Evolutionary Computation : A Unified Approach

5.2. EA-BASED OPTIMIZATION 99
of traveling from city i to city j, including the home city of the salesperson. So, for example,
the cost of Tour 1 above is given by c(home, A)+c(A, B)+c(B,C)+...+c(F, G)+c(G, home).
One can easily map permutation spaces into an unconstrained parameter space by embedding
them in the space of all combinations of N cities with the aid of a penalty function
for tours that visit a city more than once and exclude others. So, for example, strings like
AACDEFF and BBBBBBB are legal combinations of city names, but are not legal tours.
The difficulty with this approach is that, as the number of cities N increases, the ratio of
legal tours to illegal ones rapidly approaches zero and search efficiency drops off accordingly.
A plausible alternative is to modify the reproductive operators so that only legal tours
are created. For TSP problems, this is a bit of a challenge. A useful way to proceed is
to think of mutation as a perturbation operator and recombination as a mechanism for
swapping subassemblies. From this perspective the simplest perturbation is to swap the
order of two adjacent cities. More aggressive mutations might involve inverting the order of
longer segments such as:
Parent: A B C D E F G ==> Child: A B E D C F G
|_____|
|_____|
or swapping the positions of non-adjacent cities.
For recombination, the obvious subassembly to be exchanged is a subtour, i.e., a subset
of adjacent cities:
Parent 1: A B C D E F G Child 1: A B E D C F G
|_____|
|_____|
==>
Parent 2: E D C A B F G Child 2: C D E A B F G
|_____|
|_____|
With reproductive operators like this we now produce legal offspring. What remains is
to verify that these operators provide useful variations, i.e., that the offspring inherit useful
subassemblies from their parents. Formal methods for assessing this will be presented in
the next chapter. For now, an intuitive notion is that good reproductive operators induce
a correlation in fitness between parents and offspring. For these particular operators, the
correlation is not ideal and has lead to considerable exploration of other operators and
representations (see, for example, Whitley (1989b) or Manderick et al. (1991)).
It turns out that there are also one-to-one mappings between permutation spaces and
binary string spaces (De Jong, 1985). This means that a genotypic approach to solving TSP
problems is also possible, and has the distinct advantage that no new operators need to
be invented. Unfortunately, the known mappings are not distance preserving, so that small
changes in string space can result in large changes in permutation space and vice versa. As a
consequence, genotypic representations based on this mapping have only met with moderate
success.