Doshisha Univ., Kyoto Japan NCGA : Neighborhood Cultivation Genetic Algorithm for Multi-Objective Optimization Problems Intelligent Systems Design Laboratory，

Doshisha Univ., Kyoto Japan

NCGA : Neighborhood Cultivation Genetic Algorithm

for Multi-Objective Optimization Problems

Intelligent Systems Design Laboratory，Doshisha University， Kyoto Japan

○ Shinya Watanabe Tomoyuki Hiroyasu

Mitsunori Miki


Multi-objective Optimization Problems●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.

f 1(x)

f 2(x)

Design variables

Objective function

ConstraintsGi(x)<0 ( i = 1, 2, … , k)

F={f1(x), f2(x), … , fm(x)}

X={x1, x2, …. , xn} Feasible regionFeasible region

Pareto optimal solutions


• MOPs solved by Evolutionary algorithms

EMO

•VEGA 　 :Schaffer (1985)

•MOGA :Fonseca (1993)

•DRMOGA :Hiroyasu, Miki, Watanabe (2000)

• SPEA2 :Zitzler (2001)

•NPGA2 :Erickson, Mayer, Horn (2001)

•NSGA-II :Deb, Goel (2001)

Typical method on EMO

• EMOEvolutionary Multi-criterion Optimization


• The following topics are the mechanisms that the recent GA approaches have.

EMO

• Archive of the excellent solutions• Cut down (sharing) method of the reserved excellent solu

tions• An appropriate assign of fitness• Reflection to search solutions mechanism of the reserve

d excellent solutions• Unification mechanism of values of each objective


• NCGA : Neighborhood Cultivation GA

• The neighborhood crossover• Archive of the excellent solutions• Cut down (sharing) method of the reserved

excellent solutions• An appropriate assign of fitness• Reflection to search solutions mechanism of the

reserved excellent solutions• Unification mechanism of values of each objective

The features of NCGA

Neighborhood Cultivation GA (NCGA)


• A neighborhood crossover– In MOPs GA, the searching area is wide and the searc

hing area of each individuals are different.

f2(x)

f1(x)

If the distance between two selected parents is so large, cross over may have no effect for local search.

Neighborhood Cultivation GA (NCGA)


• One of the objectives is changed at every generation.

• The pair for the mating is changed based on a probabillity.

f2(x)

f1(x)

Neighborhood Cultivation GA (NCGA)• A neighborhood crossover

• Two parents of crossover are chosen from the top of the sorted individuals.

In order not to make the same couple.


• Continuous Function– ZDT4

]5,5[]1,0[

)4cos(1091)(

)(1)()(min

)(min

1

10

2

2

12

11

i

iii

xx

xxxg

xgxxgxf

xxf

Test Problems


• Continuous Function– KUR

100,,1,]5,5[

)sin(5||)(min

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

2

100

1

21

21

nnixxxxf

xxxf

i

ii

i ii

Test Problems


Objectives

Constraints

• Combination problem– KP 750-2

2,1)(750

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 ici,= capacity of knapsack i


Applied models and ParametersGA Operator•Applied models

• Crossover– One point crossover

• Mutation– bit flip

• SPEA2• NSGA-II• NCGA• non-NCGA

(NCGA except neighborhood croosover )

population size 100crossover rate 1.0mutation rate 0.01

Parameters

terminal condition 250

250

2000number of trial 30


Results (Pareto solutions of ZDT4)


Results (Pareto solutions of KUR)


Results (Pareto solutions of KP750-2)


• We proposed a new model for Multi-objective GA.– NCGA: Neighborhood Cultivation GA

Effective method for multi objective GA • Neighborhood crossover• Reservation mechanism of the excellent solutions• Reflection to search solutions mechanism of the reserved

excellent solutions• Cut down (sharing) method of the reserved excellent soluti

ons• Assignment method of fitness function• Unification mechanism of values of each objective

Conclusion


• NCGA was applied to test functions and results were compared to the other methods; those are SPEA2, NSGA-II and non-NCGA.

