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31-2 - Section3. Time Preparedby:JenniferMcNeill IEE 533 1...

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IEE 533 Scheduling 1 Section 3.1  The Total Weighted Completion  Time Prepared by: Jennifer McNeill
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IEE 533 Scheduling 2 Theorem 3.1.1   For               ,  WSPT is Optimal WSPT is Weighted Shortest Processing Time Jobs are ordered in decreasing order of wj/pj wj is the weight, or importance factor, of job j May represent holding cost/unit time or value already added to  job j WSPT is solved in O( nlog ( n )) time, the time required to sort  the jobs Proof by contradiction Suppose an optimal schedule, S, exists that is not  WSPT In S, there must be at least two adjacent jobs, jobs j and k,  such that wj/pj<wk/pk C w j j || 1
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IEE 533 Scheduling 3 Theorem 3.1.1 Proof (cont’d) Performing an adjacent pair wise interchange on jobs  j and k in schedule S yields schedule S’ All jobs other than j and k remain in their original  position Total weighted completion time of jobs before or after  jobs j and k is unaffected by the interchange t j k t+pj+pk t j k t+pj+pk Schedule S Schedule S’
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IEE 533 Scheduling 4 Theorem 3.1.1 Proof (cont’d) Under schedule S, the total weighted completion  time of jobs j and k is (t + pj)wj + (t + pj + pk)wk Under schedule S’, the same value is (t + pk)wk + (t + pk + pj)wj If wj/pj<wk/pk, the sum of the two completion times  under S’ is strictly less than under S Therefore, the optimality of S is contradicted and theorem  3.1.1 is proven
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IEE 533 Scheduling 5 Precedence Constraints and Total Weighted  Completion Time For simple precedence constraints, the total weighted  completion time problem can still be solved in  polynomial time Simplest precedence constraints are constraints in  which the jobs form parallel chains
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IEE 533 Scheduling 6 Lemma 3.1.2 Consider two chains One of jobs 1, …,k and one of jobs k+1, …,n Precedence constraints are 1   2   k and k+1 …  n Once a chain is selected for processing, all jobs on  that chain much be processed before any job from  another chain can begin processing Which chain should the scheduler process first to  minimize total weighted completion time?
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