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Modeling: Parameters

Modeling: Parameters. Typical scheduling parameters: Number of resources ( m machines, operators) Configuration and layout Resource capabilities Number of jobs ( n ) Job processing times ( pij ) Job release and due dates (resp. rij and dij ) Job weight ( wij ) or priority

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Modeling: Parameters

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  1. Modeling: Parameters • Typical scheduling parameters: • Number of resources (m machines, operators) • Configuration and layout • Resource capabilities • Number of jobs (n) • Job processing times (pij) • Job release and due dates (resp. rij and dij ) • Job weight (wij ) or priority • Setup times

  2. Modeling: Objective function • Objectives and performance measures: • Throughput, makespan (Cmax, weighted sum) • Due date related objectives (Lmax, Tmax, ΣwjTj) • Work-in-process (WIP), lead time (response time), finished inventory • Total setup time • Penalties due to lateness (ΣwjLj) • Idle time • Yield • Multiple objectives may be used with weights

  3. Modeling: Constraints • Precedence constraints (linear vs. network) • Routing constraints • Material handling constraints • (Sequence dependent) Setup times • Transport times • Preemption • Machine eligibility • Tooling/resource constraints • Personnel (capability) scheduling constraints • Storage/waiting constraints • Resource capacity constraints

  4. Machine configurations: • Single-machine vs. parallel-machine • Flow shop vs. job shop Processing characteristics: • Sequence dependent setup times and costs • length of setup depends on jobs • sijk: setup time for processing job j after k on machine i • costs: waste of material, labor • Preemptions • interrupt the processing of one job to process another with a higher priority

  5. Generic notation of scheduling problem • Machine Job Objective • characteristics characteristics function • for example: • Pm | rj, prmp | ΣwjCj (parallel machines) • 1 | sjk | Cmax (sequence dependent • setup / traveling salesman) • Q2 | prec | ΣwjTj (2 machines w. different speed, precedence rel., weighted tardiness)

  6. Scheduling models • Deterministic models • input matches realization • vs. • Stochastic models • distributions of processing times, release and due dates, etc. known in advance • outcome/realization of distribution known at completion

  7. Symbol

  8. Static V.S. Dynamic • Static Assume all the jobs are ready at the beginning which means ai=0 • Dynamic Each job with a different arrival time. Which ai≠0

  9. Large Scale Problem (man-made) approach Upper Bound (Heuristic) available solution space Optimum unavailable solution space Lower Bound (Release Constraints) approach

  10. Performance Measure • Max Completion Time (Makspan) • Cmax = Max Ci = C6 • Minimize Inventory • fi : Reduce Inv. • fi = Ci – ai ( Static Problem : ai=0) • Satisfy Due Date • Tardiness = Max(Ci-di , 0 ) • Earliness = Max(di-Ci , 0 ) • JIT = Ci-di • Bi-criteria Multi-Objective (flow time = waiting time + process time)

  11. １ ２ 　３　 4 0 5 8 12 13 Compute flow time 5 3 4 1 5 5 5 5 3 3 3 4 4 1

  12. Gantt Chart 　３　 ２ １ ５ ４ ６ jobs are ready d3 c3d1c2 d2 c1d4c5c4d5 d6 ＝ c6 flow time tardiness c2 – a2 c4 > d4

  13. Scheduling Problem Representation 4 / 1 / (n / m / o ) objective function # machine # job . . . . .

  14. Example: A factory has receive 4 different orders as follows • Please assign the production sequence of the 4 jobs to satisfy: • Due Date • Min Inventory

  15. Sol. 1. Using FCFS (First come first serve) 1-2-3-4 １ ２ 　３　 4 0 5 8 12 13

  16. Sol. • Using EDD (Earliest Due Date) 4-2-3-1 4 ２ 　３　 １ 0 1 4 8 13

  17. Sol. • Using SPT (Shortest Processing Time) The same with EDD Optimum 4-2-3-1 4 ２ 　３　 １ 0 1 4 8 13 • EDD – Due Date – Tmax • SPT – Inventory - Flow time

  18. Bi-criterion SPT Frontier EDD

  19. HW. 5 / 1 / Draw the Frontier when

  20. Dynamic Problem Example: 4 / 1 /

  21. Dynamic Problem 0

  22. Dynamic Problem

  23. Sol. Ck > ai , C1≧ a2 - no idle time Else, if ai > Ck, a2 > C1 - idle 1. Using Job index 1-2-3-4 １ ２ 　３　 4 0 3 8 11 15 16 3-5 3-2 3-4 • 5-2 5+1 5-1 =18 • 3 3 3 = 9 • 4 4 = 8 • 1 = 1 • 36 or

  24. Sol. 2. Using SPT. EDD 4-2-3-1 4 ２ 　３　 １ 0 4 5 8 12 17

  25. Sol. 3. Using FCFS then SPT (ESPT) Use SPT to arrange jobs (available jobs) 3-4-2-1 　３　 4 ２ １ 0 2 6 7 10 15 Static (SPT) After arrange job 3, the dynamic problem will become a Static one. Then use SPT.

  26. Rule ESPT

  27. Ex:ESPT • find Min • for min

  28. Sol. 1. Using Job index 1-2-3-4 １ ２ 　３　 4 0 3 8 11 15 16

  29. Sol. • Using SPT • Using EEDD (next slide) 4-2-3-1 4 ２ 　３　 １ 0 4 5 8 12 17

  30. Rule EEDD

  31. Ex:EEDD • find Min • for min let • Return 3

  32. JIT problem Slackness Rule: Find di-pi (Job j have to start before this time) a4 d4 a2 d2 a3 d3 a1 d1 ２ 4 １ 　３　 try ２ １ ２ or

  33. Ex. 2-4-3-1 ２ 4 　３　 １ 0 5 9 10 17 26

  34. HW. 1. 5 / 1 / 2. 5 / 1 / 3. 5 / 1 / Find an optimal solution!

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