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

1
The School Course Timetabling Problemor 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

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
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
Lit
Bio
Phil
Psyc
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
Lit
Bio
Phil
Psyc
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
Mat 2
Phys
Phil
Chem
Art
Lit
Bio
Phil
Psyc
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
Mat 2
Phys
Phil
Chem
Art
Block 4
Lit
Bio
Chem
Geog
Bio
Phil
Psyc
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
Mat 2
Phys
Phil
Chem
Art
Block 4
Lit
Bio
Chem
Geog
Bio
Phil
Psyc
Block 5
Eng 3
Eng 1
Psyc
12
Blocking Example
Block 1
Hist
Mat 3
Lit
Block 2
Mat 1
Eng 2
Cook
Art
Block 3
Mat 2
Phys
Phil
Block 4
Chem
Geog
Bio
Block 5
Eng 3
Eng 1
Psyc
13
Blocking Example
Block 1
Timetable
Hist
Mat 3
Lit
Block 2
Mat 1
Eng 2
Cook
Art
Block 3
Mat 2
Phys
Phil
Block 4
Chem
Geog
Bio
Block 5
Eng 3
Eng 1
Psyc
14
Blocking Example
Block 1
Hist
Mat 3
Lit
Block 2
Mat 1
Eng 2
Cook
Art
Block 3
Mat 2
Phys
Phil
Block 4
Chem
Geog
Bio
Block 5
Eng 3
Eng 1
Psyc
15
Blocking Example
Student 25
Block 1
Hist
Mat 3
Lit
Lit
Block 2
Mat 1
Eng 2
Cook
Art
Mat
Block 3
Mat 2
Phys
Phil
Phys
Block 4
Chem
Geog
Bio
Chem
Block 5
Eng 3
Eng 1
Psyc
Eng
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
• Doesnt work if teachers 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
Pure Blocking Model

• Subject Network Model

A
B
C
D
E
F
Subjects
G
H
I
J
K
L
Blocks
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Pure Blocking Model

• Subject Network Model

No. Classes
1
4
1
1
6
1
2
1
3
1
1
2
A
B
C
D
E
F
Subjects
G
H
I
J
K
L
Blocks
23
Pure Blocking Model

• Network Models for each discrete program and
  teacher.
• For example program 7 taken by 4 students

A
B
C
D
E
F
Subjects
Blocks
24
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
A
B
C
D
E
F
Subjects
Blocks
4
4
4
4
4
4
Total capacity for each Subject-Block combination
dependent on Subject-Block allocation
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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

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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

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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
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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

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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)

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Subject Distribution
Year Level Type Subjects
12 Elective 6 Electives (10)
Streamed R.E. (4)
11 Elective 6 Electives (10)
Streamed R.E. (8)
10 Elective 3 Electives (6)
Streamed Mathematics (10), Science (8)
Core English (9), R.E. (6), 2 Humanities (6 each), P.E. Practical (4), P.E. Theory (1)
9 Elective 3 Electives (6)
Streamed Mathematics (10)
Core English (9), R.E. (8), Science (8), Geography (5), History (5), P.E. Prac(4), P.E. Theory (1)
33
Pure Blocking (times/nodes)
M1 M1(ni) M2 M2 (ni)
9 (sec) 0.38 0.04 0.53 0.12
9(nodes) 10 0 10 0
10 (s) 1.01 0.07 1.99 1.17
10 (n) 30 0 32 20
11 (s) 385 26 680 70
11 (n) 280 90 376 130
12 (s) 90 2.5 71 43
12 (n) 220 0 180 20
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
Blocks for Year 11 and 12
Year 12 Blocks
Block 1
Block 2
Block 3
Block 4
Block 5
Block 6
37
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 6
Block 5
RE
38
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
Block 6
R.E.
Bl 5
R.E.
39
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 2
Block 1
Block 3
42
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
Block 6
R.E.
Bl 5
R.E.
Overlap Areas
1
2
3
4
5
8
7
6
9
1 0
Overlap variables specify the number of sessions
each lower year block has in common with
each overlap area
43
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
Block 6
R.E.
Bl 5
R.E.
Year 10
Maths
Sci 1
Sci 2
El1
El2
El3
Year 9
Maths 1
Maths 2
El1
El3
El 2
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
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

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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

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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

51
Model

• Max ?a?A ?c?C xca
• ?a?A xca ? ?c , ? c ? C
• xc1a xc2a ? 2 - ?c1c2 , ? c1, c2 ? C
• xca ? 2 - ?bc - ?ba, ? c ? C, ? b ? B
• Solutions produced in under an hour

52
And Year 10 Science?

• The subject that just refused to be timetabled
  nicely
• 8 Lessons - too small for upper block, too large
  for lower block
• Allocated to upper year blocks in CBPP, but
  requires sessions from this block to be used in
  core class timetabling
• Need to unallocate after BTP and re-allocate in
  core class timetabling problem.

53
The Final Timetable

• Timetables for the school can be produced in
  about 25 hours
• Timetables took the entire summer break to
  produce using a program that merely checked
  feasibility
• Achievements
• A model that solves the CBPP in one step
• New variables combining students with the same
  program of subjects