Modeling of TIG Welding Process Using Conventional Regression Analysis and ANN Aproaches

8/2/2019 Modeling of TIG Welding Process Using Conventional Regression Analysis and ANN Aproaches

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Journal of Materials Processing Technology 184 (2007) 56–68

Modeling of TIG welding process using conventional regressionanalysis and neural network-based approaches

Parikshit Dutta, Dilip Kumar Pratihar∗

Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721 302, India

Received 26 May 2006; received in revised form 1 November 2006; accepted 7 November 2006

Abstract

Conventional regression analysis was carried out on some experimental data of a tungsten inert gas (TIG) welding process (obtained from

published literature), to find its input–output relationships. One thousand training data for neural networks were created at random, by varyingthe input variables within their respective ranges and the responses were calculated for each combination of input variables by using the response

equations obtained through the above conventional regression analysis.Theperformances of theconventional regression analysis approach, a back-

propagation neural network (BPNN) and a genetic-neural system (GA-NN) were compared on some randomly generated test cases (experimental),

which are different from the training cases. It is interesting to note that for the said test cases, the NN-based approaches could yield predictions that

are more adaptive in nature compared to those of the more conventional regression analysis approach. It could be due to the fact that NN-based

approaches are able to bring adaptability, which is missing in the conventional regression analysis. Moreover, GA-NN was found to perform better

than the BPNN, in most of the test cases. A BPNN works based on the principle of a steepest descent method, whose solutions have the chance

of being trapped at the local minima, whereas in GA-NN, the search for a minimum deviation in prediction, is carried out using a GA. However,

their performance depends on the nature of the deviation function.

© 2006 Elsevier B.V. All rights reserved.

Keywords: TIG welding; Conventional regression analysis; BPNN; GA-NN

1. Introduction

To ensure both high productivity as well as good quality of

the products, a manufacturing process is to be automated. In

order to automate a process, a proper model has to be con-

structed and tested before implementing for on-line control.

This paper deals with modeling of a tungsten inert gas (TIG)

welding process. There is a natural quest of the researchers to

establish input–output relationships of a process. Rosenthal [1]

studied the temperature distributions on an infinite sheet, due to

a moving point heat source considering the heat dissipation by

conduction. His analysis could be related to arc welding after

making a number of assumptions. However, he never tried torelate his theoretical solution to the weld bead geometry, which

was attempted later on by Roberts and Wells [2]. Later on, a

considerable amount of work have been carried out on analyti-

calmodeling of welding process by various investigators. In this

connection, the work of Bhadeshia [3] is worth mentioning. A

∗ Corresponding author. Tel.: +91 3222 282992; fax: +91 3222 282278.

E-mail addresses: pdutta [email protected] (P. Dutta),

[email protected](D.K. Pratihar).

model was developed by Bhadeshiaet al. [4] to study theprocess

of micro-structure formation in low-alloy steel weld deposits.

Svensson et al. [5] carried out an analysis of cooling curves for

the fusion zone of steel weld deposits. The cooling curves were

obtained for a wide range of welding current, voltage, speed and

inter-pass temperature. Moreover, Bhadeshia [6] developed the

model of phase transformations and micro-structure formation

in steel welds. Two-dimensional axi-symmetric finite element

analysis of conduction heat flow in laser spot formation was

done by De et al. [7]. They could also predict the cooling rate

and micro-structure formation in laser spot welds [8]. It might

be difficult to model a complicated process like welding analyti-

cally. Realizing this fact, several attempts were made by various

investigators to model the welding process by using some con-

ventional regression analysis approaches. Both the linearas well

as non-linear conventional regression analyzes had been car-

ried out in the past, based on the experimental data collected

in a particular fashion (e.g., full factorial design of experiments,

fractional factorial design of experiments). Some of these works

are mentioned below. Yang et al. [9] used a non-linear regres-

sion analysis for modeling a submerged arc welding process.

