Applied Statistics and Probability for Engineers

14-9 FRACTIONAL REPLICATION OF THE 2kDESIGN 559

that the variability in the shrinkage of parts may be smaller when the holding time is at the

low level.

Figure 14-37 shows the data from this experiment projected onto a cube in the factors A,

B, and C. The average observed shrinkage and the range of observed shrinkage are shown at

each corner of the cube. From inspection of this figure, we see that running the process with

the screw speed (B) at the low level is the key to reducing average parts shrinkage. If Bis low,

virtually any combination of temperature (A) and holding time (C) will result in low values of

average parts shrinkage. However, from examining the ranges of the shrinkage values at each

corner of the cube, it is immediately clear that setting the holding time (C) at the low level is

the most appropriate choice if we wish to keep the part-to-part variability in shrinkage low

during a production run.

The concepts used in constructing the 2^6 ^2 fractional factorial design in Example 14-8

can be extended to the construction of any 2kpfractional factorial design. In general, a 2kfrac-

tional factorial design containing 2kpruns is called a 1 2 pfraction of the 2kdesign or, more

simply, a 2kpfractional factorial design. These designs require the selection of pindependent

generators. The defining relation for the design consists of the pgenerators initially chosen

and their 2p p 1 generalized interactions.

The alias structure may be found by multiplying each effect column by the defining rela-

tion. Care should be exercised in choosing the generators so that effects of potential interest

are not aliased with each other. Each effect has 2p 1 aliases. For moderately large values of

k, we usually assume higher order interactions (say, third- or fourth-order or higher) to be neg-

ligible, and this greatly simplifies the alias structure.

It is important to select the pgenerators for the 2kpfractional factorial design in such

a way that we obtain the best possible alias relationships. A reasonable criterion is to

select the generators so that the resulting 2kpdesign has the highest possible design res-

olution. Montgomery (2001) presents a table of recommended generators for 2kpfrac-

tional factorial designs for k 15 factors and up to as many as n 128 runs. A portion

of his table is reproduced here as Table 14-29. In this table, the generators are shown with

either  or  choices; selection of all generators as  will give a principal fraction, while

if any generators are  choices, the design will be one of the alternate fractions for the

same family. The suggested generators in this table will result in a design of the highest

possible resolution. Montgomery (2001) also gives a table of alias relationships for these

designs.

R = 11

+

y = 31.5

R = 8

y = 56.0

R = 10

y = 10.0

R = 2

y = 11.0

R = 2

y = 33.0

R = 2

y = 7.0

B, screw speed

R = 0

y = 60.0

R = 12

y = 10.0

• 
  
  
  
  • A, mold temperature
  
  
  
  

C, holding time

+

Figure 14-37 Average

shrinkage and range of

shrinkage in factors A,

B, and Cfor Example

14-8.

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