may not be independent, therefore combinations of factors may be important. Note that these interactions can easily be missed in a straight “hold all other variables constant” scientific approach. If you have interaction effects you can NOT find the global optimum using the “OFAT” (one factor at a time) approach! Replicate – performance of the basic experiment

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How Can DOE Help? run a relatively small number of tests to isolate the most important factors (screening test). determine if any of the factors interact (combined effects are as important as individual effects) and the level of interaction. predict response for any combination of factors using only empirical results optimize using only empirical results determine the design space for simulation models

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Trial and Error Problem : Current gas mileage is 23 mpg. Want 30 mpg. But what is the best we can get? We might do the following : Change brand of gas Change octane rating Drive slower Tune-up car Wash and wax car Buy new tires Change tire pressure Terminology : response variable / dependent variable - what is being measured/optimized ex. gas mileage factor / main effect / independent variable - a controlled variable being studied at 2 or more levels during the experiment ex. brand of gas

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From the previous example, here were some potential factors to investigate (factors can be either continuous or discrete) : brand of gas (discrete) octane rating (continuous) driving speed (continuous) tires (discrete) tire pressure (continuous) Terminology : factor levels / main effect levels - the values or settings of the variable being manipulated ex. levels for brand of gas are Shell & Texaco ex. levels for speed are 55 & 60 ex. levels for octane are 85 & 90

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One Factor at a Time Can you explain the results? How many more runs would you need to figure out the best combination of variables? If there were more variables, how many runs would it take to get an optimized solution? What if there is a specific combination of two or more variables that leads to the best mpg ? Problem: Gas mileage is 23 mpg Baseline =>

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Full Factorial Experiment OFAT Runs Problem: Gas Mileage is 23 mpg What conclusion do you make now?

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Full Factorial Experiment Full Factorial experiment consists of all possible combinations of the levels of the factors Design Matrix is the complete specification of the experimental test runs, as seen in the example below Treatment Combination is a specific test run set-up, consisting of a specific combination of the factor levels Design Matrix A treatment combination

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What makes up an experiment? Response Variable(s) Factors Randomization Repetition and Replication

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Response Variable The variable that is measured and the object of the characterization or optimization (the Y) Defining the response variable can be difficult Often selected due to ease of measurement Some questions to ask : How will the results be quantified/analyzed? How good is the measurement system? What are the baseline mean and standard deviation? How big of a change do we care about? Are there several response variables of interest?

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Factor A variable which is controlled or varied in a systematic way during the experiment (the X) Tested at 2 or more levels to observe its effect on the response variable(s) (Ys) Some questions to ask : what are reasonable ranges to ensure a change in Y? knowledge of relationship, i.e. linear or quadratic, etc? Examples material, supplier, EGR rate, injection timing can you think of others?

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Randomization Randomization can be done in several ways : run the treatment combinations in random order assign experimental units to treatment combinations randomly an experimental unit is the entity to which a specific treatment combination is applied Advantage of randomization is to “average out” the effects of extraneous factors (called noise) that may be present but were not controlled or measured during the experiment spread the effect of the noise across all runs these extraneous factors (noise) cause unexplained variation in the response variable(s)

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Repetition and Replication Repetition : Running several samples during one experimental setup (short-term variability) Replication : Repeating the entire experiment (long-term variability) You can use both in the same experiment Repetition and Replication provide an estimate of the experimental error this estimate will be used to determine whether observed differences are statistically significant

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Pressure : HHHH LLLL HHHH LLLL HHHH LLLL Temp: HHLL HHLL HHLL HHLL HHLL HHLL Pressure : HHHH LLLL HHHH LLLL HHHH LLLL Temp: HHLL HHLL HHLL HHLL HHLL HHLL Test Sequence Repetition

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Replication Pressure : HHHH LLLL HHHH LLLL HHHH LLLL Temp: HHLL HHLL HHLL HHLL HHLL HHLL Pressure : HHHH LLLL HHHH LLLL HHHH LLLL Temp: HHLL HHLL HHLL HHLL HHLL HHLL Test Sequence 3rd Replicate 2nd Replicate 1st Replicate

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Steps in DOE 1. Statement of the Problem 2. Selection of Response Variable 3. Choice of Factors and Levels Factors are the potential design parameters, such as angle or tire pressure Levels are the range of values for the factors, 15 degrees or 30 degrees 4. Choice of Design screening tests response prediction factor interaction 5. Perform Experiment 6. Data Analysis

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2 3 Factorial Design Example Problem Statement: A soft drink bottler is interested in obtaining more uniform heights in the bottles produced by his manufacturing process. The filling machine theoretically fills each bottle to the correct target height, but in practice, there is variation around this target, and the bottler would like to understand better the sources of this variability and eventually reduce it. Choice of Factors: The process engineer can control three variables during the filling process: (A) Percent Carbonation (B) Operating Pressure (C) Line Speed Pressure and speed are easy to control, but the percent carbonation is more difficult to control during actual manufacturing because it varies with product temperature.

