# 44 1tj fmprmupijpjtj lemma351 fmprmupijpjwjtj 1wjtj

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Unformatted text preview: .1.8 – The makespan of Fm|prmu,pij=pj|Cmax 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 1||ΣCj Fm|pij=pj|ΣCj Algorithm 3.3.1 1||ΣUj Fm|prmu,pij=pj|ΣUj Algorithm 3.2.1 1||hmax Fm|prmu,pij=pj|hmax Algorithm 3.4.4 1||ΣTj Fm|prmu,pij=pj|ΣTj Lemma 3.5.1 Fm|prmu,pij=pj|ΣwjTj 1||ΣwjTj Note: WSPT is not always optimal for Fm|prmu,pij=pj|ΣwjCj 10 Slope heuristic for Fm| prmu|Cmax 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 F3|block|Cmax is strongly 13 NP­hard Profile fitting (PF) heuristic for Fm|block|Cmax 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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