DCMOGA: Distributed Cooperation model of Multi-Objective Genetic Algorithm

Tamaki Okuda ● Tomoyuki Hiroyasu 　

Mitsunori Miki 　Shinya Watanabe 　

Doshisha University, Kyoto Japan

Multi-objective Optimization Problems(MOPs)In the optimization problems, when there are several objective functions, the problems are called multi-objective or multi-criterion problems.

Design variables

X = { x1, x2, … , xn }

Objective function

F = { f1(x), f2(x), … , fm(x) }

Constrains

Gi(x) < 0 ( i = 1, 2, … , k )

Multi-objective Optimization Problems

Feasible region

Pareto-optimal front

f1(x): Maximumf2(x): Maximum

Non-Dominated solutions

EMOs: Evolutionary Multi-criterion Optimization

Typical method on EMOs:– VEGA : Schaffer (1985)– MOGA : Fonseca (1993)– SPEA2 : Zitzler (2001)– NPGA2 : Erickson, Mayer, Horn (2001)– NSGA-II : Deb (2001)

Good non-dominated solutions:– Minimal distance to the Pareto-optimal front– Uniform distribution– Maximum spreading

>> Proposed a new model of EMOs.This model searches for non-dominated solutions, which

are widespread and closed to pareto-optimal front, and can make the existing algorithms more efficient.

DCMOGA

DCMOGA: Distributed Cooperation model

of Multi-Objective GA

DC-scheme for MOGA

The features of DC-scheme:– N+1 sub populations (when N objects)

1 group: searches for pareto optimum by MOGA

N groups: searches for optimum by SOGA

(Single Objective GA)– Cooperative search

Exchanged between each group

Adjustment of each sub population size

N+1 groups (sub populations)

MOGA group:The Pareto-optimal solutions are searched by MOGA

SOGA groups:The optimum of ith objective function is searched by SOGA.

Cooperative search (1)

Exchange of the solutions:Exchange the best solutions between the MOGA group and

each SOGA group.

Cooperative search (2)

Adjustment of each sub population size:– Some individuals move to other group.– This adjustment is according to the function values of best

solution in each group.– Each sub population size changes, but the whole population

size don’t change.

>> The difference between search ability of each group is reduce

The algorithm of DC-scheme>> N objective function1. all individuals are

initialized. 2. all individuals are divided

into N+1 groups.3. In the MOGA group, the

pareto optimal solutions are searched, and in each SOGA group the optimum is searched.

4. After some iterations, exchange the solutions between MOGA group and SOGA.

5. The exchanged solutions are compared, and sub each population size is adjusted.

6. The terminal condition is checked,

>> 2 objective functions

Combined MOGA and SOGA method

DCMOGA:Distributed Cooperation model of MOGADC-scheme

>> Combined MOGA and SOGA

DC-scheme can combine any MOGA and SOGA.

Used algorithms:

SOGA: DGA

MOGA: MOGA (Fonseca)

SPEA2 (Zitzler)

NSGA-II (Deb)

Test Problem: KP750-m

0/1 Knapsack Problem (750items, m knapsacks)

-Combination problems

Objectives:

Constraints:

3,2,1)(max 750

1,

jjjii ixpxf 　 　

750

1,

jijji cxw 　

1,0),,( 75021 jxxxxx 　 　

profit of item j according to knapsack i jip ,

weight of item j according to knapsack i jiw ,

capacity of knapsack i ic

Test Problem: KUR

KUR (2 objective function, 100 design variables )

- Continuous- Multi-modal

100

1

21

21 ))2.0exp(10()(min

i ii xxxf

38.02 )sin(5||)(min ii xxxf

100,,1,]5,5[ nnixi

Test Problem: ZDT4

ZDT4 ( 2 objective function, 10 design variables )- Continuous- Multi-modal

]5,5[]1,0[

)4cos(1091)(

)(1)()(min

)(min

1

10

2

2

12

11

i

iii

xx

xxxg

xg

xxgxf

xxf

Performance Metrics Function(C)

Coverage of two sets: C

||

|}:|{|:),(

B

baAaBbBAC

A: front1

B: front2

C(A,B) = 1/5 = 0.2

C(B,A) = 2/4 = 0.5

Applied models and Parameters

Applied models– MOGA / DCMOGA– SPEA2 / DCSPEA2– NSGA-II / DCNSGA-II

Parameters

GA operator

Crossover:

2 points crossover

Mutation:bit flip

KP750-2 KUR ZDT4

Chromosome length 750 2000 200

Population size 250 100 100

Crossover Rate 1.0

Mutation Rate 1/L (L: Chromosome length)

Terminal Condition 5 x 105 10 x 106 2.5 x 104

Number of trial 30

Results: KP750-3MOGADCMOGA

<< with DC-scheme is more widespread

<< without DC-scheme is closer to Pareto optimum

Results: KP750-3SPEA2DCSPEA2

<< with DC-scheme is more widespread

Results: KP750-3NSGA-IIDCNSGA-II

<< with DC-scheme is more widespread

<< without DC-scheme is closer to Pareto optimum

Results: KP750-3

The Coverage of two sets

>> both results are almost the same.

<< Without DC superior than with DC

<< With DC superior than without DC

<< Without DC superior than with DC

Results: KUR

MOGADCMOGA

SPEA2DCSPEA2

Results: KUR

<< With DC superior than without DC

<< With DC superior than without DC

<< With DC superior than without DC

>> With DC-scheme is superior

NSGA-IIDCNSGA-II

Results: ZDT4

Results: ZDT4

<< With DC superior than without DC

<< With DC superior than without DC

<< With DC superior than without DC

>> With DC-scheme is superior

Conclusion

We proposed a new model of EMOs.– DCMOGA (DC-scheme):

Distributed Cooperation model of MOGA

Compared the algorithms combined with DC-scheme against algorithm without DC-scheme.– The algorithms combined DC-scheme derives the efficient

results.– DC-scheme can be combined any other EMOs, and the

algorithm is more efficient algorithm than without DC-scheme.

>> DC-scheme is efficient scheme for EMOs.

Performance Metrics

RNI: Ratio of Non-dominated Individualsderived from 2 types of non-dominated solutions

Set A: front1

Set B: front2

Set A: 3/5 = 0.6Set B: 2/5 = 0.4

Results: ZDT6

Results: ZDT6

Results: KP750-2

Results: KP750-2

Performance Metrics: Coverage(D)

Coverage difference of two sets: D

)()(:),( BSBASBAD

)(AS

)(BS

)( BAS

),( BAD

),( ABD

The improved MOGA

MOGA is proposed by Fonseca

Improved MOGA:

MOGA added to the Sharing and using pareto archive.

SOGA and MOGA

DC-scheme are compared SOGA and MOGA

>> SOGA and MOGASOGA: 300 generation x 2MOGA: 400