Genetic Algorithms

1
Genetic Algorithms
Sources

• Jaap Hofstede
• Beasly, Bull, Martin

2
Introduction to Evolutionary Computation

• Evolutionary Computation is the field of study
  devoted to the design, development, and analysis
  is problem solvers based on natural selection
  (simulated evolution).
• Evolution has proven to be a powerful search
  process.
• Evolutionary Computation has been successfully
  applied to a wide range of problems including
• Aircraft Design,
• Routing in Communications Networks,
• Tracking Windshear,
• Game Playing (Checkers Fogel)

3
Introduction to Evolutionary Computation(Applicat
ions cont.)

• Robotics,
• Air Traffic Control,
• Design,
• Scheduling,
• Machine Learning,
• Pattern Recognition,
• Job Shop Scheduling,
• VLSI Circuit Layout,
• Strike Force Allocation,

4
Introduction to Evolutionary Computation(Applicat
ions cont.)

• Theme Park Tours (Disney Land/World)
  http//www.TouringPlans.com
• Market Forecasting,
• Egg Price Forecasting,
• Design of Filters and Barriers,
• Data-Mining,
• User-Mining,
• Resource Allocation,
• Path Planning,
• Etc.

5
Example of Evolutionary Algorithm

• An Example Evolutionary Computation
• Procedure EC
• t 0
• Initialize Pop(t)
• Evaluate Pop(t)
• While (Not Done)
• Parents(t) Select_Parents(Pop(t))
• Offspring(t) Procreate(Parents(t))
• Evaluate(Offspring(t))
• Pop(t1) Replace(Pop(t),Offspring(t))
• t t 1

6
Candidate Solutions CS

1. In an Evolutionary Computation, a population of
   candidate solutions (CSs) is randomly generated.
2. Each of the CSs is evaluated and assigned a
   fitness based on a user specified evaluation
   function.
3. The evaluation function is used to determine the
   goodness of a CS.
4. A number of individuals are then selected to be
   parents based on their fitness.
5. The Select_Parents method must be one that
   balances the urge for selecting the best
   performing CSs with the need for population
   diversity.

7
Parents and Generations

1. The selected parents are then allowed to create a
   set of offspring which are evaluated and assigned
   a fitness using the same evaluation function
   defined by the user.
2. Finally, a decision must be made as to which
   individuals of the current population and the
   offspring population should be allowed to
   survive.
3. Typically, in EC , this is done to guarantee that
   the population size remains constant.
4. The study of ECs with dynamic population sizes
   would make an interesting project for this course

8
Selecting and Stopping

• Once a decision is made the survivors comprise
  the next generation (Pop(t1)).
• This process of selecting parents based on their
  fitness, allowing them to create offspring, and
  replacing weaker members of the population is
  repeated for a user specified number of cycles.
• Stopping conditions for evolutionary search could
  be
• The discovery of an optimal or near optimal
  solution
• Convergence on a single solution or set of
  similar solutions,
• When the EC detects the problem has no feasible
  solution,
• After a user-specified threshold has been
  reached, or
• After a maximum number of cycles.

9
A Brief History of Evolutionary Computation

• The idea of using simulated evolution to solve
  engineering and design problems have been around
  since the 1950s (Fogel, 2000).
• Bremermann, 1962
• Box, 1957
• Friedberg, 1958
• However, it wasnt until the early 1960s that we
  began to see three influential forms of EC emerge
  (Back et al, 1997)
• Evolutionary Programming (Lawrence Fogel, 1962),
• Genetic Algorithms (Holland, 1962)
• Evolution Strategies (Rechenberg, 1965
  Schwefel, 1968),

10
A Brief History of Evolutionary
Computation(cont.)

• The designers of each of the EC techniques saw
  that their particular problems could be solved
  via simulated evolution.
• Fogel was concerned with solving prediction
  problems.
• Rechenberg Schwefel were concerned with solving
  parameter optimization problems.
• Holland was concerned with developing robust
  adaptive systems.

11
A Brief History of Evolutionary
Computation(cont.)

• Each of these researchers successfully developed
  appropriate ECs for their particular problems
  independently.
• In the US, Genetic Algorithms have become the
  most popular EC technique due to a book by David
  E. Goldberg (1989) entitled, Genetic Algorithms
  in Search, Optimization Machine Learning.
• This book explained the concept of Genetic Search
  in such a way the a wide variety of engineers and
  scientist could understand and apply.

