SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Biogeography-based optimization</title>

</head>

Biogeography-based optimization

</h1>

</header>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">Biogeography-based optimization (BBO) is an <a href="Evolutionary_algorithm" title="Evolutionary algorithm">evolutionary algorithm</a> (EA) that <a href="Optimization" class="mw-redirect" title="Optimization">optimizes</a> a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> by <a href="Stochastic" title="Stochastic">stochastically</a> and <a href="Iterative_method" title="Iterative method">iteratively</a> improving <a href="Candidate_solution" class="mw-redirect" title="Candidate solution">candidate solutions</a> with regard to a given measure of quality, or <a href="Fitness_function" title="Fitness function">fitness function</a>. BBO belongs to the class of <a href="Metaheuristic" title="Metaheuristic">metaheuristics</a> since it includes many variations, and since it does not make any assumptions about the problem and can therefore be applied to a wide class of problems.

BBO is typically used to optimize multidimensional real-valued functions, but it does not use the <a href="Gradient" title="Gradient">gradient</a> of the function, which means that it does not require the function to be <a href="Differentiable_function" title="Differentiable function">differentiable</a> as required by classic optimization methods such as <a href="Gradient_descent" title="Gradient descent">gradient descent</a> and <a href="Quasi-newton_methods" class="mw-redirect" title="Quasi-newton methods">quasi-newton methods</a>. BBO can therefore be used on <a href="Continuous_function" title="Continuous function">discontinuous functions</a>.

BBO optimizes a problem by maintaining a population of candidate solutions, and creating new candidate solutions by combining existing ones according to a simple formula. In this way the <a href="Loss_function" title="Loss function">objective function</a> is treated as a black box that merely provides a measure of quality given a candidate solution, and the function's gradient is not needed.

Like many EAs, BBO was motivated by a natural process; in particular, BBO was motivated by <a href="Biogeography" title="Biogeography">biogeography</a>, which is the study of the distribution of biological species through time and space.<a href="#cite_note-Quammen1997-1">[1]</a> BBO was originally introduced by <a rel="nofollow" class="external text" href="http://academic.csuohio.edu/simond/">Dan Simon</a> in 2008.<a href="#cite_note-Simon2008-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Underlying_principles">Underlying principles</h2></div>

Mathematical models of <a href="Biogeography" title="Biogeography">biogeography</a> describe <a href="Speciation" title="Speciation">speciation</a> (the evolution of new <a href="Species" title="Species">species</a>), the <a href="Animal_migration" title="Animal migration">migration</a> of species (animals, fish, birds, or insects) between islands, and the <a href="Extinction" title="Extinction">extinction</a> of species.<a href="#cite_note-MacArthur1967-3">[3]</a> Islands that are friendly to life are said to have a high habitat suitability index (HSI).<a href="#cite_note-Wesche1987-4">[4]</a> Features that correlate with HSI include rainfall, vegetative diversity, topographic diversity, land area, temperature, and others. The features that determine are called suitability index variables (SIVs). In terms of habitability, SIVs are the independent variables and HSI is the dependent variable.

Islands with a high HSI can support many species, and islands with a low HSI can support only a few species. Islands with a high HSI have many species that <a href="Emigration" title="Emigration">emigrate</a> to nearby habitats because of the large populations and the large numbers of species that they host. Note that emigration from an island with a high HSI does not occur because species want to leave their home; after all, their home island is an attractive place to live. Emigration occurs because of the accumulation of random effects on a large number of species with large populations. Emigration occurs as animals ride <a href="Flotsam" class="mw-redirect" title="Flotsam">flotsam</a>, swim, fly, or ride the wind to neighboring islands. When a species emigrates from an island, it does not mean that the species completely disappears from its original island; only a few representatives emigrate, so an emigrating species remains present on its original island while at the same time migrating to a neighboring island. However, in BBO it is assumed that emigration from an island results in extinction from that island. This assumption is necessary in BBO because species represent the independent variables of a function, and each island represents a candidate solution to a function optimization problem.

Islands with a high HSI not only have a high emigration rate, but they also have a low immigration rate because they already support many species. Species that migrate to such islands will tend to die in spite of the island's high HSI, because there is too much competition for resources from other species.

