Example of a 2^{3}
Factorial Experiment
Below is a
hypothetical example of a 2^{3} factorial experiment to
illustrate the application of factorial experiments in improving
processes.
In this
experiment, the process engineer's goal is to determine how the yield of
an adhesive application process can be improved by adjusting three (3) process
parameters: mixture ratio, curing temperature, and curing time.
For each of these input parameters, two levels will be defined for use
in this 2-level experiment. For the mix ratio, the high level is
set at 55%, while the low level is set at 45%. For the curing
temp., the high level is set at 150 deg C while the low level is set at
100 deg C. For the curing time, the high level is set at 90
minutes, while the low level is set at 30 minutes. As mentioned,
the output response monitored is process yield. Assume further that the
data were gathered by performing just a single replicate (n=1) per
combination treatment.
Table 1. Results of the Example 2^{3} Factorial Experiment
RUN |
Comb. |
Factors |
Yield |
Mix
Ratio |
Temp |
Time |
1 |
(1) |
45% (-) |
100C (-) |
30m (-) |
8 |
2 |
a |
55%
(+) |
100C (-) |
30m (-) |
9 |
3 |
b |
45% (-) |
150C (+) |
30m (-) |
34 |
4 |
ab |
55%
(+) |
150C (+) |
30m (-) |
52 |
5 |
c |
45% (-) |
100C (-) |
90m (+) |
16 |
6 |
ac |
55%
(+) |
100C (-) |
90m (+) |
22 |
7 |
bc |
45% (-) |
150C (+) |
90m (+) |
45 |
8 |
abc |
55%
(+) |
150C (+) |
90m (+) |
56 |
Applying
Table 6 of the article factorial
design tables to get the algebraic signs of the
coefficients of the factorial effect formulas as discussed in the article
on 2-Level
factorial experiments, the
following calculations for the main and interaction effects of these 3
factors are obtained:
A = 1/(4n) x
[-(1)+a-b+ab-c+ac-bc+abc] =
[-8+9-34+52-16+22-45+56]
= 1/4 x 36 = 9
B = 1/4 x
[-8-9+34+52-16-22+45+56] = 1/4 x 132 = 33
AB = 1/4 x
[+8-9-34+52+16-22-45+56] = 1/4 x 22 = 5.5
C = 1/4 x
[-8-9-34-52+16+22+45+56] = 1/4 x 36 = 9
AC = 1/4 x
[+8-9+34-52-16+22-45+56] = 1/4 x -2 = -0.5
BC = 1/4 x
[+8+9-34-52-16-22+45+56] = 1/4 x -6 = -1.5
ABC = 1/4 x
[-8+9+34-52+16-22-45+56] = 1/4 x -12 = -3
Based on
these calculations, the main effect of temperature (B=33)
has the greatest influence on the
process yield, although the main effects of mixture ratio (A=9) and
time (C=9) are also significant. The interaction between mixture ratio and
temperature also produces a positive effect on yield (AB=5.5), but the rest of
the factorial interactions affect the yield in the negative direction
(although to much lower degrees).
See
also:
Factorial Experiments;
2-Level Factorial Experiments;
Factorial Design Tables
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