Assessment of Adaptive One-Factor-at-a-Time
Methods vs. Fractional Factorial Methods
Using ReconfigurablePaper Aircraft
by
Jeffrey B. Persons, Jr.
Submitted to the Department of
Mechanical Engineering in Partial
Fulfillment of the Requirements for the
Degree of
Bachelor of Science
at the
Massachusetts Institute of Technology
June 2006
© 2006 Jeffrey B. Persons, Jr.
All rights reserved
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
AUG
2 2006
LIBRARIES
The author hereby grants to MIT permission to reproduce and to
Distribute publicly paper and electronic copies of this thesis document in whole or
in part in any medium now know r Iereafter created.
Signature of Author .........................
'? Depardntqf
MechanicalEngineering
May 12, 2006
Certified by ..
......................................................
Asst Proeechanica
· · /
DanielD. Frey
l Engineering & Engineering Systems
Thesis Supervisor
Accepted by.
.......................
...........................
John H. Lienhard
Professor of Mechanical Engineering
Chairman, Undergraduate Thesis Committee
ARCHVES
Assessment of Adaptive One-Factor-at-a-Time Methods vs.
Fractional Factorial Methods
Using Reconfigurable Paper Aircraft
by
Jeffrey B. Persons. Jr.
Submitted to the Department of Mechanical Engineering
On May 12, 2006 in Partial Fulfillment of the
Requirements for the Degree of Bachelor of Science
in Engineering as recommended by the
Department of Mechanical Engineering
ABSTRACT
Recent research has suggested that under certain conditions, adaptive one-factor-at-a-time
(aOFAT) methods outperform more commonly used fractional factorial methods. This
study sought to corroborate these claims by analyzing a case study of a real-life
experiment. A full factorial experiment was conducted to collect data for simulations of
fractional factorial and adaptive one-factor-at-a-time experiments. The experiment used
a reconfigurable paper aircraft template with four three-level control factors.
Results indicated that the exploitation of control factor interactions by adaptive onefactor-at-a-time occurred at similar rates as predicted by Frey and Wang (2006). AOFAT
experiments proved particularly effective at avoiding factor levels that led to poor
performance. with rates of avoidance approaching 100% for the worst levels. When bias
in the full factorial experiment was eliminated, aOFAT methods even returned a higher
(weighted average) leading quality indicator value than full factorial methods.
Thesis Supervisor: Daniel D. Frey
Title: Asst Professor of Mechanical Engineering & Engineering Systems
MOTIVATION
This paper serves as a case study for the purposes of evaluating fractional
factorial and adaptive one-factor-at-a-time
methods using a real-life example.
While
many other case studies consist of computer-simulated experiments, this paper discusses
an experiment in which performance was observed in person and in which experimental
error is real rather than simulated.
In comparing the two robust design methods, this
paper attempts to validate claims by Frey of conditional
adaptive one-factor-at-a-time
plans.
improvements
provided by
In particular, it seeks to address exploitation of
control factor interactions and performance optimization.
BACKGROUND
ROBUST DESIGN
The purpose of Robust Design, as stated by Phadke (1989), is to improve the
quality of a product by minimi ing the effect of the causes of variaction iwithout
eliminactingthe causes. In designing robustly, we usually refer to minimizing variations
in production and operation, though the technique can also be applied to maximizing (or
minimizing) performance characteristics of a product or process.
In any Robust Design operation, we must first identify our leading quality
indiccator.
This
is
the
performance
characteristic
that
we
wish
to
maximize/minimize/control the variance of. To ensure the usefulness of results, the
leading quality indicator must be easily observed and quantified.
select control factors.
The next step is to
Control factors are the variables that we adjust to optimize the
leading quality indicator.
To arrive at the best solution, we adjust the control factors,
observe the results, and keep the best value of the group. Without a method for choosing
factors and factor levels, however, this procedure
can be painfully inefficient.
To
facilitate the Robust Design process, therefore, different robust design algorithms have
been developed.
Of interest in this study are two particular algorithms, or methods, of
Robust Design: fractional factorial experimentation.
and adaptive one-factor-at-a-time
experimentation.
Fractional Factorial Experimentation
The use of statistical experimentation for practical design benefit has been around
since at least the 1920's, when Sir Ronald Fisher applied it to agricultural processes in
order to determine ideal levels of control factors for crop growth.
1950's. however,
It was not until the
that Dr. Genichi Taguchi adapted the use of fractional factorial
experiments to robust product design and developing Robust Design into its own field.
Taguchi's methods are now well-known, having become the industry standard for robust
design algorithms.
Fractional factorial experimentation uses orthogonal arrays to isolate a favorable
combination
of factor levels.
In effect, it allows us to run a relative handful of
experiments and based on the data collected to estimate a good selection of factor levels.
The exact number of' experiments required for this method depends on the number of
factors involved and the number of possible levels per factor.
For example, the
orthogonal array for an experiment testing four factors at three levels each would look
like this:
Factor:
Levels
(1,2,
or 3):
A
B
C
D
1
1
1
1
1
2
2
2
1
3
3
I
31
-3
2
3
3
1
3
2
1
3
3
'?3
2
1
1
2
2
I
4
I~~-
-~
Figure 1. The standard L9 orthogonal array.
For the matrix to be orthogonal, each level appears an equal number of times in each
column, and in any two columns every combination of factor levels appears. This feature
ensures that each factor and its various levels are equally weighted during analysis.
The process for applying the fractional factorial method is as follows:
1. Establish an orthogonal array consisting of the required experiments
2. To ensure a more robust result, cross the control factor array with an outer' noise
factor array. These noise factors are predetermined by the experimenter.
3.
Run the experiments set out in the crossed array.
4. Determine the signal-to-noise ratio, ir , for each data set. For a larger-the-better
leading quality indicator, we use the following equation:
q, =-10log1
Y
l
Where yj is the observed leading quality indicator, and qi is the signal-to-noise
ratio for set i.
5. Using averages of these signal-to-noise ratios, determine ratios for each factor and
its level. To do this, refer to the orthogonal array created earlier. For example, if
factor A was set to level 1 for data sets 1-3, then one would use the following
equation to determine the S/N ratio for Al:
1
m./ - - (71 + 7'2 +73 )
Repeat for all factors and levels (for an L9 orthogonal array, this would involve a
total of 12 m values.
