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
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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…
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• 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
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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.
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• 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
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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)
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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
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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.
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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
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The overview of MOGADESf2
(x)
f1(x)
searching direction
changed weight
neighborhood migration
non-dominated
archive
individual
island
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• 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
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• 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
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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
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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
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ZDT4
MOGADES is superior toNSGA-II and SPEA2
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KUR
MOGADES is superior toNSGA-II and SPEA2
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KP750-3
MOGADES is superior toNSGA-II and SPEA2
18
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• 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.
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Doshisha Univ., Kyoto Japan21
• 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
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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
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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.
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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)
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