Research Article
Assembly Line Balancing Based on Beam Ant Colony Optimisation
Table 1
Results of benchmark instances.
| Instances | C | Optimal | Priority rule | ACO-BS | Difference | Solution | Running time [s] | avg. | std. | avg. | std. |
| Arcus1 | 3786 | 21 | 22 | 21 | 1 | 21 | 0 | 3.437 | 0.077 | Arcus2 | 11570 | 13 | 14 | 13 | 1 | 13 | 0 | 65.417 | 60.831 | Barthol2 | 84 | 51 | 52 | 51 | 1 | 51 | 0 | 7.575 | 0.127 | Barthol2 | 85 | 50 | 51 | 51 | 0 | | 0 | 0.254 | 0.022 | Barthol2 | 87 | 49 | 50 | 49 | 1 | 49 | 0 | 7.818 | 0.161 | Barthol2 | 89 | 48 | 49 | 48 | 1 | 48 | 0 | 8.277 | 0.182 | Barthol2 | 91 | 47 | 48 | 47 | 1 | 47 | 0 | 8.333 | 0.257 | Barthol2 | 93 | 46 | 47 | 46 | 1 | 46 | 0 | 8.034 | 0.201 | Barthol2 | 95 | 45 | 46 | 45 | 1 | 45 | 0 | 8.193 | 0.153 | Barthol2 | 97 | 44 | 45 | 44 | 1 | 44 | 0 | 8.013 | 0.175 | Barthol2 | 99 | 43 | 44 | 43 | 1 | 43 | 0 | 9.656 | 3.344 | Barthol2 | 101 | 42 | 43 | 42 | 1 | 42 | 0 | 11.605 | 7.768 | Barthol2 | 104 | 41 | 42 | 41 | 1 | 41 | 0 | 8.338 | 0.182 | Barthol2 | 106 | 40 | 41 | 40 | 1 | 40 | 0 | 10.094 | 3.506 | Barthol2 | 109 | 39 | 40 | 39 | 1 | 39 | 0 | 8.675 | 0.203 | Barthol2 | 112 | 38 | 39 | 38 | 1 | 38 | 0 | 8.583 | 0.197 | Barthol2 | 115 | 37 | 38 | 37 | 1 | 37 | 0 | 8.758 | 0.174 | Barthol2 | 118 | 36 | 37 | 36 | 1 | 36 | 0 | 8.864 | 0.127 | Barthol2 | 121 | 35 | 36 | 35 | 1 | 35 | 0 | 11.590 | 4.274 | Barthol2 | 125 | 34 | 35 | 34 | 1 | 34 | 0 | 8.886 | 0.153 | Barthol2 | 129 | 33 | 34 | 33 | 1 | 33 | 0 | 9.145 | 0.144 | Barthol2 | 133 | 32 | 33 | 32 | 1 | 32 | 0 | 9.107 | 0.175 | Barthol2 | 137 | 31 | 32 | 31 | 1 | 31 | 0 | 9.190 | 0.115 | Barthol2 | 146 | 29 | 30 | 29 | 1 | 29 | 0 | 10.081 | 2.874 | Barthol2 | 152 | 28 | 29 | 28 | 1 | 28 | 0 | 9.309 | 0.130 | Barthol2 | 157 | 27 | 28 | 27 | 1 | 27 | 0 | 9.233 | 0.117 | Barthol2 | 163 | 26 | 27 | 26 | 1 | 26 | 0 | 9.232 | 0.118 | Barthol2 | 170 | 25 | 26 | 25 | 1 | 25 | 0 | 9.306 | 0.109 | Barthold | 403 | 14 | 15 | 14 | 1 | 14 | 0 | 8.093 | 0.255 | Barthold | 434 | 13 | 14 | 13 | 1 | 13 | 0 | 8.597 | 0.108 | Barthold | 470 | 12 | 13 | 12 | 1 | 12 | 0 | 