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Learn about flow shops, permutation rules, Johnson's rule, makespan computation, critical path method, and heuristic algorithms for efficient scheduling. Explore limited storage flow shops and the PF heuristic for block scheduling.
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IOE/MFG 543 Chapter 6: Flow shops Sections 6.1 and 6.2 (skip section 6.3)
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
Section 6.1. Unlimited storage - Permutation rule • 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
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
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
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
Fm|prmu|Cmax • Theorem 6.1.7 • F3|prmu|Cmax is strongly NP-hard • 3-Partition reduces to F3|prmu|Cmax
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
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
Single machine models and proportionate flow shops Note: WSPT is not always optimal for Fm|prmu,pij=pj|SwjCj
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 6.1.10
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
Computing the makespan of a sequence • The makespan of a given sequence can also be computed by a critical path method • The problem F3|block|Cmax is strongly NP-hard
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=> STOPOtherwise go to Step 2.
