Mathematical Problems in Engineering

Research Article

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

Table 1

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 −1.100 103.98 276.35 3921045 10555474 0.543 1.381 0.530 1.452 0.247 0.883
    −1.010 63.93 217.06 2380521 8139948 0.223 0.659 0.409 0.924 0.197 0.658
    −1.001 65.47 222.48 2437016 8344800 0.216 0.507 0.165 0.483 0.170 0.381
   1.010 −1.100 95.82 253.71 3623888 9719538 0.478 1.500 0.397 1.219 0.298 1.126
    −1.010 58.55 198.37 2187372 7459183 0.272 0.725 0.239 0.630 0.182 0.586
    −1.001 59.55 201.41 2223379 7575433 0.271 0.671 0.322 0.897 0.145 0.416
   1.001 −1.100 95.15 251.83 3600246 9653456 0.420 1.255 0.656 1.637 0.175 0.620
    −1.010 58.05 196.57 2169483 7394568 0.174 0.466 0.416 1.046 0.168 0.525
    −1.001 59.06 199.77 2206236 7517115 0.200 0.625 0.208 0.618 0.153 0.497
  0.50 1.100 −1.100 85.80 248.97 3205789 9430455 0.527 1.543 0.227 0.569 0.210 0.455
    −1.010 68.71 244.97 2549809 9236732 0.581 2.138 0.318 0.891 0.171 0.419
    −1.001 68.38 248.11 2551177 9366715 0.692 1.981 0.161 0.386 0.347 0.856
   1.010 −1.100 79.98 232.32 2995531 8817308 0.575 1.440 0.439 1.103 0.219 0.667
    −1.010 63.28 226.24 2354281 8550234 0.699 1.835 0.622 1.664 0.242 0.662
    −1.001 62.63 227.25 2341949 8599304 0.316 0.789 0.297 0.777 0.143 0.348
   1.001 −1.100 79.34 230.50 2972794 8751750 0.417 1.087 0.530 1.435 0.180 0.485
    −1.010 62.76 224.44 2335360 8482895 0.285 0.771 0.567 1.530 0.321 1.380
    −1.001 62.08 225.42 2321896 8531873 0.405 0.965 0.217 0.694 0.114 0.314
  0.60 1.100 −1.100 75.42 256.62 2793897 9601920 0.546 1.308 0.447 0.970 0.359 1.051
    −1.010 64.40 245.82 2384768 9203252 0.625 1.263 0.811 2.576 0.314 0.666
    −1.001 65.53 248.86 2427465 9311518 0.369 0.874 0.352 0.859 0.285 0.676
   1.010 −1.100 71.27 242.87 2648314 9106695 0.576 1.457 0.327 0.922 0.254 0.679
    −1.010 59.88 229.35 2225401 8623062 0.381 0.693 0.429 1.125 0.405 1.064
    −1.001 60.77 231.22 2258351 8680937 0.565 1.358 0.732 2.489 0.329 1.122
   1.001 −1.100 70.71 241.02 2627453 9038121 0.360 0.943 0.357 1.033 0.135 0.320
    −1.010 60.15 230.09 2237063 8644312 0.809 1.726 0.531 1.307 0.378 1.006
    −1.001 60.20 229.00 2237648 8599211 0.567 1.440 0.462 1.334 0.266 0.614
 0.50 0.40 1.100 −1.100 19.94 59.21 780723 2349939 0.312 1.111 0.276 0.711 0.141 0.387
    −1.010 30.23 77.19 1183839 3050478 0.270 0.730 0.303 1.152 0.095 0.253
    −1.001 30.91 78.69 1209380 3107819 1.058 6.974 0.139 0.420 0.927 6.386
   1.010 −1.100 18.32 54.31 721468 2166990 0.249 0.619 0.201 0.522 0.080 0.245
    −1.010 27.57 70.55 1085563 2803070 0.202 0.511 0.355 1.077 0.480 2.424
    −1.001 28.17 71.84 1108207 2852184 0.850 3.491 0.273 0.751 0.187 0.844
   1.001 −1.100 18.16 53.83 715408 2149077 0.086 0.220 0.196 0.611 0.113 0.301
    −1.010 27.30 69.90 1075579 2778464 1.006 5.165 0.230 1.018 0.821 5.100
    −1.001 27.89 71.17 1097883 2827375 1.220 8.550 0.518 2.713 0.164 0.818
  0.50 1.100 −1.100 27.86 73.51 1088932 2909727 0.557 2.021 0.244 0.553 0.132 0.344
    −1.010 27.67 74.52 1067668 2921795 0.464 1.396 0.258 0.811 0.132 0.335
    −1.001 27.89 76.10 1078229 2982686 0.417 1.217 0.607 1.722 0.096 0.331
