3. a = Cni and fl = CH2.

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4. Compute the mask, m (see below). 5. For i 1, , / , if (mi = 1) then swap the genetic material: (a) oti Cn2,i (b) & - CnLti 6. Return the offspring ai and /%. In the procedure above, m is a mask which specifies which bits of the parents should be swapped to generate offspring. Several cross-over operators have been developed to compute the mask: Uniform cross-over: For uniform cross-over the mask of length / is created at random for each pair of individuals selected for reproduction. A bit with value of 1 indicates that the corresponding allele have to be swapped between the two parents. Figure 9.2(a) illustrates uniform cross-over, while the pseudocode for generating the mask is given below: 1. mi = 0 for all i = 1, , / . 2. For each i = 1, , / : (a) Calculate a random value ~ U(0,1). (b) If ( < p x ), then mi = l. 3. Return the mask vector m. In the above px is the cross-over probability at each position in the chromosome. If, for example, px = 0.5, each bit has an equal chance to take part in cross-over.

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CHAPTERS. GENETIC ALGORITHMS

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One-point cross-over: A single bit position is randomly selected and the bit substrings after that point are swapped between the two chromosomes. One-point cross-over is illustrated in Figure 9.2(b), while the mask vector is calculated as 1. 2. 3. 4. Calculate a random value ~ U(1, I 1). mi = 0 for all i = 1, ,!. For each i + 1, , / let mi = 1. Return the mask vector ra.

Two point cross-over: In this case two bit positions are randomly selected, and the bit substrings between these points are swapped as illustrated in Figure 9.2(c). The mask vector is calculated as 1. 2. 3. 4. Compute two random variable 1,^2 ~ U(l, I). mi = 0 f o r all i = 1, - , / . For each i = 1, , 2 let mj = 1. Return the mask vector m.

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In the case of continuous-valued genes, and arithmetic cross-over can be used. Consider the two parents Cn1 and Cn2. Then, two offspring, On1 and On2 are generated using O ni>i = riC ni>i + (1.0 On^i = (1.0 - r2)Cn1,i withri,r 2

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Mutation

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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 cross-over to make sure that the full range of allele values is accessible in the search. Mutation also occurs at a certain probability pm, referred to as the mutation rate. Usually, a small value for pm [0,1] is used to ensure that good solutions are not distorted too much. However, research has shown that an initial large mutation rate that decreases exponentially as a function of the number of generations improves convergence speed and accuracy. (An annealing schedule similar to that of equation (4.22) for the learning rate can be used.) The initial large pm ensures that a large search space is covered, while the pm becomes rapidly smaller when individuals start to converge to the optimum. Considering binary representations, the following mutation schemes have been developed:

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9.5. MUTATION

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Figure 9.2: Cross-over operators

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Random mutate, where bit positions are chosen randomly and the corresponding bit values negated (as illustrated in Figure 9.3(a)). A pseudocode algorithm, for random mutation is the following: 1. For each i = 1, , I : (a) Compute a random value ~ U(0,1). (b) If < pm then C n , i = Cn,i, where C is the complement of C. Inorder mutate, where two bit positions are randomly selected and only bits between these positions are mutated (as illustrated in Figure 9.3(b)). The algorithm for inorder mutation is: 1. Select two random values 1,^2 ~ U(l, , /). 2. For each = 1, , 2' (a) Compute a random value ~ U(0,1). (b) I f < P

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Figure 9.3: Mutation operators In the case of nominal-valued variables, the above mutation operators can be adapted such that the D bits that represent a certain nominal value are replaced by the D bits representing a randomly chosen nominal value. For real-valued representations, mutation occurs by adding a random value to allele, usually sampled from a Gaussian distribution with zero mean and small variance a2: 1. For each real-valued gene

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