Branch and Bound Example Initial lower bound J

Branch and Bound Example

Initial lower bound J r p d 1 0 4 8 2 12 3 3 6 11 4 5 5 10 Use 1 machine preemptive schedule as lower bound Job 2 has a lateness of 5, this is a lower bound on Lmax J 1 J 3 4 J 4 5 J 3 10 J 2 15 17

Initial upper bound J r p d 1 0 4 8 2 12 3 3 6 11 4 5 5 10 Find some schedule. Lmax is 7. J 1 J 3 J 2 4 6 J 4 12 17

Branch on First job *, *, *, * Lower bound 5 Upper bound 7 4, *, *, * 1, *, *, * 2, *, *, * 3, *, *, * Pick one node to explore. Let’s choose the one with 2 first

Optimal Preemptive Schedule with job 2 first J r p d 1 0 4 8 2 12 3 3 6 11 4 5 5 10 Best preemptive schedule with J 2 first has Lmax of 7. The schedule is also non-preemptive, so we have upper and lower Bounds of 7 1 3 J 4 J 1 J 2 7 12 18

Explored node 2 *, *, *, * Lower bound 5 Upper bound 7 4, *, *, * 1, *, *, * 2, *, *, * 3, *, *, * Lower bound 7 Upper bound 7 Pick one node to explore. Let’s choose the one with 4 next

Optimal Preemptive Schedule with job 4 first J r p d 1 0 4 8 2 12 3 3 6 11 4 5 5 10 Best preemptive schedule with J 4 first has Lmax of 9. The schedule is also non-preemptive, so we have upper and lower Bounds of 9 PRUNE J 1 J 4 5 10 J 3 14 J 2 20 22

Explored node 4 *, *, *, * Lower bound 5 Upper bound 7 4, *, *, * 1, *, *, * 2, *, *, * 3, *, *, * Lower bound 7 Upper bound 7 Pick one node to explore. Let’s choose the one with 1 next We already know that the lower bound is 5 and is preemptive. Lower bound 9 Upper bound 9

Exploring node 1 1, *, *, * 1, 2, *, * 1, 3, *, * . 1, 4, *, *

Optimal Preemptive Schedule with job 1 and 2 first J r p d 1 0 4 8 2 12 3 3 6 11 4 5 5 10 Best preemptive schedule with J 1, J 2 first, has Lmax of 6 and Is non-preemptive J 1 J 4 J 2 4 6 J 3 11 17

Exploring node 1, 2 1, *, *, * 1, 2, *, * Lower bound 6 Upper bound 6 1, 3, *, * . let’s try node 1, 3 next 1, 4, *, *

Optimal Preemptive Schedule with job 1 and 3 first J r p d 1 0 4 8 2 12 3 3 6 11 4 5 5 10 Best preemptive schedule with J 1, J 2 first, has Lmax of 5 and Is non-preemptive J 1 4 J 2 J 4 J 3 10 15 17

Exploring node 1, 3 1, *, *, * 1, 2, *, * Lower bound 6 Upper bound 6 1, 3, *, * 1, 4, *, * Lower bound 5 Upper bound 5 We have found a schedule that matches the global lower bound are done!

summary *, *, *, * Lower bound 5 Upper bound 7 4, *, *, * 1, *, *, * 3, *, *, * 2, *, *, * Lower bound 9 Upper bound 9 Lower bound 7 Upper bound 7. 1, 2, *, * Lower bound 6 Upper bound 6 1, 3, *, * Lower bound 5 Upper bound 5 1, 4, *, *