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8/14/2019 DOE TAG Article Interactions CGG-2
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C A R L O S G O N Z L E Z G O N Z L E ZA S Q FELLOW MASTER BLACK BELT
Consultores CUAUTITLN IZCALLI EDO. DE MEX.C.G.G
e-mail: [email protected]
e-mail:[email protected]
Cuautitln Izcalli Estado de Mxico October 14 2008
Author: Carlos Gonzlez G.
Subject: DOE Theory and Practice (Confounded Interactions How to Separate
them)
PhD. Genichi Taguchi created the named Taguchi Method for designing
experiments. He was born on January the 1st. of 1924 in Japan. He graduated
from Kiryu Technical College. After serving in the Astronomical Department of
the Navigation Institute of the Imperial Japanese Navy in from 1942 to 1945, he
was working in the Ministry of Public Health and Welfare and at the Institute of
Statistical Mathematics, Ministry of Education.
1946: R. L. Plackett and J. P. Burman presented a methodology of
creation of Orthogonal Arrays to be applied to Design of Experiments writing
the article The Design of Optimal Multifactorial Experiments in the Journal
Biometrika (vol. 35), these methods were studied by G. Taguchi and the prize-
winning Japanese Statistician Matosaburo Masuyama, whom he met while he
was working at the Ministry of Public Health and Morinaga Pharmaceuticals.
1949: G. Taguchi joined the Electrical Communications Laboratory of
NTT Co. until 1961 to increase the productivity of its R&D actions, at that time
he began to develop his methodology now named Taguchi Method or Robust
Engineering. G. Taguchis first book which introduced the orthogonal arrays,
was published in 1951.
1951 and 1953: he won Deming Prize award for literature.
During 1954 and 1955 G. Taguchi met in India to Ronald A. Fisher and
Walter Andrew Shewhart. In 1957 and 1958 he published his two volume book
Design of Experiments.
1960: G. Taguchi won the Deming Application Prize.
1962: He visited USA and Princeton University visiting too AT&T Bell
Laboratories, there he met statistician John Tukey. This year too, received his
PhD in Science from Kyushu University.
1964: G. Taguchi and several coauthors wrote Management by Total
Results.
First applications outside Japan of Taguchi Methods were in Taiwan and
India during 1960s. In this period and throughout 1970s most applications were
on production processes.
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Taguchi Methods applied in product design began later in the early 1970s
G. Taguchi develop the concept of Quality Loss Function, publishing other two
books and the 3rd. edition of Design of Experiments.
In the Late 1970 he earned recognition in Japan and abroad.
1980: G. Taguchi visited again AT&T Bell Laboratories running
experiments within Bell Laboratories, after this visit more and more industriesand Universities in U.S.A. implemented the Taguchi Methodology.
1982: G. Taguchi became an advisor at the Japanese Standards
Association and Chairman of the Quality Control Research Group.
1984: G. Taguchi again won the Deming Prize for literature.
G. Taguchi received recognitions for his contributions to industries
worldwide:
The Willard F. Rockwell Jr. Medal.
The Shewhart Medal from ASQC.
The Blue Ribbon Award from the Emperor of Japan in 1990 for hiscontributions to industry.
Honorary Member in the ASQ (1997).
Induction into the Automotive Hall of Fame and the World Level of the
Hall of Fame for Engineering, Science, and Technology.
G. Taguchi is Executive Director of the American Supplier Institute Inc.
in Dearborn Michigan.
Honorary Professor at Nanjing Institute of Technology in China.
Classical experimentation is based on Analysis of Variance (ANOVA)and the Taguchi Method includes Analysis of Variance too.
Experiments:
Ch. Hicks & K. Turner define experiment as:
The experiment includes a statement of the problem to be solved. This
sounds rather obvious, but in practice it often takes quite a while to get general
agreement as to the statement of a problem. It is important to bring out all the
points of view to establish just what the experiment is intended to do. A careful
statement of the problem goes a long way toward its solution.
