Editing Chromosome (evolutionary algorithm)

{{Short description|Set of parameters for a genetic or evolutionary algorithm}}
{{Evolutionary algorithms}}

A '''chromosome''' or '''genotype''' in [[evolutionary algorithm]]s (EA) is a set of parameters which define a proposed solution of the problem that the evolutionary algorithm is trying to solve. The set of all solutions, also called ''individuals'' according to the biological model, is known as the ''[[Population model (evolutionary algorithm)|population]]''.<ref name=ga-description>{{cite web|title=Introduction to genetic algorithms: IV. Genetic Algorithm|url=http://www.obitko.com/tutorials/genetic-algorithms/ga-basic-description.php|accessdate=12 August 2015}}</ref><ref name=":0">{{Cite book |last1=Eiben |first1=A.E. |url=https://link.springer.com/10.1007/978-3-662-44874-8 |title=Introduction to Evolutionary Computing |last2=Smith |first2=J.E. |date=2015 |publisher=Springer |isbn=978-3-662-44873-1 |series=Natural Computing Series |location=Berlin, Heidelberg |pages=28–34 |language=en |chapter=Components of Evolutionary Algorithms |doi=10.1007/978-3-662-44874-8|s2cid=20912932 }}</ref> The genome of an individual consists of one, more rarely of several,<ref>{{Citation |last=Baine |first=Nicholas |title=A simple multi-chromosome genetic algorithm optimization of a Proportional-plus-Derivative Fuzzy Logic Controller |date=2008 |url=https://ieeexplore.ieee.org/document/4531273 |work=NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society |pages=1–5 |publisher=IEEE |doi=10.1109/NAFIPS.2008.4531273 |isbn=978-1-4244-2351-4 |s2cid=46591432 |url-access=subscription }}</ref><ref>{{Citation |last1=Peng |first1=Jin |last2=Chu |first2=Zhang Shu |title=A Hybrid Multi-chromosome Genetic Algorithm for the Cutting Stock Problem |date=2010 |url=https://ieeexplore.ieee.org/document/5694457 |work=3rd International Conference on Information Management, Innovation Management and Industrial Engineering |pages=508–511 |publisher=IEEE |doi=10.1109/ICIII.2010.128 |isbn=978-1-4244-8829-2 |s2cid=15608610 |url-access=subscription }}</ref> chromosomes and corresponds to the [[genetic representation]] of the task to be solved. A chromosome is composed of a set of genes, where a gene consists of one or more semantically connected [[Parameter|parameters]], which are often also called ''decision variables''. They determine one or more [[Phenotype|phenotypic]] characteristics of the individual or at least have an influence on them.<ref name=":0" />  In the basic form of genetic algorithms, the chromosome is represented as a binary [[string (computer science)|string]],<ref>{{Cite book |last=Holland |first=John H. |url= |title=Adaptation in natural and artificial systems |date=1992 |publisher=MIT Press |isbn=0-585-03844-9 |edition= |location=Cambridge, Mass. |language=en |oclc=42854623}}</ref> while in later variants<ref>{{Citation |last1=Janikow |first1=C.Z. |title=An Experimental Comparison of Binary and Floating Point Representations in Genetic Algorithms |date=1991 |url=http://www.cs.umsl.edu/~janikow/publications/1991/GAbin/text.pdf |work=Proceedings of the Fourth International Conference on Genetic Algorithms |pages=31–36 |editor-last=Belew |editor-first=Richard K. |editor2-last=Booker |editor2-first=Lashon B. |place=San Francisco, CA |publisher=Morgan Kaufmann Publishers |isbn=1-55860-208-9 |last2=Michalewicz |first2=Z.}}</ref><ref name=ga-tutorial>{{cite journal |last1=Whitley |first1=Darrell |title=A genetic algorithm tutorial |journal=Statistics and Computing |date=June 1994 |volume=4 |issue=2 |doi=10.1007/BF00175354 |citeseerx=10.1.1.184.3999| s2cid=3447126}}<!--|accessdate=12 August 2015--></ref> and in EAs in general, a wide variety of other [[data structure]]s are used.<ref name=":1">{{Cite journal |last=Whitley |first=Darrell |date=2001 |title=An overview of evolutionary algorithms: practical issues and common pitfalls |url=https://linkinghub.elsevier.com/retrieve/pii/S0950584901001884 |journal=Information and Software Technology |language=en |volume=43 |issue=14 |pages=817–831 |doi=10.1016/S0950-5849(01)00188-4|s2cid=18637958 |url-access=subscription }}</ref><ref>{{Citation |last1=Bäck |first1=Thomas |last2=Hoffmeister |first2=Frank |last3=Schwefel |first3=Hans-Paul |title=A Survey of Evolution Strategies |date=1991 |url=https://www.academia.edu/27025389 |work=Proceedings of the Fourth International Conference on Genetic Algorithms |pages=2–9 |editor-last=Belew |editor-first=Richard K. |editor2-last=Booker |editor2-first=Lashon B. |place=San Francisco, CA |publisher=Morgan Kaufmann Publishers |isbn=1-55860-208-9 }}</ref><ref name=":5">{{Cite book |last=Koza |first=John R. |url=https://www.worldcat.org/oclc/26263956 |title=Genetic programming : on the programming of computers by means of natural selection |date=1992 |publisher=MIT Press |isbn=0-262-11170-5 |location=Cambridge, Mass. |oclc=26263956}}</ref>