5
6. By comparing these factor level signal-to-noise ratios, select the highest value for
each factor.
The factor levels corresponding
to these values will be the best
estimates suggested by the fractional factorial method
Adaptive One Factor At-a-Time (AOFAT) Experimentation
One-factor-at-a-time (OFAT) experimentation is one of the oldest and most
archaic of robust design methods. OFAT involves conducting a full factorial experiment,
running through every possible combination of factor levels, changing, as the name
suggests, one factor at a time.
This method is thorough but quite cumbersome. and in
most areas of industry limited resources prevent its use.
Recently, however, researchers have begun using a modified version of OFAT
that uses fewer experiments
than standard OFAT methods and has the potential to
outperform fractional factorial experimentation. Dubbed adaptive one-factor-at-a-time
(aOFAT) experimentation, this procedure requires only as many experiments as the
ftractional factorial method, and according to recent work by Frey and Li (2004), aOFAT
can provide better results, particularly
in cases with high levels of control factor
interaction and low experimental error. aOFAT is more effective in these cases because
it can exploit the control factor interactions to which fractional factorial methods are
insensitive.
The process behind aOFAT is quite simple.
First, pick a random starting configuration.
To use an example from a four-factor, three-
level experiment as before:
A
B
1_
C
_~3 1~I
6
D
Next, select a random order in which to vary factor levels. We'll say D, A, B, C. First
change the level of D to 2.
A
B
C
D
1
2
3
2
If the signal-to-noise ratio for this configuration is higher than the first, then we keep this
new configuration.
It is important that we try all levels for each factor, so we must also
try D=3. Once we select the optimal level of D, we go on to A, then B, then C. Once we
finish adjusting a factor, we do not come back to it. This is an important difference
between aOFAT and OFAT, and it is this feature that limits the required number of
experiments.
The final configuration at which we arrive will be the best configuration as
suggested by aOFAT. For further examples of the adaptive OFAT process, refer to
Appendix D.
Another advantage of aOFAT over fractional factorial experiments is the ability
to streamline the process based on available information.
If little is known about the
system, then it makes sense to pick a random starting configuration and a random order
for varying the factor levels; however, an expert's knowledge of likely interactions may
make it possible to identify a better starting point or adjust the most influential factors
first. This ability to tailor experimentation as desired provides the added benefit of
getting closer to the best solution faster. If for some reason resources are limited or
experimentation
has to be ended early, one is more likely to have good results with
aOFAT.
METHOD
Experiment
7
In order to simulate experimentation using different algorithms, we needed a full
data set to draw from. To obtain this data, we conducted a full factorial experiment using
the aircraft template provided by Eppinger (1995) (APPENDIX A). This template can be
modified by varying four main factors: weight placement (indicated by "Weight" on the
template and labeled factor A), winglet configuration (Stabliz., B), nose length (Nose, C)
and wing angle (Wing, D).
To ensure a reliable and robust data set, the full factorial array was crossed with a
noise array consisting of four combinations of three two-level noise factors: throwing
right- or left-handed, throwing with or without a glove, and throwing while standing on
one leg or two.
By crossing the control array with a noise array, we ensure that the
solution at which we eventually arrive will be insensitive to anticipated noise in
operation. The noise conditions were set up as follows:
Throwing hand
Glove?
# of Legs
N1I
Left
Yes
2
N2
Left
No
1
N3
Right
Yes
I
N4
Right
No
2
Aside from this crossed array used in the experiment, there were other sources of noise as
well that could account for variation in performance.
These sources include minor
variations in folding the airplanes, errors in placing the weights on the airplanes, drafts or
turbulent air in the room where the experiments
took place, and the strength of the
throws.
When only one foot was used, the researcher stood on his right foot. When two
feet were used, the researcher stood with his feet slightly offset, right foot forward. The
throwing style used was that of throwing a dart - roughly level and not forced. Distances
were measured from the location of the nose of the aircraft when the aircraft touched the
ground.
8
Because time allowed it, we conducted two full factorial experiments at different
dates, conducted in different ways. These two data sets are referred to in this document
as Data Trial I and Data Trial II.
Data Trial I was a full factorial experiment conducted sequentially by Variable B
(winglet orientation), beginning with Set I. To save time, all three configurations of
Variable A (weight placement) were done on a single aircraft, so in all only 27 different
aircraft were used. Noise factors were also taken sequential, N 1 -N4 in each set.
Data Trial II was a full factorial experiment conducted randomly by set and run
through all 81 sets. Noise factors were taken N1,N2,N3,N4 for the first set, then
N2,N3,N4,NI
for the next set, and so on, with the cycle repeating every four sets. The
random ordering of sets and cycling of noise factors was expected to reduce bias in the
results and to allow better observation of control factor interactions.
Analysis
Following the conduct of the two trials, the data was subjected to simulations
using two robust design algorithms: the fractional factorial method, and the adaptive onefactor-at-a-time (aOFAT) method.
For fractional factorial simulations, a standard L9 (34) Orthogonal Array was used,
and the process of analysis was carried out as prescribed by Phadke (CITE).
To simulate the aOFAT procedure, every permutation of starting point and factor
order was assessed, for 81 x 24 = 1944 possible routes per data trial.
large data sets that information of interest to aOFAT was taken.
9
It is from these
While previous experiments have applied simulated error to their data to estimate
the effect of greater variance on their results, the author determined that the real error
contained in the experiments was sufficient for the purposes of this study.
RESULTS
Data Trial I
Fractional Factorial
For Data Trial I. the fractional factorial method returned a maximum value of
189.5 in. corresponding to the configuration A2 B3 C2 D2. However, one of the sets
tested was A2 B3 C1 D2, which delivers an average distance of 209.5 in. Neither of
these values approaches the recorded optimum. which is 224.5 in, found at A3 B1 C1 D1.
By subtracting the value n from each of the signal-to-noise ratios associated with
the factor levels recommended
by the fractional factorial method, it is possible to
estimate expected contributions
to the S/N ratio above the mean.