Response Variables:
The statements of the problem must include reference to at least one
characteristic of an experimental unit on which information is to be obtained.
Such characteristics are called response.
Independent Variables:
Many controllable experimental variables, called independent variables or
factors may contribute to the value of the response variable. Factor variables
could have two levels, these levels can be qualitative (different suppliers,
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different methods, different shifts, etc.) or quantitative (different temperatures in
degrees, speeds, weight, etc.)
The Design:
The investigator needs an experimental design for obtaining data that
provide objective results with a minimum expenditure of time and resources.
How many observations are needed?
One of the first questions we face when designing an experiment is: How
many observations are to be taken? Considerations of how large a difference is
to be detected, how much variation is present, and what size risks can be
tolerated, what kind of measurement internal or external, precision and accuracy
of readings, destructive or not destructive, are all important in answering this
question.
Sometimes there is no other option and you only have one reading as aresponse by experiment, but, it is recommended that if possible, obtain as many
replicates as can be economical or practical. You can obtain very valuable
information when you analyze more than one replication of your experiments,
especially if your software is capable of handling replicates.
Order of experimentation:
It is recommended that you randomize the sequence of the experiment
order, although it depends sometimes of the experiment logistic.
Model Description:
There are several models of experimentation where the ANOVA Method
and Yates Algorithm is applied, but G. Taguchi uses the Orthogonal Arrays L4,
L8, L12, L16, L32 for two level factors and L9, L18 and L27 for Three level
factors, to accommodate the experiments on rows and factors and levels on
columns.
I prefer to use symbols () and (+) to indicate different category of level,
low or high within the Orthogonal array, because you are going to find the
interactions between factors or columns when you simply multiply algebraicallysigns of each column, then in other column will be the resulting sign of the
interaction.
Theory and Practice Interactions 4 Factors:
We are going to run an experiment in parallel, theory and practice.
Note: This helicopter design is property of the author C.G.G. (You can
use only giving credit of it)
In this experiment we have four factors that we are going to study for
flying time of the helicopter (higher is better).
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FACTOR A: LW=Length of Wing mm. () level = 70 and (+) level = 80
FACTOR B: AW=Angle between Wings () level = 15 and (+) level = 30
FACTOR C: LF=Length of Fuselage mm. () level = 30 and (+) = 40
FACTOR D: WW=Width of Wing mm. () level = 15 and (+) level = 20
When you use an L8 Orthogonal Array to accommodate the three factors
on columns tagged as A, B, and C as it is shown in matrix fig. 1. You will find
the interactions between factors when you multiply the sign of the factor of each
Column by the sign of the factor of another Column this multiplication is named
(As an example sign of A Multiplied by sign of B = AxB) or simply AB. The
resulting sign it is located for this example in the third column row by row.
Figure 1.- L8 Orthogonal Array
Now we are going to use an Orthogonal Array L8 to accommodate Four
factors. We will have six double interactions, two in each of columns 3, 5, and 6
as it is shown in figure No. 2.We will find double interactions confounded in columns 3, 5, and 6.
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Figure No. 2 L8 Orthogonal Array, Four Factors
If you multiply the signs of columns A (1) and B (2) you will get as a
result the sign located in column 3 which it is an indication that the interaction
AxB it is located in such column.
At the same time if you multiply the signs of columns 4 and 7 which
correspond to factors C and D you will get the resulting sing in column threetoo, giving evidence that CxD double interaction it is confounded with other
double interaction AxB in the same column 3.
A similar situation it is reproduced in column 5 with two double
interactions confounded AC and BD and column 6 another two double
interactions confounded BC and AD interactions.
The L8 Orthogonal Array will allocate Four Factors A, B, C, D and three
pairs of confounded double interactions.
Construction of Helicopters:
The next model shows how the helicopters are going to be constructed.
List of Material:
1.- One Sheet of little square paper (square = 5 mm.)
2.- Two plastic straw to cut sections of fuselage.
