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Optimization Problem with Simple Genetic Algorithms 2000. 9. 27 Cho, Dong-Yeon (dycho@scai.snu.ac.kr)

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Function Optimization Problem Example

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Representation – Binary String Code length

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Mapping from a binary string to real number

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Framework of Simple GA Generate Initial Population Evaluate Fitness Select Parents Generate New Offspring Termination Condition? Yes No Fitness Function Crossover, Mutation Best Individual

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Initial Population Initial population is randomly generated.

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Fitness Evaluation Procedure: Evaluation Convert the chromosome’s genotype to its phenotype. This means converting binary string into relative real values. Evaluate the objective function. Convert the value of objective function into fitness. For the maximization problem, the fitness is simply equal to the value of objective function. For the minimization problem, the fitness is the reciprocal of the value of objective function.

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Selection Fitness proportional (roulette wheel) selection The roulette wheel can be constructed as follows. Calculate the total fitness for the population. Calculate selection probability p k for each chromosome v k. Calculate cumulative probability q k for each chromosome v k.

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Procedure: Selection Generate a random number r from the range [0,1]. If r q 1, then select the first chromosome v 1 ; else, select the kth chromosome v k (2 k pop_size) such that q k-1 < r q k.

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pkpk qkqk 10.111180 20.0975150.208695 30.0538390.262534 40.1650770.427611 50.0880570.515668 60.0668060.582475 70.1008150.683290 80.1109450.794234 90.1482110.942446 100.0575541.000000

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Genetic Operations Crossover One point crossover Crossover rate p c Procedure: Crossover Select two parents. Generate a random number r c from the range [0,1]. If r c < p c then perform undergo crossover. Mutation Mutation alters one or more genes with a probability equal to the mutation rate p m.

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Experiments Various experimental setup Termination condition: maximum_generation 2 pop_size (large, small) 5 parameter settings 10 runs Parameter setting (p c, p m ) Elitism The best chromosome of the previous population is just copied. At least two test functions Example function given here (*) - maximization Rastrigin’s function –minimization Ackley’s function – minimization Schwefel’s (sine root) function – minimization

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Test Functions Rastrigin’s function

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Ackley’s function Schwefel’s (sine root) function

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Results For each test function Result table for the best solution and your analysis f opt, (x opt, y opt ), chromosome opt among whole runs Fitness curve for the run where the best solution was found. Large (pop_size)Small (pop_size) Average SD BestWorst Average SD BestWorst Setting 1 Setting 2 Setting 3 Setting 4 Setting 5

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References Source Codes Simple GA code GA libraries Web sites Books Genetic Algorithms and Engineering Design, Mitsuo Gen and Runwei Cheng, pp. 1-15, John Wiley & Sons, 1997.

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제출 제출 마감 (10 월 25 일, 수 ): 두 가지 모두 제출 제출물 Source code, 실행 file Source 에 적절한 comment 작성 File 들은 e-mail 이나 diskette 에 제출 보고서 : 반드시 인쇄물로 제출 여러 가지 실험 설정에 대한 결과 실험 결과를 다양한 형식으로 표현하여 분석하고 그 결과 를 기술한다. 실행 환경 명시

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