Module 5 Lecture 4 Final

7/30/2019 Module 5 Lecture 4 Final

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Module

5

Design for Reliability andQuality


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Lecture

4Approach to Robust Design


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Instructional Objectives

The primary objectives of this lecture are to outline the concept of robust design and various

tools to achieve the same for typical manufacturing processes.

Defining Robust Design

Robust design is an engineering methodology for improving productivity during research and

development so that high-quality products can be produced quickly and at low cost. According to

Dr. Genichi Taguchi, a robust design is one that is created with a system of design tools to

reduce variability in product or process, while simultaneously guiding the performance towards

an optimal setting. A product that is robustly designed will provide customer satisfaction even

when subjected to extreme conditions on the manufacturing floor or in the service environment.

Tools for Robust Design

Taguchi method, design of experiments and multiple regression analysis are some of the

important tools used for robust design to produce high quality products quickly and at low cost.

Taguchi MethodTaguchi method is based on performing evaluation or experiments to test the sensitivity of a set

of response variables to a set of control parameters (or independent variables)by considering

experiments in orthogonal array with an aim to attain the optimum setting of the control

parameters. Orthogonal arrays provide a best set of well balanced (minimum) experiments [1].

Table 5.4.1 shows eighteen standard orthogonal arrays along with the number of columns at

different levels for these arrays [1]. An array name indicates the number of rows and columns it

has, and also the number of levels in each of the columns. For example array L 4 (23) has four

rows and three 2 level columns. Similarly the array L18 (213

7) has 18 rows; one 2 level

column; and seven 3 level columns. Thus, there are eight columns in the array L18. The

number of rows of an orthogonal array represents the requisite number of experiments. The

number of rows must be at least equal to the degrees of the freedom associated with the factors

i.e. the control variables. In general, the number of degrees of freedom associated with a factor


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(control variable) is equal to the number of levels for that factor minus one. For example, a case

study has one factor (A) with 2 levels (A), and five factors (B, C, D, E, F) each with 3 level.

Table 5.4.2 depicts the degrees of freedom calculated for this case. The number of columns of an

array represents the maximum number of factors that can be studied using that array.

Table 5.4.1 Standard orthogonal arrays [1]

Orthogonal

array

Number

of rows

Maximum

number of

factors

Maximum number of columns at these

levels

2 3 4 5

L

L

4

L

8

L

9

4

12

8

9

12

3

7

4

11

3

7

-

11

-

-

4

-

-

-

-

-

-

-

-

-

L

L

16

16

L

L

18

16

25

16

18

25

15

5

8

6

15

-

1

-

-

-

7

-

-

5

-

-

-

-

-

6

L

L

27

L

32

32

L

L

36

36

27

3232

36

36

13

3110

23

16

-

311

11

3

13

--

12

13

-

-9

-

-

-

--

-

-

L

L

50

L

54

L

64

64

L

50

81

54

64

64

81

12

26

63

21

40

1

1

63

-

-

-

25

-

-

40

-

-

-

21

-

11

-

-

-

-

The signal to noise ratios (S/N), which are log functions of desired output, serve as the

objective functions for optimization, help in data analysis and the prediction of the optimum

results. The Taguchi methodtreats the optimization problems in two categories: static problems


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and dynamic problems. For simplicity, the detailed explanation of only the static problems is

given in the following text. Next, the complete procedure followed to optimize a typical process

using Taguchi method is explained with an example.

Table 5.4.2 The degrees of freedom for one factor (A) in 2 levels and five factors

(B, C, D, E, F) in 3 levels

Factors Degrees of freedom

Overall mean

A

B, C, D, E, F

1

2-1 = 1

5 (3-1) = 10

Total 12

Static problems

Generally, a process to be optimized has several control factors (process parameters) which

directly decide the target or desired value of the output. The optimization then involves

determining the best levels of the control factor so that the output is at the target value. Such a

problem is called as a "STATIC PROBLEM". This can be best explained using a P-Diagram

(Figure 5.4.1) which is shown below ("P" stands for Process or Product). The noise is shown to

be present in the process but should have no effect on the output. This is the primary aim of the

Taguchi experiments - to minimize the variations in output even though noise is present in the

process. The process is then said to have become ROBUST.

Figure 5.4.1 P- Diagram for static problems [1].


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Signal to Noise (S/N) Ratio

There are three forms ofsignal to noise (S/N) ratio that are of common interest for optimization

of static problems.