Murugan et al. [10] utilized a response surface methodology to

0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.jmatprotec.2006.11.004

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58 P. Dutta, D.K. Pratihar / Journal of Materials Processing Technology 184 (2007) 56–68

Table 2

Data (as per full factorial DOE) used to carry out regression analysis

S. no. Treatment combination Level of the factors, A, B, C , D, E Responses Y values

FH (mm) FW (mm) BH (mm) BW

1 1 − − − − − −0.149 6.090 0.672 5.664

2 A + − − − − 0.357 4.982 0.001 2.255

3 B − + − − − 0.155 6.676 0.743 5.960

4 C − − + − − −0.179 7.432 0.593 7.058

5 D − − − + − 0.027 6.411 0.412 5.197

6 E − − − − + −0.599 11.348 0.805 11.679

7 AB + + − − − 0.390 4.780 0.062 1.330

8 AC + − + − − 0.088 5.020 0.281 3.302

9 AD + − − + − 0.168 4.898 0.277 2.998

10 AE + − − − + −0.217 6.092 0.359 6.419

11 BC − + + − − −0.129 7.009 0.878 6.989

12 BD − + − + − 0.099 6.824 0.803 5.732

13 BE − + − − + −0.232 9.338 0.866 10.611

14 CD − − + + − −0.306 7.287 0.630 6.895

15 CE − − + − + −0.254 11.237 0.470 12.000

16 DE − − − + + −0.745 11.491 1.100 11.848

17 ABC + + + − − 0.380 5.231 0.397 2.817

18 ABD + + − + − 0.487 4.992 0.139 1.600

19 ABE + + − − + −0.010 6.396 0.536 6.197

20 ACD + − + + − 0.090 4.423 0.420 3.172

21 ACE + − + − + −0.249 7.719 0.492 7.706

22 ADE + − − + + −0.339 7.335 0.619 7.520

23 BCD − + + + − −0.077 7.460 0.820 7.809

24 BCE − + + − + −0.623 11.767 1.128 12.860

25 BDE − + − + + −0.557 12.348 1.139 12.403

26 CDE − − + + + −0.683 12.946 0.945 13.921

27 ABCD + + + + − 0.394 5.337 0.378 3.041

28 ABCE + + + − + −0.201 7.052 0.658 7.480

29 ABDE + + − + + 0.074 6.863 0.484 6.072

30 ACDE + − + + + −0.396 7.633 0.458 7.601

31 BCDE − + + + + −0.617 12.533 1.084 13.346

32 ABCDE + + + + + −0.358 7.759 0.798 7.917

variables. This table has been prepared based on the data avail-

able in theliterature [16]. It is important tomention that therepli-

cate values (repeatability) of the responses for each of the input

combinations are not available in the above literature. It is also

important to mention that another set of 36 data (refer to Table

3) have been collected from the above literature [16], for the

purpose of testing the models. It is important to mention that the

above testcases wereobtainedthrough the realexperiments [16].

3. Modeling of the process

To determine input–output relationships in the TIG welding

process, both conventional regression analysis as well as neuralnetwork (NN)-based approaches have been developed, which

are explained below.

3.1. Conventional regression analysis

The following steps are to be considered, to carry out regres-

sion analysis for the purpose of developing each response

equation as a function of several input process parameters:

• identification of input and output variables of a process,• determination of the range for each variable,• selection of a design of experiments, such as full factorial

design and fractional factorial design,

• data collection by conducting the experiment,• developing mathematical model to derive the response equa-

tions,• conducting significance test, to check the contributions of

main factors and their interaction terms,• checking the model adequacy,• performance testing of the model using test cases.

