Research Article
Improved Combinatorial Benders Decomposition for a Scheduling Problem with Unrelated Parallel Machines
Table 1
Comparison of T&B and ICBD methods using the instances from Tran and Beck (2012).
| | | LB | T&B | ICBD | % gap | Time | # iter | # cut | Time | # iter | # cut | Time |
| 10 | 2 | 0.23 | 0.05 | 1.40 | 2.80 | 0.10 | 1.40 | 2.80 | 0.11 | 3 | 0.14 | 0.11 | 1.20 | 3.60 | 0.16 | 1.20 | 3.60 | 0.17 | 4 | 0.44 | 0.23 | 1.10 | 4.40 | 0.30 | 1.10 | 4.40 | 0.29 | 5 | 0.40 | 0.23 | 1.20 | 6.00 | 0.34 | 1.20 | 6.50 | 0.33 |
| 20 | 2 | 0.11 | 0.18 | 1.80 | 3.60 | 0.45 | 1.70 | 4.00 | 0.42 | 3 | 0.34 | 0.40 | 2.10 | 6.30 | 0.90 | 2.00 | 8.70 | 0.86 | 4 | 0.18 | 1.02 | 1.50 | 6.00 | 1.55 | 1.50 | 6.00 | 1.58 | 5 | 0.31 | 1.85 | 1.50 | 7.50 | 2.79 | 1.40 | 7.00 | 2.59 |
| 30 | 2 | 0.07 | 0.33 | 1.70 | 3.40 | 0.71 | 1.60 | 3.40 | 0.66 | 3 | 0.17 | 0.94 | 1.80 | 5.40 | 1.70 | 1.70 | 6.60 | 1.63 | 4 | 0.27 | 2.56 | 1.90 | 7.60 | 7.01 | 1.90 | 10.80 | 6.87 | 5 | 0.17 | 18.19 | 1.80 | 9.00 | 99.49 | 1.70 | 10.00 | 68.15 |
| 40 | 2 | 0.06 | 0.72 | 1.90 | 3.80 | 2.02 | 1.90 | 4.00 | 1.94 | 3 | 0.05 | 1.57 | 1.60 | 4.80 | 2.77 | 1.50 | 5.40 | 2.73 | 4 | 0.17 | 11.73 | 1.80 | 7.20 | 18.77 | 1.70 | 8.00 | 18.66 | 5 | 0.19 | 41.14 | 1.90 | 9.50 | 102.10 | 1.70 | 12.00 | 99.24 |
| 50 | 2 | 0.03 | 0.69 | 1.80 | 3.60 | 2.23 | 1.70 | 4.00 | 2.14 | 3 | 0.07 | 3.37 | 1.50 | 4.50 | 6.42 | 1.50 | 4.80 | 5.24 | 4 | 0.12 | 28.65 | 1.70 | 6.80 | 44.32 | 1.70 | 9.20 | 40.59 | 5 | 0.09 | 133.74 | 1.70 | 8.50 | 267.45 | 1.70 | 12.00 | 212.99 |
| 60 | 2 | 0.04 | 1.44 | 1.80 | 3.60 | 3.89 | 1.70 | 3.80 | 3.79 | 3 | 0.05 | 5.66 | 1.80 | 5.40 | 8.89 | 1.70 | 5.70 | 8.58 | 4 | 0.05 | 64.36 | 1.70 | 6.80 | 111.89 | 1.60 | 8.00 | 106.71 | 5 | 0.13 | 303.25 | 2.30 | 11.50 | 878.30 | 2.10 | 15.00 | 697.76 |
| Average | 0.16 | 25.93 | 1.69 | 5.90 | 65.19 | 1.62 | 6.90 | 53.50 |
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