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Find Lower Bound

Find Lower Bound. When jobs without arrival time , the idle time is null. the lower bound we can get is good. When jobs with arrival time , the idle time will be extended. the lower bound we can get is bad Symbol ：. : the i -th job in the scheduled set. Branch & Bound.

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Find Lower Bound

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  1. Find Lower Bound • When jobs without arrival time, the idle time is null. the lower bound we can get is good. • When jobs with arrival time, the idle time will be extended. the lower bound we can get is bad Symbol： : the i-th job in the scheduled set

  2. Branch & Bound xxxx 4 jobs • Complex Problem Branching strategy Separate the solution space + Bounding 2xxx 3xxx 1xxx 4xxx

  3. Best First Method Transform dynamic into static Estimate the Lower Bound (except the first job) Y(i)+P(i) feasible sol. optimum infeasible sol. LB C(n) h(n) (take unscheduled jobs as static ones, use WSPT to arrange these jobs.) unscheduled scheduled

  4. Branch and Bound with Best First Method min 1xxxx 2xxxx 3xxxx 4xxxx 5xxxx > If Z(25)> Z(3) > 21xxx 23xxx 24xxx 25xxx min If the Z in the current layer is larger than the other Z in the previous layer, go back to the previous layer and B&B else . . . Optimal !!

  5. EX. For 4 5 9 17 27 39 48 0 1 9 19 31 40

  6. EX. 1xxxx 2xxxx 3xxxx 4xxxx 5xxxx by p/w 1-2-4-5-3 2-1-4-5-3 3-1-2-4-5 4-1-2-5-3 5-1-2-4-3 min Z=67.1 Z=83.7 Z=71.9 Z=68 Z=94.7 > > Next page

  7. EX. 1xxxx 2xxxx 3xxxx 4xxxx 5xxxx Z=67.1 Z=83.7 Z=71.9 Z=68 Z=94.7 12xxx 13xxx 14xxx 15xxx Z=76.7 Z=77 Z=67.5 Z=85.1 142xx 143xx 145xx Z=67.5 Z=74.1 Z=68.3 1423x 1425x 1-4-2-5-3 is the optimum Z=71.1 Z=67.5

  8. I index • Tradition：arrange one job at a time • I index： arrange two jobs at a time Idle Time Unscheduled

  9. Ex. Stage 1. Find the first job j=2 j=3 j=4 j=5 . . . Homework. Complete the compute above then find the first & the second job.

  10. HW. 4 jobs to be scheduled with following attributes. • Formulate this problem. Using linear programming. • Use I index Heuristic. • Use B&B with Best First Lower Bound approach. (Programming will be on plus)

  11. Discrete overtime 1-2-3 3-1-2 2-3-1 1-3-2 3-2-1 2-1-3 Dynamic Programming Baker p.45 1-3-2 3-1-2 1-2-3 3-2-1 2-1-3 2-3-1

  12. Tardiness Function See p.45 Machine starting time Objective function: min cost In general: Recursive Function where Residual set

  13. Ex. Backward 10-2 10-7 10-5 10-6

  14. Ex. 7+1-2 7+2-7 G({1})

  15. Ex. So 1-3-2-4

  16. HW. • A company needs to complete 4 orders within the limited time frame. The delivery time and manufacturing time are listed as follows: Please find a minimum mean T schedule using backward DP for comparison.

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