1
Genetic Algorithms
Contents
1. Basic Concepts
2. Algorithm
3. Practical considerations
2
Basic Concepts
•
Individuals
(or members of population or chromosomes)
individuals surviving from the previous generation
+
children
generation
Simulated Annealing
Tabu Search
versus
Genetic Algorithms
•
a single solution is carried
over from one iteration
to the next
•
population based method
3
Fitness
of an individual (a schedule) is measured by the value of the
associated objective function
Representation
Example
from scheduling problems:
the order of jobs to be processed can be represented as a permutation:
[1, 2, ... ,
n
]
Initialisation
How to choose initial individuals?
•
High

quality solutions obtained from another heuristic technique
can help a genetic algorithm to find better solutions more quickly
than it can from a random start.
4
Reproduction
•
Crossover
: combine the sequence of operations on one machine
in one parent schedule with a sequence of operations on
another machine in another parent.
Example 1
. Ordinary crossover operator is not useful!
Cut Point 1
Cut Point 2
P1 = [
2
1
3
4
5
6 7]
P2 = [
4
3
1
2
5
7 6]
O1 = [
4
3 1
2
5
6 7]
O2 = [
2
1 3
4
5
7 6]
Cut Point
P1 = [
2 1 3
4 5 6 7
]
P2 = [
4 3 1
2 5 7 6
]
O1 = [
2 1 3
2 5 7 6
]
O2 = [
4 3 1
4 5 6 7
]
Example 2
. Partially Mapped Crossover
3
1
4
2
5
5
5
Example 3
. Preserves the absolute positions of the jobs taken from P1
and the relative positions of those from P2
Cut Point 1
P1 = [
2 1
3 4 5 6 7]
P2 = [
4 3
1 2
5 7 6
]
O1 = [
2 1
4 3 5 7 6
]
O2 = [4 3 2 1 5 6 7]
Example 4
. Similar to Example 3 but with 2 crossover points.
Cut Point 1
Cut Point 2
P1 = [2 1
3 4 5
6 7]
P2 = [4 3
1
2
5
7 6
]
O1 = [
3 4 5
1 2 7 6
]
6
•
Mutation
enables genetic algorithm to explore the search space
not reachable by the crossover operator.
Adjacent pairwise interchange
in the sequence
[1,2, ... ,
n
]
[2,1, ... ,
n
]
Exchange mutation
: the interchange of two randomly chosen elements
of the permutation
Shift mutation
: the movement of a randomly chosen element a
random number of places to the left or right
Scramble sublist mutation
: choose two points on the string in random
and randomly permuting the elements between these two positions.
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Selection
•
Roulette wheel
: the size of each slice corresponds to the fitness of
the appropriate individual.
slice for the 1st individual
slice for the 2nd individual
selected individual
.
.
.
Steps for the roulette wheel
1. Sum the fitnesses of all the population members,
TF
2. Generate a random number
m
, between 0 and
TF
3. Return the first population member whose fitness added to the
preceding population members is greater than or equal to
m
8
•
Tournament selection
1. Randomly choose a group of
T
individuals from the population.
2. Select the best one.
How to guarantee that the best member of a population will survive?
•
Elitist model
: the best member of the current population is set
to be a member of the next.
9
Algorithm
Step 1
.
k
=1
Select
N
initial schedules
S
1,1
,... ,
S
1,N
using some heuristic
Evaluate each individual of the population
Step 2
.
Create new individuals by mating individuals in the current population
using crossover and mutation
Delete members of the existing population to make place for
the new members
Evaluate the new members and insert them into the population
S
k+1,1
,... ,
S
k+1,N
Step 3
.
k = k
+1
If
stopping condition =
true
then return the best individual as the solution and STOP
else go to Step 2
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Example
Metric: minimize total tardiness (tardiness of a job is the amount by
which it exceeds its deadline)
•
Population size: 3
•
Selection: in each generation the single most fit individual
reproduces using adjacent pairwise interchange chosen at random
there are 4 possible children, each is chosen with probability 1/4
Duplication of children is permitted.
Children can duplicate other members of the population.
•
Initial population: random permutation sequences
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Generation 1
Individual
25314
14352
12345
Cost
25
17
16
Selected individual: 12345 with offspring 13245, cost 20
Generation 2
Individual
13245
14352
12345
Cost
20
17
16
Average fitness is improved, diversity is preserved
Selected individual: 12345 with offspring 12354, cost 17
Generation 3
Individual
12354
14352
12345
Cost
17
17
16
Selected individual: 12345 with offspring 12435, cost 11
12
Generation 4
Individual
14352
12345
12435
Cost
17
16
11
Selected individual: 12435
This is an optimal solution.
Disadvantages of this algorithm
:
•
Since only the most fit member is allowed to reproduce
(or be mutated) the same member will continue to reproduce unless
replaced by a superior child.
13
Practical considerations
•
Population size: small population run the risk of seriously
under

covering the solution space, while large populations will
require computational resources.
Empirical results suggest that population sizes around 30
are adequate in many cases, but 50

100 are more common.
•
Mutation is usually employed with a very low probability.
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