1. Self-Adaptive Simulated Binary Crossover
for Real-Parameter Optimization
Kalyanmoy Deb*, S Karthik*, Tatsuya Okabe*
GECCO 2007
*Indian Inst. India
**Honda Research Inst., Japan
Reviewed by
Paskorn Champrasert
paskorn@cs.umb.edu
http://dssg.cs.umb.edu

2. Outline
• Simulated Binary Crossover (SBX Crossover )
(revisit)
• Problem in SBX
• Self-Adaptive SBX
• Simulation Results
– Self-Adaptive SBX for Single-OOP
– Self-Adaptive SBX for Multi-OOP
• Conclusion
November 7, 08 DSSG Group Meeting 2/31

3. SBX
Simulated Binary Crossover
• Simulated Binary Crossover (SBX) was proposed in
1995 by Deb and Agrawal.
• SBX was designed with respect to the one-point
crossover properties in binary-coded GA.
– Average Property:
The average of the decoded parameter values is the same
before and after the crossover operation.
– Spread Factor Property:
The probability of occurrence of spread factor β ≈ 1 is more
likely than any other β value.
November 7, 08 DSSG Group Meeting 3/31

4. Important Property of One-Point Crossover
Spread factor:
Spread factor β is defined as the ratio of the spread of
offspring points to that of the parent points:
c1 −c2
β= | p1 −p2 |
c1 c2
- Contracting Crossover β<1 p1 p2
The offspring points are enclosed by the parent points.
- Expanding Crossover β>1 c1 c2
The offspring points enclose by the parent points. p1 p2
- Stationary Crossover β=1 c1 c2
The offspring points are the same as parent points p1 p2
November 7, 08 DSSG Group Meeting 4/31

5. Spread Factor (β) in SBX
• The probability distribution of β in SBX should be similar (e.g.
same shape) to the probability distribution of β in Binary-coded
crossover.
c(β) = 0.5(nc + 1)β nc , β ≤ 1
• The probability distribution
function is defined as: c(β) = 0.5(nc + 1) β n1+2 , β > 1
c
1.6
1.4
Probability Distribution
1.2
1.0 n=2
0.8
0.6
0.4
0.2
0.0
0 1 2 3 4 5 6 7
β
November 7, 08 DSSG Group Meeting 5/31

6. A large value of n gives a higher
probability for creating ‘near-parent’
contracting solutions.
c(β) = 0.5(n + 1)β n , β ≤ 1
expanding
1
c(β) = 0.5(n + 1) β n+2 , β > 1
Mostly, n=2 to 5
November 7, 08 DSSG Group Meeting 6/31

7. Spread Factor (β) as a random number
• β is a random number that follows the proposed
probability distribution function:
1.6
1.4
To get a β value,
Probability Distribution
1.2
1) A unified random number u
n=2
is generated [0,1]
1.0
0.8
0.6 2) Get β value that makes the
0.4
0.2
area under the curve = u
0.0
u
0 1 2 3 4 5 6 7
β Offspring:
β
c1 = x − 1 β(p2 − p1 )
¯ 2
c2 = x + 1 β(p2 − p1 )
¯ 2
November 7, 08 DSSG Group Meeting 7/31

8. SBX Summary
• Simulated Binary Crossover (SBX) is a real-parameter
combination operator which is commonly used in
evolutionary algorithms (EA) to optimization problems.
– SBX operator uses two parents and creates two offspring.
– SBX operator uses a parameter called the distribution index
(nc) which is kept fixed (i.e., 2 or 5) throughout a simulation
run.
• If nc is large the offspring solutions are close to the parent solutions
• If nc is small the offspring solutions are away from the parent solutions
– The distribution index (nc) has a direct effect in controlling the
spread of offspring solutions.
November 7, 08 DSSG Group Meeting 8/31

9. Research Issues
• Finding the distribution index (nc) to an appropriate value
is an important task.
– The distribution index (nc) has effect in
1) Convergence speed
if nc is very high the offspring solutions are very close to the parent solutions the
convergence speed is very low.
2) Local/Global optimum solutions.
Past studies of SBX crossover GA found that it cannot work well in
complicated problems (e.g., Restrigin’s function)
November 7, 08 DSSG Group Meeting 9/31

10. Self-Adaptive SBX
• This paper proposed a procedure for updating the
distribution index (nc) adaptively to solve optimization
problems.
– The distribution index (nc) is updated every generation.
– How the distribution index of the next generation (nc) is
updated (increased or decreased) depends on how the
offspring outperform (has better fitness value) than its two
parents.
• The next slides will show the process to make an
equation to update the distribution index.
November 7, 08 DSSG Group Meeting 10/31