12
A Brief History of Evolutionary
Computation(cont.)

• However, a number of other books helped fuel the
  growing interest in EC
• Lawrence Davis, Handbook of Genetic
  Algorithms, (1991),
• Zbigniew Michalewicz book (1992), Genetic
  Algorithms Data Structures Evolution
  Programs.
• John R. Kozas Genetic Programming (1992), and
• D. B. Fogels 1995 book entitled, Evolutionary
  Computation Toward a New Philosophy of Machine
  Intelligence.
• These books not only fueled interest in EC but
  they also were instrumental in bringing together
  the EP, ES, and GA concepts together in a way
  that fostered unity and an explosion of new and
  exciting forms of EC.

13
A Brief History of Evolutionary ComputationThe
Evolution of Evolutionary Computation

• First Generation EC
• EP (Fogel)
• GA (Holland)
• ES (Rechenberg, Schwefel)
• Second Generation EC
• Genetic Evolution of Data Structures
  (Michalewicz)
• Genetic Evolution of Programs (Koza)
• Hybrid Genetic Search (Davis)
• Tabu Search (Glover)

14
A Brief History of Evolutionary ComputationThe
Evolution of Evolutionary Computation (cont.)

• Third Generation EC
• Artificial Immune Systems (Forrest)
• Cultural Algorithms (Reynolds)
• DNA Computing (Adleman)
• Ant Colony Optimization (Dorigo)
• Particle Swarm Optimization (Kennedy Eberhart)
• Memetic Algorithms
• Estimation of Distribution Algorithms
• Fourth Generation ????

15
Introduction to Evolutionary ComputationA
Simple Example

• Lets walk through a simple example!
• Lets say you were asked to solve the following
  problem
• Maximize
• f6(x,y) 0.5 (sin(sqrt(x2y2))2 0.5)/(1.0
  0.001(x2y2))2
• Where x and y are take from -100.0,100.0
• You must find a solution that is greater than
  0.99754, and you can only evaluate a total of
  4000 candidate solutions (CSs)
• This seems like a difficult problem.
• It would be nice if we could see what it looks
  like!
• This may help us determine a good algorithm for
  solving it.

16
Introduction to Evolutionary ComputationA
Simple Example

• A 3D view of f6(x,y)

17
Introduction to Evolutionary ComputationA
Simple Example

• If we just look at only one dimension f6(x,1.0)

18
Introduction to Evolutionary ComputationA
Simple Example

• Lets develop a simple EC for solving this
  problem
• An individual (chromosome or CS)
• ltxi,yigt
• fiti f6(xi,yi)

19
Introduction to Evolutionary ComputationA
Simple Example

• Procedure simpleEC
• t 0
• Initialize Pop(t) / of P individuals /
• Evaluate Pop(t)
• while (t lt 4000-P)
• Select_Parent(ltxmom,ymomgt) / Randomly /
• Select_Parent(ltxdad,ydadgt) / Randomly /
• Create_Offspring(ltxkid,ykidgt)
• xkid rnd(xmom, xdad) Nx(0,?)
• ykid rnd(ymom, ydad) Ny(0,?)
• fitkid Evaluate(ltxkid,ykidgt)
• Pop(t1) Replace(worst,kid)Pop(t)-worst?
  kid
• t t 1

20
Introduction to Evolutionary ComputationA
Simple Example

• To simulate this simple EC we can use the applet
  at
• http//www.eng.auburn.edu/gvdozier/GA.html

21
Introduction to Evolutionary ComputationA
Simple Example

• To get a better understanding of some of the
  properties of ECs lets do the in class lab
  found at http//www.eng.auburn.edu/gvdozier/GA_L
  ab.html

22
Hill climbing
23
Introduction 1

• Inspired by natural evolution
• Population of individuals
• Individual is feasible solution to problem
• Each individual is characterized by a Fitness
  function
• Higher fitness is better solution
• Based on their fitness, parents are selected to
  reproduce offspring for a new generation
• Fitter individuals have more chance to reproduce
• New generation has same size as old generation
  old generation dies
• Offspring has combination of properties of two
  parents
• If well designed, population will converge to
  optimal solution

