Design of Experiments (DOE)

ME 470Fall 2013 – Day 2

We will use statistics to make good design decisions!

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.

DOE is a powerful tool for analyzing and predicting system behavior.

At the end of the DOE module, students should be able to perform the following actions:• Explain the advantages of designed experiments (DOE) over

one-factor-at-a-time• Define key terminology used in experimental design• Analyze experimental data using the DOE techniques

introduced• Make good design decisions!!!

Be prepared to define the terms below.

Factor - A controllable experimental variable thought to influence response (in the case of the Frisbee thrower: angle, motor speed, tire pressure)

Response - The outcome or result; what you are measuring (distance Frisbee goes)

Levels - Specific value of the factor (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

There are six suggested steps in DOE.

1. Statement of the Problem2. 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 Experiment6. Data Analysis

23 Factorial Design Example

#1. 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.#2. Selection of Response Variable: Variation of height of liquid from target#3. 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. It can be controlled in a lab setting.

23 Factorial Design Example

#3. Choice of Levels – Each test will be performed for both high and low levels#4. Choice of Design – Interaction effects#5. Perform Experiment

Determine what tests are required using tabular data or MinitabDetermine the order in which the tests should be performed

Stat>DOE>Factorial>Create Factorial Design

3 factors

Full Factorial

Number of replicates

Enter Information

Ask for random

runs

Term

Standardized Effect

AC

BC

ABC

AB

C

B

A

876543210

2.306Factor NameA %CarbonationB PressureC Line Speed

Pareto Chart of the Standardized Effects(response is Deviation from Target, Alpha = .05)

The Pareto Chart shows the significant effects. Anything to the right of the red line is significant at a (1-a) level. In our case a =0.05, so we are looking for significant effects at the 0.95 or 95% confidence level. So what is significant here?

Residual

Perc

ent

10-1

99

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Fitted Value

Resid

ual

6420-2

1.0

0.5

0.0

-0.5

-1.0

Residual

Freq

uenc

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1.00.50.0-0.5-1.0

6.0

4.5

3.0

1.5

0.0Observation Order

Resi

dual

16151413121110987654321

1.0

0.5

0.0

-0.5

-1.0

Normal Probability Plot of the Residuals Residuals Versus the Fitted Values

Histogram of the Residuals Residuals Versus the Order of the Data

Residual Plots for Deviation from Target

Estimated Effects and Coefficients for Deviation from Target (coded units)

Term Effect Coef SE Coef T PConstant 1.0000 0.1976 5.06 0.001%Carbonation 3.0000 1.5000 0.1976 7.59 0.000Pressure 2.2500 1.1250 0.1976 5.69 0.000Line 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 0.545Pressure*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 = 20R-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.

Mea

n of

Dev

iatio

n fr

om T

arge

t

1210

2

1

0

3025

250200

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%Carbonation Pressure

Line Speed

Main Effects Plot (data means) for Deviation from Target

Estimated Effects and Coefficients for Deviation from Target (coded units)

Term Effect Coef SE Coef T PConstant 1.0000 0.1976 5.06 0.001%Carbonation 3.0000 1.5000 0.1976 7.59 0.000Pressure 2.2500 1.1250 0.1976 5.69 0.000Line 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 0.545Pressure*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 = 20R-Sq = 93.59% R-Sq(pred) = 74.36% R-Sq(adj) = 87.98%

The statisticians at Cummins suggest that you remove all terms that have a p value greater than 0.2. This allows you to have more data to estimate the values of the coefficients.

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 PConstant 1.0000 0.2030 4.93 0.000%Carbonation 3.0000 1.5000 0.2030 7.39 0.000Pressure 2.2500 1.1250 0.2030 5.54 0.000Line Speed 1.7500 0.8750 0.2030 4.31 0.001%Carbon*Press 0.7500 0.3750 0.2030 1.85 0.092

S = 0.811844 PRESS = 15.3388R-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)

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 inchHow does this compare with the actual runs at those settings?

>Stat>DOE>Factorial>Response Optimizer

Now that we have our model, we can play with it to find items of interest.

Select C8 Deviation (tenths of an inch)

CurHighLow1.0000

DOptimal

d = 1.0000

Targ: 0.0Deviatio

y = 0.0

1.0000DesirabilityComposite

200.0250.0

25.030.0

10.012.0

Pressure Line Spe%Carbona[10.0] [29.3713] [223.2464]

The engineer wants the higher line speed and decides to put the

target slightly negative. Why??

NEVER GIVE THIS SETTING TO PRODUCTION UNTIL YOU HAVE VERIFIED THE MODEL!!!

Not too Noisy

Noise Level < 75 db

VOC

System Spec

Lawn Mower

Example

Engine Noise Blade 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

Let’s revisit the Frisbee Thrower and see what the data shows us.

BC

ABC

B

AC

AB

A

C

9876543210

Term

Standardized Effect

2.306A Speed %B Tire Pressure (psi)C Angle (Degrees)

Factor Name

Pareto Chart of the Standardized Effects(response is Distance (ft), Alpha = 0.05)

5.02.50.0-2.5-5.0

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Fitted Value

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420-2-4

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Observation OrderRe

sidua

l

Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for Distance (ft)

Factorial Fit: Distance (ft versus Speed %, Tire Pressure, Angle (Degrees)

Estimated Effects and Coefficients for Distance (ft) (coded units)

Term Effect Coef SE Coef T PConstant 51.775 0.9604 53.91 0.000Speed % 6.000 3.000 0.9604 3.12 0.014Tire Pressure (psi) -3.125 -1.562 0.9604 -1.63 0.142Angle (Degrees) -15.650 -7.825 0.9604 -8.15 0.000Speed %*Tire Pressure (psi) -5.225 -2.612 0.9604 -2.72 0.026Speed %*Angle (Degrees) -4.400 -2.200 0.9604 -2.29 0.051Tire Pressure (psi)*Angle (Degrees) 0.075 0.037 0.9604 0.04 0.970Speed %*Tire Pressure (psi)* 1.775 0.887 0.9604 0.92 0.382 Angle (Degrees)

S = 3.84171 PRESS = 472.28R-Sq = 92.02% R-Sq(pred) = 68.09% R-Sq(adj) = 85.04%

Following recommended procedures, we achieved a reduced model.

B

AC

AB

A

C

9876543210

Term

Standardized Effect

2.228A Speed %B Tire Pressure (psi)C Angle (Degrees)

Factor Name

Pareto Chart of the Standardized Effects(response is Distance (ft), Alpha = 0.05)

840-4-8

99

90

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Observation Order

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Normal Probability Plot Versus Fits

Histogram Versus Order

Residual Plots for Distance (ft)

Let’s look at both main effects and interaction effects.

STAT >DOE>FACTORIAL>FACTORIAL PLOTS

10050

60

55

50

45

4530

3015

60

55

50

45

Speed %

Mea

n

Tire Pressure (psi)

Angle (Degrees)

Main Effects Plot for Distance (ft)Data Means

4530 3015

60

50

40

60

50

40

Speed %

Tire Pressure (psi)

Angle (Degrees)

50100

Speed %

3045

(psi)Pressure

Tire

Interaction Plot for Distance (ft)Data Means