==Chromosome design==
When creating the [[genetic representation]] of a task, it is determined which decision variables and other degrees of freedom of the task should be improved by the EA and possible additional heuristics and how the [[Genetic representation#Distinction between search space and problem space|genotype-phenotype mapping]] should look like. The design of a chromosome translates these considerations into concrete data structures for which an EA then has to be selected, configured, extended, or, in the worst case, created. Finding a suitable [[Genetic representation|representation]] of the problem domain for a chromosome is an important consideration, as a good representation will make the search easier by limiting the [[Genetic representation#Distinction between search space and problem space|search space]]; similarly, a poorer representation will allow a larger search space.<ref name=ga-notes>{{cite web|title=Genetic algorithms|url=http://www.cse.unsw.edu.au/~billw/cs9414/notes/ml/05ga/05ga.html|accessdate=12 August 2015|archive-date=22 October 2019|archive-url=https://web.archive.org/web/20191022162416/http://www.cse.unsw.edu.au/~billw/cs9414/notes/ml/05ga/05ga.html|url-status=dead}}</ref> In this context, suitable [[mutation (genetic algorithm)|mutation]] and [[crossover (genetic algorithm)|crossover]] [[genetic operator|operators]]<ref name=":0" /> must also be found or newly defined to fit the chosen chromosome design. An important requirement for these operators is that they not only allow all points in the search space to be reached in principle, but also make this as easy as possible.<ref>{{Cite book |last=Rothlauf |first=Franz |url=http://link.springer.com/10.1007/978-3-642-88094-0 |title=Representations for Genetic and Evolutionary Algorithms |date=2002 |publisher=Physica-Verlag HD |isbn=978-3-642-88096-4 |series=Studies in Fuzziness and Soft Computing |volume=104 |location=Heidelberg |pages=31 |doi=10.1007/978-3-642-88094-0}}</ref><ref>{{Cite book |last1=Eiben |first1=A.E. |url=https://link.springer.com/10.1007/978-3-662-44874-8 |title=Introduction to Evolutionary Computing |last2=Smith |first2=J.E. |date=2015 |publisher=Springer |isbn=978-3-662-44873-1 |series=Natural Computing Series |location=Berlin, Heidelberg |pages=49–51 |language=en |chapter=Representation and the Roles of Variation Operators |doi=10.1007/978-3-662-44874-8|s2cid=20912932 }}</ref> 

The following requirements must be met by a well-suited chromosome:

* It must allow the accessibility of all admissible points in the search space.
* Design of the chromosome in such a way that it covers only the search space and no additional areas. so that there is no [[Genetic representation#Redundancy|redundancy]] or only as little redundancy as possible.