[1] Smaller-the-better

This is expressed as

]datameasuredofsquaresofsumofmean[Log10n 10= (1)

This is usually the chosen S/N ratio for all the undesirable characteristics like defects for which

the ideal value is zero. When an ideal value is finite and its maximum or minimum value is

defined (like the maximum purity is 100% or the maximum temperature is 92 K or the minimum

time for making a telephone connection is 1 sec) then the difference between the measured data

and the ideal value is expected to be as small as possible. Thus, the generic form of S/N ratio

becomes,

}]idealmeasured{ofsquaresofsumofmean[Log10n 10 = (2)

[2] Larger-the-better


]datameasuredofreciprocalofsquaresofsumofmean[Log10n 10= (3)

This is often converted to smaller-the-better by taking the reciprocal of the measured data and

next, taking the S/N ratio as in the smaller-the-bettercase.

[3] Nominal-the-best


=iancevar

meanofsquareLog10n 10 (4)


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This case arises when a specified value is the most desired, meaning that neither a smaller nor a

larger value is desired.

Example for application of Taguchi Method

Determine the effect of four process parameters: temperature (A), pressure (B), setting time (C),

and cleaning method (D) on the formation of surface defects in a chemical vapor deposition

(CVD) process to produce silicon wafers. Also estimate the optimum setting of the above

process parameters for minimum defects. Table 5.4.3 depicts the factors and their levels.

Table 5.4.3 Factors and their levels

Factor

Level

1 2 3

A. Temperature (0B. Pressure (mtorr)

C)

C. Settling time (min)D. Cleaning method

T0

P

25

0

t

200

None

0

T

P

0

t

0

0

CM

+8

T

2

0

P

+25

0

t

+200

0

CM

+16

3

Step 1: Select the design matrix and perform the experiments

The present example is associated with four factors with each at three levels. Table 5.4.1

indicates that the best suitable orthogonal array is L9. Table 5.4.4 shows the design matrix for

L9. Next conduct all the nine experiments and observe the surface defect counts per unit area at

three locations each on three silicon wafers (thin disks of silicon used for making VLSI circuits)

so that there are nine observations in total for each experiment. The summary statistic, i

, for an

experiment, i, is given by

i10i Clog10= (5)

where Ci refers to mean squared effect countforexperiment i and the mean square refers to the

average of the squares of the nine observations in the experiment i. Table 5.4.4 also depicts the

observed value ofi for all the nine experiments. This summary statistic i is called the signal to

noise (S/N) ratio.


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Table 5.4.4 L9

array matrix experiment table [1].

Expt

No.

Column number and factor assigned

Observation, (dB)

1

Temperature

(A)

2

Pressure

(B)

3

Settling time

(C)

4

Cleaning

method (D)

1

2

3

4

5

6

7

8

9

1

1

1

2

2

2

3

3

3

1

2

3

1

2

3

1

2

3

1

2

3

2

3

1

3

1

2

1

2

3

3

1

2

2

3

1

1

= -20

2

= -10

3

= -30

4

= -25

5

= -45

6

= -65

7

= -45

8

= -65

9 = -70

Step 2: Calculation offactor effects

The effect of a factor level is defined as the deviation it causes from the overall mean. Hence as a

first step, calculate the overall mean value of for the experimental region defined by the factorlevels in Table 5.4.4 as

( ) dB67.41.....9

1

9

1m

9

1i

921i =+++== =

(6)

The effect of the temperature at level A 1 (at experiments 1, 2 and 3) is calculated as the

difference of the average S/N ratio for these experiments (mA1

The effect of temperature at level A

) and the overall mean. The same

is given as

1 = mA1 ( ) m3

1321 ++ m = (7)

Similarly,


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The effect of temperature at level A2 = mA2 ( ) m3

1654 ++ m = (8)

The effect of temperature at level A3 = mA3 ( ) m3

1987 ++ m = (9)

Using the S/N ratio data available in Table 5.4.4 the average of each level of the four factors is

calculated and listed in Table 5.4.5. These average values are shown in Figure 5.4.2. They are

separate effect of each factor and are commonly called main effects.

Table 5.4.5 Average for different factor levels [1].

