Taguchi’s Design of Experiments and Selection of Orthogonal Array

60

CHAPTER 4 TOOLS AND TECHNIQUES USED IN THE PRESENT STUDY

4.1

INTRODUCTION

In the near-dry WEDM study, Taguchi approach and Response Surface Method (RSM) have been used to conduct the systematic design of experiments. The appropriate data collected from the experiments have been analysed by statistical methods. Multi-Objective Evolutionary Algorithm (MOEA) has been used to predict the optimum solutions by solving two conflict objective functions obtained from the regression analysis. In this chapter, the fundamentals of these three techniques have been discussed. 4.2

TAGUCHI METHOD

Genichi Taguchi (1980s) developed the fractional factorial design concept to optimize the process of engineering experimentation. Taguchi espoused an excellent philosophy for quality control in the manufacturing industries. This philosophy was first applied in Ford Motor Company to train the engineers for quality improvement. It was founded on three simple and fundamental concepts (Ross 1988 and Roy 1990). Quality should be designed for the product and not inspected into it.

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61 The cost of quality should be measured as a function of deviation from the standard and the losses should s hould be measured system-wide. Best quality is achieved by minimizing the deviations from the target. The product or process should be so designed that it is immune to uncontrollable uncontrollable environmental environmental variables. The above principles were the guidelines for developing the systems, specifying the parameters and testing the factors affecting quality improvement in place of an attempt to inspect the quality of a product on the production line. He observed that poor quality cannot be improved by the process of inspection, screening and recovering. Taguchi recommends recommends a three stage process in the engineering applications to achieve desirable product quality by the design (Ross 1988 and Roy 1990). System design is used to identify the working levels of the

design factor and parameter design seek to determine the factor levels that produce the best performance of the product/process product/process under study. study. Parametric design is used to find the optimum condition

which influences the uncontrolled factors cause the minimum variation of system performance. Tolerance design is used to fine tune the results of parameter

design by tightening the tolerance of the factors with











62 The number of parameters can influence the quality characteristic or the response of the product. The parameters can be classified into the following two classes (Phadke 1989). Control factors certain parameters can be specified freely by

the designer/operators. Multiple values of each control factor can be called as level. Noise factors certain parameters cannot be controlled by the

designer/operator. designer/operator. It is difficult to control and set their levels. 4.2.1

Loss Function

Taguchi defines the loss function which is proportional to the deviation from the target quality characteristic. At zero deviation, performance is focused on the target and loss is zero. The T he following equation represents the quality loss function (Ross 1996 and Roy 2001).

L(Y )

(Y Y 0 ) 2

where, (Y - Y 0 Y 0

(4.1) from the target

is a constant which is dependent upon the cost structure of a

manufacturing manufacturing process or an organization. The graphical representation of loss function is shown in Figure 4.1. When the quality characteristic of a product meets its target value, the loss must be zero. The magnitude of loss increases rapidly as the quality characteristics deviate from target values. The loss function must be a











63

Figure 4.1 Taguchi loss function (Ross 1988, Phadke 1989 and Roy 1990) 4.2.2

Signal to Noise Ratio

Taguchi created a transform function from the loss-function and named as Signal-to-Noise (S/N) ratio (Phadke 1989 and Barker 1990). S/N ratio was earlier specified as a concurrent statistic which is able to look at two more characteristics of a distribution and roll these characteristics into a single number. It combines both the parameters (the mean level of the quality characteristic and variance of the mean) into a single metric (Barker 1990). Thus, S/N ratio consolidates several replications (at least two data points are required) into a value. A high value of S/N indicates that the signal is much higher than the effects of noise factors. The equation for calculating S/N

S/ N HB

10 log log10 MSD MSD HB dB

(4.2)











64 and

MSD HB

MSD NB

MSD MSD LB

n

1 n

i 1

Y i

2

n

1

(Y i 2

n

Y 0 )

i 1

n

1 n

1

2

Y i i 1

where, MSD

Mean Squared Deviation, th

Y i

Response value of i experiment,

Y 0

Target response value,

N

Number of replications. replications.

