1. Doshisha Univ., Kyoto Japan NCGA : Neighborhood Cultivation Genetic Algorithm for Multi-Objective Optimization Problems Intelligent Systems Design Laboratory， Doshisha University，Kyoto Japan ○ Shinya Watanabe Tomoyuki Hiroyasu Mitsunori Miki

2. Doshisha Univ., Kyoto Japan Multi-objective Optimization Problems ● Multi-objective Optimization Problems (MOPs) In the optimization problems, when there are several objective functions, the problems are called multi-objective or multi-criterion problems. f 1 (x) f 2 Design variables Objective function Constraints G i (x)<0 ( i = 1, 2, …, k) F={f 1 (x), f 2 (x), …, f m (x)} X={x 1, x 2, …., x n } Feasible region Pareto optimal solutions

3. Doshisha Univ., Kyoto Japan MOPs solved by Evolutionary algorithms EMO VEGA :Schaffer (1985) MOGA :Fonseca (1993) DRMOGA :Hiroyasu, Miki, Watanabe (2000) SPEA2 :Zitzler (2001) NPGA2 :Erickson, Mayer, Horn (2001) NSGA-II :Deb, Goel (2001) Typical method on EMO EMO Evolutionary Multi-criterion Optimization

4. Doshisha Univ., Kyoto Japan The following topics are the mechanisms that the recent GA approaches have. EMO Archive of the excellent solutions Cut down (sharing) method of the reserved excellent solutions An appropriate assign of fitness Reflection to search solutions mechanism of the reserved excellent solutions Unification mechanism of values of each objective

5. Doshisha Univ., Kyoto Japan NCGA : N eighborhood C ultivation GA The neighborhood crossover Archive of the excellent solutions Cut down (sharing) method of the reserved excellent solutions An appropriate assign of fitness Reflection to search solutions mechanism of the reserved excellent solutions Unification mechanism of values of each objective The features of NCGA Neighborhood Cultivation GA (NCGA)

6. Doshisha Univ., Kyoto Japan A neighborhood crossover –In MOPs GA, the searching area is wide and the searching area of each individuals are different. f 2 (x) f 1 (x) If the distance between two selected parents is so large, cross over may have no effect for local search. Neighborhood Cultivation GA (NCGA)

7. Doshisha Univ., Kyoto Japan One of the objectives is changed at every generation. The pair for the mating is changed based on a probabillity. f 2 (x) f 1 (x) Neighborhood Cultivation GA (NCGA) A neighborhood crossover Two parents of crossover are chosen from the top of the sorted individuals. In order not to make the same couple.

8. Doshisha Univ., Kyoto Japan Continuous Function –ZDT4 Test Problems

9. Doshisha Univ., Kyoto Japan Continuous Function –KUR Test Problems

10. Doshisha Univ., Kyoto Japan Objectives Constraints Combination problem –KP 750-2 p i,j = profit of item j according to knapsack i Test Problems w i,j = weight of item j according to knapsack i c i, = capacity of knapsack i

11. Doshisha Univ., Kyoto Japan Applied models and Parameters GA Operator Applied models Crossover –One point crossover Mutation –bit flip SPEA2 NSGA-II NCGA non-NCGA ( NCGA except neighborhood croosover ) population size 100 crossover rate 1.0 mutation rate 0.01 Parameters terminal condition 250 2000 number of trial 30

12. Doshisha Univ., Kyoto Japan Results (Pareto solutions of ZDT4)

13. Doshisha Univ., Kyoto Japan Results (Pareto solutions of KUR)

14. Doshisha Univ., Kyoto Japan Results (Pareto solutions of KP750- 2)

15. Doshisha Univ., Kyoto Japan We proposed a new model for Multi-objective GA. –NCGA: N eighborhood C ultivation GA Effective method for multi objective GA Neighborhood crossover Reservation mechanism of the excellent solutions Reflection to search solutions mechanism of the reserved excellent solutions Cut down (sharing) method of the reserved excellent solutions Assignment method of fitness function Unification mechanism of values of each objective Conclusion