Islands with a low HSI have a high immigration rate because of their low populations. Again, this is not because species want to immigrate to such islands; after all, these islands are undesirable places to live. The reason that immigration occurs to these islands is because there is a lot of room for additional species. Whether or not the immigrating species can survive in its new home, and for how long, is another question. However, <a href="Species_diversity" title="Species diversity">species diversity</a> is correlated with HSI, so when more species arrive at a low HSI island, the island's HSI will tend to increase.<a href="#cite_note-Wesche1987-4">[4]</a>

The figure on the right illustrates an island migration model.<a href="#cite_note-MacArthur1967-3">[3]</a> The immigration rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> and the emigration rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> are functions of the number of species on the island. The maximum possible immigration rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> occurs when there are zero species on the island. As the number of species increases, the island becomes more crowded, fewer species are able to survive immigration, and the immigration rate decreases. The largest possible number of species that the habitat can support is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\max }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{\max }}</annotation>

</semantics>

</math><img src="./32eb3f46d17f063513ef1d0bf55ba85bdef51f06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.716ex; height:2.509ex;" alt="{\displaystyle S_{\max }}" loading="lazy">, at which point the immigration rate is zero. If there are no species on the island, then the emigration rate is zero. As the number of species on the island increases, it becomes more crowded, more species representatives are able to leave the island, and the emigration rate increases. When the island contains the largest number of possible species <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\max }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{\max }}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">.

In BBO, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{k}}</annotation>

</semantics>

</math><img src="./57dfa9eb1c96d16ff53f264bd9710a16c0108469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.509ex;" alt="{\displaystyle \lambda _{k}}" loading="lazy"> is the probability that a given independent variable in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-th candidate solution will be replaced; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{k}}</annotation>

</semantics>

</math><img src="./57dfa9eb1c96d16ff53f264bd9710a16c0108469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.509ex;" alt="{\displaystyle \lambda _{k}}" loading="lazy"> is the immigration probability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy">. If an independent variable is to be replaced, then the emigrating candidate solution is chosen with a probability that is proportional to the emigration probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{k}}</annotation>

</semantics>

</math><img src="./25f9c464114cfcd7e31e53de202703377b7c5ffb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.49ex; height:2.176ex;" alt="{\displaystyle \mu _{k}}" loading="lazy">. This is usually performed using <a href="Fitness_proportionate_selection" title="Fitness proportionate selection">roulette wheel selection</a>.

</mrow>

<msub>

</mrow>

</msub>

<mtext> is selected for emigration</mtext>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Prob}}(x_{j}){\text{ is selected for emigration}}={\frac {\mu _{j}}{\sum _{i=1}^{N}\mu _{i}}}}</annotation>

</semantics>

</math><img src="./dad0c5791ecc0d536a81ea33b6f9b687db3563bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.295ex; height:6.343ex;" alt="{\displaystyle {\text{Prob}}(x_{j}){\text{ is selected for emigration}}={\frac {\mu _{j}}{\sum _{i=1}^{N}\mu _{i}}}}" loading="lazy"></dd></dl></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\cdots ,N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j=1,\cdots ,N}</annotation>

</semantics>

</math><img src="./c313f0babbed55da8a662c030de66cca221175a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:12.488ex; height:2.509ex;" alt="{\displaystyle j=1,\cdots ,N}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

Like most other EAs, BBO includes <a href="Mutation_(genetic_algorithm)" class="mw-redirect" title="Mutation (genetic algorithm)">mutation</a>. A basic BBO algorithm with a population size of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> for optimizing an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-dimensional function can be described as follows.

<pre>Initialize a population of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> candidate solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{k}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x_{k}\}}</annotation>

</semantics>

</math><img src="./da303cc40bcf961218a8415e24719e51f30c224e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.743ex; height:2.843ex;" alt="{\displaystyle \{x_{k}\}}" loading="lazy">

While not(termination criterion)

For each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy">, set emigration probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{k}\propto }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{k}\propto }</annotation>

</semantics>

</math><img src="./4519339427aff2dd2a67173911dbf38d52e08835.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.944ex; height:2.176ex;" alt="{\displaystyle \mu _{k}\propto }" loading="lazy"> fitness of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>

</semantics>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{k}\in [0,1]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{k}\in [0,1]}</annotation>

</semantics>

</math><img src="./6d02276410f1d3161d652d200e1d831baf760105.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.984ex; height:2.843ex;" alt="{\displaystyle \mu _{k}\in [0,1]}" loading="lazy">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{k}}</annotation>

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy">, set immigration probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}=1-\mu _{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{k}=1-\mu _{k}}</annotation>

</semantics>

</math><img src="./b76c3e98480817e394a04fd02c5a8401d6797a19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.036ex; height:2.676ex;" alt="{\displaystyle \lambda _{k}=1-\mu _{k}}" loading="lazy"> do

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{z_{k}\}\leftarrow \{x_{k}\}}</annotation>

</semantics>

</math><img src="./cb94fcc4a5c3da0b7af121f8cd61a43ad4097f47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.852ex; height:2.843ex;" alt="{\displaystyle \{z_{k}\}\leftarrow \{x_{k}\}}" loading="lazy">

For each individual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}(k=1,\cdots ,N)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}(k=1,\cdots ,N)}</annotation>

</semantics>

</math><img src="./31b36e71ac39b2805f8a52793d01f3b9f1b88fb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.693ex; height:2.843ex;" alt="{\displaystyle z_{k}(k=1,\cdots ,N)}" loading="lazy"> do