To determine a response equation, a conventional linear

regression model (considering the main factors and their inter-

action terms) can be considered, as given below:

Y = b0 + b1X1 + b2X2 + b3X3 + b4X4 + b5X5

+ b12X1X2 + b13X1X3 + b14X1X4 + b15X1X5

+ b23X2X3 + b24X2X4 + b25X2X5 + b34X3X4

+ b35X3X5 + b45X4X5 + b123X1X2X3

+ b124X1X2X4 + b125X1X2X5 + b134X1X3X4

+ b135X1X3X5 + b145X1X4X5 + b234X2X3X4

+ b235X2X3X5 + b245X2X4X5 + b345X3X4X5

+ b1234X1X2X3X4 + b1235X1X2X3X5

+ b1245X1X2X4X5 + B1345X1X3X4X5

+ b2345

X2X

3X

4X

5+ b

12345X

1X

2X

3X

4X

5,

(1)



P. Dutta, D.K. Pratihar / Journal of Materials Processing Technology 184 (2007) 56–68 59

Table 3

Thirty-six sets of test data

S. no. Inputs Outputs

A (cm/min) B (cm/min) C (%) D (mm) E (A) FH (mm) FW (mm) BH (mm) BW (mm)

1 24 2 30 2.4 80 −0.066 6.123 0.801 5.541

2 24 2 30 3.2 80 0.114 5.979 0.682 4.633

3 24 2 70 2.4 80 −0.213 7.424 0.806 7.026

4 24 2 70 3.2 80 0.034 7.516 0.557 7.480

5 24 2 30 2.4 95 −0.167 8.481 0.713 8.340

6 24 2 30 3.2 95 −0.296 8.928 0.807 8.640

7 24 2 70 2.4 95 −0.219 9.677 0.688 9.717

8 24 2 70 3.2 95 −0.448 10.523 1.005 11.088

9 24 2 30 2.4 110 −0.281 10.871 0.713 11.142

10 24 2 30 3.2 110 −0.452 10.830 0.803 11.370

11 24 2 70 2.4 110 −0.651 13.986 1.090 14.146

12 24 2 70 3.2 110 −0.74 12.273 1.148 12.712

13 35 2 30 2.4 80 0.144 5.474 0.425 5.057

14 35 2 30 3.2 80 0.224 5.449 0.379 3.884

15 35 2 70 2.4 80 0.023 5.758 0.515 4.970

16 35 2 70 3.2 80 0.041 5.758 0.540 4.768

17 35 2 30 2.4 95 −0.094 6.665 0.613 6.304

18 35 2 30 3.2 95 −0.154 7.402 0.564 7.440

19 35 2 70 2.4 95 −0.179 7.614 0.610 7.557

20 35 2 70 3.2 95 −0.005 7.506 0.457 7.310

21 35 2 30 2.4 110 −0.433 8.011 0.868 8.047

22 35 2 30 3.2 110 −0.449 8.473 0.780 8.466

23 35 2 70 2.4 110 −0.396 9.652 0.782 10.277

24 35 2 70 3.2 110 −0.553 9.773 0.847 10.427

25 46 2 30 2.4 80 0.454 5.581 0.315 3.046

26 46 2 30 3.2 80 0.193 4.645 0.332 2.810

27 46 2 70 2.4 80 0.023 5.646 0.584 4.034

28 46 2 70 3.2 80 0.219 5.538 0.363 2.857

29 46 2 30 2.4 95 0.057 5.600 0.495 4.836

30 46 2 30 3.2 95 0.155 6.002 0.351 4.922

31 46 2 70 2.4 95 −0.189 5.859 0.729 5.201

32 46 2 70 3.2 95 −0.182 6.124 0.569 5.299

33 46 2 30 2.4 110 −0.368 6.927 0.748 6.775

34 46 2 30 3.2 110 −0.154 6.877 0.539 6.335

35 46 2 70 2.4 110 −0.35 7.630 0.650 7.869

36 46 2 70 3.2 110 −0.225 7.553 0.557 7.707

where Xis represent the coded values of the input variables, Y

indicates the response, b0, b1, . . . , b12345 represent the coeffi-

cients, whose values are to be determined using the least square

technique. The above response equation expressed in terms of

the coded values of the variables is to be written in terms of

the actual values of the variables. The relationship between the

codedandactual values ofa variablecan beexpressedas follows:

coded value = actual value−

average valueaverage variation level

.

