Editing Chromosome (evolutionary algorithm) (section)

=== Chromosomes for binary codings ===
In their classical form, GAs use bit strings and map the decision variables to be optimized onto them. An example for one Boolean and three integer decision variables with the value ranges <math>0 \leq D_1 \leq 60</math>, <math>28 \leq D_2 \leq 30</math> and <math>-12 \leq D_3 \leq 14</math> may illustrate this: 

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center;"
|+ Example representation of four decision variables in a bitstring
|-
| decision variable: ||colspan="6"| <math>D_1 = 22</math> ||colspan="5"| <math>D_2 = 29</math> ||colspan="5"| <math>D_3 = -4</math> || <math>D_4 = 0</math>
|-
|style="text-align:left"| bits: || 0 || 1 || 0 || 1 || 1 || 0 || 1 || 1 || 1 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 
|-
|style="text-align:left"| position: || 17 || 16 || 15 || 14 || 13 || 12 || 11 || 10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1
|}
Note that the negative number here is given in [[two's complement]]. This straight forward representation uses five bits to represent the three values of <math>D_2</math>, although two bits would suffice. This is a significant redundancy. An improved alternative, where 28 is to be added for the genotype-phenotype mapping, could look like this: 

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center;"
|+ Example of an improved representation of the four decision variables
|-
| decision variable: ||colspan="6"| <math>D_1 = 22</math> ||colspan="2"| <math>D'_2 = 1</math> ||colspan="5"| <math>D_3 = -4</math> || <math>D_4 = 0</math>
|-
|style="text-align:left"| bits: || 0 || 1 || 0 || 1 || 1 || 0 || 0 || 1 || 1 || 1 || 1 || 0 || 0 || 0 
|-
|style="text-align:left"| position: ||  14 || 13 || 12 || 11 || 10 || 9 || 8 || 7 || 6 || 5 || 4 || 3 || 2 || 1
|} with <math>D_2 = 28 + D'_2 = 29</math>.