• In some the test functions, NCGA derives the good results.

• Comparing to NCGA and NCGA without neighborhood crossover, the former is obviously superior to the latter in all problems.

NCGA is good model of Multi-objective GA

Conclusion


Results (RNI of KP750-2)


• EMO 全般に関して– http://www.lania.mx/~ccoello/EMOO/EMOObib.html

• 多目的 0 /1 ナップザック問題に関して– http://www.tik.ee.ethz.ch/~zitzler/

• 発表に用いたソースプログラム– http://mikilab.doshisha.ac.jp/dia/research/mop_ga/archive/

• 発表者の電子メールアドレス– [email protected]

参照 URL

http://www.lania.mx/~ccoello/EMOO/EMOObib.html

http://www.tik.ee.ethz.ch/~zitzler/


• The Ratio of Non-dominated Individuals (RNI) is derived from two types of Pareto solutions.

Performance Measure

(x)f 1

f 2(x

) Method B

(x)f 1

f 2(x

) Method A

(x)f 1

f 2(x

)

Method AMethod B

0.3330.666


Results (RNI of KUR)


Performance Assessment• The Ratio of Non-dominated Individuals :RNI

– The Performance measure perform to compare two type of Pareto solutions.

– Two types of pareto solutions derived by difference methods are compared.

• Cover Rate Index– Diversity of the Pareto optimum.

• Error – The distance between the real pareto front and

derived solutions.• Various rate

– Diversity of the pareto optimum individuals.

Measures


数値結果 ( KP750-2 )


• Continuous Function– ZDT4

]5,5[]1,0[

)4cos(1091)(

)(1)()(min

)(min

1

10

2

2

12

11

i

iii

xx

xxxg

xgxxgxf

xxf

Test Problems


Results (Pareto solutions of ZDT4)


Results (RNI of ZDT4)


数値結果 (ZDT4)


数値結果 (KUR)


数値結果 (ZDT4)


数値結果 (KUR)


数値結果 ( KP750-2 )


• パレート保存個体群の利用– 多目的では，最終的に求める解候補（パレート

解）が複数存在するため，探索途中での優良な個体の欠落を防ぐ必要がある．

f2(x

)

f1(x)

探索個体群優良個体保存群探索個体群に優良個体群を反映させることにより探索の高速化，効率化を期待することができる．

近傍培養型マスタースレーブモデル


多目的多目的 GAGA では，求める解が複数存在するためでは，求める解が複数存在するため単一目的と比較して，単一目的と比較して，十分な個体数と探索世代数十分な個体数と探索世代数が必要となる．が必要となる．

多目的 GA の問題点

・探索効率の良いアルゴリズム・探索効率の良いアルゴリズム

・膨大な評価計算回数

・非常に高い計算負荷


GA による多目的最適化への応用・多目的・多目的 GAGA

交叉・突然変異を用いてパレート最適解集合の探索を行う

ff11(x)(x)

ff 22(x)

(x)

1st generation5th generation

10th generation50th generation30th generation


Multi-Criterion Optimization Problems(2)

・・ Pareto dominant and Ranking methodPareto dominant and Ranking method

Pareto-optimal Set

Ranking

Rank ＝１＋ number of dominant individuals

The set of non-inferior individuals in each

generation.

f2

f1

1

3

1Pareto optimal solutions


クラスタシステム

Spec. of Cluster (16 nodes)Processor Pentium

(Coppermine)ⅢClock 600MHz# Processors 1 × 16Main memory 256Mbytes × 16Network Fast Ethernet (100Mbps)Communication TCP/IP, MPICH 1.2.1OS Linux 2.4Compiler gcc 2.95.4

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Doshisha Univ., Kyoto Japan NCGA : Neighborhood Cultivation Genetic Algorithm for Multi-Objective Optimization Problems Intelligent Systems Design Laboratory，