A Hybrid Genetic Algorithm to Minimize Total Tardiness for Unrelated Parallel Machine Scheduling with Precedence Constraints

<table class="table-group" id="tab3"><tr><td><table class="table"><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td align="left">  Parameter </td><td align="center"> Value</td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td align="left">Population size (PopSize) </td><td align="center"> 20 </td></tr><tr><td align="left">Crossover probability <svg height="16.4125" id="M194" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 28.15 16.4125" width="28.15" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,16.75,16.212)"><path d="M383 397q0 -32 -35 -49q-12 -6 -23 8q-37 45 -84 45t-90 -71q-40 -65 -40 -167q0 -57 22 -86t59 -29q38 0 81.5 24.5t69.5 51.5l16 -21q-44 -53 -104 -84t-109 -31q-56 0 -89.5 41t-33.5 117q0 61 30 124t79 105q33 28 81 50.5t86 22.5q34 0 59 -15.5t25 -35.5z" id="x1D450"></path></g>
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</svg></td><td align="center"> 0.9 </td></tr><tr><td align="left">Mutation probability <svg height="16.4125" id="M195" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 32.712502 16.4125" width="32.712502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
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<g transform="matrix(.012,-0,0,-.012,16.75,16.212)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
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<g transform="matrix(.017,-0,0,-.017,26.762,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg></td><td align="center"> 0.05 </td></tr><tr><td align="left">Arbitrary constant for stopping criterion <svg height="15.0875" id="M196" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 18.862499 15.0875" width="18.862499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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t-44.5 92.5q0 46 38 78t95 45v2q-35 10 -54.5 33t-19.5 52q0 55 53 88.5t122 33.5q67 0 98.5 -23.5t31.5 -49.5z" id="x1D700"></path></g><g transform="matrix(.017,-0,0,-.017,12.914,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg></td><td align="center"> 0.0001 </td></tr><tr><td align="left">Maximum number of generation <svg height="16.4125" id="M197" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 47.387501 16.4125" width="47.387501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg></td><td align="center"> 200 </td></tr><tr class="table-tr"><td colspan="2"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Mathematical Problems in Engineering

tab3

Table 3

Table 3: A Hybrid Genetic Algorithm to Minimize Total Tardiness for Unrelated Parallel Machine Scheduling with Precedence Constraints 