Based on the
cumulative effects of the configuration recommended above, one would expect a S/N
ratio of 46.39 db. The observed S/N ratio associated with the configuration A2 B3 C1 D2
is 46.42 db. much in line with expectations.
In contrast, the S/N ratio of the optimum
configuration has an expected value of 45.15 db, while the observed value is 47.02 db,
much higher than anticipated.
What this means is that, unlike the phenomenon we will
observe with the adaptive OFAT method,
the configuration
recommended
fractional factorial method does not take advantage of control factor interactions.
10
by the
Plot of factor effects
46.00000
45.00000
44.00000
m 43.00000
.. ..
..
. . I
. I
. ........
----_----
. I
--------
- - - - -
I
i
rn(nVA
AL.vvvvv
-----
41.00000
.. .
-
.. - ..
···-------.
- . .. .. ..·- --.
..---- ----... - ..· ··.
i
I
.. . .......
....
..
.
.
.1 .
....... ...
.
...
....
I. . .
*. Ii
40.00000
mAl
mA2
mA3
Weight placement
mB1
mB2
mB3
Winglet orientaion
mC1
mC2
mC3
Noselength
mD1
mD2
mD3
Wing angle
Figure 2. Factor effects from fractional factorial experiment, Data Trial I.
aOFAT
For Data Trial I, Adaptive One Factor At-a-Time experimentation led to a
weighted average distance of 198.95 in. By eliminating all starting points with B2, this
value was increased to 204.27 in. Using the modified starting points, AOFAT met or
exceeded the value returned by fractional factorial methods 58% of the time. If we also
eliminate starting points with C3, intuitively the least stable configuration of factor C, we
further increase the weighted average to 205.25 in, and aOFAT equals or betters
fiactional factorial results 62% of the time. In all three starting point groups, the optimal
solution (A3 B I C 1 D1) was selected approximately 20% of the time.
More interesting than the absolute distances suggested by aOFAT is the efficiency
with which it exploits factor effects.
In the original aOFAT simulations (no starting
points excluded), 100% of the recommended values avoided the configurations B2 and
C3, which are intuitively much less stable than other options.
Additionally, 93% of the results avoided the combination Al-Dl, and 73% avoid
B -D1. which were suggested by graphical evidence (see Figures 3 and 4) as producing a
11
decidedly unfavorable interaction.
These interactions are not intuitive, and their
placement suggests that they may be misleading. Most likely, these apparent interactions
were the result of a sequential testing procedure in which performance was anticipated by
the experimenter.
A-C Interactions
180
S 170
,
c
160
150
a
140
C
130
C1
~-· · ·--..... ..· ....
.......
W.......
C2
._ ......
.......
.
.... ..... ........
-
l, 120
,
110
100
.
Al
A2
C3
. .
A3
Factor
Figure 3. Mild synergistic interaction between factors A and C.
A-D Interactions
180
S 170
· 160
c 150
a 140
~-.
"~' ~~-~~~---···
·- ·-
D1
·--- ·-
· ··.....
- --... ..--.. ..-·--.
:
......
I .......
D2
a 130
......
:~:....D3
'= 120
M 110
100
Al
A2
Factors
Figure 4. Antisynergistic interaction between Al and DI.
Data Trial II
Fractional Factorial
12
A3
For Data Trial II, the fractional factorial method returned the configuration A2 B
C( Dl for an average distance of 211.5 in. The actual optimal configuration was A3 BI
C1 D1, returning a flight distance of 233.25 in. The fractional factorial solution failed to
capture the full effects of factor interactions; it had a signal-to-noise ratio of 46.5 db, only
slightly higher than the expected value of 45.64 db, derived from individual factor effects
above the mean.
The optimal solution indicates that there may be some three-level
interactions taking place. as it had the same C and D values as the fractional factorial
solution, yet its S/N ratio was 47.4 db, significantly higher than the expected value of
45.1 db.
Plot of factor effects
Ar,
nnnnn
-- -----
44.00000
m
---
----
------
--------
43.00000
A'
nnnr
I_._____
rnAl
I
_
mA2
mA3
Weight placement
mB1
mB2
mB3
mC1
Winglet orientaion
mC2
mC3
Nose length
mD1
mD2
mD3
Wing angle
Figure 5. Factor effects for fractional factorial experiment, Data Trial II.
aOFAT
The results for aOFAT tests of Data Trial II were even more favorable than for
Data Trial I. aOFAT returned a weighted average distance of 211.7 in. exceeding the
distance
for
the
configuration
recommended
by
fractional
factorial
methods.
Additionally. by eliminating starting points with the value B2, the weighted average
increased to 213 in, and aOFAT returned the optimal value of 233.25 in 32% of the time.
Because Data Trial II was conducted in a random order, it can be expected to
provide cleaner results more indicative of physical interactions inherent in the aircraft
13
design. For example, graphical evidence suggests a moderate negative interaction in the
configuration B3-DI (see Figure 6). This configuration corresponds to winglets down
with the highest wing angle, an arrangement that has the winglets angled outboard and
therefore channels air more tightly between the winglet and the fuselage toward the
trailing edge of the wing, predictably increasing drag.
Significantly, 70% of the time
aOFAT recommended designs that avoided this antisynergistic configuration.
created
bv Frey and Wang (2006) predicted
that for moderate
The model
interactions
and
experimental error, the largest interaction is exploited 74% of the time. The results of
this case study are slightly less favorable, but very much in line with these predictions.
The results also indicate that aOFAT exploited the second largest interaction, that
of D1-C3.
In this configuration, wing surface area is minimized, so one would expect
that performance for this arrangement would be particularly poor.
In simulations,
aOFAT avoided this combination of factor levels 100% of the time.
D-B Interactions
180
160
r.-
150
B1
140
2
B2
--
130
B3
.? 120
U- 110
100-- ·-
100
--
D1
..........
D2
Factors
Figure 6. Antisynergistic interaction between DI and B3.
14
D3
C-D Interactions
-"-"--.
zuu
a
.
180
'
c 160
.....
+D1
----- D2
u)
bi 140
-.....
" .
:
..--.-.