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2 3 Factorial Design Example Choice of Levels – Each test will be performed for both high and low levels Selection of Response Variable – Variation of height of liquid from target Choice of Design – Interaction effects Perform Experiment Determine what tests are required using tabular data Determine the order in which the tests should be performed

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Determine which experiments should be performed RunA (% C) B (Pressure) C (Speed) Treatment combinations 1---(1) 2+--a 3-+-b 4++-ab 5--+c 6+-+ac 7-++bc 8+++abc

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Determine Order of Experiments Decided to run two replicates Requires 16 tests Put 16 numbers in a hat and draw out the numbers in a random order Assume that the number 7 is pulled out first, then run test 7 first. (% C low, Pressure high, line speed high) What happens when you draw a 10? Minitab can do this for you automatically!!

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Stat>DOE>Factorial>Create Factorial Design Full Factorial Number of Replicates Number of Factors

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Data for the Fill Height Problem (Average deviation from target in tenths of an inch) Operating Pressure (B) 25 psi Line speed (C) 30 psi Line Speed (C) (A) Percent Carbonation 200 (bpm) 250 (bpm) 200 (bpm) 250 (bpm) 10-3 0 0 1111 120101 2121 2323 6565

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Enter Information Ask for random runs

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Data Gathered on Each Run DOE Run ABC Data 1 (1)----3,-1 2 (a)+--0,1 3 (b)-+--1,0 4 (ab)++-2,3 5 (c)--+-1,0 6 (ac)+-+2,1 7 (bc)-++1,1 8 (abc)+++6,5 Low, Low, Low

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Algebraic Signs for Calculating Effects Treatment Combination Factorial Effect (1)ABABCACBCABC (1)+--+-++- a++----++ b+-+--+-+ ab++++---- c+--++--+ ac++--++-- bc+-+-+-+- abc++++++++ I now know how this one is done!

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Calculate Contrast for A Procedure 1. Add all runs where A is positive. 2. Subtract all runs where A is negative. 3. The difference is called the contrast.

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Definition of terms in formulas A,B,C are factors. a, b, c are the levels of factors A, B, C. In our example we are only testing high and low, so a = b = c = 2. n = the number of replicates which happens to be 2 in our example

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ANOVA TABLE Source of variation Sum of squares Degrees of freedom Mean Square FoWho cares? A(a-1) Num = DOF A Den = DOF MSE B(b-1) Num = DOF B Den = DOF MSE C(c-1) Num = DOF C Den = DOF MSE AB(a-1)(b-1) Num = DOF AB Den = DOF MSE AC(a-1)(c-1) Num = DOF AC Den = DOF MSE BC(b-1)(c-1) Num = DOF BC Den = DOF MSE ABC (a-1)(b-1)(c-1)Num = DOF ABC Den = DOF MSE Error abc(n-1) Totalabcn-1 The effect is significant if Fo is greater than the value from the table.

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In English, y… is the sum of all data points. So SS T is the (sum of the square of each data point) - (sum of all data points) 2 /(abcn)

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Carbonation Example Source of variation Sum of squares Degrees of freedom Mean Square FoWho cares? % Carbon. (A) 36.001 57.605.32 for =.05 (95%) Pressure (B)20.251 32.4011.26 for 99% Line speed ©12.251 19.60 AB2.251 3.6 AC0.251 0.4 BC1.001 1.6 ABC1.001 1.6 Error5.0080.625 Total78.0015

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I don’t want to do all of the math Can I get Minitab to do it for me?

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>Stat>ANOVA>General Linear Model Select the response by double clicking Select model terms by double clicking or by typing label, c5*c6

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>Stat>DOE>Factorial>Analyze Factorial Design Select the Graphs tab to get the next screen Select “Deviation” As Response

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The Pareto Chart shows the significant effects. Anything to the right of the red line is significant at a (1- ) level. In our case so we are looking for significant effects at the 0.95 or 95% confidence level. So what is significant here?

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Estimated Effects and Coefficients for Deviation from Target (coded units) Term Effect Coef SE Coef T P Constant 1.0000 0.1976 5.06 0.001 %Carbonation 3.0000 1.5000 0.1976 7.59 0.000 Pressure 2.2500 1.1250 0.1976 5.69 0.000 Line Speed 1.7500 0.8750 0.1976 4.43 0.002 %Carbonation*Pressure 0.7500 0.3750 0.1976 1.90 0.094 %Carbonation*Line Speed 0.2500 0.1250 0.1976 0.63 0.545 Pressure*Line Speed 0.5000 0.2500 0.1976 1.26 0.242 %Carb*Press*Line Speed 0.5000 0.2500 0.1976 1.26 0.242 S = 0.790569 PRESS = 20 R-Sq = 93.59% R-Sq(pred) = 74.36% R-Sq(adj) = 87.98% We could construct an equation from this to predict Deviation from Target. Deviation = 1.00 + 1.50*(%Carbonation) +1.125*(Pressure) + 0.875*(Line Speed) + 0.375*(%Carbonation*Pressure) + 0.125*(%Carbonation*Line Speed) + 0.250*(Pressure*Line Speed) + 0.250*(%Carbonation*Pressure*Line Speed) We can actually get a better model, which we will discuss in a few slides.

