MOGADES: Multi-Objective Genetic Algorithm with Distributed Environment Scheme Intelligent Systems Design Laboratory ， Doshisha University ， Kyoto Japan.

MOGADES: Multi-Objective Genetic Algorithm with Distributed Environment Scheme

Intelligent Systems Design Laboratory ， Doshisha University ， Kyoto Japan

○ Jiro KAMIURATomoyuki HIROYASU,Mitsunori MIKI,Shinya WATANABE

Doshisha Univ., Kyoto Japan2

Multi-objective Optimization Problems : MOPs

In the optimization problems, when there are several objective functions, the problems are called multi-objective problems.

f 1 (x)

f 2(x

)

Objective function

Constraints

Gi(x)<0 ( i = 1, 2, … , k)

F={f1(x), f2(x), … , fm(x)}

X={x1, x2, …. , xn}

non-dominated solutions

Design variable

Solving MOPs needs huge calculation costs, so we need the parallel model for solving MOPs.

f1(x) : Minimize

f2(x) : Minimize


Multi-Objective Genetic Algorithms : MOGAs

•VEGA: Schaffer (1985)

•MOGA : Fonseca (1993)

•SPEA2 : Zitzler (2001)

•NPGA2 : Erickson, Mayer, Horn (2001)

•NSGA-II : Deb, Goel (2001)

Typical method on MOGAs

Genetic Algorithm for solving MOPs

None of all is parallel model…


• MOGADES : Multi-Objective Genetic Algorithm with Distributed Environment Scheme

• Distributed Genetic Algorithm(enable to implement on parallel computers)

• Unification of objective functions using a weighted-sum• Adaptive change of the weight parameters• Neighborhood migration• Archive of the excellent solutions

Features

Proposed method : MOGADES


Distributed Genetic Algorithm : DGA (Tanese ‘89)

Migration: Exchange of individuals among islands

DGA can show better performance than single population GAs in solving single objective problems.

A population is divided into smaller subpopulations (islands)

One of the parallel models of GAs

Canonical GA is performed in each island

Distributed Environment Scheme (Miki 1999): the environment (that is crossover rate, mutation rate, and so on) in each island are different.


• Assignment of fitness using the weighted-sum of each objective function

• Using Distributed Environment Scheme : Weight parameters are different in each island.

Unification of the objective functions

f2(x

)f1(x)

)(

1

xfw ii

k

i

k

i

ki ww1

1 ,0

Fitness value =

:the number of objective functionsk:the weight parameter of the ith objective functioniw

:the value of the ith objective function)(xif

The searching directions


Assignment of weight parameters

• The weight values are arranged equally from 0.0 to 1.0.

e.g.) 2 objective functions, 5 islands

island 1w 2w1.0 0.01

0.75 0.252

0.5 0.53

0.25 0.754

0.0 1.05

searchingdirection

f2(x

)

f1(x)


Adaptive change of the weight parameters

f2(x

)f1(x)

e.g.) 2 objective functions, 3 islands

• To get good distributed non-dominated solutions.• Performed in the migration phase.

f2(x

)

f1(x)

Change

Island 1

Island 2

Island 3

Island 2Island 1Island 3

distance

d1

d2


Neighborhood migration

• Exchange individuals with neighborhood islands.• The weight values of islands change.

iw

Step 1. Sort islands by . iw ( changes for each migration phase.)i

neighborhood

3island

2, 4

Step 2. Migrate with neighborhood islands.

Step 3. Change the weight values of each islands.


