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

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

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)

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)

Pareto dominant A F2F2 F1F1 B C better

Pareto Solutions 1 / Speed Cost better

Ranking 1 F2F2 F1F better

Genetic Algorithms Evaluation Crossover Mutation Selection Multi point searching methods

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

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

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

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

Divided Range Genetic Algorithms (DRGA) F2F2 F1F1

F2F2 F1F1

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

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

Example 1 Constraints Objective functions

Example 2 Objective functions Constraints

Example 3 Objective functions Constraints

Example 4 Objective functions Constraints

Distributed Genetic Algorithms Used parametersPopulation size and the sharing range

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)

Results(example 1) Pareto optimum solutions DGADRGA

Results(example 1) Error

Results(example 1) Number of function calls

Results(example 1) Calculation time

Results(example 2) Pareto optimum solutions DGADRGA

Results(example 2) Cover rate

Results(example 2) Number of function calls

Results(example 3) Pareto optimum solutions DGADRGA

Results(example 4) Pareto optimum solutions SGADRGA

Results(example 4) Cover rate

Results(example 4) Number of function calls

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?

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.