D3
.__-·····-·--··~
.2 120
Inn
C1
C2
C3
Factors
Figure 7. Mild antisynergistic
interaction between factors C and D.
CONCLUSIONS
It is interesting to note the difference in factor interactions between Data Trial I
and Data Trial II. The fact that the interactions that take place in Data Trial I are neither
intuitive nor easily explained suggests that the results stem not from any physical
interactions taking place, but rather from experimental bias in the conduct of the full
factorial array experiment. The sequential testing of factor configurations likely led the
experimenter to have expectations of performance for different configurations and,
consciously or unconsciously, to induce error in noise factors (i.e. throw strength). Based
on this observation, more weight should be given to the data produced by Data Trial II.
Results of aOFAT simulations using Data Trial II results confirm Frey's theory of
the superiority of aOFAT plans over fractional factorial methods in cases with significant
control factor interactions and low experimental error. Noticeably, the weighted average
of performance indicators of factor configurations recommended by aOFAT plans
exceeded those suggested by fractional factorial methods. More importantly, however,
was the manner in which the aOFAT results exploited factor interactions. The strongest
factor interaction was exploited much to the same degree as predicted by Frey and Wang
(2006) - 70% vs 74%. The second-strongest interaction was avoided 100% of the time.
even better than anticipated.
15
It is important to remember that aOFAT plans are designed to exploit factor
effects and improve performance.
but they do not offer insight into the strength of
interactions vice main effects, nor do they attempt to explain reasons for improvement.
This study corroborated
Frey and Jugulum's
(2005) observation
that fbr ce'ertain
carrangemens
of main efecls and interactions, cLdaptlive
one-/actor-at-a-timneexperiment
exploit interactioons ith high pirohahility despite the fact that these designes lack the
resolittion
to estimate interactions.
AOFAT provides good solutions for immediate
problems. particularly when time and resources are limited. If researchers wish to
understand the system better. though, fractional factorial methods remain the most
effective option.
Further Study
In the fractional factorial analysis portion of this study, we looked only at the
standard L9 orthogonal array as prescribed by Taguchi.
It would be interesting for the
purpose of comparison to rearrange the order of the factors (i.e. B,C,D,A) or to achieve
the same effect by altering the order of the columns of the orthogonal array. There are 4!
=--24 different ways to arrange the columns and maintain orthogonality,
different arrangements suggest different answers.
and these
The original labeling of the factors (A
to weight. B to winglets, and so on) was preset, and a different original ordering could
have produced different results.
It has been posited that aOFAT methods have the further advantage of offering
improvements even if the process is interrupted.
Further studies might analyze results
suggested by aOFAT simulations that are ended prematurely.
The ability to choose starting points for aOFAT experimentation is another of its
advantages over fractional factorial methods.
eliminating to
In this study, we analyzed the effects of
factor levels with obviously detrimental performance effects from the
16
starting configurations. It would be interesting to assess the effects of further limiting the
starting points on overall performance.
ACKNOWLEDGMENTS
Thanks to Prof. Frey for his continued mentorship and guidance in a field to which I am
very much a newcomer.
Thanks to Will Reichert for helping write code to analyze all the data - and saving me
countless hours in the process.
17
APPENDIX A
AIRCRAFT
TEMPLATE
-
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r.
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i .;
r r
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J'--
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tt
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,"
~"t
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as
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.is
[4
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-XI
r
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,
tr_
f-
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r
r
s;
f_
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it
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APPENDIX B.1
EXPERIMENT I DATA
Data Trial I
Order of testing:
Sets 1-81 sequentially
Full factorial experiment using mod template pic.pdf
Run 09JAN05-13JAN
05, Compiled 16JAN05
Distance (in
Configuration
Set No.
AVG.
D
156
178
168
182
172
159
184
180
176
166
206
184
160
162
170
174
164
166
186
180
166
156
204
172
146
148
136
140
152
154
146
156
142
144
146
1
1
1
1
2
2
2
2
3
3
3
3
1
1
1
1
2
2
2
2
3
3
3
3
1
1
1
1
2
2
2
2
3
3
3
19
171
1
173.75
2
183
3
166.5
174
174.5
142.5
152
143
C
D
N
Dist.
3
3
4
1
1
1
1
1
1
1
4
140
132
144
150
136
120
122
168
162
106
140
138
128
130
142
124
156
114
142
138
138
104
96
112
100
1
1
2
3
4
1
1
2
2
2
2
2
3
4
1
3
1
1
3
3
3
2
3
4
2
2
1
1
1
2
1
2
3
4
1
1
1
1
2
1
2
2
2
2
2
2
2
2
2
2
2
2
1
1
2
3
4
3
1
3
3
3
2
3
3
1
1
110
3
1
2
120
128
118
108
128
128
132
88
108
116
136
148
170
166
170
172
174
198
178
156
166
3
1
3
3
1
4
3
2
1
3
2
2
3
2
3
3
2
4
3
3
1
3
3
2
3
3
3
3
3
4
1
1
1
1
1
1
1
2
3
4
1
1
3
1
3
1
3
1
3
1
3
3
1
2
2
2
2
3
3
1
20
1
2
3
4
1
2
AVG
Set No.
140.5
10
143
11
128
12
138
13
133
14
103
15
119
16
124
17
112
18
163.5
19
180.5
20
165.5
21
A
Dist.
1
184
156
204
190
202
192
188
190
184
208
132
134
148
154
132
160
132
124
144
136
136
130
120
142
122
130
180
212
194
186
158
168
180
176
174
164
172
166
158
180
180
154
162
160
166
158
164
158
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
21
AVG
Set No.
197
22
192.5
23
142
24
137
25
136.5
26
128.5
27
193
28
170.5
29
169
30
168
31
161.5
32
169
33
A
B
Dist.
2
1
2
1
2
1
2
1
180
174
140
144
146
142
164
172
150
144
144
156
142
138
152
162
190
140
96
120
120
84
144
60
174
84
144
150
132
124
120
118
146
118
94
100
108
114
104
118
118
120
106
100
128
102
98
108
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
AVG
Set No.
