1. Kedar Nath DasKedar Nath Das
Hybrid Binary Coded GA for
Constrained Optimization
NIT SILCHAR, ASSAM,
INDIA

2. MOST GENERAL OPTIMIZATION PROBLEM
Minimize (Maximize) f (X),
where
s.t. X∈S ⊆ , where S is defined
by
( )nxxxX ...,,2,1=
RRf n
→:
n
R
.,........,2,1
,.......,2,10)(
;,......,2,10)(
niforbxa
ljforxg
mkforxh
iii
j
k
=≤≤
=≥
==

3. DETERMINISTIC
APPROACH
To Find the Global Optimal Solution
PROBABILISTIC
APPROACH
1. Genetic Algorithm
2. Memetic Algorithm
3. Random Search Methods
4. Tabu Search
5. Ant Colony Optimization
6. Particle Swarm Optimization,
etc…..
Approaches
Many

4. Working Principle of GA
 Encoding
 Selection
 Crossover
 Mutation
 Elitism (Opt.)
Repetition

5. a) Roulette Wheel
Selection
USED GA OPERATORS
b) Tournament
Selection
Mating
pool
30
23
37
24
38
26
37
20
23
24
20 26

6. BEFORE CROSS-OVER AFTER CROSS-OVER
101011
=s
001012
=s
001011
=
′
s
101012
=
′
s
c) One Point Cross-Over
d) Uniform Cross-Over
BEFORE CROSS-OVER AFTER CROSS-OVER
001002
=s
100011
=s
100012
=
′
s
001001
=
′
s

7. BEFORE MUTATION AFTER MUTATION
1 0 0 1 0 1 1 0 1 0 0 0 0 1 1 0
e) Bit-Wise Mutation
f) Elitism
12
17
18
2
45
2
12
8
20
41
2
2
8
12
20
Bigin of a
GA cycle End of the
GA cycle
Process of
Elitism
After
Mutation

8. Quadratic Approximation
(Hybridization)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 







−+−+−
−+−+−
321213132
3
2
2
2
12
2
1
2
31
2
3
2
2
)()(
)()(
RfRRRfRRRfRR
RfRRRfRRRfRR
 Find the point of minima (child) of the quadratic
surface passing through R1, R2 and R3 defined as:
Child = 0.5*
 Select the individuals R1, with the best fitness value.
Choose two random individuals R2 and R3.

9. Selection StrategySelection Strategy
forfor
ConstrainedConstrained
OptimizationOptimization

10. (A) Selection Strategy for Mating(A) Selection Strategy for Mating
PoolPool
• Roulette Wheel
Selection
• Penalty Parameter:
• Fitness:
where

11. (B) Selection Strategy for Best(B) Selection Strategy for Best
Individuals in a population:Individuals in a population:
Tournament Selection

12. 1
2
3
The feasible solution
5
6
1 4
5
x1
x2 The feasible domain
1
2
3
The feasible solution
5
6
1 4
5
x1
x2 The feasible domain
1
2
3
The feasible solution
5
6
1 4
5
x1
x2 The feasible domain
1
2
3
The feasible solution
5
6
1 4
5
1
2
3
The feasible solution
5
6
1 4
5
The feasible solution
5
6
1 4
5
x1
x2 The feasible domain
4

13. Step 1: Begin with a random population (P) of size 10*N
Step 2: Evaluation fitness of P(t)
Step3: Stop if it satisfies the stopping criteria
Step 4: Select the individuals taking the tournament
selection strategy
Step 5: Apply Single Point Crossover
Step 6: Apply Bitwise Mutation
Step 7: Hybridize with Quadratic Approximation
Step 8: Apply Complete Elitism through tournament
selection
Methodology of HBGA-Methodology of HBGA-
CC

14. BGA: Pc – Pm
performance
Pm
Pc
0.6 0.7 0.8 0.9
0.001 0.6208 0.6399 0.6293 0.6396
0.005 0.7962 0.8381 0.8111 0.8259
0.009 0.8407 0.8854 0.8613 0.8914
0.013 0.8757 0.8722 0.8887 0.9065