For each independent variable index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in [1,n]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in [1,n]}</annotation>

</semantics>

</math><img src="./204fce467241a7677c583a9510c1bd47d07158ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.816ex; height:2.843ex;" alt="{\displaystyle s\in [1,n]}" loading="lazy"> do

Use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{k}}</annotation>

</semantics>

</math><img src="./57dfa9eb1c96d16ff53f264bd9710a16c0108469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.509ex;" alt="{\displaystyle \lambda _{k}}" loading="lazy"> to probabilistically decide whether to immigrate to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}}</annotation>

</semantics>

</math><img src="./a51cdfac24f8b95ed711f11ea9502da4087b6a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.17ex; height:2.009ex;" alt="{\displaystyle z_{k}}" loading="lazy">

If immigrating then

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\mu _{i}\}}</annotation>

</semantics>

</math><img src="./090a6af4a19ec7631786786b38919dad6a169f2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.526ex; height:2.843ex;" alt="{\displaystyle \{\mu _{i}\}}" loading="lazy"> to probabilistically select the emigrating individual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation>

</semantics>

</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}(s)\leftarrow x_{j}(s)}</annotation>

</semantics>

</math><img src="./75d7725beaf5c2ebe30e2a13415c8f8d6834170d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.823ex; height:3.009ex;" alt="{\displaystyle z_{k}(s)\leftarrow x_{j}(s)}" loading="lazy">

End if

Next independent variable index: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\leftarrow s+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\leftarrow s+1}</annotation>

</semantics>

</math><img src="./7cd3a998628563f1260d40cc2b412f931eb4a0c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.798ex; height:2.343ex;" alt="{\displaystyle s\leftarrow s+1}" loading="lazy">

Probabilistically mutate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}}</annotation>

</semantics>

Next individual: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\leftarrow k+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\leftarrow k+1}</annotation>

</semantics>

</math><img src="./469b0d14d5eff3c19bde9957713c939411776b66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.039ex; height:2.343ex;" alt="{\displaystyle k\leftarrow k+1}" loading="lazy">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x_{k}\}\leftarrow \{z_{k}\}}</annotation>

</semantics>

</math><img src="./4adb09a80330f7ad2d8251c147f175408613aa2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.852ex; height:2.843ex;" alt="{\displaystyle \{x_{k}\}\leftarrow \{z_{k}\}}" loading="lazy">

Next generation

</pre>

<div class="mw-heading mw-heading2"><h2 id="Discussion_of_the_BBO_algorithm">Discussion of the BBO algorithm</h2></div>

<ul><li>The population size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is a tuning parameter. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is too small or too large, then the optimization performance of BBO will suffer. Typical implementations of BBO use a value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

<li>The initial population of candidate solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{k}\}_{k=1}^{N}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x_{k}\}_{k=1}^{N}}</annotation>

</semantics>

</math><img src="./1a88807fddedb9e504c200595ba606364c8f3677.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.933ex; height:3.176ex;" alt="{\displaystyle \{x_{k}\}_{k=1}^{N}}" loading="lazy"> is usually generated randomly. However, it could be generated in a problem-dependent way based on some reasonable guesses or previously-known good solutions to the optimization problem.</li>

<li>The termination criterion is problem-dependent, like in any other EA. In most applications the termination criterion is a generation count limit or a function evaluation limit (that is, how often the objective function is evaluated).</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{z_{k}\}}</annotation>

</semantics>

</math><img src="./f6ea18732a3b055214cf076f95a61a5cca862946.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.495ex; height:2.843ex;" alt="{\displaystyle \{z_{k}\}}" loading="lazy"> is a temporary population so that all emigrating variables can originate from the population that is in place at the beginning of the generation, which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{k}\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{x_{k}\}}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Algorithmic_variations">Algorithmic variations</h2></div>

Many variations have been proposed to the basic BBO algorithm, among which are the following.

<ul><li>Elitism is implemented in most EAs to make sure that the best candidate solution is not lost from one generation to the next. This can be implemented in a variety of ways, but one common way is to save the best candidate solutions at the beginning of each generation in a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} }</annotation>

</semantics>

</math><img src="./ad9faf1fd4a61d36d7f8a2f3204f3805a43c0d4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {E} }" loading="lazy">; then replace the worst candidate solutions with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} }</annotation>

</semantics>

</math><img src="./ad9faf1fd4a61d36d7f8a2f3204f3805a43c0d4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {E} }" loading="lazy"> at the end of the generation, after migration and mutation have completed. The size of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} }</annotation>

</semantics>

</math><img src="./ad9faf1fd4a61d36d7f8a2f3204f3805a43c0d4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {E} }" loading="lazy"> is a tuning parameter, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} }</annotation>

</semantics>

</math><img src="./ad9faf1fd4a61d36d7f8a2f3204f3805a43c0d4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {E} }" loading="lazy"> typically includes the best two individuals. Elitism was originally proposed for <a href="Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a> by DeJong.<a href="#cite_note-DeJong1975-5">[5]</a> Elitism can make a significant difference in the performance of BBO, and is highly recommended.</li>