143
34
157.5
35
145
36
161
37
105
38
115.5
39
137.5
40
125.5
41
104
42
115
43
109
44
103
45
Dist.
104
102
170
172
178
166
206
194
220
218
144
188
228
158
216
216
206
194
172
182
210
194
156
148
166
130
120
160
148
122
112
126
130
122
114
112
154
120
228
236
222
212
180
174
204
172
174
152
23
AVG
Set No.
171.5
46
209.5
47
179.5
48
208
49
189.5
50
150
51
137.5
52
122.5
53
125
54
224.5
55
182.5
56
181.5
57
A
B
C
D
N
Dist.
3
1
1
3
3
3
1
1
3
4
3
1
2
1
1
3
1
2
1
2
3
1
3
1
1
4
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
2
2
2
2
2
2
2
2
2
2
1
3
3
1
3
1
1
3
1
1
2
3
1
3
3
1
3
3
1
3
1
4
3
1
3
1
3
1
3
3
1
3
3
1
3
3
1
3
3
1
3
3
1
3
2
2
2
2
3
3
3
3
1
3
3
4
3
2
1
1
1
3
2
1
1
2
3
2
1
1
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
4
1
1
1
2
2
2
2
3
1
3
2
206
194
172
176
184
190
178
150
154
164
164
136
172
158
144
164
144
156
180
162
156
150
120
130
158
142
144
168
158
138
130
176
158
120
84
92
1
3
3
3
110
4
2
2
2
1
1
1
2
1
3
2
1
4
2
2
2
2
2
2
2
2
2
2
2
4
3
1
3
2
146
124
116
118
164
106
130
112
114
106
84
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
1
1
1
1
2
2
2
2
1
2
3
4
3
1
3
2
3
3
3
4
2
3
4
1
2
3
2
3
4
1
1
3
24
AVG
Set No.
180.5
58
161.5
59
157.5
60
152
61
162
62
137.5
63
152
64
146
65
108
66
130.5
67
115.5
68
108
69
A
C
D
N
3
2
3
3
3
2
3
4
3
1
1
1
2
1
3
1
4
3
3
3
3
3
3
3
3
3
3
2
3
3
3
2
2
2
2
3
3
3
1
3
3
3
3
3
3
3
3
3
3
3
2
3
4
1
1
1
1
1
3
1
1
2
3
3
1
1
4
3
3
3
1
2
1
2
2
2
3
1
2
4
3
1
3
3
1
2
3
3
1
3
3
3
3
3
2
2
2
2
2
3
3
3
1
1
3
3
3
3
3
2
3
3
3
3
3
3
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
1
4
1
3
1
4
1
1
1
1
2
3
1
4
2
2
2
2
3
3
3
3
1
2
3
4
1
2
3
4
1
1
3
1
3
3
1
2
3
3
3
1
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
2
3
3
3
3
1
2
3
4
1
2
25
Dist.
122
120
106
110
134
128
94
114
120
124
92
106
120
100
182
172
190
186
266
128
224
196
196
206
210
162
226
190
224
208
198
194
178
184
144
122
148
140
186
186
134
148
148
134
140
124
122
160
AVG
Set No.
119.5
70
113
71
104.5
72
182.5
73
203.5
74
193.5
75
212
76
188.5
77
138.5
78
163.5
79
136.5
80
129
81
A
B
C
D
N
Dist.
3
3
3
3
3
124
3
3
3
3
4
110
A-D are as dictated on the template. Noise conditions (N) are as follows:
N1: Left hand, gloved, both feet on ground
N2: Left hand, barehand, one foot on ground
N3: Right hand, gloved, one foot on ground
N4: Right hand, barehand, both feet on ground
26
AVG
Set No.
APPENDIX B.2
EXPERIMENT II DATA
Data Trial II
Order of Testing:
Sets 4, 1, 20, 33, 64, 28, 65, 35, 5, 34, 60, 79, 7, 73, 9, 10,42, 24, 2, 16, 41, 6,
49, 48, 29, 27, 44, 43, 45, 80, 8, 70, 13, 50, 54, 19, 36, 74, 69, 72, 77, 47. 68,
39, 59, 58, 81. 51, 15, 32, 67, 38, 18, 23, 57, 25, 12, 53, 21, 14, 62, 63. 66, 75.
71, 31, 56, 40, 22, 76, 46, 37. 11, 52, 61. 78, 3, 30, 17, 26, 55
With noise factors rotated: 1, 2, 3, 4, then 2, 3, 4, 1, then 3, 4, 1. 2, then 4. 1.
2, 3. repeated.
Full factorial experiment using Template 27 APR.doc
Run 27APR06, Compiled
01MAY05
Configuration
Distance (in)
A
1
186
183
204
189
159
159
177
174
153
180
201
180
183
165
198
165
144
138
1
141
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
27
Set No.
AVG.
190.5
1
167.25
2
178.5
3
177.75
4
139.5
5
C
D
2
Dist.
135
AVG.
2
2
2
2
2
3
3
3
3
144
1 43.25
3
141
3
3
1
150
138
138
144
153
147
3
2
2
2
2
3
3
3
3
3
3
3
3
1
1
1
2
3
141
4
138
105
96
117
117
117
135
144
132
138
129
135
3
3
1
1
1
1
1
1
1
2
2
2
2
1
132
147
132
108
132
126
132
114
156
165
153
126
132
111
114
78
123
105
108
138
159
126
147
117
129
3
1
1
141
3
1
1
N
1
3
1
1
3
3
1
3
1
2
2
2
2
2
2
2
2
2
2
2
2
2
3
1
1
2
3
1
2
2
1
3
1
4
2
3
3
3
3
2
3
2
2
2
2
2
2
2
2
2
1
1
1
2
2
2
2
3
1
3
2
3
3
3
4
2
1
2
2
3
2
28
Set No.
6
145.5
7
138
8
124.5
9
147
10
120.75
11
103.5
12
142.5
13
131.25
14
108.75
15
132
16
133.5
17
Dist.