24
Algorithm

• BEGIN
• Generate initial population
• Compute fitness of each individual
• REPEAT / New generation /
• FOR population_size / 2 DO
• Select two parents from old generation
• / biased to the fitter ones /
• Recombine parents for two offspring
• Compute fitness of offspring
• Insert offspring in new generation
• END FOR
• UNTIL population has converged
• END

25
Example of convergence
26
Introduction 2

• Reproduction mechanism has no knowledge of the
  problem to be solved
• Link between genetic algorithm and problem
• Coding
• Fitness function

27
Basic principles 1

• Coding or Representation
• String with all parameters
• Fitness function
• Parent selection
• Reproduction
• Crossover
• Mutation
• Convergence
• When to stop

28
Basic principles 2

• An individual is characterized by a set of
  parameters Genes
• The genes are joined into a string Chromosome
• The chromosome forms the genotype
• The genotype contains all information to
  construct an organism the phenotype
• Reproduction is a dumb process on the
  chromosome of the genotype
• Fitness is measured in the real world (struggle
  for life) of the phenotype

29
Coding

• Parameters of the solution (genes) are
  concatenated to form a string (chromosome)
• All kind of alphabets can be used for a
  chromosome (numbers, characters), but generally a
  binary alphabet is used
• Order of genes on chromosome can be important
• Generally many different codings for the
  parameters of a solution are possible
• Good coding is probably the most important factor
  for the performance of a GA
• In many cases many possible chromosomes do not
  code for feasible solutions

30
Example of coding for TSP

• Travelling Salesman Problem
• Binary
• Cities are binary coded chromosome is string of
  bits
• Most chromosomes code for illegal tour
• Several chromosomes code for the same tour
• Path
• Cities are numbered chromosome is string of
  integers
• Most chromosomes code for illegal tour
• Several chromosomes code for the same tour
• Ordinal
• Cities are numbered, but code is complex
• All possible chromosomes are legal and only one
  chromosome for each tour
• Several others

31
Reproduction

• Crossover
• Two parents produce two offspring
• There is a chance that the chromosomes of the two
  parents are copied unmodified as offspring
• There is a chance that the chromosomes of the two
  parents are randomly recombined (crossover) to
  form offspring
• Generally the chance of crossover is between 0.6
  and 1.0
• Mutation
• There is a chance that a gene of a child is
  changed randomly
• Generally the chance of mutation is low (e.g.
  0.001)

32
Crossover

• One-point crossover
• Two-point crossover
• Uniform crossover

33
One-point crossover 1

• Randomly one position in the chromosomes is
  chosen
• Child 1 is head of chromosome of parent 1 with
  tail of chromosome of parent 2
• Child 2 is head of 2 with tail of 1

Randomly chosen position
Parents 1010001110 0011010010 Offspring
0101010010 0011001110
34
One-point crossover 2
35
Two-point crossover

• Randomly two positions in the chromosomes are
  chosen
• Avoids that genes at the head and genes at the
  tail of a chromosome are always split when
  recombined

Randomly chosen positions
Parents 1010001110 0011010010 Offspring
0101010010 0011001110
36
Uniform crossover

• A random mask is generated
• The mask determines which bits are copied from
  one parent and which from the other parent
• Bit density in mask determines how much material
  is taken from the other parent (takeover
  parameter)

Mask 0110011000 (Randomly
generated) Parents 1010001110 0011010010 Offsp
ring 0011001010 1010010110
37
Problems with crossover

• Depending on coding, simple crossovers can have
  high chance to produce illegal offspring
• E.g. in TSP with simple binary or path coding,
  most offspring will be illegal because not all
  cities will be in the offspring and some cities
  will be there more than once
• Uniform crossover can often be modified to avoid
  this problem
• E.g. in TSP with simple path coding
• Where mask is 1, copy cities from one parent
• Where mask is 0, choose the remaining cities in
  the order of the other parent

38
Fitness Function

• Purpose
• Parent selection
• Measure for convergence
• For Steady state Selection of individuals to die
• Should reflect the value of the chromosome in
  some real way
• Next to coding the most critical part of a GA