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3.- One bar of plastiline (clay) to be used as ballast or dead weight inside
fuselage.
4.- One stick of glue (pritt) to fix wings to fuselage.
5.- One Chronometer capable to read seconds and centesimal of seconds.
6.- Scissors to cut paper and straws.
7.- One plastic rule of 20 or 30 centimeters.
Photo No. 1 Set of materials and 8 already constructed helicopters
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Photo No. 2. Look at the helicopters, #1 and #4
Helicopter #1 has 70 mm of length of wings and 15 as an angle between wings,
#4 has 80 mm of length of wings and 30 as an angle between wings also you
can see the ballast (plastiline or clay), dead weight inside the fuselage that you
can not see clearly in helicopter #1, then can be good to find and use transparent
straws as fuselage, but it is not indispensable. In both you can see how the
folded paper of the wings pass through the middle of the fuselage vertically for
about of 8 mm., in that section you need to use glue to fix the wings to the
fuselage, taking care to maintain the straightness of the vertical axis
symmetrically with wings and collinear with the axis of the fuselage.
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Photo No. 3 Note: Inside the lower side of fuselage (straw) it is the ballast (5 or
6 mm of plastiline) also you see the folded paper passing (about 8 mm.)
vertically through the upper end of the fuselage (straw).
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Figure No. 1 Sheet of paper to build the wings and two straws made of plastic to
build the fuselage.
Please see photos before you build the helicopters
How are you going to fly the helicopters?
Once you have the 8 helicopters already built you should fly the
helicopters taken the time of flying them when you let it down from an altitude
of 2.5 meters (8 feet 4 inches), 5 times each one to get Mean and Std. Dev..
Note: As you can see on photos 1 through 4, you should perform a very tiny
loop on each wing of the helicopter
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Photo No. 4 Note: You can see the gently form of the wings (half loop) given to
the paper sliding each wing (paper) pressed gently by your fingers thumb and
index
Table 1 shows results obtained by the author reproducing the procedure
indicated previously.Table 1.-Run LW AW AxB
CxD
LF AxC
BxD
BxC
AxD
WW R1 R2 R3 R4 R5 Xbar Sig.
1 70 15 + 30 + + 15 2.36 2.17 2.53 2.57 2.30 2.39 .165
2 80 15 30 + 20 2.63 2.64 2.83 2.82 2.67 2.72 .099
3 70 30 30 + 20 2.60 2.57 2.60 2.66 2.58 2.60 .035
4 80 30 + 30 15 2.99 2.63 2.76 2.76 2.70 2.77 .135
5 70 15 + 40 20 2.38 2.45 2.44 2.60 2.50 2.47 .082
6 80 15 40 + 15 2.39 2.51 2.41 2.42 2.38 2.42 .052
7 70 30 40 + 15 2.40 2.35 2.53 2.53 2.41 2.44 .082
8 80 30 + 40 + + 20 2.85 3.08 2.99 3.11 3.03 3.01 .101M + 2.73 2.70 2.66 2.59 2.60 2.64 2.70
M 2.48 2.50 2.55 2.62 2.60 2.56 2.50
M 0.25 0.20 0.11 0.03 0.00 0.08 0.20
S + .097 .088 .121 .079 .088 .111 .079
S .091 .099 .067 .108 .099 .076 .108
S .006 .011 .054 .029 .011 .035 .029
You can use the software DOETAG_EN.exe that you can download from site:
www.spc-inspector.com/cgg
To be used for analysis of data for each column, response lines and ANOVA
which includes the percentage of contributions as are shown now (I can tell you
http://www.spc-inspector.com/cgghttp://www.spc-inspector.com/cgghttp://www.spc-inspector.com/cgg8/14/2019 DOE TAG Article Interactions CGG-2
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that this software separates the confounded double interactions) later I will
explain how they are separated.
Figure No. 2, Lines of Response for the seven columns including calculations
for Means and Standard Deviations of factors and double interactions separated.