* Observance of [[Causality conditions|strong causality]]: small changes in the chromosome should only lead to small changes in the phenotype.<ref>{{Cite book |last1=Galván-López |first1=Edgar |last2=McDermott |first2=James |last3=O'Neill |first3=Michael |last4=Brabazon |first4=Anthony |title=Proceedings of the 12th annual conference on Genetic and evolutionary computation |chapter=Towards an understanding of locality in genetic programming |date=2010-07-07 |chapter-url=https://dl.acm.org/doi/10.1145/1830483.1830646 |language=en |location=Portland Oregon USA |publisher=ACM |pages=901–908 |doi=10.1145/1830483.1830646 |isbn=978-1-4503-0072-8|s2cid=15348983 |url=https://mural.maynoothuniversity.ie/15390/1/EG_towards.pdf }}</ref> This is also called [[Genetic representation#Locality|locality]] of the relationship between search and problem space.
* Designing the chromosome in such a way that it excludes prohibited regions in the search space completely or as much as possible.

While the first requirement is indispensable, depending on the application and the EA used, one usually only has to be satisfied with fulfilling the remaining requirements as far as possible. The evolutionary search is supported and possibly considerably accelerated by a fulfillment as complete as possible.

== Examples of chromosomes ==

=== 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>.

=== Chromosomes with real-valued or integer genes ===
For the processing of tasks with real-valued or mixed-integer decision variables, EAs such as the [[evolution strategy]]<ref name=":3">{{Cite book |last=Schwefel |first=Hans-Paul |url=https://www.researchgate.net/publication/220690578 |title=Evolution and optimum seeking |date=1995 |publisher=John Wiley & Sons |isbn=0-471-57148-2 |location=New York |oclc=30701094}}</ref> or the real-coded GAs<ref>{{Citation |last1=Eshelman |first1=Larry J. |title=Real-Coded Genetic Algorithms and Interval-Schemata |date=1993 |url=https://linkinghub.elsevier.com/retrieve/pii/B9780080948324500180 |work=Foundations of Genetic Algorithms |volume=2 |pages=187–202 |publisher=Elsevier |language=en |doi=10.1016/b978-0-08-094832-4.50018-0 |isbn=978-0-08-094832-4 |access-date=2023-01-26 |last2=Schaffer |first2=J. David|url-access=subscription }}</ref><ref>{{Cite book |last=Michalewicz |first=Zbigniew |url= |title=Genetic Algorithms + Data Structures = Evolution Programs |date=1996 |publisher=Springer |others=Third, revised and extended edition |isbn=978-3-662-03315-9 |edition= |location=Berlin, Heidelberg |language=en |oclc=851375253}}</ref><ref>{{Cite journal |last1=Deep |first1=Kusum |last2=Singh |first2=Krishna Pratap |last3=Kansal |first3=M.L. |last4=Mohan |first4=C. |date=June 2009 |title=A real coded genetic algorithm for solving integer and mixed integer optimization problems |url=https://linkinghub.elsevier.com/retrieve/pii/S0096300309001830 |journal=Applied Mathematics and Computation |language=en |volume=212 |issue=2 |pages=505–518 |doi=10.1016/j.amc.2009.02.044|url-access=subscription }}</ref> are suited. In the case of mixed-integer values, rounding is often used, but this represents some violation of the [[Genetic representation#Relationships between search space and problem space|redundancy requirement]]. If the necessary precisions of the real values can be reasonably narrowed down, this violation can be remedied by using integer-coded GAs.<ref>{{Citation |last1=Wang |first1=Fuchang |title=Decimal-Integer-Coded Genetic Algorithm for Trimmed Estimator of the Multiple Linear Errors in Variables Model |date=2011 |url=http://link.springer.com/10.1007/978-3-642-25255-6_46 |work=Information Computing and Applications |pages=359–366 |editor-last=Liu |editor-first=Baoxiang |series=LNCS 7030 |place=Berlin, Heidelberg |publisher=Springer |doi=10.1007/978-3-642-25255-6_46 |isbn=978-3-642-25254-9 |access-date=2023-01-23 |last2=Cao |first2=Huirong |last3=Qian |first3=Xiaoshi |editor2-last=Chai |editor2-first=Chunlai|url-access=subscription }}</ref><ref>{{Cite journal |last1=Cheng |first1=Xueli |last2=An |first2=Linchao |last3=Zhang |first3=Zhenhua |date=2019 |title=Integer Encoding Genetic Algorithm for Optimizing Redundancy Allocation of Series-parallel Systems |journal=Journal of Engineering Science and Technology Review |volume=12 |issue=1 |pages=126–136|doi=10.25103/JESTR.121.15 |s2cid=149497992 |doi-access=free }}</ref> For this purpose, the valid digits of real values are mapped to integers by multiplication with a suitable factor. For example, 12.380 becomes the integer 12380 by multiplying by 1000. This must of course be taken into account in genotype-phenotype mapping for evaluation and result presentation. A common form is a chromosome consisting of a list or an array of integer or real values.

