An Introduction to Genetic Algorithms - Boente

2.
3.
Calculate the fitness f(x) of each string x in the population.
Choose (with replacement) two parents from the current population with probability proportional to
each string's relative fitness in the population.
Cross over the two parents (at a single randomly chosen point) with probability pc to form two
offspring. (If no crossover occurs, the offspring are exact copies of the parents.) Select one of the
offspring at random and discard the other.
4.
Mutate each bit in the selected offspring with probability pm, and place it in the new population.
5.
Go to step 2 until a new population is complete.
6.
Go to step 1.
Chapter 4: Theoretical Foundations of Genetic Algorithms
The only difference between this and the standard simple GA is that only one offspring from each crossover
survives. Thus, for population size n, a total of n recombination events take place. (This modification
simplifies parts of the formalization.)
In the formal model of Vose and Liepins, each string in the search space is represented by the integer i
between 0 and 2 l 1 encoded by the string. For example, for l = 8, the string 00000111 would be represented
by the integer 7. The population at generation t is represented by two real−valued vectors, and , each
of length 2 l . The ith component of (denoted pi(t)) is the proportion of the population at generation t
consisting of string i, and the ith component of (denoted si(t)) is the probability that an instance of string i,
will be selected to be a parent at step 2 in the simple GA given above. For example, if l = 2 and the population
consists of two copies of 11 and one copy each of 01 and 10,
If the fitness is equal to the number of ones in the string,
(For the purpose of matrix multiplication these vectors will be assumed to be column vectors, though they will
often be written as row vectors.)
The vector exactly specifies the composition of the population at generation t, and reflects the
selection probabilities under the fitness function. These are connected via fitness: let F be a two−dimensional
matrix such that Fi,j = 0 for i `j and Fi,i = f(i). That is, every entry of F is 0 except the diagonal entries ((i,i)),
which give the fitness of the corresponding string i. Under proportional selection,
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