Genetic Algorithms

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Genetic Algorithms
Contents 1. Basic Concepts 2. Algorithm 3.
Practical considerations
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Literature 1. Modern Heuristic Techniques for
Combinatorial Problems, (Ed) C.Reeves 1995,
McGraw-Hill. Chapter 4. 2. Operations Scheduling
with Applications in Manufacturing and Services,
Michael Pinedo and Xiuli Chao, McGraw Hill,
2000, Chapter 3.7. or Scheduling, Theory,
Algorithms, and Systems, Second
Addition, Michael Pinedo, Prentice Hall, 2002,
Chapter 14.5
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Basic Concepts
Simulated AnnealingTabu Search
Genetic Algorithms
versus

• a single solution is carried over from one
  iteration to the next

• population based method

• Individuals (or members of population or
  chromosomes)

individuals surviving from the previous
generation children
generation
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• Fitness of an individual (a schedule) is measured
  by the value of the associated objective
  function
• Representation
• Example.
• the order of jobs to be processed can be
  represented as a permutation1, 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.

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• 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
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
Cut Point 1
Cut Point 2
3?1 4?2 5?5
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
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Example 3. Preserves the absolute positions of
the jobs taken from P1and 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
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• 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 elementsof 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
randomand 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.

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
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• 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.

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Algorithm Step 1. k1 Select N initial
schedules S1,1 ,... , S1,N using some
heuristic Evaluate each individual of the
population Step 2. Create new individuals by
mating individuals in the current
populationusing crossover and mutation Delete
members of the existing population to make place
forthe new members Evaluate the new members and
insert them into the population Sk1,1 ,... ,
Sk1,N Step 3. k k1 If stopping condition
true then return the best individual as the
solution and STOP else go to Step 2
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Example

• 1 ?Tj
• 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

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• 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.

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• 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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Summary

• Meta-heuristic methods are designed to escape
  local optima.
• They work on complete solutions. However,
  they introduce parameters (such as temperature,
  rate of reduction of the temperature, memory,
  ...) How to choose the parameters?