   1.010 −1.100 25.46 67.07 1000555 2665211 0.250 0.875 0.285 0.839 0.083 0.190
    −1.010 25.43 68.23 985378 2685319 0.512 1.747 0.380 1.148 0.158 0.697
    −1.001 25.70 69.91 997591 2751685 0.507 1.384 0.392 1.143 0.293 1.141
   1.001 −1.100 25.23 66.50 992131 2644048 0.681 2.156 0.217 0.807 0.216 0.815
    −1.010 25.20 67.61 977049 2661873 0.519 1.392 0.264 0.703 0.289 0.982
    −1.001 25.43 69.16 987380 2722447 0.311 0.823 0.311 0.963 0.296 1.136
  0.60 1.100 −1.100 32.49 83.06 1259437 3245114 0.381 1.107 0.432 1.539 0.118 0.302
    −1.010 22.45 74.47 847123 2812785 0.718 1.975 0.345 1.111 0.090 0.230
    −1.001 22.85 75.78 861566 2861344 0.295 0.705 0.598 1.635 0.151 0.324
   1.010 −1.100 29.96 76.43 1163916 2987954 0.535 1.425 0.378 1.250 0.098 0.269
    −1.010 21.02 69.88 797393 2652794 0.386 1.012 0.479 1.419 0.207 0.757
    −1.001 21.34 70.85 808645 2687918 0.848 2.490 0.454 1.468 0.147 0.469
   1.001 −1.100 29.76 75.79 1156347 2963377 0.319 0.960 0.360 0.984 0.296 0.930
    −1.010 20.92 69.31 793687 2631768 0.455 1.079 0.642 1.865 0.182 0.696
    −1.001 21.16 70.27 802222 2666815 0.468 1.233 0.514 1.464 0.232 0.842
 0.60 0.40 1.100 −1.100 1.95 5.15 79358 214347 0.311 1.439 0.105 0.295 0.124 0.376
    −1.010 1.73 5.65 70241 235697 0.125 0.309 0.105 0.336 0.165 0.827
    −1.001 1.73 5.70 70552 237669 0.144 0.366 0.147 0.368 0.094 0.322
   1.010 −1.100 1.82 4.83 74060 201604 0.330 1.453 0.262 1.441 0.125 0.389
    −1.010 1.57 5.10 64068 213383 0.266 1.477 0.146 0.361 0.076 0.197
    −1.001 1.56 5.14 64003 215085 0.120 0.289 0.128 0.334 0.079 0.273
   1.001 −1.100 1.81 4.82 73630 200978 0.150 0.360 0.144 0.324 0.102 0.292
    −1.010 1.55 5.05 63473 211343 0.220 0.706 0.131 0.391 0.106 0.279
    −1.001 1.55 5.10 63463 213337 0.326 1.471 0.172 0.508 0.045 0.151
  0.50 1.100 −1.100 2.32 7.36 95199 309557 0.159 0.569 0.141 0.528 0.069 0.220
    −1.010 3.06 9.20 124875 380846 0.223 0.713 0.225 0.656 0.136 0.610
    −1.001 3.11 9.33 127086 386185 0.363 1.193 0.334 1.068 0.089 0.289
   1.010 −1.100 2.11 6.73 86838 283413 0.270 0.875 0.132 0.481 0.179 0.805
    −1.010 2.78 8.44 114224 351088 0.385 1.169 0.135 0.402 0.048 0.127
    −1.001 2.83 8.56 116276 355790 0.233 0.853 0.258 0.800 0.054 0.180
   1.001 −1.100 2.09 6.66 85990 280750 0.373 1.450 0.186 0.809 0.062 0.210
    −1.010 2.75 8.36 113098 347937 0.112 0.354 0.229 0.793 0.046 0.125
    −1.001 2.80 8.48 115164 352785 0.158 0.643 0.397 1.297 0.052 0.197
  0.60 1.100 −1.100 3.44 10.66 139504 436210 0.145 0.546 0.293 0.936 0.059 0.206
    −1.010 4.91 13.22 202484 547733 0.086 0.310 0.121 0.260 0.072 0.197
    −1.001 5.02 13.44 207022 556511 0.327 1.314 0.304 1.301 0.126 0.423
   1.010 −1.100 3.12 9.69 127253 398528 0.206 0.902 0.230 0.934 0.097 0.324
    −1.010 4.47 12.10 185230 502913 0.303 1.447 0.079 0.212 0.047 0.187
    −1.001 4.57 12.29 189209 510614 0.427 2.235 0.350 2.226 0.038 0.130
   1.001 −1.100 3.09 9.61 126170 395114 0.175 0.618 0.219 0.518 0.096 0.328
    −1.010 4.43 11.99 183664 498778 0.356 1.516 0.213 0.493 0.035 0.090
    −1.001 4.53 12.19 187563 506370 0.306 1.461 0.428 2.235 0.087 0.286
Average  32.64 103.83 1234705 3960746 0.402 1.356 0.326 1.003 0.184 0.668