Research Article
Minimizing the Number of Tardy Jobs on a Single Machine with an Availability Constraint
| Series | Number of jobs1 | Number of optimal instances | Comp. time of BB (s) | Number of optimal states in H | Ave. percent fathomed nodes | BB | H | Min | Ave. | Max | D1 | C4 | Others | D2 | D3 | D4 | LB |
| | 10 | 20 | 20 | | >0 | | 0 | 0 | 2 | 1.09 | >0 | 14.23 | 41.84 | 20 | 20 | 20 | | 0.01 | | 0 | 0 | 9 | 0.35 | >0 | 6.12 | 35.29 | 30 | 20 | 20 | | 5.57 | | 0 | 0 | 2 | >0 | >0 | 7.04 | 44.98 | 40 | 20 | 20 | | 736.26 | | 0 | 0 | 6 | 1.47 | >0 | 4.83 | 37.41 | 50 | 11 | 11 | | >0 | | 0 | 0 | 4 | >0 | >0 | >0 | 63.64 |
|
| 10 | 20 | 19 | >0 | >0 | >0 | 0 | 0 | 0 | 10.20 | >0 | 9.92 | 61.45 | 20 | 20 | 20 | >0 | >0 | 0.02 | 0 | 0 | 12 | 25.21 | >0 | 11.32 | 24.82 | 30 | 20 | 19 | >0 | 0.28 | 5.52 | 4 | 0 | 14 | 17.58 | >0 | 1.18 | 57.42 |
*40 | 20 | 20 | >0 | >0 | >0 | 15 | 0 | 5 | — | — | — | — |
*50 | 20 | 20 | >0 | >0 | >0 | 19 | 0 | 1 | — | — | — | — |
*70 | 20 | 20 | >0 | >0 | 0.02 | 19 | 0 | 1 | — | — | — | — |
*100 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*300 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*500 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*700 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*1000 | 20 | 20 | >0 | >0 | 0.03 | 20 | 0 | 0 | — | — | — | — |
*3000 | 20 | 20 | >0 | 0.01 | 0.06 | 20 | 0 | 0 | — | — | — | — |
*5000 | 20 | 20 | >0 | 0.01 | 0.06 | 20 | 0 | 0 | — | — | — | — |
*7000 | 20 | 20 | >0 | 0.01 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*10000 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
|
| 10 | 20 | 19 | | >0 | | 0 | 0 | 3 | 1.64 | >0 | 1.57 | 67.69 | 20 | 20 | 20 | | >0 | | 0 | 0 | 4 | >0 | >0 | >0 | 80.00 | 30 | 20 | 20 | | 0.18 | | 0 | 0 | 4 | >0 | >0 | 1.16 | 76.61 | 40 | 20 | 20 | | 100.23 | | 0 | 0 | 7 | 1.35 | >0 | 3.16 | 38.99 | 50 | 12 | 12 | | 160.56 | | 0 | 0 | 5 | >0 | >0 | 0.41 | 51.81 |
|
| 10 | 20 | 20 | >0 | >0 | >0 | 0 | 0 | 1 | 1.07 | 7.92 | 3.05 | 72.85 | 20 | 20 | 19 | >0 | >0 | >0 | 0 | 0 | 5 | >0 | 2.50 | 0.24 | 68.69 | 30 | 20 | 20 | >0 | >0 | 0.02 | 0 | 0 | 2 | >0 | >0 | >0 | 90.00 | 40 | 20 | 20 | >0 | >0 | 0.02 | 0 | 0 | 8 | >0 | >0 | >0 | 60.00 | 50 | 20 | 20 | >0 | 0.01 | 0.02 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 70 | 20 | 20 | 0.02 | 0.03 | 0.05 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 100 | 20 | 20 | 0.03 | 0.10 | 0.16 | 0 | 0 | 3 | >0 | >0 | >0 | 85.00 | 300 | 20 | 20 | 0.09 | 5.71 | 8.23 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 500 | 20 | 20 | 29.92 | 49.08 | 79.83 | 0 | 0 | 4 | >0 | >0 | >0 | 80.00 | 700 | 20 | 20 | 130.91 | 198.40 | 301.61 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 1000 | 20 | 20 | 503.92 | 692.73 | 1178.98 | 0 | 0 | 2 | >0 | >0 | >0 | 90.00 |
|
| 10 | 20 | 20 | >0 | >0 | 0.02 | 0 | 0 | 2 | 3.96 | >0 | 2.29 | 74.30 | 20 | 20 | 20 | >0 | 0.02 | 0.11 | 0 | 0 | 5 | 5.82 | >0 | 7.93 | 50.63 | 30 | 20 | 20 | >0 | 10.48 | 47.77 | 0 | 0 | 5 | 9.06 | >0 | 8.49 | 43.20 | 40 | 10 | 9 | >0 | 624.11 | 3815.59 | 0 | 0 | 2 | 8.59 | >0 | 2.23 | 54.43 | 50 | 12 | 12 | >0 | >0 | 0.02 | 0 | 0 | 6 | >0 | >0 | >0 | 50.00 |
