1. Branch and Bound Example

2. Initial lower bound J r p d 1 4 8 2 12 3 6 11 5 104 8 2 12 3 6 11 5 10
Use 1 machine preemptive schedule as lower bound
Job 2 has a lateness of 5, this is a lower bound on Lmax J1 J3 J4 J3 J2 4 5 10 15 17

3. Initial upper bound J r p d 1 4 8 2 12 3 6 11 5 10 Find some schedule.4 8 2 12 3 6 11 5 10
Find some schedule.
Lmax is 7. J1 J2 J3 J4 4 6 12 17

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

5. Optimal Preemptive Schedule with job 2 first1 4 8 2 12 3 6 11 5 10
Best preemptive schedule with J2 first has Lmax of 7.
The schedule is also non-preemptive, so we have upper and lower
Bounds of 7 J2 J1 J4 J3 1 3 7 12 18

6. Explored node 2 Lower bound 5 *,*,*,* Upper bound 7 1,*,*,* 4,*,*,*
2,*,*,*
3,*,*,*
Lower bound 7
Upper bound 7
No need to explore this node further
Pick another node to explore. Let’s choose the one with 4 next

7. Optimal Preemptive Schedule with job 4 first1 4 8 2 12 3 6 11 5 10
Best preemptive schedule with J4 first has Lmax of 9.
The schedule is also non-preemptive, so we have upper and lower
Bounds of 9. The lower bound of 9 means that we should
PRUNE J4 J1 J3 J2 5 10 14 20 22

8. Explored node 4 Lower bound 5 *,*,*,* Upper bound 7 1,*,*,* 4,*,*,*
2,*,*,*
3,*,*,*
Lower bound 9
Upper bound 9
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.

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

10. Optimal Preemptive Schedule with job 1 and 2 first4 8 2 12 3 6 11 5 10
Best preemptive schedule with J1, J2 first, has Lmax of 6 and
Is non-preemptive J1 J2 J4 J3 4 6 11 17

11. Exploring node 1,2 1,*,*,* 1,2,*,* 1,3,*,* 1,4,*,* Lower bound 6
Upper bound 6
No need to explore further
let’s try node 1,3 next

12. Optimal Preemptive Schedule with job 1 and 3 first4 8 2 12 3 6 11 5 10
Best preemptive schedule with J1, J3 first, has Lmax of 5 and
Is non-preemptive J2 J1 J3 J4 4 10 15 17

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

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