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7-11 Jan 2013, Aussois
A New Lower Bound for Curriculum-Based Course Timetabling
- V. Cacchiani, A. Caprara, R. Roberti, P. Toth
7-11 Jan 2013, Aussois
A New Lower Bound for Curriculum-Based Course Timetabling V. - - PowerPoint PPT Presentation
A New Lower Bound for Curriculum-Based Course Timetabling V. - - PowerPoint PPT Presentation
A New Lower Bound for Curriculum-Based Course Timetabling V. Cacchiani, A. Caprara, R. Roberti, P. Toth Universita` di Bologna 7-11 Jan 2013, Aussois Dedicated to the memory of Alberto Caprara (1968-2012) 7-11 Jan 2013, Aussois Outline
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- Problem description
- Integer Linear Programming (ILP) models with exponentially many variables
- A good ILP model with exponentially many variables
- Computational Results
- Conclusions and Future Research
Outline
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Curriculum-Based Course Timetabling Determine the best scheduling of university course lectures in a given time horizon (5 or 6 working days).
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Curriculum-Based Course Timetabling Determine the best scheduling of university course lectures in a given time horizon (5 or 6 working days). International timetabling competitions ITC2002 and ITC2007 have stated a commonly accepted definition of the problem. Real-world benchmark instances are available
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Curriculum-Based Course Timetabling Determine the best scheduling of university course lectures in a given time horizon (5 or 6 working days). International timetabling competitions ITC2002 and ITC2007 have stated a commonly accepted definition of the problem. Real-world benchmark instances are available Wide amount of research on heuristics Fairly large gap between upper and lower bounds
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Given:
- a set C of courses:
- number of lectures
- minimum working days
- number of students
- unavailable time periods
Curriculum-Based Course Timetabling
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Given:
- a set C of courses:
- number of lectures
- minimum working days
- number of students
- unavailable time periods
- a set R of rooms:
- capacity
Curriculum-Based Course Timetabling
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7-11 Jan 2013, Aussois
Given:
- a set C of courses:
- number of lectures
- minimum working days
- number of students
- unavailable time periods
- a set R of rooms:
- capacity
- a set H of time periods
Curriculum-Based Course Timetabling
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7-11 Jan 2013, Aussois
Given:
- a set C of courses:
- number of lectures
- minimum working days
- number of students
- unavailable time periods
- a set R of rooms:
- capacity
- a set H of time periods
- a set Q of curricula
- courses in each curriculum
Curriculum-Based Course Timetabling
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7-11 Jan 2013, Aussois
Given:
- a set C of courses:
- number of lectures
- minimum working days
- number of students
- unavailable time periods
- a set R of rooms:
- capacity
- a set H of time periods
- a set Q of curricula
- courses in each curriculum
- a set T of teachers
- courses of the teacher
Curriculum-Based Course Timetabling
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Hard Constraints
- All the lectures of each course must be scheduled
Curriculum-Based Course Timetabling
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Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period Curriculum-Based Course Timetabling
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Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period
- Each teacher can teach at most one lecture per time period
Curriculum-Based Course Timetabling
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7-11 Jan 2013, Aussois
Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period
- Each teacher can teach at most one lecture per time period
- Each room can host at most one lecture per time period
Curriculum-Based Course Timetabling
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Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period
- Each teacher can teach at most one lecture per time period
- Each room can host at most one lecture per time period
- Unavailable time periods for a course (or for the teacher of the
course) cannot be used for scheduling a lecture of that course Curriculum-Based Course Timetabling
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Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period
- Each teacher can teach at most one lecture per time period
- Each room can host at most one lecture per time period
