Completion Time Scheduling

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Completion Time Scheduling
Notes from Hall, Schulz, Shmoys and Wein,
Mathematics of Operations Research, Vol 22,
513-544, 1997
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One LP formulation for 1SwjCj
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Other ways to bound Cj
Smiths rule Scheduling jobs by wj/pj is
guaranteed to be optimal
wj
pj
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Think of all jobs as having wj pj
Smiths rule Any order is then equivalent since
wj/pj 1 for all jobs
pj
pj
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(No Transcript)
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1prec SwjCj

• LP minimize SwjCj
• Subject to
• Ck Cj pk (if job j precedes job k)
• Let Ci denote the LP optimal values
• Let Ci denote the true optimal values

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Algorithm and Analysis

• Let Ci denote the LP optimal values
• Let Ci denote the true optimal values
• Greedy Algorithm
• Solve LP for Ci
• Solvable in poly time despite exponential size
• Prioritize the jobs by Ci values
• Let Gi be the resulting completion times
• Key result Gi 2Ci 2Ci
• From Lemma 2.1

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1rj, prec SwjCj

• LP minimize SwjCj
• Subject to
• Cj rj pj (all jobs)
• Ck Cj pk (if job j precedes job k)
• Let Ci denote the LP optimal values
• Let Ci denote the true optimal values

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Algorithm

• Let Ci denote the LP optimal values
• Let Ci denote the true optimal values
• Greedy Algorithm
• Solve LP for Ci
• Solvable in poly time despite exponential size
• Prioritize the jobs by Ci values
• Let Gi be the resulting completion times

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Analysis

• Key result Gi 3Ci 3Ci
• Fix j and define S 1, , j
• Gj rmax(S) p(S)
• Cj rmax(S)
• Thus Gj Cj p(S) 3Cj
• From Lemma 2.1

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Extending to parallel machines
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Extending to parallel machines
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Prj SwjCj

• LP minimize SwjCj
• Subject to
• Cj rj pj
• Let Ci denote the LP optimal values
• Let Ci denote the true optimal values

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Algorithm

• Let Ci denote the LP optimal values
• Let Ci denote the true optimal values
• Greedy Algorithm
• Solve LP for Ci
• Solvable in poly time despite exponential size
• Prioritize the jobs by Ci values
• Let Gi be the resulting completion times

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Analysis

• Key result Gi (4-1/m)Ci 4Ci
• 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

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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 a of the job is complete
• List schedule these jobs nonpreemptively using
  this priority

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1 machine conversion

• Let Cpi denote any preemptive values
• Let Cni denote the nonpreemptive values
• Cni 2 Cpi

CPj

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1rj SCj

• 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)