1. Edge Assembly Crossover
An Analysis of
Edge Assembly Crossover
for the Traveling Salesman Problem
Yuichi Nagata and Shigenobu Kobayashi
IEEE, Conference on Evolutionary Computation, 1999

2. Genetic Algorithm Holland, 1975 -
Imitation of Evolution of life form in the Nature
individuals - members of species - in the nature
model of evolution processes
where the basic operations are natural selection, crossover and mutation
Schema Theorem - analysis of reproduction model
Nothing to do with the real genetic organism

3. Problem Solving Search Space No Calculation  Search for the Solution
Polynomial time function
y = ax2 + b
Given constant a, b, know x, calculate y
Find x that gives Maximum y in a given range of x
No Calculation  Search for the Solution
Search Space
For small space, use classical exhaustive techniques
For larger space, need special techniques
Analysis of space
Global Optima vs Local Optima

4. Stochastic Search search space Local Search Techniques
Adaptive Search Techniques
Random search
Ant Colony
Simulated Annealing
Neural Network
search space

5. Standard Genetic Algorithm
Step 0: Initialization
Step 1: Selection
Step 2 : Crossover
Step 3 : Mutation
Step 5 : Termination Test
Step6: End
Step 4: Evaluation
Procedure GA
begin
t := 0 ;
initialize P(t) ;
evaluate P(t) ;
while (not termination-condition) do
t := t + 1 ;
select P(t) from P(t-1) ;
alter P(t) ;
end

6. GAs: Terminology Representation : gene, chromosome, Population
Step 0: Initialization
Step 1: Selection
Step 2 : Crossover
Step 3 : Mutation
Step 5 : Termination Test
Step6: End
Step 4: Evaluation
Representation : gene, chromosome, Population
Evaluation : objective function, fitness function
Selection
Operator : crossover, mutation
Replacement : new Generation
Termination : Generation count, Convergence

7. Representation Very crucial step
representation should satisfy the presumption that the whole chromosome is decomposable to building blocks
String of genes of given alphabet:
Binary
Float
Integer
More complex representation
matrices
rules
trees

8. Initialization of the Starting Population
Aspects affecting a performance of GA
schemata sampled in the initial population
Initialization mechanisms
random
informed - uses prior knowledge of the desired solution shape
Pre-processing
runs several short pre-processing runs
samples the promising areas of the search space identified during the foregone pre-processing runs

9. Selection Models nature’s survival-of-the-fittest principle
Selection strategies:
Roulette wheel (proportionate)
Ranking
Tournament
Selection process:
determination of Expected values: EVi = fitnessi / fitnessavg
sampling algorithm - conversion of EVi to the actual number of individuals

10. Roulette Wheel Selection

11. Crossover
Provides random information exchange - works on couples of individuals
Simple 1-point crossover

12. Mutation Mutation - preserves population diversity
works on single individual

13. Replacement Strategy Replacement strategy defines:
how big portion of the old population will be replaced in each generation of the new population, and
the rule that determines which individuals from the old population will be replaced and which individuals will be placed in the new population
Generational - the old population is entirely rebuilt in each generation (short-lived species)
Steady-state - just a few individuals are replaced in each generation (longer-lived species)

14. Premature Convergence
The ratio of the best-fit individual’s reproduction rate to the average reproduction rate is too high
selection kills ‘worse’ individuals too early

15. Theory of GAs

16. Schema Theorem Schema = Pattern Schema Theorem
Short, low-order, above-average schemata
receive exponentially increasing trials in subsequent generations of a genetic algorithm
Building Block Hypothesis
GA seeks near-optimal performance through short, low-order, high-performance schemata

17. Schema In binary representation - 2L strings, 3L schemata
L = 7, S = (**0*1*1) - covers 24 strings
{0,1, *}
S = {*1*01***, 1*0*11*0, , *******1, ****0*** }
Fitness of a schema - average fitness computed over all covered strings
Schema property
order
the length of string minus the number of *
defining the specialty of a schema
8 bits : , schema and building block 1*010*1*
defining length
the distance between the first and the last fixed string positions
defining the compactness of information contained in a schema
(*11**1*0) = 6, (1******1) = 7

18. Selection
eval(S,t) is the average fitness of all strings in the population matched by the schema S at time t ;
Expecting to have strings matched by schema S
the average fitness of the population
becomes ;

19. Crossover survives destroyed
the string is matched by these two schemata
survives
destroyed

20. the probability of destruction of a schema S :
the probability of survival of a schema S :
(S) = 7, m = 8 ?

21. Selective probability of crossover
The combined effect of selection and crossover
a new schema growth equation :

22. Mutation
All of the fixed positions of a schema must remain unchanged to survive mutation
mutate at least one of these bits would destroy the schema
the probability of destruction of a schema S :
the probability of survival of a schema S :

23. TSP
with GA

24. Path representation Crossover operators Mutation operators
( ) 
Crossover operators
Node Orientation vs Edge Orientation
Mutation operators
insertion 
Reciprocal Exchange 
Inversion 
Information of the parents transferred to offsprings
Node crossover = simple but information discarded
Edge crossover = tough but information enclosed

25. TSP: Edge-Recombination Operator
a-b-c-d-e
b-d-e-c-a
b-c-e-a-d

26. Edge Assembly Crossover

27. Edge Assembly Crossover

28. Previous work
Exx crossover
Ex crossover
Test Library

29. EXX
Crossover

30. EX
Crossover

31. EXX
Crossover

32. att532