An Introduction to Genetic Algorithms - Boente

Defining ri,j(k) and is somewhat tricky. Vose and Liepins first defined a simpler matrix M whose
elements Mi,j give the probability ri,j(0) that string 0 (i.e., the string of all zeros) will be produced by a
recombination event between string i and string j, given that i and j are selected to mate. I will go through this
construction in detail so readers less familiar with probability theory can see how such constructions are done.
(Other readers may wish to attempt it themselves before reading the following.) Once ri,j(0) is defined, it can
be used in a clever way to define the general case.
The expression for ri,j(0) is equal to the sum of two terms: the probability that crossover does not occur
between i and j and the selected offspring (i or j) is mutated to all zeros (first term) and the probability that
crossover does occur and the selected offspring is mutated to all zeros (second term).
If i and j are selected to mate, the probability that crossover occurs between them is pc and the probability that
it does not occur is 1 p c. Likewise, the probability that mutation occurs at each bit in the selected offspring
is pm and the probability that it does not occur is 1 p m. If | i| is the number of ones in a string iof length l, the
probability that i will be mutated to all zeros is the probability that all of the |i| ones will be mutated times the
probability that none of the (l |i|) zeros will be mutated:
The first term in the expression for ri,j(0) translates to
Recall that, in this model, only one offspring is selected for the next population. The factor ½ indicates that
each of the two offspring has equal probability of being selected.
For the second term, let h and k denote the two offspring produced from a crossover at point c(counted from
the right−hand side of the string; see figure 4.3). Note that there are l 1 possible crossover points, so the
probability of choosing point c is 1/(l 1). The second term can then be written as
Figure 4.3: Illustration of c, i1, i2, j1, and j2.
Chapter 4: Theoretical Foundations of Genetic Algorithms
Again, the factor 1/2 indicates that one of the two offspring is selected, with equal probability for each.
To complete this, we need only the expressions for |h| and |k|. Let i1 be the substring of i consisting of the l c
bits to the left of point c, let i2 be the substring consisting of the c bits to the right of point c, and let j1 and j2
106