FactorLevel

1 2 3

A. TemperatureB. PressureC. Settling timeD. Cleaning method

-20-30

-50

-45

-45-40

-35

-40

-60-55

-40

-40

Figure 5.4.2 Plots of factor effects


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Step 3: Selecting optimum factor levels

Our goal in this experiment is to minimize the surface defect counts to improve the quality of the

silicon wafers produced through the chemical vapor deposition process. Since log depicts a

monotonic decreasing function [equation (5)], we should maximize . Hence the optimum level

for a factor is the level that gives the highest value of in the experimental region. From Figure

5.4.2 and the Table 5.4.5, it is observed that the optimum settings of temperature, pressure,

settling time and cleaning method are A1, B1, C2 and D2 or D3. Hence we can conclude that the

settings A1B1C2D2 and A1B1C2D3

can give the highest or the lowest surface defect count.

Step 4: Developing the additive model for factor effects

The relation between and the process parameters A, B, C and D can be approximated

adequately by the following additive model:

edcbam)D,C,B,A( lkjilkji +++++= (10)

where the term m refers to the overall mean (that is the mean of for the experimental region).

The terms ai, bj, ck and dl refer to the deviations from caused by the setting A i, Bj, Ck, and Dl

of factors A, B, C and D, respectively. The term e stands for the error. In additive model the

cross- product terms involving two or more factors are not allowed. Equation (10) is utilized in

predicting the S/N ratio at optimum factor levels.

Step 5: Analysis of Variance (ANOVA)

Different factors affect the surface defects formation to a different degree. The relative

magnitude of the factor effects are listed in Table 5.4.5. A better feel for the relative effect of the

different factors is obtained by the decomposition of variance, which is commonly called as

analysis of variance (ANOVA). This is obtained first by computing the sum of squares.

Total sum of squares = 22229

1i

2i )dB(19425)70(.....)10()20( =+++=

=

(11)


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Sum of squares due to mean = 222 )dB(1562567.419ms)experimentofnumber( == (12)

Total sum of squares = 29

1i

2i )dB(3800)m( =

=

(13)

Sum of squares due to factor A

= [(number of experiments at level A1) (mA1-m)2

[(number of experiments at level A

] +

2) (mA2-m)2

[(number of experiments at level A

] + (14)

3) (mA3-m)2

= [3 (-20+41.67)

]

2] + [3 (-45+41.67)

2] + [3 (-60+41.67)

2] = 2450 (dB)

2

.

Similarly the sum of squares due to factor B, C and D can be computed as 950, 350 and 50 (dB)2

Table 5.4.6 ANOVA table for [1].

,

respectively. Now all these sum of squares are tabulated in Table 5.4.6. This is called as the

ANOVA table.

FactorDegree of

freedom

Sum of

squares

Mean square =

sum of squares/degree of freedomF

A. TemperatureB. PressureC. Settling timeD. Cleaning method

2

2

2

2

2450

950

350

50

*

1225

*

475

175

25

12.25

4.75

Error 0 0 -

Total 8 3800

(Error) (4) (400) (100)

*Indicates sum of squares added together to estimate the pooled error sum of squares shown within

parenthesis. F ratio is calculated as the ratio of factor mean square to the error mean square.

Degrees of freedom:

The degrees of freedom associated with the grand total sum of squares are equal to thenumber of rows in the design matrix.

The degree of freedom associated with the sum of squares due to mean is one.


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The degrees of freedom associated with the total sum of squares will be equal to thenumber of rows in the design matrix minus one.

The degrees of freedom associated with the factor will be equal to the number of levelsminus one.

The degrees of freedom for the error will be equal to the degrees of freedom for the totalsum of squares minus the sum of the degrees of freedom for the various factors.

In the present case-study, the degrees of freedom for the error will be zero. Hence an

approximate estimate of the error sum of squares is obtained by pooling the sum of squares

corresponding to the factors having the lowest mean square. As a rule of thumb, the sum of

squares corresponding to the bottom half of the factors (as defined by lower mean square) are

used to estimate the error sum of squares. In the present example, the factors C and D are used to

estimate the error sum of squares. Together they account for four degrees of freedom and their

sum of squares is 400.

Step 6: Interpretation of ANOVA table.

The major inferences from the ANOVA table are given in this section. Referring to the sum of

squares in Table 5.4.6, the factor A makes the largest contribution to the total sum of squares

[(2450/3800) x 100 = 64.5%]. Thefactor B makes the next largest contribution (25%) to the totalsum of squares, whereas thefactors C and D together make only 10.5% contribution. The larger

the contribution of a particular factor to the total sum of squares, the larger the ability is of that

factor to influence . Moreover, the larger the F-value, the larger will be the factor effect in

comparison to the error mean square or the error variance.