16. Doshisha Univ., Kyoto Japan NCGA was applied to test functions and results were compared to the other methods; those are SPEA2, NSGA-II and non-NCGA. In some the test functions, NCGA derives the good results. Comparing to NCGA and NCGA without neighborhood crossover, the former is obviously superior to the latter in all problems. NCGA is good model of Multi-objective GA Conclusion

17. Doshisha Univ., Kyoto Japan Results (RNI of KP750-2)

18. Doshisha Univ., Kyoto Japan EMO 全般に関して –http://www.lania.mx/~ccoello/EMOO/EMO Obib.htmlhttp://www.lania.mx/~ccoello/EMOO/EMO Obib.html 多目的 0 /1 ナップザック問題に関して –http://www.tik.ee.ethz.ch/~zitzler/http://www.tik.ee.ethz.ch/~zitzler/ 発表に用いたソースプログラム –http://mikilab.doshisha.ac.jp/dia/research/mop_ga /archive/ 発表者の電子メールアドレス –sin@mikilab.doshisha.ac.jp 参照 URL

19. Doshisha Univ., Kyoto Japan The Ratio of Non-dominated Individuals (RNI) is derived from two types of Pareto solutions. Performance Measure (x) f 1 f 2 Method B (x) f 1 f 2 Method A (x) f 1 f 2 Method A Method B 0.333 0.666

20. Doshisha Univ., Kyoto Japan Results (RNI of KUR)

21. Doshisha Univ., Kyoto Japan Performance Assessment The Ratio of Non-dominated Individuals :RNI –The Performance measure perform to compare two type of Pareto solutions. –Two types of pareto solutions derived by difference methods are compared. Cover Rate Index –Diversity of the Pareto optimum. Error –The distance between the real pareto front and derived solutions. Various rate –Diversity of the pareto optimum individuals. Measures

22. Doshisha Univ., Kyoto Japan 数値結果 ( KP750-2 )

23. Doshisha Univ., Kyoto Japan Continuous Function –ZDT4 Test Problems

24. Doshisha Univ., Kyoto Japan Results (Pareto solutions of ZDT4)

25. Doshisha Univ., Kyoto Japan Results (RNI of ZDT4)

26. Doshisha Univ., Kyoto Japan 数値結果 (ZDT4)

27. Doshisha Univ., Kyoto Japan 数値結果 (KUR)

28. Doshisha Univ., Kyoto Japan 数値結果 (ZDT4)

29. Doshisha Univ., Kyoto Japan 数値結果 (KUR)

30. Doshisha Univ., Kyoto Japan 数値結果 ( KP750-2 )

31. Doshisha Univ., Kyoto Japan パレート保存個体群の利用 – 多目的では，最終的に求める解候補 （パレート 解）が複数存在するため，探索途中での優良な個 体の欠落を防ぐ必要がある． f 2 (x) f 1 (x) 探索個体群 優良個体保存群 探索個体群に優 良個体群を反映 させることによ り探索の高速化， 効率化を期待す ることができる． 近傍培養型マスタースレーブモデル

32. Doshisha Univ., Kyoto Japan 多目的 GA では，求める解が複数存在するため 単一目的と比較して，十分な個体数と探索世代数 が必要となる． 多目的 GA の問題点 ・探索効率の良いアルゴリズム ・膨大な評価計算回数 ・非常に高い計算負荷

33. Doshisha Univ., Kyoto Japan GA による多目的最適化への応用 ・多目的 GA 交叉・突然変異を用いて パレート最適解集合の探索を行うf1 (x) f 2 (x) 1 st generation 5 th generation 10 th generation 50 th generation 30 th generation

34. Doshisha Univ., Kyoto Japan Multi-Criterion Optimization Problems(2) ・ Pareto dominant and Ranking method Pareto-optimal Set Ranking Rank ＝ １＋ number of dominant individuals The set of non-inferior individuals in each generation. f 2 f 1 1 3 1 Pareto optimal solutions

35. Doshisha Univ., Kyoto Japan クラスタシステム Spec. of Cluster (16 nodes) Processor Pentium Ⅲ (Coppermine) Clock 600MHz # Processors 1 × 16 Main memory 256Mbytes × 16 Network Fast Ethernet (100Mbps) Communication TCP/IP, MPICH 1.2.1 OS Linux 2.4 Compiler gcc 2.95.4