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(a) Compute a random value $ ~ 7(0,1). (b) Compute a step size r/j ~ N(0, cr 2 ). (c) If & < Pm then Cn,i+ = rjiUsually, the variance a2 is a function of the fitness of the individual. Individuals with a good fitness value will be mutated less, while a bad fitness value will lead to large mutations.
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While GAs with single populations are a form of parallelization, more benefits can be obtained by evolving a number of subpopulations in parallel, which is a cooperative model [Grosso 1985]. In this GA model we have islands, where each island represents one population. Selection, cross-over and mutation occur in each subpopulation independently from the other subpopulations. In addition, individuals are allowed to migrate to another island, or subpopulation, as illustrated in Figure 9.4. In this way genetic material is shared among the subpopulations.
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Figure 9.4: An island GA system
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Island GA models introduce interesting questions, one of which is how to initialize the subpopulations. Of course a pure random approach can be used, which will cause different populations to share the same parts of the search space. A better approach would be to initialize subpopulations to cover different parts of the search space, thereby covering a larger search space and facilitating a kind of niching by individuals islands. Also, in multicriteria optimization, each subpopulation can be allocated the task to optimize one criterion. A meta-level step is then required to combine the solutions from each island. Another question is, when can individuals migrate, and from where, to where The simplest approach is to let migration occur at random. Individuals are selected randomly, and the destination subpopulation is also selected randomly. Alternatively, tournament selection can be used to select migrated individuals, as well as the destination. Intuitively, the best individuals of a poor island may want to migrate to a better island. However, individuals from a poor island may introduce bad genetic material into a good island. Acceptance of an immigrant can then be based on a probability as a function of the immigrant's fitness value compared to that of the intended destination island, or acceptance if the immigrant has the ability to increase the diversity in genetic material, that is, if the individual is maximally different. It is also possible that a highly fit individual from a prosperous island visits islands, taking part in reproduction. Culling of weak individuals in subpopulations can also be modeled. Immigrants can be allowed, for example, if that immigrant can replace an individual of the destination island with a lower fitness than the immigrant. In doing so, the weaker individual is killed and removed from the population. It should be noted at this point that the EA employed for islands does not necessarily need to be a GA. It can be any of the other EAs. A different kind of "island" GA model is the Cooperative Coevolutionary GA of Potter [Potter 1997]. In this case, instead of distributing entire individuals over several subpopulations, each subpopulation is given one gene (one parameter of the optimization problem) to optimize. This is a coevolutionary process, discussed in more detail in chapter 15.
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This section studies a real-world application of GAs. The problem is the optimization of routes in a telecommunications network [Sevenster and Engelbrecht 1996]. Given a network of M switches, an origin switch and a destination switch, the objective is to find the best route to connect a call between the origin and destination switches. The design of the GA is done in the following steps:
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1. Chromosome representation: A chromosome consists of a maximum of M switches. Chromosomes can be of variable length, since telecommunication routes can differ in length. Each gene represents one switch. Integer values representing switch numbers are used as gene values - no binary encoding is used. The first gene represents the origin switch and the last gene the destination switch. Example chromosomes are
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( 1 3 6 10)
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Duplicate switches are ignored. The first chromosome represents a route from switch 1 to switch 3 to switch 6 to switch 10. 2. Initialization of population: Individuals are generated randomly, with the restriction that the first gene represents the origin switch and the last gene represents the destination switch. For each gene, the value of that gene is selected as a uniform random value in the range [1, M]. 3. Fitness function: The multi-criteria objective function
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