Fitness Function

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In the Darwinian model of evolution, individuals with the best characteristics have the best chance to survive and to reproduce In order to determine the ability of an individual of an EA to survive, a mathematical function is used to quantify how good the solution represented by a chromosome is The fitness function, f , maps a chromosome representation into a scalar value: f : nx R (86)

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where represents the data type of the elements of an nx -dimensional chromosome The tness function represents the objective function, , which describes the optimization problem It is not necessarily the case that the chromosome representation corresponds to the representation expected by the objective function In such cases, a more detailed description of the tness function is f : SC SX R R+

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(87)

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where SC represents the search space of the objective function, and , and respectively represent the chromosome decoding function, the objective function, and the scaling function The (optional) scaling function is used in proportional selection to ensure positive tness values (refer to Section 85) As an example, f : {0, 1}nb Rnx R R+

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(88)

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where an nb -bitstring representation is converted to a oating-point representation using either equation (82) or (85) For the purposes of the remainder of this part on EC, it is assumed that SC = SX for which f = Usually, the tness function provides an absolute measure of tness That is, the solution represented by a chromosome is directly evaluated using the objective function For some applications, for example game learning (refer to 11) it is not possible to nd an absolute tness function Instead, a relative tness measure is used

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8 Introduction to Evolutionary Computation

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to quantify the performance of an individual in relation to that of other individuals in the population or a competing population Relative tness measures are used in coevolutionary algorithms (refer to 15) It is important to realize at this point that di erent types of optimization problems exist (refer to Section A3), which have an in uence on the formulation of the tness function: Unconstrained optimization problems as de ned in De nition A4, where, assuming that SC = SX , the tness function is simply the objective function Constrained optimization problems as de ned in De nition A5 To solve constrained problems, some EAs change the tness function to contain two objectives: one is the original objective function, and the other is a constraint penalty function (refer to Section A6) Multi-objective optimization problems (MOP) as de ned in De nition A10 MOPs can be solved by using a weighted aggregation approach (refer to Section A8), where the tness function is a weighted sum of all the sub-objectives (refer to equation (A44)), or by using a Pareto-based optimization algorithm Dynamic and noisy problems, where function values of solutions change over time Dynamic tness functions are time-dependent whereas noisy functions usually have an added Gaussian noise component Dynamic problems are de ned in De nition A16 Equation (A58) gives a noisy function with an additive Gaussian noise component As a nal comment on the tness function, it is important to emphasize its role in an EA The evolutionary operators, eg selection, crossover, mutation and elitism, usually make use of the tness evaluation of chromosomes For example, selection operators are inclined towards the most- t individuals when selecting parents for crossover, while mutation leans towards the least- t individuals

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Selection

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Selection is one of the main operators in EAs, and relates directly to the Darwinian concept of survival of the ttest The main objective of selection operators is to emphasize better solutions This is achieved in two of the main steps of an EA: Selection of the new population: A new population of candidate solutions is selected at the end of each generation to serve as the population of the next generation The new population can be selected from only the o spring, or from both the parents and the o spring The selection operator should ensure that good individuals do survive to next generations Reproduction: O spring are created through the application of crossover and/or mutation operators In terms of crossover, superior individuals should have more opportunities to reproduce to ensure that o spring contain genetic material of the best individuals In the case of mutation, selection mechanisms

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