Finally, the fitness values are stored in a vector of length Nind. In all of these data structures, each row corresponds to a particular individual.
In this section we outline the major procedures of the GA Toolbox. Population representation and initialisation: crtbase, crtbp, crtrp The GA Toolbox supports binary, integer and floating-point chromosome representations.
Binary and integer populations may be initialised using the Toolbox function to create binary populations, crtbp. An additional function, crtbase, is provided that builds a vector describing the integer representation used. Real-valued populations may be initialised using crtrp. Conversion between binary and real-values is provided by the routine bs2rv that also supports the use of Gray codes and logarithmic scaling.
Fitness assignment: ranking, scaling The fitness function transforms the raw objective function values into non-negative figures of merit for each individual. The Toolbox supports the offsetting and scaling method of Goldberg [9] and the linear-ranking algorithm 1. In addition, non-linear ranking is also supported in the routine ranking. Selection functions: reins, rws, select, sus These functions select a given number of individuals from the current population, according to their fitness, and return a column vector to their indices.
Currently available routines are roulette wheel selection [9], rws, and stochastic universal sampling [11], sus. A high-level entry function, select, is also provided as a convenient interface to the selection routines, particularly where multiple populations are used. In cases where a generation gap is required, i. Crossover operators: recdis, recint, reclin, recmut, recombin, xovdp, xovdprs, xovmp, xovsh, xovshrs, xovsp, xovsprs The crossover routines recombine pairs of individuals with given probability to produce offspring.
Single-point, double-point [12] and shuffle crossover [13] are implemented in the routines xovsp, xovdp and xovsh respectively. Reduced surrogate [13] crossover is supported with both single-, xovsprs, and double-point, xovdprs, crossover and with shuffle, xovshrs. A general multi-point crossover routine, xovmp, that supports uniform crossover [14] is also provided. To support real-valued chromosome representations, discrete, intermediate and line recombination are supplied in the routines, recdis, recint and reclin respectively [15].
The routine recmut performs line recombination with mutation features [15]. A high-level entry function to all the crossover operators supporting multiple subpopulations is provided by the function recombin. Mutation operators: mut, mutate, mutbga Binary and integer mutation are performed by the routine mut.
Real-value mutation is available using the breeder GA mutation function [15], mutbga. Again, a high-level entry function, mutate, to the mutation operators is provided. Multiple subpopulation support: migrate The GA Toolbox provides support for multiple subpopulations through the use of high-level genetic operator functions and a function for exchanging individuals amongst subpopulations, migrate.
A single population is divided into a number of subpopulations by modifying the data structures used by the Toolbox routines such that subpopulations are stored in contiguous blocks within each data element. The high-level routines, such as select and reins, operate independently on each subpopulation contained in a data structure allowing each subpopulation to evolve in isolation from the others.
Based on the Island or Migration model [16], migrate allows individuals to be transferred between subpopulations. Uni- and bi-directional ring topologies as well as a fully interconnected network are selectable via option settings as well as fitness-based and uniform selection and re-insertion strategies. The first few lines of the code set the parameters that the GA uses, such as the number and length of the chromosomes, the crossover and mutation rates, the number of generations and, in this case, the binary representation scheme.
The genetic algorithm uses the following options to determine when to stop. MaxGenerations — The algorithm stops when the number of generations reaches MaxGenerations. MaxTime — The algorithm stops after running for an amount of time in seconds equal to MaxTime. FitnessLimit — The algorithm stops when the value of the fitness function for the best point in the current population is less than or equal to FitnessLimit. MaxStallGenerations — The algorithm stops when the average relative change in the fitness function value over MaxStallGenerations is less than Function tolerance.
MaxStallTime — The algorithm stops if there is no improvement in the objective function during an interval of time in seconds equal to MaxStallTime.
FunctionTolerance — The algorithm runs until the average relative change in the fitness function value over MaxStallGenerations is less than Function tolerance. ConstraintTolerance — The ConstraintTolerance is not used as stopping criterion. It is used to determine the feasibility with respect to nonlinear constraints. Also, max sqrt eps ,ConstraintTolerance determines feasibility with respect to linear constraints. The selection function chooses parents for the next generation based on their scaled values from the fitness scaling function.
The scaled fitness values are called the expectation values. An individual can be selected more than once as a parent, in which case it contributes its genes to more than one child. The default selection option, selectionstochunif , lays out a line in which each parent corresponds to a section of the line of length proportional to its scaled value.
The algorithm moves along the line in steps of equal size. At each step, the algorithm allocates a parent from the section it lands on. A more deterministic selection option is selectionremainder , which performs two steps:. In the first step, the function selects parents deterministically according to the integer part of the scaled value for each individual.
For example, if an individual's scaled value is 2. In the second step, the selection function selects additional parents using the fractional parts of the scaled values, as in stochastic uniform selection. The function lays out a line in sections, whose lengths are proportional to the fractional part of the scaled value of the individuals, and moves along the line in equal steps to select the parents. Note that if the fractional parts of the scaled values all equal 0, as can occur using Top scaling, the selection is entirely deterministic.
For details and more selection options, see Selection Options. Reproduction options control how the genetic algorithm creates the next generation. The options are. EliteCount — The number of individuals with the best fitness values in the current generation that are guaranteed to survive to the next generation.
These individuals are called elite children. When EliteCount is at least 1, the best fitness value can only decrease from one generation to the next. This is what you want to happen, since the genetic algorithm minimizes the fitness function.
Setting EliteCount to a high value causes the fittest individuals to dominate the population, which can make the search less effective. CrossoverFraction — The fraction of individuals in the next generation, other than elite children, that are created by crossover.
Setting the Crossover Fraction describes how the value of CrossoverFraction affects the performance of the genetic algorithm. Because elite individuals have already been evaluated, ga does not reevaluate the fitness function of elite individuals during reproduction.
This behavior assumes that the fitness function of an individual is not random, but is a deterministic function. To change this behavior, use an output function. The genetic algorithm uses the individuals in the current generation to create the children that make up the next generation.
Besides elite children, which correspond to the individuals in the current generation with the best fitness values, the algorithm creates. Crossover children by selecting vector entries, or genes, from a pair of individuals in the current generation and combines them to form a child.
Mutation children by applying random changes to a single individual in the current generation to create a child. Both processes are essential to the genetic algorithm.
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