<li>Duplicate replacement is often implemented in BBO. This is a procedure at the end of each generation that replaces duplicate individuals in the population. Scanning for duplicates can be computationally intensive because it is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(N^{2})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O(N^{2})}</annotation>

</semantics>

</math><img src="./e5d43a3df904fa4d7220f5b86285298aa36d969b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.76ex; height:3.176ex;" alt="{\displaystyle O(N^{2})}" loading="lazy"> operation, so it is often performed only every few generations, rather than every generation.</li>

<li>Blending can be implemented in BBO. With blending, instead of replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}(s)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}(s)}</annotation>

</semantics>

</math><img src="./68435002c07ec12e541eaeecc51bf2fd3e75414f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.07ex; height:2.843ex;" alt="{\displaystyle z_{k}(s)}" loading="lazy"> in an immigrating candidate solution with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}(s)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j}(s)}</annotation>

</semantics>

</math><img src="./fab037cb35b83f4c9e814869a54003a72952894a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.139ex; height:3.009ex;" alt="{\displaystyle x_{j}(s)}" loading="lazy"> from the emigrating candidate solution, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{k}(s)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}(s)}</annotation>

</semantics>

</math><img src="./68435002c07ec12e541eaeecc51bf2fd3e75414f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.07ex; height:2.843ex;" alt="{\displaystyle z_{k}(s)}" loading="lazy"> is set equal to a linear combination of its original value and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}(s)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{j}(s)}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z_{k}(s)\leftarrow \alpha z_{k}(s)+(1-\alpha )x_{j}(s)}</annotation>

</semantics>

</math><img src="./2d40c0dd05d3af4be99dba85ce52626e2c33e689.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.52ex; height:3.009ex;" alt="{\displaystyle z_{k}(s)\leftarrow \alpha z_{k}(s)+(1-\alpha )x_{j}(s)}" loading="lazy"></dd></dl></dd>

<dd>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in [0,1]}</annotation>

</semantics>

</math><img src="./daf3c62599ea71319c85f715c9e590d2bab2d036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.981ex; height:2.843ex;" alt="{\displaystyle \alpha \in [0,1]}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation>

</semantics>

</math><img src="./30cc00f65bbc630448311dd2dc82e7ce5e90985a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}" loading="lazy"> corresponds to standard migration as shown in the algorithm above. Blended BBO is based on blended crossover in genetic algorithms,<a href="#cite_note-Muhlenbein1993-6">[6]</a> and has been shown to outperform standard BBO.<a href="#cite_note-Ma2011b-7">[7]</a></dd></dl>

<ul><li>The BBO algorithm presented above is called partial immigration-based BBO because the immigrating candidate solution is selected before the emigrating candidate solution is selected, and migration for each independent variable in the immigrating candidate solution is performed independently of all other independent variables. Other approaches for selecting the immigrating and emigrating candidate solutions have also been proposed.<a href="#cite_note-Simon2013-8">[8]</a><a href="#cite_note-Kundra2010-9">[9]</a></li>

<li>The migration curves in the above figure are linear, but nonlinear migration curves often give better performance.<a href="#cite_note-Ma2010-10">[10]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Hybridization">Hybridization</h2></div>

<ul><li>BBO has been hybridized with several other EAs, including <a href="Particle_swarm_optimization" title="Particle swarm optimization">particle swarm optimization</a>,<a href="#cite_note-Kundra2010-9">[9]</a><a href="#cite_note-11">[11]</a> <a href="Differential_evolution" title="Differential evolution">differential evolution</a>,<a href="#cite_note-Bhattacharya2010-12">[12]</a> <a href="Evolution_strategy" title="Evolution strategy">evolution strategy</a>,<a href="#cite_note-Du2009-13">[13]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130301111839/http://tizhoosh.uwaterloo.ca/Research/opposition_based_learning.htm">opposition-based computing</a>,<a href="#cite_note-Ergezer2009-14">[14]</a> <a href="Case-based_reasoning" title="Case-based reasoning">case-based reasoning</a>,<a href="#cite_note-Panchal2009-15">[15]</a> <a href="Artificial_bee_colony_algorithm" title="Artificial bee colony algorithm">artificial bee colony algorithm</a>, bacterial foraging optimization,<a href="#cite_note-Lohokare2009-16">[16]</a> <a href="Harmony_search" class="mw-redirect" title="Harmony search">harmony search</a>,<a href="#cite_note-Wang2013-17">[17]</a> and the <a href="Simplex_algorithm" title="Simplex algorithm">simplex algorithm</a>.<a href="#cite_note-Wang2011-18">[18]</a></li>

