1. Operational Research & ManagementOperations Scheduling Introduction Operations Scheduling 1.Setting up the Scheduling Problem 2.Single Machine Problems 3.Solving Scheduling Problems

2. Operational Research & ManagementOperations Scheduling2 Program

3. Operational Research & ManagementOperations Scheduling3 We are very grateful for the slides prepared by colleagues from other universities, in particular the slides of Siggi Olafsson has been used in our course extensively. Acknowledgement

4. Operational Research & ManagementOperations Scheduling Topic 1 Setting up the Scheduling Problem

5. Operational Research & ManagementOperations Scheduling5 Scheduling  Scheduling concerns optimal allocation or assignment of resources, over time, to a set of tasks or activities. – Machines M i, i=1,...,m(i th machine) – Jobs J j, j=1,...,n (j th job)  Schedule may be represented by Gantt charts.

6. Operational Research & ManagementOperations Scheduling6 Notation  Static data: – Processing time (p ij )on machine i – Release date (r j ) – Due date (d j ) – Weight (w j )  Dynamic data: – Completion time(C ij ) on machine i – Flow Time (F j = C j – r j )

7. Operational Research & ManagementOperations Scheduling7 Modeling  Three components for any machine scheduling model: – Machine configuration – Constraints and processing characteristics – Objective and performance measures  Notation:  |  |   Characteristics for  obviously present because of , are NOT mentioned.

8. Operational Research & ManagementOperations Scheduling8  : Machine Configuration  Standard machine configurations: – (1)Single-Machine models – (Pm)Parallel-Machine models – (Jm)Job Shop modelsjobs have different routes – (Fm)Flow Shop models:jobs have same order and same machines – (Om)Open Shop:routing also to be determined  Real world always more complicated: – (FJc)Flexible Job Shop:with parallel machines at each workstation – (FFc)Flexible Flow Shop:with parallel machines at each stage

9. Operational Research & ManagementOperations Scheduling9  : Constraints  (r j )Release dates  (prec)Precedence constraints  (s jk )Sequence dependent setup times  (prmp) Preemptions (resume or repeat)  (block)Storage / waiting constraints  (M j )Machine eligibility  (circ)Recirculation  Tooling / resource constraints  Personnel scheduling constraints

10. Operational Research & ManagementOperations Scheduling10  : Objectives and Performance Measures 1.Throughput (TP) and makespan (C max ) 2.Due date related objectives 3.Work-in-process (WIP), lead time (response time), finished inventory 4.(Setup Times)

11. Operational Research & ManagementOperations Scheduling11 1. Throughput and Makespan  Throughput – Defined by bottleneck machines  Makespan  Minimizing makespan tends to maximize throughput and balance load

12. Operational Research & ManagementOperations Scheduling12 2. Due Date Related Objectives  Lateness Minimize maximum lateness  Tardiness Minimize the weighted tardiness  Tardy job Minimize the number of tardy jobs

13. Operational Research & ManagementOperations Scheduling13 Due Date Penalties In practice Tardiness Late or Not Lateness

14. Operational Research & ManagementOperations Scheduling14 3. WIP and Lead Time  Work-in-Process (WIP) inventory cost  Minimizing WIP also minimizes average lead time (throughput time)  Minimizing lead time tends to minimize the average number of jobs in system  Equivalently, we can minimize sum of the completion times: or

15. Operational Research & ManagementOperations Scheduling Topic 2 Single-Machine Scheduling Problems

16. Operational Research & ManagementOperations Scheduling16 Classic Scheduling Theory  Look at a specific machine environment with a specific objective  Analyze to prove an optimal policy or to show that no simple optimal policy exists  Thousands of problems have been studied in detail with mathematical proofs! 3 Examples: single machine

17. Operational Research & ManagementOperations Scheduling17 1. Completion Time Models  Lets say we have – Single machine (1), where – the total weighted completion time should be minimized (  w j C j )  We denote this problem as