Archive of the excellent solutions

Archive of• the non-dominated solutions• the solutions which have good fitness

: non-dominated solutions

: solutions which have good fitness

: searching direction

f2(x

)

f1(x)

: individuals


The overview of MOGADESf2

(x)

f1(x)

searching direction

changed weight

neighborhood migration

non-dominated

archive

individual

island


• ZDT4– Continuous– 2 objective functions– 10 design variables– Multi-modal

]5,5[]1,0[

)4cos(1091)(

)(1)()(min

)(min

1

10

2

2

12

11

i

iii

xx

xxxg

xg

xxgxf

xxf

Test Problems


• KUR– Continuous– 2 objective functions– 100 design variables– Multi-modal

100,,1,]5,5[

)sin(5||)(min

))2.0exp(10()(min38.0

2

100

1

21

21

nnix

xxxf

xxxf

i

ii

i ii

Test Problems


Objectives

Constraints

• 0/1 Knapsack Problem (750items 3knapsacks)– Combination problem

3,2,1)(max750

1,

ixpxfj

jjii

750

1,

jijji cxw

1,0),,,( 75021 jxxxxx pi,j = profit of item j according to knapsack i

Test Problems

wi,j = weight of item j according to knapsack i

ci,= capacity of knapsack i


Applied models and Parameters

Applied models • Crossover– 2 points crossover

• Mutation– bit flip

• Migration Interval– 10 generations

• SPEA2• NSGA-II• MOGADES

population size 100(10islands)crossover rate 1.0mutation rate 1/(chromosome length)

Parameters

terminal condition 50000

250(25islands)

1000000number of trials 30

ZDT4 KP750-3

chromosome length

KUR

200 2000 750

100000


ZDT4

MOGADES is superior toNSGA-II and SPEA2


KUR



KP750-3


18


• We proposed a new model of MOGA.– MOGADES: Multi-Objective Genetic Algorithm with

Distributed Environment Scheme

Conclusion

MOGADES was compared to SPEA2 and NSGA-II in 3 test functions.

In all of the test functions in which we compared to,MOGADES derives the good results.

MOGADES is good model for solving MOPs.

MOGADES is based on Distributed Genetic Algorithm which is one of the parallel models, so MOGADES is the parallel model, too.



• ZDT6– Continuous– 2 objective functions– 10 design variables– Non-convex

]5,5[]1,0[

191)(

1)()(min

)6(sin)4exp(1)(min

1

25.010

2

2

2

12

16

11

i

ii

xx

N

xxg

g

fxgxf

xxxf

Test Problems


Assignment of weight parameters

• As many as possible, the weight values are arranged equally from 0.0 to 1.0.

• In the rest of the islands, the weight values are assigned randomly.

e.g.) 3 objective functions

6 islands

1w

2w3w

8 islands 10 islands

Random

0 10

10

1

0.5, 0.0, 0.5

0.5, 0.5, 0.00.0, 0.5, 0.50.0, 1.0, 0.0

1.0, 0.0, 0.00.0, 0.0, 1.0


The flow of MOGADES

initialization

evaluation ( includes reservation of the excellent solutions)

selection for reproduction

crossover

mutation

selection for survival

evaluation

neighborhood migration

migration interval

terminal check

end

0P

0E

: populationtPtE : excellent solutions

tC

ttt CEP 2 individuals are selected from Pt + Et by tournament selection

: parents

tC ' : offsprings1tE

1' ttt PCC

tC '

2 individuals are sampled withoutreplacement from Ct + C’tand replace bad 2 individuals of Pt.


Adaptive change of the weight parameters

• Weight values are changed by following equation

)1,(),1(

),1()1(

)1,(),1(

)1,()1('

nnnn

nnn

nnnn

nnnn

dd

dw

dd

dww

nw),( bad : distance between islands a and b.

: weight value of nth island.

f2(x

)

f1(x)

f2(x

)

f1(x)

Change

Island 1

Island 2Island 3

d(1,2)

d(2,3)


MOGADES: Multi-Objective Genetic Algorithm with Distributed Environment Scheme Intelligent Systems Design Laboratory ， Doshisha University ， Kyoto Japan.

Documents

multiobjective problems

weight values of islands

ith objective functionthe

weight parametersf2xf1xto

weight of item j

neighborhood islands

mogasgenetic algorithm

good fitness