1. Chapter 6: Flow shops Sections 6.1 and 6.2 (skip section 6.3)
IOE/MFG 543
Chapter 6: Flow shops
Sections 6.1 and 6.2
(skip section 6.3)

2. Flow shop (Fm) m machines, n jobs
Jobs are processed on the machines in series
Processing time of job j on machine i is pi,j
Buffers between machines
Unlimited
Limited => blocking

3. Section 6.1. Unlimited storage - Permutation rule
All jobs are processed in the same order on the machines
Equivalent to a FCFS rule
For F2||Cmax and F3||Cmax there exists a permutation schedule that is optimal
It is much harder to minimize the makespan when the sequencing is not restricted to the permutation rule

4. Computing the makespan for a given permutation
Let j1,…,jn be a given permutation
i.e., job jk is the kth job on all the machines
Ci,j=completion time of job j on machine i i
Ci,j1=
S pl,j1
i=1,…,m
l=1 k
C1,jk=
S p1,jl
k=1,…,n
Ci,jk=
max(Ci-1,jk, Ci,jk-1)+pi,jk
i=2,…,m; k=2,…,n

5. Computing the makespan for a given permutation (2)
Instead of solving the recursive equations on the previous slide the makespan can be computed by a critical path method
Example 6.1.1
job j 1 2 3 4 5
p1j 6
p2j
p3j
p4j

6. Johnson’s rule for F2||Cmax
Set I: All jobs such that p1j<p2j
Set II: All jobs such that p1j>p2j
Jobs with p1j=p2j can be put in either set
SPT(1) – LPT(2) schedule (Johnson’s rule):
Jobs in Set I go first and in an increasing (non-decreasing) order of p1j => SPT(1)
Jobs in Set II go last and in a decreasing (non-increasing) order of p2j => LPT(2)
Theorem 6.1.4
Any SPT(1)-LPT(2) schedule is optimal for F2||Cmax

7. Fm|prmu|Cmax Theorem 6.1.7 F3|prmu|Cmax is strongly NP-hard
3-Partition reduces to F3|prmu|Cmax

8. Mixed integer programming formulation of Fm|prmu|Cmax
Notation
xjk=1 if job j is the kth job in the sequence and 0 otherwise
Iik is the idle time on machine i between jobs in the kth and (k+1)th position
Wik is the waiting time after it has finished on the ith machine of the job in the kth position
Dik is the difference between the time when the job in the (k+1)th position starts on machine i+1 and the time the job in the kth position finishes on machine i
pi(k) is the processing time on machine i of the job in the kth position

9. Proportionate flow shops
The processing time (work) for job j is pij=pj
Theorem 6.1.8
The makespan of Fm|prmu,pij=pj|Cmax is
Cmax=Spj+(m-1)max(p1,…,pn)
and is independent of the schedule

10. Single machine models and proportionate flow shops
Rule/algorithm
Single machine
Proportionate flow shop
SPT rule
1||SCj
Fm|pij=pj|SCj
Algorithm 3.3.1
1||SUj
Fm|prmu,pij=pj|SUj
Algorithm 3.2.1
1||hmax
Fm|prmu,pij=pj|hmax
Algorithm 3.4.4
1||STj
Fm|prmu,pij=pj|STj
Lemma 3.5.1
1||SwjTj
Fm|prmu,pij=pj|SwjTj
Note: WSPT is not always optimal
for Fm|prmu,pij=pj|SwjCj

11. Slope heuristic for Fm|prmu|Cmax
Slope index of job j
The slope index is large if the processing times on the downstream machines are large relative to the processing times on the upstream machines
Heuristic rule
Sequence jobs in decreasing order of the slope index
Example m
Aj= -
S (m-(2i-1))pij
j=1,…,n
i=1

12. Section 6.2 Limited storage flow shops
Only need to consider the case where the storage between machines is zero
New notation
Dij is the time when job j departs machine i
D0j is the time when job j starts processing on machine 1
Note that Cij≤Dij

13. Computing the makespan of a sequence
Di,j1=
S pl,j1
i=1,…,m
l=1
Di,jk=
max(Di-1,jk+pi,jk, Di+1,jk-1)
i=1,…,m-1; k=2,…,n
Dm,jk=
Dm-1,jk+pm,jk
k=2,…,n
The makespan of a given sequence can also be computed by a critical path method
The problem F3|block|Cmax is strongly NP-hard

14. Profile fitting (PF) heuristic for Fm|block|Cmax
A job j1 is selected to go first
Try all the other jobs as the next job
Use the equations on the previous slide to compute the departure times
Compute a penalty as the sum of idle times and blocked times on all machines
Choose the job with the lowest penalty to go next
If all jobs have been scheduled=> STOP Otherwise go to Step 2.

15. Example 6.2.5
job j 1 2 3 4 5
p1j 6
p2j
p3j
p4j