Doshisha Univ., Japan 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 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. Background (1) ● ＥＭＯ・・・・ Ｅｖｏｌｕｔｉｏｎａｒｙ Ｍｕｌｔｉ－ｃｒｉｔｅｒｉｏｎ Optimizations （ Ex. VEGA,MOGA,NPGA…etc ） Parallel Computing SW-HUB

Doshisha Univ., Japan 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. Background (2) ● Parallel ＥＭＯ Algorithms DRMOGA hasn’t been applied to discrete problems. Purpose The purpose of this study is to find the effectiveness of DRMOGA.

Doshisha Univ., Japan Multi-Criterion Optimization Problems(1) ● Multi-Criterion Optimization Problems (MOPs) 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 } In the optimization problems, when there are several objective functions, the problems are called multi-objective or multi-criterion problems. f 2 (x) Feasible region f 1 (x) Weak pareto optimal solutions Pareto optimal solutions

Doshisha Univ., Japan ・ Multi-objective GA Like single objective GA, genetic operations such as evaluation, selection, crossover, and mutation, are repeated.f1 (x) 1 st generation 5 th generation 10 th generation f 2 (x) Pareto optimal solutions 50 th generation Multi-objective GA (1) 30 th generation

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

Doshisha Univ., Japan f (x) f 1 2 Division 1 Division 2 Max Pareto Optimum solution Min f 1 (x) f 2 Division 1 f 1 (x) f 2 Division 2 Divided Range Multi-Objective GA(1) 1 st The individuals are sorted by the values of focused objective function. 2 nd The N/m individuals are chosen in sequence. 3 rd SGA is performed on each sub population. 4 th After some generations, the step is returned to first Step

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

Doshisha Univ., Japan Application models –SGA, DGA, DRMOGA Layout problems –13, 27blocks Parameter Numerical Example Block No. verticalhorizontal ex)13 blocks GA parametervalue mutation rate 0.05 migration interval (resorted interval) 20 migration rate 0.2 crossover rate 1.0 Number of individuals 100 (total 1600) terminal condition300generation 20

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

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

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

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

Doshisha Univ., Japan AB (f_1, f_2) = ( 38446, ) (f_1, f_2) = ( 42739, ) 27

Doshisha Univ., Japan 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. These problems have little trade-off relationships between the objective functions.

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

Doshisha Univ., Japan 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. Conclusion The results of DRMOGA were compared with those of SGA and DGA

Doshisha Univ., Japan ~ アルゴリズムの流れ ~ ⑦へ⑦へ

Doshisha Univ., Japan f 2 (x) f 1 (x) f 2 (x) f 1 (x) ・ DGA( Island model) ・ DRMOGA f 2 (x) f 1 (x) f 2 (x) f 1 (x) + = f 2 (x) f 1 (x) f 2 (x) f 1 (x) + = Divided Range Multi-Objective GA(2)

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

Doshisha Univ., Japan AB

DGA SGA Results of 10 Blocks case

Doshisha Univ., Japan 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 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 Multi-objective GA (2) Squire EMO

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

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

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

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