Mathematical Problems in Engineering

Research Article

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

Table 3

Performance of the branch-and-bound and SA algorithms ( , pro = 0.4).
Pro Branch-and-bound algorithmSA1SA2SA3
CPU timeNumber of nodesError percentages
MeanStdMeanStdMeanStdMeanStdMeanStd
0.40.40 0.40 1.100 −1.100 311.88 414.22 11758285 15903115 0.837 1.546 0.887 1.652 0.293 0.483
    −1.010 191.71 349.53 7136396 13128691 0.456 0.679 0.745 1.130 0.541 1.062
    −1.001 196.36 358.32 7305845 13462940 0.519 0.683 0.491 0.745 0.397 0.537
   1.010 −1.100 287.40 379.73 10866955 14623008 0.321 0.367 0.821 1.332 0.279 0.467
    −1.010 175.58 319.29 6557095 12024063 0.543 0.917 0.347 0.504 0.443 0.911
    −1.001 178.59 324.07 6665084 12209228 0.497 0.661 0.915 1.382 0.359 0.655
   1.001 −1.100 285.39 376.85 10796044 14522042 0.855 1.659 1.354 2.175 0.220 0.281
    −1.010 174.07 316.36 6503448 11918702 0.401 0.694 1.049 1.572 0.379 0.764
    −1.001 177.12 321.43 6613663 12115285 0.320 0.576 0.313 0.531 0.401 0.804
  0.50 1.100 −1.100 257.34 384.96 9612742 14644726 1.302 2.391 0.531 0.862 0.474 0.639
    −1.010 206.07 398.32 7644147 15064816 0.744 1.464 0.588 1.103 0.267 0.370
    −1.001 205.08 404.74 7648212 15313361 1.871 3.131 0.438 0.558 0.433 0.707
   1.010 −1.100 239.89 359.34 8982129 13695175 1.282 1.814 0.821 1.005 0.600 1.058
    −1.010 189.79 368.05 7057756 13951685 1.599 2.800 1.104 2.457 0.612 1.004
    −1.001 187.82 370.71 7020711 14059100 0.580 1.139 0.729 1.177 0.305 0.446
   1.001 −1.100 237.98 356.54 8913945 13593996 0.803 1.032 1.179 1.838 0.451 0.742
    −1.010 188.22 365.16 7000983 13842236 0.633 1.160 1.384 2.292 0.945 2.301
    −1.001 186.16 367.78 6960609 13950622 0.999 1.431 0.214 0.254 0.236 0.457
  0.60 1.100 −1.100 226.20 413.41 8376937 15502069 0.799 1.222 0.921 1.162 0.507 0.894
    −1.010 193.13 404.47 7149216 15170536 1.688 1.715 1.225 1.728 0.636 0.847
    −1.001 196.52 409.14 7277271 15334414 0.781 1.150 0.665 1.036 0.736 0.988
   1.010 −1.100 213.76 391.38 7940293 14704311 1.473 2.165 0.742 1.324 0.371 0.619
    −1.010 179.59 377.57 6671217 14223196 0.831 0.898 1.027 1.720 0.526 0.744
    −1.001 182.23 380.25 6770069 14300776 1.536 2.013 0.802 1.033 0.511 0.739
   1.001 −1.100 212.06 388.42 7877739 14594634 0.999 1.440 0.665 1.477 0.328 0.477
    −1.010 180.39 378.71 6706222 14252125 1.393 1.856 0.981 1.287 0.632 0.764
    −1.001 180.54 376.59 6707971 14165604 0.962 1.708 1.115 2.115 0.580 0.853
 0.50 0.40 1.100 −1.100 59.79 92.20 2340463 3674295 0.742 1.792 0.599 1.025 0.334 0.513
    −1.010 90.67 113.76 3549507 4515071 0.599 0.900 0.461 0.727 0.264 0.388
    −1.001 92.71 115.83 3626112 4594477 0.440 0.675 0.393 0.663 0.261 0.351
   1.010 −1.100 54.94 84.53 2162741 3386252 0.581 0.882 0.424 0.726 0.187 0.339
    −1.010 82.68 104.10 3254738 4152699 0.479 0.703 0.728 1.174 0.235 0.437
    −1.001 84.48 105.83 3322651 4219459 2.067 5.768 0.738 1.153 0.142 0.179
   1.001 −1.100 54.44 83.79 2144562 3358398 0.196 0.280 0.575 0.964 0.274 0.430
    −1.010 81.88 103.15 3224787 4117010 0.760 1.991 0.586 1.682 0.138 0.224
    −1.001 83.65 104.87 3291685 4183898 0.281 0.436 0.496 0.936 0.177 0.290
  0.50 1.100 −1.100 83.54 109.89 3264865 4371726 0.624 0.706 0.660 0.789 0.350 0.522
    −1.010 82.99 112.28 3200949 4429808 1.369 2.181 0.660 1.275 0.215 0.348
