IOEMFG 543

1
IOE/MFG 543

• Chapter 8 Open shops
• Section 8.1 (you may skip Sections 8.2 8.5)

2
Open shop (Om)

• m machines, n jobs
• The routing of each job is up to the scheduler
  (i.e., it is open)
• Nondelay schedules
• If there is a job waiting for processing when a
  machine is free, then that machine is not allowed
  to remain idle
• See also Definition 2.3.1 on page 22
• Here we only consider nondelay schedules

3
Minimizing the makespan on two machines O2Cmax

• The makespan must be at least the total
  processing time on each machine
• This gives the lower bound
• The open shop scheduling is flexible so this
  bound is typically attained
• In an optimal schedule at most 1 machine idles

4
LAPT rule

• Whenever a machine is freed, start processing
  among the jobs that have not yet received
  processing on either machine the job with the
  longest processing time on the other machine
• Longest Alternate Processing Time first
• If a job has the longest processing times on both
  machines and if both machines are freed at the
  same time it does not matter on which machine the
  job is processed first

5
LAPT rule example
6
Theorem 8.1.1

• The LAPT rule yields an optimal schedule for
  O2Cmax with makespan

7
Minimizing the makespan on m machines OmCmax

• Theorem 8.1.2
• The problem O3Cmax is NP-hard
• PARTITION reduces to O3Cmax
• The LTRP-OM is a reasonable heuristic
• Whenever a machine is freed process the job that
  has the highest total remaining processing time
  on other machines is put on the machine
• Longest Total Remaining Processing on Other
  Machines first

8
Summary of other open shop models

• Om prmp Cmax is solvable in polynomial time
• O2 Lmax is strongly NP-hard
• Om rj , prmp Lmax is solvable in polynomial
  time
• O2 prmp SUj is NP-hard (not in the text)
• O2 SCj is strongly NP-hard
• O3 prmp SCj is strongly NP-hard