INTRODUCTION TO TAGUCHI TECHNIQUES

By

Dr. K. KRISHNAIAH

FORMER PROFESSOR

INDUSTRIAL ENGINEERING DEPARTMENT

ANNA UNIVERSITY : CHENNAI 600 025

1. Introduction to Analysis of Variance

1.1 The purpose of product or process development is to improve the performance characteristics of the

product or process relative to customer needs and expectations.

The purpose of experimentation should be to reduce and control variation of a product or process and

decide which parameters affect the performance of a product or process.

Analysis of variance (ANOVA) is a statistical method used to interpret experimented data and make

decisions about the parameters.

ANOVA is technique, which breaks total variation down into accountable sources; total variation is

decomposed into its appropriate components. The basic equation of analysis of variance is given by:

Total sum of Sum of squares Sum of squares

Squares (SST) = due to factors + due to error

1.2 Degrees of Freedom :

A degree of freedom in a statistical sense is associated with each piece of information that is

estimated from the data. For instance, the mean is estimated from all the data and requires one degree of

freedom (d.f) for that purpose. Another way to think of the concept of degree of freedom is to allow 1 d.f.

for each fair(independent) comparison that can be made in the data. Similar to the sum of squares, or

summation can be made for degrees of freedom.

Total d.f. = d.f. associated with factors + error d.f.

1.3 Variance: The variance (V) of any component is given by its sum of squares divided by its degree of

freedom.

1.4 F - test for variance comparison: F - test is used to test whether the effects due the factors are significantly

different or not. F - is computed by dividing the factor variance with the error variance.

1.5 ANOVA Table : A summary of all analysis of variance computations are shown in the ANOVA Table

below :

Source of

variation

Sum of squares Degrees of

freedom

Variance F

Factor SSF K-1 VF=SSF/K-1 VF / Ve

Error SSe N-K Ve=SSe/N-K

Total SST N-1

Where N = Total number of observations

K = number of levels of the factor

2. INTRODUCTION TO ORTHOGONAL ARRAYS

2.1 Before the discussion of orthogonal arrays (OAs), it would be desirable to review some of the test

strategies.

The most common test plan is to evaluate the effect of one parameter on product performance. In this

approach, a test is run at two different conditions of that parameter (Table 2.1)

Table 2.1 One - factor Experiment

------------------------------------------------------------------------------------

Trial No. Factor level Test results

-------------------------------------------------------------------------------------

1. 1 * *

2. 2 * *

------------------------------------------------------------------------------------

If the first factor chosen fails to produce the expected result, some other factor would be tested,

resulting the following test plan. Let the experimenter looked at four different factors labeled A, B, C and

D, each evaluated one at a time (Table 2.2).

Table 2.2 Several factors at a time

-------------------------------------------------------------------------------

Trial No. Factor and factor level Test results

A B C D

-------------------------------------------------------------------------------

1 1 1 1 1 * *

2 2 1 1 1 * *

3 1 2 1 1 * *

4 1 1 2 1 * *

5 1 1 1 2 * *

-------------------------------------------------------------------------------

One can see in Table 2.2 that the first trial is a base line condition. The results of trial 2 can be

compared to trial 1 to estimate the effect of factor A on product performance. Thus, each factor level is

changed one at a time, holding all others constant. This is the traditional scientific approach to

experimentation.

The third and most urgent situation finds the experimenter changing several things all at the same

time with a hope that at least one of the changes will improve the situation sufficiently (Table 2.3)

Table 2.3 Several factors all at the same time

------------------------------------------------------------------------------------------

Traial No. Factor and factor levels Test results

A B C D

------------------------------------------------------------------------------------------

1 1 1 1 1 * *

2 2 2 2 2 * *

------------------------------------------------------------------------------------------

Again one can see that the first trial represents the base line conditions. The average of data of

trial 1 is compared to the average of data of trial 2 to estimate the combined effect of all factors.

