A genetic algorithm approach to the optimization

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –

6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME

459

A GENETIC ALGORITHM APPROACH TO THE OPTIMIZATION OF

PROCESS PARAMETERS IN LASER BEAM WELDING

Dr. G HARINATH GOWD1*

Professor, Department of Mechanical Engineering

Madanapalle Institute of Technology & Science, Madanaaplle

Andhra Pradesh., INDIA.

Email: [email protected]

E VENUGOPAL GOUD

Associate Professor, Department of Mechanical Engineering

G. Pullareddy Engineering college, Kurnool

1*

Corresponding author Email: [email protected] ABSTRACT

Laser beam welding (LBW) is a field of growing importance in industry with respect

to traditional welding methodologies due to lower dimension and shape distortion of

components and greater processing velocity. Because of its high weld strength to weld size

ratio, reliability and minimal heat affected zone, laser welding has become important for

varied industrial applications. LBW process is so complex in nature that the selection of

appropriate input parameters (Pulse duration, Pulse frequency, Welding speed and Pulse

energy) is not possible by the trial-and-error method. So there is a need to develop a

methodology to find the optimal process parameters in ND-YAG Laser beam welding

process thereby producing sound welded joints at a low cost. In view of this, research is

carried on INCONEL to find the optimal process parameters. Accurate prediction

mathematical models to estimate Bead width, Depth of Penetration & Bead Volume were

developed from experimental data using Response Surface Methodology (RSM). These

predicted mathematical models are used for optimization of the process. Total volume of the

weld bead, an important bead parameter, is optimized (minimized), keeping the dimensions

of the other important bead parameters as constraints, to obtain sound and superior quality

welds. As the amount of data generated in the iterative process for optimization is enormous

and each design cycle requires substantial calculations, the popular evolutionary algorithm

Genetic Algorithm is used for the optimization. In summary, the proposed methodology

enables the manufacturing engineers to compute the optimal control factor settings depending

upon the production requirements. Consequently, the process could be automated based on

the optimal settings.

INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING

AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 3, Issue 3, September - December (2012), pp. 459-470 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2012): 3.8071 (Calculated by GISI) www.jifactor.com

IJMET

© I A E M E



460

Keywords: ND-YAG Laser Beam welding, Modeling, Genetic algorithm, Optimization.

1. INTRODUCTION

Laser Beam Welding (LBW) processes is a welding technique used to join multiple

pieces of metal through the heating effect of a concentrated beam of coherent monochromatic

light. Light amplification by stimulated emission of radiation (LASER) is a mechanism

which emits electromagnetic radiation, through the process of simulated emission. Lasers

generate light energy that can be absorbed into materials and converted into heat

energy.LBW is a high-energy-density welding process and well known for its deep

penetration, high speed, small heat-affected zone, fine welding seam quality, low heat input

per unit volume, and fiber optic beam delivery [1]. The energy input in laser welding is

controlled by the combination of focused spot size, focused position, shielding gas, laser

beam power and welding speed. Because of the above advantages, LBW is widely used. For

the laser beam welding of butt joint, the parameters of joint fit-up and the laser beam to joint

alignment [2] becomes important. An inert gas, such as helium or argon, is used to protect the

weld bead from contamination, and to reduce the formation of absorbing plasma. Depending

upon the type of weld required a continuous or pulsed laser beam may be used. There are

three basic types of lasers viz., the solid state laser, the gas laser and the semi conductor laser.

Among all these variants Nd:YAG lasers are being used most extensively for industrial

applications because they are capable of durable multikilowatt operation.

