1. Chapter 14. Genetic Algorithms
Comp3710 Artificial Intelligence
Computing Science
Thompson Rivers University

2. Course Outline Part I – Introduction to Artificial Intelligence
Part II – Classical Artificial Intelligence, and Searching
Knowledge Representation
Searching
Search Methodologies
Advanced Search
Genetic Algorithms (relatively new study area)
Knowledge Represenation and Automated Reasoning
Propositinoal and Predicate Logic
Inference and Resolution for Problem Solving
Rules and Expert Systems
Part III – Machine Learning
Part IV – Advanced Topics
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Genetic Algorithms

3. Chapter Objectives
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Genetic Algorithms

4. Chapter Outline Introduction – Why do we use Genetic Algorithms?
Representations – How to represent a problem
The Algorithm
Fitness – How to evaluate if a chromosome is good or bad
Crossover – How to generate the next generation
Mutation – How to escape local maxima
Termination Criteria – When do we have to stop the algorithm?
Examples – What can we learn from experiences?
Why genetic algorithms work?
Non-standard GAs
Messy genetic algorithms
Predators and coevolution
Project ideas
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Genetic Algorithms

5. Reference
Artificial Intelligence Illuminated, Ben Coppin, Jones and Bartlett Illuminated Series
Introduction to Evolutionary Computing Genetic Algorithms, A. E. Eiben and J. E. Smith
Java Genetic Programming Package
GA Playground
Very interesting
Demo
Traveling salesman problem
Genetic programming Maze Marchers –
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6. 1. Introduction
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7. Natural Evolution Darwin : The Origin of Species
Natural evolution has produced many optimal forms
sharks, ichthyosaurs, and dolphins – very good at swimming
albatross wings – very good at soaring
Important principles used by natural evolution
Assortment (diversity): recombination of genetic material
Selection (cumulative improvement): survival of the fittest
Random mutation (random walk)
In this class, I am not taking any side of Evolutionism and Creationism.
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8. WordEvol: Cumulative Selection
Cumulative selection with simple random selection: randomly chooses a character for the first letter in the string, then randomly choose a character for second letter, and so on.
[Q] What do you think of this idea?
[Q] When we consider one trial as one generation, how many generations would it take to find the correct word?
COMPUTER D C K P O R
MPUTER D
A S Z
B F Y
T X Q
I P K R
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9. COMPUTER FHGLJGHP FHGLJG P A FHCLJGHP FHGLTGHP . . . FHGLJGHR
Cumulative selection with randomness over multiple offspring:
1. generates a random string
2. if it doesn’t match,
2.1 generates a new generation of mutant offspring
3. selects the single offspring most like the target
4. if it doesn’t match, go to 2.1
COMPUTER
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
FHGLJGHP 7
FHGLJG P A
[Q] What do you think of this idea?
[Q] When we consider the steps 2 and 3 as one generation, how many generations would it take to find the correct word?
[Q] Diversity? Cumulative improvement? Random walk? 3
FHCLJGHP 5
FHGLTGHP 8
FHGLJGHR
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10. Genetic Algorithm – quick view
Topics
Genetic Algorithm – quick view
Developed: USA in the 1970’s
Early names: J. Holland, K. DeJong, D. Goldberg
Typically applied to:
discrete optimization
Attributed features:
not too fast
good heuristic for combinatorial problems
Special Features:
Traditionally emphasizes combining information from good parents (crossover)
many variants, e.g., reproduction models, operators
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11. 2. Representation
How to represent a solution (or state) for a given problem?
Genetic techniques can be applied with a range of representations.
Usually, we have a population of chromosomes (individuals) that are
Usually a string of bits or numbers
A complete representation of solution
Each chromosome (individual) consists of a number of genes.
Other representations are equally valid. Nowadays it is generally accepted that it is better to encode numerical variables directly as
Integers
Floating point variables
Permutation representation: sorting; traveling salesman problem
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12. Topics
Encoding (representation of a solution) is very important.
Genotype space = {0,1}L
Phenotype space
Encoding
(representation)
Decoding
(inverse representation)
Actual observed properties
Full hereditary information
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13. Each individual is a goal candidate.
Topics
3. The Algorithm
The algorithm is used as follows:
Generate a random population of chromosomes (, the first generation);
While (the termination criteria are not satisfied)
Determine the fitness of each chromosome (individual);
Apply crossover to selected chromosomes from the current generation to generate a new population of chromosomes (the next generation);
Fitter chromosomes have more chances to survive to the next generation. (Survival of the fittest)
Each pair of parents produces two offspring and those offspring replace the parents in the next generation.
