At times, the factors to be investigated in screening experiments are
so large that even running a fractional factorial design is impractical.
This can be partially solved by using resolution III fractional factorial
designs in the cases where three factor and higher order interactions
can be assumed to be unimportant. Resolution III designs, such as the
2 design, can be used to estimate main effects using just runs. In these designs, the main effects
are aliased with two factor interactions. Once the results from these
designs are obtained, and knowing that three factor and higher order interactions
are unimportant, the experimenter can decide if there is a need to run
a fold-over design to de-alias the main effects from the two factor interactions.
Thus, the 2 design can be used to investigate
three factors in four runs, the 2 design can be used to investigate
seven factors in eight runs, the 2 design can be used to investigate
fifteen factors in sixteen runs and so on.
Example 7.8
A baker wants to investigate the factors that most affect the taste
of the cakes made in his bakery. He chooses to investigate seven factors,
each at two levels: flour type (factor ), conditioner type (factor ), sugar quantity (factor ), egg quantity (factor ), preservative type (factor ), bake time (factor ) and bake temperature (factor ). The baker expects most of these
factors and all higher order interactions to be inactive. On the basis
of this, he decides to run a screening experiment using a 2 design that requires just 8 runs.
The cakes are rated on a scale of 1 to 10. The design properties for the
2 design (with generators , , and ) are shown in Figure 7.44.
The resulting design along with the rating of the cakes corresponding
to each run is shown in Figure 7.45.
Figure 7.44:
Design properties for the experiment in Example 7.8.
Figure 7.45:
Experiment design for Example 7.8.
The normal probability plot of effects for the unreplicated design shows
main effects , and to be significant (see Figure 7.46). However, for this design, the following alias
relations exist for the main effects:
Figure 7.46: Normal probability plot of effects for the
experiment in Example 7.8.
Based on the alias structure, three separate possible conclusions can
be drawn. It can be concluded that effect is active instead of so that effects , and their interaction, , are the significant effects. Another
conclusion can be that effect is active instead of so that effects , and their interaction, , are significant. Yet another conclusion
can be that effects , and their interaction, , are significant. To accurately discover
the active effects, the baker decides to a run a fold-over of the present
design and base his conclusions on the effect values calculated once results
from both the designs are available. Using the alias relations, the effects
obtained from DOE++
for the present design (Figure 7.47) can be expressed
as:
Figure 7.47:
Effect values for the experiment in Example 7.8.
The fold-over design for the experiment is obtained by reversing the
signs of the columns , , and . The generators to be used are , , and . The resulting design and the corresponding
response values obtained are shown in Figure 7.48.
The effect values obtained from DOE++ for this design (Figure 7.49)
can be expressed as:
Figure 7.48:
Fold-over design for the experiment in Example 7.8.
Figure 7.49:
Effect values for the fold-over design in Example 7.8.
Using the effect values from both the designs, the effects can be separated
(using addition and subtraction of the effect equations) as follows:
Comparing the absolute values of the effects, the largest effects are
, and the interaction . Therefore, the most important factors
affecting the taste of the cakes in the present case are sugar quantity,
egg quantity and their interaction.
design, can be used to estimate 
main effects using just 
runs. In these designs, the main effects
are aliased with two factor interactions. Once the results from these
designs are obtained, and knowing that three factor and higher order interactions
are unimportant, the experimenter can decide if there is a need to run
a fold-over design to de-alias the main effects from the two factor interactions.
Thus, the 2
design can be used to investigate
three factors in four runs, the 2
design can be used to investigate
seven factors in eight runs, the 2
design can be used to investigate
fifteen factors in sixteen runs and so on.
