Parallel Evolutionary Multi-Criterion Optimization for Block Layout Problems

1. Parallel Evolutionary Multi-Criterion Optimization for Block Layout Problems ○ Shinya Watanabe Tomoyuki Hiroyasu Mitsunori Miki Intelligent Systems Design Laboratory， Doshisha University，Japan Doshisha Univ., Japan

2. SW-HUB Parallel Computing Background (1) ●ＥＭＯ・・・・ • Some of EMO can derive the good pareto optimum solutions. • EMO need high calculation cost. • Evolutionary algorithms have potential parallelism. • PC Cluster Systems become very popular. Ｅｖｏｌｕｔｉｏｎａｒｙ　Ｍｕｌｔｉ－ｃｒｉｔｅｒｉｏｎ　Optimizations （Ex. VEGA,MOGA,NPGA…etc） Doshisha Univ., Japan

3. DRMOGA hasn’t been applied to discrete problems. Background (2) ●Parallel ＥＭＯ Algorithms • Some parallel models for EMO are proposed • There are few studies for the validity on parallel model. • Divided Range Multi-Objective Genetic Algorithms (DRMOGA) • it is applied to some test functions and it is found that this model is effective model for continuous multi-objective problems. Purpose The purpose of this study is to find the effectiveness of DRMOGA. Doshisha Univ., Japan

4. Weak pareto optimal solutions (x) f 2 Feasible region Pareto optimal solutions f (x) 1 Multi-Criterion Optimization Problems(1) ●Multi-Criterion Optimization Problems (MOPs) In the optimization problems, when there are several objective functions, the problems are called multi-objective or multi-criterion problems. Design variables X={x1, x2, …. , xn} Objective function F={f1(x), f2(x), … , fm(x)} Constraints Gi(x)<0 ( i = 1, 2, … , k) Doshisha Univ., Japan

5. (x) f 2 f (x) 1 Multi-objective GA (1) ・Multi-objective GA Like single objective GA , genetic operations such as evaluation, selection, crossover, and mutation, are repeated. 1st generation 5thgeneration 10th generation 30th generation Pareto optimal solutions 50th generation Doshisha Univ., Japan

6. DGA model Migration 1 island / 1PE • Distributed GAs • A population is divided into subpopulations (islands) • SGA is performed on each subpopulation • Migration is performed for some generations Exchange of individuals Doshisha Univ., Japan

7. (x) (x) Division 1 f 2 f 2 Division 1 Division 2 f (x) 1 (x) f 2 Division 2 Pareto Optimum solution Min Max f (x) 1 f (x) 1 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 Doshisha Univ., Japan

8. n n f1=ΣΣcijdij i=1 j=1 Formulation of Block Layout Problems ・Block Layout Problems with Floor Constraints (Sirai 1999) Objects Block Packing method ３ ４ ６ １ ７ f2=Total Area S where n:number of blocks cij: flow from block i to block j dij: distance from block i to block j ５ ２ ：Dead Space Doshisha Univ., Japan

9. ex)13 blocks Block No. vertical horizontal 1 18 24 2 36 18 3 18 42 4 42 18 5 36 42 6 24 36 7 24 54 8 30 36 9 48 18 10 36 24 11 36 36 12 54 24 13 36 30 Numerical Example • Application models • SGA , DGA , DRMOGA • Layout problems • 13, 27blocks • Parameter GA parameter value Number of individuals 100 (total 1600) crossover rate 1.0 mutation rate 0.05 migration interval (resorted interval) 20 20 migration rate 0.2 terminal condition 300generation Doshisha Univ., Japan

10. Cluster system for calculation Spec. of Cluster(16 nodes) Processor PentiumⅡ(Deschutes) Clock 400MHz # Processors 1 × 16 Main memory 128Mbytes × 16 Network Fast Ethernet (100Mbps) Communication TCP/IP, MPICH 1.1.2 OS Linux 2.2.10 Compiler gcc (egcs-2.91.61) Doshisha Univ., Japan