=== Chromosomes for permutations ===
[[Combinatorial optimization|Combinatorial problems]] are mainly concerned with finding an optimal sequence of a set of elementary items. As an example, consider the problem of the [[Travelling salesman problem|traveling salesman]] who wants to visit a given number of cities exactly once on the shortest possible tour. The simplest and most obvious mapping onto a chromosome is to number the cities consecutively, to interpret a resulting sequence as [[permutation]] and to store it directly in a chromosome, where one gene corresponds to the ordinal number of a city.<ref>{{Cite book |last1=Eiben |first1=A.E. |url=https://link.springer.com/10.1007/978-3-662-44874-8 |title=Introduction to Evolutionary Computing |last2=Smith |first2=J.E. |date=2015 |publisher=Springer |isbn=978-3-662-44873-1 |series=Natural Computing Series |location=Berlin, Heidelberg |pages=67–74 |language=en |chapter=Permutation Representation |doi=10.1007/978-3-662-44874-8|s2cid=20912932 }}</ref> Then, however, the [[Genetic operator|variation operators]] may only change the gene order and not remove or duplicate any genes.<ref name=":2">{{Cite journal |last1=Larrañaga |first1=P. |last2=Kuijpers |first2=C.M.H. |last3=Murga |first3=R.H. |last4=Inza |first4=I. |last5=Dizdarevic |first5=S. |date=1999 |title=Genetic Algorithms for the Travelling Salesman Problem: A Review of Representations and Operators |url=http://link.springer.com/10.1023/A:1006529012972 |journal=Artificial Intelligence Review |volume=13 |issue=2 |pages=129–170 |doi=10.1023/A:1006529012972|s2cid=10284682 |url-access=subscription }}</ref> The chromosome thus contains the path of a possible tour to the cities. As an example the sequence <math>3,5,7,1,4,2,9,6,8</math> of nine cities may serve, to which the following chromosome corresponds:
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none; text-align:center;"
|-
|| 3 || 5 || 7 || 1 || 4 || 2 || 9 || 6 || 8 
|} 
In addition to this encoding frequently called ''path representation'', there are several other ways of representing a permutation, for example the ''ordinal representation'' or the ''matrix representation''.<ref name=":2" /><ref>{{Cite book |last=Whitley |first=Darrell |url= |title=Evolutionary computation. Vol. 1, Basic algorithms and operators |date=2000 |publisher=Institute of Physics Pub |isbn=0-585-30560-9 |editor-last=Fogel |editor-first=David B. |location=Bristol |pages=139–150 |language=en |chapter=Permutations |oclc=45730387 |editor-last2=Bäck |editor-first2=Thomas |editor-last3=Michalewicz |editor-first3=Zbigniew}}</ref>