- Unavailable time periods for a course (or for the teacher of the
course) cannot be used for scheduling a lecture of that course Curriculum-Based Course Timetabling Soft Constraints (objective function)
- room capacity: a penalty for each student that cannot have a seat in
the room assigned to each course
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7-11 Jan 2013, Aussois
Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period
- Each teacher can teach at most one lecture per time period
- Each room can host at most one lecture per time period
- Unavailable time periods for a course (or for the teacher of the
course) cannot be used for scheduling a lecture of that course Curriculum-Based Course Timetabling Soft Constraints (objective function)
- room capacity: a penalty for each student that cannot have a seat in
the room assigned to each course
- minimum number of working days: a penalty for each day below the
minimum number of working days for each course
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Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period
- Each teacher can teach at most one lecture per time period
- Each room can host at most one lecture per time period
- Unavailable time periods for a course (or for the teacher of the
course) cannot be used for scheduling a lecture of that course Curriculum-Based Course Timetabling Soft Constraints (objective function)
- room capacity: a penalty for each student that cannot have a seat in
the room assigned to each course
- minimum number of working days: a penalty for each day below the
minimum number of working days for each course
- curriculum compactness: a penalty for isolated lectures
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7-11 Jan 2013, Aussois
Hard Constraints
- All the lectures of each course must be scheduled
- Courses belonging to the same curriculum cannot be scheduled in the
same time period
- Each teacher can teach at most one lecture per time period
- Each room can host at most one lecture per time period
- Unavailable time periods for a course (or for the teacher of the
course) cannot be used for scheduling a lecture of that course Curriculum-Based Course Timetabling Soft Constraints (objective function)
- room capacity: a penalty for each student that cannot have a seat in
the room assigned to each course
- minimum number of working days: a penalty for each day below the
minimum number of working days for each course
- curriculum compactness: a penalty for isolated lectures
- room stability: a penalty for each additional room used for a course
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Formulations
- Basic formulation (UD1): room stability is not taken into account
- Extended formulation (UD2)
They have different weights for the penalties of soft constraints violations Introduced for the ITC2002 and ITC2007 (McCollum et al [2010]) Curriculum-Based Course Timetabling
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Curriculum-Based Course Timetabling Literature review
- Many works on heuristic algorithms
- Works on lower bounds:
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Curriculum-Based Course Timetabling Literature review
- Many works on heuristic algorithms
- Works on lower bounds:
- three-index ILP model by Burke et al [2010, 2011]:
prc
x
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Curriculum-Based Course Timetabling Literature review
- Many works on heuristic algorithms
- Works on lower bounds:
- three-index ILP model by Burke et al [2010, 2011]:
- Surface and Surface2 models by Burke et al [2010]:
(additional constraints to respect the number of rooms; in Surface all the rooms are joined into a single room of multiplicity |R| and the largest capacity; in Surface2 rooms are divided into two groups: larger and smaller rooms)
prc
x
pc
w
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Curriculum-Based Course Timetabling Literature review
- Many works on heuristic algorithms
- Works on lower bounds:
- three-index ILP model by Burke et al [2010, 2011]:
- Surface and Surface2 models by Burke et al [2010]:
(additional constraints to respect the number of rooms; in Surface all the rooms are joined into a single room of multiplicity |R| and the largest capacity; in Surface2 rooms are divided into two groups: larger and smaller rooms)
- decomposition of the model into two stages by Lach and
Luebbecke [2012]: first stage: second stage: minimum weight bipartite perfect matching problems or ILP it is exact for the Basic formulation
prc
x
pc
w
pc
w
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7-11 Jan 2013, Aussois
Curriculum-Based Course Timetabling Literature review
- Problem partitioned into k subproblems and relaxation of soft
