C A R L O S G O N Z Á L E Z G O N Z Á L E Z

A S Q FELLOW – MASTER BLACK BELT

Consultores CUAUTITLÁN IZCALLI EDO. DE MEX.

e-mail: cgg1@hotmail.com

e-mail: cgg111@gmail.com

Cuautitlán Izcalli Estado de México October 14 2008

Author: Carlos González 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 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. Taguchi’s 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 1960’s. In this period and throughout 1970s most applications

were on production processes.

C.G.G

.

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 industries

and 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 his

contributions 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,

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 a

response 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

algebraically signs 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).

FACTOR A: LW=Length of Wing mm. (−) level = 70 and (+) level = 80

FACTOR B: AW=Angle between Wings (−) level = 15º and (+) level = 30º

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.

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 three

too, 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.

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

Photo No. 2. Look at the helicopters, #1 and #4

Helicopter #1 has 70 mm of length of wings and 15º as an angle between

wings, #4 has 80 mm of length of wings and 30º 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.

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).

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

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 .101

M + 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

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

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 in

columns 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 = 0.62182

P%CD = CD/TABCD

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.

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 into

Products and Processes. Tokyo Japan, Asian Productivity Organization, 1986.

CGG-SOFT: DOETAG_EN-CGG-3.1, Carlos González González, México City

México.

Author: Carlos González González

ASQ Fellow

Master Black Belt

ASQ Press Reviewer

MBA National University, San Diego Ca. U.S.A.

E-mail: cgg1@hotmail.com E-mail: cgg111@gmail.com