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Unformatted text preview: IEE 533 1 Section 3.1 The Total Weighted Completion Time Prepared by: Jennifer McNeill IEE 533 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 IEE 533 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’ IEE 533 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 IEE 533 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 IEE 533 6 Lemma 3.1.2 • Consider two chains – One of jobs 1, …,k and one of jobs k+1, …,n – Precedence constraints are 1 à 2 à …à k and k+1à …à 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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This note was uploaded on 10/07/2011 for the course IEE 533 taught by Professor Johnfowler during the Spring '10 term at ASU.
 Spring '10
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