Divided Range Genetic Algorithms in Multiobjective Optimization Problems
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, …. , xn} Objective function F Constraints Gi(x)<0 ( i = 1, 2, … , k)
Multi objective optimization problems Design variables X={x 1, x 2, …. , xn} Objective function F={f 1(x), f 2(x), … , fm(x)} Constraints Gi(x)<0 ( i = 1, 2, … , k)
Pareto dominant C A better F 2 B F 1 better
1 / Speed better Pareto Solutions Cost better
Ranking 5 2 F 2 better 3 1 1 1 F 1 better
Genetic Algorithms Evaluation Crossover Selection Mutation Multi point searching methods
I=0 better I=K+1 I=K I=1 F 2 F 1 better
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 Micro-grained model • Population Coarse-grained model Island model
Island 1 f 2 (x) Distributed Genetic Algorithm f 1(x) Island 2 f 1(x) ・Cannot perform the efficient search ・Need a big population size in each island
Divided Range Genetic Algorithms (DRGA) F 2 F 1
Divided Range Genetic Algorithms (DRGA) F 2 F 1
Genetic Algorithms in Multi objective optimization • Expression of genes Vector • Crossover Gravity crossover • Selection Rank 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 Objective functions Constraints
Example 2 Objective functions Constraints
Example 3 Objective functions Constraints
Example 4 Objective functions Constraints
Distributed Genetic Algorithms Used parameters Population size and the sharing range
Evaluation methods • Pareto optimum individuals • Error • Cover rate (smaller values arebetter ( E>0) (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 DGA DRGA
Results(example 1) Error
Results(example 1) Number of function calls
Results(example 1) Calculation time
Results(example 2) Pareto optimum solutions DGA DRGA
Results(example 2) Cover rate
Results(example 2) Number of function calls
Results(example 3) Pareto optimum solutions DGA DRGA
Results(example 4) Pareto optimum solutions SGA DRGA
Results(example 4) Cover rate
Results(example 4) Number of function calls
How DRGA works well? + f 2(x) f 1(x) = f 1(x) f 2(x) ・DRGA = + f 1(x) f 2(x) ・DGA f 1(x)
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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