11. Results of 13 Blocks case 13 Real weak pareto solutions DGA DRMOGA Doshisha Univ., Japan

12. Results of 13 Blocks case (SGA) 13 Local optimum solutions Real weak pareto solutions Doshisha Univ., Japan

13. Results of 27 Blocks case 27 A B DGA DRMOGA Doshisha Univ., Japan

14. 27 (f_1, f_2) = (38446, 49920) (f_1, f_2) = (42739, 43264) A B Doshisha Univ., Japan

15. These problems have little trade-off relationships between the objective functions. Results • Most of the solutions were weak-pareto solutions. • SGA, DGA and DRMOGA are applied to the layout problems • There are small difference between the results of three methods. • When results of DRMOGA compared with those of DGA, there isn’t big advantage. • SGA sometimes could not find the real weak pareto solutions. Doshisha Univ., Japan

16. Results Division 1 (x) 2 f Division 2 f (x) 1 (x) 2 f The individuals can’t be divided into two division by the value of the focused objective function(f2(x)). Can’t exchange individuals enough. f (x) Doshisha Univ., Japan 1

17. The results of DRMOGA were compared with those of SGA and DGA Conclusion • The DRMOGA was applied to discrete problems ; The block layout problems. • The test problems didn’t have definitely pareto solutions. • The searching ability of DGA and DRMOGA were almost same in numerical examples. • The mechanism of DRMOGA didn’t work effectively in these problems. • SGA may be caught by local minimum. Doshisha Univ., Japan

18. ~アルゴリズムの流れ~ ②個体を各島に分配 ⑤総個体数を調べる ④評価・選択・交叉 ⑥全体シェアリング ①初期個体生成 ③終了判定 ⑦終了 ⑦へ Doshisha Univ., Japan

19. f2(x) f2(x) f2(x) f1(x) f1(x) f1(x) f2(x) f2(x) f2(x) f1(x) f1(x) f1(x) Divided Range Multi-Objective GA(2) ・DGA( Island model) = + ・DRMOGA = + Doshisha Univ., Japan

20. Results of 10 Blocks case (DRMOGA) A Local optimum set B Real weak pareto set Doshisha Univ., Japan

21. A B Doshisha Univ., Japan

22. Results of 10 Blocks case DGA SGA Doshisha Univ., Japan

23. Why are the results in this presentation different from the results in the paper? • In first, we selected GA parameters with no consideration. But we investigated more suitable GA parameters, and in this presentation, we used new GA parameters. That’s why this results Is different from results in paper. • What do you aim in this presentation? • Main purpose in this study is to investigate the effectiveness of DRMOGA for Block layout problems. To my regret, this problem isn’t suitable for multi-criterion problems and we can’t get good results. • How do you think about meaning of this presentation? • In other discrete problem, the effectiveness of DRMOGA hasn’t been researched yet. And I think that in the problem that has obviously trade-off relationships, DRMOGA will get good results. Because in that problems , the characteristics of DRMOGA can work effectively. Doshisha Univ., Japan

24. Multi-objective GA (2) Squire EMO • VEGASchaffer (1985) • VEGA+Pareto optimum individuals Tamaki(1995) • RankingGoldberg (1989) • MOGA Fonseca (1993) • Non Pareto optimum Elimination Kobayashi (1996) • Ranking + sharing Srinvas (1994) • Others Doshisha Univ., Japan

25. 13 (f_1, f_2) = (838544, 14238) (f_1, f_2) = (879179, 13560) Doshisha Univ., Japan

26. Pareto reservation strategy Selection PMX method Crossover 2 bit substitution method Mutation Configuration of GA • Expression of solutions • Genetic operations Doshisha Univ., Japan

27. Calculation Time Calculation time(sec) method Case 1.08E+03 SGA DGA 1.73E+01 13blocks 2.02E+01 DRMOGA 1.37E+03 SGA 27blocks 5.28E+01 DGA 5.61E+01 DRMOGA Doshisha Univ., Japan

28. Results of 27 Blocks case 27 SGA Doshisha Univ., Japan