Genetic Algorithm for Combinatorial Path Planning: The Subtour Problem

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="483643.fig.0026a" src="https://static.hindawi.com/articles/mpe/volume-2011/483643/figures/483643.fig.0026a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Double cutting-point crossover: parents are cut at two different points</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="483643.fig.0026b" src="https://static.hindawi.com/articles/mpe/volume-2011/483643/figures/483643.fig.0026b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>Double cutting-point crossover: halves have different size</td></tr></table></td></tr></table>

<table>Double cutting-point crossover: a simple recombination is not possible, since the new offsprings <svg height="11.575" id="M313" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 12.3 11.575" width="12.3" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.575)">
<g transform="translate(72,-62.74)">
<text transform="matrix(1,0,0,-1,-71.95,66.04)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐬</tspan>
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<text transform="matrix(1,0,0,-1,-67.09,62.91)">
<tspan style="font-size: 8.75px; " x="0" y="0">3</tspan>
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</svg> and <svg height="11.425" id="M314" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 12.3 11.425" width="12.3" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.425)">
<g transform="translate(72,-62.86)">
<text transform="matrix(1,0,0,-1,-71.95,66.04)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐬</tspan>
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<text transform="matrix(1,0,0,-1,-67.09,62.91)">
<tspan style="font-size: 8.75px; " x="0" y="0">4</tspan>
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</svg> can have different lengths. For clearness, only the indexes of the targets are reported and agent <svg height="7.1750002" id="M315" style="vertical-align:-0.1254pt" version="1.1" viewbox="0 0 7.9749999 7.1750002" width="7.9749999" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.175)">
<g transform="translate(72,-66.26)">
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<tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan>
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</g>
</svg> is not shown.</table>

Mathematical Problems in Engineering

fig26

Figure 26

Figure 26: Genetic Algorithm for Combinatorial Path Planning: The Subtour Problem 