1. Production SchedulingP.C. Chang, IEM, YZU. 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. Production SchedulingP.C. Chang, IEM, YZU. 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. Production SchedulingP.C. Chang, IEM, YZU. 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. Production SchedulingP.C. Chang, IEM, YZU. 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 –s ijk : 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. Production SchedulingP.C. Chang, IEM, YZU. 5 Generic notation of scheduling problem MachineJobObjective characteristicscharacteristicsfunction 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. Production SchedulingP.C. Chang, IEM, YZU. 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. Production SchedulingP.C. Chang, IEM, YZU. 7 Symbol : Job number : Machine number : Arrival time : Processing time of job : Completion time of job : due date T: Tardiness E: Earliness

8. Production SchedulingP.C. Chang, IEM, YZU. 8 Static V.S. Dynamic  Static Assume all the jobs are ready at the beginning which means a i =0  Dynamic Each job with a different arrival time. Which a i ≠0

9. Production SchedulingP.C. Chang, IEM, YZU. 9 Large Scale Problem (man-made) available solution space unavailable solution space Upper Bound Lower Bound approach Optimum (Heuristic) (Release Constraints)

10. Production SchedulingP.C. Chang, IEM, YZU. 10 Performance Measure 1.Completion Time Cmax = Max C i = C 6 指工件集合 s 中，最晚之完工時間，即指 Cmax. (Makespan) 2.Minimize Inventory fi : 庫存降低 fi = C i – a i ( Static Problem : ai=0) 3.Satisfy Due Date Tardiness = Max(Ci-di, 0 ) Earliness = Max(di-Ci, 0 ) JIT = Ci-di 4.Bi-criteria Multi-Objective (flow time = waiting time + process time)

11. Production SchedulingP.C. Chang, IEM, YZU. 11 Compute flow time 4 １２ ３ 0 5 8 12 13 5 5 3 3 3 4 1 5 3 4 1 第 i 順位之 P i

12. Production SchedulingP.C. Chang, IEM, YZU. 12 Gantt Chart ６４５１２ ３ d 3 c 3 d 1 c 2 d 2 c 1 d 4 c 5 c 4 d 5 d 6 c6c6 ＝ tardiness c4 > d4 jobs are ready flow time c2 – a2

13. Production SchedulingP.C. Chang, IEM, YZU. 13 Scheduling Problem Representation 4 / 1 / (n / m / o ) # job # machine objective function..........

14. Production SchedulingP.C. Chang, IEM, YZU. 14 Example: A factory has receive 4 different orders as follows ipidi 159 234 347 413 Please assign the production sequence of the 4 jobs to satisfy: 1.Due Date 2.Min Inventory

15. Production SchedulingP.C. Chang, IEM, YZU. 15 Sol. 1. Using FCFS (First come first serve) 4 １２ ３ 0 5 8 12 13 1-2-3-4

16. Production SchedulingP.C. Chang, IEM, YZU. 16 Sol. 2.Using EDD (Earliest Due Date) 4 １２ ３ 0 1 4 8 13 4-2-3-1

17. Production SchedulingP.C. Chang, IEM, YZU. 17 Sol. 3.Using SPT (Shortest Processing Time) The same with EDD Optimum 4 １２ ３ 0 1 4 8 13 4-2-3-1  EDD – Due Date – Tmax  SPT – Inventory - Flow time

18. Production SchedulingP.C. Chang, IEM, YZU. 18 Bi-criterion Frontier EDD SPT

19. Production SchedulingP.C. Chang, IEM, YZU. 19 HW. 5 / 1 / ipidi 1313 228 359 447 5610 Draw the Frontier when

20. Production SchedulingP.C. Chang, IEM, YZU. 20 Dynamic Problem Example: 4 / 1 / iaiai pidi 1359 2534 3247 4413

21. Production SchedulingP.C. Chang, IEM, YZU. 21 Sol. 4 １２ ３ 0 3 8 11 15 16 1. Using Job index1-2-3-4 Ck > ai, C1 ≧ a2 - no idle time Else, if ai > Ck, a2 > C1 - idle 5 5 -2 5 +1 5 -1 =18 3 3 3 = 9 4 4 = 8 1 = 1 36 or 3-5 3-2 3-4

22. Production SchedulingP.C. Chang, IEM, YZU. 22 Sol. 4 １２ ３ 0 4 5 8 12 17 2. Using SPT. EDD4-2-3-1

23. Production SchedulingP.C. Chang, IEM, YZU. 23 Sol. 4 １２ ３ 0 2 6 7 10 15 3. Using FCFS then SPT (ESPT) 從 Available jobs 找 SPT 3-4-2-1 Static (SPT) 排了工件 3 之後， Dynamic 問題變 為 Static ，所以 SPT 每個工件輪流排第一來比較

24. Production SchedulingP.C. Chang, IEM, YZU. 24 Rule ESPT

25. Production SchedulingP.C. Chang, IEM, YZU. 25 Ex:ESPT 1.find Min 2. 3.for min

26. Production SchedulingP.C. Chang, IEM, YZU. 26 Sol. 4 １２ ３ 0 3 8 11 15 16 1. Using Job index1-2-3-4

27. Production SchedulingP.C. Chang, IEM, YZU. 27 Sol. 4 １２ ３ 0 4 5 8 12 17 2.Using SPT 3.Using EEDD (next slide) 4-2-3-1

28. Production SchedulingP.C. Chang, IEM, YZU. 28 Rule EEDD

29. Production SchedulingP.C. Chang, IEM, YZU. 29 Ex:EEDD 1.find Min 2. 3.for min let 4.Return 3

30. Production SchedulingP.C. Chang, IEM, YZU. 30 HW. 1. 5 / 1 / 2. 5 / 1 / iaiai pipi didi 12615 27213 35825 41530 59328 Find an optimal solution!