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1 IOE/MFG 543 Chapter 8: Open shops Section 8.1 (you may skip Sections 8.2 – 8.5)

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2 Open shop (Om) m machines, n jobs m machines, n jobs The routing of each job is up to the scheduler (i.e., it is open) The routing of each job is up to the scheduler (i.e., it is open) Nondelay schedules Nondelay schedules –If there is a job waiting for processing when a machine is free, then that machine is not allowed to remain idle –See also Definition 2.3.1 on page 22 –Here we only consider nondelay schedules

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3 Minimizing the makespan on two machines O2||C max The makespan must be at least the total processing time on each machine The makespan must be at least the total processing time on each machine –This gives the lower bound The open shop scheduling is flexible so this bound is typically attained The open shop scheduling is flexible so this bound is typically attained –In an optimal schedule at most 1 machine idles C max ≥max ( n n ) p 1j, p 2j j=1j=1

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4 LAPT rule Whenever a machine is freed, start processing among the jobs that have not yet received processing on either machine the job with the longest processing time on the other machine Whenever a machine is freed, start processing among the jobs that have not yet received processing on either machine the job with the longest processing time on the other machine => Longest Alternate Processing Time first If a job has the longest processing times on both machines and if both machines are freed at the same time it does not matter on which machine the job is processed first If a job has the longest processing times on both machines and if both machines are freed at the same time it does not matter on which machine the job is processed first

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5 LAPT rule example job j p 1j p 2j 134 256 342 413

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6 Theorem 8.1.1 The LAPT rule yields an optimal schedule for O2||C max with makespan The LAPT rule yields an optimal schedule for O2||C max with makespan C max =max ( max j {1,…,n} (p 1j +p 2j ), n n ) p 1j, p 2j j=1j=1

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7 Minimizing the makespan on m machines Om||C max Theorem 8.1.2 Theorem 8.1.2 –The problem O3||C max is NP-hard –P ARTITION reduces to O3||C max The LTRP-OM is a reasonable heuristic The LTRP-OM is a reasonable heuristic –Whenever a machine is freed process the job that has the highest total remaining processing time on other machines is put on the machine –Longest Total Remaining Processing on Other Machines first

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8 Summary of other open shop models Om | prmp | C max is solvable in polynomial time Om | prmp | C max is solvable in polynomial time O2 || L max is strongly NP-hard O2 || L max is strongly NP-hard Om | r j, prmp | L max is solvable in polynomial time Om | r j, prmp | L max is solvable in polynomial time O2 | prmp | U j is NP-hard (not in the text) O2 | prmp | U j is NP-hard (not in the text) O2 || C j is strongly NP-hard O2 || C j is strongly NP-hard O3 |prmp| C j is strongly NP-hard O3 |prmp| C j is strongly NP-hard C max =max ( max j {1,…,n} m n ) p ij, max i {1,…,m} p ij i=1j=1

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Approximation Algorithms

Approximation Algorithms

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