Solving the Travelling Salesman Problem Using a Genetic Algorithm

1
Solving the Travelling Salesman Problem Using a
Genetic Algorithm

• Lappeenranta University of Technology
• Ti5261300 Evolutionary Algorithms
• Project work
• Toni Jaakkola

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The Travelling Salesman Problem

• A salesman has to visit a certain number of
  cities, where every city is visited once, and
  only once, and the whole trip is as short as
  possible. Which route should he choose?

• An NP-compelete combinatorial problem

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TSP as an optimization problem

• Combinatorial optimization
• The objective function is the sum of the
  euclidian lengths of all edges among the
  salesman's route
• The search space is a discrete space of all
  permutationsof N cities
• Causes challenges and limitations for the genetic
  algorithm!

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The genetic algorithm in pseudo code

• Create a random map of N cities
• Create a random population of solutions
• Repeat
• Repeat
• Select two parents using Tournament Selection
• Generate two offspring using Partially Matched
  Crossover
• Mutate Offspring 1 with a given probability
  using Swap Mutation
• Mutate Offspring 2 with a given probability
  using Swap Mutation
• Add the resulting offspring to the new generation
• Until (population in new generation population
  in parent generation)
• Replace the parent generation using Elitist
  Replacement
• Until (no improvement to the best solution has
  occurred in G generations)

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The structure of the population

• Cities numbered by integers, coordinates stored
  in arrays xcity_number and ycity_number
• Routes represented as sequences of city numbers.
  The population is implemented as a 2-dimensional
  arraygenomegenome_numberroute_index
  city_numberwhere genome_number is an integer
  used for identifying the genome, route_index is
  the location of the city city_number on the
  salesman's route.

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Parent selection

• The Tournament SelectionRepeat Pick k random
  individuals from the population From those k
  individuals, pick one with the best fitness If
  the same individual chosen as both parents,
  discard the second oneUntil (two parents
  chosen)
• The parameter k is the tournament size

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Crossover

• Partially Matched Crossover (PMX)
• Choose two random crossing sitesParent 1 2 5
  14 0 93 7 8 6Parent 2 5 9 27 6 83 4 0 1
• Exchange the segment between the crossing
  sitesOffspring 1 2 5 17 6 83 7 8 6Offspring
  2 5 9 24 0 93 4 0 1
• Exchange the duplicate allelesOffspring 1 2 5
  17 6 83 4 9 0Offspring 2 5 8 24 0 93 7 6 1

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Mutation

• Swap mutation
• Two positions of the offspring are picked
  randomly, and their values swapped4 2 6 5 8 9 0
  1 37 ? 4 2 0 5 8 9 6 1 3 7
• Mutation takes place in each offspring with a
  given probability

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Replacement

• The Elitist replacement
• From the combined population of parents and
  offspring, a number e of elites is chosen. The
  elites are the e individuals with the best
  fitness. All the elites will be transferred into
  a new parent population, and the rest will be
  chosen randomly from the offspring.
• The number of elites is given as a percentage of
  population size.

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The test runs Convergence
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The effect of the number of cities
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The effect of the population size