A
B
C
D
1
2
3
2
1
2
3
3
1
2
3
3
1
2
3
3
1
2
3
3
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
1
1
3
1
2
1
3
1
2
1
3
1
2
1
3
1
2
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
2
1
132
102
105
111
93
189
222
225
225
207
186
216
210
201
165
177
138
120
1
3
2
1
141
138
153
156
135
165
159
138
117
168
135
144
138
144
132
144
153
126
138
120
126
135
147
201
225
225
195
183
180
222
1
3
2
1
1
3
2
1
1
3
2
2
1
3
2
2
1
3
2
2
1
3
2
2
1
3
2
3
1
3
3
1
3
2
2
1
3
2
3
1
3
3
1
1
3
3
1
1
3
3
1
1
3
3
1
1
3
3
2
1
3
3
2
1
3
3
2
1
3
3
2
1
3
3
3
1
3
3
3
1
3
3
3
1
3
3
3
2
1
1
1
2
1
1
1
2
2
2
2
1
1
1
1
1
1
1
1
2
1
1
2
1
1
2
2
3
29
AVG.
Set No.
102.75
18
215.25
19
204.75
20
170.25
21
138
22
153.75
23
139.5
24
139.5
25
140.25
26
132
27
211.5
28
198
29
A
B
C
D
N
2
1
1
2
4
207
2
1
1
3
1
171
2
1
1
3
2
144
2
1
1
3
3
189
2
1
1
3
4
156
2
1
2
1
1
150
2
1
2
1
2
153
2
1
2
1
3
165
2
1
2
1
4
171
2
1
2
2
1
141
2
1
2
2
2
129
138
Dist.
2
1
2
2
3
2
1
2
2
4
141
2
1
2
3
1
159
2
1
2
3
2
135
2
1
2
3
3
156
2
1
2
3
4
150
2
1
3
1
1
150
2
1
3
1
2
144
2
1
3
1
3
144
2
1
3
1
4
138
2
1
3
2
1
117
2
1
3
2
2
129
2
1
3
2
3
132
2
1
3
2
4
120
2
1
3
3
1
126
2
1
3
3
2
123
2
1
3
3
3
132
2
1
3
3
4
147
2
2
1
1
1
144
2
2
1
1
2
147
2
2
1
1
3
144
2
2
1
1
4
114
2
2
1
2
1
120
2
2
1
2
2
141
2
2
1
2
3
159
2
2
1
2
4
156
2
2
1
3
1
117
2
2
1
3
2
123
2
2
1
3
3
93
2
2
1
3
4
147
2
2
2
1
1
126
2
2
2
1
2
150
2
2
2
1
3
147
2
2
2
1
4
147
2
2
2
2
1
117
2
2
2
2
2
108
2
2
2
2
3
135
30
AVG.
165
Set No.
30
159.75
31
137.25
32
150
33
144
34
124.5
35
132
36
137.25
37
144
38
120
39
142.5
40
125.25
41
A
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
B
C
D
N
2
2
2
4
141
2
2
2
3
1
2
3
2
2
2
3
3
2
2
3
4
2
3
1
1
2
3
1
2
2
3
1
3
2
3
1
4
2
3
2
1
2
3
2
2
2
3
2
3
2
3
2
4
2
3
3
1
2
3
3
2
2
3
3
3
2
3
3
4
3
1
1
1
3
1
1
2
3
1
1
3
3
1
1
4
93
111
114
126
102
183
123
129
117
114
132
126
102
120
114
120
189
174
183
183
3
1
2
1
201
3
1
2
2
3
1
2
3
3
1
2
4
3
1
3
1
3
1
3
2
3
1
3
3
3
1
3
4
3
2
1
1
3
2
1
2
3
2
1
3
3
2
1
4
3
2
2
1
3
2
2
2
3
2
2
3
3
2
2
4
3
2
3
1
3
3
2
3
3
141
3
2
2
2
162
210
153
159
183
174
168
168
174
177
183
129
144
195
150
123
132
3
4
141
3
3
1
1
141
3
3
1
2
147
3
3
1
3
141
3
3
1
4
3
3
2
1
3
3
2
2
3
3
2
3
147
123
132
159
3
31
Dist.
AVG.
Set No.
111
42
134.25
43
122.25
44
114
45
182.25
46
181.5
47
171
48
175.5
49
154.5
50
134.25
51
144
52
132.75
53
B
D
3
2
N
4
Dist.
117
3
3
1
132
3
3
2
132
3
3
3
165
3
3
4
129
1
1
1
198
1
1
2
228
1
1
3
261
1
1
4
246
1
1
201
2
174
3
186
1
2
2
2
2
3
1
1
1
1
4
159
1
150
3
2
150
3
3
3
171
1
4
129
1
1
1
168
1
1
1
1
2
3
4
177
183
150
1
144
1
1
1
1
1
2
2
2
2
4
144
1
3
1
159
1
2
138
3
129
1
3
3
3
4
144
1
1
1
150
1
1
2
171
1
1
3
147
1
1
4
144
1
2
2
2
2
3
3
3
3
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
153
3
165
1
114
2
150
3
141
4
138
1
141
2
150
3
132
4
129
1
1
135
1
168
180
141
1
2
3
4
2
2
1
135
2
105
2
3
123
1
32
AVG.
Set No.
139.5
54
233.25
55
180
56
150
57
169.5
58
151.5
59
142.5
60
153
61
135.75
62
138
63
156
64
123
65
A
B
C
D
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
N
4
Dist.
129
1
3
1
129
1
3
2
135
1
3
3
123
1
3
4
156
2
1
1
135
2
2
2
2
2
2
2
2
2
2
2
3
3
1
1
2
3
4
156
153
150
2
2
2
2
1
132
2
3
4
156
144
132
3
1
108
3
2
105
3
3
3
4
144
123
1
1
123
1
2
132
3
1
3
147
3
1
4
144
3
2
2
2
1
129
2
129
3
138
2
4
132
3
3
1
99
3
3
2
111
3
3
3
138
3
3
4
114
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
1
3
3
1
1
1
162
3
3
1
1
2
186
1
1
3
159
3
3
3
3
1
1
4
204
3
3
1
201
3
1
2
186
3
1
3
195
3
1
2
2
2
2
1
3
3
3
3
1
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
180
3
1
132
1
3
1
3
3
1
3
2
3
4
216
189
126
3
2
2
2
2
2
2
2
1
1
126
1
2
3
4
156
156
186
1
135
2
129
3
141
3
1
1
2
2
2
33
AVG.