=== Chromosomes for co-evolution ===
When a genetic representation contains, in addition to the decision variables, additional information that influences evolution and/or the mapping of the genotype to the phenotype and is itself subject to evolution, this is referred to as ''co-evolution''. A typical example is the [[evolution strategy]]  (ES), which includes one or more mutation step sizes as strategy parameters in each chromosome.<ref name=":3" /> Another example is an additional gene to control a selection heuristic for resource allocation in a scheduling tasks.<ref name=":6">{{Cite journal |last1=Jakob |first1=Wilfried |last2=Strack |first2=Sylvia |last3=Quinte |first3=Alexander |last4=Bengel |first4=Günther |last5=Stucky |first5=Karl-Uwe |last6=Süß |first6=Wolfgang |date=2013-04-22 |title=Fast Rescheduling of Multiple Workflows to Constrained Heterogeneous Resources Using Multi-Criteria Memetic Computing |department=p.253-255 |journal=Algorithms |language=en |volume=6 |issue=2 |pages=245–277 |doi=10.3390/a6020245 |issn=1999-4893|doi-access=free }}</ref>

This approach is based on the assumption that good solutions are based on an appropriate selection of strategy parameters or on control gene(s) that influences genotype-phenotype mapping. The success of the ES gives evidence to this assumption.

=== Chromosomes for complex representations ===
The chromosomes presented above are well suited for processing tasks of continuous, mixed-integer, pure-integer or combinatorial optimization. For a combination of these optimization areas, on the other hand, it becomes increasingly difficult to map them to simple strings of values, depending on the task. The following extension of the gene concept is proposed by the EA GLEAM (General Learning Evolutionary Algorithm and Method) for this purpose:<ref name=":4">{{Citation |last1=Blume |first1=Christian |last2=Jakob |first2=Wilfried |title=GLEAM - An Evolutionary Algorithm for Planning and Control Based on Evolution Strategy |date=2002 |url=https://publikationen.bibliothek.kit.edu/170053025/3814288 |work=Conf. Proc. of Genetic and Evolutionary Computation Conference (GECCO 2002) |volume=Late Breaking Papers |pages=31–38 |access-date=2023-01-01 }}</ref> A gene is considered to be the description of an element or elementary trait of the phenotype, which may have multiple parameters. For this purpose, gene types are defined that contain as many parameters of the appropriate data type as are required to describe the particular element of the phenotype. A chromosome now consists of genes as data objects of the gene types, whereby, depending on the application, each gene type occurs exactly once as a gene or can be contained in the chromosome any number of times. The latter leads to chromosomes of dynamic length, as they are required for some problems.<ref>{{Cite journal |last1=Pawar |first1=Sunil Nilkanth |last2=Bichkar |first2=Rajankumar Sadashivrao |date=June 2015 |title=Genetic algorithm with variable length chromosomes for network intrusion detection |journal=International Journal of Automation and Computing |language=en |volume=12 |issue=3 |pages=337–342 |doi=10.1007/s11633-014-0870-x |s2cid=255346767 |issn=1476-8186|doi-access=free }}</ref><ref>{{Citation |last=Blume |first=Christian |title=Optimized Collision Free Robot Move Statement Generation by the Evolutionary Software GLEAM |date=2000 |url=http://link.springer.com/10.1007/3-540-45561-2_32 |work=Real-World Applications of Evolutionary Computing |series=Lecture Notes in Computer Science |volume=1803 |pages=330–341 |editor-last=Cagnoni |editor-first=Stefano |access-date=2023-06-25 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/3-540-45561-2_32 |isbn=978-3-540-67353-8|url-access=subscription }}</ref> The gene type definitions also contain information on the permissible value ranges of the gene parameters, which are observed during chromosome generation and by corresponding mutations, so they cannot lead to lethal mutations. For tasks with a combinatorial part, there are suitable [[Genetic operator|genetic operators]] that can move or reposition genes as a whole, i.e. with their parameters.
[[File:Genmodell Chromosombeispiel.png|thumb|212x212px|Three exemplary genes matching the adjacent gene type definitions in a chromosome organized as a list]]
[[File:Gene model gene types.png|left|thumb|224x224px|Three exemplary genes matching the adjacent gene type definitions in a chromosome organized as a list]]
A [[Scheduling (computing)|scheduling]] task is used as an illustration, in which [[Workflow|workflows]] are to be scheduled that require different numbers of heterogeneous resources. A workflow specifies which work steps can be processed in parallel and which have to be executed one after the other. In this context, heterogeneous resources mean different processing times at different costs in addition to different processing capabilities.<ref name=":6" /> Each scheduling operation therefore requires one or more parameters that determine the resource selection, where the value ranges of the parameters depend on the number of alternative resources available for each work step. A suitable chromosome provides one gene type per work step and in this case one corresponding gene, which has one parameter for each required resource. The order of genes determines the order of scheduling operations and, therefore, the precedence in case of allocation conflicts. The exemplary gene type definition of work step 15 with two resources, for which there are four and seven alternatives respectively, would then look as shown in the left image. Since the parameters represent indices in lists of available resources for the respective work step, their value range starts at 0. The right image shows an example of three genes of a chromosome belonging to the gene types in list representation. 

