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Lecture 3: Manufacturing Scheduling Concepts

Lecture 3: Manufacturing Scheduling Concepts. Outline. Jobs & Operations Characteristics & notation Resources/machines Setup/transition cost Objective functions Complexity. M 1. M 2. M 3. S ij. C ij. S ij – starting time of job j on machine i C ij – completion time of job j.

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Lecture 3: Manufacturing Scheduling Concepts

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  1. Lecture 3: Manufacturing Scheduling Concepts © J. Christopher Beck 2005

  2. Outline • Jobs & Operations • Characteristics & notation • Resources/machines • Setup/transition cost • Objective functions • Complexity © J. Christopher Beck 2005

  3. M1 M2 M3 Sij Cij Sij – starting time of job j on machine i Cij – completion time of job j Jobs rj dj wj pij pij – processing time of job j on machine i rj – release date of job j dj – due date of job j wj – weight of job j © J. Christopher Beck 2005

  4. Jobs & Operations • Often jobs are made up of a set of operations rj dj wj p2j p0j p3j p1j precedence constraints © J. Christopher Beck 2005

  5. Example: House Building Excavate Floor joists Foundations … 4 wks 2 wks 3 wks Exterior plumbing … 3 wks © J. Christopher Beck 2005

  6. Resources/Machines • Jobs may need resources • Mixing machine, back-hoe, cement mixer • May be multiple similar resources are available and you need to choose one © J. Christopher Beck 2005

  7. House Building Resources Excavate Floor joists Foundations … 4 wks 2 wks 3 wks requires Carpenter Exterior plumbing Backhoe Backhoe operator Dump truck … … 3 wks © J. Christopher Beck 2005

  8. Resources & Setup • If 2 jobs need the same resource (and the resource can only do 1 thing at a time), then the jobs must be sequenced • May be a time or cost for a resource to change jobs (“sequence dependent setup”) © J. Christopher Beck 2005

  9. Objectives • Minimize maximum completion time (aka “makespan”) • Min Cmax • Cmax = max(C1, … Cn) • Minimize maximum lateness • Min Lmax • Lmax = max(C1 – d1, … Cn – dn) © J. Christopher Beck 2005

  10. Objectives • Minimize total weighted tardiness • Min ΣwjTj • Tj = max(Cj – dj, 0) © J. Christopher Beck 2005

  11. Exercise 2.1 a) Draw precedence graph b) Calculate makespan

  12. Hard Problems vs. Easy Problems • Exercise 2.1b was “easy” • Adding resources would have made it hard • Hard & easy have precise mathematical definitions • You need to have, at least, an intuitive understanding of what this means © J. Christopher Beck 2005

  13. Hard vs Easy • Easy: • Sort n numbers • Solve a system of linear equations • Hard: • Schedule a factory, deliver packages, schedule buses, … © J. Christopher Beck 2005

  14. Hard vs Easy • f (n): the number of “basic operations” needed to solve the problem with input size n • Easy: f (n) is polynomial in n • O(n), O(n log n), O(n2), … • Hard: f (n) is exponential in n • O(2n), … © J. Christopher Beck 2005

  15. Hard vs Easy

  16. Hard vs Easy

  17. Hard vs Easy

  18. Hard vs Easy

  19. Hard vs Easy • 10301 operations required in worst case • Age of universe: 1018 seconds • Fastest Computer today: 1014 op/sec • Let’s say we get a computer 1018 times faster (a sextillion times faster) • 1033 op/sec • It may still take 10250 times longer than the age of the universe to solve the problem! © J. Christopher Beck 2005

  20. Hard vs Easy • If it is going to take 10250 times the age of the universe to schedule a factory, why bother? © J. Christopher Beck 2005

  21. Hard vs Easy • If it is going to take 10250 times the age of the universe to schedule a factory, why bother? • May be we can do it in a reasonable time in most cases? • May be we can get a good (but not necessarily best possible) solution in a reasonable amount of time? © J. Christopher Beck 2005

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