Doe Tag Article Interactions Cgg-2

C.G.G

Consultores

CARLOS GONZÁLEZ GONZÁLEZ A S Q FELLOW – MASTER BLACK BELT CUAUTITLÁN IZCALLI EDO. DE MEX.

e-mail: [email protected] e-mail: [email protected]

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 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 prizewinning 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.

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. FACTOR B: AW=Angle between Wings FACTOR C: LF=Length of Fuselage mm. FACTOR D: WW=Width of Wing mm.

(−) level = 70 and (+) level = 80 (−) level = 15º and (+) level = 30º (−) level = 30 and (+) = 40 (−) 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

1 2 3 4 5 6 7 8 M+ M− UM S+ S− US

70 80 70 80 70 80 70 80 2.73 2.48 0.25 .097 .091 .006

15 15 30 30 15 15 30 30 2.70 2.50 0.20 .088 .099 .011

AxB CxD + − − + + − − + 2.66 2.55 0.11 .121 .067 .054

LF 30 30 30 30 40 40 40 40 2.59 2.62 0.03 .079 .108 .029

AxC BxD + − + − − + − + 2.60 2.60 0.00 .088 .099 .011

BxC AxD + + − − − − + + 2.64 2.56 0.08 .111 .076 .035

WW

R1

R2

R3

R4

R5

Xbar

Sig.

15 20 20 15 20 15 15 20 2.70 2.50 0.20 .079 .108 .029

2.36 2.63 2.60 2.99 2.38 2.39 2.40 2.85

2.17 2.64 2.57 2.63 2.45 2.51 2.35 3.08

2.53 2.83 2.60 2.76 2.44 2.41 2.53 2.99

2.57 2.82 2.66 2.76 2.60 2.42 2.53 3.11

2.30 2.67 2.58 2.70 2.50 2.38 2.41 3.03

2.39 2.72 2.60 2.77 2.47 2.42 2.44 3.01

.165 .099 .035 .135 .082 .052 .082 .101

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: [email protected] E-mail: [email protected]