39
Parent selection

• Chance to be selected as parent proportional to
  fitness
• Roulette wheel
• To avoid problems with fitness function
• Tournament
• Not a very important parameter

40
Roulette wheel

• Sum the fitness of all chromosomes, call it T
• Generate a random number N between 1 and T
• Return chromosome whose fitness added to the
  running total is equal to or larger than N
• Chance to be selected is exactly proportional to
  fitness
• Chromosome 1 2 3 4 5 6
• Fitness 8 2 17 7 4 11
• Running total 8 10 27 34 38 49
• N (1 ? N ? 49) 23
• Selected 3

41
Tournament

• Binary tournament
• Two individuals are randomly chosen the fitter
  of the two is selected as a parent
• Probabilistic binary tournament
• Two individuals are randomly chosen with a
  chance p, 0.5ltplt1, the fitter of the two is
  selected as a parent
• Larger tournaments
• n individuals are randomly chosen the fittest
  one is selected as a parent
• By changing n and/or p, the GA can be adjusted
  dynamically

42
Problems with fitness range

• Premature convergence
• ?Fitness too large
• Relatively superfit individuals dominate
  population
• Population converges to a local maximum
• Too much exploitation too few exploration
• Slow finishing
• ?Fitness too small
• No selection pressure
• After many generations, average fitness has
  converged, but no global maximum is found not
  sufficient difference between best and average
  fitness
• Too few exploitation too much exploration

43
Solutions for these problems

• Use tournament selection
• Implicit fitness remapping
• Adjust fitness function for roulette wheel
• Explicit fitness remapping
• Fitness scaling
• Fitness windowing
• Fitness ranking

44
Fitness scaling

• Fitness values are scaled by subtraction and
  division so that worst value is close to 0 and
  the best value is close to a certain value,
  typically 2
• Chance for the most fit individual is 2 times the
  average
• Chance for the least fit individual is close to 0
• Problems when the original maximum is very
  extreme (super-fit) or when the original minimum
  is very extreme (super-unfit)
• Can be solved by defining a minimum and/or a
  maximum value for the fitness

45
Example of Fitness Scaling
46
Fitness windowing

• Same as window scaling, except the amount
  subtracted is the minimum observed in the n
  previous generations, with n e.g. 10
• Same problems as with scaling

47
Fitness ranking

• Individuals are numbered in order of increasing
  fitness
• The rank in this order is the adjusted fitness
• Starting number and increment can be chosen in
  several ways and influence the results
• No problems with super-fit or super-unfit
• Often superior to scaling and windowing

48
Other parameters of GA 1

• Initialization
• Population size
• Random
• Dedicated greedy algorithm
• Reproduction
• Generational as described before (insects)
• Generational with elitism fixed number of most
  fit individuals are copied unmodified into new
  generation
• Steady state two parents are selected to
  reproduce and two parents are selected to die
  two offspring are immediately inserted in the
  pool (mammals)

49
Other parameters of GA 2

• Stop criterion
• Number of new chromosomes
• Number of new and unique chromosomes
• Number of generations
• Measure
• Best of population
• Average of population
• Duplicates
• Accept all duplicates
• Avoid too many duplicates, because that
  degenerates the population (inteelt)
• No duplicates at all

50
Example run

• Maxima and Averages of steady state and
  generational replacement

51
Introduction to Evolutionary ComputationReading
List

• Bäck, T., Hammel, U., and Schwefel, H.-P. (1997).
  Evolutionary Computation Comments on the
  History and Current State, IEEE Transactions on
  Evolutionary Computation, VOL. 1, NO. 1, April
  1997.
• Spears, W. M., De Jong, K. A., Bäck, T., Fogel,
  D. B., and de Garis, H. (1993). An Overview of
  Evolutionary Computation, The Proceedings of the
  European Conference on Machine Learning, v667,
  pp. 442-459. (http//www.cs.uwyo.edu/wspears/pape
  rs/ecml93.pdf)
• De Jong, Kenneth A., and William M. Spears
  (1993). On the State of Evolutionary
  Computation, The Proceedings of the Int'l
  Conference on Genetic Algorithms, pp. 618-623.
  (http//www.cs.uwyo.edu/wspears/papers/icga93.pdf
  )