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Branch and Bound Example

Branch and Bound Example. Initial lower bound. 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. 10. 5. 4. 17. 15. Initial upper bound. Find some schedule. Lmax is 7. J3. J4. J1. J2. 12. 6. 4. 17.

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Branch and Bound Example

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  1. Branch and Bound Example

  2. Initial lower bound 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 10 5 4 17 15

  3. Initial upper bound Find some schedule. Lmax is 7. J3 J4 J1 J2 12 6 4 17

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

  5. Optimal Preemptive Schedule with job 2 first 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 J3 J4 J1 J2 3 12 1 7 18

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

  7. Optimal Preemptive Schedule with job 4 first 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 PRUNE J1 J4 J3 J2 5 10 14 20 22

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

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

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

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

  13. Exploring node 1,3 1,*,*,* 1,2,*,* 1,4,*,* 1,3,*,* 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 4,*,*,* 1,*,*,* 3,*,*,* 2,*,*,* Lower bound 9 Upper bound 9 Lower bound 7 Upper bound 7 1,3,*,* 1,2,*,* 1,4,*,* . Lower bound 5 Upper bound 5 Lower bound 6 Upper bound 6

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