ncga : neighborhood cultivation genetic algorithm for multi-objective optimization problems

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

Constraints

Gi(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


• NCGA : Neighborhood Cultivation GA

• The neighborhood crossover.• Archive of excellent solutions.• A Method which cuts down reserved excellent

solutions.• Use of the reserved excellent solutions for

searching solutions.• Unification mechanism of the 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

searching area of each individual is different.

f2(x

)

f1(x)

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



• One of the objectives is changed at each generation.

• The sorting of a population includes a little probabilistic change.

f2(x

)

f1(x)

Neighborhood Cultivation GA (NCGA)• A neighborhood crossover

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

In order not to make the same couple,



• NCGA has the neighborhood crossover mechanism.

• NCGA has only one selection in one generation.• Many methods have two types of selection

(the environment selection and the mating selection). But, NCGA has the environment selection only.

•The differences from the recent major algorithms like SPEA2 and NSGA-II.


• Sampling of the Pareto frontier Lines of

Intersection (ILI) (Knowles and Corne 2000)

Comparison method

= 5/12=0.42= 7/12=0.58


• SPEA2• NSGA-II• NCGA• non-NCGA

(NCGA except neighborhood crossover )

Applied models and Parameters

GA OperatorApplied models• Crossover

– One point crossover

• Mutation– Bit flip

population size 100crossover rate 1.0mutation rate 0.01

Parameters

terminal condition 250

250

2000number of trials 30


• Discontinuous Function– Fdiscon (Deb’00)

Test Problems

100,,1,]1,0[

)10sin(11

)(101)(min

))2.0exp(10()(min

111 1

2

100

1

21

211

NNix

fg

f

N

xxf

xxxxf

i

N

i

i ii


Pareto solutions of Fdiscon


Comparison result of Fdiscon (ILI)


• Continuous Function– KUR

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


Pareto solutions of KUR


Comparison result of KUR (ILI)


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


Pareto solutions of KP750-2


Comparison result of KP750-2 (ILI)


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

Effective method for multi objective GA • The neighborhood crossover• Archive of excellent solutions.• A Method which cuts down reserved excellent solutions.• Use of the reserved excellent solutions for searching soluti

ons.• Unification mechanism of the values of each objective.

Conclusion


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

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

• Comparing NCGA to NCGA without neighborhood crossover, NCGA is obviously superior to in all problems.

NCGA is an effective algorithm for multi-objective problems.

Conclusion


• Continuous Function– ZDT4

]5,5[]1,0[

)4cos(1091)(

)(1)()(min

)(min

1

10

2

2

12

11

i

iii

xx

xxxg

xg

xxgxf

xxf

Test Problems


Pareto solutions of ZDT4


Comparison result of ZDT4 (ILI)


ILI of KP750-2


• About EMO– http://www.lania.mx/~ccoello/EMOO/EMOObib.ht

ml

• About 0/1 Knapsack problem– http://www.tik.ee.ethz.ch/~zitzler/

• NCGA source program– http://mikilab.doshisha.ac.jp/dia/research/mop_ga/

archive/

• My e-mail address– [email protected]

URL of reference


• 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


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

reserved excellent solutions• Unification mechanism of values of each objective


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


Cluster System

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

ncga : neighborhood cultivation genetic algorithm for multi-objective optimization problems

Documents

ncga ncga

reserved excellent solutions

multi objective ga

objective functions

multiobjective gas

multicriterion problems

comparison method

environment selection