18. Operational Research & ManagementOperations Scheduling18 Optimal Solution  Theorem: Weighted Shortest Processing Time first - called the WSPT rule - is optimal for  Note: The SPT rule starts with the job that has the shortest processing time, moves on the job with the second shortest processing time, etc.  WSPT starts with job with largest

19. Operational Research & ManagementOperations Scheduling19 Proof (by contradiction) Suppose it is not true and schedule S is optimal – Then there are two adjacent jobs, say job j followed by job k such that Do a pairwise interchange to get schedule S’ jk kj

20. Operational Research & ManagementOperations Scheduling20 Proof (continued)  The weighted completion time of the two jobs under S is  The weighted completion time of the two jobs under S’ is  Now:  Contradicting that S is optimal.

21. Operational Research & ManagementOperations Scheduling21 More Completion Time Models SPT rule WSPT rule NP-hard preemptive SPT rule NP-hard polynomial algorithm

22. Operational Research & ManagementOperations Scheduling22 2. Lateness Models  Lets say we have – Single machine (1), where – the maximum cost for late jobs should be minimized (h max ) – subject to precedence constraints  We denote this problem as h j (C j ) denotes the cost for completing job j at time C j e.g. h j (C j ) = C j -d j (than h max = L max and EDD optimal)

23. Operational Research & ManagementOperations Scheduling23 Optimal Solution  Theorem: Lawler’s algorithm is optimal for  Lawler’s Backwards recursive algorithm (Minimizing Maximum Cost) : 1. Determine makespan 2. Determine job j * with smallest 3. Schedule job j * as last job in the sequence 4. Repeat same procedure with one job less (j * )  Proof by contradiction

24. Operational Research & ManagementOperations Scheduling24 More Lateness Models Lawler’s algorithm EDD rule B&B algorithm (App B2) same B&B procedure preemptive EDD rule Job j is interrupted when job k arrives with d k < d j

25. Operational Research & ManagementOperations Scheduling25 3. Tardiness Models  Lets say we have – Single machine (1), where – the number of late jobs should be minimized (  U j )  We denote this problem as

26. Operational Research & ManagementOperations Scheduling26 Optimal Solution  Theorem: Moore’s algorithm is optimal for  EDD rule with modification: 1.Three sets: J = empty; (complement of J) J C = {1..n}; (tardy jobs) J D 2.Determine job j * with smallest d j and add to J and delete from J C 3.If then remove job k from J with largest p j and add k to J D 4.Repeat step 2 and 3 until J C is empty  Proof by induction

27. Operational Research & ManagementOperations Scheduling27 More Tardiness Models Moore’s algorithm NP-hard special cases: d j = 0WSPT d j looseMS otherwise: apparent tardiness heuristic

28. Operational Research & ManagementOperations Scheduling Topic 3 Solving Scheduling Problems (Appendix C)

29. Operational Research & ManagementOperations Scheduling29 General Purpose Scheduling Procedures  Some scheduling problems are easy – Simple priority rules – Complexity: polynomial time  However, most scheduling problems are hard – Complexity: NP-hard, strongly NP-hard – Finding an optimal solution is infeasible in practice  heuristic methods

30. Operational Research & ManagementOperations Scheduling30 Different Methods  Basic Dispatching Rules  Composite Dispatching Rules  Branch and Bound  Beam Search  Simulated Annealing  Tabu Search  Genetic Algorithms Construction Methods Improvement Methods

31. Operational Research & ManagementOperations Scheduling31 Dispatching Rules  Other names: list scheduling, priority rules  Prioritize all waiting jobs – job attributes – machine attributes – current time  Whenever a machine becomes free: select the job with the highest priority  Static or dynamic

32. Operational Research & ManagementOperations Scheduling32 Release/Due Date Related  Earliest release date first (ERD) rule – variance in throughput times (flow times)  Earliest due date first (EDD) rule – maximum lateness  Minimum slack first (MS) rule – maximum lateness Current Time Processing Time Deadline

33. Operational Research & ManagementOperations Scheduling33 Processing Time Related  Longest Processing Time first (LPT) rule – balance load on parallel machines – makespan  Shortest Processing Time first (SPT) rule – sum of completion times – WIP  Weighted Shortest Processing Time first (WSPT) rule

