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Given a clearly defined problem to be solved and a bit string representation for candidate solutions, a simple GA works as follows:

Start with a randomly generated population of n 7-bit chromosomes (candidate solutions to a problem).

Calculate the fitness f(x) of each chromosome x in the population.

Repeat the following steps until n offspring have been created: a.

Select a pair of parent chromosomes from the current population, the probability of selection being an increasing function of fitness. Selection is done "with replacement," meaning that the same chromosome can be selected more than once to become a parent.

With probability pc (the "crossover probability" or "crossover rate"), cross over the pair at a randomly chosen point (chosen with uniform probability) to form two offspring. If no crossover takes place, form two offspring that are exact copies of their respective parents. (Note that here the crossover rate is defined to be the probability that two parents will cross over in a single point. There are also "multi-point crossover" versions of the GA in which the crossover rate for a pair of parents is the number of points at which a crossover takes place.)

Mutate the two offspring at each locus with probability pm (the mutation probability or mutation rate), and place the resulting chromosomes in the new population.

If n is odd, one new population member can be discarded at random.

Replace the current population with the new population.

Each iteration of this process is called a generation. A GA is typically iterated for anywhere from 50 to 500 or more generations. The entire set of generations is called a run. At the end of a run there are often one or more highly fit chromosomes in the population. Since randomness plays a large role in each run, two runs with different random-number seeds will generally produce different detailed behaviors. GA researchers often report statistics (such as the best fitness found in a run and the generation at which the individual with that best fitness was discovered) averaged over many different runs of the GA on the same problem.

The simple procedure just described is the basis for most applications of GAs. There are a number of details to fill in, such as the size of the population and the probabilities of crossover and mutation, and the success of the algorithm often depends greatly on these details. There are also more complicated versions of GAs (e.g., GAs that work on representations other than strings or GAs that have different types of crossover and mutation operators). Many examples will be given in later chapters.

As a more detailed example of a simple GA, suppose that l (string length) is 8, that f(X) is equal to the number of ones in bit string x (an extremely simple fitness function, used here only for illustrative purposes), that n(the population size)is 4, that pc = 0.7, and that pm = 0.001. (Like the fitness function, these values of l and n were chosen for simplicity. More typical values of l and n are in the range 50-1000. The values given for pc and pm are fairly typical.)

The initial (randomly generated) population might look like this:

Chromosome label |
Chromosome string |
Fitness |

A |
00000110 |
2 |

B |
11101110 |
6 |

C |
00100000 |
1 |

D |
00110100 |
3 |

A common selection method in GAs is fitness-proportionate selection, in which the number of times an individual is expected to reproduce is equal to its fitness divided by the average of fitnesses in the population. (This is equivalent to what biologists call "viability selection.")

A simple method of implementing fitness-proportionate selection is "roulette-wheel sampling" (Goldberg 1989a), which is conceptually equivalent to giving each individual a slice of a circular roulette wheel equal in area to the individual's fitness. The roulette wheel is spun, the ball comes to rest on one wedge-shaped slice, and the corresponding individual is selected. In the n = 4 example above, the roulette wheel would be spun four times; the first two spins might choose chromosomes B and D to be parents, and the second two spins might choose chromosomes B and C to be parents. (The fact that A might not be selected is just the luck of the draw. If the roulette wheel were spun many times, the average results would be closer to the expected values.)

Once a pair of parents is selected, with probability pc they cross over to form two offspring. If they do not cross over, then the offspring are exact copies of each parent. Suppose, in the example above, that parents B and D cross over after the first bit position to form offspring E = 10110100 and F = 01101110, and parents B and C do not cross over, instead forming offspring that are exact copies of B and C. Next, each offspring is subject to mutation at each locus with probability pm. For example, suppose offspring E is mutated at the sixth locus to form E' = 10110000, offspring F and C are not mutated at all, and offspring B is mutated at the first locus to form B' = 01101110. The new population will be the following:

Chromosome label |
Chromosome string |
Fitness |

E' |
10110000 |
3 |

F |
01101110 |
5 |

C |
00100000 |
1 |

B' |
01101110 |
5 |

Note that, in the new population, although the best string (the one with fitness 6) was lost, the average fitness rose from 12/4 to 14/4. Iterating this procedure will eventually result in a string with all ones.

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