3.2. Back-propagation neural network [21]

The proposed architecture of neural network (NN) consists

of three layers—input layer, hidden layer and output layer. Fig.

3 shows the schematic diagram of the NN used to model the

TIG welding process. Five inputs (i.e., J = 5), namely welding

speed,wire feed rate, % cleaning,gapandwelding current, were

fed to the network. The hidden layer contains K neurons, a suit-

able value of K is to be selected through a careful parametric

study. The output layer consists of four neurons (i.e., M = 4),

Fig. 3. A schematicdiagram showing theneural network used to model theTIG

welding process.




Fig. 4. A schematic diagram showing a specific neuron in each layer.

Fig. 5. Flowchart of the GA-NN system.

to represent four outputs, viz. FH, FW, BH and BW. Transfer

functions of all the neurons were assumed to be tan-sigmoidal

and a fixed value of bias b had been added to all the neurons.

The connecting weights between the input and hidden layerswere represented by [V ] and those between the hidden and out-

put layers were denoted by [W ]. The values of [V ] and [W ] lie

in the range of −1.0 to 1.0, whose initial values were generated

at random. Computation involved in both the forward as well as

backward directions is explained below, in brief.

3.2.1. Forward step calculations

Fig. 4 shows jth, k th and mth neurons lying on the input,

hidden and output layers, respectively.

The following steps are to be considered to carry out compu-

tation in the forward direction.

• Step 1: Input layer computation: The output of jth neuron

lying on the input layer can be determined as follows:

OIj =ea1(I Ij+b)

− e−a1(I Ij+b)

ea1(I Ij+b)+ e−a1(I Ij+b)

, (2)

where a1 is the constant of transfer function, b represents the

bias value and I Ij indicates input of jth neuron lying on the

input layer.• Step 2: Hidden layer computation: The input of k th neuron

lying on the hidden layer can be computed as follows:

I Hk =

J j=1

OIjvjk + b, (3)

where vjk indicates the connectingweight between jth neuron

of input layer and k th neuron of hidden layer. Now, the output

of k th hidden neuron can be determined like the following:

OHk =ea2(I Hk+b)

− e−a2(I Hk+b)

ea2(I Hk+b) + e−a2(I Hk+b), (4)


bias value and I Hk indicates input of k th neuron lying on the

hidden layer.• Step 3: Output layer computation: The input of mth neuron

lying on the output layer can be calculated as follows:

I Om =

Kk=1

OHkwkm + b, (5)

where wkm indicates the connecting weight between k th neu-

ron ofhidden layer and mth neuronof outputlayer. Theoutput

of mth neuron can be calculated like the following:

OOm =ea3(I Om+b)

− e−a3(I Om+b)

ea3(I Om+b) + e−a3(I Om+b), (6)


bias value and I Om indicates input of mth neuron lying on the

output layer.

This completes the forward step calculations.

3.2.2. Backward step calculations

A batch mode of training has been adopted in the present

work.Meansquared deviation (MSD denoted by E ) in prediction

Fig. 6. Pareto-chart and normal probability plot for the response—FH.




was calculated as follows:

E =1

S

S s=1

1

M

M m=1

1

2(T sOm − Os

Om)2, (7)

where S indicates the number of training cases, M represents

the number of outputs, T s

Om is the target output of mth neuronlying on the output layer corresponding to sth training case, and

OsOm represents the calculated output of mth neuron lying on the

output layer corresponding to sth training case.

The connecting weights [V ] and [W ] were updated to reduce

the error E , as explained below.

The change in w, i.e., w at t th iteration was calculated like

the following:

wkm(t ) = −η

∂E

∂wkm(t ) + αwkm(t − 1), (8)

Fig. 7. Results of the parametric study to determine the optimal NN.




