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This document outlines the application of Smith's algorithm, originally proposed in 1956, for creating efficient schedules subject to deadline constraints (Tmax). It examines the methodology of building schedules by evaluating unscheduled jobs based on their processing times and due dates. Additionally, the paper discusses the evolution of this algorithm through the extensions by Wassenhove and Gelders, optimizing the scheduling process while minimizing tardiness (Fbar) under constraints of Tmax. The effectiveness of different schedules and the concept of efficiency in scheduling are analyzed to find dominant solutions.
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Outline • Questions? • Exam Results • Go over Exam • New homework • Lecture • Efficiency • Smith’s algorithm • Wassenhove and Gelders
Min Fbar subject to Tmax =0 • This algorithm is due to Smith (1956) and builds the schedule from the back as does Lawler’s • Tmax = 0, of course, must be possible • Again we build the schedule from the back: • a. Look at unscheduled jobs, • Find all the jobs that are due at or after Tau • Select the one with the longest processing time • (arbitrary for ties) and place it last • b. Reduce Tau and the set of unscheduled jobs and repeat
Efficiency • What is meant by efficiency? (for nonequivalent measures) • Given two measures R1 and R2, an efficient schedule is one when we can find no better schedule such that • R1 <= R1’ and R2 <= R2’ • In particular, we will consider the measures Fbar and Tmax and find the schedule with the best combination of the two measures under a given relationship between the two measures, e.g., 2Tmax + (Fbar)^2 • This relationship, of course, must make sense for our particular situation
Efficiency (continued) • Rephrasing: • S produces: Tmax and Fbar • S’ produces: T’max and Fbar’ • Then S is better than, or dominates S’ if • Tmax <= T’max • Fbar <= Fbar’ • The comparisons must be strict in at least one • Tmax <= T’max or Tmax < T’max orTmax < T’max • Fbar < Fbar’ Fbar <= Fbar’ Fbar < Fbar’
Efficiency (continued) • Smith’s algorithm has been extended by Wassenhove and Gelders: • In Smith algorithm we arbitrarily break ties when two jobs have due dates greater than Tau and have equal processing times. • In the extension, the job with the later due date is selected
Efficiency (continued) • Instead of minimizing Fbar subject to Tmax =0, which frequently is not achievable, let’s minimize it subject to Tmax<= delta • Our task will be to find all the schedules for all the possible values of delta. (We won’t actually have to do this) • We begin with stating that this is no different than increasing all the due dates by delta and using Smith’s algorithm • We want to find all the efficient schedules • We need to assume that all p and d are integral (always possible - using shorter time measurements)
Efficiency (continued) • What is the tardiest that a job can be? • When its due date is 0 and it is processed last • Therefore Tmax <= the sum of the processing times • First let us consider a small problem of four jobs. There are 24 possible schedules and we could try each one, finding both Fbar and Tmax and find the best one -- not the best approach, but let us look at how we would find the efficient ones from the 24 that we had generated
Efficiency (continued) Fbar • Each point represents the result of a schedule. There are fewer than 24 points because several schedules yield identical results. The corners of the shaded area are the dominant schedules Tmax
Efficiency (continued) • Better in both worse in Fbar worse in both worse in Tmax
Efficiency (continued) • Outlining the algorithm: • 1. Find delta = sum of the processing times • 2. Add delta to all the due dates times • 3. Apply Smith’s algorithm to get S1 with Fbar1 and Tmax1 • 4. Set delta to Tmax - 1 • 5. Repeat from step 2 until there is no schedule with Tmax <=delta