The principle of operation is that the laser beam is pointed on to a joint and the beam

is moved along the joint. The process will melt the metals in to a liquid, fuse them together

and then make them solid again thereby joining the two pieces. The beam provides a

concentrated heat source, allowing for narrow, deep welds and high welding rates. The

process is frequently used in high volume applications, such as in the automotive industry. In

any welding process, bead geometrical parameters play an important role in determining the

mechanical properties of the weld and hence quality of the weld [3]. In Laser Beam welding,

bead geometrical variables are greatly influenced by the process parameters such as Pulse

frequency, Welding speed, Input energy, Shielding gas [4] and [5]. Therefore to accomplish

good quality it is imperative to setup the right welding process parameters. Quality can be

assured with embracing automated techniques for welding process. Welding automation not

only results in high quality but also results in reduced wastage, high production rates with

reduce cost to make the product. Some of the significant works in literature regard to the

modeling and optimization studies of welding are as follows: Yang performed regression

analysis to model submerged arc welding process [6]. Gunaraj and Murugan minimized weld

volume for the submerged arc welding process using an optimization tool in Matlab [7]. Bead

height, bead width and bead penetration were taken as the constraints.

The Taguchi method was utilized by Tarng and Yang to analyze the affect of welding

process parameter on the weld-bead geometry [8]. Casalino has studied the effect of welding

parameters on the weld bead geometry in laser welding using statistical and taguchi

approaches [9]. Nagesh and Datta developed a back-propagation neural network, to establish

the relationships between the process parameters and weld bead geometric parameters, in a

shielded metal arc welding process [10]. Young whan park has applied Genetic algorithms

and Neural network for process modeling and parameter optimization of aluminium laser

welding automation [11]. Mishra and Debroy showed that multiple sets of welding variables

capable of producing the target weld geometry could be determined in a realistic time frame

by coupling a real-coded GA with and neural network model for Gas Metal Arc Fillet



461

Welding [12]. Saurav data has applied RSM to modeling and optimization of the features of

bead geometry including percentage dilution in submerged arc welding using mixture of fresh

flux and fused slag [13].

The literature shows that the most dominant modeling tools used till date are Taguchi

based regression analysis and artificial neural networks. However, the accuracy and

possibility of determining the global optimum solution depends on the type of modeling

technique used to express the objective function and constraints as functions of the decision

variables. Therefore effective, efficient and economic utilization of laser welding necessitates

an accurate modeling and optimization procedure. In the present work, RSM is used for

developing the relationships between the weld bead geometry and the input variables. The

models derived by RSM are utilized for optimizing the process by using the Genetic

Algorithm.

2. EXPERIMENTAL WORK

The experiments are conducted on High peak power pulsed Nd:YAG Laser welding

system with six degrees of freedom robot delivered through 300 um Luminator fiber as

shown in Figure 1.

Fig.1. Nd:YAG Robotic Laser Beam welding equipment

In this research Butt welding of Inconel 600 is carried out at by varying the input

parameters. The size of each plate welded is 30mm long x 30mm width with thickness of

2.5mm. The laser beam is focused at the interface of the joints. An inert gas such as argon is

used to protect the weld bead from contamination, and to reduce the formation of absorbing

plasma. Based on the literature survey and the trial experiments, it was found that the process

parameters such as pulse duration (x1), pulse frequency (x2), speed (x3), and energy (x4) have

significant effect on weld bead geometrical features such as penetration (P), bead width (W),

and bead volume (V).In the present work, they are considered as the decision variables and

trial samples of butt joints are performed by varying one of the process variables to determine

the working range of each process variable. Absent of visible welding defects and at least half

depth penetrations were the criteria of choosing the working ranges.



462

After conducting the experiments as per the design matrix, for measuring the output

responses i.e bead geometry features such as Bead penetration & Bead width, welded joint is

sectioned perpendicular to the weld direction. The specimens are then prepared by the usual

metallurgical polishing methods and then etched. Then the bead dimensions were measured

using Toolmaker’s microscope. For each response the readings were measured at three

different sections of the weld joint and the average value is taken. The study is focused to

investigate the effects of process variables on the structures of the welds. An average of three

measurements taken at three different places and the output responses are recorded for each

set. The output responses recorded are shown in the Table 1.