Step 7: Prediction of under optimum conditions

In the present example, the identified optimum condition or the optimum level of factors is

A1B1C2D2

(step 3). The value of under the optimum condition is predicted using the additive

model [equation (10)] as

dB33.8)67.4130()67.4120(67.41)mm()mm(m 1B1Aopt =++++=++= (15)


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Since the sum of squares due to the factors C and D are small as well as used to estimate the

error variance, these terms are not included in equation (15). Further using equations (5) and

(15), the mean square count at the optimum condition is calculated as 10

opt

10y

= = 100.833 = 6.8

(defects/unit area)2

6.28.6 =. The corresponding root-mean square defect count is defects/unit

area.

Design of Experiments

A designed experiment is a test or series of tests in which purposeful changes are made to the

input variables of a process or system so that we may observe and identify the reasons for

changes in the output response. For example, Figure 5.4.3 depicts a process or system under

study. The process parameters x1, x2, x3, , xp are controllable, whereas other variables z1, z2,

z3, ,zq

Determining which variables are most influential on the response, y.

are uncontrollable. The term y refers to the output variable. The objectives of the

experiment are stated as:

Determining where to set the influential xs so that y is almost always near the desirednominal value.

Determining where to set the influential xs so that variability in y is small. Determining where to set the influential xs so that the effects of the uncontrollable z 1, z2

zq

are minimized.

Figure 5.4.3 General model of a process or system [2].


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Experimental design is used as an important tool in numerous applications. For instance it is used

as a vital tool in improving the performance of a manufacturing process and in the engineering

design activities. The use of the experimental design in these areas results in products those are

easier to manufacture, products that have enhanced field performance and reliability, lower

product cost, and short product design and development time.

Guidelines for designing experiments

Recognition and statement of the problem. Choice of factors and levels. Selection of the response variable. Choice of experimental design. Performing the experiment. Data analysis. Conclusions and recommendations.

Factorial designs

Factorial designs are widely used in experiments involving several factors where it is necessary

to study the joint effect of the factors on a response. For simplicity and easy understanding, in

the present section the design matrix of the 22

The 2

factorial design is presented with subsequent

explanation on the calculation of the main effects, interaction effects and the sum of squares. The

two level design matrices are very famous and used in the daily life engineering applications

very frequently.

2

The 2

design

2design is the first design in the 2

kfactorial design. This involves two factors (A and B),

each run at two levels. Table 5.4.7 depicts the 22

design matrix, where refers to the low leveland + refers to the high level. These are also called as non-dimensional or coded values of the

process parameters. The relation between the actual and the coded process parameters is given as


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2

xx

2

xxx

xlowhigh

lowhigh

i

+

= (16)

where xi

is the coded value of the process parameter (x). The term y refers to the response

parameter.

Table 5.4.7 22

Expt. No.

factorial design matrix.

Factors Response

A B AB y

1

2

3

4

-1

+1

-1

+1

-1

-1

+1

+1

1

-1

-1

1

y

y

1

y

2

y

3

4

Similarly, the main effect of factor B is calculated as

+ BB yy = r2yy

r2

yy 2143 +

+(18)

The interaction effect of AB is calculated as

+ ABAB yy =r2

yy

r2

yy 3241 ++

(19)

The next step is to compute the sum of squares of the main and interaction factors. Before doingthat, the contrast of the factors need to be calculated as follows.

(Contrast)A = (y2 + y4)-(y1 + y3

(Contrast)

) (20)

B = (y3 + y4)-(y1 + y2

(Contrast)

) (21)

AB = (y1 + y4)-(y2 + y3) (22)


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Further these contrasts are utilized in the calculation of sum of squares as follow.

(Sum of squares)A = SSA rowsofnumberr

])contrast[( 2A

= (23)

(Sum of squares)B = SSBrowsofnumberr

])contrast[( 2B

= (24)

(Sum of squares)AB = SSABrowsofnumberr

])contrast[( 2AB

= (25)

Total sum of squares = SST = = =

2

1i

2

1j

r

1k

2

avg2

ijkrowsofnumberr

yy= (26)

In general, SST

has [(r number of rows)-1] degrees of freedom (dof). The error sum of squares,

with r [number of rows-1] is calculates as

Error sum of squares = SSE = SST SSA SSB - SSAB

(27)

Moreover each process parameter is associated with a single degree of freedom. Further, the

complete analysis of variance is summarized in Table 5.4.8. This is called as analysis of variance

(ANOVA) table. The term F0

The main drawback with the two level designs is the failure to capture the non linear influence of

the process parameters on the response. Three level designs are used for this purpose. The

explanation about the three level designs is given elsewhere [2].

refers to the F ratio and the same is calculated as the ratio of factor

mean square to the error mean square. The interpretation of the ANOVA table can be done

similar to the one as explained in the Taguchi method, step 6.


