Mathematical Problems in Engineering

Research Article

A Two-Agent Single-Machine Scheduling Problem with Learning and Deteriorating Considerations

Table 4

Performance of the branch-and-bound and SA algorithms ( , pro = 0.6).
Pro Branch-and-bound algorithmSA1SA2SA3
CPU timeNumber of nodesError percentages
MeanStdMeanStdMeanStdMeanStdMeanStd
0.60.40 0.40 1.100 −1.100 508.76 747.32 15421452 22053238 0.962 1.783 0.916 1.513 0.151 0.221
    −1.010 419.13 702.28 12621515 20637904 0.944 1.289 0.655 1.047 0.516 0.838
    −1.001 423.19 705.68 12735894 20730667 0.864 1.318 0.479 0.725 0.453 0.721
   1.010 −1.100 464.44 681.30 14073410 20085625 0.556 0.892 0.396 0.477 0.421 0.871
    −1.010 386.03 646.22 11628829 18977908 1.008 1.105 0.885 1.240 0.158 0.233
    −1.001 387.96 639.10 11682914 18744229 1.092 2.516 0.810 1.574 0.248 0.474
   1.001 −1.100 460.68 675.83 13961961 19925977 0.459 0.816 0.786 1.068 0.329 0.833
    −1.010 382.96 639.75 11537878 18783161 1.160 2.383 0.759 1.344 0.288 0.527
    −1.001 384.61 632.89 11584157 18561838 1.290 2.378 0.538 1.226 0.232 0.481
  0.50 1.100 −1.100 444.01 741.94 13626115 22119158 0.878 1.042 0.949 1.204 0.399 0.667
    −1.010 405.16 754.23 12130822 22295569 0.458 0.568 0.486 0.672 0.241 0.277
    −1.001 404.01 756.97 12077970 22363737 0.864 2.460 1.275 3.130 0.218 0.328
   1.010 −1.100 409.60 686.34 12555127 20418894 0.713 0.937 0.648 0.968 0.600 1.020
    −1.010 372.23 685.16 11147650 20223236 0.667 0.883 0.429 0.645 0.336 0.431
    −1.001 372.98 691.25 11157930 20400814 0.846 0.881 0.988 1.254 0.420 0.469
   1.001 −1.100 405.58 679.26 12430454 20200090 0.845 1.118 0.705 0.787 0.163 0.225
    −1.010 371.52 687.89 11126894 20304074 0.743 0.970 0.796 1.162 0.222 0.314
    −1.001 370.77 689.06 11089959 20330734 1.036 1.723 0.844 0.984 0.216 0.228
  0.60 1.100 −1.100 407.95 829.01 12323340 24833584 0.477 0.681 0.558 0.713 0.173 0.274
    −1.010 428.20 868.38 12887492 25900159 0.322 0.471 0.732 1.121 0.338 0.491
    −1.001 423.31 850.13 12725974 25328476 0.944 0.908 0.493 0.828 0.324 0.451
   1.010 −1.100 371.63 747.82 11207421 22320678 1.601 3.184 0.469 1.084 0.147 0.271
    −1.010 384.51 766.39 11551885 22789194 0.784 1.413 0.410 0.668 0.166 0.236
    −1.001 387.24 774.19 11631830 23028425 0.788 1.386 0.514 0.841 0.168 0.240
   1.001 −1.100 369.58 746.26 11142589 22266752 0.446 0.570 1.112 2.195 0.229 0.396
    −1.010 380.45 757.19 11429538 22512411 0.489 0.591 1.124 1.760 0.230 0.329
    −1.001 381.94 761.87 11468577 22651834 0.633 0.831 0.521 0.692 0.152 0.236
 0.50 0.40 1.100 −1.100 152.89 208.18 4712381 6298903 0.599 1.252 0.439 0.590 0.242 0.411
    −1.010 134.40 206.96 4074096 6124008 0.559 0.843 0.669 0.919 0.311 0.582
    −1.001 134.81 207.42 4083266 6129901 0.365 0.507 0.521 0.845 0.237 0.527
   1.010 −1.100 136.30 180.49 4209770 5461139 0.520 0.850 0.375 0.559 0.257 0.466
    −1.010 121.37 183.77 3689527 5448570 0.501 0.968 0.576 0.846 0.374 0.664
    −1.001 121.14 182.92 3677737 5412926 0.470 0.787 0.590 0.844 0.163 0.257
   1.001 −1.100 135.09 178.19 4174596 5394367 0.224 0.239 0.406 0.611 0.232 0.453
    −1.010 120.38 182.07 3661654 5401469 0.899 1.257 0.604 0.969 0.315 0.747
    −1.001 120.00 181.39 3645728 5371004 0.536 0.712 0.585 0.862 0.280 0.460
  0.50 1.100 −1.100 144.86 255.28 4384365 7682740 0.587 1.003 0.817 1.479 0.184 0.358
    −1.010 145.08 245.30 4385018 7249908 0.727 1.340 0.297 0.467 0.218 0.521
