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Section X, Improvement

Techniques

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Will help you gain knowledge in:◦ Improving performance

characteristics◦ Reducing costs◦ Understand regression analysis◦ Understand relationships between

variables◦ Understand correlation◦ Understand how to optimize

processes

So you can:◦ Recognize opportunities◦ Understand terminology◦ Know when to get help

Objectives in Using DOE

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Let’s Start with an Example:

Plot a histogram and calculate the average and standard deviation

Data

18 16 30 29 28 21 17 41 8 1732 26 16 24 27 17 17 33 19 1831 27 23 38 33 14 13 26 11 2821 19 25 22 17 12 21 21 25 2623 20 22 19 21 14 45 15 24 34

Fuel Economy of 50 automobiles (in mpg)

Fuel Economy

0

2

4

6

8

10

12

14

16

0 to <6 6 to <12 12 to <18 18 to <24 24 to <30 30 to <36 36 to <42 42 to <48 48 to <54 54 to <=60mgp

Nu

mb

er

of

Ca

rs

7266.7

88.22

S

X

X-2

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Experimental design (a.k.a. DOE) is about discovering and quantifying the magnitude of cause and effect relationships.

We need DOE because intuition can be misleading.... but we’ll get to that in a minute.

Regression can be used to explain how we can model data experimentally.

What Might Explain the Variation?

METHOD

MOTHERNATUREMEASUREMENT

MANPOWER MACHINE

MATERIAL

X-2

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Let’s take a look at the mileage data and see if there’s a factor that might explain some of the variation.

Draw a scatter diagram for the following data:

Mileage Data with Vehicle Weight:

X - Weight (lbs) Y - Mileage(mpg)3000 182800 212100 322900 172400 313300 142700 213500 122500 233200 14

Observation1

10

6789

2345

Y=f(X)X Y

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If you draw a best fit line and figure out an equation for that line, you would have a ‘model’ that represents the data.

Regression Analysis

Scatter Chart (Weight vs mpg)

y = -0.0152x + 63.507

R2 = 0.9191

05

101520253035

1900 2400 2900 3400 3900

Weight

mp

g

Y=f(X)

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There are basically three ways to understand a process you are working on.

Classical 1FAT experiments ◦ One factor at a time (1FAT) focuses on one

variable at two or three levels and attempts to hold everything else constant (which is impossible to do in a complicated process).

Mathematical model◦ Express the system with a mathematical

equation. DOE

◦ When properly constructed, it can focus on a wide range of key input factors and will determine the optimum levels of each of the factors.

Each have their advantages and disadvantages. We’ll talk about each.

Understanding a System

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Let’s consider how two known (based on years of experience) factors affect gas mileage, tire size (T) and fuel type (F).

1FAT Example

Fuel Type Tire size

F1 T1

F2 T2

Y=f(X)

T(1,2)Y

F(1,2)

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Step 1:Select two levels of tire

size and two kinds of fuels.Step 2:

Holding fuel type constant (and everything else), test the car at both tire sizes.

One –at –a-time Design

Fuel Type

Tire size Mpg

F1 T1 20

F1 T2 30

X-2

10101010

Since we want to maximize mpg the more desirable response happened with T2

Step 3: Holding tire size at T2, test the car at both fuel types.

One –at –a-time Design

Fuel Type

Tire size Mpg

F1 T2 30

F2 T2 40

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Looks like the ideal setting is F2 and T2 at 40mpg.

This is a common experimental method.

One –at –a-time Design

Fuel Type

Tire size Mpg

F1 T2 30

F2 T2 40

What about the possible interaction effect of tire size and fuel type. F2T1

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Suppose that the untested combination F2T1 would produce the results below.

There is a different slope so there appears to be an interaction. A more appropriate design would be to test all four combinations.

One –at –a-time Design

0

10

20

30

40

50

60

70

T1 T2

Tire Size

mpg

F2

F1

X-2

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We need a way to ◦ investigate the relationship(s) between variables◦ distinguish the effects of variables from each other (and

maybe tell if they interact with each other)◦ quantify the effects...

...So we can predict, control, and optimize processes.

What About Other Factors – and Noise?

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The Other Two Possibilities

We can see some problems with 1FAT. Now let’s go back and talk about the statapult.

We can do a mathematical model or we could do a DOE.

DOE will build a ‘model’ - a mathematical representation of the behavior of measurements.

or…

You could build a “mathematical model” without DOE and it might look something like...

X-2

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A Mechanistic Model for the Statapult

X-2

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DOE to the Rescue!!

