1 / 18

Completion Time Scheduling

Completion Time Scheduling. Notes from Hall, Schulz, Shmoys and Wein, Mathematics of Operations Research, Vol 22, 513-544, 1997. One LP formulation for 1||Σw j C j. Other ways to bound C j. Smith’s rule: Scheduling jobs by w j /p j is guaranteed to be optimal. w j. p j.

paley
Download Presentation

Completion Time Scheduling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Completion Time Scheduling Notes from Hall, Schulz, Shmoys and Wein, Mathematics of Operations Research, Vol 22, 513-544, 1997

  2. One LP formulation for 1||ΣwjCj

  3. Other ways to bound Cj Smith’s rule: Scheduling jobs by wj/pj is guaranteed to be optimal wj pj

  4. Think of all jobs as having wj = pj Smith’s rule: Any order is then equivalent since wj/pj = 1 for all jobs pj pj

  5. 1|prec| ΣwjCj • LP: minimize ΣwjCj • Subject to Ck ≥ Cj + pk (if job j precedes job k) Let C’i denote the LP optimal values Let C*i denote the true optimal values

  6. Algorithm and Analysis • Let C’i denote the LP optimal values • Let C*i denote the true optimal values • Greedy Algorithm: • Solve LP for C’i • Solvable in poly time despite exponential size • Prioritize the jobs by C’i values • Let Gi be the resulting completion times • Key result: Gi ≤ 2C’i ≤ 2C*i • From Lemma 2.1

  7. 1|rj, prec| ΣwjCj • LP: minimize ΣwjCj • Subject to Cj ≥ rj + pj (all jobs) Ck ≥ Cj + pk (if job j precedes job k) Let C’i denote the LP optimal values Let C*i denote the true optimal values

  8. Algorithm • Let C’i denote the LP optimal values • Let C*i denote the true optimal values • Greedy Algorithm: • Solve LP for C’i • Solvable in poly time despite exponential size • Prioritize the jobs by C’i values • Let Gi be the resulting completion times

  9. Analysis • Key result: Gi ≤ 3C’i ≤ 3C*i • Fix j and define S = {1, …, j} • Gj ≤ rmax(S) + p(S) • C’j ≥ rmax(S) • Thus Gj ≤ C’j + p(S) ≤ 3C’j • From Lemma 2.1

  10. Extending to parallel machines

  11. Extending to parallel machines

  12. P|rj| ΣwjCj • LP: minimize ΣwjCj • Subject to Cj ≥ rj + pj Let C’i denote the LP optimal values Let C*i denote the true optimal values

  13. Algorithm • Let C’i denote the LP optimal values • Let C*i denote the true optimal values • Greedy Algorithm: • Solve LP for C’i • Solvable in poly time despite exponential size • Prioritize the jobs by C’i values • Let Gi be the resulting completion times

  14. Analysis • Key result: Gi ≤ (4-1/m)C’i ≤ 4C*i • Fix j and define S = {1, …, j} • Gj ≤ rmax(S) + 1/m p(S – {j}) + pj • = rmax(S) + 1/m p(S) + (1-1/m)pj

  15. Preemption vs Nonpreemption • Method for converting preemptive schedules into non-preemptive schedules • Effective for minimizing Cj objectives • Prioritize jobs by their preemptive completion times Cjp • Generalization: When α of the job is complete • List schedule these jobs nonpreemptively using this priority

  16. 1 machine conversion • Let Cpi denote any preemptive values • Let Cni denote the nonpreemptive values • Cni ≤ 2 Cpi CPj …

  17. 1|rj| ΣCj • With preemption, we have an optimal solution, SRPT • Nonpreemptive online: • Simulate SRPT and when a job is completed in SRPT, start it in the non-preemptive (or add it to the list to start)

More Related