In order to continue evolution, genetic algorithms need a method that allows them to form a new genetic code from the code taken from two parents.  Some features should come from one parent; other features should come from the other parent.  For a continuous chromosome composed of binary bits, the procedure involves dividing the string of 0s and 1s at a random place and taking the left fragment from one parent and the right fragment from the other one.  This process, called crossover, is illustrated in Figure A.1.

Figure A.1:  This diagram displays crossover for 6-bit continuous chromosomes.  Vertical bars denote the crossing point.

The simple crossover operator described above produced one offspring from two parents, with each parent having one chromosome.  What about individuals with many chromosomes?  One possibility is to merge all of the chromosomes of an individual into a single bit string and treat this string as one large chromosome.  In this case, we apply the crossover operator as if we had individuals with one large chromosome.  The second possibility is to treat all of the chromosomes independently.  In this case, we apply the crossover operator to each pair of corresponding chromosomes of two mates and produce the chromosomes of a new individual one by one.  These two procedures are illustrated by figures A.2 and A.3 respectively.

Figure A.2 The figure displays an example of a crossover operator for individuals with four 4-bit chromosomes.  All chromosomes are merged into one long 16-bit string.  Thick lines denote chromosome boundaries; vertical bars mark crossover points.

Figure A.3  The figure displays an example of the crossover operator for individuals with four 4-bit chromosomes each.  All chromosomes are treated independently, and crossover is applied individually to each pair of 4-bit chromosomes.  Thick lines denote chromosome boundaries; thin vertical bars mark crossover points.

As seen from the figures, the second variant of the crossover gives a more intense mixing of genetic material.  The number of crossover points is equal to the number of chromosomes in an individual.  The probability that one of the chromosomes will be copied as a whole from a single parent is relatively low.