    −1.001 83.65 115.22 3232620 4539750 1.125 1.877 1.135 2.213 0.153 0.205
   1.010 −1.100 76.35 100.18 2999791 3999506 0.423 0.698 0.837 1.304 0.240 0.269
    −1.010 76.27 102.66 2954127 4065085 1.511 2.811 1.110 1.802 0.416 1.179
    −1.001 77.07 105.75 2990752 4183939 0.973 1.764 1.021 1.797 0.412 1.167
   1.001 −1.100 75.65 99.35 2974524 3968512 1.006 1.300 0.245 0.309 0.346 0.479
    −1.010 75.59 101.72 2929144 4029183 1.105 1.705 0.672 1.040 0.745 1.588
    −1.001 76.26 104.60 2960124 4138909 0.786 1.267 0.804 1.519 0.442 1.168
  0.60 1.100 −1.100 97.45 122.49 3776266 4802973 1.069 1.723 0.649 0.835 0.273 0.393
    −1.010 67.32 119.31 2539341 4508240 1.642 2.544 1.020 1.764 0.243 0.349
    −1.001 68.51 121.41 2582654 4586272 0.708 0.920 1.773 2.473 0.406 0.452
   1.010 −1.100 89.87 112.61 3489778 4415326 1.060 1.675 0.695 1.240 0.286 0.411
    −1.010 63.04 112.01 2390213 4253672 1.072 1.535 1.014 1.487 0.604 1.236
    −1.001 63.99 113.55 2423954 4309165 1.790 3.392 1.329 2.338 0.405 0.753
   1.001 −1.100 89.25 111.59 3467072 4375682 0.662 1.008 0.744 0.983 0.361 0.630
    −1.010 62.73 111.02 2379099 4216867 1.321 1.547 1.916 2.870 0.545 1.139
    −1.001 63.45 112.61 2404690 4275428 1.255 1.815 1.098 1.540 0.690 1.367
 0.60 0.40 1.100 −1.100 5.85 7.71 236881 324192 0.296 0.468 0.224 0.323 0.283 0.505
    −1.010 5.16 9.04 209445 379285 0.329 0.464 0.291 0.539 0.175 0.331
    −1.001 5.16 9.12 210367 382778 0.384 0.530 0.403 0.553 0.231 0.498
   1.010 −1.100 5.44 7.26 221007 305841 0.295 0.405 0.146 0.226 0.289 0.553
    −1.010 4.68 8.14 190945 342880 0.763 2.524 0.354 0.540 0.189 0.295
    −1.001 4.66 8.23 190746 346282 0.312 0.426 0.328 0.502 0.232 0.440
   1.001 −1.100 5.40 7.24 219717 305198 0.392 0.510 0.299 0.354 0.173 0.306
    −1.010 4.64 8.06 189163 339587 0.361 0.627 0.360 0.618 0.226 0.397
    −1.001 4.62 8.16 189126 343494 0.928 2.476 0.467 0.805 0.113 0.241
  0.50 1.100 −1.100 6.94 11.68 284387 494376 0.477 0.920 0.423 0.860 0.206 0.347
    −1.010 9.15 14.39 373361 598325 0.587 1.147 0.589 1.004 0.402 1.021
    −1.001 9.32 14.58 380130 606005 1.007 1.916 0.949 1.703 0.257 0.463
   1.010 −1.100 6.31 10.69 259327 453058 0.502 0.637 0.389 0.784 0.228 0.344
    −1.010 8.33 13.23 341436 552785 1.149 1.823 0.394 0.628 0.139 0.191
    −1.001 8.49 13.40 347732 559407 0.687 1.390 0.730 1.267 0.150 0.291
   1.001 −1.100 6.25 10.59 256783 448847 0.621 1.179 0.247 0.368 0.183 0.337
    −1.010 8.24 13.11 338060 547960 0.329 0.561 0.680 1.277 0.112 0.182
    −1.001 8.40 13.28 344399 554855 0.445 1.069 1.128 2.085 0.151 0.324
  0.60 1.100 −1.100 10.29 16.81 417314 689695 0.410 0.900 0.796 1.484 0.174 0.333
    −1.010 14.72 19.92 606522 827274 0.225 0.496 0.303 0.348 0.179 0.276
    −1.001 15.05 20.21 620128 838524 0.965 2.173 0.872 2.173 0.238 0.361
   1.010 −1.100 9.34 15.30 380584 630421 0.599 1.508 0.636 1.551 0.267 0.519
    −1.010 13.41 18.26 554783 760674 0.865 2.444 0.231 0.319 0.141 0.308
    −1.001 13.70 18.52 566713 770556 1.279 3.789 1.013 3.830 0.111 0.210
   1.001 −1.100 9.26 15.17 377334 625018 0.509 1.005 0.630 0.744 0.282 0.528
    −1.010 13.29 18.10 550084 754492 1.052 2.524 0.591 0.707 0.104 0.132
    −1.001 13.58 18.37 561776 764270 0.919 2.456 1.241 3.799 0.261 0.455
Average　　97.89 164.20 3701438 6277631 0.832 1.464 0.730 1.253 0.335 0.588