1
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
Suppose that three factors, A, B, and C, each at two levels, are of interest. The design is called a 23 factorial design and the eight treatment combinations can now be displayed geometrically as a cube.
2
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
There are seven degrees of freedom between the eight treatment combinations in the 23 design. Three degrees of freedom are associated with the main effects of A, B, and C. Four degrees of freedom are associated with interactions; one each with AB, AC, and BC and one with ABC.
3
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
4
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
5
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
Sums of squares for the effects are easily computed, because each effect has a corresponding single-degree-of-freedom contrast. In the 23 design with n replicates, the sum of squares for any effect is
Algebraic Signs for Calculating Effects in the 23 Design
6
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
A soft drink bottler is interested in obtaining more uniform fill 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. The process engineer can control three variables during the filling process: the percent carbonation (A), the operating pressure in the filler (B), and the bottles produced per minute or the line speed (C). Suppose that only two levels of carbonation are used so that the experiment is a 23 factorial design with two replicates. The data, deviations from the target fill height, are shown in Table 6-4, and the design is shown geometrically
7
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
8
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
9
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
10
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
The largest effects are for carbonation (A = 3.00), pressure (B = 2.25), speed (C = 1.75) and the carbonation-pressure interaction (AB = 0.75), although the interaction effect does not appear to have as large an impact on fill height deviation as the main effects.
11
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
12
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
1-1
2
1
0
1-1
1-1
2
1
0
A Carbonizat ion
Me
an
B Pressure
C Speed
Main Effects Plot for Fill Hight DeviationFitted Means
13
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
10-1
99
90
50
10
1
Residual
Pe
rce
nt
6420-2
1.0
0.5
0.0
-0.5
-1.0
Fitted Value
Re
sid
ua
l1.00.50.0-0.5-1.0
6.0
4.5
3.0
1.5
0.0
Residual
Fre
qu
en
cy
16151413121110987654321
1.0
0.5
0.0
-0.5
-1.0
Observation Order
Re
sid
ua
l
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for Fill Hight Deviation
14
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
1-1 1-1
4
2
0
4
2
0
A Carbonization
B Pressure
C Speed
-1
1
A C arb o n izatio n
-1
1
B P ressu re
Interaction Plot for Fill Hight DeviationFitted Means
15
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
1-1 1-1
4
2
0
4
2
0
A Carbonization
B Pressure
C Speed
-1
1
A C arb o n izatio n
-1
1
B P ressu re
Interaction Plot for Fill Hight DeviationFitted Means
16
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results follow:
a. Estimate the factor effectsb. Prepare an analysis of variance table, and determine which factors are
important in explaining yieldc. Plot the residuals versus the predicted yield and on a normal probability
scale. Does the residual analysis appear satisfactory?
17
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
N=3 , ɑ = 104 , b = 119 , c = 127, ɑb = 148, ɑc= 113, bc= 164, ɑbc= 127, (-1) = 78
18
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
N=3 , ɑ = 104 , b = 119 , c = 127, ɑb = 148, ɑc= 113, bc= 164, ɑbc= 127, (-1) = 78
19
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
Contrasts
What is SST
20
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
21
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
22
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
50
40
30
1-1
1-1
50
40
30
1-1
50
40
30
Cutting Speed
Tool Geometry
Cutting Angle
-1
1
S p eed
C u ttin g
-1
1
G eo m etry
To o l
-1
1
A n g le
C u ttin g
Interaction Plot for Life in hoursFitted Means
23
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
1-1
45.0
42.5
40.0
37.5
35.0
1-1
1-1
45.0
42.5
40.0
37.5
35.0
Cutt ing Speed
Me
an
Tool Geometry
Cutt ing Angle
Main Effects Plot for Life in hoursFitted Means
24
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
1050-5-10
99
90
50
10
1
Residual
Pe
rce
nt
5448423630
10
5
0
-5
Fitted Value
Re
sid
ua
l
10.07.55.02.50.0-2.5-5.0
6.0
4.5
3.0
1.5
0.0
Residual
Fre
qu
en
cy
24222018161412108642
10
5
0
-5
Observation OrderR
esid
ua
l
Normal Probability Plot Versus Fits
Histogram Versus Order
Residual Plots for Life in hours
25
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
151050-5-10-15
99
95
90
80
70
60
50
40
30
20
10
5
1
RESI1
Pe
rce
nt
M ean 5.181041E -15
S tDev 4.581
N 24
A D 0.674
P -Va lu e 0.069
Probability Plot of RESI1Normal - 95% CI
26
23 Factorial DesignIndustrial Engineering
Example
23 Factorial Design
Introduction
55504540353025
55504540353025
12.5
10.0
7.5
5.0
2.5
0.0
-2.5
-5.0
12.5
10.0
7.5
5.0
2.5
0.0
-2.5
-5.0
FITS1
RE
SI1
Scatterplot of RESI1 vs FITS1