4. Genetic Algorithm

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4. Genetic Algorithm

• Genetic algorithms are a part of evolutionary
  computing, which is a rapidly growing area of
  artificial intelligence.
• Based on Darwinian principles of biological
  evolution. ????,????
• First proposed by Prof. John Holland and his
  colleague at Univ. of Michigan.

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Biological Background 1

• Chromosomes are strings of DNA and serves as a
  model for the whole organism.
• A chromosome consists of genes.
• Each gene encodes a trait.
• Complete set of genetic material (all
  chromosomes) is called genome.
• Particular set of genes in genome is called
  genotype.

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Biological Background 2

• During reproduction, first occurs recombination
  (or crossover).
• Genes from parents form in some way the whole new
  chromosome.
• The new created offspring can then be mutated.
  Mutation means, that the elements of DNA are a
  bit changed. This changes are mainly caused by
  errors in copying genes from parents.
• The fitness of an organism is measured by success
  of the organism in its life.

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Search Space

• The space of all feasible solutions (it means
  objects among those the desired solution is) is
  called search space (also state space).
• Each point in the search space represent one
  feasible solution.
• The looking for a solution is then equal to a
  looking for some extreme (minimum or maximum) in
  the search space.
• Search methods hill climbing, tabular search,
  simulated annealing and genetic algorithm.

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Basic Description of GA

• Algorithm is started with a set of solutions
  (represented by chromosomes) called population.
• Solutions from one population are taken and used
  to form a new population.
• The new population (offspring) will be better
  than the old one (parent).
• Solutions which are selected to form new
  solutions are selected according to their fitness
  - the more suitable they are the more chances
  they have to reproduce.

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Basic Genetic Algorithm
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Basic GA part 1

• Step 1 Start Generate random population of n
  chromosomes (suitable solutions for the problem.)
• Step 2 Fitness Evaluate the fitness f(x) of
  each chromosome x in the population.
• Step 3 New population Create a new population
  by repeating following steps until the new
  population is complete.

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Basic GA part 2

• Step 3 Create a new population
• Selection Select two parent chromosomes from a
  population according to their fitness.
• Crossover With a crossover probability cross
  over the parents to form a new offspring
  (children). If no crossover was performed,
  offspring is an exact copy of parents.
• Mutation With a mutation probability mutate new
  offspring at each locus.
• Accepting Place new offspring in a new
  population

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Basic GA part 3

• Step 4 Replace Use new generated population
  for a further run of algorithm
• Step 5 Test If the end condition is satisfied,
  stop, and return the best solution in current
  population
• Step 6 Loop Go to step 2.

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Operators of GA Encoding

• The chromosome should in some way contain
  information about solution which it represents.
• The most used way of encoding is a binary string.
  
• Chromosome 1 ?1101100100110110
• Chromosome 2 ?1101111000011110
• Each bit in this string can represent some
  characteristic of the solution.
• One can encode directly integer or real numbers.

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Operators of GA Selection

• Chromosomes are selected from the population to
  be parents to crossover.
• According to Darwin's evolution theory the best
  ones should survive and create new offspring.
• For example roulette wheel selection, Boltzman
  selection, tournament selection, rank selection,
  steady state selection and some others.

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Roulette Wheel Selection

• Parents are selected according to their fitness.
• The better the chromosomes are, the more chances
  to be selected they have.

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Example

• fitness values f(x)x2

x string fitness f(x) of total
13 01101 169 14.4
24 11000 576 49.2
8 01000 64 5.5
19 10011 361 30.9
total total 1170 100.0
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Rank Selection 1

• Rank selection first ranks the population and
  then every chromosome receives fitness from this
  ranking.

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Rank Selection 2

• After this all the chromosomes have a chance to
  be selected.
• But this method can lead to slower convergence,
  because the best chromosomes do not differ so
  much from other ones.

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Operators of GA Crossover

• Crossover selects genes from parent chromosomes
  and creates a new offspring.
• Chromosome 1?11011 00100110110
• Chromosome 2?11011 11000011110
• Offspring 1 ?11011 11000011110
• Offspring 2 ?11011 00100110110
• is the crossover point

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Crossover

• Single point crossover 1100101111011111
  11001111
• Two point crossover 11001011 11011111
  11011111
• Uniform crossover 11001011 11011101 11011111
  
• Arithmetic crossover 11001011 11011111
  11001001
  (AND)

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Operators of GA Mutation

• Prevent falling all solutions in population into
  a local optimum of solved problem
• Mutation changes randomly the new offspring.
• Original offspring 1?1101111000011110
• Mutated offspring 1?1100111000011110
• Original offspring 2?1101100100110110
• Mutated offspring 2?1101101100110110

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Control Parameters

• GA has two control parameters crossover rate
  (pc) and mutation rate (pm).
• pc (0.51.0) The higher the value of pc, the
  quicker are the new solutions introduced into the
  population.
• pm (0.0050.05) Large values of pm transform the
  GA into a purely random search algorithm, while
  some mutations are required to prevent the
  premature convergence of the GA to suboptimal
  solutions.

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Numerical Example
No. string x fitness of total
1 01101 13 169 14.4
2 11000 24 576 (max) 49.2
3 01000 8 64 5.5
4 10011 19 361 30.9
total total 1170 100.0
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Numerical Example
No. actual count
1 0.14 0.58 1
2 0.49 (max) 1.97 (max) 2 (max)
3 0.06 0.22 0
4 0.31 1.23 1
average 0.25 1.0 1.0
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Numerical Example
No. string mate crossover site new string
1 01101 2 4 01100
2 11000 1 4 11001
3 11000 4 2 11011
4 10011 3 2 10000
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Numerical Example
No. new population x f(x)x2
1 01100 12 144
2 11001 25 625
3 11011 27 729 (max)
4 10000 16 256
1 01101 13 169
2 11000 24 576 (max)
3 01000 8 64
4 10011 19 361
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Real-Valued GA

• Basic (binary) GA????????,??(encoding)???(decoding
  )??????,??????????,????????????????,??????????????
  ???
• ??????????????????,?????????????????????????,?????
  ???????

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?????????

• ??P1?P2???,C1?C2???
• ????C1 ? ? P1 (1??) ? P2C2 ? ? P1 (1?
  ?) ? P2?? ? ? ? ????,? ?, ? ? 0, 1
• ????C C random_noise?? random_noise
  ?????????

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????????

• ????????????????,???????????
• ??????? 0, ?,?? 4 ????,?????????15??

??? 0000 0001 --- 1000 --- 1111
?? 0 1 --- 8 --- 15
?? 0 ? / 15 --- 8? / 15 --- ?
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?????????

• ?????Binary GA ??
• ???????????? ? ????? 123456 ? 125436