Genetic Algorithms in .NET framework

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9 Genetic Algorithms
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to as the mutation rate, is usually a small value, pm [0, 1], to ensure that good solutions are not distorted too much Given that each gene is mutated at probability pm , the probability that an individual will be mutated is given by P rob( i (t) is mutated) = 1 (1 pm )nx x where the individual contains nx genes Assuming binary representations, if H( i (t), xi (t)) is the Hamming distance between x o spring, xi (t), and its mutated version, xi (t), then the probability that the mutated version resembles the original o spring is given by P rob(xi (t)) xi (t)) = pm
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H( i (t),xi (t)) x x (1 pm )nx H( i (t),xi (t))
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This section describes mutation operators for binary and oating-point representations in Sections 931 and 932 respectively A macromutation operator is described in Section 933
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Binary Representations
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For binary representations, the following mutation operators have been developed: Uniform (random) mutation [376], where bit positions are chosen randomly and the corresponding bit values negated as illustrated in Figure 94(a) Uniform mutation is summarized in Algorithm 96 Inorder mutation, where two mutation points are randomly selected and only the bits between these mutation points undergo random mutation Inorder mutation is illustrated in Figure 94(b) and summarized in Algorithm 97 Gaussian mutation: For binary representations of oating-point decision variables, Hinterding [366] proposed that the bitstring that represents a decision variable be converted back to a oating-point value and mutated with Gaussian noise For each chromosome random numbers are drawn from a Poisson distribution to determine the genes to be mutated The bitstrings representing these genes are then converted To each of the oating-point values is added the stepsize N (0, j ), where j is 01 of the range of that decision variable The mutated oating-point value is then converted back to a bitstring Hinterding showed that Gaussian mutation on the oating-point representation of decision variables provided superior results to bit ipping For large dimensional bitstrings, mutation may signi cantly add to the computational cost of the GA In a bid to reduce computational complexity, Birru [69] divided the bitstring of each individual into a number of bins The mutation probability is applied to the bins, and if a bin is to be mutated, one of its bits are randomly selected and ipped
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93 Mutation
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Before Mutation
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Before Mutation
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mutation points After Mutation
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mutation points After Mutation
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(a) Random Mutate
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(b) Inorder Mutate
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Figure 94 Mutation Operators for Binary Representations Algorithm 96 Uniform/Random Mutation for j = 1, , nx do if U (0, 1) pm then x xij (t) = ij (t), where denotes the boolean NOT operator; end end
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Algorithm 97 Inorder Mutation Select mutation points, 1 , 2 U (1, , nx ); for j = 1 , , 2 do if U (0, 1) pm then x xij (t) = ij (t); end end
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Floating-Point Representations
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As indicated by Hinterding [366] and Michalewicz [586], better performance is obtained by using a oating-point representation when decision variables are oating-point values and by applying appropriate operators to these representations, than to convert to a binary representation This resulted in the development of mutation operators for oating-point representations One of the rst proposals was a uniform mutation, where [586] xij (t) = xij (t) + (t, xmax,j xij (t)) xij (t) + (t, xij (t) xmin,j (t)) if a random digit is 0 if a random digit is 1 (923)
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where (t, x) returns random values from the range [0, x]
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9 Genetic Algorithms
Any of the mutation operators discussed in Sections 1121 (for EP) and 1243 (for ES) can be applied to GAs
Macromutation Operator Headless Chicken
Jones [427] proposed a macromutation operator, referred to as the headless chicken operator This operator creates an o spring by recombining a parent individual with a randomly generated individual using any of the previously discussed crossover operators Although crossover is used to combine an individual with a randomly generated individual, the process cannot be referred to as a crossover operator, as the concept of inheritence does not exist The operator is rather considered as mutation due to the introduction of new randomly generated genetic material