1 Jim Grayson, PhD MSC 605 MSC 605 Design of Experiments.

1 Jim Grayson, PhD MSC 605

MSC 605MSC 605

Design of ExperimentsDesign of Experiments


DESIGN OF EXPERIMENTSDESIGN OF EXPERIMENTS

Purposeful changes of the inputs (factors) to a process in order to observe corresponding changes in the output (response).

ProcessInputs Outputs

Douglas Montgomery, Design and Analysis of Experiments


Why use DOE ?Why use DOE ?

• A basis of action -- allows purposeful changes.

• An analytic study -- one in which action will be taken on a cause-and-effect system to improve performance of a product or process in the future.

• Follows the scientific approach to problem solving.

• Provides a way to measure natural variation.

• Permits the clear analysis of complex effects.

• Most efficient way to derive the required information at the least expenditure of resources.

Moen, Nolan and Provost, Improving Quality Through Planned Experimentation


InteractionsInteractionsVarying factors together vs. one at a time.

B

U

C

K

D O E- -

+

+

George Box, Do Interactions Really Matter, Quality Engineering, 1990.


B

U

C

K

D O E- -

+

+

Voila!



• Experiment run at SKF -- largest producer of rolling bearing in the world.

• Looked at three factors: heat treatment, outer ring osculation and cage design.

• Results:

•choice of cage design did not matter (contrary to previously accepted folklore -- considerable savings)

•life of bearing increased five fold if osculation and heat treatment are increased together -- saved millions of dollars !



• Bearings like this have been made for decades. Why did it take so long to discover this improvement ? One factor vs. interaction effects !

Osculation

Cage

Heat

12816

19 21

26 85

17 25



10621

18 23

Osculation

Heat

The Power of Interactions !



2 Design Example2

Consider an investigation into the effect of the concentration of the reactant and the amount of catalyst on the reaction time of a chemical process.

L H

reactant (factor A) 15% 25%

catalyst (factor B) 1 bag 2 bags



Design Matrix for 22

A B AB Total Average

- - +

+ - -

- + -

+ + +

Main effects Interaction


Factor A - B - 28 25 27 80

Settings A + B - 36 32 32 100

A - B + 18 19 23 60

A + B + 31 30 29 90

I II III Total

Replicates



An effect is the difference in the average response at one level of the factor versus the other level of the factor.

- +

A

60 90

80 100

A effect =

( [90 + 100] - [60 + 80] )

/ 2(3) = 8.33



Use a matrix to find the effects of each factor, including the interaction effect between the two factors.


- - + 80 26.7

+ - - 100 33.3

- + - 60 20

+ + + 90 30

Avg + 31.7

Avg - 23.3

Effect 8.4Douglas Montgomery, Design and Analysis of Experiments



- - + 80 26.7

+ - - 100 33.3

- + - 60 20

+ + + 90 30

Avg + 31.7 25 28.3

Avg - 23.3 30 26.7

Effect 8.4 -5 1.7

Completing the matrix with the effect calculations:



-10 -5 0 5 10

B AB A

Dot Diagram



35

30

25

20

- +

A

Response Plots

35

30

25

20

- +

B



35

30

25

20

- +

A

B -

B +

B - B +

A -

A +

26.7 20

33.3 30

Interaction

Response

Plot



Normal Probability PlotsNormal Probability Plots Effects are the differences between Effects are the differences between

two averages.two averages. As we know, the distribution of As we know, the distribution of

averages are approximately normal.averages are approximately normal. NPP can be used to identify the effects NPP can be used to identify the effects

that are different from noise.that are different from noise.

Soren Bisgaard, A Practical Introduction to Experimental Design


Construction of NPPConstruction of NPP Can be constructed with effects on Can be constructed with effects on

horizontal and cumulative percentages on horizontal and cumulative percentages on vertical -- but this requires normal vertical -- but this requires normal probability paper.probability paper.

Can also be constructed using the inverse Can also be constructed using the inverse standard normal of the plotting point ( (i - .5) standard normal of the plotting point ( (i - .5) / n )./ n ).

Look for effects that are different from Look for effects that are different from plotted ‘vertical’ reference line.plotted ‘vertical’ reference line.



Steps in constructing NPP

1. Compute effects.

2. Order effects from smallest to largest.

3. Let i be the order number (1 to n).

4. Calculate probability plotting position of the ordered effect using the formula ( p = [i - .5]/n).

5. Using a standard normal table determine the Z value corresponding to each left tail probability of step 4.

6. Plot the effects on horizontal axis and Z on vertical.

7. Fit a line through the most points.

8. Those ‘off the line’ are significant effects.



3 4 5 2


Plot reference linethrough the majorityof points. Look for effects which are offthis line.

6

7

8


Exercise

You will conduct a 22 experiment with 2 replicates.

Factors: L H

A -- Tower 3 5

B -- Front Stop 0 2

C -- Back Stop 5 7


Requirements:

1. Collect data -- total of 16 observations (random order).

2. Fill in matrix and compute effects.

3. Put averages on a cube plot.

4. Plot effects on dot plot and normal probability plot.

5. Create appropriate response plots for significant interactions and main effects.

6. Interpret results and make recommendations to management.


Design Matrix

ABCABACBCABC

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

Avg +Avg -

Effect

ABCABACBCABC

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

Avg +Avg -

Effect

ABCABACBCABC

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

Avg +Avg -

Effect

ABCABACBCABC

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

Avg +Avg -

Effect

Replicates

A B C AB AC BC ABC I II Total Average

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

Avg +Avg -

Effect


Cube Plot


Response Plots

- +

-

+


i P Z1 0.07 -1.52 0.21 -0.83 0.36 -0.44 0.50 05 0.64 0.46 0.79 0.87 0.93 1.5

Z

Effect

Normal

Probability

Plot


Why use 2Why use 2kk designs ? designs ?• Easy to use and data analysis can be performed using graphical methods.

• Relatively few runs required.

• 2k designs have been found to meet the majority of the experimental needs of those involved in the improvement of quality.

• 2k designs are easy to use in sequential experimentation.

• Fractions of the 2k (fractional factorials) can be used to further reduce the experiment size.Moen, Nolan and Provost, Improving Quality Through Planned Experimentation

1 Jim Grayson, PhD MSC 605 MSC 605 Design of Experiments.

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