The New Model of Parallel Genetic Algorithm in MultiObjective Optimization Problems

1
The New Model of Parallel Genetic Algorithm in
Multi-Objective Optimization Problems
- Divided Range Multi-Objective Genetic
Algorithms -
?Tomoyuki Hiroyasu Mitsunori Miki Shinya
Watanabe
Intelligent Systems Design Laboratory, Doshisha
University,Japan
2
Background (1)
?Multi-criterion Optimizations
?Genetic Algorithms
Evolutionary Multi-criterion Optimizations
(Ex. VEGA,MOGA,NPGAetc)
High computation cost
3
Background (2)
?Parallel EMOs Algorithms

• Distributed GA model
• Master slave model
• Cellular GA model

4
Multi-Criterion Optimization Problems(1)
?Multi-Criterion Optimization Problems (MOPs)
Design variables
Xx1, x2, . , xn
Objective function
Ff1(x), f2(x), , fm(x)
Constraints
Gi(x)lt0 ( i 1, 2, , k)
5
Multi-Criterion Optimization Problems(2)
Pareto dominant and Ranking method
Pareto-optimal Set
The set of non-inferior individuals in
each generation.
Ranking
number of dominant individuals
Rank 1
6
Genetic Algorithms
crossover
Start (Initialization)
individuals
mutation
population

• Features
• From a metaphor of the same mechanism of the
  evolution in nature
• Stochastic searching
• Multi-point searching
• High calculation cost

Iteration
End (Find solution)
7
Multi-objective GA (1)
Multi-objective GA
8
Multi-objective GA (2)
Squire EMO

• VEGA Schaffer (1985)
• VEGAPareto optimum individuals Tamaki (1995)
• Ranking Goldberg (1989)
• MOGA Fonseca (1993)
• Non Pareto optimum Elimination Kobayashi (1996)
• Ranking sharing Srinvas (1994)
• Others

9
Parallerization of Genetic Algorithms

• Distributed GA model
• Island model (Free topology)
• Master slave model
• Global Parallelization
• (Only evaluate in parallel)
• Cellular GA model
• Neighborhood model
• (Mainly Grid topology)

10
DGA model
Island 1
(x)
f
2
f
(x)
1
(x)
Island 2
f
2
f
(x)
1
Cannot perform the efficient search
Need a big population size in each island
11
Divided Range Multi-Objective GA(1)
1st The individuals are sorted by the values of
focused objective function. 2nd The N/m
individuals are chosen in sequence. 3rd SGA
is performed on each sub population. 4th After
some generations, the step is returned to first
Step
12
Divided Range Multi-Objective GA(2)
DGA( Island model)


DRMOGA


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Configuration of GA (1)
Vector
Expression of genes
a1 0.02, 10.03, , 7.52
Crossover
Center Neighborhood Crossover
Selection
Rank 1 selection with sharing
Roulette selection
Roulette selection sharing
None
Mutation
When the movement of the Pareto frontier is very
small
Terminal condition
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Configuration of GA (2)
Center Neighbored Crossover (CNX)
1st N1 parent Individuals are selected
randomly. 2nd The vector of the gravity is
derived with the following equation. 3rd New
individual is generated.
15
Configuration of GA (3)
Normal Distribution
a 3
a 6
16
Parameter (1)

• Application models
• SGA , DGA , DRMOGA
• Parameter

GA parameter
value
SGA DGADRMOGA
Population size
100 500
crossover rate
1.0
mutation rate
0
migration interval (sort interval)
5
5
migration rate
0.2
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Parameter (2)

• Cases

18
Evaluation methods

• Matrix - Evaluation methods -

• Pareto optimum individuals
• Error (smaller values arebetter ( Egt0)
• Cover rate (index of diversity, 0ltClt1)
• Generation (smaller values are better)

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Cover rate

• Cover rate

cover rate(f1)8/100.8
cover rate(f2)9/100.9
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Cluster system for calculation
Spec. of Cluster (5 nodes)
Processor Pentium?(Deschutes) Clock
400MHz Processors 1 5 Main memory
128Mbytes 5 Network Fast Ethernet
(100Mbps) Communication TCP/IP, MPICH
1.1.2 OS Linux 2.2.10 Compiler gcc
(egcs-2.91.61)
21
Numerical Example

• Tamaki et al. (1995)
• Veldhuizen and Lamount (1999)
• K. Deb (1999)

22
Example 1
Objective functions
Constraints
23
Example 2
Objective functions
Constraints
f3
f1
f2
24
Example 3
Objective functions
f2
f1
25
Example 4
Objective functions
f2
f1
å
N
x
i



