Transcript Balanced Incomplete Block Designs

Lecture 16
• Today: 10.6-10.9
• Next day:
Two-Step Optimization Procedures
• Nominal the best problem:
– Select the levels of the dispersion factors to minimize the dispersion
– The select the levels of the adjustment factors to move the process on
target
• Larger (Smaller) the better problem:
– Select levels of location factors to optimize process mean
– Select levels of dispersion factors that are not location factors to minimize
dispersion
• Leaf Spring Example was a nominal the best problem
Response Modeling
• There may be several noise factors and control factors in the
experiment
• The cross array approach identifies control factors to help adjust the
dispersion and location models, but does not identify which noise
factors interact with which control factors
• Cannot deduce the relationships between control and noise factors
• The response model approach explicitly model both control and noise
factors in a single model (called the response model)
Response Modeling
• Steps:
– Model response, y, as a function of both noise and control factors (I.e.,
compute regression model with main effects and interactions of both types
of factors)
– To adjust variance:
• make control by noise interaction plots for the significant control by noise
interactions. The control factor setting that results in the flattest relationship
gives the most robust setting.
• construct the variance model, and choose control factor settings that minimize
the variance
Example: Leaf Spring Experiment (p. 438)
• 25-1 fractional factorial design was performed: I=BCDE
• Experiment has 3 replicates
Control Factors
B
C
D
E
-1
-1
-1
-1
+1
-1
-1
+1
-1
+1
-1
+1
+1
+1
-1
-1
-1
-1
+1
+1
+1
-1
+1
-1
-1
+1
+1
-1
+1
+1
+1
+1
Q Level
7.78
8.15
7.50
7.59
7.94
7.69
7.56
7.56
-1
7.78
8.18
7.56
7.56
8.00
8.09
7.62
7.81
7.81
7.88
7.50
7.75
7.88
8.06
7.44
7.69
7.50
7.88
7.50
7.63
7.32
7.56
7.18
7.81
+1
7.25
7.88
7.56
7.75
7.44
7.69
7.18
7.50
7.12
7.44
7.50
7.56
7.44
7.62
7.25
7.59
Example: Leaf Spring Experiment (p. 438)
• 25-1 fractional factorial design was performed: I=BCDE
B
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
C
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
D
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
E
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
Q
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
Y
7.79
8.07
7.52
7.63
7.94
7.95
7.54
7.69
7.29
7.73
7.52
7.65
7.40
7.62
7.20
7.63
Example: Leaf Spring Experiment (p. 438)
Effect
B
C
D
E
Q
BC=DE
BD=CE
BE=CD
BQ
CQ
DQ
EQ
BCQ=DEQ
BDQ=CEQ
BEQ=CDQ
Estimate
0.221
-0.176
-0.029
0.104
-0.260
-0.017
-0.020
-0.035
0.085
0.165
-0.054
0.027
-0.010
0.040
-0.047
E
f
e
c
t
s
i
m
a
-0.2 -0.1 0. 0.1 0.2
Example: Leaf Spring Experiment (p. 438)
- 1
0
1
Q u a n tile s
o f
Example: Leaf Spring Experiment (p. 438)
• Response Model:
Example: Leaf Spring Experiment (p. 438)
m
e
a
n
o
f
Y
7.5 7.6 7. 7.8 7.9
C
- 1
1
- 1
1
Q
Example: Leaf Spring Experiment (p. 438)
• Variance Model:
Design Strategy for the Response Model