34. Operational Research & ManagementOperations Scheduling34 Processing Time Related  Critical Path (CP) rule – precedence constraints – makespan  Largest Number of Successors (LNS) rule – precedence constraints – makespan

35. Operational Research & ManagementOperations Scheduling35 Other Dispatching Rules  Service in Random Order (SIRO) rule  Shortest Setup Time first (SST) rule – makespan and throughput  Least Flexible Job first (LFJ) rule – makespan and throughput  Shortest Queue at the Next Operation (SQNO) rule – machine idleness

36. Operational Research & ManagementOperations Scheduling36 Discussion  Very simple to implement  Optimal for special cases  Only focus on one objective  Limited use in practice  Combine several dispatching rules: Composite Dispatching Rules

37. Operational Research & ManagementOperations Scheduling Example of Composite Dispatching Rule Single Machine with Weighted Total Tardiness

38. Operational Research & ManagementOperations Scheduling38 Setup  Problem:  No efficient algorithm (NP-Hard)  Branch and bound can only solve very small problems (<30 jobs)  Are there any special cases we can solve?

39. Operational Research & ManagementOperations Scheduling39 Case 1: Tight Deadlines  Assume d j =0 Then  We know that WSPT is optimal for this problem!

40. Operational Research & ManagementOperations Scheduling40 Case 2: “Easy” Deadlines  Theorem: If the deadlines are sufficiently spread out then the MS rule is optimal (proof a bit harder)  Conclusion: The MS rule should be a good heuristic whenever deadlines are widely spread out

41. Operational Research & ManagementOperations Scheduling41 Composite Rule  Two good heuristics – Weighted Shorted Processing Time (WSPT )  Optimal with due dates zero – Minimum Slack (MS)  Optimal when due dates are “spread out” – Any real problem is somewhere in between  Combine the characteristics of these rules into one composite dispatching rule

42. Operational Research & ManagementOperations Scheduling42 Apparent Tardiness Cost (ATC)  New ranking index  When machine becomes free: – Compute index for all remaining jobs – Select job with highest value Scaling constant

43. Operational Research & ManagementOperations Scheduling43 Special Cases (Check)  If K is very large: – ATC reduces to WSPT  If K is very small and no overdue jobs: – ATC reduces to MS  If K is very small and overdue jobs: – ATC reduces to WSPT applied to overdue jobs

44. Operational Research & ManagementOperations Scheduling44 Choosing K  Value of K determined empirically  Related to the due date tightness factor and the due date range factor

45. Operational Research & ManagementOperations Scheduling45 Beam Search  Is B&B with restricted branching 1. Quick evaluation of all candidates 2. Choose the F best options (filter width) 3. Evaluate F options more thoroughly 4. Choose the B best options (beam width) for branching

46. Operational Research & ManagementOperations Scheduling46 Local Search N(S) = Neighborhood of a solution S = set containing all solutions that can be obtained by a simple modification of S Step 1: Choose a starting solution S 1 with value c(S 1 ); k = 1; Step 2: Evaluate all solutions S in N(S k ) Step 3: Choose as S k+1 the best solution only if c(S k+1 ) < c(S k ) and k = k+1; go to step 2 Otherwise stop (local optimum found)

47. Operational Research & ManagementOperations Scheduling47 Tabu Search  TS = LS with worse solutions allowed  Step 3:- Choose as S k+1 the best solution within N(S k ) unless the associated modification is on the Tabu List. - Add the modification S k -> S k+1 on the Tabu List. - Remove oldest entry of the Tabu List.

48. Operational Research & ManagementOperations Scheduling48 Graphically S k+2 SkSk S k+1 If c(S k+2 ) > c(S k+1 ) then S k+1 is a strong candidate for S k+3

49. Operational Research & ManagementOperations Scheduling49 Exercises  From chapter 3: 3.2, 3.4  From appendix C:C.1, C.4, C.6 (1 step), C.9  Single Machine Problems (see Blackboard)  Proof algorithm of Moore optimal for  Proof EDD optimal for