constraints of curriculum compactness and hard constraints of room
- ccupancy and time period conflict: Hao and Benlic [2011]
each subproblem formulated as in Lach and Luebbecke [2012] Iterated Tabu Search for an effective partition
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7-11 Jan 2013, Aussois
Curriculum-Based Course Timetabling Literature review
- Problem partitioned into k subproblems and relaxation of soft
constraints of curriculum compactness and hard constraints of room
- ccupancy and time period conflict: Hao and Benlic [2011]
each subproblem formulated as in Lach and Luebbecke [2012] Iterated Tabu Search for an effective partition
- Techniques based on propositional satisfiability solvers: Asin Acha
and Nieuwenhuis [2012]
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7-11 Jan 2013, Aussois
Curriculum-Based Course Timetabling Literature review
- Problem partitioned into k subproblems and relaxation of soft
constraints of curriculum compactness and hard constraints of room
- ccupancy and time period conflict: Hao and Benlic [2011]
each subproblem formulated as in Lach and Luebbecke [2012] Iterated Tabu Search for an effective partition
- Techniques based on propositional satisfiability solvers: Asin Acha
and Nieuwenhuis [2012]
- Only column generation based approach by Qualizza and Serafini
[2005] on a different version of the problem (objective function with preferences to time periods due to teachers, no soft constraints)
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Input: Courses: <CourseID> <Teacher> <# Lectures> <MinWorkingDays> <# Students> Rooms: <RoomID> <Capacity> Curricula: <CurriculumID> <# Courses> <CourseID> ... <CourseID> Unavailability_Constraints: <CourseID> <Day> <Day_Period>
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Output:
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ILP models with exponentially many variables
- Curriculum Schedule ILP
- Time Period Schedule ILP
- 3 Schedule Types ILP: Time Period Schedule, Course Pattern, Curriculum Pattern
- 2 Schedule Types ILP: Time Period Schedule, Course-Curriculum Pattern
- 2 Weekly Schedule Types ILP
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables
- Curriculum Schedule ILP
- Time Period Schedule ILP
- 3 Schedule Types ILP: Time Period Schedule, Course Pattern, Curriculum Pattern
- 2 Schedule Types ILP: Time Period Schedule, Course-Curriculum Pattern
- 2 Weekly Schedule Types ILP
For the basic and the extended formulations
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ILP models with exponentially many variables: Curriculum Schedule ILP (CS)
sq
x
Course 1, Room A Course 2, Room A
…
Schedule s for curriculum q for the time horizon, specifying which courses of q are given in each room, in each time period
Time period 1 Time period 2
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: Curriculum Schedule ILP (CS)
sq
x
Course 1, Room A Course 2, Room A
…
Schedule s for curriculum q for the time horizon, specifying which courses of q are given in each room, in each time period
Time period 1 Time period 2
chr
y
=1 if course c is given in time period h in room r
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: Time Period Schedule ILP (TS)
hs
x
Course 1 Course 2
…
Time period h Room A Room B
Schedule s for time period h, specifying for each room which course is given
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: Time Period Schedule ILP (TS)
hs
x
Course 1 Course 2
…
Schedule s for time period h, specifying for each room which course is given
Time period h Room A Room B
c
z
Counts the number of days below the minimum working days for course c
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: Time Period Schedule ILP (TS)
hs
x
Course 1 Course 2
…
Schedule s for time period h, specifying for each room which course is given
Time period h Room A Room B
c
z
Counts the number of days below the minimum working days for course c
hq
δ
Assumes value 1 if curriculum q has an isolated lecture in time period h
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 3 Schedule Types ILP: Time Period Schedule, Course Pattern, Curriculum Pattern (3ST)
hs
x
Course 1 Course 2
…
Schedule s for time period h, specifying for each room which course is given
Time period h Room A Room B
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 3 Schedule Types ILP: Time Period Schedule, Course Pattern, Curriculum Pattern (3ST)
hs
x
Course 1 Course 2
…
Schedule s for time period h, specifying for each room which course is given
Time period h Room A Room B
cp
z
3 2 …
Course c Day 1 Day 2
Pattern p of a feasible distribution
- f the lectures of
course c throughout the days: it indicates for each day the number of lectures of course c given in that day