All these methods have some type of limitations. The major limitation is that the interaction

between factors can not be studied.

2.2 Better test strategies

A full factorial design with two factors A and B each with two levels will appears as shown in Table 2.4.

Table 2.4 Full factorial experiment

-------------------------------------------------------------------------------------

Trial No. Factor and factor level Response

A B

------------------------------------------------------------------------------------

1 1 1 * *

2 1 2 * *

3 2 1 * *

4 2 2 * *

------------------------------------------------------------------------------------

Here, one can see that the full factorial experiment is orthogonal. There is an equal number of test

data points under each level of each factor. Because of this balanced arrangement, factor A does not

influence the estimate of the effect of factor B and vice versa. Using these information, both factor and

interaction effects can be estimated.

A full factorial experiment is acceptable when only a few factors are to be investigated. When

several factors are to be investigated, the number of experiments to be run under full factorial design is very

large.

2.3 Efficient test strategies:

Statisticians have developed more efficient test plans, which are referred to as fractional factorial

experiments. These designs use only a portion of the total possible combinations to estimate the main

factor effects and some, not all, of the interactions. The treatment conditions are chosen to maintain the

orthogonality among the various factors and interactions.

Taguchi has developed a family of fractional factorial experimental matrices, called Orthogonal

Arrays (OAs) which can be utilized under various situations. One such design is an L8 OA, with only 8 of

the possible 128 treatment combinations. This is actually a one-sixteenth fractional factorial design.

2.4 Steps in designing, conducting and analyzing an experiment.

The major steps are:

1. Selection of factors and / or interactions to be evaluated.

2. Selection of number of levels for the factors.

3. Selection of the appropriate OA.

4. Assignment of factors and / or interactions to columns.

5. Conduct Tests.

6. Analyze results

7. Confirmation experiment.

Selection of factors and interactions:

This involves the determination of the factors which influence the product or process performance.

Here the customer expectations should be kept in mind. The following methods may be used:

a) Brainstorming

b) Flowcharting (especially for processes)

c) Cause-effect diagrams

Often, the experience and the specialized knowledge of engineers and scientists dominate the selection of

factors.

Selection of Number of levels:

Initial rounds of experimentation should involve many factors at few levels; two are recommended

to minimize the size of the beginning experiment. Initial experiments will eliminate many factors from

contention and the few remaining can then be investigated with multiple levels.

Two kinds of parameters exist which may influence a product response, continuous and discrete

parameters.

If continuous parameters (Temp, pressure, speed, etc.) are being used, then the initial experiment should

be at two levels only; interpolation or extrapolation may be used to predict other levels.

If discrete factors (types of materials, types of tools, etc.) are studied, then more than two levels may be

required in initial experiments.

Conducting the experiment

It is to the noted that the test sheets should show only the main factor levels required for each trial.

Only the analysis is concerned with the interaction columns.

The order of performing the tests of the various trials should be randomized. This will even out the

effect of unknown and uncontrolled factors that may vary during the experimentation period.

Randomization may be complete, simple repetition or complete within blocks.

Complete randomization means any trial has an equal chance of being selected for the first test. Using

random numbers, a trial can be selected. This method is used when test set up change is easy and

inexpensive.

Simple repetition means that any trial has an equal chance of being selected for the first test, but once that

trial is selected all the repetitions are tested for that trial. This method is used if test set ups are very

difficult or expensive to change.

Complete randomization within blocks is used where one factor may be very difficult or expensive to

change the test set up for, but others are very easy. If factor A were difficult to change, then the

experiment could be completed in two blocks. All A1 trials could be selected randomly and then all A2

trials could be randomly selected.

The different methods of randomization affect error variance in different ways. In simple repetition, trial

to trial variation is large and repetition variation is less. This may cause certain factors in ANOVA to be

significant when in fact they are not. Hence, complete randomization is recommended.