11. Probability density per offspring
Spread factor distribution
1.6
1.4
(1) c(β) = 0.5(nc + 1)β nc , β ≤ 1
Probability Distribution
1.2 (1) (2)
1.0 nc = 2
(2) c(β) = 0.5(nc + 1) β n1+2 , β > 1
c
0.8
0.6 ¯
An example, when p1=2, p2=5 β>1
0.4
0.2
0.0 u
0 1 2 3 4 5 6 7
β
β (1) (2)
Offspring:
¯ 2¯
c1 = x − 1 β(p2 − p1 ) (1) (2)
¯ 2¯
c2 = x + 1 β(p2 − p1 )
November 7, 08 DSSG Group Meeting 11/31

12. Self-Adaptive SBX (1)
Consider the case that β > 1:
c1 c2
β>1
p1 p2
c2 −c1
β= | p2 −p1 |
(c2 −p2 )+(p2 −p1 )+(p1 −c1 )
β= p2 −p1
(c2 − p2 ) = (p1 − c1 )
2(c2 −p2 )
β =1+ p2 −p1
--- (1)
November 7, 08 DSSG Group Meeting 12/31

13. Self-Adaptive SBX (2)
1.6
1.4
(1) c(β) = 0.5(nc + 1)β nc , β ≤ 1
Probability Distribution
1.2 (1) (2)
1.0
(2) c(β) = 0.5(nc + 1) β n1+2 , β > 1
c
0.8
0.6
0.4
0.2
0.0
u
0 1 2 3 4 5 6 7
β
β
(2)
Rβ 0.5(nc +1)
= (u − 0.5) --- (2)
1 β nc +2 dβ
November 7, 08 DSSG Group Meeting 13/31

14. (2)
Self-Adaptive SBX (3)
Rβ 0.5(nc +1)
--- (2)
1 β nc +2 dβ = (u − 0.5)
Rβ
1
0.5(nc + 1)β −(nc +2) dβ = (u − 0.5)
−0.5β −nc −1 − (−0.5) = (u − 0.5)
−0.5β −nc −1 = (u − 1)
log(β −nc −1 ) = log − (u−1)
0.5
(−nc − 1) log β = log(−2(u − 1))
log 2(1−u)
(−nc − 1) = log β
log 2(1−u)
nc = −(1 + log β ) --- (3)
November 7, 08 DSSG Group Meeting 14/31

15. Self-Adaptive SBX (4)
c1 c2 c2
ˆ
β>1
p1 p2
• If c2 is better than both parent solutions,
– the authors assume that the region in which the child
solutions is created is better than the region in which
the parents solutions are.
– the authors intend to extend the child solution further
away from the closet parent solution (p2) by a factor α
(α >1)
(c2 − p2 ) = α(c2 − p2 )
ˆ --- (4)
November 7, 08 DSSG Group Meeting 15/31

16. β Update
c1 c2 c2
ˆ
β>1 p1 p2
2(c2 −p2 )
β =1+ p2 −p1
--- (1) (c2 − p2) = α(c2 − p2) --- (4)
ˆ
from (1) ˆ
β =1+ 2(c2 −p2 )
ˆ
p2 −p1
2((α(c2 −p2 )+p2 )−p2 )
=1+ p2 −p1
α2(c2 −p2 )
=1+ p2 −p1
β−1
ˆ
β = 1 + α(β − 1) --- (5)
November 7, 08 DSSG Group Meeting 16/31

17. Distribution Index (nc) Update
nc = −(1 + log 2(1−u)
) --- (3) ˆ
β = 1 +α(β −1) --- (5)
log β
log 2(1−u)
nc = −(1 +
ˆ ˆ
log β
)
log 2(1−u)
nc = −(1 +
ˆ log(1+α(β−1)) )
log 2(1 − u) = −(nc + 1) log β
(nc +1) log β
nc = −1 +
ˆ log(1+α(β−1)) --- (6)
November 7, 08 DSSG Group Meeting 17/31

18. Distribution Index (nc) Update
When c is better than p1 and p2 then the distribution index for the
next generation is calculated as (6). The distribution index will be
decreased in order to create an offspring (c’) further away from its
parent(p2) than the offspring (c) in the previous generation.
(nc +1) log β
nc = −1 +
ˆ log(1+α(β−1)) --- (6) [0,50]
α is a constant (α >1 )
November 7, 08 DSSG Group Meeting 18/31