Table 1. Experimental Observations

Experiment

No. x1 (µs)

x2

(Hz)

x3

(mm/min)

x4

(J)

Penetration

(mm)

Bead

width

(mm)

Bead

Volume

(mm3)

1 2 10 300 12 1.800 1.200 0.469

2 4 10 300 12 2.230 1.020 0.460

3 2 18 300 12 1.900 1.150 0.500

4 4 18 300 12 2.280 1.210 0.500

5 2 10 700 12 1.700 0.819 0.330

6 4 10 700 12 2.070 0.910 0.340

7 2 18 700 12 1.800 0.805 0.495

8 4 18 700 12 2.010 0.856 0.500

9 2 10 300 18 1.840 1.010 0.444

10 4 10 300 18 2.240 0.990 0.486

11 2 18 300 18 1.950 1.015 0.531

12 4 18 300 18 2.180 1.015 0.580

13 2 10 700 18 1.770 0.768 0.318

14 4 10 700 18 2.180 0.950 0.352

15 2 18 700 18 2.010 0.756 0.500

16 4 18 700 18 2.155 0.978 0.520

17 1 14 500 15 1.500 0.900 0.400

18 5 14 500 15 2.260 1.060 0.420

19 3 6 500 15 1.912 0.940 0.150

20 3 22 500 15 2.170 1.015 0.540

21 3 14 100 15 2.250 1.269 0.555

22 3 14 900 15 1.940 0.701 0.400

23 3 14 500 9 1.800 1.015 0.390

24 3 14 500 21 2.250 0.984 0.500

25 3 14 500 15 2.050 0.980 0.512

26 3 14 500 15 2.070 0.958 0.500

27 3 14 500 15 2.080 0.950 0.520

28 3 14 500 15 2.105 0.936 0.491

29 3 14 500 15 2.098 0.928 0.487

30 3 14 500 15 2.050 0.916 0.490

31 3 14 500 15 1.961 0.900 0.490



463

3. DEVELOPMENT OF EMPIRICAL MODELS The need in developing the mathematical relationships from the experimental data is

to relate the measure output responses Penetration, Bead width and Bead volume to the input

process parameters such as pulse duration (x1), pulse frequency(x2), speed (x3), and energy

(x4) thereby facilitating the optimization of the welding process. RSM is used to predict the

accurate models.

1 2 3 4 1 2

2

1 3 1 4 2 3 2 4 3 4 1

2 2 2

2 3 4

2 . 0 6 0 . 3 4 0 . 0 8 1 - 0 . 1 1 0 . 1 2 - 0 . 1 6

- 0 . 0 7 6 - 0 . 0 5 1 0 . 0 1 4 0 . 0 1 9 0 . 1 3 - 0 . 1 8

- 0 . 0 2 0 0 . 0 3 4 - 0 . 0 3 6

P e n e t r a t i o n x x x x x x

x x x x x x x x x x x

x x x

= + + +

+ + +

+

Eq. (1)

1 2 3 4 1 2

1 3 1 4 2 3 2 4 3 4

0 . 9 6 0 . 0 6 0 0 . 0 2 2 - 0 . 2 4 0 . 0 4 6 0 . 0 6 5

0 . 1 7 0 . 0 9 1 0 . 0 5 5 0 . 0 0 6 5 0 . 1 5

B e a d w i d t h x x x x x x

x x x x x x x x x x

= + + − +

+ + − − +

Eq. (2)

1 2 3 4

1 2 1 3 1 4 2 3 2 4

2 2 2 2

3 4 1 2 3 4

0 . 5 0 0 . 0 1 6 0 . 1 4 - 0 . 0 7 7 0 . 0 3 0

- 0 . 0 0 0 7 5 - 0 . 0 0 3 2 5 0 . 0 3 5 0 . 1 1 0 . 0 3 4

0 . 0 2 2 - 0 . 0 6 3 - 0 . 1 3 0 . 0 0 4 5 5 - 0 . 0 2 8

B e a d v o l u m e x x x x

x x x x x x x x x x

x x x x x x

= + + +

+ + +

− +

Eq. (3)

The developed mathematical models are checked for their adequacy using ANNOVA

and normal probability plot of residuals. Then these models are used for Optimization of

process parameters using Genetic Algorithms.

