Page 479 - Design for Six Sigma a Roadmap for Product Development

438 Chapter Twelve

experiments is wasted. In summary, for full factorial experiments, as
the number of factors k increases, the number of runs will increase at
an exponential rate that leads to extremely lengthy and costly experi-
ments. On the other hand, as k increases, most of data obtained in the
full factorial are used to estimate higher-order interactions, which are
most likely to be insignificant.
Fractional factorial experiments are designed to greatly reduce the
number of runs and to use the information from experimental data
wisely. Fractional experiments run only a fraction of the runs of a full
1
1
1
factorial; for two-level experiments, they use only , , ,... of runs
8
4
2
from a full factorial. Fractional factorial experiments are designed to
estimate only the main effects and two-level interactions, and not
three-factor and other higher-order interactions.
3
12.4.1 A 2 3 1 design (half of a 2 )
Consider a two-level, full factorial design for three factors, namely, the
3
3
2 design. Suppose that the experimenters cannot afford to run all 2
8 treatment combinations, but they can afford four runs. If a subset of
four runs is selected from the full factorial, then it is a 2 3 1 design.
Now let us look at Table 12.12, where the original analysis matrix of
3
a 2 design is divided into two portions.
In Table 12.12 we simply rearrange the rows such that the highest
interaction, ABC contrast coefficients, are all 1s in the first four rows
and all 1s in the second four rows. The second column in this table is
called the identity column, or I column, because it is a column with
all 1s.
If we select the first four runs as our experimental design, this is
called a fractional factorial design with the defining relation I ABC,
where ABC is called the generator.

TABLE 12.12 2 3 1 Design
Factorial effects
Treatment
combination I A B C AB AC BC ABC
a 1 1 1 1 1 1 1 1
b 1 1 1 1 1 1 1 1
c 1 1 1 1 1 1 1 1
abc 1 1 1 1 1 1 1 1
ab 1 1 1 1 1 1 1 1
ac 1 1 1 1 1 1 1 1
bc 1 1 1 1 1 1 1 1
(1) 1 1 1 1 1 1 1 1