IOE/MFG 543
Chapter 8: Open shops
Section 8.1
(you may skip Sections 8.2 – 8.5)
1
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
2
Minimizing the makespan on
two machines O2||Cmax

The makespan must be at least the total
processing time on each machine
– This gives the lower bound
n
Cmax≥max(
Sp1j , Sp2j )
j=1

n
j=1
The open shop scheduling is flexible so this
bound is typically attained
– In an optimal schedule at most 1 machine idles
3
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

4
LAPT rule example
job j
1
2
3
4
p1j
3
5
4
1
p2j
4
6
2
3
5
Theorem 8.1.1

The LAPT rule yields an optimal
schedule for O2||Cmax with makespan
n
Cmax=max(maxj{1,…,n}(p1j+p2j),
n
Sp1j , Sp2j )
j=1
j=1
6
Minimizing the makespan
on m machines Om||Cmax

Theorem 8.1.2
– The problem O3||Cmax is NP-hard
– PARTITION reduces to O3||Cmax

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
7
Summary of other open
shop models

Om | prmp | Cmax is solvable in polynomial
time
m
n
Cmax=max(maxj{1,…,n}





Spij , maxi{1,…,m} Spij )
i=1
j=1
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
8