Apply mutation to a new population;
A little bit of mutation is applied to offspring.
[Q] The representation of chromosomes (individuals)?
[Q] The size of the population?
[Q] The crossover rate?: ~0.8 => the parents can survive in the n.g. ?
[Q] The mutation rate?: ~0.001
Each individual is a goal candidate.
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14. 4. Fitness and Selection
Fitness is an important concept in genetic algorithms.
The fitness of a chromosome determines how likely it is that it will reproduce.
Fitness is usually measured in terms of how well the chromosome solves some goal problem.
Fitness can also be subjective.
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15. E. g. , sorting (Just for explanation
E.g., sorting (Just for explanation. There are many good deterministic sorting algorithms.)
[Q] Chromosome for the sorting problem of numbers? Representation of solutions?
( ), ( ), ...
If the genetic algorithm is to be used to sort numbers, then the fitness of a chromosome will be determined by how close to a correct sorting it produces.
Fitness idea 1:
For each element in a list, count the number of right side elements that are bigger than the element; and add the counts.
( ) -> = 3; ( ) -> ??? = 4
Fitness idea 2:
Count the number of pairs of adjacent elements, in which the left element is smaller than the right element.
( ) -> 1; ( ) -> ???
[Q] Which idea looks better?
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16. Selection B A C Fitness(A) = 3 Fitness(B) = 1 Fitness(C) = 2
Roulette wheel selection
Age based selection … A C
1/6 = 17%
3/6 = 50% B
2/6 = 33%
Fitness(A) = 3
Fitness(B) = 1
Fitness(C) = 2
[Q] How to implement?
[50 | 67(=50+17) | 100=(67+33)]
Random floating point number in [0, 100)
[0, 50) => A, [50, 67) => ?, [67, 100) => C
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17. What is roulette_wheel[fitnesses.length-1]?
var fitnesses = []; ... var total = 0; for (var i = 0; i < fitnesses.length; i++) total += fitnesses[i]; var fitness_ratios = []; for (var i = 0; i < fitnesses.length; i++); fitness_ratios[i] = ???; var roulette_wheel = []; roulette_wheel[0] = fitness_ratios[0]; for (var i = 1; i < fitness_ratios.length; i++) ???;
What is roulette_wheel[fitnesses.length-1]?
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18. Topics
var r; r = Math.random(); // r is in [0, 1); var i; for (i = 0; i < roulette_wheel.length; i++) ???
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19. 5. Crossover Single-point crossover is applied as follows:
Select a random crossover point.
Break each chromosome into two parts, splitting at the crossover point.
Recombine the broken chromosomes by combining the front of one with the back of the other, and vice versa, to produce two new chromosomes.
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20. Single-point crossover
More likely to keep together genes that are near each other.
Can never keep together genes from opposite ends of string.
This is known as Positional Bias.
[Q] What if all the individuals have wrong genes at the left side?
Can be exploited if we know about the structure of our problem, but this is not usually the case.
[Q] How to solve?
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21. [Q] Can this idea solve the positional bias problem?
Idea 1: Usually, crossover is applied with one crossover point, but can be applied with more, such as in two-point crossover that has two crossover points:
[Q] Can this idea solve the positional bias problem?
Idea 2: Uniform crossover involves using a probability to select which genes to use from chromosome 1, and which from chromosome 2.
Single offspring can be produced by uniform crossover …
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22. [Q] Do we always have to generate new offspring?
Topics
[Q] Do we always have to generate new offspring?
Cloning is possible with the crossover rate of ~0.8. [Q] what does this mean?
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23. 6. Mutation A unary operator – applies to one chromosome.
Topics
6. Mutation
A unary operator – applies to one chromosome.
Randomly selects some bits (genes) to be “flipped”
1 => 0 and 0 =>1
Mutation is usually applied with a low probability, such as 1 in 1000.
[Q] Why is mutation necessary???
Mutation-only-GA is possible, crossover-only-GA would not work.
Crossover makes a big jump to an area somewhere “in between” two (parent) areas.
Mutation creates random small diversions, thereby staying near (in the area of ) the parent.
To hit the optimum you often need a ‘lucky’ mutation.
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24. Topics
7. Termination Criteria
A genetic algorithm is run over a number of generations until the termination criteria are reached.
Typical termination criteria are:
Stop after a fixed number of generations.
Stop when a chromosome reaches a specified fitness level.
Stop when a chromosome succeeds in solving the problem, within a specified tolerance.
Human judgement can also be used in some more subjective cases.