<li>BBO can be combined with local search to create a <a href="Memetic_algorithm" title="Memetic algorithm">memetic algorithm</a> that performs much better than BBO alone.<a href="#cite_note-Simon2013b-19">[19]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Software">Software</h2></div>

<div class="mw-heading mw-heading3"><h3 id="MATLAB">MATLAB</h3></div>

<ul><li>The following MATLAB code gives a BBO implementation for minimizing the 20-dimensional <a href="Rosenbrock_function" title="Rosenbrock function">Rosenbrock function</a>. Note that the following code is very basic, although it does include elitism. A serious BBO implementation should include some of the variations discussed above, such as duplicate replacement, blending, nonlinear migration, and local optimization.</li></ul>

<div class="mw-highlight mw-highlight-lang-matlab mw-content-ltr" dir="ltr"><pre>function BBO

% Biogeography-based optimization (BBO) to minimize a continuous function

% This program was tested with MATLAB R2012b

GenerationLimit = 50; % generation count limit

PopulationSize = 50; % population size

ProblemDimension = 20; % number of variables in each solution (i.e., problem dimension)

MutationProbability = 0.04; % mutation probability per solution per independent variable

NumberOfElites = 2; % how many of the best solutions to keep from one generation to the next

MinDomain = -2.048; % lower bound of each element of the function domain

MaxDomain = +2.048; % upper bound of each element of the function domain

% Initialize the population

rng(round(sum(100*clock))); % initialize the random number generator

x = zeros(PopulationSize, ProblemDimension); % allocate memory for the population

for index = 1 : PopulationSize % randomly initialize the population

x(index, :) = MinDomain + (MaxDomain - MinDomain) * rand(1, ProblemDimension);

end

Cost = RosenbrockCost(x); % compute the cost of each individual

[x, Cost] = PopulationSort(x, Cost); % sort the population from best to worst

MinimumCost = zeros(GenerationLimit, 1); % allocate memory

MinimumCost(1) = Cost(1); % save the best cost at each generation in the MinimumCost array

disp(['Generation 0 min cost = ', num2str(MinimumCost(1))]);

z = zeros(PopulationSize, ProblemDimension); % allocate memory for the temporary population

% Compute migration rates, assuming the population is sorted from most fit to least fit

mu = (PopulationSize + 1 - (1:PopulationSize)) / (PopulationSize + 1); % emigration rate

lambda = 1 - mu; % immigration rate

for Generation = 1 : GenerationLimit

% Save the best solutions and costs in the elite arrays

EliteSolutions = x(1 : NumberOfElites, :);

EliteCosts = Cost(1 : NumberOfElites);

% Use migration rates to decide how much information to share between solutions

for k = 1 : PopulationSize

% Probabilistic migration to the k-th solution

for j = 1 : ProblemDimension

if rand < lambda(k) % Should we immigrate?

% Yes - Pick a solution from which to emigrate (roulette wheel selection)

RandomNum = rand * sum(mu);

Select = mu(1);

SelectIndex = 1;

while (RandomNum > Select) && (SelectIndex < PopulationSize)

SelectIndex = SelectIndex + 1;

Select = Select + mu(SelectIndex);

end

z(k, j) = x(SelectIndex, j); % this is the migration step

else

z(k, j) = x(k, j); % no migration for this independent variable

end

% Mutation

for ParameterIndex = 1 : ProblemDimension

z(k, ParameterIndex) = MinDomain + (MaxDomain - MinDomain) * rand;

end

x = z; % replace the solutions with their new migrated and mutated versions

Cost = RosenbrockCost(x); % calculate cost

[x, Cost] = PopulationSort(x, Cost); % sort the population and costs from best to worst

x(PopulationSize-k+1, :) = EliteSolutions(k, :);

Cost(PopulationSize-k+1) = EliteCosts(k);

end

MinimumCost(Generation+1) = Cost(1);

disp(['Generation ', num2str(Generation), ' min cost = ', num2str(MinimumCost(Generation+1))])

end

% Wrap it up by displaying the best solution and by plotting the results

disp(['Best solution found = ', num2str(x(1, :))])

close all

plot(0:GenerationLimit, MinimumCost);

xlabel('Generation')

ylabel('Minimum Cost')

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [x, Cost] = PopulationSort(x, Cost)

% Sort the population and costs from best to worst

[Cost, indices] = sort(Cost, 'ascend');

x = x(indices, :);

return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [Cost] = RosenbrockCost(x)

% Compute the Rosenbrock function value of each element in x

NumberOfDimensions = size(x, 2);

Cost = zeros(size(x, 1), 1); % allocate memory for the Cost array

for PopulationIndex = 1 : length(x)

for i = 1 : NumberOfDimensions-1

Temp1 = x(PopulationIndex, i);

Temp2 = x(PopulationIndex, i+1);

