A genetic algorithm approach to the optimization

INTERNATIONALMechanical Engineering and Technology (IJMET), ISSN 0976 –
International Journal of JOURNAL OF MECHANICAL ENGINEERING
6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME
AND TECHNOLOGY (IJMET)

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

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: gowdmits@gmail.com

E VENUGOPAL GOUD
Associate Professor, Department of Mechanical Engineering
G. Pullareddy Engineering college, Kurnool
1*
Corresponding author Email: gowdmits@gmail.com

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.

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

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

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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.

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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

Bead Bead
Experiment x2 x3 x4 Penetration
x1 (µs) width Volume
No. (Hz) (mm/min) (J) (mm)
(mm) (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

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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.

P e n e tr a tio n = 2 .0 6 + 0 .3 4 x1 + 0 .0 8 1 x 2 - 0 .1 1 x 3 + 0 .1 2 x 4 - 0 .1 6 x1 x 2
- 0 . 0 7 6 x 1 x 3 - 0 . 0 5 1 x 1 x 4 + 0 . 0 1 4 x 2 x 3 + 0 . 0 1 9 x 2 x 4 + 0 . 1 3 x 3 x 4 - 0 . 1 8 x 12
- 0 . 0 2 0 x 22 + 0 . 0 3 4 x 32 - 0 . 0 3 6 x 42
Eq. (1)

B e a d w id th = 0 .9 6 + 0 .0 6 0 x1 + 0 .0 2 2 x 2 - 0 .2 4 x 3 − 0 .0 4 6 x 4 + 0 .0 6 5 x1 x 2
+ 0 .1 7 x1 x 3 + 0 .0 9 1 x1 x 4 − 0 .0 5 5 x 2 x 3 − 0 .0 0 6 5 x 2 x 4 + 0 .1 5 x 3 x 4
Eq. (2)

B e a d v o lu m e = 0 .5 0 + 0 .0 1 6 x1 + 0 .1 4 x 2 - 0 .0 7 7 x 3 + 0 .0 3 0 x 4

-0 .0 0 0 7 5 x1 x 2 - 0 .0 0 3 2 5 x1 x 3 + 0 .0 3 5 x1 x 4 + 0 .1 1 x 2 x 3 + 0 .0 3 4 x 2 x 4

− 0 .0 2 2 x 3 x 4 - 0 . 0 6 3 x 12 - 0 . 1 3 x 2
2 + 0 .0 0 4 5 5 x 2
3 - 0 .0 2 8 x 2
4

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

B e a d v o lu m e = 0 .5 0 + 0 .0 1 6 x1 + 0 .1 4 x 2 - 0 .0 7 7 x 3 + 0 .0 3 0 x 4

-0 .0 0 0 7 5 x1 x 2 - 0 .0 0 3 2 5 x1 x 3 + 0 .0 3 5 x1 x 4 + 0 .1 1 x 2 x 3 + 0 .0 3 4 x 2 x 4
2 2 2 2
− 0 .0 2 2 x 3 x 4 - 0 .0 6 3 x 1 - 0 .1 3 x 2 + 0 .0 0 4 5 5 x 3 - 0 .0 2 8 x 4

Subject to:
P e n e tr a tio n = 2 .0 6 + 0 .3 4 x1 + 0 .0 8 1 x 2 - 0 .1 1 x 3 + 0 .1 2 x 4 - 0 .1 6 x1 x 2
- 0 . 0 7 6 x 1 x 3 - 0 . 0 5 1 x 1 x 4 + 0 . 0 1 4 x 2 x 3 + 0 . 0 1 9 x 2 x 4 + 0 . 1 3 x 3 x 4 - 0 . 1 8 x 12
- 0 . 0 2 0 x 22 + 0 . 0 3 4 x 32 - 0 . 0 3 6 x 4 ≥ 2 . 2 5
2

&
B e a d w id th = 0 .9 6 + 0 .0 6 0 x1 + 0 .0 2 2 x 2 - 0 .2 4 x 3 − 0 .0 4 6 x 4 + 0 .0 6 5 x1 x 2
+ 0 .1 7 x1 x 3 + 0 .0 9 1 x1 x 4 − 0 .0 5 5 x 2 x 3 − 0 .0 0 6 5 x 2 x 4 + 0 .1 5 x 3 x 4 ≤ 0 .7

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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

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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

Expected Cumulative Random Selected string
S.No Probability
Count Probability number 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

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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

S.N Mating pool Crossover? Crossover Offspring Mutation Mutated chromosome
Chromosomes site site
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

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

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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

468


Table 5. Optimal values

x1 (Pulse x2 (Pulse x3 x4 (Pulse
Bead Bead
Variable Duration) frequency) (Welding Energy) Penetration
Width Volume
(µs) (Hz) Speed) (J) (mm/min)

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.

REFERENCES

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A genetic algorithm approach to the optimization

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