MSD is a statistical quantity that reveals the deviation of each value

from the target value. The expressions for MSD are varying with respect to the quality expectation of response characteristics. In order to minimize the deviations, MSD characteristics, the standard deviation is used as MSD of each large value becomes a small value and the un-stated target value is zero. Thus for all three expressions, the smallest magnitude of MSD is being expected (Ross 1996). 4.2.3

Steps of Taguchi Design of Experiments











65 parameters on the selected quality characteristics have been investigated by the plots of main effects based on raw data. The optimum condition for each of quality characteristics has been established through S/N data analysis aided by the raw data analysis. If no outer array has been used, the experiments experiments have been replicated three or more times at each experimental condition. The flow diagram for the Taguchi experimental design and analysis are shown in Figure 4.2. 4.2.4

Selection of Orthogonal Array

An appropriate orthogonal array has been selected based on the number of parameters, their levels and desire to study of particular interactions. The selection parameters and levels are very important for the Taguchi design of experiment. Brainstorming, flowchart and cause-effect methods were suggested by Taguchi to identify the parameters which influence the output responses (Ross 1988 and Roy 1990). The levels of each parameter have been selected based on the possible range of a parameter. In this study, the possible parameters and their levels were selected from exploratory experiments. The standard two and three level arrays are given below (Phadke (Phadke 1989). 1989). Two level arrays arra ys

:

L4, L8, L12, L16, L32

Three level arrays

:

L9, L18, L27

In this study, three levels have been selected for all five parameters with three levels from exploratory experiments. If the higher-order polynomial polynomial relationship between the parameters and response is expected, at least three levels for each parameter must be considered (Barker 1990). Thus,













66 orthogonal array is selected for an experiment, the following inequality must also be satisfied (Ross 1988).

Selection of Orthogonal Array (OA)

Assignment of Parameters and Interaction Parameters

Based on Number of factors Number of levels of each factor Number of interactions of interest Degrees Of Freedom (DOF)

Based on Linear graphs / Triangular tables

Number of Repetitions of Experiments Experiments

More than two repetitions / Based on number of noise factors if considered

ANOVA test for the Raw data

Identify process parameters which affect the mean of response characteristics

ANOVA test for the S/N Ratio

Identify control parameters which affect mean and variation of the response characteristics Significant and Insignificant Parameters

Significant Parameters Identification

contribution. Insignificant parameters are pooled











67

Table 4.1 Taguchi's L27 standard orthogonal array Column No. Trial No.

1

2

3

4

5

6

7

8

9

10

11

12

13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2

1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3

1 1 1 2 2 2 3 3 3 2 2 2 3 3 3 1 1 1

1 1 1 2 2 2 3 3 3 3 3 3 1 1 1 2 2 2

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

1 2 3 1 2 3 1 2 3 2 3 1 2 3 1 2 3 1

1 2 3 1 2 3 1 2 3 3 1 2 3 1 2 3 1 2

1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2

1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 1 2 3

1 2 3 2 3 1 3 1 2 3 1 2 1 2 3 2 3 1

1 2 3 3 1 2 2 3 1 1 2 3 3 1 2 2 3 1

1 2 3 3 1 2 2 3 1 2 3 1 1 2 3 3 1 2

1 2 3 3 1 2 2 3 1 3 1 2 2 3 1 1 2 3











68 Total degree of freedom (DOF) and DOF of each parameter are depending upon the number of levels. While increasing the parameters and levels, the number of trials of the experiment is also increased. As per Taguchi experimental design concept, DOF of three levels assigned to each process parameter is two. Total DOF is equal to the number of trial experiment minus one (Ross 1988). Thus, total DOF is 26 (= 27-1). For the 3 level parameter, DOF is 2 (= 3-1). This gives a total of 10 DOF for five process parameters selected in this study. DOF of two-factor interaction is

of three levels of two factors is 4 (= (3-1) × (3-1)). Thus, L27 orthogonal arrays were selected by satisf ying above mentioned mentioned DOF criterion. 4.2.5