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Table 5.4.8 Analysis of Variance (ANOVA) table.

Source of

variationSum of squares

Degree of

freedomMean square F0

A

B

AB

Error

Total

SS

SS

A

SS

B

SS

AB

SS

E

(dof)

T

(dof)

A

(dof)

B

(dof)

AB

(dof)

E

SS

T

A/(dof)

SS

A

B/(dof)

SS

B

AB/(dof)

SS

AB

E/(dof)

E

(F0)

(F

A

0)

(F

B

0)AB

Central composite rotatable design

Even though three level designs help in understanding the non linear influence of the process

parameters on the response, the number of experiments increases tremendously with the increase

in number of process parameters. For example, the number of experiments involved in three

level designs with three, four and five factors is twenty seven (33=27), eighty one (3

4=81) and

two hundred and forty three (35

The principle of central composite rotatable design includes 2f numbers of factorial experiments

to estimate the linear and the interaction effects of the independent variables on the responses,

where f is the number of factors or independent process variables. In addition, a number (n

=243), respectively. The principle of central composite rotatable

design (CCD) reduces the total number of experiments without a loss of generality [2]. This is

widely used as it can provide a second order multiple regression model as a function of the

independent process parameters with the minimum number of experimental runs [2].

C) of

repetitions [nC

The choice of the distance of the axial points () from the centre of the design is important to

make a central composite design (CCD) rotatable. The value of for rotatability of the design

scheme is estimated as = (2

> f] are made at the center point of the design matrix to calculate the model

independent estimate of the noise variance and 2f number of axial runs are used to facilitate the

incorporation of the quadratic terms into the model. The term rotatable indicates that the variance

of the model prediction would be the same at all points located equidistant from the center of thedesign matrix.

f)

1/4

[2]. The number of experiments is estimated as


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C

f n)f2(2 ++ (28)

The intermediate coded values are calculated as [2]

2

xx

2

xxx

xminmax

minmax

i

+

= (29)

where xi is the coded value of a process variable (x) between xmax and xmin

124)22(22 =++

. For example the

number of experiments in a CCD matrix corresponding to two process variables is calculated as

and the distance of the axial points from the center is calculated as =

(2*2)1/4

= 1.414. Hence Table 5.4.9 depicts the CCD for a two process parameter application.

Table 5.4.9 Central composite design (CCD) for a two process parameter application.

Expt. No.Process parameters (coded) Response variable

x X1 y2

1

23

4

5

6

7

8

9

10

11

12

-1

+1-1

+1

-1.414

+1.414

0

0

0

0

0

0

-1

-1+1

+1

0

0

-1.414

+1.414

0

0

0

0

y

y

1

y

2

y

3

y

4

y

5

y

6

y

7

y

8

y

9

y

10

y

11

12


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Regression modeling

Regression models are the mathematical estimation equations with response variable as a

function of process parameters. These models are developed statistically by utilizing the

information of the measured response variable and the corresponding design matrix. Consideringthe f number of independent process parameters, a generalized regression model can be

represented as

++

++=

= ===

f

1i

f

1jjiijjjj

f

1j

f

1j

* **** xxxxy

2

jj0m (30)

where my is a response variable in non-dimensional form, xi* and xj

* refer to the independent

variables in non-dimensional form, s refer to the regression coefficients and is the error

term.

Calculation of the regression coefficients and ANOVA terms

The coefficients, s, in the regression model [equation (5.4.30)] are calculated based on the

minimization of the error between the experimentally measured and the corresponding estimated

values of the response variables. The least square function, S, to be minimized can be expressed

as [3]

2f

1i

f

1jijjj

f

1jjj

f

1j

u

1s0 ji

2

jmij10 xx)x(xy),,,(S*

=

=

= ===

(31)

The estimated second order response surface model is represented as

= === +++=

f

1i

f

1jijjj

f

1jjj

f

1jji

2

jp xx)(xxy

*

0

(32)

Further the adequacy of the developed estimation model is tested using Analysis of Variance

(ANOVA) as shown in Table 5.4.10.


