1. Divided Range Genetic Algorithms in Multiobjective Optimization Problems Tomoyuki HIROYASU Mitsunori MIKI Sinya WATANBE Doshisha University

2. Topics Multi objective optimization problems Genetic Algorithms Parallel Processing Divided Range Genetic Algorithms (DRGAs)

3. What is Optimization Problems ? Design variables X={x 1, x 2, …., x n } Objective function F Constraints G i (x)<0 ( i = 1, 2, …, k)

4. Multi objective optimization problems Design variables X={x 1, x 2, …., x n } Objective function F={f 1 (x), f 2 (x), …, f m (x)} Constraints G i (x)<0 ( i = 1, 2, …, k)

5. Pareto dominant A F2F2 F1F1 B C better

6. Pareto Solutions 1 / Speed Cost better

7. Ranking 1 F2F2 F1F1 1 1 2 3 5 better

8. Genetic Algorithms Evaluation Crossover Mutation Selection Multi point searching methods

9. I=KI=K+1 I=0 I=1 better F2F2 F1F1

10. GAs in multi objective optimization VEGA Schaffer (1985) VEGA+Pareto optimum individuals Tamaki (1995) Ranking Goldberg (1989) MOGA Fonseca (1993) Non Pareto optimum Elimination Kobayashi (1996) Ranking + sharing Srinvas (1994) Others

11. Parallerization of Genetic Algorithms Evaluation Population Micro-grained model Coarse-grained model Island model

12. Distributed Genetic Algorithm ・ Cannot perform the efficient search ・ Need a big population size in each island f 1 (x) f 2 f 1 f 2 f 1 f 2 Island 1 Island 2

13. Divided Range Genetic Algorithms (DRGA) F2F2 F1F1

14. F2F2 F1F1

15. Genetic Algorithms in Multi objective optimization Expression of genesVector CrossoverGravity crossover SelectionRank 1 selection with sharing Terminal condition When the movement of the Pareto frontier is very small

16. Numerical examples Tamaki et al. (1995) Veldhuizen and Lamount (1999)

17. Example 1 Constraints Objective functions

18. Example 2 Objective functions Constraints

19. Example 3 Objective functions Constraints

20. Example 4 Objective functions Constraints

21. Distributed Genetic Algorithms Used parametersPopulation size and the sharing range

22. Evaluation methods Pareto optimum individuals Error (smaller values arebetter ( E>0) Cover rate( index of diversity, 0<C<1) Number of function calls(smaller values are better) Calculation time(smaller values are better)

23. Results(example 1) Pareto optimum solutions DGADRGA

24. Results(example 1) Error

25. Results(example 1) Number of function calls

26. Results(example 1) Calculation time

27. Results(example 2) Pareto optimum solutions DGADRGA

28. Results(example 2) Cover rate

29. Results(example 2) Number of function calls

30. Results(example 3) Pareto optimum solutions DGADRGA

31. Results(example 4) Pareto optimum solutions SGADRGA

32. Results(example 4) Cover rate

33. Results(example 4) Number of function calls

34. f 2 (x) f 1 (x) f 2 (x) f 1 (x) ・ DGA ・ DRGA f 2 (x) f 1 (x) f 2 (x) f 1 (x) + = f 2 (x) f 1 (x) f 2 (x) f 1 (x) + = How DRGA works well?

35. 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.