31-2 - 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 •

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 &#224; 2 &#224; …&#224; k and k+1&#224; …&#224; 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?...
View Full Document

This note was uploaded on 10/07/2011 for the course IEE 533 taught by Professor Johnfowler during the Spring '10 term at ASU.

Page1 / 24

31-2 - 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 •

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online