An Introduction to Genetic Algorithms - Boente

could be cut after the second locus to yield two strings: {(2, 0) (3,0)} and {(1,1) (4,1) (6,0)}. The splice
operator takes two strings and splices them together. For example,
could be spliced together to form
Chapter 4: Theoretical Foundations of Genetic Algorithms
Under the messy encoding, cut and splice always produce perfectly legal strings. The hope is that the
primordial phase will have produced all the building blocks needed to create an optimal string, and in
sufficient numbers so that cut and splice will be likely to create that optimal string before too long. Goldberg
and his colleagues did not use mutation in the experiments they reported.
Goldberg, Korb, and Deb (1989) performed a very rough mathematical analysis of this algorithm to argue
why it should work better than a simple GA, and then showed empirically that it performed much better than a
simple GA on a 30−bit deceptive problem. In this problem, the fitness function took a 30−bit string divided
into ten adjacent segments of three bits each. Each three−bit segment received a fixed score: 111 received the
highest score, but 000 received the second highest and was a local optimum (thus making the problem
deceptive). The score 5 of each three−bit segment was as follows: S(000) = 28; S(001) = 26; S(010) = 22;
S(011) = 0; S(100) = 14; S(101) = 0; S(110) = 0; S(111) = 30. The fitness of a 30−bit string was the sum of the
scores of each three−bit segment. The messy GA was also tried successfully on several variants of this fitness
function (all of roughly the same size).
Two immediate problems with this approach will jump out at the reader: (1) One must know (or guess) ahead
of time the minimum useful schema order k, and it is not clear how one can do this a priori for a given fitness
function. (2) Even if one could guess k, for problems of realistic size the combinatorics are intractable. For
Goldberg, Korb, and Deb's (1989) k = 3, l = 30 problem, the primordial stage started off with a complete
enumeration of the possible three−bit schemas in a 30−bit string. In general there are
such schemas, where k is the order of each schema and l is the length of the entire string. (The derivation of
this formula is left as an exercise.) For l = 30 and k = 3 and , n = 32, 480, a reasonably tractable number to
begin with. However, consider R1 (defined in chapter 4), in which l = 64 and k = 8 (a very reasonably sized
problem for a GA). In that case, n H 10 12 . If each fitness evaluation took a millisecond, evaluating the initial
population would take over 30 years! Since most messy−GA researchers do not have that kind of time, a new
approach had to be found.
Goldberg, Deb, Kargupta, and Harik (1993) refer—a bit too calmly—to this combinatorial explosion as the
"initialization bottleneck." Their proposed solution is to dispense with the complete enumeration of order−k
schemas, and to replace it by a "probabilistically complete initialization." The combinatorics can be overcome
in part by making the initial strings much longer than k (though shorter than l0, so that implicit parallelism
provides many order−k schemas on one string. Let the initial string length be denoted by l', and let the initial
population size be denoted by ng. Goldberg et al. calculate what pairs of l', and ng values will ensure that, on
average, each schema of order k will be present in the initial population. If l' is increased, ng can be greatly
decreased for the primordial stage.
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