1. Introduction to Taguchi Techniques

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Dr. K. KRISHNAIAHFORMER 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 Squares (SST) 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 Factor Error Total Sum of squares SSF SSe SST Degrees of freedom K-1 N-K N-1 Variance VF=SSF/K-1 Ve=SSe/N-K F VF / Ve Sum of squares due to factors Sum of squares due to error



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.


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.


Steps in designing, conducting and analyzing an experiment. The major steps are: 1. 2. 3. 4. 5. 6. 7. Selection of factors and / or interactions to be evaluated. Selection of number of levels for the factors. Selection of the appropriate OA. Assignment of factors and / or interactions to columns. Conduct Tests. Analyze results 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.


L a b c

= = = =

Latin square no. of rows no. of levels no. of columns (Factors)

Degrees of freedom = a-1

2 - level 3 - level 4 - level Mixed - level series series series series ------------------------------------------------------------------------------------------------+ L4(23) L9(34) L15(45) L18(21,37) 7 13 21 L8(2 ) L27(3 ) L64(4 ) L36(211,312) 11 40 *L12(2 ) L81(3 ) (or) (23,313) 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 3 4 5. 6 7 8 1 1 1 2 2 2 2 1 2 2 1 1 2 2 1 2 2 2 2 1 1 2 1 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 1 2 2 1 2 2 1 2 1 1 2

--------------------------------------------------------------------------------------------------------------NOTE: 1. 2. 3. 4. 5. 6. Eight experimental runs. Balanced numbers of 1s and 2s. Any pair of columns have only Four combinations (1,1); (1,2); (2,1); (2,2) If the same number of combinations occur, them the columns are orthogonal. In the L8, any pair of columns is orthogonal. L8 can be applied to 7 or less factors.

LOCATION OF INTERACTIONS 1. 2. LINEAR GRAPHS 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 1 3 3 2 6 FIG 1 (a) FIG 1 (b) 5 7 6 7 5 4 4 2

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. 2 B 1 2 3 4 5 6 A 3 AB 3 2 1

Column No. 4 5 6 7 5 4 7 6 1 6 7 4 5 2 3 7 6 5 4 3 2 1


SELECTION OF OA 1. 2. 3. Determine the d.f. required. Note the levels of reach factor and decide the type of OA. (2-level or 3-level) 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 Draw the required linear graph. Compare with the standard linear graph of the chosen OA. 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. 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

4. 5. 6.


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.


Degrees of Freedom

Factor Levels D.F -----------------------------------------------------------------------------A 2 2 1 =1 B 2 21 =1 C 2 21 =1 D 2 21 =1 AB (2-1) (2-1) =1 AC (2-1) (2-1) = 1 AD (2-1) (2-1) = 1 -------------------------------------------------------------------------------Total d.f =7 --------------------------------------------------------------------------------


Levels of Factors - All at 2-levels Therefore, choose a 2-level OA 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. Required Linear Graph B A




D 5. 6. The required LG is similar to the standard LG (b) of L8 OA shown in FIGURE 1 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. DESIGN LAYOUT FACTORS A 1 1 2 3 4 5 6 7 8 1 1 1 1 2 2 2 2 B 2 1 1 2 2 1 1 2 2 AB 3 1 1 2 2 2 2 1 1 C 4 1 2 1 2 1 2 1 2 AC 5 1 2 1 2 2 1 2 1 AD 6 1 2 2 1 1 2 2 1 D 7 1 2 2 1 2 1 1 2 * * * * * * * * * * * * * * * * Resp. Y


Trial No.

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


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