1. MOGADES: Multi-Objective Genetic Algorithm with Distributed Environment Scheme Intelligent Systems Design Laboratory ， Doshisha University ， Kyoto Japan ○ Jiro KAMIURA Tomoyuki HIROYASU, Mitsunori MIKI, Shinya WATANABE

2. Doshisha Univ., Kyoto Japan 2 Multi-objective Optimization Problems : MOPs In the optimization problems, when there are several objective functions, the problems are called multi-objective problems. f 1 (x) f 2 Objective function Constraints Gi(x)<0 ( i = 1, 2, …, k) F={f1(x), f2(x), …, fm(x)} X={x1, x2, …., xn} non-dominated solutions Design variable Solving MOPs needs huge calculation costs, so we need the parallel model for solving MOPs. f1(x) : Minimize f2(x) : Minimize

3. Doshisha Univ., Kyoto Japan 3 Multi-Objective Genetic Algorithms : MOGAs VEGA: Schaffer (1985) MOGA: Fonseca (1993) SPEA2: Zitzler (2001) NPGA2: Erickson, Mayer, Horn (2001) NSGA-II: Deb, Goel (2001) Typical method on MOGAs Genetic Algorithm for solving MOPs None of all is parallel model…

4. Doshisha Univ., Kyoto Japan 4 MOGADES : Multi-Objective Genetic Algorithm with Distributed Environment Scheme Distributed Genetic Algorithm (enable to implement on parallel computers) Unification of objective functions using a weighted-sum Adaptive change of the weight parameters Neighborhood migration Archive of the excellent solutions Features Proposed method : MOGADES

5. Doshisha Univ., Kyoto Japan 5 Distributed Genetic Algorithm : DGA (Tanese ‘89) Migration: Exchange of individuals among islands DGA can show better performance than single population GAs in solving single objective problems. A population is divided into smaller subpopulations (islands) One of the parallel models of GAs Canonical GA is performed in each island Distributed Environment Scheme (Miki 1999): the environment (that is crossover rate, mutation rate, and so on) in each island are different.

6. Doshisha Univ., Kyoto Japan 6 Assignment of fitness using the weighted-sum of each objective function Using Distributed Environment Scheme : Weight parameters are different in each island. Unification of the objective functions f 2 (x) f 1 (x) Fitness value = :the number of objective functions :the weight parameter of the ith objective function :the value of the ith objective function The searching directions

7. Doshisha Univ., Kyoto Japan 7 Assignment of weight parameters The weight values are arranged equally from 0.0 to 1.0. e.g.)2 objective functions, 5 islands island 1.00.01 0.750.252 0.5 3 0.250.754 0.01.05 searching direction f 2 (x) f 1 (x)

8. Doshisha Univ., Kyoto Japan 8 Adaptive change of the weight parameters f 2 (x) f 1 (x) e.g.)2 objective functions, 3 islands To get good distributed non-dominated solutions. Performed in the migration phase. f 2 (x) f 1 (x) Change Island 1 Island 2 Island 3 Island 2 Island 1 Island 3 distance d1 d2

9. Doshisha Univ., Kyoto Japan 9 Neighborhood migration Exchange individuals with neighborhood islands. The weight values of islands change. Step 1. Sort islands by. ( changes for each migration phase.) neighborhood 3 island 2, 4 Step 2. Migrate with neighborhood islands. Step 3. Change the weight values of each islands.

10. Doshisha Univ., Kyoto Japan 10 Archive of the excellent solutions Archive of the non-dominated solutions the solutions which have good fitness : non-dominated solutions : solutions which have good fitness : searching direction f 2 (x) f 1 (x) : individuals

11. Doshisha Univ., Kyoto Japan 11 The overview of MOGADES f 2 (x) f 1 (x) searching direction changed weight neighborhood migration non-dominated archive individual island

12. Doshisha Univ., Kyoto Japan 12 ZDT4 –Continuous –2 objective functions –10 design variables –Multi-modal Test Problems

13. Doshisha Univ., Kyoto Japan 13 KUR –Continuous –2 objective functions –100 design variables –Multi-modal Test Problems

14. Doshisha Univ., Kyoto Japan 14 Objectives Constraints 0/1 Knapsack Problem (750items 3knapsacks) –Combination problem 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

15. Doshisha Univ., Kyoto Japan 15 Applied models and Parameters Applied models Crossover –2 points crossover Mutation –bit flip Migration Interval –10 generations SPEA2 NSGA-II MOGADES population size 100(10islands) crossover rate 1.0 mutation rate 1/(chromosome length) Parameters terminal condition 50000 250(25islands) 1000000 number of trials 30 ZDT4KP750-3 chromosome length KUR 2002000750 100000

16. Doshisha Univ., Kyoto Japan 16 ZDT4 MOGADES is superior to NSGA-II and SPEA2

17. Doshisha Univ., Kyoto Japan 17 KUR MOGADES is superior to NSGA-II and SPEA2

18. Doshisha Univ., Kyoto Japan 18 KP750-3 MOGADES is superior to NSGA-II and SPEA2 18

19. Doshisha Univ., Kyoto Japan 19 We proposed a new model of MOGA. –MOGADES: Multi-Objective Genetic Algorithm with Distributed Environment Scheme Conclusion MOGADES was compared to SPEA2 and NSGA-II in 3 test functions. In all of the test functions in which we compared to, MOGADES derives the good results. MOGADES is good model for solving MOPs. MOGADES is based on Distributed Genetic Algorithm which is one of the parallel models, so MOGADES is the parallel model, too.

20. Doshisha Univ., Kyoto Japan 20

21. Doshisha Univ., Kyoto Japan 21 ZDT6 –Continuous –2 objective functions –10 design variables –Non-convex Test Problems

22. Doshisha Univ., Kyoto Japan 22 Assignment of weight parameters As many as possible, the weight values are arranged equally from 0.0 to 1.0. In the rest of the islands, the weight values are assigned randomly. e.g.)3 objective functions 6 islands8 islands 10 islands Random 0 1 0 1 0 1 0.5, 0.0, 0.5 0.5, 0.5, 0.0 0.0, 0.5, 0.5 0.0, 1.0, 0.0 1.0, 0.0, 0.0 0.0, 0.0, 1.0

23. Doshisha Univ., Kyoto Japan 23 The flow of MOGADES initialization evaluation ( includes reservation of the excellent solutions) selection for reproduction crossover mutation selection for survival evaluation neighborhood migration migration interval terminal check end : population : excellent solutions 2 individuals are selected from Pt + Et by tournament selection : parents : offsprings 2 individuals are sampled without replacement from Ct + C’t and replace bad 2 individuals of Pt.

24. Doshisha Univ., Kyoto Japan 24 Adaptive change of the weight parameters Weight values are changed by following equation : distance between islands a and b. : weight value of nth island. f 2 (x) f 1 (x) f 2 (x) f 1 (x) Change Island 1 Island 2 Island 3 d(1,2 ) d(2,3 )

25. Doshisha Univ., Kyoto Japan 25