Cost(PopulationIndex) = Cost(PopulationIndex) + 100 * (Temp2 - Temp1^2)^2 + (Temp1 - 1)^2;

end

end

return

</pre></div>

<ul><li>"bbo: Biogeography-Based Optimization" is an <a href="R_(programming_language)" title="R (programming language)">R</a> package for continuous BBO.<a href="#cite_note-rPackage-20">[20]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2></div>

BBO has been extended to noisy functions (that is, functions whose fitness evaluation is corrupted by noise);<a href="#cite_note-Ma2013-21">[21]</a> constrained functions;<a href="#cite_note-Roy2010a-22">[22]</a> combinatorial functions;<a href="#cite_note-Song2010-23">[23]</a> and multi-objective functions.<a href="#cite_note-Roy2010b-24">[24]</a><a href="#cite_note-DiBarba2016-25">[25]</a>

Moreover, a micro biogeography-inspired multi-objective optimization algorithm (μBiMO) was implemented: it is suitable for solving multi-objective optimisations in the field of industrial design because it is based on a small number of islands (hence the name μBiMO), i.e. few objective function calls are required.<a href="#cite_note-26">[26]</a>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_analyses">Mathematical analyses</h2></div>

BBO has been mathematically analyzed using Markov models<a href="#cite_note-Simon2011b-27">[27]</a> and dynamic system models.<a href="#cite_note-Simon2011c-28">[28]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Scholars have applied BBO into various academic and industrial applications. They found BBO performed better than state-of-the-art global optimization methods.

For example, Wang et al. proved BBO performed equal performance with FSCABC but with simpler codes.<a href="#cite_note-29">[29]</a>

Yang et al. showed BBO was superior to GA, PSO, and ABC.<a href="#cite_note-30">[30]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFQuammen1997" class="citation book cs1">Quammen, D. (1997). The Song of the Dodo: Island Biogeography in an Age of Extinction. Scribner.</cite>

</li>

<cite id="CITEREFSimon2008" class="citation journal cs1">Simon, D. (2008). <a rel="nofollow" class="external text" href="http://academic.csuohio.edu/simond/bbo/BBO_Simon.pdf">"Biogeography-based optimization"</a> (PDF). IEEE Transactions on Evolutionary Computation. 12 (6): 702–713. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftevc.2008.919004">10.1109/tevc.2008.919004</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8319014">8319014</a>.</cite>

</li>

<cite id="CITEREFMacArthurWilson1967" class="citation book cs1">MacArthur, R.; Wilson, E. (1967). The Theory of Island Biogeography. Princeton University Press.</cite>

</li>

<cite id="CITEREFWescheGoertlerHubert1987" class="citation journal cs1">Wesche, T.; Goertler, G.; Hubert, W. (1987). "Modified habitat suitability index model for brown trout in southeastern Wyoming". North American Journal of Fisheries Management. 7 (2): 232–237. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987NAJFM...7..232W">1987NAJFM...7..232W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1577%2F1548-8659%281987%297%3C232%3Amhsimf%3E2.0.co%3B2">10.1577/1548-8659(1987)7<232:mhsimf>2.0.co;2</a>.</cite>

</li>

<cite id="CITEREFDe_Jong1975" class="citation thesis cs1">De Jong, K. (1975). An Analysis of the Behaviour of a Class of Genetic Adaptive Systems (Ph.D.). University of Michigan.</cite>

</li>

<cite id="CITEREFMuhlenbeinSchlierkamp-Voosen1993" class="citation journal cs1">Muhlenbein, H.; Schlierkamp-Voosen, D. (1993). "Predictive models for the breeder genetic algorithm: I. Continuous parameter optimization". Evolutionary Computation. 1 (1): 25–49. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2Fevco.1993.1.1.25">10.1162/evco.1993.1.1.25</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16085506">16085506</a>.</cite>

</li>

<cite id="CITEREFMaSimon2011" class="citation journal cs1">Ma, H.; Simon, D. (2011). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/MaSimonEAAI2011.pdf">"Blended biogeography-based optimization for constrained optimization"</a> (PDF). Engineering Applications of Artificial Intelligence. 24 (3): 517–525. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.engappai.2010.08.005">10.1016/j.engappai.2010.08.005</a>.</cite>

</li>

<cite id="CITEREFSimon2013" class="citation book cs1">Simon, D. (2013). <a rel="nofollow" class="external text" href="http://academic.csuohio.edu/simond/EvolutionaryOptimization/">Evolutionary Optimization Algorithms</a>. Wiley.</cite>

</li>

<cite id="CITEREFKundraSood2010" class="citation journal cs1">Kundra, H.; Sood, M. (2010). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/Kundra2010.pdf">"Cross-Country Path Finding using Hybrid approach of PSO and BBO"</a> (PDF). International Journal of Computer Applications. 7 (6): 15–19. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5120%2F1167-1370">10.5120/1167-1370</a>.</cite>

