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

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

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

4. 4 Computing the makespan for a given permutation Let j 1,…,j n be a given permutation Let j 1,…,j n be a given permutation –i.e., job j k is the kth job on all the machines C i,j =completion time of job j on machine i C i,j =completion time of job j on machine i i C i,j 1 =  p l,j 1 i=1,…,m l=1 k C 1,j k =  p 1,j l k=1,…,n l=1 C i,j k = max( C i-1,j k, C i,j k-1 ) +p i,j k i=2,…,m; k=2,…,n

5. 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 Instead of solving the recursive equations on the previous slide the makespan can be computed by a critical path method Example 6.1.1 Example 6.1.1 job j 12345 p 1j 55363 p 2j 44244 p 3j 44341 p 4j 36325

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

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

8. 8 Mixed integer programming formulation of Fm|prmu|C max Notation Notation –x jk =1 if job j is the kth job in the sequence and 0 otherwise –I ik is the idle time on machine i between jobs in the kth and (k+1)th position –W ik is the waiting time after it has finished on the ith machine of the job in the kth position –  ik 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 –p i(k) is the processing time on machine i of the job in the kth position

9. 9 Proportionate flow shops The processing time (work) for job j is p ij =p j The processing time (work) for job j is p ij =p j Theorem 6.1.8 Theorem 6.1.8 –The makespan of Fm|prmu,p ij =pj|C max is C max =  p j +(m-1)max(p 1,…,p n ) and is independent of the schedule

10. 10 Single machine models and proportionate flow shops Rule/algorithm Single machine Proportionate flow shop SPT rule 1||  C j Fm|p ij =p j |  C j Algorithm 3.3.1 1||  U j Fm|prmu,p ij =p j |  U j Algorithm 3.2.1 1||h max Fm|prmu,p ij =p j |h max Algorithm 3.4.4 1||  T j Fm|prmu,p ij =p j |  T j Lemma 3.5.1 1||  w j T j Fm|prmu,p ij =p j |  w j T j Note: WSPT is not always optimal for Fm|prmu,p ij =p j |  w j C j

11. 11 Slope heuristic for Fm|prmu|C max Slope index of job j 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 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 Heuristic rule –Sequence jobs in decreasing order of the slope index Example 6.1.10 Example 6.1.10 m A j = -  (m-(2i-1))p ij j=1,…,n i=1

12. 12 Section 6.2 Limited storage flow shops Only need to consider the case where the storage between machines is zero Only need to consider the case where the storage between machines is zero New notation New notation –D ij is the time when job j departs machine i –D 0j is the time when job j starts processing on machine 1 –Note that C ij ≤D ij

13. 13 Computing the makespan of a sequence i D i,j 1 =  p l,j 1 i=1,…,m l=1 D i,j k = max( D i-1,j k +p i,j k, D i+1,j k-1 ) i=1,…,m-1; k=2,…,n D m,j k = D m-1,j k +p m,j k k=2,…,n The makespan of a given sequence can also be computed by a critical path method The makespan of a given sequence can also be computed by a critical path method The problem F3|block|C max is strongly NP-hard The problem F3|block|C max is strongly NP-hard

14. 14 Profile fitting (PF) heuristic for Fm|block|C max 1. A job j 1 is selected to go first 2. 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 3. If all jobs have been scheduled=> STOP Otherwise go to Step 2.

15. 15 Example 6.2.5 job j 12345 p 1j 55363 p 2j 44244 p 3j 44341 p 4j 36325