4. FORMULATION OF OPTIMIZATION PROBLEM In the present work, the bead geometrical parameters were chosen to be the

constraints and the minimization of volume of the weld bead was considered to be the

objective function. Minimizing the volume of the weld bead reduces the welding cost through

reduced heat input and energy consumption and increased welding production through a high

welding speed [14]. The present problem is formulated an optimization model as shown

below:

Minimize

1 2 3 4

1 2 1 3 1 4 2 3 2 4

2 2 2 2

3 4 1 2 3 4

0 . 5 0 0 . 0 1 6 0 . 1 4 - 0 . 0 7 7 0 . 0 3 0

- 0 . 0 0 0 7 5 - 0 . 0 0 3 2 5 0 . 0 3 5 0 . 1 1 0 . 0 3 4

0 . 0 2 2 - 0 . 0 6 3 - 0 . 1 3 0 . 0 0 4 5 5 - 0 . 0 2 8

B e a d v o l u m e x x x x

x x x x x x x x x x

x x x x x x

= + + +

+ + +

− +

Subject to:

1 2 3 4 1 2

2

1 3 1 4 2 3 2 4 3 4 1

2 2 2

2 3 4

2 . 0 6 0 . 3 4 0 . 0 8 1 - 0 . 1 1 0 . 1 2 - 0 . 1 6

- 0 . 0 7 6 - 0 . 0 5 1 0 . 0 1 4 0 . 0 1 9 0 . 1 3 - 0 . 1 8

- 0 . 0 2 0 0 . 0 3 4 - 0 . 0 3 6 2 . 2 5

P e n e t r a t i o n x x x x x x

x x x x x x x x x x x

x x x

= + + +

+ + +

+ ≥

&

1 2 3 4 1 2

1 3 1 4 2 3 2 4 3 4

0 . 9 6 0 . 0 6 0 0 . 0 2 2 - 0 . 2 4 0 . 0 4 6 0 . 0 6 5

0 . 1 7 0 . 0 9 1 0 . 0 5 5 0 . 0 0 6 5 0 . 1 5 0 . 7

B e a d w i d t h x x x x x x

x x x x x x x x x x

= + + − +

+ + − − + ≤



464

With the parameter feasible ranges:

1 µs ≤ x1 ≤ 5 µs,

6 Hz ≤ x2 ≤ 22 Hz,

100 mm/min ≤ x3 ≤ 900 mm/min,

9 J ≤ x4 ≤ 21 J

The bead parameters and the feasible ranges of the input variables were established with a

view to have defect-free welded joint.

Once the optimization problem is formulated, then it is solved using Genetic

algorithms (GA). The GA optimization module available in MATLAB is used to find out the

optimal parameters. Tables 2, 3 and 4 exhibit the implementation of GA for minimizing the

Bead volume as objective. Sample calculations are shown for one iteration of the algorithm.

The bit lengths chosen for x1, x2, x3 and x4 are chosen 4, 4, 5 and 4 respectively. As a first

step, an initial population of 40 chromosomes is generated randomly as shown in Table 2.

Chromosome strings of individual input variables are decoded and substituted to determine

the objective function value of Bead volume. From Table 2, the first string (0000 1101 11110

1101) is decoded to values equal to x1=1, x2=20, x3=874 and x4=19 using linear mapping

rule. Then the objective function value is computed which is obtained as 0.4874. The fitness

final value at this point using the transformation rule F(x(1)

) = 1.0/(1.0+0.4874) is obtained as

0.6723. This fitness function value is used in the reproduction operation of GA. Similarly,

other strings in the population are evaluated and fitness values are calculated. Table 2 shows

the objective function value and the fitness value for all the 40 strings in the initial

population.

