Evolution and Optimum Seeking

Genetic Algorithms 153
mean objective function value of the generation (proportional selection) orby
their rank (e.g., linear ranking selection).
Step 2: (Recombination)
Two di erent preliminary o spring are produced by recombination of two
parental genotypes by means of crossover at a given recombination probability
pc only one of those o spring (at random) is actually taken into further
consideration.
Steps 1 and 2 are repeated until individuals represent the (next) generation.
Step 3: (Mutation)
The o spring eventually (with a given xed and small probability pm) underly
further modi cation by means of point mutations working on individual bits,
either by reversing a one to a zero, or vice versa or by throwing a dice for
choosing a zero or a one, independent of the original value.
At rst glance, this scheme looks very similar to that of a multimembered ES with
discrete recombination. To reveal the di erences one has to take a closer look at the
so-called operators, \selection (S)", \mutation (M)", and \recombination (R)." The GA
sequence of events, i.e., S { R { M, as opposed to M { R { S within ESs, should not matter
signi cantly since the whole process is a circular one, and whether one likes to reverse the
order of mutation and recombination is a matter of avoiding unnecessary operations or
not. In applications, the evaluation of the individuals with respect to their corresponding
objective function values normally dominates all other operations. Canonical values for
the recombination probability arepc =0:6, for the number of crossover points nc =2,
and for the mutation probability pm =0:001.
5.3.2 Representation of Individuals
One of the most apparent di erences between GAs and ESs is the fact, that completely
di erent representations of the object variables are used. Organic evolution uses four
di erent nucleotides to encode the genotype in pairs of triplets. By means of the genetic
code these are translated to 20 di erent amino acids. Since there are 4 3 = 64 di erent
triplets, the genetic code is largely redundant. A closer look reveals its property of
maintaining similarity on the amino acid level despite most of the small variations on the
level of single nucleotides. Similar transmission laws between chains of amino acids and
proteins, proteins and higher aggregates like cells and organs, up to the overall phenotype
are called the epigenetic apparatus (Riedl, 1976). As a matter of fact, biologists as
well as behaviorists report that di erences among several children of the same parents as
well as di erences between two consecutive generations can well be described by normal
distributions with zero mean and characteristic, probably genetically coded, variances.
That is why ESs, when used for seeking optimal values for continuous variables use the
more aggregate model of normal distributions for mutations and discrete or intermediary
recombination as described in Sections 5.1 and 5.2.
GAs, however, rely on binary representations of the object variables. One might call
this genotypic modelling of the variation process, instead of phenotypic modelling as is