15. HBGA: Pc – Pm
performance
Pm
Pc
0.6 0.7 0.8 0.9
0.001 0.8000 0.8455 0.8426 0.8258
0.005 0.9175 0.9149 0.8896 0.8761
0.009 0.9030 0.9062 0.9144 0.8960
0.013 0.8864 0.8974 0.9095 0.8944

16. 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pc=0.6 Pc=0.7 Pc=0.8 Pc=0.9
Performance
Pm=0.001 Pm=0.005
Pm=0.009 Pm=0.013
Finetune for BGA-C

17. 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pc=0.6 Pc=0.7 Pc=0.8 Pc=0.9
Performance
Pm=0.001 Pm=0.005
Pm=0.009 Pm=0.013
Finetune for HBGA-C

18. Recommended Values
for Pc and Pm:
Method Pc Pm
BGA-C 0.9 0.013
HBGA-C 0.6 0.005

19. RESULTRESULT
SS

20. P. N. BGA-C HBGA-C P. N. BGA-C HBGA-C
1 98 100 14 5 39
2 4 51 15 2 2
3 9 5 16 71 15
4 9 92 17 12 7
5 3 7 18 5 41
6 5 7 19 49 74
7 16 17 20 49 65
8 11 49 21 77 100
9 100 100 22 99 100
10 0 4 23 2 2
11 100 100 24 26 9
12 70 100 25 8 4
13 11 27
Success RateSuccess Rate

21. P. N. BGA-C HBGA-C P. N. BGA-C HBGA-C
1 30967 27046 14 10436 10168
2 38400 33494 15 12380 10420
3 11155 10392 16 10492 10258
4 50835 53256 17 29875 27142
5 35120 34602 18 36516 32784
6 10660 10337 19 41669 39396
7 10901 10261 20 78114 56984
8 25356 25491 21 10296 10234
9 10246 10093 22 10675 10042
10 * 10760 23 10250 10150
11 10540 10176 24 12620 11526
12 13649 10807 25 87642 87945
13 10709 11455
No. of Function CallsNo. of Function Calls

22. P. N. BGA-C HBGA-C P. N. BGA-C HBGA-C
1 -320.000 -320.000 14 -0.096 -0.096
2 -308.484 -309.045 15 -6995.746 -6933.303
3 13.595 13.638 16 1.001 1.003
4 685.126 682.839 17 -17.000 -17.000
5 -11.983 -11.966 18 -211.557 -212.007
6 -5.480 -5.474 19 -10.965 -10.969
7 -16.727 -16.770 20 -39.000 -39.000
8 5136.792 5135.532 21 0.000 0.000
9 -1.000 -1.000 22 -2.214 -2.214
10 * -0.995 23 0.126 0.126
11 -1.000 -1.000 24 0.082 0.082
12 0.250 0.250 25 8810.958 8846.122
13 0.501 0.501
Mean FunctionMean Function
ValuesValues

23. P. N. BGA-C HBGA-C P. N. BGA-C HBGA-C
1 0.000 0.000 14 0.001 0.000
2 0.936 0.852 15 27.801 24.068
3 0.034 0.054 16 1.001 0.003
4 1.573 1.464 17 0.000 0.000
5 0.065 0.059 18 0.563 0.634
6 0.013 0.012 19 0.031 0.030
7 0.099 0.077 20 0.000 0.000
8 18.396 18.598 21 0.000 0.000
9 0.000 0.000 22 0.001 0.000
10 * 0.002 23 0.000 0.001
11 0.000 0.000 24 0.000 0.000
12 0.000 0.000 25 48.682 43.279
13 0.002 0.001
S.S.
D.D.

24. P. N. BGA-C HBGA-C P. N. BGA-C HBGA-C
1 2.5799 2.3244 14 0.3376 0.4536
2 3.66 3.5825 15 0.422 0.422
3 0.3766 0.3812 16 0.3294 0.3687
4 6.1008 7.3675 17 2.4725 2.3703
5 3.3747 4.0133 18 3.5562 3.4954
6 0.3622 0.4153 19 4.0959 4.1448
7 0.3672 0.4098 20 13.2076 9.7502
8 1.7356 1.9778 21 0.3442 0
9 0.3717 0.4044 22 0.3527 0.3674
10 * 0.4258 23 0.336 0.3675
11 0.3705 0.4106 24 0.4267 0.4479
12 0.4632 0.4177 25 16.791 17.3633
13 0.3639 0.4681
TimTim
ee