8.691 | 0.130 | Barthold | 513 | 11 | 12 | 11 | 1 | 11 | 0 | 8.618 | 0.131 | Barthold | 626 | 9 | 10 | 9 | 1 | 9 | 0 | 8.423 | 0.103 | Buxey | 41 | 8 | 9 | 8 | 1 | 8 | 0 | 0.812 | 0.014 | Buxey | 47 | 7 | 8 | 7 | 1 | 7 | 0 | 16.977 | 4.762 | Gunther | 54 | 9 | 10 | 9 | 1 | 9 | 0 | 1.035 | 0.011 | Heskiaoff | 205 | 5 | 6 | 5 | 1 | 5 | 0 | 1.050 | 0.031 | Jackson | 10 | 5 | 6 | 5 | 1 | 5 | 0 | 0.081 | 0.015 | Kilbridge | 69 | 8 | 9 | 8 | 1 | 8 | 0 | 1.784 | 0.065 | Kilbridge | 79 | 7 | 8 | 7 | 1 | 7 | 0 | 1.758 | 0.022 | Kilbridge | 92 | 6 | 7 | 6 | 1 | 6 | 0 | 1.736 | 0.050 | Kilbridge | 138 | 4 | 5 | 4 | 1 | 4 | 0 | 1.567 | 0.035 | Lutz2 | 11 | 49 | 50 | 49 | 1 | 49 | 0 | 8.561 | 6.649 | Lutz2 | 12 | 44 | 47 | 44 | 3 | | 0.527 | 99.253 | 84.952 | Lutz2 | 13 | 40 | 42 | 40 | 2 | 40 | 0 | 7.149 | 6.477 | Lutz2 | 14 | 37 | 38 | 37 | 1 | | 0.483 | 61.574 | 105.705 | Lutz2 | 15 | 34 | 35 | 34 | 1 | 34 | 0 | 7.567 | 4.152 | Lutz2 | 16 | 31 | 33 | 31 | 2 | 31 | 0 | 8.813 | 5.541 | Lutz2 | 17 | 29 | 30 | 29 | 1 | 29 | 0 | 14.619 | 6.265 | Lutz2 | 18 | 28 | 29 | 28 | 1 | 28 | 0 | 27.185 | 11.602 | Lutz2 | 19 | 26 | 27 | 26 | 1 | 26 | 0 | 8.493 | 5.250 | Lutz2 | 20 | 25 | 26 | 25 | 1 | 25 | 0 | 2.688 | 0.081 | Lutz2 | 21 | 24 | 25 | 24 | 1 | 24 | 0 | 2.748 | 0.078 | Lutz3 | 110 | 15 | 16 | 15 | 1 | | 0.422 | 31.939 | 42.531 | Lutz3 | 118 | 14 | 15 | 14 | 1 | | 0.516 | 82.378 | 107.850 | Mansoor | 62 | 3 | 4 | 3 | 1 | 3 | 0 | 0.130 | 0.007 | Mukherje | 201 | 22 | 23 | 22 | 1 | 22 | 0 | 4.698 | 0.094 | Sawyer | 41 | 8 | 9 | 8 | 1 | 8 | 0 | 0.959 | 0.020 | Sawyer | 47 | 7 | 8 | 7 | 1 | | 0.316 | 225.300 | 145.082 | Tonge | 251 | 14 | 15 | 14 | 1 | | 0.483 | 17.315 | 36.100 | Tonge | 320 | 11 | 12 | 11 | 1 | 11 | 0 | 3.575 | 0.079 | Warnecke | 54 | 31 | 32 | 31 | 1 | 31 | 0 | 50.685 | 47.707 | Warnecke | 56 | 29 | 30 | 29 | 1 | 29 | 0 | 20.740 | 14.768 | Warnecke | 60 | 27 | 29 | 28 | 1 | | 0 | 7.998 | 5.720 | Warnecke | 62 | 27 | 28 | 27 | 1 | 27 | 0 | 21.701 | 16.420 | Warnecke | 65 | 25 | 27 | 25 | 1 | 25 | 0 | 4.152 | 2.172 | Warnecke | 68 | 24 | 25 | 24 | 1 | 24 | 0 | 13.874 | 11.894 | Warnecke | 71 | 23 | 24 | 23 | 1 | 23 | 0 | 14.076 | 13.767 | Warnecke | 82 | 20 | 21 | 20 | 1 | 20 | 0 | 3.393 | 2.080 | Warnecke | 92 | 17 | 18 | 17 | 1 | 17 | 0 | 4.433 | 3.124 | Warnecke | 104 | 15 | 16 | 15 | 1 | 15 | 0 | 2.266 | 0.037 | Warnecke | 111 | 14 | 15 | 14 | 1 | 14 | 0 | 3.053 | 1.544 | Wee-mag | 30 | 62 | 63 | 62 | 1 | 62 | 0 | 6.444 | 2.356 | Wee-mag | 46 | 34 | 35 | 34 | 1 | 34 | 0 | 8.894 | 6.803 | Wee-mag | 47 | 32 | 33 | 33 | 0 | | 0 | 10.019 | 5.176 | Wee-mag | 52 | 31 | 32 | 31 | 1 | 31 | 0 | 4.056 | 0.040 | Wee-mag | 56 | 30 | 31 | 30 | 1 | 30 | 0 | 4.113 | 0.034 | Scholl | 1394 | 50 | 51 | 51 | 0 | | 0 | 7.531 | 9.124 | Scholl | 1452 | 48 | 49 | 48 | 1 | | 0.316 | 24.050 | 7.363 | Scholl | 1483 | 47 | 48 | 48 | 0 | | 0 | 12.256 | 19.041 | Scholl | 1515 | 46 | 47 | 47 | 0 | | 0 | 13.600 | 10.953 | Scholl | 1584 | 44 | 45 | 44 | 1 | | 0.316 | 75.227 | 118.097 | Scholl | 1659 | 42 | 43 | 43 | 0 | | 0 | 3.568 | 0.032 | Scholl | 1742 | 40 | 41 | 40 | 1 | | 0.422 | 47.947 | 66.518 | Scholl | 1787 | 39 | 40 | 39 | 1 | | 0.422 | 174.619 | 130.467 | Scholl | 1834 | 38 | 39 | 38 | 1 | | 0.422 | 200.751 | 111.410 | Scholl | 1883 | 37 | 38 | 38 | 0 | | 0 | 149.149 | 160.662 | Scholl | 1935 | 36 | 37 | 37 | 0 | | 0 | 185.443 | 149.573 | Scholl | 1991 | 35 | 36 | 35 | 1 | | 0.316 | 200.308 | 113.994 | Scholl | 2049 | 34 | 35 | 35 | 0 | | 0 | 45.256 | 18.299 | Scholl | 2111 | 33 | 34 | 34 | 0 | | 0 | 50.165 | 74.052 | Scholl | 2177 | 32 | 33 | 32 | 1 | | 0.483 | 156.656 | 103.493 | Scholl | 2247 | 31 | 32 | 32 | 0 | | 0 | 47.580 | 24.497 | Scholl | 2322 | 30 | 31 | 30 | 1 | | 0.316 | 75.187 | 89.942 | Scholl | 2402 | 29 | 30 | 29 | 1 | | 0.516 | 141.810 | 110.054 | Scholl | 2488 | 28 | 29 | 28 | 1 | | 0.483 | 60.101 | 69.595 | Scholl | 2580 | 27 | 28 | 27 | 1 | | 0.422 | 142.472 | 102.649 | Scholl | 2680 | 26 | 27 | 26 | 1 | 26 | 0 | 35.804 | 16.163 | Scholl | 2787 | 25 | 26 | 25 | 1 | 25 | 0 | 49.832 | 28.863 |
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Note.indicates that an optimal solution is found; beside the average of solution indicates trickiness. |