Unformatted text preview: .1.8 – The makespan of Fmprmu,pij=pjCmax is Cmax=Σpj+(m1)max(p1,…,pn)
and is independent of the schedule 9 Single machine models and proportionate flow shops Rule/algorithm Single machine Proportionate flow shop SPT rule 1ΣCj Fmpij=pjΣCj Algorithm 3.3.1 1ΣUj Fmprmu,pij=pjΣUj Algorithm 3.2.1 1hmax Fmprmu,pij=pjhmax Algorithm 3.4.4 1ΣTj Fmprmu,pij=pjΣTj Lemma 3.5.1 Fmprmu,pij=pjΣwjTj 1ΣwjTj Note: WSPT is not always optimal for Fmprmu,pij=pjΣwjCj 10 Slope heuristic for Fm
prmuCmax Slope index of job j
m Aj= Σ (m(2i1))pij
(m(2i1))p j=1,…,n i=1 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 6.1.10 11 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 jobj starts processing on machine 1
– Note that Cij≤Dij
12 Computing the makespan of a sequence
Di,j1=
Di,jk= Dm,jk= i Σ pl,j1 l=1 max(Di1,jk+pi,jk, Di+1,jk1)
Dm1,jk+pm,jk i=1,…,m
i=1,…,m1; k=2,…,n
k=2,…,n The makespan of a given sequence can also be computed by a critical path method
The problem F3blockCmax is strongly 13
NPhard Profile fitting (PF) heuristic for FmblockCmax
1.
2. A job j1 is selected to go first
Try all the other jobs as the next job –
–
– 1. 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.
14 Example 6.2.5
job j 1 2 3 4 5 p1j 5 5 3 6 3 p2j 4 4 2 4 4 p3j 4 4 3 4 1 p4j 3 6 3 2 5
15...
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 Spring '10
 King
 flow shops

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