1. DESIGN OF EXPERIMENTS

2. Designed Experiment
..is a test in which purposeful changes are made to the inputs of a system so that one may observe and quantify the changes in the outputs.
To obtain unambiguous results at minimum cost;
To learn about interactions between variables
To measure experimental error and confidence

3. Categories of Designed Experiments
Screening-determine the important factors
Comparative and Evaluative-make a statement about a process variable
Optimization-determine the settings of some factors to minimize/maximize an output
Regression-obtain a mathematical relationship between factors and outputs

4. Uses of DOE Problem Data ANOVA-to determine the important factors DOE
DOE-limited to determine the important factors
Experiment
DOE-expanded

5. Sample Problem

6. Factorial Design
Factorial Design is one in which there is a test condition for each combination of levels for the factors considered
For the sample:
Two-Level - 2x2x2 = 8
Three-Level - 3x3x3 = 27

7. Levels: Experimental Design
Two-level
Reduces Number of Tests
Assumes a Linear Relationship
Is true ONLY over chosen range of factors
2k = number of tests
Three-Level
Reduces Number of Tests
Non-linear relationships
3k = number of tests

8. Comparison of levels 2-Level Higher Levels
Good for demonstrating the concepts
Cube
Table
Equation
Tools not applicable for higher levels
Higher Levels
General approach
Uses
Regression
ANOVA

9. Two-Level Experimental DesignB1
Output B1 B2 A1 A2
Interacting Factors
Output B2 A1 A2
Non-interacting Factors

10. Three-Level Experimental DesignB1
Output
Output B1 B2 B2 A1 A2 A1 A2
Non-Interacting Non-linear
Interacting Non-linear

11. Tread Width-W Tread Gap Size-G Tire Pressure-P Y8=36 Y7=18 Y3=23 Y4=26

12. Main Effects
Main Effect for a single factor expresses the average change in the output response associated with changing the factor from its (-) level to its (+) level.
Ew = 1/4(Y2 + Y4 + Y6 + Y8)- 1/4(Y1 + Y3 + Y5 + Y7)
Ew = 1/4( )- 1/4( )
Ew = = 12.25
This estimates that the effect from increase the tire width from 0.15 to increases the tire life by thousand miles.

13. Interaction Effects WxG - first with the gap at the (-) level
Interaction Effects quantify the interrelationship of two or more factors. For two factors, its half the difference of the average effects for one factor with the other at its two levels.
WxG - first with the gap at the (-) level
WxG- = 1/2(Y2 + Y6)-1/2 (Y1 + Y5)=14.0
WxG+ = 1/2(Y4 + Y8)-1/2 (Y3 + Y7)=10.5
WxG = 1/2( )= -1.75

14. Tire Experiment Effects

15. Regression Equation
Y = Ym +0.5[Effectwxw+ Effectpxp+ Effectwxw+ Effectwgxwg+ Effectwpxwp+ Effectgpxgp+ Effectwpgxwpg]
Y= [12.25xw+ .75xp-4.75xw-1.75xwg+ 8.75xwp+ 1.75xgp-1.25xwpg]

16. Constructing Confidence Intervals
Response variance can be estimated from sample results from replicates. Factor statistics:

17. Response Statistics
The average of the factor variances provides the estimate for the response statistics.
Pooled Estimate of the Overall Variance in a 2-Level Replicated Factorial Design:

18. Tire Example
The last two columns provide the remaining lifetime from each pair of cars for NT=8:
The response variance is
sy2 = 64/2(8) = sy = 2

19. Confidence Interval of an Effect
Standard Error of an Effect
seffect = 2( sy/NT).5
True Effect = Computed Effect +ta/2seffect
Tire Example - seffect = 2(2/8) .5 = 1
The 99% confidence interval for tire width effect:
W = (1.0) =

20. Determination of Significant Factors
Comparison of factor mean and the response confidence interval

21. ANOVA- 3-Factor,2-Level

22. 3k Design: Example Example from Montgomery, C.
Filling 5 gallon metal containers with soft drink syrup.
Factors that are thought to influence frothing:
Nozzle configuration (A)
Operator (B)
Operating Pressure (C)
33 factorial experiment with Two replicates

23. Data

24. Analysis: 1. Significant parameters
ANOVA to test which Factors and Interactions are significant
Parameters that are not significant are not included in regression model
Excel for k<2
Montgomery, C. or other books on DOE for k>2
Alternatively, MATLAB for k>2

26. ANOVA Summary
At the 0.05 level, B,C and AB, BC, AC interactions are significant
B,C and B-C interaction are significant at 0.01 level
Only the significant factors are included in the regression model
Reduced effort in fitting model to data
Model is easier to use