Initial lower bound J r p d 1 0 4 8 2 1 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 J1 4 ID: 177181
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Slide1
Branch and Bound ExampleSlide2
Initial lower bound
J
r
pd1048212123361145510
Use 1 machine preemptive schedule as lower boundJob 2 has a lateness of 5, this is a lower bound on Lmax
J1
4
J3
5
J4
J3
J2
10
15
17Slide3
Initial upper bound
J
r
pd1048212123361145510
Find some schedule.Lmax is 7.
J1
4
6
J4
J3
J2
12
17Slide4
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 firstSlide5
Optimal Preemptive Schedule with job 2 first
J
r
pd1048212123361145510
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
1Slide6
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 7Slide7
Optimal Preemptive Schedule with job 4 first
J
r
pd1048212123361145510
Best preemptive schedule with J4 first has Lmax of 9.The schedule is also non-preemptive, so we have upper and lower
Bounds of 9PRUNE
J1
10
14
J4
J3
J2
22
5
20Slide8
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 9Upper bound 9
Lower bound 7Upper bound 7Slide9
Exploring node 1
1
,*,*,*
1,4,*,*1,3,*,*1,2,*,*.Slide10
Optimal Preemptive Schedule with job 1 and 2 first
J
r
pd1048212123361145510
Best preemptive schedule with J1, J2 first, has Lmax of 6 and
Is non-preemptive
J1
4
11
J4
J3
J2
6
17Slide11
Exploring node 1,2
1
,*,*,*
1,4,*,*1,3,*,*1,2,*,*.let’s try node 1,3 next
Lower bound 6Upper bound 6Slide12
Optimal Preemptive Schedule with job 1 and 3 first
J
r
pd1048212123361145510
Best preemptive schedule with J1, J2 first, has Lmax of 5 and
Is non-preemptive
J1
4
15
J4
J3
J2
10
17Slide13
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 6Upper bound 6
Lower bound 5
Upper bound 5Slide14
summary
*,*,*,*
3,*,*,*
4,*,*,*2,*,*,*1,*,*,*
Lower bound 5
Upper bound 7
.
Lower bound 9
Upper bound 9Lower bound 7Upper bound 7
1,2,*,*
1,3,*,*
1,4,*,*
Lower bound 6
Upper bound 6
Lower bound 5
Upper
bound 5