    −1.001 133.52 240.54 4089473 7123797 0.477 0.789 0.784 1.369 0.274 0.754
   1.010 −1.100 129.39 222.62 3909542 6663581 0.724 1.174 0.855 1.298 0.331 0.781
    −1.010 128.34 212.22 3874419 6246018 1.114 1.876 1.283 1.748 0.198 0.435
    −1.001 118.00 207.86 3606887 6129381 0.704 0.987 0.822 1.424 0.226 0.476
   1.001 −1.100 128.07 219.79 3869681 6575772 0.215 0.208 0.614 1.188 0.160 0.190
    −1.010 127.12 210.58 3838378 6199175 0.548 1.256 0.731 1.386 0.239 0.472
    −1.001 116.43 204.61 3558106 6030835 1.064 1.816 0.919 1.717 0.111 0.338
  0.60 1.100 −1.100 159.25 305.68 4859947 9196852 0.533 0.741 0.935 1.465 0.258 0.714
    −1.010 142.16 277.02 4328607 8163288 0.417 0.676 0.571 0.712 0.138 0.199
    −1.001 141.57 275.89 4305544 8117557 0.724 0.597 0.381 0.678 0.202 0.322
   1.010 −1.100 141.16 266.80 4292168 7969042 1.210 2.149 0.640 1.138 0.288 0.739
    −1.010 126.62 238.58 3845431 6989210 1.435 3.493 1.171 3.340 0.079 0.141
    −1.001 126.70 238.03 3845492 6967975 0.523 0.745 0.959 2.018 0.044 0.112
   1.001 −1.100 139.34 263.09 4235574 7853433 0.859 1.420 0.629 1.110 0.191 0.568
    −1.010 125.43 235.35 3810783 6895185 0.870 1.285 0.385 0.569 0.045 0.084
    −1.001 125.08 234.49 3795435 6861682 0.553 1.263 0.563 1.247 0.110 0.167
 0.60 0.40 1.100 −1.100 28.56 30.78 904301 959846 0.332 0.719 0.777 1.560 0.647 1.437
    −1.010 28.18 34.87 899501 1104145 0.285 0.463 0.220 0.386 0.269 0.536
    −1.001 29.79 34.57 956497 1095900 0.650 1.593 0.394 0.805 0.090 0.145
   1.010 −1.100 25.62 27.50 815854 864274 0.741 1.199 0.594 1.163 0.450 1.150
    −1.010 25.00 30.48 802771 973960 1.041 2.791 0.214 0.289 0.141 0.189
    −1.001 26.43 30.25 853055 968158 0.424 0.648 0.200 0.265 0.108 0.139
   1.001 −1.100 25.29 27.15 805707 853584 0.304 0.445 0.502 0.886 0.122 0.178
    −1.010 24.70 30.06 793335 961321 0.881 2.390 0.813 2.275 0.105 0.160
    −1.001 26.15 29.84 844493 955428 0.845 1.978 0.576 0.928 0.186 0.203
  0.50 1.100 −1.100 37.75 48.89 1206598 1540373 0.205 0.261 0.181 0.302 0.095 0.199
    −1.010 39.24 51.55 1246699 1622670 0.426 0.637 0.289 0.306 0.167 0.299
    −1.001 39.60 51.93 1257479 1634715 0.318 0.558 0.496 0.822 0.117 0.176
   1.010 −1.100 33.68 43.19 1080679 1369768 0.547 1.110 0.109 0.141 0.090 0.101
    −1.010 34.55 44.66 1101296 1415690 0.240 0.413 0.762 1.152 0.168 0.399
    −1.001 34.69 44.80 1105004 1420683 0.532 0.737 0.280 0.323 0.239 0.479
   1.001 −1.100 33.38 42.89 1071473 1360381 0.334 0.669 0.810 1.721 0.118 0.296
    −1.010 34.08 43.99 1086439 1395117 0.585 0.916 0.383 0.587 0.183 0.424
    −1.001 34.24 44.12 1091361 1400239 0.320 0.531 0.550 1.006 0.177 0.413
  0.60 1.100 −1.100 51.00 72.66 1619576 2279361 0.427 0.609 0.301 0.360 0.234 0.456
    −1.010 48.94 73.35 1532443 2267957 0.687 0.778 0.367 0.596 0.157 0.362
    −1.001 48.77 73.14 1524128 2255341 0.530 0.750 0.351 0.477 0.238 0.387
   1.010 −1.100 45.80 64.27 1454993 2015529 0.342 0.579 0.472 0.701 0.247 0.650
    −1.010 43.98 64.28 1379773 1998789 0.467 0.735 0.473 0.653 0.203 0.326
    −1.001 43.52 63.35 1361289 1959494 0.542 0.742 0.332 0.513 0.410 0.634
   1.001 −1.100 45.40 63.51 1442718 1991271 0.344 0.400 0.650 0.924 0.146 0.206
    −1.010 43.38 63.29 1360805 1967468 0.479 0.758 0.611 0.744 0.146 0.244
    −1.001 42.97 62.32 1344046 1927063 0.361 0.658 0.730 1.147 0.294 0.578
Average  190.83 332.10 5785994 9867958 0.655 1.104 0.615 1.029 0.232 0.429