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Branch and Bound Example

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Initial lower bound Jrpd 1048 21212 33611 45510 Use 1 machine preemptive schedule as lower bound Job 2 has a lateness of 5, this is a lower bound on Lmax J1 4 J3 5 J4 J3 J2 10 15 17

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Initial upper bound Jrpd 1048 21212 33611 45510 Find some schedule. Lmax is 7. J1 4 6 J4 J3 J2 12 17

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

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Optimal Preemptive Schedule with job 2 first Jrpd 1048 21212 33611 45510 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 J1 3 7 J4 J3 J2 12 18 1

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

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Optimal Preemptive Schedule with job 4 first Jrpd 1048 21212 33611 45510 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 1014 J4 J3 J2 22 520

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

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

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Optimal Preemptive Schedule with job 1 and 2 first Jrpd 1048 21212 33611 45510 Best preemptive schedule with J1, J2 first, has Lmax of 6 and Is non-preemptive J1 411 J4 J3 J2 617

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

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Optimal Preemptive Schedule with job 1 and 3 first Jrpd 1048 21212 33611 45510 Best preemptive schedule with J1, J2 first, has Lmax of 5 and Is non-preemptive J1 415 J4 J3 J2 1017

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

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

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Branch & Bound 34 8 7 12 10 9 3 6 8 2 7 0 0 UPPER = LOWER = 0.

Branch & Bound 34 8 7 12 10 9 3 6 8 2 7 0 0 UPPER = LOWER = 0.

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