In the next step, good strings in the population are to be selected to form the mating

pool. In this work, roulette-wheel selection procedure is used to select the good strings. As a

part of this procedure, average fitness [15] of the population is calculated by adding the

fitness values of all strings and dividing the sum by the population size and the average

fitness of the population )F(_

is obtained as 0.7772. The expected count is subsequently

calculated by dividing each fitness value with the average fitness;

_

F

)x(F

For the first string, the expected count is (0.6723/0.7772) = 0.8649. Similarly, the

expected count values are calculated for all other strings in the population and shown in

Table 3. Then, the probability of each string being copied in the mating pool can be computed

dividing the expected count values with the population size. For instance, the probability of

first string is (0.8649/40) = 0.02. Similarly, the values of probability of selection for all the

strings are calculated and cumulative probability is henceforward computed. The

probabilities of selection are listed in Table 3. Next random numbers between zero and one

are generated in order to form the mating pool.

From Table 3, random number generated for the first string is 0.30 which means the

twelfth string from the population gets a copy in the mating pool, because that string occupies

the probability interval (0.27, 0.30) as shown in the column of cumulative probability in the

Table 3. In a similar manner, other strings are selected according to the random numbers

generated in Table 3 and the complete mating pool is formed. The mating pool is displayed in



465

Table 7.14. By adopting the reproduction operator, the inferior points have been

automatically eliminated from further consideration. As a next step in the generation, the

strings in the mating pool are used for the crossover operation.