19. Distribution Index (nc) Update
When c is worse than p1 and p2 then the distribution index for the
next generation is calculated as (7). The distribution index will be
increased in order to create an offspring (c’) closer to its
parent(p2) than the offspring (c) in the previous generation.
The 1/α is used instead of α
(nc +1) log β
nc = −1 +
ˆ log(1+α(β−1)) --- (6)
(nc +1) log β
nc = −1 +
ˆ log(1+(β−1)/α) --- (7)
α is a constant (α >1 )
November 7, 08 DSSG Group Meeting 19/31

20. Distribution Index (nc) Update
So, we are done with the expanding case (2)
How about contracting case (1) ?
The same process is applied for (1)
1.6
1.4
(1) c(β) = 0.5(nc + 1)β nc , β ≤ 1
Probability Distribution
1.2
1.0
(2) c(β) = 0.5(nc + 1) β n1+2 , β > 1
c
0.8 (1) (2)
0.6
0.4
0.2
0.0
u
0 1 2 3 4 5 6 7
β
β (1)
Rβ
0
0.5(nc + 1)β nc dβ = u --- (x)
November 7, 08 DSSG Group Meeting 20/31

21. Distribution Index (nc) Update
for contracting case
When c is better than p1 and p2 then the distribution index for the
next generation is calculated as (8). The distribution index will be
decreased in order to create an offspring (c’) father away from its
parent(p2) than the offspring (c) in the previous generation.
1+nc --- (8)
nc =
ˆ α −1
When c is worse than p1 and p2, 1/α is used instead of α
nc = α(1 + nc ) − 1
ˆ --- (9)
α is a constant (α >1 )
November 7, 08 DSSG Group Meeting 21/31

22. Distribution Index Update Summary
Expanding Case Contracting Case
(u > 0.5) (u<0.5)
c is better than parents
1+nc
nc = −1 +
ˆ (nc +1) log β ˆ
nc = α −1
log(1+α(β−1))
c is worse than parents
(nc +1) log β
nc = −1 +
ˆ log(1+(β−1)/α)
nc =α(1+nc) −1
ˆ
(u = 0.5), nc = nc
ˆ
November 7, 08 DSSG Group Meeting 22/31

23. Self-SBX Main Loop
Update nc
If rand < pc
Parent 1 Offspring C1
Pt
Selection Crossover
initial Parent 2 Offspring C2
population
If rand > pc
N = 150 Mutation
nc = 2
α = 1.5
Pt+1
Population
at t+1
N = 150
α = 1.5
November 7, 08 DSSG Group Meeting 23/31

24. Simulation Results
Sphere Problem:
SBX
• Initial nc is 2.
• Population size =150
• Pc = 0.9
• The number of variables (n) =30
• Pm =1/30
• A run is terminated when a solution having a function value = 0.001
• 11 runs are applied.
Self-SBX
• Pc = 0.7
• α = 1.5
November 7, 08 DSSG Group Meeting 24/31

25. Fitness Value
• Self-SBX can
find better
solutions
November 7, 08 DSSG Group Meeting 25/31

26. α Study
• α is varied in [1.05,2]
• The effect of α is not
significant since the
average number of
function evaluations is
similar.
November 7, 08 DSSG Group Meeting 26/31

27. Simulation Results
Rastrigin’s Function:
Many local optima, one global minimum (0)
The number of variables = 20
SBX
• Initial nc is 2.
• Population size =100
• Pc = 0.9
• The number of variables (n) =30
• Pm =1/30
• A run is terminated when a solution having a function value = 0.001 or the maximum
of 40,000 generations is reached.
• 11 runs are applied.
Self-SBX
• Pc = 0.7
• α = 1.5
November 7, 08 DSSG Group Meeting 27/31

28. Fitness Value
Best Median Worst
• Self-SBX can find a very close global optimal
November 7, 08 DSSG Group Meeting 28/31

29. α Study
• α is varied in
[1.05,10]
• The effect of α is
not significant since
the average number
of function
evaluations is
similar.
• α = 3 is the best
November 7, 08 DSSG Group Meeting 29/31

30. Self-SBX in MOOP
• Self-SBX is applied in
NSGA2-II.
• Larger hyper-volume is
better
November 7, 08 DSSG Group Meeting 30/31

31. Conclusion
• Self-SBX is proposed in this paper.
– The distribution index is adjust adaptively on the run
time.
• The simulation results show that Self-SBX can
find the better solutions in single objective
optimization problems and multi-objective
optimization problems.
• However, α is used to tune up the distribution
index.
November 7, 08 DSSG Group Meeting 31/31