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8 The Mathematics of Scheduling

8 The Mathematics of Scheduling. 8.1 The Basic Elements of Scheduling 8.2 Directed Graphs (Digraphs) 8.3 Scheduling with Priority Lists 8.4 The Decreasing-Time Algorithm 8.5 Critical Paths 8.6 The Critical-Path Algorithm 8.7 Scheduling with Independent Tasks. The Critical-Path Algorithm.

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8 The Mathematics of Scheduling

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  1. 8 The Mathematics of Scheduling 8.1 The Basic Elements of Scheduling 8.2 Directed Graphs (Digraphs) 8.3 Scheduling with Priority Lists 8.4 The Decreasing-Time Algorithm 8.5 Critical Paths 8.6 The Critical-Path Algorithm 8.7 Scheduling with Independent Tasks

  2. The Critical-Path Algorithm The concept of critical paths can be used to create very good (although not necessarily optimal) schedules. The idea is to use critical times rather than processingtimes to prioritize the tasks. The priority list we obtain when we write the tasks indecreasing order of critical times(with ties broken randomly) is called thecritical-time priority list, and the process of creating a schedule using the critical-time priority list is called the critical-path algorithm.

  3. CRITICAL-PATH ALGORITHM Step 1(Find critical times).Using the backflow algorithm, find the criticaltime for every task in the project. Step 2 (Create priority list).Using the critical times obtained in Step 1, createa priority listwith the tasks listed in decreasing order of critical times (i.e., a critical-time priority list). Step 3 (Create schedule).Using the critical-time priority list obtained inStep 2, create the schedule.

  4. Example 8.12 Building That Dream Home on Mars: Part 7 We will now schedule the Martian Habitat Unit building project with N = 2processors using the critical-path algorithm. We took care of Step 1 in Example 8.11. The critical times for each task areshown in red.

  5. Example 8.12 Building That Dream Home on Mars: Part 7 Step 2 follows directly from Step 1. The critical-time priority list for the project is AP[34], AF[32], AW[28], IF[27], IW[22], AD[18], IP[15], PL[11], HU[11], ID[10], IC[7], FW[6], PU[5], PD[3], EU[2].

  6. Example 8.12 Building That Dream Home on Mars: Part 7 Step 3 is a lot of busy work– not complex, just tedious. We will skip right to the end. The timeline for the resulting schedule is shown. The project finishingtime is Fin = 36hours. This is a very good schedule, but it is not an optimalschedule.

  7. Example 8.12 Building That Dream Home on Mars: Part 7 This figure shows the timeline for an optimalschedule finishingtime is Opt = 35hours.

  8. Critical-Path Algorithm The critical-path algorithm is an excellent approximate algorithm for scheduling a project, but as Example 8.12 shows, it does not always give an optimalschedule. In this regard, scheduling problems are like traveling salesman problems (Chapter 6) and shortest network problems (Chapter 7)– there are efficientapproximate algorithms for scheduling, but no efficient optimalalgorithm is currently known.

  9. Critical-Path Algorithm Of the standard scheduling algorithms, the critical-path algorithmis by far the most commonly used. Other, more sophisticated algorithms havebeen developed in the last 40 years and under specialized circumstances they canoutperform the critical-path algorithm, but as an all-purpose algorithm for scheduling, the critical-path algorithm is hard to beat.

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