Table 4.
Algorithms’ results for two levels
| Problem # | Level | CPLEX | LR1 | LR2 | CPU time (s) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| LB | UB | Gap (%) | ITR | LB | UB | Gap (%) | ITR | CPLEX | LR1 | LR2 | |||
| 1 | 2 | 415.5 | 415.5 | 415.5 | 0.000 | 11 | 415.5 | 415.5 | 0.000 | 15 | 2 | 466 | 6 |
| 2 | 2 | 398.3 | 398.3 | 398.3 | 0.000 | 17 | 398.3 | 398.3 | 0.000 | 9 | 0.5 | 87 | 3 |
| 3 | 2 | 696.0 | 693.9 | 696.2 | 0.331 | 13 | 660.7 | 661.7 | 0.159 | 8 | 1 | 29 | 24 |
| 4 | 2 | 506.4 | 506.4 | 506.4 | 0.000 | 18 | 506.4 | 506.4 | 0.000 | 18 | 0.5 | 46 | 5 |
| 5 | 2 | 570.1 | 570.1 | 570.1 | 0.000 | 12 | 570.1 | 570.1 | 0.000 | 11 | 0.2 | 24 | 21 |
| 6 | 2 | 479.0 | 479.0 | 479.0 | 0.000 | 7 | 479.0 | 479.0 | 0.000 | 6 | 0.5 | 38 | 2 |
| 7 | 2 | 618.9 | 604.9 | 633.5 | 4.728 | 13 | 537.2 | 538.1 | 0.172 | 9 | 0.4 | 1061 | 24 |
| 8 | 2 | 450.5 | 450.5 | 450.5 | 0.000 | 18 | 450.5 | 450.5 | 0.000 | 19 | 0.5 | 116 | 5 |
| 9 | 2 | 591.7 | 591.7 | 591.7 | 0.000 | 13 | 574.2 | 591.7 | 3.040 | 13 | 0.5 | 63 | 4 |
| 10 | 2 | 587.3 | 587.1 | 587.3 | 0.034 | 14 | 583.8 | 583.8 | 0.000 | 17 | 0.7 | 1002 | 23 |
| 11 | 2 | 730.9 | 723.4 | 741.1 | 2.447 | 29 | 720.0 | 740.6 | 2.865 | 38 | 13 | 3196 | 35 |
| 12 | 2 | 869.3 | 845.5 | 907.6 | 7.345 | 20 | 864.5 | 892.2 | 3.212 | 87 | 6 | 382 | 66 |
| 13 | 2 | 885.5 | 885.5 | 885.5 | 0.000 | 87 | 866.5 | 884.5 | 2.084 | 36 | 26 | 1743 | 55 |
| 14 | 2 | 1004.4 | 993.12 | 1061.9 | 6.926 | 21 | 981.3 | 1023.1 | 4.263 | 48 | 56 | 5168 | 58 |
| 15 | 2 | 1104.1 | 1087.4 | 1120.9 | 3.081 | 49 | 1103.3 | 1167.9 | 5.855 | 92 | 6 | 3497 | 54 |
| 16 | 2 | 1031.8 | 1020.9 | 1107.6 | 8.493 | 96 | 1019.0 | 1060.3 | 4.056 | 99 | 4 | 2964 | 55 |
| 17 | 2 | 1161.1 | 1149 | 1196.1 | 4.099 | 38 | 1126.1 | 1201.2 | 6.669 | 52 | 11 | 4215 | 29 |
| 18 | 2 | 1016.4 | 1014.9 | 1053.9 | 3.843 | 44 | 1007.3 | 1048.4 | 4.086 | 105 | 18 | 5109 | 52 |
| 19 | 2 | 1095.0 | 1091.8 | 1145.1 | 4.882 | 51 | 1090.9 | 1134.9 | 4.035 | 51 | 6 | 4238 | 44 |
| 20 | 2 | 1219.9 | 1204 | 1253.8 | 4.136 | 37 | 1189.7 | 1234.8 | 3.784 | 54 | 108 | 9481 | 59 |
| 21 | 2 | 2873.1 | 2861.3 | 2885.6 | 0.849 | 41 | 2949.5 | 2987.7 | 1.295 | 95 | 246 | 71,876 | 98 |
| 22 | 2 | 3008.2 | NA** | NA | – | – | 2792.7 | 2850.0 | 2.052 | 90 | 411 | – | 78 |
| 23 | 2 | 2934.2 | NA | NA | – | – | 2811.8 | 2829.9 | 0.642 | 45 | 435 | – | 76 |
| 24 | 2 | 2921.8 | NA | NA | – | – | 2809.4 | 2825.4 | 0.568 | 58 | 428 | – | 81 |
| 25 | 2 | 2867.5 | NA | NA | – | – | 2657.5 | 2734.3 | 2.890 | 22 | 247 | – | 85 |
| 26 | 2 | 2613.4 | NA | NA | – | – | 2591.8 | 2626.2 | 1.326 | 11 | 52 | – | 88 |
| 27 | 2 | 3040.6 | NA | NA | – | – | 2964.1 | 2992.0 | 0.944 | 80 | 302 | – | 90 |
| 28 | 2 | 2617.3 | NA | NA | – | – | 2561.5 | 2712.3 | 5.885 | 96 | 197 | – | 90 |
| 29 | 2 | 3176.6 | NA | NA | – | – | 3080.2 | 3197.1 | 3.793 | 78 | 354 | – | 77 |
| 30 | 2 | 2873.4 | NA | NA | – | – | 2791.6 | 2810.5 | 0.676 | 111 | 261 | – | 98 |
| 31 | 2 | 6059.0* | NA | NA | – | – | 5393.7 | 5430.4 | 0.679 | 97 | 960 | – | 150 |
| 32 | 2 | 5474.4* | NA | NA | – | – | 4791.9 | 4867.2 | 1.572 | 161 | 960 | – | 270 |
| 33 | 2 | 6305.0* | NA | NA | – | – | 4605.0 | 4765.4 | 3.483 | 85 | 960 | – | 283 |
| 34 | 2 | 6159.0* | NA | NA | – | – | 4547.8 | 4653.3 | 2.319 | 72 | 960 | – | 702 |
| 35 | 2 | 6173.8* | NA | NA | – | – | 4991.5 | 5161.7 | 3.410 | 126 | 960 | – | 627 |
| 36 | 2 | 5519.8* | NA | NA | – | – | 4907.0 | 4946.3 | 0.801 | 72 | 960 | – | 750 |
| 37 | 2 | 6763.0* | NA | NA | – | – | 5348.3 | 5469.2 | 2.260 | 190 | 960 | – | 604 |
| 38 | 2 | 6374.0* | NA | NA | – | – | 4959.1 | 5088.2 | 2.603 | 168 | 960 | – | 459 |
| 39 | 2 | 6191.5* | NA | NA | – | – | 4902.7 | 5105.2 | 4.132 | 151 | 960 | – | 360 |
| 40 | 2 | 4756.6* | NA | NA | – | – | 4355.4 | 4393.5 | 0.875 | 52 | 960 | – | 487 |
*The solutions are not optimal, but the best possible solution found in 16 min
**NA not available