|
| 10 | 20 | 19 | >0 | >0 | >0 | 0 | 0 | 1 | 1.98 | >0 | 2.81 | 82.47 | 20 | 20 | 18 | >0 | >0 | 0.02 | 0 | 0 | 8 | 2.60 | >0 | 1.04 | 50.00 | 30 | 20 | 20 | >0 | 0.02 | 0.47 | 2 | 0 | 16 | 13.31 | >0 | 2.64 | 6.17 | 40 | 20 | 13 | >0 | >0 | >0 | 14 | 0 | 5 | >0 | >0 | >0 | 5.00 |
*50 | 20 | 20 | >0 | >0 | >0 | 17 | 0 | 3 | — | — | — | — |
*70 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*100 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*300 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*500 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*700 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*1000 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*3000 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*5000 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*7000 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*10000 | 20 | 20 | >0 | 0.01 | 0.02 | 20 | 0 | 0 | — | — | — | — |
|
| 10 | 20 | 20 | >0 | >0 | >0 | 0 | 2 | 0 | >0 | >0 | >0 | 90.00 | 20 | 20 | 20 | >0 | >0 | >0 | 0 | 8 | 0 | >0 | >0 | >0 | 60.00 | 30 | 20 | 20 | >0 | >0 | >0 | 0 | 7 | 0 | >0 | >0 | >0 | 65.00 | 40 | 20 | 20 | >0 | >0 | 0.02 | 0 | 7 | 0 | >0 | >0 | >0 | 65.00 | 50 | 20 | 20 | >0 | >0 | 0.02 | 0 | 6 | 0 | >0 | >0 | >0 | 70.00 | 70 | 20 | 20 | >0 | 0.01 | 0.06 | 0 | 7 | 0 | >0 | >0 | >0 | 65.00 | 100 | 20 | 20 | >0 | 0.05 | 0.13 | 0 | 4 | 0 | >0 | >0 | >0 | 80.00 | 300 | 20 | 20 | >0 | 2.54 | 6.67 | 0 | 6 | 0 | >0 | >0 | >0 | 70.00 | 500 | 20 | 20 | >0 | 15.26 | 55.81 | 0 | 9 | 0 | >0 | >0 | >0 | 55.00 | 700 | 20 | 20 | >0 | 96.49 | 197.95 | 0 | 5 | 0 | >0 | >0 | >0 | 75.00 | 1000 | 20 | 20 | 0.02 | 135.76 | 742.70 | 0 | 11 | 0 | >0 | >0 | >0 | 45.00 |
|
| 10 | 20 | 20 | >0 | >0 | >0 | 0 | 0 | 0 | >0 | 5.99 | 1.17 | 87.05 | 20 | 20 | 18 | >0 | >0 | >0 | 0 | 0 | 2 | >0 | 0.29 | 0.90 | 83.31 | 30 | 20 | 18 | >0 | >0 | 0.02 | 0 | 0 | 2 | >0 | 0.96 | 1.60 | 73.71 | 40 | 20 | 20 | >0 | >0 | 0.02 | 0 | 0 | 2 | >0 | >0 | >0 | 90.00 | 50 | 20 | 19 | >0 | 0.01 | 0.06 | 0 | 0 | 3 | >0 | 0.11 | 0.36 | 78.18 | 70 | 20 | 20 | >0 | 0.01 | 0.03 | 0 | 0 | 3 | >0 | >0 | >0 | 85.00 | 100 | 20 | 20 | 0.03 | 0.06 | 0.13 | 0 | 0 | 3 | >0 | >0 | >0 | 85.00 | 300 | 20 | 20 | 0.02 | 3.75 | 7.00 | 0 | 0 | 7 | >0 | >0 | >0 | 65.00 | 500 | 20 | 20 | >0 | 26.79 | 55.25 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 700 | 20 | 20 | 60.48 | 124.32 | 212.69 | 0 | 0 | 5 | >0 | >0 | >0 | 75.00 | 1000 | 20 | 20 | 0.94 | 514.41 | 871.05 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 |
|
| 10 | 20 | 20 | >0 | >0 | >0 | 0 | 0 | 0 | >0 | >0 | >0 | 100.00 | 20 | 20 | 17 | >0 | >0 | >0 | 0 | 0 | 3 | 0.87 | >0 | 3.21 | 70.62 | 30 | 20 | 19 | >0 | >0 | >0 | 0 | 0 | 4 | 1.36 | >0 | 0.07 | 76.25 | 40 | 20 | 20 | >0 | >0 | >0 | 0 | 0 | 9 | >0 | >0 | >0 | 55.00 | 50 | 20 | 20 | >0 | >0 | 0.02 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 70 | 20 | 20 | >0 | 0.01 | 0.03 | 0 | 0 | 5 | >0 | >0 | >0 | 75.00 | 100 | 20 | 20 | >0 | 0.02 | 0.05 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 300 | 20 | 20 | 0.02 | 1.26 | 2.81 | 0 | 0 | 10 | >0 | >0 | >0 | 50.00 | 500 | 20 | 20 | 0.02 | 7.04 | 16.78 | 0 | 0 | 10 | >0 | >0 | >0 | 50.00 | 700 | 20 | 20 | 7.41 | 37.80 | 72.25 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 | 1000 | 20 | 20 | 0.02 | 149.77 | 290.13 | 0 | 0 | 6 | >0 | >0 | >0 | 70.00 |