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25. 8. Examples Optimization of a mathematical function
Traveling salesman problem
Maze marcher
Sorting
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26. E.g. - optimizing a mathematical function
A genetic algorithm can be used to find the highest integer value for
f(x) = sin (x) within [0, 15].
[Q] How do you want to encode the problem? Variables? Representation of solutions?
Each chromosome consists of 4 bits, to represent the values of x from 0 to 15.
E.g., (1,0,0,1)
[Q] How to measure fitness of a chromosome?
0 (f(x) = –1) to 100 (f(x) = 1). [Q] How?
Fitness(x) = 50 * (f(x) + 1) = 50 * (sin (x) + 1)
By applying the genetic algorithm it takes just a few generations to find the optimal solution for f(x).
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27. The first generation The fitness function
The fitness function
Fitness(x) = 50 * (f(x) + 1) = 50 * (sin(x) + 1), and fitness values are …
Using the random numbers between 0 and 100, and the fitness ratio, chromosomes are selected to produce offspring. [Q] HOW???
Roulette wheel selection
[Q] What if only the fittest chromosome is used to produce the next generation?
Chromosome
Fitness(x)
Fitness ratio
1001
70.61
0.463
0011
57.06
0.374
1010
22.80
0.149
0101
2.05
0.134
70.61 / ( )
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28. Selection based on fitness ratios8.
Selection based on fitness ratios
Crossover
10| |11 =>
10| |11 => ???
[Q] The next generation and fitness values?
Chromosome
Fitness(x)
Fitness ratio
1011
0001
92.07
0.481
1000
99.47
0.519
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29. E.g. - traveling salesman problem
Given n cities,
find a complete tour with the minimal length.
[Q] Is the problem a CSP?
[Q] How to encode? Representation of solutions?
Label the cities 1, 2, … , n, and ?
One complete tour is one permutation.
(But, for n = 4, [1,2,3,4] and [3,?,?,?] are
the same.)
Search space is really BIG:
For 30 cities there are 30!  1032
possible tours.
Demo
Many web sites;
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30. 1 2 3 4 5 Fitness [Q] Do we know a minimal path?
[Q] Then, what kind of fitness function can be used?
Let’s assume that for each pair of two cities, the shortest path is already determined.
[Q] Can we determine the length of a tour?
For (1, 2, 3, 4, 5), the length = D(1, 2) + D(2, 3) + D(3, 4) + D(4, 5) + D(5, 1)
The shorter, the better.
[Q] Let a, b, c, d are the lengths of 4 tours. Then fitness ratios???
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31. Fitness
[Q] Let a, b, c, d are the lengths of 4 tours. Then fitness ratios???
E.g., a = 10, b=20, c=30, d=40 (The smaller, the better.)
Idea 1: Like the example of n-queens puzzle
Let x = a + b + c + d; y = (x – a) + (x – b) + (x – c) + (x – d)
The fitness ratio of a can be defined as (x – a) / y.
=> x = 100; y = = 300
=> 9/30, 8/30, 7/30, 6/30 not much distinctive
Idea 2:
Let x = a^2 + b^2 + c^2 + d^2; y = (x – a^2) + (x – b^2) + (x – c^2) + (x – d^2)
The fitness ratio of a can be defined as (x – a^2) / y.
=> x = 3000; y = = 9000
=> 29/90, 26/90, 21/90, 16/90 still not much distinctive
Idea 3:
Let x = 1/a + 1/b + 1/c + 1/d
The fitness ratio of a can be defined as (1/a) / x. (The larger the better)
=> 12/25, 6/25, 4/25, 3/25
=> a has better chance than the ideas 1 and 2.
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32. Example 4 cities – 1, 2, 3, 4 Distances
4 individuals – [1, 2, 3, 4], [1, 3, 2, 4], [2, 4, 3, 1], [4, 1, 2, 3]
Fitness using the Idea 3 in the previous slide?
Lengths – 70, 100, 70, 70
x = 1/70 + 1/ /70 + 1/70 = 37/700
fitness1 = (1/70)/(37/700) = 10/37, fitness2 = 7/37, fitness3 = 10/37,
fitness4 = 10/37
Roulette wheel? 10 20 30
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33. Crossover
“Normal” crossover operators will often lead to inadmissible solutions.
Idea 1: a) Copy randomly selected set from first parent
b) Copy rest from second parent while preserving the order in second parent
Any problem???
Why???
[Q] The other one???
