Applied Statistics and Probability for Engineers

556 CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS

To illustrate the 14 fraction, consider an experiment with six factors and suppose that the

engineer is primarily interested in main effects but would also like to get some information

about the two-factor interactions. A 2^6 ^1 design would require 32 runs and would have

31 degrees of freedom for estimating effects. Since there are only six main effects and 15 two-

factorinteractions, the one-half fraction is inefficient—it requires too many runs. Suppose we

consider a 14 fraction, or a 2^6 ^2 design. This design contains 16 runs and, with 15 degrees of

freedom, will allow all six main effects to be estimated, with some capability for examining

the two-factor interactions.

To generate this design, we would write down a 2^4 design in the factors A, B, C, and Das

the basic design and then add two columns for Eand F. To find the new columns we could se-

lect the two design generatorsI ABCEand I BCDF. Thus, column Ewould be found

from E ABC, and column Fwould be F BCD. That is, columns ABCEand BCDFare

equal to the identity column. However, we know that the product of any two columns in the

table of plus and minus signs for a 2kdesign is just another column in the table; therefore, the

product of ABCEand BCDFor ABCE(BCDF)  AB^2 C^2 DEF ADEFis also an identity col-

umn. Consequently, the complete defining relationfor the 2^6 ^2 design is

We refer to each term in a defining relation (such as ABCEabove) as a word.To find the alias

of any effect, simply multiply the effect by each word in the foregoing defining relation. For

example, the alias of Ais

The complete alias relationships for this design are shown in Table 14-27. In general, the res-

olution of a 2kpdesign is equal to the number of letters in the shortest word in the complete

defining relation. Therefore, this is a resolution IV design; main effects are aliased with three-

factor and higher interactions, and two-factor interactions are aliased with each other. This

design would provide good information on the main effects and would give some idea about

the strength of the two-factor interactions. The construction and analysis of the design are

illustrated in Example 14-8.

EXAMPLE 14-8 Parts manufactured in an injection-molding process are showing excessive shrinkage, which

is causing problems in assembly operations upstream from the injection-molding area. In an

effort to reduce the shrinkage, a quality-improvement team has decided to use a designed

experiment to study the injection-molding process. The team investigates six factors—mold

temperature (A), screw speed (B), holding time (C), cycle time (D), gate size (E), and holding

ABCEABCDFDEF

IABCEBCDFADEF

Table 14-27 Alias Structure for the Design with IABCE

BCDF ADEF

A BCE DEF ABCDF AB CE ACDF BDEF

B ACE CDF ABDEF AC BE ABDF CDEF

C ABE BDF ACDEF AD EF BCDE ABCF

D BCF AEF ABCDE AE BC DF ABCDEF

E ABC ADF BCDEF AF DE BCEF ABCD

F BCD ADE ABCEF BD CF ACDE ABEF

ABD CDE ACF BEF BF CD ACEF ABDE

ACD BDE ABF CEF

(^2) IV^6 ^2
c 14 .qxd 5/9/02 7:54 PM Page 556 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH112 FIN L: