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The School Course Timetabling Problem or How the Messiness of Real Life Can Obscure a Nice Model

The School Course Timetabling Problem or How the Messiness of Real Life Can Obscure a Nice Model. Presentation by Liam Merlot joint work with Natashia Boland, Barry Hughes and Peter Stuckey. First Some Terminology. Session Subject Content, Year Level, Capacity, No. Classes, No. Lessons

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The School Course Timetabling Problem or How the Messiness of Real Life Can Obscure a Nice Model

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  1. The School Course Timetabling ProblemorHow the Messiness of Real Life Can Obscure a Nice Model Presentation by Liam Merlot joint work with Natashia Boland, Barry Hughes and Peter Stuckey

  2. First Some Terminology • Session • Subject • Content, Year Level, Capacity, No. Classes, No. Lessons • Class • Union of Subject, Teacher and Students • Lesson • Each individual meeting of a class

  3. Problem Description (1) • Create a timetable for a School • find sessions for all lessons of all classes of all subjects • Initial data: • student subject choices • teacher subject allocations • all information about subjects (capacity, no. classes, no. lessons)

  4. Problem Description (2) • Some subjects have multiple classes • What English class is Fred in? • Who is teaching Fred English? • 2 Sub-Problems: • The Student Population Problem • The Class Timetabling Problem • Both problems combined: The Population and Class Timetabling Problem (PCTP)

  5. Blocking Decomposition • Class Blocking and Population Problem (CBPP): • Students are allocated to classes • Classes are allocated to blocks • Block Timetabling Problem (BTP): • Blocks allocated to sessions What are Blocks? • Sets of classes for which all lessons will be allocated to the same set of sessions in the timetable.

  6. Hist Eng 1 Phys Eng 2 Mat 1 Eng 3 Mat 2 Geog Mat 3 Cook Chem Art Bio Lit Psyc Phil Blocking Example:

  7. Blocking Example: Block 1 Hist Eng 1 Hist Mat 3 Lit Phys Eng 2 Mat 1 Eng 3 Mat 2 Geog Mat 3 Cook Chem Art Bio Lit Psyc Phil

  8. Blocking Example: Block 1 Hist Eng 1 Hist Mat 3 Lit Phys Eng 2 Block 2 Mat 1 Eng 3 Mat 1 Eng 2 Cook Art Mat 2 Geog Mat 3 Cook Chem Art Bio Lit Psyc Phil

  9. Blocking Example: Block 1 Hist Eng 1 Hist Mat 3 Lit Phys Eng 2 Block 2 Mat 1 Eng 3 Mat 1 Eng 2 Cook Art Mat 2 Geog Block 3 Mat 3 Cook Phys Mat 2 Phil Chem Art Bio Lit Psyc Phil

  10. Blocking Example: Block 1 Hist Eng 1 Hist Mat 3 Lit Phys Eng 2 Block 2 Mat 1 Eng 3 Mat 1 Eng 2 Cook Art Mat 2 Geog Block 3 Mat 3 Cook Phys Mat 2 Phil Chem Art Block 4 Bio Lit Bio Geog Chem Psyc Phil

  11. Blocking Example: Block 1 Hist Eng 1 Hist Mat 3 Lit Phys Eng 2 Block 2 Mat 1 Eng 3 Mat 1 Eng 2 Cook Art Mat 2 Geog Block 3 Mat 3 Cook Phys Mat 2 Phil Chem Art Block 4 Bio Lit Bio Geog Chem Psyc Phil Block 5 Psyc Eng 1 Eng 3

  12. Blocking Example: Block 1 Hist Mat 3 Lit Block 2 Mat 1 Eng 2 Cook Art Block 3 Phys Mat 2 Phil Block 4 Bio Geog Chem Block 5 Psyc Eng 1 Eng 3

  13. M T W T F Blocking Example: Block 1 Timetable Hist Mat 3 Lit Block 2 Mat 1 Eng 2 Cook Art Block 3 Phys Mat 2 Phil Block 4 Bio Geog Chem Block 5 Psyc Eng 1 Eng 3

