The mutation of bit strings ensue through bit flips at random positions.

Example:

The probability of a mutation of a bit is ${\displaystyle {\frac {1}{l}}}$ , where ${\displaystyle l}$  is the length of the binary vector. Thus, a mutation rate of ${\displaystyle 1}$  per mutation and individual selected for mutation is reached.

Many EAs, such as the evolution strategy^[9][10] or the real-coded genetic algorithms,^[11][12][8] work with real numbers instead of bit strings. This is due to the good experiences that have been made with this type of coding.^[8][13]

The value of a real-valued gene can either be changed or redetermined. A mutation that implements the latter should only ever be used in conjunction with the value-changing mutations and then only with comparatively low probability, as it can lead to large changes.

In practical applications, the decision variables to be changed of the optimisation problem to be solved are usually limited. Accordingly, the values of the associated genes are each restricted to an interval ${\displaystyle [x_{\min },x_{\max }]}$ . Mutations may or may not take these restrictions into account. In the latter case, suitable post-treatment is then required as described below.

Mutation without consideration of restrictions edit

A real number ${\displaystyle x}$  can be mutated using normal distribution ${\displaystyle {\mathcal {N}}(0,\sigma )}$  by adding the generated random value to the old value of the gene, resulting in the mutated value ${\displaystyle x'}$ :

${\displaystyle x'=x+{\mathcal {N}}(0,\sigma )}$ 

In the case of genes with a restricted range of values, it is a good idea to choose the step size of the mutation ${\displaystyle \sigma }$  so that it reasonably fits the range ${\displaystyle [x_{\min },x_{\max }]}$  of the gene to be changed, e.g.:

${\displaystyle \sigma ={\frac {x_{\text{max}}-x_{\text{min}}}{6}}}$ 

The step size can also be adjusted to the smaller permissible change range depending on the current value. In any case, however, it is likely that the new value ${\displaystyle x'}$  of the gene will be outside the permissible range of values. Such a case must be considered a lethal mutation, since the obvious repair by using the respective violated limit as the new value of the gene would lead to a drift. This is because the limit value would then be selected with the entire probability of the values beyond the limit of the value range.

The evolution strategy works with real numbers and mutation based on normal distribution. The step sizes are part of the chromosome and are subject to evolution together with the actual decision variables.

Mutation with consideration of restrictions edit

One possible form of changing the value of a gene while taking its value range ${\displaystyle [x_{\min },x_{\max }]}$  into account is the mutation relative parameter change of the evolutionary algorithm GLEAM (General Learning Evolutionary Algorithm and Method),^[14] in which, as with the mutation presented earlier, small changes are more likely than large ones.

First, an equally distributed decision is made as to whether the current value ${\displaystyle x}$  should be increased or decreased and then the corresponding total change interval is determined. Without loss of generality, an increase is assumed for the explanation and the total change interval is then ${\displaystyle [x,x_{\max }]}$ . It is divided into ${\displaystyle k}$  sub-areas of equal size with the width ${\displaystyle \delta }$ , from which ${\displaystyle k}$  sub-change intervals of different size are formed:

${\displaystyle i}$ -th sub-change interval: ${\displaystyle [x,x+\delta \cdot i]}$  with

${\displaystyle \delta ={\frac {(x_{\text{max}}-x)}{k}}}$  and ${\displaystyle i=1,\dots ,k}$ 

Subsequently, one of the ${\displaystyle k}$  sub-change intervals is selected in equal distribution and a random number, also equally distributed, is drawn from it as the new value ${\displaystyle x'}$  of the gene. The resulting summed probabilities of the sub-change intervals result in the probability distribution of the ${\displaystyle k}$  sub-areas for the exemplary case of ${\displaystyle k=10}$  shown in the adjacent figure. This is not a normal distribution as before, but this distribution also clearly favours small changes over larger ones.

This mutation for larger values of ${\displaystyle k}$ , such as 10, is less well suited for tasks where the optimum lies on one of the value range boundaries. This can be remedied by significantly reducing ${\displaystyle k}$  when a gene value approaches its limits very closely.

Mutations of permutations are specially designed for genomes that are themselves permutations of a set. These are often used to solve combinatorial tasks.^[8][15][16] In the two mutations presented, parts of the genome are rotated or inverted.

Rotation to the right edit

The presentation of the procedure^[16] is illustrated by an example on the right:

Inversion edit

The presentation of the procedure^[15] is illustrated by an example on the right:

Variants with preference for smaller changes edit

The requirement raised at the beginning for mutations, according to which small changes should be more probable than large ones, is only inadequately fulfilled by the two permutation mutations presented, since the lengths of the partial lists and the number of shift positions are determined in an equally distributed manner. However, the longer the partial list and the shift, the greater the change in gene order.

This can be remedied by the following modifications. The end index ${\displaystyle j}$  of the partial lists is determined as the distance ${\displaystyle d}$  to the start index ${\displaystyle i}$ :

${\displaystyle j=(i+d){\bmod {\left|P_{0}\right|}}}$ 

where ${\displaystyle d}$  is determined randomly according to one of the two procedures for the mutation of real numbers from the interval ${\displaystyle \left[0,\left|P_{0}\right|-1\right]}$  and rounded.

For the rotation, ${\displaystyle k}$  is determined similarly to the distance ${\displaystyle d}$ , but the value ${\displaystyle 0}$  is forbidden.

For the inversion, note that ${\displaystyle i\neq j}$  must hold, so for ${\displaystyle d}$  the value ${\displaystyle 0}$  must be excluded.

• Evolutionary algorithms
• Genetic algorithms

• John Holland (1975). Adaptation in Natural and Artificial Systems, PhD thesis, University of Michigan Press, Ann Arbor, Michigan. ISBN 0-262-58111-6.
• Schwefel, Hans-Paul (1995). Evolution and Optimum Seeking. New York: John Wiley & Sons. ISBN 0-471-57148-2.
• Davis, Lawrence (1991). Handbook of genetic algorithms. New York: Van Nostrand Reinhold. ISBN 0-442-00173-8. OCLC 23081440.
• Eiben, A.E.; Smith, J.E. (2015). Introduction to Evolutionary Computing. Natural Computing Series. Berlin, Heidelberg: Springer. doi:10.1007/978-3-662-44874-8. ISBN 978-3-662-44873-1. S2CID 20912932.
• Yu, Xinjie; Gen, Mitsuo (2010). Introduction to Evolutionary Algorithms. Decision Engineering. London: Springer. doi:10.1007/978-1-84996-129-5. ISBN 978-1-84996-128-8.
• De Jong, Kenneth A. (2006). Evolutionary computation : a unified approach. Cambridge, Mass.: MIT Press. ISBN 978-0-262-25598-1. OCLC 69652176.
• Fogel, David B.; Bäck, Thomas; Michalewicz, Zbigniew, eds. (1999). Evolutionary computation. Vol. 1, Basic algorithms and operators. Bristol: Institute of Physics Pub. ISBN 0-585-30560-9. OCLC 45730387.