Distributed Genetic Algorithms with a New Sharing Approach in

Multiobjective Optimization Problems

Tomoyuki HIROYASU

Mitsunori MIKI

Sinya WATANABE

Doshisha UniversityKyoto, Japan

1. Introduction

Introduction (No.1)

Genetic Algorithms Multiobjective Optimization Problems

Need a lot of iterations

Need a large memory

Real world problem

Many objects

The Pareto optimum solutions

Parallel Processing

Introduction (No.2)

Low cost

High performanceCommodity hardware

PC Clusters

Introduction (No.3)

Evaluation fitness

Crossover, selection

Population

Makinen, et. al. ， Parallel CFD96, (1996)

Rowe, et. al.， 2NWGA, (1996)

Hiyane, No. 9 Automatic system symposium(1997)

Distributed Genetic Algorithms

Introduction (No. 4)

Distributed Genetic Algorithms

Hiyane (1997) concluded that DGAs are the powerful tool for MOPs.

The diversity of solutions becomes low

Sharing to total population

Aim of this studyPreliminary study of parallel genetic algorithms

Single processor

Introduced simple algorithms of Distributed Genetic Algorithm with sharing for total population

Effects of sharing in distributed genetic algorithms

2. Distributed Genetic Algorithms with Sharing

Divide population into sub populations

island

Distributed Genetic Algorithms

Genetic operations in each island

Migration

Migration interval

Migration rate

Genetic operations in each island

Distributed Genetic Algorithms with Sharing

divide population into islands

Genetic operations in each island

migrationgather populations from islandsTotal sharing

F1

F2Divide population into islandsTotal sharing

F1

F2

Evaluation methods

•The number of solutions

•Error

•Cover rate of solutions

•Coefficient of variation

Evaluation method (Error)

E =di

2i = 1

n

N

d1

d2

d3 d4

F1

F2

Evaluation method (Cover rate)Min Max

Evaluation method (Coefficient of variation)

F1

F2

1) Count the number of solutions in the certain radius for each solution2) Derive the coefficient of variation of the numbers

3) Derive the average

4) It shows the diversity of the solutions ( 1.0 is the best)

3. Numerical Examples

Test Function

fi = – xi i = 1,2, n

g j = –x j j = 1,2, ,n

gn+k = xk – 6 k = 1,2, ,n

g2n + 1 = 1 – x1x2 xn

Objective function

Constraints

In this study, we used 4 objectives.

Test functions

-6 -5 -4 -3 -2 -1

-6

-5

-4

-3

-2

-1

-6

-4

-2

0

-6

-4

-2

0

-1.5

-1

-0.5

0

-1.5

-1

-0.5

0

2 objectives

3 objectives

Coding

Design variables　→　 real values

keep good heredity

phenotype 　ｘ

genotype　 X

=

X={1.23, 34.2, 4.23, 8.29}

x={1.23, 34.2, 4.23, 8.29}

Parameters

initial population size

crossover rate

mutation rate

migration rate

migration interval

island number

1000

1.0

0.0

0.1

2

10

parameter value

Effect of distribution

1 island

10 islands

number of

solutions errorcoverratio

generations calculation

time [sec]

1980

2690

0.191

0.196

0.856

0.853

6

6

194.9

34.3

coefficient of variation

2.46

3.10

Terminal condition = function call (1000)

DGA

DGA with sharing

number ofsolutions error

coverratio

coefficient of variation generations

functioncall

3422

1581

0.182

0.226

0.856

0.847

3.65

2.15

7.8

3.0

18998

4985

DGA

DGA withsharing

number ofsolutions

errorcoverratio

coefficient of variation

generationscalculationtime [sec]

3888

3079

0.171

0.153

0.855

0.855

4.11

3.10

8.7

10.1

91.0

563.1

Termination condition= number of function call

Termination condition= calculation time

Errors

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12E

rror

0.000 0.025 0.050 0.075 0.100 0.125

Sleep time

DGA with sharing

DGA

Cover ratio

0.850

0.875

0.900

0.925

0.950C

over

rat

io

0.000 0.025 0.050 0.075 0.100 0.125

Sleep time

DGA with sharing

DGA

Hybrid sharing method

divide population into small islands

genetic operation in each island migration sharing in each island

total sharing

gather populations from islands

Results of hybrid method

DGA

DGA withsharing

number ofsolutions

errorcoverratio

coefficient of variation generations

CalculationTime [sec]

3888

3079

0.171

0.153

0.855

0.855

4.11

3.10

8.7

10.1

91.0

563.1

Hybridsharing

2922 0.183 0.858 2.43 10.0 275.5

4. Conclusions

Conclusions

The proposed approach is especially useful when it takes much time to evaluate objective functions

Distributed genetic algorithm is good method for parallel processing but it reduces the diversity of solutions.

To increase the diversity of solutions, the sharing is necessary even in distributed genetic algorithm.

DGA with sharing to total population

The proposed approach increase the diversity and the accuracy of solutions

Another approach where the sharing is performed in islands and in total population is proposed and this approach reduces the calculation time and makes some increase in the diversity while the accuracy of the solutions is decreased.

Conclusions (future work)

Larger problems, something from real applications

Applying to another test functions

Parallel processing Sorting in parallel

Crossover

G

C =G + N(0, i

2)GPi

GPi

i = 1

n + 1

Constraints

c2p1

c1

c3

Feasible region