9 Injection Molding
EXAMPLE 14-9 Injection Molding
Parts manufactured in an injection-molding process are show-
process should produce an average shrinkage of around 10
ing excessive shrinkage, which is causing problems in assem-
regardless of the temperature level chosen.
bly operations upstream from the injection-molding area. In
Based on this initial analysis, the team decides to set
an effort to reduce the shrinkage, a quality-improvement team
both the mold temperature and the screw speed at the low
has decided to use a designed experiment to study the injection-
level. This set of conditions should reduce the mean shrink-
molding process. The team investigates six factors—mold
age of parts to around 10. However, the variability in
temperature (A), screw speed (B), holding time (C), cycle time
shrinkage from part to part is still a potential problem. In
(D), gate size (E), and holding pressure (F )— each at two lev-
effect, the mean shrinkage can be adequately reduced by
els, with the objective of learning how each factor affects
the above modifications; however, the part-to-part variabil-
shrinkage and obtaining preliminary information about how
ity in shrinkage over a production run could still cause
the factors interact.
problems in assembly. One way to address this issue is to
The team decides to use a 16-run two-level fractional fac-
see if any of the process factors affect the variability in
torial design for these six factors. The design is constructed by
parts shrinkage.
writing down a 2 4 as the basic design in the factors A, B, C,
Figure 14-39 presents the normal probability plot of the
and D and then setting E ABC and F BCD as discussed
residuals. This plot appears satisfactory. The plots of residuals
above. Table 14-29 shows the design, along with the observed
versus each factor were then constructed. One of these plots,
shrinkage (
10) for the test part produced at each of the 16
that for residuals versus factor C (holding time), is shown in
runs in the design.
Fig. 14-40. The plot reveals much less scatter in the residuals
A normal probability plot of the effect estimates from this
at the low holding time than at the high holding time. These
experiment is shown in Fig. 14-37. The only large effects are A
residuals were obtained in the usual way from a model for pre-
(mold temperature), B (screw speed), and the AB interac-
dicted shrinkage
tion. In light of the alias relationship in Table 14-28, it seems reasonable to tentatively adopt these conclusions.
ˆ y
ˆ 0 ˆ 1 x 1 ˆ 2 x 2 ˆ 12 x 1 x 2
The plot of the AB interaction in Fig. 14-38 shows that the
27.3125 6.9375x 1 17.8125x 2 5.9375x 1 x 2
process is insensitive to temperature if the screw speed is at the low level but sensitive to temperature if the screw speed
where x 1 ,x 2 , and x 1 x 2 are coded variables that correspond to
is at the high level. With the screw speed at a low level, the
the factors A and B and the AB interaction. The regression
JWCL232_c14_551-636.qxd 11610 9:57 AM Page 611
14-7 FRACTIONAL REPLICATION OF THE 2 k DESIGN
20 Normal probability 10 Shrinkage (
Mold temperature, A
Figure 14-37 Normal probability plot of effects
Figure 14-38 Plot of AB (mold temperature–
for Example 14-9.
screw speed) interaction for Example 14-9.
model used to produce the residuals essentially removes
From inspection of this figure, we see that running the
the location effects of A, B, and AB from the data; the
process with the screw speed (B) at the low level is the key to
residuals therefore contain information about unexplained
reducing average parts shrinkage. If B is low, virtually any
variability. Figure 14-40 indicates that there is a pattern in
combination of temperature (A) and holding time (C ) will re-
the variability and that the variability in the shrinkage of
sult in low values of average parts shrinkage. However, from
parts may be smaller when the holding time is at the low
examining the ranges of the shrinkage values at each corner
level.
of the cube, it is immediately clear that setting the holding
Practical Interpretation: Figure 14-41 shows the data
time (C ) at the low level is the most appropriate choice if we
from this experiment projected onto a cube in the factors A,
wish to keep the part-to-part variability in shrinkage low during
B, and C. The average observed shrinkage and the range of
a production run.
observed shrinkage are shown at each corner of the cube.
70 0 50 Residuals 30 –2
20 Normal probability 10
Holding time (C)
Figure 14-39 Normal probability plot of resid-
Figure 14-40 Residuals versus holding time
uals for Example 14-9.
(C ) for Example 14-9.
JWCL232_c14_551-636.qxd 11610 9:57 AM Page 612
CHAPTER 14 DESIGN OF EXPERIMENTS WITH SEVERAL FACTORS
y = 33.0 R=2
y = 60.0 R=0
B, screw speed
y = 10.0 R = 12
y = 10.0 R = 10
Figure 14-41 +
Average
y = 7.0
shrinkage and range of
R=2
y = 11.0 R=2
–
C, holding time
shrinkage in factors A, B,
and C for Example 14-9.
A, mold temperature
The concepts used in constructing the 2 6 2 fractional factorial design in Example 14-9
can be extended to the construction of any 2 k p fractional factorial design. In general, a 2 k fractional factorial design containing 2 k p runs is called a 1 p fraction of the 2 k 兾2 design or, more simply, a 2 k p fractional factorial design. These designs require the selection of p inde- pendent generators. The defining relation for the design consists of the p generators initially chosen and their 2 p p 1 generalized interactions.
The alias structure may be found by multiplying each effect column by the defining relation. Care should be exercised in choosing the generators so that effects of potential inter- est are not aliased with each other. Each effect has 2 p
1 aliases. For moderately large values
of k, we usually assume higher order interactions (say, third- or fourth-order or higher) to be negligible, and this greatly simplifies the alias structure.