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Algorithm: Annealing-PAES
Data:
Mt is the archive of solution vectors discovered
x is the current solution vector
x is the mutant solution vector
T is the temperature parameter and Tstart is its initial value
Functions:
Terminate(t Mt ) returns TRUE if a termination condition based on the archive
or number of iterations is satised
Init() returns a solution vector in x 2 X
Mutate(x) returns a neighbour of x
Grpop(x) returns the population of x's grid region
ND(X ) returns the nondominated vectors from a set X
Sm(Mt T ) takes a sample of the solutions in Mt . The size of the sample depends
on the temperature, T
Cool(T Mt ) returns a new value of T depending on the current value of T , and
possibly the state of Mt
0
0
x 2 X Init()
t t +1
T Tstart
Mt Mt 1 fxg
while (Terminate(t Mt ) 6= TRUE)
t t+1
x 2 X Mutate(x)
if (x < x)
xx
Mt ND(fx g Mt 1 )
/* Initialization */
t
Mt
;
/* Main Loop */
0
0
0
0
else
if (x
t 1)
ND
(fx g t 1 )
t
xx
else if (x Sm( t 1 ) )
if (j t 1 j a
)
t fx g t 1
else if (x increases the extent of the grid)
cr
cr
t fx g t 1 n fx g where x is one
randomly selected solution from the
set fxi 2 t 1 j 8xj 2 t 1 Grpop(xi ) Grpop(xj )g
else if (9xi 2 t 1 such that Grpop(x ) Grpop(xi ))
cr
t fx g t 1 n fx g
if (Grpop(x ) Grpop(x))
xx
Cool(
t)
return M( final )
/* Termination */
0
< M
;
;
0
M
M
;
0
0
M
M
<
;
;
0
M
T
rcsize
M
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0
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Algorithm: Tabu-PAES
Data:
Mt is the archive of solution vectors discovered
x is the current solution vector
is the number of neighbour solutions generated at each iteration
N is the set of neighbour solutions of x
B is the set of best solutions from among N
Temp is a temporary set used for updating the archive with the new solutions
t abu l is the set of tabu lists
a ttr l is the set of attribute lists
Functions:
Terminate(t Mt ) returns TRUE if a termination condition based on the archive
or number of iterations is satised
Init() returns a solution vector in x 2 X
Mutate(x) returns a neighbour of x
ND(X ) returns the nondominated vectors from a set X
Tabu(N t abu l a ttr l) returns the subset of N allowed by the current tabu lists
and attribute lists
Eq best(X Y ) returns the set of solutions in Y which are in the best nondominated sorting equivalence class of X
Low pop(X Y ) returns the solutions in Y the from the least crowded regions
in X
Rand(X ) returns a uniformly randomly selected solution from X
Most crowded(X ) returns the solutions in X from amongst the most crowded
regions
t0
Mt x 2 X Init()
tt+1
T Tstart
Mt Mt;1 fxg
/* Initialization */
while (Terminate(t Mt ) 6= TRUE)
t t+1
N foreach(j 2 1::)
xj 2 X Mutate(x)
N N fxj g
B Tabu(N t abu l a ttr l)
B Eq best(B Mt;1 B )
B Low pop(B Mt;1 B )
x Rand(B )
Temp ND(B Mt;1 )
while (jTempj > a rcsize)
Temp Temp n Most crowded(Temp)
Update tabu l
Update attr l
return M(tfinal )
/* Main Loop */
/* Termination */
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40
30
20
10
0
-6
-4
-2
y = f21
y = f220000
1111
4
x
6
U U , %0
,J%0
+ (+* 1 ( ( ( N % K J9
111
000
Pareto optimal range 111
000
y
y
8
6
y
4
2
0
-2
-2
1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0
2
x
= f31
= f32
111
000
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
111
000
111
000
4
6
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N09
y = f41
y = f42
y
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
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Best
1
1
NSGA+ARC
NPGA+ARC
PAES(1+1)
0.8
NSGA+ARC
NPGA+ARC
PAES(1+1)
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Worst
1
NSGA+ARC
NPGA+ARC
PAES(1+1)
0.8
0.6
0.4
0.2
0
0
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0.6
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10000
2-500
20000
Total profit (knapsack 2)
Total profit (knapsack 2)
19500
9500
9000
8500
19000
18500
18000
17500
17000
8000
8000
8500
9000
9500
10000
16500
15500
16000
16500
Total profit (knapsack 1)
17000
17500
18000
18500
19000
19500
20000
Total profit (knapsack 1)
2-750
29000
Total profit (knapsack 2)
28500
Key :
28000
27500
27000
26500
26000
MPAES
(1+1)-PAES
SPEA(ZT)
SPEA(KC)
SO_5
25500
25000
25000
25500
26000
26500
27000
27500
28000
28500
29000
Total profit (knapsack 1)
7 9 - ;& (+*
)
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2-250
2-500
20000
10000
9800
Total profit (knapsack 2)
Total profit (knapsack 2)
19500
9600
9400
9200
9000
19000
18500
18000
8800
8600
8000
8500
9000
9500
Total profit (knapsack 1)
10000
17500
17000
17500
18000
18500
19000
Total profit (knapsack 1)
19500
20000
2-750
29000
28500
Total profit (knapsack 2)
28000
27500
Key:
27000
M-PAES
RD-MOGLS
26500
(1+1)-PAES
26000
25500
25000
25000 25500 26000 26500 27000 27500 28000 28500 29000
Total profit (knapsack 1)
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W
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solution that minimizes
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Tabular chromosome
vertex, v
l1 l 2 l 6 l 7
3
1 2 3 4 5 6 7
dv = 1 1, 2, 1, 3, 2, 2, 3
dv = 2 2, 2, 1, 1, 2, 1, 1
2
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no correlation
correlation=-0.95
w2
ij
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
w2
ij
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w1
w1
ij
ij
correlation=0.95
w2
ij
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w1
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correlation = 0.95
20
W
2
correlation = 0.7
20
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15
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1
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25
0
5
10
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15
20
25
1
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correlation = -0.95
correlation = -0.7
20
20
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20
15
2
10
5
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1
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0
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20
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lightest
heaviest
7 PNN9 4 N9 7 Edge weights key:
lightest
heaviest
7 PN%9 4 %9 ! N ( ++
Edge weights key:
lightest
heaviest
7 PN;9 4 ;9 4 <= - 4
Edge weights key:
lightest
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7 PNK9 4 K9 ! ( N
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