ISyE 6414 Exam #2(a) – DOE-I –with Complete
Solution
1. Consider a 16-run design, 26−2 (4 = −156 and 3 = 246).
a) In constructing the layout of this design, what are the four generating columns? [6 points]
Sol: There are many alternative solutions for this problem. One of them is
X1, X2, X5 and X6 are the standard (−1, +1, …) generating columns.
b) How do you construct the two generated-columns? Please describe it without
working out the layout itself. [6 points]
Sol: Based on the answer given in (a), the following two are columns generated from
the two generators.
X3 = X2 * X4 * X6 and X4 = −X1*X5*X6.
c) Derive its confounding patterns for the 16 estimable effects. [25 points]
Note: Ignore all 4, 5 and 6-factor-interactions. Moreover, to keep the presentation brief, also ignore the
3-factor-interactions with main effects, e.g., −156 can be ignored in the 4 = −156 confounding
pattern. But, present the confounding patterns among 3-factor-interactions, e.g., 124 = −256
=…
Sol: I = −1456 = 2346 = −1235
Since it is a Resolution IV design, all six main effects, X1, X2, …, X6, will be
confounded with 3-factor-interactions (3-fi’s), and thus, we do not need to write
their confounding patterns according to the note above.
The following are 7 sets of confounded two-factor-interactions.
12 = −35, 13 = −25, 14 = −56, 15 = −46 = −23, 16 = −45,
24 = 36, 26 = 34.
The following are the remaining two estimable effects at three-factor-interactions.
124 = −256 = 136 = −345 and 126 = −245 = 134 = −356.
1
, Students wasted time in writing down unnecessary three-factor-interactions confounded with main
effects.
2
Solution
1. Consider a 16-run design, 26−2 (4 = −156 and 3 = 246).
a) In constructing the layout of this design, what are the four generating columns? [6 points]
Sol: There are many alternative solutions for this problem. One of them is
X1, X2, X5 and X6 are the standard (−1, +1, …) generating columns.
b) How do you construct the two generated-columns? Please describe it without
working out the layout itself. [6 points]
Sol: Based on the answer given in (a), the following two are columns generated from
the two generators.
X3 = X2 * X4 * X6 and X4 = −X1*X5*X6.
c) Derive its confounding patterns for the 16 estimable effects. [25 points]
Note: Ignore all 4, 5 and 6-factor-interactions. Moreover, to keep the presentation brief, also ignore the
3-factor-interactions with main effects, e.g., −156 can be ignored in the 4 = −156 confounding
pattern. But, present the confounding patterns among 3-factor-interactions, e.g., 124 = −256
=…
Sol: I = −1456 = 2346 = −1235
Since it is a Resolution IV design, all six main effects, X1, X2, …, X6, will be
confounded with 3-factor-interactions (3-fi’s), and thus, we do not need to write
their confounding patterns according to the note above.
The following are 7 sets of confounded two-factor-interactions.
12 = −35, 13 = −25, 14 = −56, 15 = −46 = −23, 16 = −45,
24 = 36, 26 = 34.
The following are the remaining two estimable effects at three-factor-interactions.
124 = −256 = 136 = −345 and 126 = −245 = 134 = −356.
1
, Students wasted time in writing down unnecessary three-factor-interactions confounded with main
effects.
2