|
| 10 | 20 | 19 | >0 | >0 | >0 | 0 | 0 | 2 | 0.99 | >0 | 1.03 | 90.50 | 20 | 20 | 18 | >0 | >0 | >0 | 0 | 7 | 4 | 3.06 | >0 | 0.95 | 79.33 | 30 | 20 | 19 | >0 | >0 | >0 | 3 | 14 | 2 | 10.20 | >0 | 8.16 | 16.33 | 40 | 20 | 19 | >0 | >0 | >0 | 11 | 7 | 1 | >0 | >0 | >0 | 100.00 |
*50 | 20 | 20 | >0 | >0 | >0 | 18 | 2 | 0 | — | — | — | — |
*70 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*100 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*300 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*500 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*700 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*1000 | 20 | 20 | >0 | >0 | >0 | 20 | 0 | 0 | — | — | — | — |
*3000 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*5000 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*7000 | 20 | 20 | >0 | >0 | 0.02 | 20 | 0 | 0 | — | — | — | — |
*10000 | 20 | 20 | >0 | 0.01 | 0.02 | 20 | 0 | 0 | — | — | — | — |
|
|
*10 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*20 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*30 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*40 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*50 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*70 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*100 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*300 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*500 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*700 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*1000 | 20 | 20 | 0.02 | 0.02 | 0.03 | 0 | 20 | 0 | — | — | — | — |
*3000 | 20 | 20 | 0.14 | 0.16 | 0.22 | 0 | 20 | 0 | — | — | — | — |
*5000 | 20 | 20 | 0.44 | 0.45 | 0.50 | 0 | 20 | 0 | — | — | — | — |
*7000 | 20 | 20 | 0.86 | 0.88 | 0.92 | 0 | 20 | 0 | — | — | — | — |
*10000 | 20 | 20 | 1.78 | 1.84 | 2.03 | 0 | 20 | 0 | — | — | — | — |
|
| 10 | 20 | 20 | >0 | >0 | >0 | 0 | 15 | 0 | >0 | >0 | >0 | 25.00 | 20 | 20 | 20 | >0 | >0 | >0 | 0 | 18 | 0 | >0 | >0 | >0 | 10.00 |
*30 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*40 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*50 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*70 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*100 | 20 | 20 | >0 | >0 | >0 | 0 | 20 | 0 | — | — | — | — |
*300 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*500 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*700 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*1000 | 20 | 20 | >0 | >0 | 0.02 | 0 | 20 | 0 | — | — | — | — |
*3000 | 20 | 20 | 0.11 | >0 | 0.16 | 0 | 20 | 0 | — | — | — | — |
*5000 | 20 | 20 | 0.34 | >0 | 0.38 | 0 | 20 | 0 | — | — | — | — |
*7000 | 20 | 20 | 0.70 | >0 | 0.72 | 0 | 20 | 0 | — | — | — | — |
*10000 | 20 | 20 | 1.45 | 0.01 | 1.50 | 0 | 20 | 0 | — | — | — | — |
|
|
Not reported instances could not be solved by BB approach in 4000 seconds.
|