An Introduction to Genetic Algorithms - Boente

Inversion works by choosing two points in the string and reversing the order of the bits between them—in the
example just given, bits 3–6 were reversed. This does not change the fitness of the chromosome, since to
calculate the fitness the string is ordered by the indices. However, it does change the linkages: the idea behind
inversion is to produce orderings in which beneficial schemas are more likely to survive. Suppose that in the
original ordering the schema 00**01** is very important. Under the new ordering, that schema is 0010****.
Given that this is a high−fitness schema and will now tend to survive better under single−point crossover, this
permutation will presumably tend to survive better than would the original string.
The reader may have noticed a hitch in combining inversion with single−point crossover. Suppose, for
example, that
crosses with
after the third bit. The offspring are
and
The first offspring has two copies each of bits 1 and 6 and no copies of bits 3 and 5. The second offspring has
two copies of bits 3 and 5 and no copies of bits 1 and 6. How can we ensure that crossover will produce
offspring with a full set of loci? Holland proposed two possible solutions: (1) Permit crossover only between
chromosomes with the same permutation of the loci. This would work, but it severely limits the way in which
crossover can be done. (2) Employ a "master/slave" approach: choose one parent to be the master, and
temporarily reorder the other parent to have the same ordering as the master. Use this ordering to produce
offspring, returning the second parent to its original ordering once crossover has been performed. Both
methods have been used in experiments on inversion.
Inversion was included in some early work on GAs but did not produce any stunning improvements in
performance (Goldberg 1989a). More recently, forms of inversion have been incorporated with some success
into GAs applied to "ordering problems" such as the DNA fragment assembly problem (Parsons, Forrest, and
Burks, in press). However, the verdict on the benefits of inversion to GAs is not yet in; more systematic
experimental and theoretical studies are needed. In addition, any performance benefit conferred by inversion
must be weighed against the additional space (to store indices for every bit) and additional computation time
(e.g., to reorder one parent before crossover) that inversion requires.
Evolving Crossover "Hot Spots"
Chapter 4: Theoretical Foundations of Genetic Algorithms
A different approach, also inspired by nature, was taken by Schaffer and Morishima (1987). Their idea was to
evolve not the order of bits in the string but rather the positions at which crossover was allowed to occur
(crossover "hot spots"). Attached to each candidate solution in the population was a second string—a
"crossover template"—that had a 1 at each locus at which crossover was to take place and a 0 at each locus at
which crossover was not to take place. For example, 10011111:00010010 (with the chromosome preceding
and the crossover template following the colon) meant that crossover should take place after the fourth and
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