10
1
)
,
,
(
2
x
x
g
i

2
N
-
1
N
26
Results (Example1)
f2
f2
f1
f1
DRMOGA (Case5)
DGA (Case5)
27
Results (Example1)
Case
cover rate
generations
number of solutions
error
Simple
436
0.00
1.00
799
Case1
382
0.03
1.00
1000
Case2
471
0.00
1.00
35
Case3
444
0.00
1.00
367
Case4
461
0.00
1.00
39
Case5
330
0.00
1.00
1000
Case6
436
0.01
1.00
43
island
Case1
438
0.01
1.00
59
Case2
423
0.01
1.00
273
Case3
435
0.01
1.00
44
Case4
431
0.01
1.00
66
Case5
404
0.01
1.00
927
Case6
DR
500
0.00
1.00
40
Case1
500
0.00
1.00
48
Case2
494
0.00
1.00
105
Case3
494
0.00
1.00
548
Case4
495
0.00
1.00
199
Case5
494
0.00
1.00
814
Case6
28
Results (Example2)
DRMOGA (Case1)
DGA (Case1)
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Results (Example2)
x1
x1
x2
x2
DRMOGA (Case1)
DGA (Case1)
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Results (Example2)
Case
cover rate
generations
number of solutions
Simple
Case1
500
0.75
15
Case2
500
0.74
18
Case3
491
0.51
19
Case4
485
0.50
30
Case5
316
0.48
19
Case6
207
0.47
198
island
Case1
428
0.79
19
Case2
426
0.79
36
Case3
434
0.76
22
Case4
403
0.77
55
Case5
6
0.04
1000
Case6
125
0.43
943
0.95
DR
Case1
386
44
0.96
Case2
330
256
0.92
Case3
429
82
0.85
Case4
255
277
0.88
Case5
337
66
0.53
Case6
90
117
31
Results (Example3)
f2
f2
f1
f1
DRMOGA (Case4)
DGA (Case4)
32
Results (Example3)
f1
f1
x1
x1
DRMOGA (Case4)
DGA (Case4)
33
Results (Example3)
Case
number of solutions
error
cover rate
generations
Case1
Simple
500
7.21
0.41
394
Case2
456
5.92
0.32
612
Case3
469
3.75
0.14
1000
Case4
374
2.37
0.47
1000
Case5
482
3.84
0.48
1000
Case6
423
2.60
0.43
1000
Case1
island
345
6.41
0.46
570
Case2
322
5.88
0.48
919
Case3
301
3.70
0.22
1000
Case4
220
2.60
0.35
1000
Case5
283
3.39
0.43
1000
Case6
240
2.41
0.31
1000
Case1
DR
412
6.87
0.38
533
Case2
363
5.38
0.28
774
Case3
425
4.53
0.40
780
0.99
Case4
0.01
293
1000
Case5
393
3.92
0.41
692
0.94
Case6
0.14
254
971
34
Results (Example3)
DRMOGA (Case4) - Object sharing -
DRMOGA (Case4)-Plan sharing-
35
Results (Example3)
DRMOGA (Case4) - Object sharing -
DRMOGA (Case4)-Plan sharing-
36
Results (Example4)
f2
f2
f1
f1
DRMOGA (Case6)
DGA (Case6)
37
Results (Example4)
f1
f1
x1
x1
DRMOGA (Case6)
DGA (Case6)
38
Results (Example4)
Case
error
cover rate
number of solutions
generations
Simple
Case1
500
1.70
0.31
209
Case2
500
1.71
0.38
358
Case3
470
0.32
0.22
1000
0.03
Case4
477
0.40
1000
Case5
492
0.34
0.58
855
0.08
0.60
Case6
493
899
island
Case1
385
1.89
0.40
333
Case2
409
1.75
0.53
403
Case3
304
0.31
0.33
1000
0.24
Case4
361
0.46
1000
Case5
376
0.27
0.60
1000
0.25
0.60
Case6
365
1000
DR
Case1
494
1.93
0.37
212
Case2
457
3.10
0.34
54
Case3
460
0.39
0.30
262
0.03
Case4
451
0.52
387
Case5
442
0.39
0.47
291
0.07
0.61
Case6
402
654
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Conclusion

• In this study, we introduced the new model of
  genetic algorithm in the multi objective
  optimization problems Distributed Genetic
  Algorithms (DRGAs).
• DRGA is the model that
• is suitable for the parallel processing.
• can derive the solutions with short time.
• can derive the solutions that have high accuracy.
• can sometimes derive the better solutions
  compared to the single island model.

40
Conclusions

• In this study, we introduced the new model of
  genetic algorithm in the multi objective
  optimization problems Distributed Genetic
  Algorithms (DRGAs).
• DRGA is the model that
• is suitable for the parallel processing.
• can derive the solutions with short time.
• can derive the solutions that have high accuracy.
• can sometimes derive the better solutions
  compared to the single island model.

41
Results of 27 Blocks case
27
SGA