</li>

<cite id="CITEREFMa2010" class="citation journal cs1">Ma, H. (2010). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/HaipingMa2.pdf">"An analysis of the equilibrium of migration models for biogeography-based optimization"</a> (PDF). Information Sciences. 180 (18): 3444–3464. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ins.2010.05.035">10.1016/j.ins.2010.05.035</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFZhang2015" class="citation journal cs1">Zhang, Y. (2015). <a rel="nofollow" class="external text" href="http://www.jpier.org/PIER/pier152/04.15040602.pdf">"Pathological Brain Detection in Magnetic Resonance Imaging Scanning by Wavelet Entropy and Hybridization of Biogeography-based Optimization and Particle Swarm Optimization"</a> (PDF). Progress in Electromagnetics Research. 152: 41–58. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2528%2Fpier15040602">10.2528/pier15040602</a>.</cite>

</li>

<cite id="CITEREFBhattacharyaChattopadhyay2010" class="citation journal cs1">Bhattacharya, A.; Chattopadhyay, P. (2010). "Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch". IEEE Transactions on Power Systems. 25 (4): 1955–1964. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010ITPSy..25.1955B">2010ITPSy..25.1955B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftpwrs.2010.2043270">10.1109/tpwrs.2010.2043270</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:30052218">30052218</a>.</cite>

</li>

<cite id="CITEREFDuSimonErgezer2009" class="citation conference cs1">Du, D.; Simon, D.; Ergezer, M. (2009). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/mbbo.pdf">"Biogeography-based optimization combined with evolutionary strategy and immigration refusal"</a> (PDF). IEEE Conference on Systems, Man, and Cybernetics. San Antonio, Texas. pp. 1023–1028.</cite>

</li>

<cite id="CITEREFErgezerSimonDu2009" class="citation conference cs1">Ergezer, M.; Simon, D.; Du, D. (2009). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/OBBO_SMC.pdf">"Oppositional biogeography-based optimization"</a> (PDF). IEEE Conference on Systems, Man, and Cybernetics. San Antonio, Texas. pp. 1035–1040.</cite>

</li>

<cite id="CITEREFKundraKaurPanchal2009" class="citation journal cs1">Kundra, H.; Kaur, A.; Panchal, V. (2009). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/Kundra.pdf">"An integrated approach to biogeography based optimization with case-based reasoning for exploring groundwater possibility"</a> (PDF). The Delving: Journal of Technology and Engineering Sciences. 1 (1): 32–38.</cite>

</li>

<cite id="CITEREFLohokarePattnaikDeviPanigrahi2009" class="citation conference cs1">Lohokare, M.; Pattnaik, S.; Devi, S.; Panigrahi, B.; Das, S.; Bakwad, K. (2009). "Intelligent biogeography-based optimization for discrete variables". World Congress on Nature and Biologically Inspired Computing. Coimbatore, India. pp. 1088–1093. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FNABIC.2009.5393808">10.1109/NABIC.2009.5393808</a>.</cite>

</li>

<cite id="CITEREFWangGuoDuanWang2013" class="citation journal cs1">Wang, G.; Guo, L.; Duan, H.; Wang, H.; Liu, L.; Shao, M. (2013). "Hybridizing harmony search with biogeography based optimization for global numerical optimization". Journal of Computational and Theoretical Nanoscience. 10 (10): 2312–2322. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013JCTN...10.2312W">2013JCTN...10.2312W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1166%2Fjctn.2013.3207">10.1166/jctn.2013.3207</a>.</cite>

</li>

<cite id="CITEREFWangXu2011" class="citation journal cs1">Wang, L.; Xu, Y. (2011). "An effective hybrid biogeography-based optimization algorithm for parameter estimation of chaotic systems". Expert Systems with Applications. 38 (12): 15103–15109. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.eswa.2011.05.011">10.1016/j.eswa.2011.05.011</a>.</cite>

</li>

<cite id="CITEREFSimonOmranClerc" class="citation web cs1">Simon, D.; Omran, M.; Clerc, M. <a rel="nofollow" class="external text" href="http://academic.csuohio.edu/simond/bbo/linearized/">"Linearized Biogeography-Based Optimization with Re-initialization and Local Search"</a>. Retrieved 6 September 2013.</cite>

</li>

<li id="cite_note-rPackage-20"><a href="#cite_ref-rPackage_20-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/bbo/">"Bbo: Biogeography-Based Optimization"</a>. 2014-09-18.</cite>

</li>

<cite id="CITEREFMaFeiSimonYu" class="citation web cs1">Ma, H.; Fei, M.; Simon, D.; Yu, M. <a rel="nofollow" class="external text" href="http://academic.csuohio.edu/simond/bbo/noisy/">"Biogeography-Based Optimization for Noisy Fitness Functions"</a>. Retrieved 7 September 2013.</cite>

