In terms of this crossover operator function, the conditional probability of produc-
ing, via reproduction and crossover, a solution k e S given a current population described
byn S'is
P2'(k I n) =p, x P(i I ) x P (jI n) x l(ij, k, s)
iE SjE S L s
+(1 c) x Pi(k I n) Eq. 4.22
1 s=L
=P x-x Y P1(i n)x Pi(j n)xI(i,j,k,s)
L ie Sj Ss=l
+(1 pc) x P,(k I n)
where P,(i n) is as defined as in Eq. 4.2 and where P2'(i I n) refers to the two-operator
algorithm consisting of reproduction and crossover without mutation. This result assumes
uniformly distributed crossover site selection.
The array of conditional probabilities [P2'(i I n)] plays a role in the three-operator
simple genetic algorithm very analogous to the role played by the array [P,(i I n)] in the
two-operator variant. In fact, the [P2'(i I n)] array can be used as counterparts of Eq. 4.2 to
develop results exactly analogous to Eq. 4.3 and Eq. 4.6. Further, for n SA', Eq. 4.22
reduces to
P2'(k nA)= P(k I nh), Eq. 4.23
and consequently this (fictitious) two-operator algorithm (reproduction and crossover)
demonstrates the same sort of absorbing state behavior as the one-operator algorithm.
From Eq. 4.22, the three-operator conditional probabilities and state transition
matrix are expressible as
P3(i | n)= x Y a"('J) X P2'(j I n), Eq. 4.24
(1+a)L jes