1. A comparative analysis of selection schemes used in genetic algorithms
David E. Goldberg
Kalyanmoy Deb

2. What is the paper about? Defines and compare four selection schemes
Presents a technique for comparisons:
Produce a difference/differential equation modeling the selection scheme
Test computer implementation against diff. equation model
Defines criteria for comparison:
Convergence time
Schema growth ratios
Conclusions: practical applications of analysis
Genetic Algorithms

3. Where are we now?
Many papers claim the superiority of this or that selection scheme
But many of these claims are based on limited (and uncontrolled experiments).
Little analysis has been done
This paper attempts to provide the needed analysis
Genetic Algorithms

4. What selection strategies?
Proportionate reproduction
Ranking selection
Tournament selection
Genitor (“steady state”) selection
Genetic Algorithms

5. Birth, life, and death m(i, t+1) = m(i, t) + m(i, t, b) – m(i,t,d)
Ex: in non-overlapping population models:
m(i,t+1) = m(i,t,b) ; m(i,t,d) = m(i,t)
We can also do proportions:
P(i,t+1) = P(i,t) + P(i,t,b) – P(i,t,d)
Genetic Algorithms

6. Proportionate Reproduction
Probability of selection:
prob(i,t) = f(i)/∑m(j,t) f(j)
Various methods for implementation:
Roulette wheel
Stochastic remainder
Stochastic universal
Genetic Algorithms

7. How many in next generation?
m(i,t+1) = m(i,t) * n * prob(i,t)
m(i,t+1) = m(i,t) * f(i)/f(avg,t)
Proportion(i,t+1) = Proportion(i,t) * f(i)/f(avg,t)
Genetic Algorithms

8. Graph of Eqn, implementation
Genetic Algorithms
Convergence behavior

9. Takeover time
How many individuals between specified values of x in objective function f(x)?
Let p0(x) be uniform, integral  1
Consider f(x) = xc and f(x) = ecx
Limits x and x – 1/n
Genetic Algorithms

10. Behavior of f(x) = x^c Integrate with limits x & x – 1/n
x = 1 is best, x = 0 is worst individual
Compare theory and experiment for f(x) = x
Genetic Algorithms

11. Takeover time for f(x) = x^c
Solving for t and approximating
Setting x = 1, we get proportion of best individual
Setting P = n-1/n, we calculate when population contains n-1 best individuals
Thus the takeover time for a polynomially distributed objective function is O(nlogn)
Genetic Algorithms

12. Takeover time for f(x) = e^cx
The takeover time for a polynomially and exponentially distributed objective function is O(nlogn)
Genetic Algorithms

13. Time complexity of Proportionate Reproduction
Roulette Wheel
O(n2) or O(nlogn) with binary search
Stochastic remainder selection
floor(f(i)/favg) number of copies
Remainder = flip(fractional(f(i)/favg))
O(n) without replacement or O(n2) with
Genetic Algorithms

14. Ranking Sort from best to worst
Create a transformation function called an assignment function that converts a rank to an equivalent “fitness”
assignFunction(rank)
Proportionate reproduction on assignFunction(rank)
Genetic Algorithms

15. Tournament Selection Binary N-ary
Randomly choose N individuals from population
Select best for further processing
Genetic Algorithms

16. Binary Tournament
Tournament size = 3
Genetic Algorithms

17. Tournaments
Genetic Algorithms

18. Genitor Choose an offspring based on ranking
Replace worst individual in population
Genetic Algorithms

19. Genitor
Genetic Algorithms

20. Growth Comparison
Genetic Algorithms

21. Takeover time comparison
Genetic Algorithms

22. Time complexity
Genetic Algorithms

23. Conclusions
The paper provides a framework for comparing selection operators
Compares selection “pressure” for each type of selection
Introduces the concept of takeover time to help us understand the exploration/exploitation tradeoff
Provides takeover time estimates for different types of selection
Implications for genetic search
The models provide us with theory necessary to compare selection methods and
Balance growth ratio (quick convergence) with higher crossover/mutation (more exploration)
Genetic Algorithms