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Table 5.4.10 Analysis of variance (ANOVA) method for testing the significance of regression

model [3].

Source of variationSum of

squares

Degree of freedom

(dof)

Mean

square

F-statistic

(F)P-value

Regression SS m-1R MS FR PR R

Linear terms SS m-1-mR_L MS FR_L PR_L R_L

Non-linear terms SS mR_NL MS FR_NL PR_NL R_NL

Residual SS u-mRes MS

Res

Lack of fit SS u-m-nLOF C MS+1 LOF

Pure error SS nPE C MS-1 PE

Total SS u-1T

2AdjR

The terms in ANOVA table are calculated in the following manner.

2u

1sspsm

s

s

u

1sm

s )y()y(SS;u

)y(

)y(SS sRe

2

u

1

pR

==

=

=

=

(33, 34)

2

LL_R

2

T

u

1s

u

1ssm

s_p

u

1s

u

1ssm

smu

)y(

)y(SS;u

)y(

)y(SS

=

=

=

=

=

=

(35, 36)

=

=

==

u

43s

u

43ssm

sm

2

PEL_RRNL_R

44u

)y(

)y(SS;SSSSSS (37, 38)

'm1m

SSMS;

mu

SSMS

;1m

SSMS;SSSSSS

L_RL_R

sResRe

RRPEsReLOF

=

==

=

(39, 40)


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==

==

=

=

+==

1u

SS

mu

SS

1R;MS

MSF

;MS

MSF;

MS

MSF

;MS

MSF;

1n

SSMS

;1nmu

SSMS;

'm

SSMS

T

sRe

2Adj

PE

LOFLOF

sRe

NL_R

NL_R

sRe

L_R

sRe

R

RC

PE

PE

C

LOFLOF

NL_R

NL_R

L_R(41 48)

where

(a) SSR, SSRes, and SST

(b) SS

refer to the regression sum of squares, residual sum of squares and

total sum of squares with degrees of freedom m1 (m is the number of terms in the

regression model), um and u1 respectively.

R_L, SSR_NL, SSPE and SSLOF refer to the regression sum of squares of the model

having only linear terms, regression sum of squares of the model having only non-

linear terms, pure error sum of squares and the lack of fit sum of squares with degrees

of freedom m1m, m (number of non-linear terms in response surface model), nC1

and umnC

(c)

1 respectively.L_py refers to the regression model with only linear terms.

(d) MSR and MSRes

(e) MS

refer to the regression mean squares and the residual mean squares

respectively.

R_L, MSR_NL, MSPE and MSLOF

(f) F

refer to the regression mean squares of the model

having only linear terms, regression mean squares of the model having only non-linear

terms, pure error mean squares and lack of fit mean squares respectively.

R, FR_L, FR_NL and FLOF

(g) P

refer to the F-statistic required for the hypothesis testing of

the regression model, model with only linear terms, model with only quadratic terms

and the lack of fit of regression model respectively.

R, PR_L, PR_NL and PLOF refer to the P-value of the regression model, model with only

linear terms, model with only non-linear terms, and the lack of fit of second order


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response surface model respectively. The term P-value refers to the smallest

significance level at which the data lead to rejection of the null hypothesis. In other

words, if the P-value is less than level of significance () then the null hypothesis is

rejected. These values are calculated using the corresponding F-statistic value and the

F-distribution table.

(h) 2AdjR refers to the adjusted coefficient of determination.

Model adequacy checking

The various steps followed to check the adequacy of the regression model are

[1] Step 1

Initially, the lack of fit test is performed to check the lack of fit for the regression model. The

appropriate hypothesis considered for testing is

H0

H

: The regression model is adequate (Null hypothesis) (49)

1

: The regression model is not adequate (Alternate hypothesis) (50)

For a given significance level (), the null hypothesis is rejected if

1cn,1cnmu,FFLOF +> and >LOFP (51)

The terms 1cn,1cnmu,F + and PLOF

are calculated from the F-distribution table. The value of

is considered as 0.1 in the present study [3]. If the equation (51) is not satisfied then the null

hypothesis is accepted, implying that there is no evidence of lack of fit for the regression model

and the same model can be used for further analysis.

