Evolutionary Computation : A Unified Approach

142 CHAPTER 6. EVOLUTIONARY COMPUTATION THEORY
P (t +1)=evolve(P (t)) = reproduction(parent selection(P (t)))
We can further isolate the effects of reproductive variation by assuming uniform (neutral)
selection; i.e., the probability of selecting a particular genotype to be a parent depends only
on its frequency in the population:
P (t +1)=evolve(P (t)) = reproduction(uniform selection(P (t)))
As before, this can viewed as comprising r − 1 equations of the form:
c i (t +1) = reproduce i (uniform selection(P (t)))
each of which specifies the number of genotypes of type i that will be generated by the reproductive
operators. For single parent (i.e., asexual) reproductive operators, this corresponds
to equations of the form:
c i (t +1) =
∑ r
j=1 reproduce ij(t)(select j (t))
in which select j (t) represents the number of times genotype j is selected to be a parent at
time t, andreproduce ij (t), the number of times an asexual reproductive process converts
genotype j into genotype i. Ifreproduce ij (t) is independent of time, this simplifies further
to:
c i (t +1) =
∑ r
j=1 reproduce ij(select j (t))
If parent selection is assumed to be deterministic and uniform, then select j (t) = c j (t), and
c i (t +1) =
∑ r
j=1 reproduce ij(c j (t))
Multi-parent (sexual) reproductive operators are handled in a similar fashion. For example,
two-parent reproduction is given by:
c i (t +1) =
∑ r
j=1
∑ r
k=1 reproduce ijk(t)(select j (t), select k (t))
which again simplifies to:
c i (t +1) =
∑ r
j=1
∑ r
k=1 reproduce ijk(select j (t), select k (t))
when reproduce ijk is independent of time, and further to
c i (t +1) =
∑ r
j=1
∑ r
k=1 reproduce ijk(c j (t),c k (t))
with deterministic-uniform parent selection.
Clearly, the required reproduce elements of these models will be highly dependent on
the genome representation and the particular choices of reproductive operators. These
dependencies are explored in the following subsections.