1. 1 Set #1 Dr. LEE Heung Wing Joseph Email : majlee@polyu.edu.hk Phone: 2766 6951 Office : HJ639

2. 2 Subject Code: AMA522 Location and Time: Wednesday,  18:30, at DE303 Recommended background knowledge: Calculus, Linear Algebra, Probability, Fortran or C Programming, Combinatorics. Assessment: 2 Assignments, 10% Project, 15% Test, 15% Final Examination 60%

3. 3 Aim To provide a sound understanding of the fundamental techniques and algorithms for scheduling problems from a range of commercial and service sectors. Objectives * To give an understanding of the methods and techniques that are available for building scheduling systems. * To introduce modern approaches for dealing with scheduling problems including treating uncertainties.

4. 4 Adopted Text 1. Scheduling, Theory, Algorithms, and Systems, Michael Pinedo, Prentice Hall, 1995. NEW: Second edition, 2002

5. 5 * Lecture Notes will be available for download online at: http://www.acad.polyu.edu.hk/~majlee/ama522.html * All announcements for the subject will be made in lectures and put on the web site

6. 6 INTRODUCTION TO SCHEDULING Contents 1. Definition of Scheduling 2. Examples 3. Terminology 4. Classification of Scheduling Problems 5. P and NP problems

7. 7 Gantt Chart

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10. 10 Definition of Scheduling Scheduling concerns optimal allocation or assignment of resources, over time, to a set of tasks or activities. machines M i, i=1,...,m(ith machine) jobs J j, j=1,...,n (jth job) Schedule may be represented by Gantt charts. J3J3 J2J2 J2J2 J3J3 J1J1 J1J1 J1J1 J3J3 J4J4 M3M3 M2M2 M1M1 Machine oriented Gantt chart M1M1 M2M2 M2M2 M1M1 M3M3 M3M3 M2M2 J1J1 J2J2 J3J3 Job oriented Gantt chart M1M1 M1M1 J4J4 t t

11. 11 Bicycle Assembly 3 workers within a team each task has its own duration precedence constraints no preemption T7T7 T2T2 T1T1 T5T5 T4T4 T6T6 T9T9 T8T8 T 10 T3T3 Examples

12. 12 T1T1 T6T6 T4T4 T8T8 T2T2 T 10 T7T7 T3T3 7 T9T9 T5T5 1421 39 24614 256 21 Task assignment

13. 13 T1T1 T6T6 T4T4 T8T8 T2T2 T 10 T3T3 7 T9T9 T5T5 1416 34 2469 29 17 25 Improved task assignment 16 T7T7

14. 14 An optimal task assignment T1T1 T6T6 T4T4 T8T8 T2T2 T 10 T3T3 7 T9T9 T5T5 14 32 25714322416 T7T7

15. 15 Scheduling Systems Enterprise Resource Planning (ERP) –Common for larger businesses Materials Requirement Planning (MRP) –Very common for manufacturing companies Advanced Planning and Scheduling (APS) –Most recent trend –Considered “advanced feature” of ERP

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17. 17 Scheduling Problem Allocate scarce resources to tasks Combinatorial optimization problem Maximize profit Subject to constraints Mathematical techniques and heuristics

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24. 24 Classroom Assignment one day seminar 14 seminars 5 rooms 8:00 - 5:00pm no seminars during the lunch hour 12:00 - 1:00pm

25. 25 Soft Drink Bottling single machine 4 flavours each flavour has its own filling time cleaning and changeover time between the bottling of successive flavours aim: to minimise cycle time, sufficient: to minimise the total changeover time f 1 - f 2 - f 3 - f 4 - f 1 2+3+2+50 = 57 f 2 - f 3 - f 4 - f 1 - f 2 3+2+50+2 = 57 f 3 - f 4 - f 2 - f 1 - f 3 2+5+6+70 = 83 f 4 - f 2 - f 3 - f 1 - f 4 4+3+8+50 = 66 f 1 - f 2 - f 4 - f 3 - f 1 2+4+6+8 = 20 optimal:

26. 26 Terminology Scheduling is the allocation, subject to constraints, of resources to objects being placed in space-time, so that the total cost is minimised. Schedule includes the spacial and temporal information. Sequencing is the construction, subject to constraints, of an order in which activities are to be carried out. Sequence is an order in which activities are carried out. Timetabling is the allocation, subject to constraints, of resources to objects being placed in space-time, so that the set of objectives are satisfied as much as possible. Timetable shows when particular events are to take place. Rostering is the placing, subject to constraints, of resources into slots in a pattern. Roster is a list of people's names that shows which jobs they are to do and when.

27. 27 Example 1.1.1 A Paper Bag Factory 1. Printing of the logo 2. gluing of the side of the bag 3. sewing of one end of the bag Colours, size may affect processing speed. Late delivery implies loss of goodwill. Sequence dependent setup time.