25. Analysis of ResultsAnalysis of Results
Sl. Sense Better Tie Worse
1 Success Rate 17 4 4
2 Ave. Fun. Calls 21 0 4
3 Mean Obj. fun. Value 11 8 6
4 S. D. 14 5 6
5 Time 7 1 17
HBGA-C Vs. BGA-CHBGA-C Vs. BGA-C
HBGA-C is………HBGA-C is………
………….than BGA-C.than BGA-C

26. ConclusioConclusio
nn
 HBGA-C >>> BGA-CHBGA-C >>> BGA-C
(in more percentage of success)(in more percentage of success)
 HBGA-C >>> BGA-CHBGA-C >>> BGA-C
(in less no. of function evaluation)(in less no. of function evaluation)
 HBGA-C >>> BGA-C (in less S. D.)HBGA-C >>> BGA-C (in less S. D.)
 HBGA-C >>> BGA-CHBGA-C >>> BGA-C
(in better obj. fun. value)(in better obj. fun. value)
 HBGA-C <<< BGA-C (in time)HBGA-C <<< BGA-C (in time)

27. References:
[1] A. Osyczka, S. Krenich and S. Kundu. Proportional and Tournament
Selections for Constrained Optimization Problems using GAs. Evolutionary
Optimization, an Int. Jr. on the internet, 1(1): pp. 89-92, 1999.
[2] A. Osyczka. Evolutionary Algorithms for Single and Multi-criteria Design
Optimization, Physica-Verlag Heidelberg, New York, 2002.
[3] C. A. Coella and M. E. Mezura. Constraint-Handling in Genetic Algorithms
through the use of dominance-based tournament selection. Advance Engineering
Informatics, 16: pp. 193-203, 2002.
[4] D. Orvosh and L. Davis. Using a Genetic Algorithm to Optimize problems
with Feasibility Constraints. Proceeding of the Sixth Int. Conf. on Gas, Echelman,
L. J. Ed., pp. 548-552, 1995.
[5] H. Myung and J. H. Kim. Hybrid Evolutionary Programming for Heavily
Constrained Problems. Bio-Systems, 38, pp. 29-43, 1996.
[6] J. H. Kim and H. Myung. A Two Phase Evolutionary Programming for
general Constrained Optimization Problem. Proceedings of the Fifth Annual Conf.
on Evolutionary Programming, San Diego, 1996.
[7] K. Deb and S. Agarwal. A Niched-Penalty Approach for Constraint
Handling GAs, Proceeding of the ICANNGA, Portoroz, Slovenia, 1999.
[8] K. Deb. A Robust Optimal Design Technique Component Design in
Evolutionary Algorithms in Engineering Applications. Springer Verlag, pp. 497-514,
1997.

28. [9] K. Deb. Optimization for Engineering Design: Algorithms and
Examples, Prentice-Hall of India, NewDelhi, 1995.
[10] K. Deep and K. N. Das. Choice of selection and crossover on some
Benchmark problems. Int. Jr. of Computer, Mathematical Sciences and
Applications, Vol.1, No. 1, 99-117, 2007.
[11] K. Deep and K. N. Das. Quadratic approximation based Hybrid
Genetic Algorithm for Function Optimization. AMC, Elsevier, Vol. 203: 86-98,
2008.
[12] K. N. Das. Design and Applications of Hybrid Genetic Algorithms for
Function Optimization. PhD thesis, Indian Institute of Technology, Roorkee,
India, Dec. 2007 .
[13] S. Akhtar, K. Tai and T. Ray. A Socio-Behavioural Simulation Model
for Engineering Design Optimization, 34(4): pp.341-354, 2002.
[14] S. Kundu and A. Osyczka. Genetic Multi-criteria Optimization of
structural systems. Proceedings of the 19th ICTAM, Kyoto, Japan, IUTAM,
272, 1996.
[15] Z. Michalewicz. Genetic Algorithms, Numerical Optimization and
Constraints. Proceedings of Sixth Int. Conf. on Genetic Algorithms, Echelman
L. J. Ed., pp. 151-158, 1995.