Fig. 7. (Continued ).

where η and α denote the learning rate and momentum constant,

respectively. Now, ∂E/∂wkm can be determined by using the

chain rule of differentiation as given below:

∂E

∂wkm

=∂E

∂Es

∂Es

∂Em

∂Em

∂OOm

∂OOm

∂I Om

∂I Om

∂wkm

. (9)

Similarly, the change in v, i.e., v at t th iteration was deter-

mined as follows:

vjk(t ) = −η∂E

∂vjk(t ) + αvjk(t − 1), (10)




where ∂E/∂vjk = (1/M )M

m=1(∂Em/∂vjk). Now, ∂Em/∂vjkcan be calculated as follows:

∂Em

∂vjk=

∂Em

∂Ems

∂Ems

∂OOm

∂OOm

∂I Om

∂I Om

∂OHk

∂OHk

∂I Hk

∂I Hk

∂vjk. (11)

3.3. Genetic-neural system (GA-NN)

In genetic-neural (GA-NN) system [22], an optimal network

was evolved by minimizing the deviation in prediction of the

outputs, using a genetic algorithm (GA) (in place of a steepest

descent method utilized in BPNN). A GA-string carries infor-

mation of the NN, such as weight values—[V ], [W ], coefficients

of transfer functionsused in input,hiddenandoutput layers, bias

value, which will look as follows:

1001101011 v1,1

· · · 0100110101 v5,K

1100101001 w1,1

· · · 0110110100 wK,4

× 110 . . . 1

a1

001 . . . 0

a2

010 . . . 0

a3

100 . . . 1

b

where K indicates the number of neurons lying in the hidden

layer. There are 5K and 4K values of [V ] and [W ], respectively;

three values of coefficients of the transfer functions (a1, a2 and

a3) and one bias value (i.e., b). It is to be noted that 10-bits were

used to represent each of the above real variables.

Thus, a population of GA-strings represents a numberof can-

didateNNs(whosenumber equals to the population size). Fig. 5

shows the schematic diagram of the developed GA-NN system.

As the batch mode of training was adopted, the whole train-

ing set was passed through a neural network represented by a

GA-string. The fitness f of a GA-string was made equal to the

mean squared deviation E in prediction of the responses, and

was represented by using the following expression:

E =1

S

S s=1

1

M

M m=1

1

2(T sOm − Os

Om)2, (12)

where the symbols carry the same meaning as described earlier.The fitness values were determined for all the strings lying in the

GA-population. The population of GA-strings (i.e., neural net-

works) was then modified by using the operators reproduction,

crossover and mutation.

4. Results and discussion

Results of the conventional regression analysis, back-

propagation neural network and genetic-neural system, used to

establish input–output relationships in TIG welding process, are

shown and discussed below.

Conventionalregression analysis wascarried outby using thedata collected as per full factorial DOE (refer to Table 2). Four

different responses (i.e., FH, FW, BH, BW) were found to be as

follows, in the un-coded form:

FH = −17.2504 + 0.62018A + 4.6762B + 0.086647C

+ 7.4479D + 0.043108E − 0.18695AB

− 0.005792AC − 0.22099AD − 0.0029123AE

+ 0.0018129BC − 1.8396BD + 0.019139BE

− 0.058577CD + 0.0017885CE − 0.035219DE

+ 0.0014061ABC + 0.062296ABD

Fig. 8. Results of GA-parametric study.




+ 0.00020568ABE + 0.0022313ACD − 6.76

× 10−6ACE + 0.0011409ADE + 0.0060975BCD

− 0.0013628BCE − 0.0030BDE − 0.00027533CDE

− 0.00042377ABCD + 0.0189 × 10−3ABCE

− 0.0459 × 10−3ABDE − 8.76 × 10−7ACDE

−

0.00037623BCDE−

6.87×

10

−6

ABCDE, (13)

