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MOGADES: Multi-Objective Genetic Algorithm
with Distributed Environment Scheme
○ Jiro KAMIURA
Tomoyuki HIROYASU,
Mitsunori MIKI,
Shinya WATANABE
Intelligent Systems Design Laboratory，Doshisha University，Kyoto Japan
Multi-objective Optimization Problems : MOPs
Design variable
X={x1, x2, …. , xn}
f2 (x)
In the optimization problems, when there are several objective
functions, the problems are called multi-objective problems.
f1(x) : Minimize
f2(x) : Minimize
Objective function
F={f1(x), f2(x), … , fm(x)}
non-dominated solutions
Constraints
Gi(x)<0 ( i = 1, 2, … , k)
Solving MOPs needs huge calculation costs,
so we need the parallel model for solving MOPs.
2
f 1 (x)
Doshisha Univ., Kyoto Japan
Multi-Objective Genetic Algorithms : MOGAs
Genetic Algorithm for solving MOPs
Typical method on MOGAs
•VEGA
: Schaffer (1985)
•MOGA
: Fonseca (1993)
•SPEA2
: Zitzler (2001)
•NPGA2
: Erickson, Mayer, Horn (2001)
•NSGA-II : Deb, Goel (2001)
None of all is parallel model…
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Proposed method : MOGADES
• MOGADES : Multi-Objective Genetic Algorithm
with Distributed Environment Scheme
Features
• 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
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Distributed Genetic Algorithm : DGA
(Tanese ‘89)
One of the parallel models of GAs
A population is divided into smaller
subpopulations (islands)
Canonical GA is performed in each
island
Migration: Exchange of individuals
among islands
Distributed Environment Scheme
(Miki 1999): the environment (that
is crossover rate, mutation rate, and
so on) in each island are different.
DGA can show better performance than single population GAs
in solving single objective problems.
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Unification of the objective functions
k
Fitness value =
w
i 1
i
fi ( x )
f2(x)
• Assignment of fitness using the weighted-sum of each
objective function
• Using Distributed Environment Scheme :
Weight parameters are different in each island.
The searching
directions
k
w i  0,  w k  1
i 1
k
wi
:the number of objective functions
f1(x)
:the weight parameter of the ith objective function
fi ( x ) :the value of the ith objective function
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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
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w1 w2
1
1.0
0.0
2
0.75 0.25
3
0.5
4
0.25 0.75
5
0.0
f2(x)
island
searching
direction
0.5
1.0
f1(x)
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Adaptive change of the weight parameters
• To get good distributed non-dominated solutions.
• Performed in the migration phase.
f2(x)
f2(x)
e.g.) 2 objective functions, 3 islands
Island 1
d1
Island 2
Island 3
Change
d2
f1(x)
Island 1
Island 3
Island 2
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f1(x)
distance
Doshisha Univ., Kyoto Japan
Neighborhood migration
• Exchange individuals with neighborhood islands.
• The weight values of islands change.
Step 1. Sort islands by
wi
wi . ( i changes for each migration phase.)
island
neighborhood
3
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
f2(x)
Archive of
• the non-dominated solutions
• the solutions which have good fitness
: individuals
: non-dominated solutions
: solutions which have good fitness
: searching direction
f1(x)
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The overview of MOGADES
f2(x)
searching direction
changed weight
neighborhood
migration
island
non-dominated
archive
f1(x)
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individual
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Test Problems
• ZDT4
–
–
–
–
Continuous
2 objective functions
10 design variables
Multi-modal
min
f1 ( x)  x1
min

f 2 ( x)  g ( x)  1 

10
x1 

g ( x) 
g ( x)  91   xi  10 cos(4xi )
2
i2
x1  [0,1]
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xi  [5,5]
Doshisha Univ., Kyoto Japan
Test Problems
• KUR
–
–
–
–
Continuous
2 objective functions
100 design variables
Multi-modal
min
min
f1 ( x)  i 1 (10exp(0.2 xi  xi 1 ))
100
2
2
f 2 ( x)  | xi |0.8 5 sin(xi )3 
xi  [5,5] , i  1,, n n  100
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Test Problems
• 0/1 Knapsack Problem (750items 3knapsacks)
– Combination problem
Objectives
max f i ( x) 
750
p
j 1
i, j
750
Constraints
w
j 1
x  ( x1 , x2 , , x750 )
i, j
 xj
i  1,2,3
 x j  ci
x j  0,1
pi,j = profit of item j according to knapsack i
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
• SPEA2
• NSGA-II
• MOGADES
Parameters
• Crossover
– 2 points crossover
• Mutation
– bit flip
• Migration Interval
– 10 generations
ZDT4
chromosome length
population size
crossover rate
mutation rate
terminal condition
number of trials
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KUR
KP750-3
200
2000
750
100(10islands) 250(25islands)
1.0
1/(chromosome length)
50000 100000 1000000
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ZDT4
MOGADES is superior to
NSGA-II and SPEA2
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KUR
MOGADES is superior to
NSGA-II and SPEA2
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KP750-3
MOGADES is superior to
NSGA-II and SPEA2
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Conclusion
• We proposed a new model of MOGA.
– MOGADES: Multi-Objective Genetic Algorithm with
Distributed Environment Scheme
MOGADES is based on Distributed Genetic Algorithm
which is one of the parallel models, so MOGADES is
the parallel model, too.
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.
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Test Problems
• ZDT6
–
–
–
–
Continuous
2 objective functions
10 design variables
Non-convex
min
f1 ( x)  1  exp(4 x1 ) sin 6 (6x1 )
min
  f 2 
f 2 ( x)  g ( x)  1   1  
 g 


0.25

2 
  xi 
g ( x)  1  9 i  2 
 N  1




x1  [0,1] xi  [5,5]
10
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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
0.0, 1.0, 0.0
0
0.0, 0.5, 0.5
1
0.5, 0.5, 0.0
w3
1
0
0.0, 0.0, 1.0
w2
1
w1
6 islands
22
0
1.0, 0.0, 0.0
0.5, 0.0, 0.5
Random
8 islands
10 islands
Doshisha Univ., Kyoto Japan
The flow of MOGADES
initialization
Pt
Et
P0
: population
: excellent solutions
evaluation ( includes reservation of the excellent solutions)
selection for reproduction
mutation
C' t
evaluation
Et
neighborhood migration
end
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Ct
C' t
 1
selection for survival
terminal check
Pt  Et  Ct
2 individuals are selected from Pt + Et
by tournament selection
migration interval
crossover
E0
Ct  C ' t  Pt
: parents
: offsprings
 1
2 individuals are sampled without
replacement from Ct + C’t
and replace bad 2 individuals of Pt.
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Adaptive change of the weight parameters
• Weight values are changed by following equation
wn : weight value of nth island.
: distance between islands a and b.
f2(x)
d ( n, n  1)
w' n  w( n  1)
d ( n  1, n )  d ( n, n
Island 1
d(1,2
)
Island 2
Island 3
 1)
 w( n
f2(x)
d (a, b)
 1)
d (n
d (n
 1, n )
 1, n )
 d ( n, n
 1)
Change
d(2,3
)
f1(x)
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f1(x)
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