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

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

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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)

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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)

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Pareto dominant A F2F2 F1F1 B C better

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Pareto Solutions 1 / Speed Cost better

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Ranking 1 F2F2 F1F1 1 1 2 3 5 better

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Genetic Algorithms Evaluation Crossover Mutation Selection Multi point searching methods

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

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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

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

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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

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Divided Range Genetic Algorithms (DRGA) F2F2 F1F1

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F2F2 F1F1

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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

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

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Example 1 Constraints Objective functions

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Example 2 Objective functions Constraints

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Example 3 Objective functions Constraints

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Example 4 Objective functions Constraints

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Distributed Genetic Algorithms Used parametersPopulation size and the sharing range

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Evaluation methods Pareto optimum individuals Error (smaller values arebetter ( E>0) Cover rate( index of diversity, 0

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Results(example 1) Pareto optimum solutions DGADRGA

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Results(example 1) Error

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Results(example 1) Number of function calls

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Results(example 1) Calculation time

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Results(example 2) Pareto optimum solutions DGADRGA

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Results(example 2) Cover rate

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Results(example 2) Number of function calls

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Results(example 3) Pareto optimum solutions DGADRGA

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Results(example 4) Pareto optimum solutions SGADRGA

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Results(example 4) Cover rate

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Results(example 4) Number of function calls

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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?

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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.

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