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>Stat>DOE>Factorial>Factorial Plots Go to set up

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Practical Application Carbonation has a large effect, so try to control the temperature more precisely There is less deviation at low pressure, so use the low pressure Although the slower line speed yields slightly less deviation, the process engineers decided to go ahead with the higher line speed - WHY???

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We can also use Minitab to construct a predictive model!! Estimated Effects and Coefficients for Deviation from Target (coded units) Term Effect Coef SE Coef T P Constant 1.0000 0.1976 5.06 0.001 %Carbonation 3.0000 1.5000 0.1976 7.59 0.000 Pressure 2.2500 1.1250 0.1976 5.69 0.000 Line Speed 1.7500 0.8750 0.1976 4.43 0.002 %Carbonation*Pressure 0.7500 0.3750 0.1976 1.90 0.094 %Carbonation*Line Speed 0.2500 0.1250 0.1976 0.63 0.545 Pressure*Line Speed 0.5000 0.2500 0.1976 1.26 0.242 %Carb*Press*Line Speed 0.5000 0.2500 0.1976 1.26 0.242 S = 0.790569 PRESS = 20 R-Sq = 93.59% R-Sq(pred) = 74.36% R-Sq(adj) = 87.98% It is recommended to delete items with P > 0.200

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>Stat>DOE>Factorial>Analyze Factorial Design Select this arrow to remove the 3-way interaction term.

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Estimated Effects and Coefficients for Deviation from Target (coded units) Term Effect Coef SE Coef T P Constant 1.0000 0.2041 4.90 0.001 %Carbonation 3.0000 1.5000 0.2041 7.35 0.000 Pressure 2.2500 1.1250 0.2041 5.51 0.000 Line Speed 1.7500 0.8750 0.2041 4.29 0.002 %Carbonation*Pressure 0.7500 0.3750 0.2041 1.84 0.099 %Carbonation*Line Speed 0.2500 0.1250 0.2041 0.61 0.555 Pressure*Line Speed 0.5000 0.2500 0.2041 1.22 0.252 S = 0.816497 PRESS = 18.9630 R-Sq = 92.31% R-Sq(pred) = 75.69% R-Sq(adj) = 87.18% Next term to remove

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Here is the final model from Minitab with the appropriate terms. Estimated Effects and Coefficients for Deviation from Target (coded units) Term Effect Coef SE Coef T P Constant 1.0000 0.2030 4.93 0.000 %Carbonation 3.0000 1.5000 0.2030 7.39 0.000 Pressure 2.2500 1.1250 0.2030 5.54 0.000 Line Speed 1.7500 0.8750 0.2030 4.31 0.001 %Carbon*Press0.7500 0.3750 0.2030 1.85 0.092 S = 0.811844 PRESS = 15.3388 R-Sq = 90.71% R-Sq(pred) = 80.33% R-Sq(adj) = 87.33% Deviation from Target = 1.000 + 1.5*(%Carbonation) + 1.125*(Pressure) + 0.875*(Line Speed) + 0.375*(%Carbonation*Pressure)

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Estimated Effects and Coefficients for Deviation from Target (coded units). The term coded units means that the equation uses a -1 for the low value and a +1 for the high value of the data. Deviation from Target = 1.000 + 1.5*(%Carbonation) + 1.125*(Pressure) + 0.875*(Line Speed) + 0.375*(%Carbonation*Pressure) Let’s check this for %Carbonation = 10, Pressure = 30 psi, and Line Speed = 200 BPM %Carbonation is at its low value, so it gets a -1. Pressure is at its high value, so it gets +1, Line Speed is at its low value, so it gets a -1. Deviation from Target = 1.000 + 1.5*(-1) + 1.125*(1)+ 0.875*(-1)+ 0.375*(-1*-1) Deviation from Target = -0.625 tenths of an inch How does this compare with the actual runs at those settings?

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NEVER GIVE THIS SETTING TO PRODUCTION UNTIL YOU HAVE VERIFIED THE MODEL! The engineer wants the higher line speed and decides to put the target slightly negative. Why??

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Not too Noisy Noise Level < 75 db VOC System Spec Lawn Mower Example Engine NoiseBlade Assy Noise Combustion Noise Muffler Noise Muffler Volume Hole Area Diameter Blade Speed Blade Area Blade Width Blade Length Grass Height Blade to Hsg Clearance

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Individual Homework (30 pts): Due Monday, Tuesday 19, 20-Mar-2012 Use the Frisbee Thrower file that is supplied to you. Do a DOE analysis of the data to determine the significant factors. – Include a Pareto chart, and a 4 way Residual Plots Chart. Remove the appropriate terms in order to get the best model that you can. – Show the equation with the appropriate terms. – Include a main effects plot and an interaction plot. – Calculate the predicted value (distance) at one setting and compare to the experimental values. Discuss the significance of your results. What do these results mean?

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