The selection of sample size (number of repetitions) is also important. A minimum of one test result for

each trial is required. More than one test per trial increases the sensitivity of experiment to detect small

changes in averages of populations. An economic consideration can also be used to determine the

number of repetitions.

NOMENCLATURE OF ARRAYS

La (bc) L = Latin square

a = no. of rows

b = no. of levels

c = no. of columns (Factors)

Degrees of freedom = a-1

2 - level 3 - level 4 - level Mixed - level

series series series series

-------------------------------------------------------------------------------------------------

L4(23) L9(3

4) L15(4

5)

+L18(2

1,3

7)

L8(27) L27(3

13) L64(421) L36(211,312)

*L12(211) L81(3

40) (or) (2

3,3

13)

L16(215)

L32(231)

-------------------------------------------------------------------------------------------------

* Interactions cannot be studied

+ Can study 1 interaction between the 2-level factor and one 3-level factor.

L8 (27) ORTHOGONAL ARRAY

--------------------------------------------------------------------------------------------------------------

COLUMNS

EXPERIMENT

1 2 3 4 5 6 7

---------------------------------------------------------------------------------------------------------------

1 1 1 1 1 1 1 1

2 1 1 1 2 2 2 2

3 1 2 2 1 1 2 2

4 1 2 2 2 2 1 1

5. 2 1 2 1 2 1 2

6 2 1 2 2 1 2 1

7 2 2 1 1 2 2 1

8 2 2 1 2 1 1 2

---------------------------------------------------------------------------------------------------------------

NOTE:

1. Eight experimental runs.

2. Balanced numbers of 1s and 2s.

3. Any pair of columns have only Four combinations (1,1); (1,2); (2,1); (2,2)

4. If the same number of combinations occur, them the columns are orthogonal.

5. In the L8, any pair of columns is orthogonal.

6. L8 can be applied to 7 or less factors.

LOCATION OF INTERACTIONS

1. LINEAR GRAPHS

2. TRIANGULAR TABLES

- Each OA has a set of linear graphs and a triangular Table associated with it.

LINEAR GRAPHS

- Taguchi devised this technique.

- Graphic representation of Interaction information in a matrix experiment.

- Helps to assign main factors and interactions to the different columns of an OA.

Example of standard linear graphs: Figure 1 (a) and (b)

Linear graph of the L8 OA

•

FIG 1 (a) FIG 1 (b)

Main Factors A, B, C & D are assigned to columns 1, 2, 4 & 7. Interactions AB, AC & BC should be assigned

to columns 3, 5 & 6.

TRIANGULAR TABLE

- These tables give all of the possible interacting column relationships that exist for a given OA.

Column No.

Column No.

2 B 3 4 5 6 7

1 A 3 AB 2 5 4 7 6

2 1 6 7 4 5

3 7 6 5 4

4 1 2 3

5 3 2

6 1

L8 TRIANGULAR TABLE

1

2 4

53

6

7

2

4

7

3

6

5

SELECTION OF OA

1. Determine the d.f. required.

2. Note the levels of reach factor and decide the type of OA. (2-level or 3-level)

3. Select the particular OA which satisfies the following conditions.

a) d.f.(OA) > d.f. required for the experiment

b) Interactions possible (OA) > the interactions required

4. Draw the required linear graph.

5. Compare with the standard linear graph of the chosen OA.

6. Superimpose the required LG on the standard LG to find the location of factor columns and interaction

columns.

The remaining columns (if any) are left out.

7. Draw the layout indicating the assignment of factors and interactions.

The rows will indicate the number of experiments (trials) to be conducted.

