IOEMFG 543

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IOE/MFG 543

• Chapter 5 Parallel machine models
• (Sections 5.1-5.2)

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Two-step process

• Determine which jobs have to be allocated to
  which machines
• Determine the sequence of the jobs allocated to
  each machine
• In some cases use single machine models

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Makespan Cmax

• Single machine models
• Sum of processing times of all jobs
• Exception Release dates or sequence dependent
  setup times
• Parallel machine models
• Minimizing the makespan ensures a good load
  balance on the machines

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Section 5.1 Makespan without preemptions PmCmax

• PmCmax is NP-hard gt Heuristics
• Easy to compute lower bound
• Cmax Cmax maxpmax, (Spj)/m
• where pmaxmaxpj
• Longest Processing Time first (LPT)
• Largest m jobs are put on the machines first
• When a machine is freed, the longest job among
  the jobs not yet processed is put on the machine

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Worst case bound for the LPT rule

• Let
• Cmax(LPT) be the makespan under the LPT rule
• Cmax(OPT) be the optimal makespan
• Theorem 5.1.1

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LPT Example

• 4 machines and 9 jobs
• Compute Cmax(LPT)
• Pair the jobs 1-5, 2-6, 4-5, 7-8-9
• What is Cmax?
• Is this Cmax(OPT)? Why?
• What is Cmax(LPT)/Cmax(OPT)

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Precedence constraints PmprecCmax

• At least as hard as PmCmax
• Strongly NP-hard in general
• Special case P8precCmax
• unlimited resources (or mn, more resources than
  jobs)
• For P8precCmax the Critical Path Method (CPM)
  minimizes the makespan

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Algorithm 5.1.3Critical path method

• Start a job as soon as all its preceding jobs
  have been completed
• Example 5.1.4

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Precedence constraints PmprecCmax (2)

• A critical path rule is optimal for Pmpj1,
  treeCmax (Theorem 5.1.5)
• Largest Number of Successors rule often used for
  Pmpj1, precCmax
• Not optimal (see pg. 102)
• Largest total amount of processing rule for
  PmprecCmax
• Not optimal

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PmMjCmax

• For interest See pages 103-104

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Section 5.2 Makespan with preemptions PmprmpCmax

• Preemptions are useful when there are multiple
  machines even if there are no release dates
• Assumption A job cannot be processed on 2
  machines at the same time

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Linear program for PmprmpCmax
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Linear program for PmprmpCmax (2)

• xij is the amount of processing of job j on
  machine i
• Cmax is a decision variable
• The solution of the LP does not give a feasible
  schedule, although a feasible schedule can be
  constructed given the solution

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Algorithm 5.2.3 for PmprmpCmax

• Create a single machine schedule of the n jobs in
  any order without preemptions
• Compute Cmax maxpmax, (Spj)/m.
• Cut the single machine schedule in m intervals
  I10, Cmax, I2 Cmax, 2Cmax, ,
  Im(m-1)Cmax,mCmax
• Take the schedule for machine i of the parallel
  machines to be the processing sequence in
  interval Ii

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PmprmpCmax example

• Use Algorithm 5.2.3 to determine a schedule that
  minimizes the makespan of the following jobs on 3
  machines

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Longest Remaining Processing Time rule

• Analogous to the LPT rule
• LRPT minimizes the makespan
• What happens while a job is processed?
• RPT decreases gt switch to a new job
• Results in infinitely many switchovers!
• LRPT-FM (FMfastest machine) is optimal for
  QmprmpCmax