FW = −329.6758 + 8.2539A + 167.1041B + 5.8187C

+ 101.4624D + 3.9953E − 4.0707AB

− 0.14141AC − 2.5489AD − 0.099144AE

− 2.9150BC − 54.1378BD − 1.9883BE

− 1.8510CD − 0.068644CE − 1.2150DE

+ 0.069989ABC + 1.3175ABD + 0.048568ABE

+ 0.044112ACD + 0.0016977ACE + 0.03081ADE

+ 0.93986BCD + 0.034547BCE + 0.6524BDE

+ 0.022294CDE − 0.02226ABCD − 0.83924

× 10−3ABCE − 0.015943ABDE − 0.54259

× 10−3ACDE − 0.011294BCDE + 0.27183

× 10−3ABCDE, (14)

BH = 20.7999 − 0.38305A − 3.5745B + 0.10795C

− 9.3284D − 0.092436E + 0.0058295AB

− 0.0054309AC + 0.16652AD + 0.00049091AE

− 0.11145BC + 2.2936BD − 0.016822BE

− 0.009152CD − 0.0036834CE + 0.057568DE

+ 0.0044163ABC − 0.023731ABD+ 0.0014568ABE + 0.0015578ACD

+ 0.00012402ACE − 0.00055232ADE

+ 0.026228BCD + 0.0023696BCE − 0.0041BDE

+ 0.00095CDE − 0.0013485ABCD − 0.0769

× 10−3ABCE − 0.00031487ABDE − 3.93

× 10−5ACDE − 0.00068428BCDE + 0.0250

× 10−3ABCDE, (15)

BW = −179.4354 + 4.1209A + 104.7708B + 4.1113C

+ 52.8753D + 2.4368E − 2.5474AB − 0.094617AC

− 1.2695AD − 0.057292AE − 2.2272BC

− 34.1677BD − 1.2973BE − 1.3856CD

Fig. 9. Scatter plots to study the target vs. predicted values of different responses.




− 0.050824CE − 0.71979DE + 0.052044ABC

+ 0.81877ABD + 0.032125ABE + 0.031846ACD

+ 0.0012161ACE + 0.01818ADE + 0.76844BCD

+ 0.026839BCE + 0.4353BDE + 0.017557CDE

− 0.017848ABCD − 0.64265 × 10−3ABCE

−

0.010785ABDE−

0.00042119ACDE− 0.0093502BCDE + 0.22457 × 10−3ABCDE,

(16)

where A, B, C , D and E represent different input parameters,

namely welding speed, wire feed rate, cleaning percentage, arc

gap and welding current, respectively, in the un-coded form.

Significance tests were carried outforeach response,to deter-

mine the significant terms by using both the Pareto-chart as well

as normal probability plot. Fig. 6 shows the Pareto-chart and

normal probability plot for the response FH.

It is interesting to note that the main factors A, B, C , E have

significant influence on FH.Similarly, the response FW depends

mainly on the main factors A, C , D, E and the interaction termAE. The main factors A, B, E were found to be significant for

the response BH. The response BW are dependent mainly on

A, C , E and AE . It is also to be noted that ANOVA (analysis of

variance) could not be carried out, to check the adequacy of the

model, as replicates of the responses were not available in the

literature [16], for each combination of input process parame-

ters. The performance of conventional regression analysis was

tested using some test cases, which is discussed at the end of

this section.

Duringthelearningof neuralnetworks, a batch mode of train-

ing had been adopted. As the set of 32 input–output data (with

the help of which the regression analysis had been carried out)

might not be sufficient to provide proper training to thenetwork,

one thousand input–output data had been generated at random

by using the above regression equations. Results of the neural

network-based approaches developed to model the input–output

relationships in TIG welding process are stated and explained

below.