COMPUTATION OF DEGREE OF FREEDOM

Degrees of Freedom (d.f)

- Maximum number of independent pair wise comparisons

- D.f. For each factor with 'a' levels = a-1

- D.f of an interaction = Product of d.f of Interacting factors

- For Factor 'A' with 'a' levels and Factor 'B' with 'b' levels

d.f of 'AB' interaction = (a-1) (b-1)

- D.f for an experimental design = sum of d.f's of Factors and interactions

- D.f available in an OA = no. of trials - 1 For a L8 OA , d.f = 8-1 = 7

SELECTION OF OA - EXAMPLE

An experiment has to be conducted with 4 factors (A,B,C & D) each at two levels. Also, the interactions AB,

AC and ADare to be studied.

1. Degrees of Freedom

Factor Levels D.F

------------------------------------------------------------------------------

A 2 2 – 1 = 1

B 2 2 – 1 = 1

C 2 2 – 1 = 1

D 2 2 – 1 = 1

AB (2-1) (2-1) = 1

AC (2-1) (2-1) = 1

AD (2-1) (2-1) = 1

--------------------------------------------------------------------------------

Total d.f = 7

--------------------------------------------------------------------------------

2. Levels of Factors

- All at 2-levels

Therefore, choose a 2-level OA

3. Selection of required OA

(a) The OA which satisfies the required d.f is OA, L8.

(b) Interactions required = 3

Interactions possible in L8 =3

Therefore, the best OA would be L8.

4. Required Linear Graph

5. The required LG is similar to the standard LG (b) of L8 OA shown in FIGURE 1

6. Superimpose the required LG with the standard LG and match the factors and columns.

The assignment of factors and interactions are shone in the design layout below.

7. DESIGN LAYOUT

FACTORS

A B AB C AC AD D

Trial No.

1 2 3 4 5 6 7

Resp.

Y

1 1 1 1 1 1 1 1 * *

2 1 1 1 2 2 2 2 * *

3 1 2 2 1 1 2 2 * *

4 1 2 2 2 2 1 1 * *

5 2 1 2 1 2 1 2 * *

6 2 1 2 2 1 2 1 * *

7 2 2 1 1 2 2 1 * *

8 2 2 1 2 1 1 2 * *

NOTE: For conducting the experiment test sheet may be prepared without the interacting columns. Interactions are

dependent on the main factors and hence can not be controlled during experimentation.

B

A C

D

TAGUCHI TECHNIQUE

(A CASE STUDY)

A tyre manufacturer was trying to maximize the water dispersion from a tyre tread. In an experimental

investigation prior to developing a super-dispersing tyre.

The following 6 Factors were identified, each at two levels.

A - Tread Pattern : Chevron, Lattice

B - Rolling Speed : 85, 25

C - Tread Diameter : Normal, Wide

D - Road Surface : Rough, Smooth

E - Type of Surface : Compact, Loose

F - Wet Surface : Low, High

Here, only main effects are considered

Total dof = 6 ( 6 * (2-1) )

L8 OA was selected

The 6 - Factors were assigned to the first 6 - columns (Table)

Response variable (Y) : FLUID RETENTION

Only one repetition was performed

ASSIGNMENT OF FACTORS TO L8 OA

FACTORS

A B C D E F e

COLUMNS

TRIAL

1 2 3 4 5 6 7

(Y)

Fluid

retention

1 1 1 1 1 1 1 1 42

2 1 1 1 2 2 2 2 48

3 1 2 2 1 1 2 2 44

4 1 2 2 2 2 1 1 49

5 2 1 2 1 2 1 2 47

6 2 1 2 2 1 2 1 45

7 2 2 1 1 2 2 1 47

8 2 2 1 2 1 1 2 46

RESPONSE TOTALS

FACTOR LEVEL TOTAL MEAN

A 1

2

183 / 4

185

45.8

46.3

B 1

2

182

186

45.5

46.5

C 1

2

183

185

45.8

46.3

D 1

2

180

188

45.0

47.0

E 1

2

177

191

44.3

47.8

F 1

2

184

184

46.0

46.0

Error 1

2

183

185

Total 368 / 8 46.0

SUM OF SQUARES

ST = 422 + 48

2 + ----- + 46

2 - (368)