(Screen of the software)
Figure No. 3, ANOVA for Medias which shows the percentage of contribution
by column. For example, Factor A (Length of Wing), contributes with 31.94%;
Factor B (Angle between Wings), contributes with 21.02% and AxB combined
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contributes with 5.98%, Factor D (Width of Wing) contributes with 18.98%
and, the error contributes with 20.45%.
Figure No. 4, ANOVA for Standard Deviations showing too the percentage of
contribution by column. Where it is shown that Column 3 where two Double
Interactions combined contributes with 47.46% and AxB% is 11.10% and
CxD% is 36.36%, Column 6 where two Double Interactions combined
contributes with 20.67% and BxC% is 10.62% and AxD% is 10.05%.
How I and the software can separate the confounded double interactions incolumns 3, 5 and 6.
Here I am going to explain how I separate the confounded interactions:
First.- I am going to consider that Main Effects are alone each in one column
by itself and it is not confounded with any other double interaction.
Double interactions are confounded in only one column of the several we have,
and are not confounded with Main Effects.
Second.- Triple or major interactions are not considered, following the same
opinion of PhD. G. Taguchi that says: Main Effects are bigger than double
interactions than triple or than quadruple interactions.
Example:
I will consider columns: (CALCULATIONS MADE WITH EFFECTS)
Column
1; Effect A, = 0.2535
2; Effect B, = 0.2065
3; Total Effect = 0.1135 (AxB Effect and CxD Effect confounded)AB = (0.2535^2 + 0.2065^2)^0.5 = (0.06426 + 0.04264 )^0.5
AB = (0.1069)^0.5 = 0.326955
4; Effect C, = -0.0305
7; Effect D, = 0.1965
CD = (0.0305^2 + 0.1965^2)^0.5 = (0.00093025 + 0.03861)^0.5
CD = (0.03954025)^0.5 = 0.198847
TABCD = AB + CD = 0.326955 + 0.198847 = 0.525802
P%AB = AB/TABCD
P%AB = 0.326955/0.525802 = 0.62182P%CD = CD/TABCD
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P%CD = 0.198847/0.525802 = 0.378178
Effect AxB = (Effect Col. 3)*(P%AB)
Effect AxB = 0.1135 * 0.62182 = 0.07057
Effect CxD = (Effect Col. 3)*(P%CD)
Effect CxD = 0.1135*0.378178 = 0.04360
Same procedure it is applied to calculate % of contribution when the
effects are calculated inside the ANOVA Tables for Means and Standard
Deviations to separate the confounded interactions.
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Figure No. 5 Effects vs MR of means and standards deviations by columns.
Note: You can download this software in six different languages Spanish
(DOETAG_ES), English (DOETAG_EN), Deutsch (DOETAG_GR), French
DOETAG_FR), Italian (DOETAG_IT), Portuguese (DOETAG_PT). from site:
www.spc-inspector.com/cgg
Bibliography:
ASI, Special Information Package American Supplier Institute, 1987, 1988.
Dearborne Michigan 48126, U.S.A.
Hicks R. Charles, Turner V. Kenneth. Fundamental Concepts in the Design of
Experiments Fifth Edition, New York NY Oxford University Press Inc., 1999.
Ross J. Phillip. Taguchi Techniques for Quality Engineering Loss Function,
Orthogonal Experiments, Parameter and Tolerance Design. New York NY,
McGraw-Hill, Inc. 1996.
Taguchi Genichi. Introduction to Quality Engineering Designing Quality intoProducts and Processes. Tokyo Japan, Asian Productivity Organization, 1986.
CGG-SOFT: DOETAG_EN-CGG-3.1, Carlos Gonzlez Gonzlez, Mxico City
Mxico.
Author: Carlos Gonzlez Gonzlez
ASQ Fellow
Master Black Belt
ASQ Press Reviewer
MBA National University, San Diego Ca. U.S.A.
E-mail: [email protected] E-mail: [email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]