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92 Crossover
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(a) UNDX Operator
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Figure 92 Illustration of Multi-parent Center of Mass Crossover Operators (dots represent potential o psring) over these distances is calculated, ie = O spring is generated using
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n 2 xi (t) = xi (t) + N (0, 1 )|di (t)| + l=1,i=l 2 N (0, 2 ) el (t) n l=1,l=i l
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(918)
where xi (t) is the randomly selected parent of o spring xi (t), and el (t) are the n 1
9 Genetic Algorithms
orthonormal bases that span the subspace perpendicular to di (t) The e ect of the PCX operator is illustrated in Figure 92(c) Eiben et al [231, 232, 233] developed a number of gene scanning techniques as multiparent generalizations of n-point crossover For each o spring to be created, the gene scanning operator is applied as summarized in Algorithm 95 The algorithm contains two main procedures: A scanning strategy, which assigns to each selected parent a probability that the o spring will inherit the next component from that parent The component under consideration is indicated by a marker A marker update strategy, which updates the markers of parents to point to the next component of each parent Marker initialization and updates depend on the representation method For binary representations the marker of each parent is set to its rst gene The marker update strategy simply advances the marker to the next gene Eiben et al proposed three scanning strategies: Uniform scanning creates only one o spring The probability, ps (xl (t)), of inheriting the gene from parent xl (t), l = 1, , n , as indicated by the marker of that parent is computed as ps (xl (t + 1)) = 1 n (919)
Each parent has an equal probability of contributing to the creation of the o spring Occurrence-based scanning bases inheritance on the premise that the allele that occur most in the parents for a particular gene is the best possible allele to inherit by the o spring (similar to the majority mating operator) Occurrencebased scanning assumes that tness-proportional selection is used to select the n parents that take part in recombination Fitness-based scanning, where the allele to be inherited is selected proportional to the tness of the parents Considering maximization, the probability to inherit from parent xl (t) is ps (xl (t)) =
n i=1
f (xl (t)) f (xi (t))
(920)
Roulette-wheel selection is used to select the parent to inherit from For each of these scanning strategies, the o spring inherits ps (xl (t + 1))nx genes from parent xl (t)
93 Mutation
x1(t)
1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 0000000000000000000001111 1111111111111111111110000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000
1 2 3
x2(t)
x3(t)
1111111111111111111111111 0000000000000000000000000 1111111111111111111110000 0000000000000000000001111 1111111111111111111110000 0000000000000000000001111
Multiple o spring
Single o spring
Figure 93 Diagonal Crossover
Algorithm 95 Gene Scanning Crossover Operator Initialize parent markers; for j = 1, , nx do Select the parent, xl (t), to inherit from; xj (t) = xlj (t); Update parent markers; end
The diagonal crossover operator developed by Eiben et al [232] is a generalization of n-point crossover for more than two parents: n 1 crossover points are selected and applied to all of the n = n + 1 parents One or n + 1 o spring can be generated by selecting segments from the parents along the diagonals as illustrated in Figure 93, for n = 2, n = 3
Mutation
The aim of mutation is to introduce new genetic material into an existing individual; that is, to add diversity to the genetic characteristics of the population Mutation is used in support of crossover to ensure that the full range of allele is accessible for each gene Mutation is applied at a certain probability, pm , to each gene of the o spring, xi (t), to produce the mutated o spring, xi (t) The mutation probability, also referred
1111 0000 1111 0000 1111 0000 11111 00000 11111 00000 11111 00000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000
1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000
11111 00000 11111 00000 11111 00000
1111 11111 0000x (t) 00000 1111 0000 1111 00000 0000 11111 11111 1111 0000 1111 0000 0000000000000000000001111 11111111111111110000 11111111111111111111 00000000000000000000 x (t) 11111111111111111111 00000000000000000000 1111 0000 00000000000000001111 11111111111111110000 111111111111111111111 000000000000000000000 111111111111111111111 x (t) 000000000000000000000 111110000000000000000 000001111111111111111