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 3 Schedule Types ILP: Time Period Schedule, Course Pattern, Curriculum Pattern (3ST)
hs
x
Course 1 Course 2
…
Schedule s for time period h, specifying for each room which course is given
Time period h Room A Room B
cp
z
3 2 …
Course c Day 1 Day 2
qp
y
Pattern p of time periods where lectures of any course of curriculum q are given
1 …
Curriculum q Time period 1 Time period 2
Pattern p of a feasible distribution
- f the lectures of
course c throughout the days: it indicates for each day the number of lectures of course c given in that day
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Schedule Types ILP: Time Period Schedule, Course-Curriculum Pattern (2ST)
hs
x
Course 1 Course 2
…
Schedule s for time period h, specifying for each room which course is given
Time period h Room A Room B
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Schedule Types ILP: Time Period Schedule, Course-Curriculum Pattern (2ST)
qp
y
Pattern p for curriculum q specifying for each time period which course is given
Course 1 Course 1 Course 2 … hs
x
Course 1 Course 2
…
Schedule s for time period h, specifying for each room which course is given
Time period h Room A Room B Curriculum q Time period 1 Time period 2
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ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one! (2WST)
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ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one! (2WST)
s
x
Schedule s represents a feasible assignment of lectures to rooms and time periods for the time horizon
Course 1, Room A Course 2, Room B
…
Course 1, Room B
Time period 1 Time period 2
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one! (2WST)
s
x
Course 1, Room A Course 2, Room B
…
Course 1, Room B
Time period 1 Time period 2
p
y
Pattern p represents a feasible assignment of lectures to time periods for the time horizon
Course 1 Course 2
…
Course 1
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
s
x
Takes into account: Hard constraints:
- at most one lecture per room per time period
- at most one lecture of courses belonging to a certain curriculum per time period
- at most one lecture per teacher per time period
- all lectures must be scheduled for each course
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
s
x
Takes into account: Hard constraints:
- at most one lecture per room per time period
- at most one lecture of courses belonging to a certain curriculum per time period
- at most one lecture per teacher per time period
- all lectures must be scheduled for each course
Soft constraints:
- room capacity
- room stability
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
p
y
Takes into account: Hard constraints:
- at most one lecture of courses belonging to a certain curriculum per time period
- at most one lecture per teacher per time period
- all lectures must be scheduled for each course
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
p
y
Takes into account: Hard constraints:
- at most one lecture of courses belonging to a certain curriculum per time period
- at most one lecture per teacher per time period
- all lectures must be scheduled for each course
Soft constraints:
- minimum working days
- curriculum compactness
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
∑ ∑
∈ ∈
+ =
P p p p S s s s
y c x c WST z min ) 2 (
1 =
∑
∈S s s
x 1 =
∑
∈P p p
y
H h C c y b x a
p P p ch p s S s ch s
∈ ∈ = −∑
∑
∈ ∈
, ,
S s xs ∈ ∈ }, 1 , {
P p y p ∈ ∈ }, 1 , {
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
∑ ∑
∈ ∈
+ =
P p p p S s s s
y c x c WST z min ) 2 (
1 =
∑
∈S s s
x 1 =
∑
∈P p p
y
H h C c y b x a
p P p ch p s S s ch s
∈ ∈ = −∑
∑
∈ ∈
, ,
S s xs ∈ ∈ }, 1 , {
P p y p ∈ ∈ }, 1 , {
SLIDE 52
7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
∑ ∑
∈ ∈
+ =
P p p p S s s s
y c x c WST z min ) 2 (
1 =
∑
∈S s s
x 1 =
∑
∈P p p
y
H h C c y b x a
p P p ch p s S s ch s
∈ ∈ = −∑
∑
∈ ∈
, ,
S s xs ∈ ∈ }, 1 , {
P p y p ∈ ∈ }, 1 , {
SLIDE 53
7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
∑ ∑
∈ ∈
+ =
P p p p S s s s
y c x c WST z min ) 2 (
1 =
∑
∈S s s
x 1 =
∑
∈P p p
y
H h C c y b x a
p P p ch p s S s ch s
∈ ∈ = −∑
∑
∈ ∈
, ,
S s xs ∈ ∈ }, 1 , {
P p y p ∈ ∈ }, 1 , {
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7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: solution method
We relax the linking constraints. We solve separately the subproblem for the x variables and the subproblem for the y variables.
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
∑
∈
=
S s s s x
x c WST z min ) 2 (
1 =
∑
∈S s s
x 1 =
∑
∈P p p
y
S s xs ∈ ∈ }, 1 , {
P p y p ∈ ∈ }, 1 , {
∑
∈
=
P p p p y
y c WST z min ) 2 (
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7-11 Jan 2013, Aussois
ILP models with exponentially many variables: 2 Weekly Schedule Types ILP: the good one!