The performance of a back-propagation neural network

(BPNN)dependsonthequalityandquantityof data used intrain-

ing. It is also dependent on its architecture, connecting weights,

learning rate, momentum constant, coefficients of transfer func-

tions (TFs), bias value. To determine an optimal set of the above

parameters, a study was carried out, by varying oneparameter at

a time and keeping the other parameters unaltered. Fig. 7 shows

the results of the parametric study conducted to determine the

optimal network. In the first stage, the number of neurons to

be lying in the hidden layer was varied in the range of 5–30,

keeping the other parameters, viz. learning rate η, momentumconstant α, coefficient of transfer function of the input neurons

a1, coefficient of transfer function of the hidden neurons a2,

coefficient of transfer function of the output neurons a3, and

bias value fixed to 0.5, 0.5, 1.0, 1.0, 1.0 and 0.0005, respec-

tively. It is interesting to note that the NN with 22 neurons

lying in its hidden layer showed the best performance in terms

of mean squared deviation in prediction. Thus, in the second

stage and on-wards, the number of hidden neurons was kept

fixed to 22. In the similar way, the optimal/near-optimal values

Fig. 10. Deviation in prediction of different responses using different approaches.




of η, α, a1, a2, a3 and bias value were determined in stages.

As the parameters were determined in stages (but not simulta-

neously), there is no guarantee that the obtained network will

be globally optimal but it could be a near-optimal one. The

parameters of the optimal network were found to be as follows:

No. of neurons in the hidden layer 22Learning rate, η 0.35

Momentum constant, α 0.40

Coefficient of TF of the input neurons 0.15

Coefficient of TF of the hidden neurons 1.6

Coefficient of TF of the output neurons 0.90

Bias value 0.0005

In the proposed genetic-neural (GA-NN) system, the num-

ber of hidden neurons had been kept fixed to 22 (as found to be

optimal in the previous approach on BPNN). The other param-

eters, like the coefficients of TFs of the neurons lying on the

input, hidden and output layers and the bias value were varied in

the ranges (0.01, 2.0), (0.0001, 0.001), respectively, during opti-

mization (training). It is to be noted that the ranges of the above

parameters were decided after carrying out a careful study with

different values of them. As the performance of a GA depends

on its parameters, such as crossover probability (pc), mutation

probability (pm) and population size, an extensive study was

conducted to determine the optimal set of GA-parameters. A

uniform crossover scheme with a probability of 0.5 was used

and the optimal values of other GA-parameters were decided

through a careful study. The probability of mutation pm was

varied in the range of 0.1/L to 1.0/L (where L indicates the

string length and here it is equal to 2020), after keeping the

Fig. 11. Comparisons of three approaches in terms of % deviation in prediction of different responses.




Table 4

The values of average absolute % deviation in prediction of different responses obtained by regression analysis, BPNN and GA-NN

Approach FH FH (after removing some bad cases) FW BH BW

Regression analysis 544.6 144.1 25.1 34.1 36.6

BPNN 168.7 94.8 14.1 15.1 16.2

GA-NN 137.2 56.9 11.1 14.3 14.8

other parameters crossover probability pc and population size

fixed to 0.5 and 180, respectively. The best performance was

observed with pm = 1.0/2020. In the second stage of the study,

experiments were conducted with different values of population

size starting from 50 to 520 and pc and pm were kept fixed to

0.5 and 1.0/2020, respectively. Results of the GA-parametric

study are shown in Fig. 8. The following GA-parameters were

found to give the best performance in terms of the mean squared

deviation in prediction of the responses.

Crossover probability, pc 0.5

Mutation probability, pm 1/2020Population size 500

Results of the conventional regression analysis and two NN-

based approaches were compared using the36 randomtest cases

(refer to Table 3), with the help of scatter plots (refer to Fig. 9)

and in terms of deviation (i.e., the difference between the target

and predicted values of a response) in prediction of different

responses—FH, FW, BH and BW (refer to Fig. 10). It is to be

noted that the said test cases were collected through the real

experiments by Juang et al. [16]. Fig. 11(a) shows a compari-

son of the above three approaches in terms of % deviation in

prediction of FH. Both the NN-based approaches were found

to perform better than the conventional regression analysis for

this response. Moreover, GA-NN showed a slightly better per-

formance compared to that of BPNN. A large amount of %

deviation in prediction had been recorded for 4th, 20th, 27th test

casesandthus thosewere removedandcomparisonsof theabove

approaches are shown again in Fig. 11(b). The large amount of

% deviation values for these test cases could be due to the errors

in experiment and measurement. The above three approaches

were compared in terms of % deviation in prediction of weld

bead front width, as shown in Fig. 11(c). Similar comparisons of

the above three approaches are made in Fig. 11(d), with respect

to % deviation in prediction of bead back height. The values of

% deviation in prediction of bead back width, as obtained byusing the above three approaches, are compared in Fig. 11(e).