[[File:Genetic Program Tree.png|thumb|214x214px|Syntax tree of a formula example]]
=== Chromosomes for tree representations ===
Tree representations in a chromosome are used by [[genetic programming]], an EA type for generating [[Computer program|computer programs]] or [[Electronic circuit|circuits]].<ref name=":5" /> The trees correspond to the [[Parse tree|syntax trees]] generated by a [[compiler]] as internal representation when translating a computer program. The adjacent figure shows the syntax tree of a mathematical expression as an example. Mutation operators can rearrange, change or delete subtrees depending on the represented syntax structure. Recombination is performed by exchanging suitable subtrees.<ref>{{Cite book |last1=Eiben |first1=A.E. |url=https://link.springer.com/10.1007/978-3-662-44874-8 |title=Introduction to Evolutionary Computing |last2=Smith |first2=J.E. |date=2015 |publisher=Springer |isbn=978-3-662-44873-1 |series=Natural Computing Series |location=Berlin, Heidelberg |pages=75–78 |language=en |chapter=Tree Representation |doi=10.1007/978-3-662-44874-8|s2cid=20912932 }}</ref>

== Bibliography ==
* Thomas Bäck (1996): ''[https://books.google.com/books?id=htJHI1UrL7IC Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms]'', Oxford Univ. Press. {{ISBN|978-0-19-509971-3}}
* Wolfgang Banzhaf, P. Nordin, R. Keller, F. Francone (1998): ''Genetic Programming - An Introduction'', Morgan Kaufmann, San Francisco. {{ISBN|1-55860-510-X}}
* Kenneth A. de Jong (2006): ''Evolutionary Computation: A Unified Approach.'' MIT Press, Cambridge, MA. {{ISBN|0-262-04194-4}}
* Melanie Mitchell (1996): ''An Introduction to Genetic Algorithms.'' MIT Press, Cambridge MA. {{ISBN|978-0-262-63185-3}}
* Hans-Paul Schwefel (1995): ''[https://www.researchgate.net/publication/220690578_Evolution_and_Optimum_Seeking Evolution and Optimum Seeking]''. Wiley & Sons, New York. {{ISBN|0-471-57148-2}}

== References ==
<references />

{{DEFAULTSORT:Chromosome (Genetic Algorithm)}}
[[Category:Evolutionary algorithms]]