1. © J. Christopher Beck 20051 Lecture 3: Manufacturing Scheduling Concepts

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

3. © J. Christopher Beck 2005 3 Jobs p ij – processing time of job j on machine i r j – release date of job j d j – due date of job j w j – weight of job j p ij rjrj djdj wjwj M1M1 M2M2 M3M3 S ij C ij S ij – starting time of job j on machine i C ij – completion time of job j

4. © J. Christopher Beck 2005 4 Jobs & Operations Often jobs are made up of a set of operations p 2j rjrj djdj wjwj p 0j p 3j p 1j precedence constraints

5. © J. Christopher Beck 2005 5 Example: House Building … Excavate Foundations Floor joists … Exterior plumbing 4 wks 2 wks3 wks

6. © 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

7. © J. Christopher Beck 2005 7 House Building Resources … Excavate Foundations Floor joists … Exterior plumbing 4 wks 2 wks3 wks Backhoe Backhoe operator Dump truck … requiresCarpenter

8. © 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”)

9. © J. Christopher Beck 2005 9 Objectives Minimize maximum completion time (aka “makespan”) Min C max C max = max(C 1, … C n ) Minimize maximum lateness Min L max L max = max(C 1 – d 1, … C n – d n )

10. © J. Christopher Beck 2005 10 Objectives Minimize total weighted tardiness Min Σw j T j T j = max(C j – d j, 0)

11. #JobDuration (weeks)Predecessor(s) 1Excavation4– 2Foundations21 3Floor joists32 4Exterior Plumbing31 5Floor23,4 6Power On12 7Walls105 8Wiring26,7 9Communication Lines18 10Inside Plumbing57 11Windows210 12Doors210 13Sheetrock39,10 14Interior Trim512,13 15Exterior Trim412 16Painting311,14,15 17Carpeting116 18Inspection117 Exercise 2.1 a) Draw precedence graph b) Calculate makespan

12. © J. Christopher Beck 2005 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

13. © 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, …

14. © 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(n 2 ), … Hard: f (n) is exponential in n O(2 n ), …

15. Hard vs Easy O(n)O(n log n)O(n 2 )O(2 n ) 101 10 100 2026400 50852500 10020010000 100030001,000,000

16. Hard vs Easy O(n)O(n log n)O(n 2 )O(2 n ) 1012 10 1001024 2026400 50852500 10020010000 100030001,000,000

17. Hard vs Easy O(n)O(n log n)O(n 2 )O(2 n ) 1012 10 1001024 20264001048576 508525001,125,899,906,842,624 10020010000 100030001,000,000

18. Hard vs Easy O(n)O(n log n)O(n 2 )O(2 n ) 1012 10 1001024 20264001048576 508525001,125,899,906,842,624 100200100001.268 X 10 30 100030001,000,0001.072 X 10 301

19. © J. Christopher Beck 2005 19 Hard vs Easy 10 301 operations required in worst case Age of universe: 10 18 seconds Fastest Computer today: 10 14 op/sec Let’s say we get a computer 10 18 times faster (a sextillion times faster) 10 33 op/sec universe It may still take 10 250 times longer than the age of the universe to solve the problem!

20. © J. Christopher Beck 2005 20 Hard vs Easy If it is going to take 10 250 times the age of the universe to schedule a factory, why bother?

21. © J. Christopher Beck 2005 21 Hard vs Easy If it is going to take 10 250 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?