Assignment of Parameters to the Orthogonal Array

Assignment of parameters in the orthogonal array column is mainly related with the number of parameters and desired interactions. Taguchi gave two tools for the assignment of parameters and interactions in orthogonal arrays (Ross 1988, Phadke 1989 and Roy 1990). Triangular tables Linear graphs











69 Each OA has a particular set of linear graphs and a triangular table associated with it. Linear graphs for L27 orthogonal array with three way and four way interactions are given in Figure 4.3. In the linear graph, each node indicates the allocation of the main parameters and each line indicates the interaction terms. If the node is not assigned by the parameter, the column corresponding nodes nodes are not included for the experimentation experimentation and analysis. 4.2.6

Data Analysis

The appropriate response data have been collected during the experimentation. Collected raw and S/N data have been analysed through Analysis of Variance (ANOVA) test. The formulae of the ANOVA test for the L27 design of experiments with five parameters and three interactions are shown in Table 4.2 (Roy 1990). A pictorial representation of each parameter and their level influences on the responses are plotted based on the mean values. The response changes with respect to the levels of each parameter can easily be visualized from these curves. The S/N ratio is used to measure of variation within replications when noise factors present. ANOVA test of S/N ratio and raw data have been conducted to identify significant process parameters which affect the variance











70 Table 4.2

ANOVA

formulae

for

L27

orthogonal

array

with

interaction (Roy 1990)

Parameter

A

B

C

D

Degree of freedom ( f )

f A

f B

f C

SS A

( N L 1)

SS B

( N L 1)

( N L

SS C

1)

SS D

( N L 1)

f D

Sequential Sum of Square (SS ( SS))

A12

A22

A32

N A1

N A2

N A3

B12

B22

B32

N B1

N B 2

N B 3

C 12

C 22

C 32

N C 1

N C 2

N C 3

D12

D22

D32

N D1

N D 2

N D3

2

E

f E

SS E

( N L 1)

Interaction A×B

f E

f A

f B

SS E

Interaction B×C

f E

f B

f C

SS E

Interaction A×C

Error

f E

f A

f C

f error f T

( f A

f C

f D

f A B

f E

f B C

f A C )

CF MS A

E 2

E 3

N E 1

N E 2

N E 3

( A B) 22

N ( A B)1

N ( A B) 2

( B C )12

( B C ) 22

N ( B C )1

N ( B C )2

( A C )12

( A C ) 22

N ( A C )1

N ( A C ) 2

SS error

SS T

(SS A

SS C

SS D

SS E

SS B

CF MS D CF

( A B)12

SS A B

CF MS C

C

SS A

MS E

F-Test

SS A

MS A

f A

MS error

SS B

MS B

f B

MS error

SS C

MS C

f C

MS error

SS D

MS D

f D

MS error

SS E

MS E

f E

MS error

SS A B

MS ( A B )

f ( A B )

MS error

SS B

MS ( B

CF MS A

2

E 1

SS E f B

2

Variance or Mean Sum of Square ( MS) MS)

CF MS E

CF MS E

CF MS E

f ( B

C C )

SS A C

MS ( A

f ( A C )

MS error

SS B

SS error

MS errore

f error

-

C

27

Total

f

N 1

SS T

Y i CF

C )

MS error

-

-

C )













71 4.2.7

Estimation of Optimum Values

The mean response at the optimum condition is predicted after determination of optimum levels of parameters. The mean is estimated from the significant parameters and interaction terms. For example, parameters A, B and C are significant and A 1 B3 and C2 (first level of A =A 1, third level of B=B3 and second level of C=C 2) are the optimal treatment condition. Then, the optimal value of response characteristic (

opt)

is estimated as (Phadke

1989 and Roy 1990)

opt

where,

T

= T + (A1 - T) + (B3 - T) + (C2 - T)

is the overall mean response, A , 1

B3 and C 2 are

(4.5)

the average response

value of the first, third and second level of parameters A, B and C respectively.