  14. Blocking Example: Block 1 Hist Mat 3 Lit Block 2 Mat 1 Eng 2 Cook Art Block 3 Phys Mat 2 Phil Block 4 Bio Geog Chem Block 5 Psyc Eng 1 Eng 3

  15. Blocking Example: Student 25 Block 1 Lit Hist Mat 3 Lit Block 2 Mat Mat 1 Eng 2 Cook Art Block 3 Phys Phys Mat 2 Phil Block 4 Chem Bio Geog Chem Block 5 Eng Psyc Eng 1 Eng 3

  16. Advantages and Disadvantages • Allows some information from timetabling to be used in student allocation • Greatly reduced search space • Problem becomes difficult across year levels • Doesn’t work if teacher’s have significant time restrictions

  17. PCTP Literature • Very few papers: • School problems normally solved with blocking (CBPP, BTP) • University problems normally a different decomposition (CTP, SPP) • CBPP is further decomposed (non-linear model)

  18. Pure Blocking • All students do the same number of subjects • All subjects require the same number of lessons (classes can be allocated to any block) • All classes must be blocked • The number of blocks is equal to the number of subjects that each student is taking

  19. Innovations in Our Model • Students with the same set of subjects (program) are combined • CBPP is solved in one step

  20. Pure Blocking Model • Multiple related Network Models: • One for the Subjects • One for each discrete student program • One for each teacher • Classes of subjects are allocated to blocks • Students and teachers are allocated to take subjects in blocks

  21. 1 2 3 4 5 6 Pure Blocking Model • Subject Network Model: Subjects A B C D E F G H I J K L Blocks

  22. 1 2 3 4 5 6 1 3 Pure Blocking Model • Subject Network Model: No. Classes 1 4 1 1 6 1 2 1 3 1 1 2 Subjects A B C D E F G H I J K L Blocks

  23. 1 2 3 4 5 6 Pure Blocking Model • Network Models for each discrete program and teacher. • For example: program 7 taken by 4 students: Subjects A B C D E F Blocks

  24. 1 2 3 4 5 6 3 2 4 2 1 4 1 4 1 2 Pure Blocking Model • Network Models for each discrete program and teacher. • For example: program 7 taken by 4 students: 4 4 4 4 4 4 Subjects A B C D E F Blocks 4 4 4 4 4 4 Total capacity for each Subject-Block combination dependent on Subject-Block allocation

  25. Model Data • p  P - discrete student programs • t  T - teachers • s  S - subjects • b  B - blocks • s - number of classes of subject s • s - capacity of each class of subject s • p - number of students taking discrete program p • ts - number of classes of subject s taken by teacher t

  26. Model Variables • xsb -integer variable: number of classes of s allocated to b • wp – integer variable: number of students taking p with a ‘full allocation’ • ypsb - integer variable: number of students studying p allocated to take subject s in block b • ztsb - binary variable whether teacher t takes subject s in block b

  27. Model 1 Max p wp + t s b ztsb Subject to: bB xsb = s,  s  S wp  p,  p  P bB ypsb = wp,  s  p,  p  P sp ypsb = wp,  b  B,  p  P pP ypsb  sxsb,  b  B,  s  P bB ztsb  ts,  s  t,  t  T st ztsb  1,  b  B,  t  T tT zpsb  xsb,  b  B,  s  T

  28. Model 2 Max p s b ypsb + t s b ztsb Subject to: bB xsb = s,  s  S bB ypsb  p,  s  p,  p  P sp ypsb  p,  b  B,  p  P pP ypsb  sxsb,  b  B,  s  P bB ztsb  ts,  s  t,  t  T st ztsb  1,  b  B,  t  T tT zpsb  xsb,  b  B,  s  T

  29. Integrality • Network (and other) problems often have integrality property • Student and teacher variables do NOT have the integrality property (although network-like) • Will this matter? • Will we be lucky?

