85 Selection

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should focus on weak individuals The hope is that mutation of weak individuals will result in introducing better traits to weak individuals, thereby increasing their chances of survival Many selection operators have been developed A summary of the most frequently used operators is given in this section Preceding this summary is a discussion of selective pressure in Section 851

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Selective Pressure

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Selection operators are characterized by their selective pressure, also referred to as the takeover time, which relates to the time it requires to produce a uniform population It is de ned as the speed at which the best solution will occupy the entire population by repeated application of the selection operator alone [38, 320] An operator with a high selective pressure decreases diversity in the population more rapidly than operators with a low selective pressure, which may lead to premature convergence to suboptimal solutions A high selective pressure limits the exploration abilities of the population

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Random Selection

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Random selection is the simplest selection operator, where each individual has the 1 same probability of ns (where ns is the population size) to be selected No tness information is used, which means that the best and the worst individuals have exactly the same probability of surviving to the next generation Random selection has the lowest selective pressure among the selection operators discussed in this section

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Proportional Selection

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Proportional selection, proposed by Holland [376], biases selection towards the most t individuals A probability distribution proportional to the tness is created, and individuals are selected by sampling the distribution, s (xi (t)) = f (xi (t)) ns l=1 f (xl (t)) (89)

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where ns is the total number of individuals in the population, and s (xi ) is the probability that xi will be selected f (xi ) is the scaled tness of xi , to produce a positive oating-point value For minimization problems, possible choices of scaling function, , are f (xi (t)) = (xi (t)) = fmax f (xi (t)) where f (xi (t)) = (xi (t)) is the raw tness value of xi (t) However, knowledge of fmax (the maximum possible tness) is usually not available An alternative is to use fmax (t), which is the maximum tness observed up to time step t

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8 Introduction to Evolutionary Computation

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1 f (xi (t)) = (xi (t)) = 1+f (xi (t)) fmin (t) , where fmin (t) is the minimum observed tness up to time step t Here, f (xi (t)) (0, 1]

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In the case of a maximization problem, the tness values can be scaled to the range (0,1] using 1 (810) f (xi (t)) = (xi (t)) = 1 + fmax (t) f (xi (t)) Two popular sampling methods used in proportional selection is roulette wheel sampling and stochastic universal sampling Assuming maximization, and normalized tness values, roulette wheel selection is summarized in Algorithm 82 Roulette wheel selection is an example proportional selection operator where tness values are normalized (eg by dividing each tness by the maximum tness value) The probability distribution can then be seen as a roulette wheel, where the size of each slice is proportional to the normalized selection probability of an individual Selection can be likened to the spinning of a roulette wheel and recording which slice ends up at the top; the corresponding individual is then selected Algorithm 82 Roulette Wheel Selection Let i = 1, where i denotes the chromosome index; Calculate s (xi ) using equation (89); sum = s (xi ); Choose r U (0, 1); while sum < r do i = i + 1, ie advance to the next chromosome; sum = sum + s (xi ); end Return xi as the selected individual;

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When roulette wheel selection is used to create o spring to replace the entire population, ns independent calls are made to Algorithm 82 It was found that this results in a high variance in the number of o spring created by each individual It may happen that the best individual is not selected to produce o spring during a given generation To prevent this problem, Baker [46] proposed stochastic universal sampling (refer to Algorithm 83), used to determine for each individual the number of o spring, i , to be produced by the individual with only one call to the algorithm Because selection is directly proportional to tness, it is possible that strong individuals may dominate in producing o spring, thereby limiting the diversity of the new population In other words, proportional selection has a high selective pressure

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85 Selection Algorithm 83 Stochastic Universal Sampling for i = 1, , ns do i (t) = 0; end 1 r U (0, ), where is the total number of o spring; sum = 00; for i = 1, , ns do sum = sum + s (xi (t)); while r < sum do i + +; 1 r = r + ; end end return = ( 1 , , ns );

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