Run X1 X2 X3 X4 Y1 Y2 Y3 Y-bar SY

1 - - - -2 - - + +3 - + - +4 - + + -5 + - - +6 + - + -7 + + - -8 + + + +

DOE uses purposeful changes of the inputs (factors) in order to observe corresponding changes to the outputs (response).

Remember the IPO’s we did – they are real important here.

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Set objectives (Charter)◦ Comparative

Determine what factor is significant◦ Screening

Determine what factors will be studied◦ Model – response surface method

Determine interactions and optimize Select process variables (C&E) and

levels you will test at Select an experimental design Execute the design CONFIRM the model!! Check that

the data are consistent with the experimental assumptions

Analyze and interpret the results Use/present the results

Planning - DOE Steps

X-4

18181818

Planning - Charter

X-8

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Planning - Charter

http://jimakers.com/downloads/DOE_Setup.docxX-8

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To ‘design’ an experiment, means to pick the points that you’ll use for a scatter diagram.

See DOE terms X-9 through X-15

The Basics

Run A B

1 - -2 - +3 + -4 + +

In tabular form, it would look like:

High (+)

Low (-)

Fa

cto

r B

Se

ttin

gs

Factor A Settings High (+)Low (-)

(-,+)

(+,-)

(+,+)

(-,-)

YA

B

X1

X2

X-9

21212121

A full factorial is an experimental design which contains all levels of all factors. No possible treatments are omitted. ◦ The preferred (ultimate) design◦ Best for modeling

A fractional factorial is a balanced experimental design which contains fewer than all combinations of all levels of all factors.◦ The preferred design when a full

factorial cannot be performed due to lack of resources

◦ Okay for some modeling ◦ Good for screening

Full vs.Fractional Factorial

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Full factorial◦ 2 level◦ 3 factors◦ 8 runs◦ Balanced

(orthogonal)

Fractional factorial◦ 2 level◦ 3 factors◦ 4 runs - Half

fraction◦ Balanced

(orthogonal)

2 Level Designs

runs 823

runs 42 13

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Res

pons

e - Y

Factor ALow High

Average Y when A was set ‘high’

Average Y when A was set ‘low’

The difference in the average Y when A was ‘high’ from the average Y when A was ‘low’ is the ‘factor effect’

The differences are calculated for every factor in the experiment

Measuring An “Effect”

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When the effect of one factor changes due to the effect of another factor, the two factors are said to ‘interact.’

more than two factors can interact at the same time, but it is thought to be rare outside of chemical reactions.

Res

pons

e -

Y

Factor ALow High

B = High

B = Low

Slight

Res

pons

e -

Y

Factor ALow High

B = High B = Low

Strong

Looking For Interactions

Res

pons

e -

Y

Factor ALow High

B = High

B = LowNone

X-16

25252525

Using the statapult, we will experiment with some factors to “model” the process.

We will perform a confirmation run to determine if the model will help us predict the proper settings required to achieve a desired output.

Let’s Try This Out!

What design should we use?

YAB

X1

X2

CD

X3X4

Y=f(X1, X2, X3, X4)

X-17

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Too much variation in the response

Measurement error Poor experimental discipline Aliases (confounded) effects Inadequate model Something changed

- And: -

Reasons Why a Model Might Not Confirm:

There may not be a true cause-and-effect relationship.

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Remember?

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Factor A B C D Response #1

Row #A -

B -

C -

D - Y1 Y2 Y3

1 -1 -1 -1 -1

2 -1 -1 -1 1

3 -1 -1 1 -1

4 -1 -1 1 1

5 -1 1 -1 -1

6 -1 1 -1 1

7 -1 1 1 -1

8 -1 1 1 1

9 1 -1 -1 -1

10 1 -1 -1 1

11 1 -1 1 -1

12 1 -1 1 1

13 1 1 -1 -1

14 1 1 -1 1

15 1 1 1 -1

16 1 1 1 1

Experiment Factorial Full

run 1624

X-17

Confirmation runs

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Example Pages - Data

X-17

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Marginal Means Plot

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Regression Table

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Prediction and Confirmation

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Full factorial◦ 3 level◦ 3 factors◦ 27 runs◦ Balanced

(orthogonal)◦ Used when it is expected the

response in non-linear

3 Level Designs

runs 2733

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Useful to see how factors effect the response and to determine what other settings provide the same response

2D Contour Plot

X-30

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Helpful in reaching the optimal result

3D Response Surface Plot

X-30