  30. Xavier College Senior School • 4 Year Levels • 68 Sessions (7 per day) • 922 students • 571 programs • 91 subjects • 242 classes • 112 teachers

  31. Types of Subject • Elective: • students choose a specified number with no restrictions, classes need to be populated, same number of sessions required • Streamed: • students already allocated to classes, all lessons of all classes held in same set of sessions • Core: • students already allocated to classes (same sets for each subject in each year level)

  32. Subject Distribution

  33. Pure Blocking (times/nodes)

  34. Applying Pure Blocking • Works well for the elective subjects for each year level • Problems arise over multiple year levels • Subjects do not all require same number of lessons

  35. Adapting Pure Blocking • Temporarily ignore core subjects • One complex blocking scheme for years 11 and 12 • One simple blocking scheme for each of years 9 and 10 • Allow the three blocking schemes to overlap

  36. RE Blocks for Year 11 and 12 Year 12 Blocks: Block 1 Block 2 Block 3 Block 4 Block 5 Block 6

  37. RE Blocks for Year 11 and 12 Year 12 Blocks: Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 Year 11 Blocks (Ideal Case): Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 RE

  38. RE Blocks for Year 11 and 12 Year 12 Blocks: Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 Year 11 Blocks (Real case): Block 1 Block 2 Block 3 Block 4 Bl 5 R.E. Block 6 R.E. Bl 5

  39. K A B C D E I J F G Blocks for Year 11 and 12 Year 12 Blocks: A B C D E H Year 11 Blocks (Real case): Dummy subjects for year 12 added to blocks F, G, I, J Dummy subjects for year 11 added to blocks H, K, (E, F), (G, I) Constraints added to prevent teachers from taking classes in blocks which overlap

  40. Years 9 and 10 • Streamed subjects allocated in year 11 and 12 blocking scheme • Year 10 Maths: one block • Year 10 Science: two blocks • Year 9 Maths: two blocks • Elective subjects: • 3 block model for each year level • Variables determine overlap between blocking scheme

  41. Year 9 and 10 Blocking Scheme Year 10 Blocks: Block 1 Block 2 Block 3 Year 9 Blocks: Block 1 Block 2 Block 3

  42. RE Overlap Areas Year 11 and 12 Blocks: Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 Block 1 Block 2 Block 3 Block 4 Bl 5 R.E. Block 6 R.E. Bl 5 Overlap Areas: 1 2 3 4 5 6 7 8 9 1 0 Overlap variables specify the number of sessions each lower year block has in common with each overlap area

  43. RE Blocks for Year 9 and 10 Year 11 and 12 Blocks: Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 Block 1 Block 2 Block 3 Block 4 Bl 5 R.E. Block 6 R.E. Bl 5 Year 10: Maths Sci 1 El1 Sci 2 El2 El3 Year 9: El1 El 2 Maths 1 2 El3 Maths 2

  44. Results for Blocking • Years 9 to 12 from a Victorian secondary school • 922 students, 571 programs • 91 subjects, 242 classes • 112 teachers • Modeled in AMPL • Solves in ~24 hours (8 hours with non-integer variables) • Takes 2 months to solve by hand currently

  45. New Overlap Areas (15): But we are only half done …. • Need to allocate blocks to sessions • First construct new overlap areas:

  46. Allocate overlap areas to sessions • One idea for BTP is to allocate overlap areas to sessions:

  47. Model • a  A - sessions • o  O - overlap areas • b  B - blocks • xoa - binary variable if overlap area o allocated to session a • xba - binary variable if block b is allocated to session a • o - number of lessons required by overlap area o • b - number of lessons required by block b • Fo - set of blocks in overlap area o

  48. Model Max aA oO xoa aA xoa  o ,  o  O oO xoa  1,  a  A aA xba  b ,  b  B xoa  xba ,  b  Fo,  o  O • xba variables allow the model to keep track of blocks to allow constraints for doubles, weekly balance, etc. • Solutions produced in about 10 minutes

  49. But what about the Core Subjects? • A third timetabling problem is solved after the other two problems have been solved. • Core classes are allocated to sessions which are not used by Year 9 or 10 blocked subjects • Clashing constraints are required • Data: 130 classes for 474 students

  50. Model • a  A - sessions • c  C - classes of core subjects • b  B - blocks • xca - binary variable if class c is allocated to session a • c - number of lessons required by class c • c1c2 - binary parameter, 1 if classes c1, c2 have a student or teacher in common • bc - binary parameter, 1 if class c and block b have a student or teacher in common • ba - binary parameter, 1 if block b was allocated to session a

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