Set No.
135.75
66
148.5
67
141
68
120
69
136.5
70
132
71
115.5
72
177.75
73
190.5
74
165.75
75
156
76
133.5
77
Dist.
C
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
A-D are as dictated on the template.
129
114
144
138
147
96
117
138
156
153
150
159
138
120
135
114
129
Noise conditions (N) are as follows:
N1: Left hand, gloved, both feet on ground
N2: Left hand, barehand, one foot on ground
N3: Right hand, gloved, one foot on ground
N4: Right hand, barehand, both feet on
ground
34
AVG.
Set No.
135.75
78
126.75
79
150
80
124.5
81
APPENDIX C
Fractional Factorial Analysis
Data Trial I
Orthogonal array
i
A
B
C
D
yl
y2
y3
1
1
1
1
1
2
1
2
2
2
3
1
3
3
4
2
1
2
3
3
5
2
2
3
1
156
114
120
164
104
168
138
122
180
118
6
2
3
1
2
206
7
3
1
3
8
3
2
1
2
3
180
84
9
3
3
2
1
226
178
142
142
158
118
194
162
92
190
220
156
110
224
I
in = 9 i=,
1
m 11 =
i3~77+z2
;· +173)
·
43.03637
44.02823
43.57004
44.42292
41.22008
44.99163
43.70663
44.45364
42.47437
44.08193
44.29851
42.25420
nl
44.61321
mAl
n2
n3
n4
n5
n6
n7
n8
n9
42.37396
42.12193
44.52436
41.16996
46.39036
44.13120
40.11632
46.46260
m
43.54488
mA2
mA3
mB1
mB2
mB3
mC1
mC2
mC3
mD1
mD2
mD3
1:3
The fractional factorial method suggests the following values:
A
B
C
D
2
3
2
2
35
y4
182
138
130
174
120
218
150
146
208
Which returns an average flight distance of 189.5 in (45.55 db).
However, one of the values tested was:
A
B
C
D
2
3
1
2
Which returns an average flight distance of 209.5 in (46.42 db).
36
Data Trial II
Orthogonal array
A
B
C
D
yl
y2
1
1
1
1
1
2
3
4
1
2
2
1
3
2
2
2
1
2
3
2
3
1
2
183
129
126
135
183
162
150
135
156
5
6
7
3
8
3
9
3
1
3
2
2
1
3
3
2
1
186
117
120
159
102
201
114
129
126
+
3 --11 + q2
113 )
2
3
3
3
1
(kYJ
I17=:-
F7,
9
:=1
nl
45.57561
mAl
n2
n3
n4
42.29234
42.33626
43.46882
42.00370
44.94002
42.51424
42.55410
mA2
mA3
mB1
mB2
mB3
mC1
mC2
mC3
mD1
mD2
mD3
n5
n6
n7
n8
n9
m
43.61321
43.25537
43.40140
43.47085
42.89385
43.85289
42.28338
43.62983
44.35658
43.12479
42.28473
43.73084
43.24887
42.78639
The fractional factorial method suggests the following values:
A
B
C
2
1
1
Which returns an average flight distance of 211.5 in (46.5 db).
37
D
y3
204
141
135
156
123
210
141
123
156
y4
189
138
147
150
129
153
138
156
186
APPENDIX
D
aOFAT Analysis (Examples)
I)ata Trial I
Order of factor variation: D. A, B, C
1
A
Set
67
Try D2
68
Try D3
69
C
B
D
Dist
N
3
2
2
1
1
124
3
2
2
1
2
3
2
2
1
3
3
2
2
1
4
116
118
164
3
2
2
3
2
2
3
2
2
3
2
2
2
2
2
2
2
3
2
2
2
3
2
2
AVG
130.5
keep 67
1
2
3
4
106
130
112
114
115.5
106
84
122
120
108
130
142
124
156
138
144
150
132
124
137.5
168
208
keep 67
3
3
Try Al
208
D,A,B,C
2
2
3
3
3
3
1
2
3
4
keep 13
13
Try A2
40
Try B1
31
Try B3
49
1
1
1
2
1
3
1
4
keep 13
2
2
2
1
2
4
2
3
keep 31
2
1
2
2
2
2
4
158
180
180
154
3
keep 49
2
3
2
1
1
216
2
3
2
1
2
2
3
2
1
3
2
3
2
1
4
216
206
194
38
Try C1
46
Try C3
52
keep 49
2
2
2
2
3
1
1
1
3
3
3
1
1
1
1
2
3
1
1
4
KEEP
49
2
2
2
2
3
3
3
3
3
3
3
3
1
1
1
1
2
3
1
4
Recommended solution:
A
B
C
D
2
3
2
1
Which returns an average flight distance of 209 in (46.34 db).
39
170
172
178
166
171.5
120
160
148
122
137.5
APPENDIX E
aOFAT Results
January aOFAT Breakdown
450
An
t4UU
.
..
..
...
....
...-..-......
.
I
..
350
300
350
..
Er250
00
. .....
.....
..
i
_
[ Seriesl
!~.
L....
~
. _......... ...
50
1 3 5 7 9 111315171921232527293133353739414345 4749515355 57596163656769717375777981
Ending
sets
Figure 8. aOFAT results for Data Trial I
A
B
C
1
1
1
1
1
17
171
3
1
1
1
3
243
183
5
1
1
2
2
32
174
6
1
1
2
3
48
174.5
20
1
3
1
2
41
180.5
22
1
3
2
1
112
197
24
1
3
2
3
24
142
28
2
1
1
1
98
193
29
33
47
49
2
2
2
2
1
1
3
3
1
2
1
2
2
3
2
1
15
6
268
6
170.5
169
209.5
208
55
3
1
1
1
391
224.5
56
58
74
76
3
3
3
3
1
1
3
3
1
2
1
2
2
1
2
1
Total:
94
5
159
385
1944
182.5
180.5
203.5
193.5
Frequency
D
Distance (in)
Weighted
100% avoid B2
100% avoid C3
83% avoid D3
93% avoid A1-D1 (second strongest interaction) (misleading?)
