Optimization Problem with Simple Genetic Algorithms Cho, Dong-Yeon

Function Optimization Problem Example

Representation – Binary String Code length

Mapping from a binary string to real number

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

Initial Population Initial population is randomly generated.

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.

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.

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.

pkpk qkqk

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.

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

Test Functions Rastrigin’s function

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

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

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.

제출 제출 마감 (10 월 25 일, 수 ): 두 가지 모두 제출 제출물  Source code, 실행 file  Source 에 적절한 comment 작성  File 들은 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