Table 2. Initial population with fitness values in GA

S.No Chromosomes x1 x2 x3 x4 Objective Fitness values

1 0000 1101 11110 1101 1 20 874 19 0.4874 0.6723

2 0100 0111 10101 1001 2 13 642 16 0.2485 0.8010

3 1101 1101 10111 1011 4 20 694 18 0.3236 0.7555

4 1010 0111 11010 0111 4 13 771 15 0.3429 0.7447

5 1110 0010 11100 0100 5 8 823 12 0.3397 0.7464

6 0111 0111 10110 1101 3 13 668 19 0.2598 0.7937

7 1100 1011 11100 1000 4 18 823 15 0.4195 0.7045

8 0110 0111 11001 1001 3 13 745 16 0.3225 0.7562

9 1111 0100 01101 0100 5 10 435 12 0.1123 0.8991

10 1011 1111 01101 0110 4 22 435 14 0.1642 0.8589

11 0110 0111 10000 0111 3 13 513 15 0.1690 0.8554

12 0011 1110 11110 1010 2 21 874 15 0.4997 0.6668

13 0101 1011 01101 1101 2 18 435 19 0.1425 0.8753

14 1001 0010 10011 0101 3 8 590 13 0.1827 0.8455

15 1001 0110 10110 0010 3 12 668 11 0.2643 0.7909

16 1010 0111 01111 1000 4 13 487 15 0.1521 0.8680

17 1110 0100 11100 0010 5 10 823 11 0.3615 0.7345

18 1001 0111 10101 1110 3 13 642 20 0.2399 0.8065

19 1100 0000 10011 0000 4 6 590 9 0.1729 0.8526

20 0111 1111 10001 0010 3 22 539 11 0.2353 0.8095

21 0010 0110 11111 0000 2 12 900 9 0.4605 0.6847

22 1101 0000 10011 0010 4 6 590 11 0.1703 0.8545

23 1010 1111 11101 0111 4 22 848 15 0.4836 0.6740

24 0110 1001 11111 1110 3 16 900 20 0.4657 0.6823

25 0001 0110 10001 0001 1 12 539 10 0.1860 0.8432

26 0100 0111 01110 1110 2 13 461 20 0.1358 0.8804

27 1111 1011 11110 1011 5 18 874 18 0.4597 0.6851

28 0100 0111 11001 0111 2 13 745 15 0.3264 0.7539

29 0000 1011 11101 1101 1 18 848 19 0.4439 0.6926

30 0101 0110 11010 0110 2 12 771 14 0.3388 0.7469

31 1100 1011 01101 0111 4 18 435 15 0.1446 0.8737

32 1001 0000 11011 0100 3 6 797 12 0.3047 0.7664

33 0011 1111 11101 0101 2 22 848 13 0.4917 0.6704

34 0010 1000 11110 0100 2 15 874 12 0.4507 0.6893

35 1111 1111 11001 0001 5 22 745 10 0.3936 0.7175

36 0011 1111 01100 1010 2 22 410 17 0.1493 0.8701

37 1111 1000 11010 0011 5 15 771 11 0.3537 0.7387

38 0111 1011 10001 1011 3 18 539 18 0.2034 0.8309

39 0101 0101 10011 0110 2 12 590 14 0.2105 0.8261

40 1110 1011 10110 0011 5 18 668 11 0.2970 0.7710



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Table 3. Selection in GA

S.No Expected

Count Probability

Cumulative

Probability

Random

number

Selected string

number

1 0.8649 0.02 0.02 0.30 12

2 1.0305 0.03 0.05 0.68 27

3 0.9720 0.02 0.07 0.91 36

4 0.9580 0.02 0.09 0.61 24

5 0.9603 0.02 0.12 0.32 13

6 1.0212 0.03 0.14 0.70 28

7 0.9063 0.02 0.17 0.73 30

8 0.9728 0.02 0.19 0.56 22

9 1.1567 0.03 0.22 0.92 37

10 1.1050 0.03 0.25 0.36 14

11 1.1005 0.03 0.27 0.25 10

12 0.8579 0.02 0.30 0.44 17

13 1.1261 0.03 0.32 0.17 7

14 1.0878 0.03 0.35 0.40 16

15 1.0176 0.03 0.38 0.38 15

16 1.1167 0.03 0.40 0.94 38

17 0.9449 0.02 0.43 0.75 30

18 1.0376 0.03 0.45 0.52 20

19 1.0969 0.03 0.48 0.47 19

20 1.0415 0.03 0.51 0.54 21

21 0.8809 0.02 0.53 0.80 32

22 1.0993 0.03 0.56 0.89 36

23 0.8672 0.02 0.58 0.85 34

24 0.8778 0.02 0.60 0.99 40

25 1.0848 0.03 0.63 0.59 24

26 1.1327 0.03 0.66 0.65 26

27 0.8814 0.02 0.68 0.62 25

28 0.9699 0.02 0.70 0.10 4

29 0.8910 0.02 0.72 0.39 16

30 0.9610 0.02 0.75 0.33 13

31 1.1240 0.03 0.78 0.86 35

32 0.9861 0.02 0.80 0.96 39

33 0.8625 0.02 0.82 0.11 5

34 0.8868 0.02 0.85 0.51 20

35 0.9231 0.02 0.87 0.08 3

36 1.1194 0.03 0.90 0.05 2

37 0.9504 0.02 0.92 0.34 14

38 1.0690 0.03 0.95 0.15 6

39 1.0628 0.03 0.97 0.37 15

40 0.9919 0.02 1.00 0.64 25



467

In the crossover operation, two strings are selected at random and crossed at a random site.

Since the mating pool contains strings at random, pairs of strings are picked up form top of

the list as shown in Table 4.

Table 4. Crossover and Mutation in GA

Thus strings 12 and 27 participate in the first crossover operation. In this work, two

point crossover [15] is adopted with the probability, Pc = 0.85 to check whether a crossover is

desired or not. To perform crossover, a random number is generated with crossover

probability (Pc) of 0.85. If the random number is less than Pc then the crossover operation is

performed, otherwise the strings are directly placed in an intermediate population for

subsequent genetic operation. When crossover is required to be performed then crossover

sites are to be decided at random by creating random numbers between (0, l-1), where l

represents the total length of the string. For Example, when crossover is required to be

performed for the strings 3, 4 two sites of crossover are to be selected randomly. Here, the

random sites are happened to be 6, 12. Thus the portions between sites 6 and 12 of the strings

3 and 4 are swapped to create the new offspring as shown in Table 4. However with the

random sites, the children strings produced may or may not have a combination of good

