GENETIC ALGORITHMS in Visual Studio .NET

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a tness of F(c) = 0, as this is problematic for some selection operators, as will be seen later 653 Reproduction The operators are the instruments with which a GA explores the search space of a problem Crossover and mutation are reproduction operators that create new chromosomes from existing ones Their design is closely tied to the chosen chromosome representation Since a solution represented by a chromosome should be feasible otherwise the search space becomes even larger operators must be de ned correspondingly Crossover The crossover operator is the more signi cant one of the two It implements the principle of evolution New chromosomes are created with this operator by combining two randomly selected parent chromosomes, thereby inheriting the genetic material of its ancestors How this is done depends on the encoding of the chromosome In Section 652 two indirect chromosome encodings are presented, whereby the direct representation is a combination of the two Since the two encodings differ signi cantly, two different operators are necessary
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Processor Allocation For the chromosome encoding of the processor allocation, quite simple crossover operators can be employed The processor allocation chromosome is a value encoded chromosome, where each gene can assume the same range of values (1 to |P|) Furthermore, the value of one gene has no impact on the possible values of the other genes Therefore, a simple two-point crossover operator works as follows Given two randomly chosen chromosomes c1 and c2 , two new chromosomes c3 and c4 are generated by swapping a randomly determined gene interval Let the interval range from i to j Outside this interval, that is, [1, i 1] and [ j + 1, |V|], the genes of c3 have the values of c1 and inside those of c2 For c4 it is the converse Figure 618 illustrates this crossover operator The simpler and more common single-point crossover operator is a special case of this, achieved by setting j = |V| Figure 619 visualizes a single-point crossover operator The extension of this concept to multiple crossover points is straightforward Note that the generated new chromosomes are always valid chromosomes; that is, they represent a valid processor allocation of the tasks V
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Figure 618 Illustration of two-point crossover of processor allocation chromosomes (i = 4, j = 7)
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Figure 619 Illustration of single-point crossover of processor allocation chromosomes (i = 6)
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Node List The chromosome encoding of a node list is a permutation encoding of the node set V Each element of the permutation, that is, each node n V, can appear only once in the chromosome Thus, the simple crossover operator presented for the processor allocation encoding cannot be employed To understand this, image two chromosomes c1 and c2 in which node f is at position 5 in c1 and position 6 in c2 In the case where a single-point crossover operator divides the chromosomes at, for example, position 6, the resulting chromosome c3 will have node f twice, at position 5 and at position 6, while there will be no f in chromosome c4 This is depicted in Figure 620 The problem is overcome with the following single-point permutation crossover operator Given two randomly chosen chromosomes c1 and c2 , a crossover point i, 1 i < |V|, is selected randomly The genes [1, i] of c1 and c2 are copied to the genes [1, i] of the new chromosomes c3 and c4 , respectively To ll the remaining genes [i + 1, |V|] of c3 (c4 ), chromosome c2 (c1 ) is scanned from the rst to the last gene and each node that is not yet in c3 (c4 ) is added to the next empty position of c3 (c4 ) in the order that it is discovered Figure 621 illustrates the procedure of this operator Under the condition that the node lists of chromosomes c1 and c2 are in precedence order, this operator even guarantees that the node lists of c3 and c4 also are It is easy to see this for the genes [1, i] of both c3 and c4 as they are only copied from c1 and c2 The remaining genes of c3 and c4 are lled in the same relative order in which they appear in c2 and c1 , respectively Hence, among themselves, these remaining nodes must also be in precedence order Furthermore, there cannot be a precedence con ict between the nodes on the left side of the crossover point with those on the right side of the crossover point,
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Figure 620 Simple single-point crossover as in Figure 619 fails for node list chromosomes (i = 6)
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