The values of average absolute % deviation in prediction of

different responses obtained by using different approaches viz.

conventional regression analysis, BPNN and GA-NN are shown

in Table 4. From the above study, it has been observed that both

theNN-basedapproaches outperformed theconventionalregres-

sion analysis andGA-NN approach was seen to perform slightly

better than the BPNN. The NN-based approaches were able to

bring adaptability, which was absent in conventional regression

analysis. It happened due to the fact that the errors generated for

1000 training data were averaged out during the training of the

NNs. Thus, the NN-based approaches were found to be more

Fig. 12. Comparisons of BPNN and GANN in terms of search speed.

adaptive compared to conventional regression analysis, for the

test cases (which are different from the data used to carry out

the regression analysis). It happened due to the fact that the NN-

based approaches might have carried out interpolation within

the specific ranges of the data. The BPNN works based on the

principle of steepest descent algorithm, which might have the

local minima problem. On the other hand, the chance of GA-

solutions getting trapped into the local minima is smaller. Thus,

GA-NN approach slightly outperformed BPNN approach for all

the responses. Fig. 12 shows the nature of search carried out

by BPNN and GA-NN, iteration-wise, to minimize the mean

squared deviation in prediction of the responses. Initially, the

BPNN showed a better performance compared to that of the

GA-NN, but after about 60,000 iterations, the latter took over.Being a gradient-based search, the BPNN was able to reduce the

deviation suddenly at the initial stage of its search and after that,

it was unable to make further improvement. It could be due to

the fact that the solutions of the steepest descent method have

reached the local minima. On the other hand, the GA through its

exhaustive search, is able to yield the global optimal solution.

However, the performances of BPNN and GA-NN will depend

on the nature of the deviation surface. If the deviation surface

is found to be uni-modal in nature, BPNN may find the optimal

solution faster than the GA-NN does. On the other hand, GA-

NN may outperform the BPNN, while carrying out search on a

multi-modal deviation surface.




5. Conclusions

To determine input–output relationships in TIG welding pro-

cess, conventional regression analysis was carried out based

on full factorial design of experiments (DOE) and two neu-

ral network-based approaches (i.e., back-propagation algorithm

and genetic-neural system) were developed. Comparisons were

made of the above approaches, after testing their performances

on 36 randomly generated test cases. From the above study,

conclusions have been made as follows.

• Both the NN-based approaches were seen to be more adap-

tive compared to the conventional regression analysis, for the

test cases. It could be due to the reason that the NN-based

approaches were able to carry out interpolation within the

ranges of the variables.• Genetic-neural (GA-NN) system outperformed the BPNN in

most of the test cases (but not all). It is so, because a GA-

based search was used in the former approach, in place of a

gradient-based search utilized in the latter approach. Beinga gradient-based search, the back-propagation algorithm had

a natural tendency to get stuck at the local minima, whereas

a more exhaustive search was carried out by the GA, in the

genetic-neural system.• BPNN showed a slightly better performance compared to the

genetic-neural system initially, but after about 60,000 itera-

tions, the latter started to perform better than the former. It

might be due to the fact that the solutions of BPNN were

still lying on the local basin of the deviation (error) function,

whereas the GA continued its search on a wider space, to

reachthe global optimal solution.However, the performances

of these two approaches were dependent on the nature of the

deviation function.

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Modeling of TIG Welding Process Using Conventional Regression Analysis and ANN Aproaches

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