Genetic Algorithm for Combinatorial Path Planning: The Subtour Problem

<table>The locations of the targets <svg height="14.6625" id="M59" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 71.599998 14.6625" width="71.599998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6625)">
<g transform="translate(72,-60.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.45)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑇</tspan>
<tspan style="font-size: 12.50px; " x="10.339981" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.380369" y="0">{</tspan>
<tspan style="font-size: 12.50px; " x="28.381809" y="0">𝑡</tspan>
</text>
<text transform="matrix(1,0,0,-1,-39.65,60.32)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-34.77,63.45)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="5.2012482" y="0">𝑡</tspan>
</text>
<text transform="matrix(1,0,0,-1,-25.65,60.32)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-20.78,63.45)">
<tspan style="font-size: 12.50px; " x="0" y="0">}</tspan>
</text>
</g>
</g>
</svg> and the agent <svg height="7.1750002" id="M60" 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)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑎</tspan>
</text>
</g>
</g>
</svg> are specified by the vectors <svg height="14.6" id="M61" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 28.4625 14.6" width="28.4625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐫</tspan>
<tspan style="font-size: 12.50px; " x="5.551332" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="9.7148314" y="0">𝑡</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.32,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-53.45,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
</text>
</g>
</g>
</svg>, <svg height="14.6" id="M62" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 28.4625 14.6" width="28.4625" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
<g transform="translate(72,-60.32)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐫</tspan>
<tspan style="font-size: 12.50px; " x="5.551332" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="9.7148314" y="0">𝑡</tspan>
</text>
<text transform="matrix(1,0,0,-1,-58.32,60.37)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-53.45,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
</text>
</g>
</g>
</svg>, and <svg height="13.45" id="M63" style="vertical-align:-2.21957pt" version="1.1" viewbox="0 0 25.3125 13.45" width="25.3125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.45)">
<g transform="translate(72,-61.24)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐫</tspan>
<tspan style="font-size: 12.50px; " x="5.551332" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="9.7148314" y="0">𝑎</tspan>
<tspan style="font-size: 12.50px; " x="15.991337" y="0">)</tspan>
</text>
</g>
</g>
</svg>, respectively.</table>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Genetic Algorithm for Combinatorial Path Planning: The Subtour Problem 