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34. Idea 2:
Random single point crossover
Other part comes from the other parent while preserving the order following the last city in the selected part
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35. How to mutate? Idea 1: Insert Mutation for permutations
Pick two cities at random
Move the second to follow the first, shifting the rest along to accommodate
Note that this preserves most of the order and the adjacency information
Idea 2: Swap mutation for permutations
Pick two cities at random and swap their positions
Preserves most of adjacency information (4 links broken), disrupts order more
Inversion
Scramble
...
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36. Termination criterion?8.
Termination criterion?
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37. 8.
E.g. – maze marchers
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38. E.g. – sorting for experiment
Chromosome ???
Lists
E.g., ( ), ( ), ...
Fitness ???
Fitness idea 1:
For each element in a list, count the number of right side elements that are bigger than the element; and add the counts.
( ) -> = 3; ( ) -> = 4
Fitness idea 2:
Count the number of pairs of adjacent elements, in which the left element is smaller than the right element.
( ) -> 1; ( ) -> 1
[Q] Crossover ???
[Q] Mutation ???
Termination criteria ???
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39. [Q] Crossover ??? [Q] Mutation ??? Termination criteria ???
[Q] Simple crossover ???
( ) and ( ) -> ???
( ) and ( )?
[Q] Then?
Idea 1: Can we use the crossover in the traveling salesman problem?
( ) and ( ) // problem?
Idea 2: A bit different from the above crossover, so that the elements coming from the spouse have the same order.
( ) and ( )
Can you evaluate them to see an improvement?
Idea …
[Q] Mutation ???
Termination criteria ???
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40. Mutation ??? Termination criteria ??? 8. Topics
Idea 1: Exchange of adjacent elements that are randomly chosen
( ) -> ???
Mutation rate ???
Termination criteria ???
Idea 1: Until the algorithm finds the sorted list
Idea 2: Limited number of generations
Idea 3: Until the overall fitness reaches a certain threshold
Idea 3: Until the difference of overall fitness becomes smaller than a
threshold
Idea 4: Combination of the above ideas
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41. 9. Why does GA work well?
In many cases, genetic algorithms quickly produce optimal or near-optimal solutions to combinatorial problems that would otherwise be impractical to solve.
Why do genetic algorithms work well?
It is possible to explain genetic algorithms by comparison with natural evolution: small changes that occur on a selective basis combined with reproduction will tend to improve the fitness of the population over time.
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42. The Building Block Hypothesis9.
The Building Block Hypothesis
Topics
Genetic algorithms manipulate short, low-order, high fitness schemata in order to find optimal solutions to problems.
These short, low-order, high fitness schemata are known as building blocks.
Hence genetic algorithms work well when small groups of genes that are close together represent useful features in the chromosomes. (Try to recall TSP.)
This tells us that it is important to choose a correct representation.
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43. 10. Non-Standard GAs
Genetic algorithms can be deceived by fit building blocks that happen not to combine to give the optimal solutions.
Deception can be avoided by inversion: this involves reversing the order of a randomly selected group of bits within a chromosome.
The inversion rate is ~0.001.
Other ideas
Messy GAs
Co-evolution
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44. Messy Genetic Algorithms
An alternative to standard genetic algorithms, which avoids deception.
Each bit in the chromosome is represented as a (position, value) pair. For example:
((1,0), (2,1), (4,0))
In this case, the third bit is undefined, which is allowed with MGAs.
Underspecified bits are filled with bits taken from a template chromosome. The template chromosome is usually the best performing chromosome from the previous generation.
A bit can also overspecified:
((1,0), (2,1), (3,1), (3,0), (4,0))
Overspecified bits are usually dealt with by working from left to right and using the first value specified for each bit.
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45. Messy Genetic Algorithms
10.
Messy Genetic Algorithms
MGAs use splice and cut instead of crossover.
Splicing involves simply joining two chromosomes together:
((1,0), (3,0), (4,1), (6,1))
((2,1), (3,1), (5,0), (7,0), (8,0))
((1,0), (3,0), (4,1), (6,1), (2,1), (3,1), (5,0), (7,0), (8,0))
Cutting involves splitting a chromosome into two:
((1,0), (3,0), (4,1))
((6,1), (2,1), (3,1), (5,0), (7,0), (8,0))
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46. 10.
Co-Evolution
Topics
In the real world, the presence of predators is responsible for many evolutionary developments.
Similarly, in many artificial life systems, introducing “predators” produces better results.
Example
This process is known as co-evolution.
For example, Ramps, which were evolved to sort numbers: “parasites” were introduced which produced sets of numbers that were harder to sort, and the ramps produced better results.
Fish tank
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47. Project idea Topics Genetic algorithm for Local search for
The n-queens problem
The class scheduling problem
Local search for
The traveling salesman problem
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