Flow Shop Scheduling
Contents
1. Problem Statement
2. Johnson's rule
Literature
Heuristic Scheduling Systems with Applications to Production
Systems and Project Management,
Thomas Morton and David Pentico,
A Volume in the Wiley Series in Engineering and Technology
Management, (Ed.) D.Kocaoglu, 1993, Chapter 13.2.2
1
Problem Statement
F2 | | Cmax
Flow shop environment:
• 2 machines, n jobs
• objective function: makespan
• arrival times of jobs rj = 0
 Johnson's rule gives an optimal schedule.
2
Algorithm
Step 1. Schedule the group of jobs U that are shorter on the
first machine than the second.
U = { j | p1j < p2j }
Step 2. Schedule the group of jobs V that are shorter on the
second machine than the first.
V = { j | p1j  p2j }
Step 3. Arrange jobs in U in non-decreasing order by their
processing times on the first machine.
Step 4. Arrange jobs in V in non-increasing order by their
processing times on the second machine.
Step 5. Concatenate U and V and that is the processing order
for both machines.
3
idle time
...
first machine
...
second machine
idle time
4
Example.
jobs 1
p1j 5
p2j 2
2
2
6
3
1
2
4
7
5
5
6
6
6
3
7
7
7
2
8
5
1
5
6
6
12
22
4
7
5
19
27
7
7
2
26
29
1
5
2
31
33
8
5
1
36
37
U = {2, 3, 6}
V = {1, 4, 5, 7, 8}
jobs
p1j
p2j
C1j
C2j
3
1
2
1
3
2
2
6
3
9
Cmax = 37
6
3
7
6
16
5