</li>

<cite id="CITEREFRoyGhoshalThakur2010" class="citation journal cs1">Roy, P.; Ghoshal, S.; Thakur, S. (2010). "Biogeography based optimization for multi-constraint optimal power flow with emission and non-smooth cost function". Expert Systems with Applications. 37 (12): 8221–8228. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.eswa.2010.05.064">10.1016/j.eswa.2010.05.064</a>.</cite>

</li>

<cite id="CITEREFSongLiuWang2010" class="citation conference cs1">Song, Y.; Liu, M.; Wang, Z. (2010). "Biogeography-based optimization for the traveling salesman problems". International Joint Conference on Computational Science and Optimization. Huangshan, Anhui, China. pp. 295–299.</cite>

</li>

<cite id="CITEREFRoyGhoshalThakur2010" class="citation journal cs1">Roy, P.; Ghoshal, S.; Thakur, S. (2010). "Multi-objective optimal power flow using biogeography-based optimization". Electric Power Components and Systems. 38 (12): 1406–1426. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F15325001003735176">10.1080/15325001003735176</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:109069222">109069222</a>.</cite>

</li>

<cite id="CITEREFDi_BarbaDughieroMognaschiSavini2016" class="citation journal cs1">Di Barba, P.; Dughiero, F.; Mognaschi, M.E.; Savini, A.; Wiak, S. (2016). "Biogeography-Inspired Multiobjective Optimization and MEMS Design". IEEE Transactions on Magnetics. 52 (3): 1–4. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016ITM....5288982D">2016ITM....5288982D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTMAG.2015.2488982">10.1109/TMAG.2015.2488982</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17355264">17355264</a>.</cite>

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <cite id="CITEREFMognaschi2017" class="citation journal cs1">Mognaschi, M.E. (2017). "Micro biogeography-inspired multi-objective optimisation for industrial electromagnetic design". Electronics Letters. 53 (22): 1458–1460. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017ElL....53.1458M">2017ElL....53.1458M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fel.2017.3072">10.1049/el.2017.3072</a>.</cite>

</li>

<cite id="CITEREFSimonErgezerDuRarick2011" class="citation journal cs1">Simon, D.; Ergezer, M.; Du, D.; Rarick, R. (2011). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/Markov9.pdf">"Markov models for biogeography-based optimization"</a> (PDF). IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics. 41 (1): 299–306. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftsmcb.2010.2051149">10.1109/tsmcb.2010.2051149</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20595090">20595090</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11852624">11852624</a>.</cite>

</li>

<cite id="CITEREFSimon2011" class="citation journal cs1">Simon, D. (2011). <a rel="nofollow" class="external text" href="http://embeddedlab.csuohio.edu/BBO/BBO_Papers/BBODyn.pdf">"A dynamic system model of biogeography-based optimization"</a> (PDF). Applied Soft Computing. 1 (8): 5652–5661. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.asoc.2011.03.028">10.1016/j.asoc.2011.03.028</a>.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFWang2015" class="citation journal cs1">Wang, S. (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe17085711">"Fruit Classification by Wavelet-Entropy and Feedforward Neural Network trained by Fitness-scaled Chaotic ABC and Biogeography-based Optimization"</a>. Entropy. 17 (8): 5711–5728. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015Entrp..17.5711W">2015Entrp..17.5711W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fe17085711">10.3390/e17085711</a>.</cite>

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite id="CITEREFYangYang2015" class="citation journal cs1">Yang, G.; Yang, J. (2015). "Automated classification of brain images using wavelet-energy and biogeography-based optimization". Multimedia Tools and Applications. 75 (23): 15601–15617. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11042-015-2649-7">10.1007/s11042-015-2649-7</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:254825916">254825916</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Major_subfields_of_optimization78" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Major_subfields_of_optimization78" style="font-size:114%;margin:0 4em"><a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">Major subfields of optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_programming" class="mw-redirect" title="Convex programming">Convex programming</a></li>

<li><a href="Fractional_programming" title="Fractional programming">Fractional programming</a></li>

<li><a href="Integer_programming" title="Integer programming">Integer programming</a></li>

<li><a href="Quadratic_programming" title="Quadratic programming">Quadratic programming</a></li>

<li><a href="Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a></li>

<li><a href="Stochastic_programming" title="Stochastic programming">Stochastic programming</a></li>

<li><a href="Robust_optimization" title="Robust optimization">Robust optimization</a></li>

<li><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial optimization</a></li>

<li><a href="Infinite-dimensional_optimization" title="Infinite-dimensional optimization">Infinite-dimensional optimization</a></li>

<li><a href="Metaheuristic" title="Metaheuristic">Metaheuristics</a></li>

<li><a href="Constraint_satisfaction" title="Constraint satisfaction">Constraint satisfaction</a></li>

<li><a href="Multiobjective_optimization" class="mw-redirect" title="Multiobjective optimization">Multiobjective optimization</a></li>

<li><a href="Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-16" href="https://en.wikipedia.org/wiki/?title=Biogeography-based_optimization&oldid=1285913813">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>