72 The confidence interval for the sample group (CIE):

Around the estimated average of a treatment condition used in a confirmatory experiment to verify the predictions. It is used for only a sample group made under the specif ied conditions. The confidence interval for the population (CI P): Around

the estimated average of a treatment condition predicted from experiment. It is used for entire population i.e., all parts ever made under the specified conditions c onditions.. In this study, 95% of confidence level has been considered. The 95% confidence intervals of confirmation experiments (CI E) and population (CI p) are calculated by using following f ollowing equations (Roy 1990). 1990).

CI E =

F (1, f )V

1

1

(4.6)











73 4.2.9

Confirmation Experiments

The final step of the Taguchi design is the confirmation of experiments in which estimated optimum response has been evaluated using significant parameters. More than two experiments have been conducted under specified conditions. The average of the confirmation experiment experiment result compares with an anticipated average based on the parameters and levels tested. The confirmation experiment is an important step and is highly recommended to verify the predicted results by the Taguchi method (Ross 1988 and Roy 1990). 4.3

RESPONSE SURFACE METHOD

Response surface method (RSM) is a combination of statistical and mathematical technique useful for modeling and optimization of the engineering problems in which the relationship between several input











74 polynomial polynomial model gives the better correlation between input and output variables, and also fit with response surfaces. The second-order polynomial model is widely used and shown in Equation (4.9) (Montgomery 2005). k

y

xi

0

i i 1

2

k ii

x

2 i

i 1

xi x j

ij

(4.9)

i j 2

contains linear, squared and cross product terms of variables xi and x j. The number of experiments design techniques is used to estimate the regression co-efficient (

0

i

ii and

ij).

In this present

work, second-order response surface model is used to build the correlation and analysis the data. 4.3.1

Central Composite Design

The most popular class of second-order model of the response surface method is a central composite rotatable design. The CCD design











75 Table 4.3

Half-fractional central composite second-order rotatable design for five parameters

Std. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1

B -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1

C -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1

D -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1

E 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1

Comment

s t n i o P l a i r o t c a F













76

in the CCD. When numbers of factors are greater than or equal to five, the k

experimental size can be reduced by using half replication of 2 factorial design (Akhanazarova and Kafarov 1982). The design matrix for five independent variables of half fractional central composite design is shown in Table 4.3. In this study, st udy, totally five process parameters have been considered. The estimation of the number of runs has been described as follows. Star points position

=

= (2(k-1))1/4 = 24/4 = 2











77 and y1

0

1

y2

1

2

[Y ]

[ ]

[ ] y N

C

N

1 x11

x21

xk 1

2 x11

x11 x21

1 x

x

x

x 2

x x











78 4.3.3

Analysis of Variance Test

Analysis of variance is done by the total sum of squares test which consists of first and second-order terms, lack of fit test and estimation of experimental error. In order to find the individual coefficients for -

f

f freedom has been used and n0 is -

value is compared with the theoretical value at 95% of confidence level. If -

terms are











79 The sequential sum of square tests, lack of fit tests and model summary statistics analysis has been performed to select the appropriate model to be fitted. The linear, two-factor interaction, quadratic and cubic models were compared to select the adequacy of the model. The complexity of increasing the terms of the model has been estimated by the sequential sum of square test of the model. The lack of fit test used to estimate the residual error from replication of the experiments in central design points. ANOVA -

ermination

coefficients (R 2), adjusted R 2, predicted R 2 and the Adequate Precision (AP).











80 optimal decisions need to be taken into the presence of trade-offs between two or more conflicting objectives (Coello and Toscano 2005). Generally, such problems consist consist of conflicting objectives objectives so that it is not possible to obtain obtain an individual solution which is optimized for all objectives. As an alternative of a single optimal solution, a set of optimum solutions called the Pareto optimal set exists within such cases. Members of the Pareto-set are such that there is no solution in the set which is better than the others in all the objectives and neither does a solution exist in the set which is worse than the others in all the objectives. By using evolutionary algorithms, one can













81 frontiers representing the trade-off between the criteria, simultaneously providing providing a link with the decision variables variables (Deb 2001). 2001).

Taguchi’s Design of Experiments and Selection of Orthogonal Array

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