2

-------

8

SA = (Σx1 - x2)2 / nr (r = replications =1 here)

= (183 - 185)2 / 8 = 0.5

SB = (182 - 186)2 / 8 = 2.0

SC = (183 - 185)2 / 8 = 0.5

SD = (180 - 188)2 / 8 = 8.0

SE = (177 - 191)2 / 8 = 24.5

SF = (184 - 184)2 / 8 = 0

Se = ST - SA - SB - SC - SD - SE - SF = 0.5

(OR)

Se = (183 - 185)2 / 8 = 0.5

n= no. of experiments in OA

r= no. of replications

INITIAL ANOVA

------------------------------------------------------------------------------------------------

FACTOR DOF S V F

------------------------------------------------------------------------------------------------

A 1 0.5 0.5 1.0

B 1 2.0 2.0 4.0

C 1 0.5 0.5 1.0

D 1 8.0 8.0 16.0

E 1 24.5 24.5 49.0

F 1 0.0 0.0 0.0

Error 1 0.5 0.5

-----------------------------------------------------------------------------------------------

TOTAL 7 36.0

-----------------------------------------------------------------------------------------------

NOTE :

DOF (Total ) = (No. of trials -1) = 8-1 = 7

DOF (Factor) = (No. of levels-1) = 2-1 = 1

DOF (Error) = DOF (Total) - DOF (all factors)

= 7 - 6 = 1

V = Variance = S/DOF

F - Statistic (F) = V (Factor) / V (error)

POOLING RULES APPLIED IN TAGUCHI DESIGNED EXPERIMENTS

To obtain a better estimate of the relative importance of each of the significant factors, non-significant factors

are removed especially when more factors are involved. This is known as pooling.

1. If the F - Statistic for any factor is less than one, then pool the 'SS' of that factor into the error term.

2. If the number of factors is not approximately one-half or less the number of columns in the table, pool the

factors with the smallest F - Statistic until the no. of factors remaining is approximately one-half the non. Of

columns.

FINAL ANOVA

-----------------------------------------------------------------------------------------------

Factor dof S V S1 P% F

-----------------------------------------------------------------------------------------------

B 1 2.0 2.0 1.625 5 5.33*

D 1 8.0 8.0 7.625 21 21.33

E 1 24.5 24.5 24.125 67 65.33

Error 4 1.5 0.375 2.6257

Total 7 36.0 100

-------------------------------------------------------------------------------------------------

F = V (Factor) / Ve * NS

PURE SUM OF SEQUARES (S1)

S1B = SB - (DOFB * Ve)

= 2 - ( 1 x 0.375) = 1.625

S1D = 8 - (1 x 0.375) = 7.625

S1E = 24.5 - (1 x 0.375 ) = 24.125

S1e = Se + (DOFT - DOFe) Ve

= 1.5 + (7 - 4) 0.375 = 2.625

PERCENTAGE CONTRIBUTION (P)

PB = S1B / ST = 1.625/36 = 4.51

PD = S1D / ST = 7.625/36 = 21.18

PE = S1E / ST = 24.125/36 = 67.01

Pe = S1 e/ ST =2.625/36 = 7.29

INTERPRETATION OF ANOVA

- Factors D and E : Significant

- D and E Contributes about 80% of the variation

- To minimize Fluid retention, Level 1 of Factors D and E are selected

- Mean value of fluid retention

D1 = 45; E1 = 44.3

- The outcome expected for D1 and E1 can be predicted.

- Insignificant factors are used to obtain cost benefits. That is, select the levels of these factors which cost

less.

PREDICTED OPTIMUM CONDITION

*U = T1 + (D

1I - T

1) + (E

1I - T

1)

T1 = Grand Mean

D1I = Mean for level 1 of Factor D

E1I = Mean for level 1 of Factor E

U = 46 + (45 - 46) + (44.3 - 46)

= 43.3