S.N Mating pool

Chromosomes

Crossover? Crossover

site

Offspring Mutation

site

Mutated chromosome

1 0011 1110 11110 1010 No -- 0011 1110 11110 1010 8, 13 0011 1111 11111 1010

2 1111 1011 11110 1011 No -- 1111 1011 11110 1011 6 1111 1111 11110 1011

3 0011 1111 01100 1010 Yes 6, 12 0011 1001 11110 1010 8 0011 1000 11110 1010

4 0110 1001 11111 1110 Yes 6, 12 0110 1111 01101 1110 -- 0110 1111 01101 1110

5 0101 1011 01101 1101 No -- 0101 1011 01101 1101 -- 0101 1011 01101 1101

6 0100 0111 11001 0111 No -- 0100 0111 11001 0111 8 0100 0110 11001 0111

7 0101 0110 11010 0110 No -- 0101 0110 11010 0110 -- 0101 0110 11010 0110

8 1101 0000 10011 0010 No -- 1101 0000 10011 0010 7 1101 0010 10011 0010

9 1111 1000 11010 0011 No -- 1111 1000 11010 0011 -- 1111 1000 11010 0011

10 1001 0010 10011 0101 No -- 1001 0010 10011 0101 -- 1001 0010 10011 0101

11 1011 1111 01101 0110 No -- 1011 1111 01101 0110 -- 1011 1111 01101 0110

12 1110 0100 11100 0010 No -- 1110 0100 11100 0010 -- 1110 0100 11100 0010

13 1100 1011 11100 1000 Yes 9, 11 1100 1011 01100 1000 -- 1100 1011 01100 1000

14 1010 0111 01111 1000 Yes 9, 11 1010 0111 11111 1000 -- 1010 0111 11111 1000

15 1001 0110 10110 0010 No -- 1001 0110 10110 0010 3, 7 1011 0100 10110 0010

16 0111 1011 10001 1011 No -- 0111 1011 10001 1011 4 0110 1011 10001 1011

17 0101 0110 11010 0110 Yes 15, 17 0101 0110 11010 0010 -- 0101 0110 11010 0010

18 0111 1111 10001 0010 Yes 15, 17 0111 1111 10001 0110 -- 0111 1111 10001 0110

19 1100 0000 10011 0000 No -- 1100 0000 10011 0000 -- 1100 0000 10011 0000

20 0010 0110 11111 0000 No -- 0010 0110 11111 0000 -- 0010 0110 11111 0000

21 1001 0000 11011 0100 No -- 1001 0000 11011 0100 12 1001 0000 11001 0100

22 0011 1111 01100 1010 No -- 0011 1111 01100 1010 8 0011 1110 01100 1010

23 0010 1000 11110 0100 Yes 9, 12 0010 1000 10110 0100 3, 14 0000 1000 10110 1100