73% avoid B1-D1 (strongest interaction) (misleading?)
40
average:
198.9522
MayaOFATBreakdown
800
700
600
, 500
= 400
o Sedesl
ut 300
.. .
200
[
...
..
100
. vf--- A''n''
'''''''''''''
-s -
.. ..._____
ii
n
.....
_______
__ ,._ __ ...
.__.____...
__
0 :u._
---------i1 3 5 7 9 11 1315 1719 21 2325 27 29 31 33 35 3739 41 43 45 47 49 51 53 55 57 59 6163 65 67 69 71 73 75 77 7981
^
-
-
-
l
-
-
-
-
n
Finalsets
Figure 9. aOFAT results for Data Trial 11
A
B
C
D
Distance (in)
Frequency
1111
1
4 1 1 2
1
190.5
177.5
215.25
204.75
120
12
393
132
24
159
19
1
3
1
1
20
26
1
1
3
3
1
3
2
2
28
2
1
1
1
29
2
1
1
2
31
2
1
2
1
46
47
2
2
3
3
1
1
1
2
171
9
181.5
49
2
3
2
1
55
3
1
1
1
15
714
233.25
58
74
80
3
3
3
1
3
3
2
1
3
1
2
2
Total:
140.25
211.5
45
3
198
159.75
182.25
175.5
42
169.5
190.5
150
93
12
1944
100% avoid B2
98% avoid C3
100% avoid D3
70% avoid B3-D1 (strongest interaction)
41
Weighted
average:
211.729
Breakdown January, no B2 in starting set
400
300
, 200
O Seriesl
n
100
..........'.
_'.
..'
nl
n
U
fl
1 4
7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79
Final set
Figure 10. aOFAT results for Data Trial 1, excluding starting sets with B2
A
B
C
D
1
1
1
1
1
2
3
5
6
1
1
1
1
1
1
1
2
2
3
2
3
182
11
39
183
174
174.5
22
1
3
2
1
64
197
23
1
3
2
2
17
192.5
28
2
1
1
1
12
193
29
33
2
2
1
1
1
2
2
3
12
6
170.5
169
47
2
3
1
2
49
2
3
2
1
55
3
1
1
1
252
224.5
56
3
1
1
2
73
182.5
74
76
3
3
3
3
1
2
2
1
112
300
203.5
212
Total:
1296
Frequency
Distance (in)
Weighted
208
171
209.5
6
208
42
average:
204.272
I
Breakdown May, noB2 in starting set
i
I
450
400 .
350 300 .
' 250
I
I
I
I
11Sedesl:
r 200.
,
150
--- -
100
50:
-------
1.:.._. _.. ....
1
4
....
6 ..
._.
7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79
Finalsets
Figure I 1. aOFAT results for Data Trial 11, excluding starting points with B2
ABCD
Frequency
Distance (in)
Weighted
1
1
1
1
19
1
3
1
20
1
3
1
28
2
1
1
1
1
2
1
29
2
1
1
2
3
46
2
3
1
1
140
47
2
3
1
2
55
3
1
1
74
3
3
1
1
2
T¢otal:
97
328
106
134
190.5
215.25
204.75
211.5
189
182.25
8
181.5
420
60
1296
233.25
190.5
43
average:
21 3.0046
i
BreakdownJan, no B2or C3 in startingset
250
200 --. 150 :
O Sedesl
2 100
50-
0
-__
B-- -
--
n
_
Y
_
~
E
___~
...n................
~~~~
---
S
. ...
·L
.
II
-----
--------1--~~~~I
II
-
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81
Finalset
Figure 12. aOFAT results for Data Trial , excluding starting points with B2 or C3
AB CD
3 1 1 1
6 112
3
3
Frequency
Distance (in)
157
30
53
183
174.5
197
192.5
169
22
23
33
1
1
2
3
3
1
2
2
2
1
2
3
47
2
3
1
2
49
2
3
2
1
6
170
6
55
3
1
1
1
171
74
76
3
3
3
3
1
2
2
1
Total:
192
864
11
209.5
208
224.5
203.5
68
212
44
Weighted
average:
205.2465
BreakdownMay,no B2or C3in startingset
300
250
>, 200
, 150
o Series1
LL 100
50
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81
Finalset
Figure 13. aOFAT results for Data Trial 1I,excluding starting points with B2 or C3
AB C
111 1
Distance
D
1
Frequency
87
(in)
190.5
247
67
105
215.25
204.75
211.5
2
3
87
8
198
182.25
181.5
1
257
233.25
2
3
864
190.5
19
1
3
1
1
20
1
3
1
2
28
2
1
1
1
29
2
1
1
2
46
2
3
1
1
47
2
3
1
55
3
1
1
74
3
3
1
Total:
45
Weighted
average:
213.0608
REFERENCES
Eppinger, S. D., 1995, "Taguchi Airplanes." in Games and Exercises for Operations
Management J. N. Heineke and L. C. Meile, (eds.), Prentice Hall, Englewood
Cliffs, NJ, pp. 213-224. 1995.
Frey, D. D., and Jugulum, R. (2005), "The Mechanisms by which Adaptive One-Factor-
at-a-Time Experimentation Leads to Improvement," accepted for ASME Journal of
Mechanical Design.
Frey. D. D., and Li, X., 2004, "Validating Robust Parameter Design Methods,"
DETC2004-57518, ASME Design Engineering Technical Conferences, September
28 to October 2, Salt Lake City, Utah.
Frey, D.D., N. Sudarsanam, and J.B. Persons, 2006, "An Adaptive One-Factor-at-a-time
Method for Robust Parameter Design: Comparison with Crossed Arrays via Case
Studies." DETC2006-99593, ASME Design and Engineering Technical
Conferences, September 10-13, Philadelphia, PA.
Frey, D. D., and Wang, H., 2006, "Adaptive One-factor-at-a-time Experimentation and
Expected Value of Improvement", accepted to Technometrics.
Phadke, Madhav S., 1989, Quality Engineering Using Robust Design. Prentice Hall,
Englewood Cliffs, NJ.
46