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Introduction Operations Scheduling

Introduction Operations Scheduling. 1. Setting up the Scheduling Problem 2. Single Machine Problems 3. Solving Scheduling Problems. Program. Acknowledgement.

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Introduction Operations Scheduling

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  1. Introduction Operations Scheduling 1. Setting up the Scheduling Problem 2. Single Machine Problems 3. Solving Scheduling Problems

  2. Program

  3. Acknowledgement 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.

  4. Topic 1 Setting up the Scheduling Problem

  5. Scheduling • Scheduling concerns optimal allocation or assignment of resources, over time, to a set of tasks or activities. • Machines Mi, i=1,...,m (ith machine) • Jobs Jj, j=1,...,n (jth job) • Schedule may be represented by Gantt charts.

  6. Notation • Static data: • Processing time (pij) on machine i • Release date (rj) • Due date (dj) • Weight (wj) • Dynamic data: • Completion time (Cij) on machine i • Flow Time (Fj = Cj – rj)

  7. 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. : Machine Configuration • Standard machine configurations: • (1) Single-Machine models • (Pm) Parallel-Machine models • (Jm) Job Shop models jobs 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. : Constraints • (rj) Release dates • (prec) Precedence constraints • (sjk) Sequence dependent setup times • (prmp) Preemptions (resume or repeat) • (block) Storage / waiting constraints • (Mj) Machine eligibility • (circ) Recirculation • Tooling / resource constraints • Personnel scheduling constraints

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

  11. 1. Throughput and Makespan • Throughput • Defined by bottleneck machines • Makespan • Minimizing makespan tends to maximize throughput and balance load

  12. 2. Due Date Related Objectives • Lateness Minimize maximum lateness • Tardiness Minimize the weighted tardiness • Tardy job Minimize the number of tardy jobs

  13. Tardiness Lateness Late or Not In practice Due Date Penalties

  14. 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. Topic 2 Single-Machine Scheduling Problems

  16. 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. 1. Completion Time Models • Lets say we have • Single machine (1), where • the total weighted completion time should be minimized (SwjCj) • We denote this problem as

  18. 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. 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’ j k k j

  20. 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. More Completion Time Models SPT rule WSPT rule NP-hard preemptive SPT rule NP-hard NP-hard polynomial algorithm

  22. 2. Lateness Models • Lets say we have • Single machine (1), where • the maximum cost for late jobs should be minimized (hmax) • subject to precedence constraints • We denote this problem as hj(Cj) denotes the cost for completing job j at time Cj e.g. hj(Cj) = Cj-dj (than hmax = Lmax and EDD optimal)

  23. Optimal Solution • Theorem: Lawler’s algorithm is optimal for • Lawler’s Backwards recursive algorithm (Minimizing Maximum Cost): • Determine makespan • Determine job j* with smallest • Schedule job j* as last job in the sequence • Repeat same procedure with one job less (j*) • Proof by contradiction

  24. 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 dk < dj

  25. 3. Tardiness Models • Lets say we have • Single machine (1), where • the number of late jobs should be minimized (SUj) • We denote this problem as

  26. Optimal Solution • Theorem: Moore’s algorithm is optimal for • EDD rule with modification: 1. Three sets: J = empty; (complement of J) JC = {1..n}; (tardy jobs) JD 2. Determine job j* with smallest dj and add to J and delete from JC 3. If then remove job k from J with largest pj and add k to JD 4. Repeat step 2 and 3 until JC is empty • Proof by induction

  27. More Tardiness Models Moore’s algorithm NP-hard NP-hard NP-hard special cases: dj = 0 WSPT dj loose MS otherwise: apparent tardiness heuristic

  28. Topic 3 Solving Scheduling Problems (Appendix C)

  29. 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. Different Methods • Basic Dispatching Rules • Composite Dispatching Rules • Branch and Bound • Beam Search • Simulated Annealing • Tabu Search • Genetic Algorithms Construction Methods Improvement Methods

  31. 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. 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. 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. Processing Time Related • Critical Path (CP) rule • precedence constraints • makespan • Largest Number of Successors (LNS) rule • precedence constraints • makespan

  35. 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. 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. Example of Composite Dispatching Rule Single Machine with Weighted Total Tardiness

  38. 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. Case 1: Tight Deadlines • Assume dj=0 Then • We know that WSPT is optimal for this problem!

  40. 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. 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. 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. 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. Choosing K • Value of K determined empirically • Related to the due date tightness factor and the due date range factor

  45. Beam Search • Is B&B with restricted branching • Quick evaluation of all candidates • Choose the F best options (filter width) • Evaluate F options more thoroughly • Choose the B best options (beam width) for branching

  46. 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 S1 with value c(S1); k = 1; Step 2: Evaluate all solutions S in N(Sk) Step 3: Choose as Sk+1 the best solution only if c(Sk+1) < c(Sk) and k = k+1; go to step 2 Otherwise stop (local optimum found)

  47. Tabu Search • TS = LS with worse solutions allowed • Step 3: - Choose as Sk+1 the best solution within N(Sk) unless the associated modification is on the Tabu List. - Add the modification Sk -> Sk+1 on the Tabu List. - Remove oldest entry of the Tabu List.

  48. Graphically If c(Sk+2) > c(Sk+1) then Sk+1 is a strong candidate for Sk+3 Sk Sk+1 Sk+2

  49. 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

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