[2] Step 2

The significance of this quadratic model is checked by conducting hypothesis testing. The

appropriate hypothesis considered for testing is

hypothesisAlternate;elmodregressiontheintermoneatleastfor0:H

hypothesisNull;0:H

1

ji

ij1312jj2211j210

========= and PR -value < (53)

The terms mu,1m,F and PR

are calculated from the F-distribution table. If the equation (53) is

satisfied then the null hypothesis is rejected, implying that at least one of the regressors in the

model are non zero or significant.

[3] Step 3

The contribution of the linear and non-linear terms to the model is tested. For a given

significance level (), the linear terms contribute significantly when

mu,'m1m,FF L_R > and corresponding PR_L -value < ; (54)

and the quadratic terms contribute significantly when

mu,'m,FF NL_R > and corresponding PR_NL -value < (55)

[4] Step 4

The coefficient of determination 2AdjR is calculated. This represents the proportion of the

variation in the response explained by the regression model. If the value of 2AdjR is close to 1.0

then most of the variability in response is explained by the model.

[5] Step 5

The t-statistic and P-value of all the coefficients in regression model are calculated. If the P-

value of any term in the model is greater than then the same are insignificant.

[6] Step 6

The significant terms in the regression model are identified using the step wise regression

analysis. Step wise regression analysis involves multiple steps of regression, where in each step a


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single variable having low P-value (


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Utilizing the measured values of the weld bead width at different welding conditions [4], develop

the regression model of the weld bead width as a function of welding condition?

Table 5.4.11 Process parameters and their limits [4].

Process parameters NotationFactor levels

2.3784 1 0 1 2.3784

Leading wire current ILE 300(A) 384 445 506 590

Trailing wire +ve pulse

current

+TRI (A) 319 343 360 377 401

Trailing wire ve pulse

current

TRI (A) 401 562 680 797 958

Trailing wire negative

pulse time

TRt (s) 0.00835 0.00956 0.01044 0.01132 0.01253

Welding speed v (mm/s) 7 10 12.23 14.45 17.45

Solution

Sequentially following the steps explained under the section regression modeling, the weld bead

width regression model as a function of process parameters is developed as

****

****

****

TRTRTRTR

22

TR

2

TR

TRTRTRLEp

tI043.0II0370.0

)v(0620.0)I(0430.0)I(0320.0v3900.0

t0710.0I1570.0I0450.0I1200.0655.2w*

+

+

+

+

+++=

(57)

where the term *pw refers to the predicted non-dimensional weld bead width. Table 5.4.12

depicts the corresponding ANOVA table. This ANOVA tables explain the contribution of the

linear and non-linear terms, and the proportion of variation in the predicted weld bead width

form the measured.


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Table 5.4.12 ANOVA table for the weld bead width regression model.

Source of variationSum of

squares

Degree of

freedomMean square

F-statistic

(F)P-value

Regression 9.2087 20 0.4604 36.77 0.00

Linear terms 8.5678 5 1.7136 137.09 0.00

Non linear terms 0.6409 15 0.0427 3.4181 0.002

Residual 0.3631 29 0.0125

1.4737 0.292Lack of fit 0.2985 22 0.0136

Pure error 0.0646 7 0.0092

Total 9.5718 49

2AdjR = 0.94

The adjusted coefficient of determination ( 2AdjR ) corresponding to the equations (5.4.41is

calcualted as 0.94 (table 5.4.12). The adjusted coefficient of determination represents the

proportion of the variation in the response explained by the regression model [3]. It is thus

envisaged that equation (5.4.41) can capture 94% of the variation in the measured values of weld

width as function of the five independent welding conditions within the ranges considered in the

present study.


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Exercise

1. Develop the design matrix for three factors operating at three levels.2. Develop the regression model for the penetration as a function of process parameters

using the data published in the reference 4.

Reference

[1] M. S. Phadke, Quality engineering using robust design, 2nd

[2] D. C. Montgomery, Design and analysis of experiments, 3

edition, Pearson, 2009.

rd

[3] D. C. Montgomery, E. A. Peck and G. G. Vining, Introduction to linear regression

analysis, 4

edition, John wiley and sons,

1991.

th

[4] D. V. Kiran, B. Basu and A. De, Influence of process variables on weld bead quality in two

wire tandem submerged arc welding of HSLA steel, Journal of Materials Processing

Technology, 2010, doi:10.1016/jmatprotec.2012.05.008.

edition, , John wiley and sons, 2006.

Module 5 Lecture 4 Final

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Transcript of Module 5 Lecture 4 Final