24 1110 1011 10110 0011 Yes 9, 12 1110 1011 11110 0011 -- 1110 1011 11110 0011

25 0110 1001 11111 1110 No -- 0110 1001 11111 1110 17 0110 1001 11111 1111

26 0100 0111 01110 1110 No -- 0100 0111 01110 1110 -- 0100 0111 01110 1110

27 0001 0110 10001 0001 No -- 0001 0110 10001 0001 -- 0001 0110 10001 0001

28 1010 0111 11010 0111 No -- 1010 0111 11010 0111 -- 1010 0111 11010 0111

29 1010 0111 01111 1000 No -- 1010 0111 01111 1000 1, 13 0010 0111 01110 1000

30 0101 1011 01101 1101 No -- 0101 1011 01101 1101 12 0101 1011 01111 1101

31 1111 1111 11001 0001 No -- 1111 1111 11001 0001 -- 1111 1111 11001 0001

32 0101 0101 10011 0110 No -- 0101 0101 10011 0110 -- 0101 0101 10011 0110

33 1110 0010 11100 0100 Yes 12, 17 1110 0010 11101 0010 -- 1110 0010 11101 0010

34 0111 1111 10001 0010 Yes 12, 17 0111 1111 10000 0100 -- 0111 1111 10000 0100

35 1101 1101 10111 1011 No -- 1101 1101 10111 1011 13 1101 1101 10110 1011

36 0100 0111 10101 1001 No -- 0100 0111 10101 1001 -- 0100 0111 10101 1001

37 1001 0010 10011 0101 No -- 1001 0010 10011 0101 17 1001 0010 10011 0100

38 0111 0111 10110 1101 No -- 0111 0111 10110 1101 -- 0111 0111 10110 1101

39 1001 0110 10110 0010 Yes 4, 6 1001 0110 10110 0010 -- 1001 0110 10110 0010

40 0001 0110 10001 0001 Yes 4, 6 0001 0110 10001 0001 8, 13 0001 0111 10000 0001



468

strings from parent strings, depending on whether or not the crossing sites fall in the

appropriate locations. If good strings are not created by crossover, they will not survive too

long because reproduction will select against those chromosomes in subsequent generation.

In order to preserve some of the good chromosomes that are already present in the mating

pool, all the chromosomes are not used in crossover operation. When a crossover probability

Pc is used, the expected number of strings that will be subjected to crossover is only 100Pc

and the remaining percent of the population remains as they are in the current population. The

calculations of intermediate population are shown in the Table 4. The crossover is mainly

responsible for the creation of new strings.

The third operator, mutation, is then applied on the intermediate population. Mutation

is basically intended for local search around the current solution. Bit-wise mutation is

performed with a probability, Pm = 0.10. A random number is generated with Pm; if random

number is less than Pm then the bit is altered form 1 to 0 or 0 to 1 depending on the bit value

otherwise no action is taken. Mutation is implemented with the probability, Pm=0.10 as

shown in Table 4. The procedure is repeated for all the strings in the intermediate population.

This completes one iteration of the GA. The above procedure is continued until the maximum

number of generations is completed. For better convergence of the present problem, the

Genetic algorithm is run for 120 generations. GA narrows down the search space as the

search progresses and the algorithm is converged to the objective function value of 0.2688.

The convergence graph is displayed in Fig.2 and the optimal values of the control factors are

listed in Table 5.

The following inference discusses the performance of proposed methodology: From

the experimental observations presented in the Table 1, the 10th

experiment resulted for 0.486

for Bead volume and Bead penetration as 2.24. After optimization using GA, it is observed

from Table 5, that Bead volume can be decreased to 0.2688 (by 55 %) for the same Bead

penetration.

Fig. 2. Convergence graph for minimization of Bead volume



469

Table 5. Optimal values

Variable

x1 (Pulse

Duration)

(µs)

x2 (Pulse

frequency)

(Hz)

x3

(Welding

Speed) (J)

x4 (Pulse

Energy)

(mm/min)

Penetration Bead

Width

Bead

Volume

Value 3.929 6.31 761.444 15.932 2.24 0.7 0.2688

5. CONCLUSIONS

In the present study, Design based experiments and analysis have been carried out in

order to optimize the bead volume considering the effects of bead geometrical parameters

like: Bead penetration and Bead width in butt welding of INCONEL 600 using ND:YAG

Laser beam welding setup. First Experiments were carried out by as per Central Composite

Rotatable factorial design to substantially reduce the number of experiments. Then RSM is

used to develop second order polynomial models between the bead geometrical parameters:

Bead volume, Bead width, Depth of penetration and the chosen control variables: Pulse

duration, Pulse frequency, Welding speed and the Pulse energy. Later A constrained

optimization problem is then formulated to minimize the bead volume subject to the bead

width and bead penetration as constraints. A binary coded Genetic algorithm was used to

solve the above said problem. The genetic algorithm was able to reach near the globally

optimal solution, after satisfying the above constraints. The optimal values obtained by the

proposed methodology could serve as a ready reckoner to conduct the welding operations

with great ease to achieve the quality and the production rate demanded by the consumers. In

summary, the proposed work enables the manufacturing engineers to select the optimal

values depending on the production requirements and as a consequence, automation of the

process could be done based on the optimal values.

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A genetic algorithm approach to the optimization