28. 28 Example 1.1.2 Gate Assignments at an Airport Dozens of gates and hundreds of airplanes arriving and departing according to a schedule each day. Gates as well as the airplanes are not all identical. Some gates are in locations where it is difficult to bring in the planes Certain planes may have to be towed to their gates. Weather, or events at other airports may cause randomness in the schedule.

29. 29 Example 1.1.3 Scheduling Tasks in CPU Multi-tasking computer. Exact processing time are not known in advance, only the distributions may be known. Priority level. The operation system slices the tasks into little pieces. Preemption allowed.

30. 30 Classification of Scheduling Problems machines j=1,…,m jobs i =1,…,n (i, j) processing step, or operation, of job j on machine i Job data Processing time p ij - processing time of job j on machine i Release date r j - earliest time at which job j can start its processing Due date d j - committed shipping or completion date of job j Weight w j - importance of job j relative to the other jobs in the system

31. 31 Scheduling problem:  |  |   machine environment  job characteristics  optimality criteria

32. 32 Machine characteristics Single machine 1 Identical machines in parallel P m m machines in parallel Job j requires a single operation and may be processed on any of the m machines If job j may be processed on any one machine belonging to a given subset M j P m | M j |... Machines in parallel with different speeds Q m Unrelated machines in parallel R m machines have different speeds for different jobs

33. 33 Machine Configurations Single-MachineParallel-Machine

34. 34 Flow shop F m m machines in series all jobs have the same routing each job has to be processed on each one of the m machines (permutation) first in first out (FIFO) F m | prmu |... Flexible flow shop FF s s stages in series with a number of machines in parallel at each stage job j requires only one machine FIFO discipline is usually between stages

35. 35 Machine Configurations Flow Shop Job Shop

36. 36 Open shop O m m machines each job has to be processed on each of the m machines scheduler determines the route for each job Job shop J m m machines each job has its own route job may visit a machine more then once (recirculation) F m | recrc |...

37. 37 Job characteristics Release date r j - earliest time at which job j can start its processing Sequence dependent setup times s jk - setup time between jobs j and k s ijk - setup time between jobs j and k depends on the machine Preemptions prmp - jobs can be interrupted during processing

38. 38 Precedence constraints prec - one or more jobs may have to be completed before another job is allowed to start its processing may be represented by an acyclic directed graph G=(V,A) V={1,…,n} corresponds to the jobs (j, k)  A iff jth job must be completed before kth chains each job has at most one predecessor and one successor outree each job has at most one predecessor intree each job has at most one successor

39. 39 Breakdowns brkdwn - machines are not continuously available Machine eligibility restrictions M j - M j denotes the set of machines that can process job j Permutation prmu - in the flow shop environment the queues in front of each machine operates according to the FIFO discipline Blocking block - in the flow shop there is a limited buffer in between two successive machines, when the buffer is full the upstream machine is not allowed to release a completed job. No wait no-wait- jobs are not allowed to wait between two successive machines Recirculation recrc - in the job shop a job may visit a machine more than once

40. 40 Constraints  blocking  Machine Eligibility Completion time Start timeBuffer Space JobsMachines

41. 41 Optimality criteria We define for each job j: C ij completion time of the operation of job j on machine i C j time when job j exits the system L j = C j - d j lateness of job j T j = max(C j - d j, 0)tardiness of job j unit penalty of job j

42. 42 Due Date Penalties In practice Tardiness Unit Penalty (Late or Not) Lateness

43. 43 Possible objective functions to be minimised: Makespan C max - max (C 1,...,C n ) Maximum lateness L max - max (L 1,...,L n ) Total weighted completion time  w j C j - weighted flow time Total weighted tardiness  w j T j Weighted number of tardy jobs  w j U j Examples Bicycle assembling: precedence constrained parallel machines P3 | prec | C max

44. 44 Summary * Scheduling is a decision making process with the goal of optimising one or more objectives * Production scheduling problems are classified based on machine environment, job characteristics, and optimality criteria.

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48. 48 Classes of Schedules Nondelay Schedule: A feasible schedule is called nondelay if no machine is kept idle while on operation is waiting for processing. Active Schedule: A feasible schedule is called active if it is not possible to construct another schedule by changing the order of processing on the machines and having at least one operation finishing earlier and no operation finishing later.

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50. 50 Semi-Active Schedule: A feasible schedule is called semi-active if no operation can be completed earlier without changing the order of processing on any one of the machines.

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53. 53 Complexity Hierarchies

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55. 55 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!

56. 56 Example: single machine 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

57. 57 Optimal Solution Theorem: Weighted Shortest Processing time first - called the WSPT rule - is optimal for In decreasing order of w j /p j. 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.

58. 58 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

59. 59 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.