∑
∈
=
S s s s x
x c WST z min ) 2 (
1 =
∑
∈S s s
x 1 =
∑
∈P p p
y
S s xs ∈ ∈ }, 1 , {
P p y p ∈ ∈ }, 1 , {
∑
∈
=
P p p p y
y c WST z min ) 2 (
We next present:
- an ILP model for the subproblem on the x variables
- an ILP model for the subproblem on the y variables
- an ILP model for the subproblem on the y variables with exponentially many variables
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7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
=1 if course c is scheduled in time period h in room r
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7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
=1 if course c is scheduled in room r
SLIDE 59
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
} ), ( ) ( max{( r cap c stud W RCap −
SLIDE 60
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
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7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
SLIDE 62
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
SLIDE 63
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
SLIDE 64
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for x variables
∑∑ ∑∑ ∑
∈ ∈ ∈ ∈ ∈
− + =
C c H h C c R r cr RStb R r chr cr x
C W d WST z ) | | ( min ) 2 ( η ξ
Q q H h
q C c R r chr
∈ ∈ ≤
∑ ∑
∈ ∈
, , 1
) (
ξ
R r H h
C c chr
∈ ∈ ≤
∑
∈
, , 1 ξ
∑ ∑
∈ ∈
∈ ∈ ≤
) (
, , 1
t C c R r chr
T t H h ξ
∑∑
∈ ∈
∈ =
H h R r chr
C c c lect ), ( ξ R r C c c lect
cr H h chr
∈ ∈ ≤
∑
∈
, , ) ( η ξ
R r H h C c
chr
∈ ∈ ∈ ∈ , , }, 1 , { ξ
R r C c
cr
∈ ∈ ∈ , }, 1 , { η
SLIDE 65
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
Curricula that do not share any course can be dealt with separately in this model. Connected components of curricula: groups of curricula that share at least one course. Solve separately the model for each connected component.
SLIDE 66
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
∑∑ ∑
∈ ∈ ∈
+ =
H h q q hq CComp q C c c Mwd y
W z W WST z
~ ) ~ (
min ) 2 ( δ
q q H h
q C c ch
~ , , 1
) (
∈ ∈ ≤
∑
∈
ξ ) ~ ( , , 1
) (
q T t H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( q C c c lect
H h ch
∈ =
∑
∈
ξ D d q C c
cd d H h ch
∈ ∈ ≥
∑
∈
), ~ ( ,
) (
φ ξ ) ~ ( ), ( q C c c mwd zc
D d cd
∈ ≥ +
∑
∈
φ
q q H h
f hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q H H H h
l f hq q C c ch ch ch
~ }, { \ , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
U δ ξ ξ ξ
q q H h
l hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
H h q C c
ch
∈ ∈ ∈ ), ~ ( }, 1 , { ξ
D d q C c
cd
∈ ∈ ∈ ), ~ ( }, 1 , { φ
) ~ ( , q C c zc ∈ Ζ ∈
+
q q H h
hq
~ , }, 1 , { ∈ ∈ ∈ δ
Counts the number of days below the min working days
SLIDE 67
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
∑∑ ∑
∈ ∈ ∈
+ =
H h q q hq CComp q C c c Mwd y
W z W WST z
~ ) ~ (
min ) 2 ( δ
q q H h
q C c ch
~ , , 1
) (
∈ ∈ ≤
∑
∈
ξ ) ~ ( , , 1
) (
q T t H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( q C c c lect
H h ch
∈ =
∑
∈
ξ D d q C c
cd d H h ch
∈ ∈ ≥
∑
∈
), ~ ( ,
) (
φ ξ ) ~ ( ), ( q C c c mwd zc
D d cd
∈ ≥ +
∑
∈
φ
q q H h
f hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q H H H h
l f hq q C c ch ch ch
~ }, { \ , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
U δ ξ ξ ξ
q q H h
l hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
H h q C c
ch
∈ ∈ ∈ ), ~ ( }, 1 , { ξ
D d q C c
cd
∈ ∈ ∈ ), ~ ( }, 1 , { φ
) ~ ( , q C c zc ∈ Ζ ∈
+
q q H h
hq
~ , }, 1 , { ∈ ∈ ∈ δ
=1 if there is an isolated lecture for curriculum q in time period h
SLIDE 68
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
∑∑ ∑
∈ ∈ ∈
+ =
H h q q hq CComp q C c c Mwd y
W z W WST z
~ ) ~ (
min ) 2 ( δ
q q H h
q C c ch
~ , , 1
) (
∈ ∈ ≤
∑
∈
ξ ) ~ ( , , 1
) (
q T t H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( q C c c lect
H h ch
∈ =
∑
∈
ξ D d q C c
cd d H h ch
∈ ∈ ≥
∑
∈
), ~ ( ,
) (
φ ξ ) ~ ( ), ( q C c c mwd zc
D d cd
∈ ≥ +
∑
∈
φ
q q H h
f hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q H H H h
l f hq q C c ch ch ch
~ }, { \ , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
U δ ξ ξ ξ
q q H h
l hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
H h q C c
ch
∈ ∈ ∈ ), ~ ( }, 1 , { ξ
D d q C c
cd
∈ ∈ ∈ ), ~ ( }, 1 , { φ
) ~ ( , q C c zc ∈ Ζ ∈
+
q q H h
hq
~ , }, 1 , { ∈ ∈ ∈ δ
=1 if course c is given in time period h
SLIDE 69
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
∑∑ ∑
∈ ∈ ∈
+ =
H h q q hq CComp q C c c Mwd y
W z W WST z
~ ) ~ (
min ) 2 ( δ
q q H h
q C c ch
~ , , 1
) (
∈ ∈ ≤
∑
∈
ξ ) ~ ( , , 1
) (
q T t H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( q C c c lect
H h ch
∈ =
∑
∈
ξ D d q C c
cd d H h ch
∈ ∈ ≥
∑
∈
), ~ ( ,
) (
φ ξ ) ~ ( ), ( q C c c mwd zc
D d cd
∈ ≥ +
∑
∈
φ
q q H h
f hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q H H H h
l f hq q C c ch ch ch
~ }, { \ , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
U δ ξ ξ ξ
q q H h
l hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
H h q C c
ch
∈ ∈ ∈ ), ~ ( }, 1 , { ξ
D d q C c
cd
∈ ∈ ∈ ), ~ ( }, 1 , { φ
) ~ ( , q C c zc ∈ Ζ ∈
+
q q H h
hq
~ , }, 1 , { ∈ ∈ ∈ δ
SLIDE 70
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
∑∑ ∑
∈ ∈ ∈
+ =
H h q q hq CComp q C c c Mwd y
W z W WST z
~ ) ~ (
min ) 2 ( δ
q q H h
q C c ch
~ , , 1
) (
∈ ∈ ≤
∑
∈
ξ ) ~ ( , , 1
) (
q T t H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( q C c c lect
H h ch
∈ =
∑
∈
ξ D d q C c
cd d H h ch
∈ ∈ ≥
∑
∈
), ~ ( ,
) (
φ ξ ) ~ ( ), ( q C c c mwd zc
D d cd
∈ ≥ +
∑
∈
φ
q q H h
f hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q H H H h
l f hq q C c ch ch ch
~ }, { \ , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
U δ ξ ξ ξ
q q H h
l hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
H h q C c
ch
∈ ∈ ∈ ), ~ ( }, 1 , { ξ
D d q C c
cd
∈ ∈ ∈ ), ~ ( }, 1 , { φ
) ~ ( , q C c zc ∈ Ζ ∈
+
q q H h
hq
~ , }, 1 , { ∈ ∈ ∈ δ
SLIDE 71
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
∑∑ ∑
∈ ∈ ∈
+ =
H h q q hq CComp q C c c Mwd y
W z W WST z
~ ) ~ (
min ) 2 ( δ
q q H h
q C c ch
~ , , 1
) (
∈ ∈ ≤
∑
∈
ξ ) ~ ( , , 1
) (
q T t H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( q C c c lect
H h ch
∈ =
∑
∈
ξ D d q C c
cd d H h ch
∈ ∈ ≥
∑
∈
), ~ ( ,
) (
φ ξ ) ~ ( ), ( q C c c mwd zc
D d cd
∈ ≥ +
∑
∈
φ
q q H h
f hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q H H H h
l f hq q C c ch ch ch
~ }, { \ , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
U δ ξ ξ ξ
q q H h
l hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
H h q C c
ch
∈ ∈ ∈ ), ~ ( }, 1 , { ξ
D d q C c
cd
∈ ∈ ∈ ), ~ ( }, 1 , { φ
) ~ ( , q C c zc ∈ Ζ ∈
+
q q H h
hq
~ , }, 1 , { ∈ ∈ ∈ δ
=1 if a lecture of course c is given on a time period of day d
SLIDE 72
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
∑∑ ∑
∈ ∈ ∈
+ =
H h q q hq CComp q C c c Mwd y
W z W WST z
~ ) ~ (
min ) 2 ( δ
q q H h
q C c ch
~ , , 1
) (
∈ ∈ ≤
∑
∈
ξ ) ~ ( , , 1
) (
q T t H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( q C c c lect
H h ch
∈ =
∑
∈
ξ D d q C c
cd d H h ch
∈ ∈ ≥
∑
∈
), ~ ( ,
) (
φ ξ ) ~ ( ), ( q C c c mwd zc
D d cd
∈ ≥ +
∑
∈
φ
q q H h
f hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q H H H h
l f hq q C c ch ch ch
~ }, { \ , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
U δ ξ ξ ξ
q q H h
l hq q C c ch ch
~ , , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
H h q C c
ch
∈ ∈ ∈ ), ~ ( }, 1 , { ξ
D d q C c
cd
∈ ∈ ∈ ), ~ ( }, 1 , { φ
) ~ ( , q C c zc ∈ Ζ ∈
+
q q H h
hq
~ , }, 1 , { ∈ ∈ ∈ δ
SLIDE 73
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables
The presented ILP model is solved by Cplex but it turned out that it is computationally heavy. Thus, we use it only for the connected components that contain at most 10 curricula and at most 10 courses globally.
SLIDE 74
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables: column generation based ILP
For the remaining cases we solve the LP-relaxation of an ILP model with exponentially many variables
dp
y
Time period 1 Time period 2
Course 1 Course 2
…
Course 1
Pattern p represents a feasible assignment of lectures to time periods for the day d
SLIDE 75
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables: column generation based ILP
For the remaining cases we solve the LP-relaxation of an ILP model with exponentially many variables
dp
y
Time period 1 Time period 2
Course 1 Course 2
…
Course 1 c
z
Counts the number of days below the minimum working days for course c Pattern p represents a feasible assignment of lectures to time periods for the day d
SLIDE 76
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables: column generation based ILP
∑ ∑ ∑
∈ ∈ ∈
+ =
D d q C c c Mwd d P p dp dp y
z W y c WST z
) ~ ( ) (
min ) 2 ( D d y
d P p dp
∈ =
∑
∈
, 1
) (
∑ ∑
∈ ∈
∈ =
D d d P p dp cd p
q C c c lect y a ) ~ ( ), (
) (
) ~ ( ), (
) (
q C c c mwd z y b
c D d d P p dp cd p
∈ ≥ +
∑ ∑
∈ ∈
) ( , }, 1 , { d P p D d ydp ∈ ∈ ∈
) ~ ( , q C c zc ∈ Ζ ∈
+
We solve the LP-relaxation of the following model:
SLIDE 77
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables: column generation based ILP
∑ ∑ ∑
∈ ∈ ∈
+ =
D d q C c c Mwd d P p dp dp y
z W y c WST z
) ~ ( ) (
min ) 2 ( D d y
d P p dp
∈ =
∑
∈
, 1
) (
∑ ∑
∈ ∈
∈ =
D d d P p dp cd p
q C c c lect y a ) ~ ( ), (
) (
) ~ ( ), (
) (
q C c c mwd z y b
c D d d P p dp cd p
∈ ≥ +
∑ ∑
∈ ∈
) ( , }, 1 , { d P p D d ydp ∈ ∈ ∈
) ~ ( , q C c zc ∈ Ζ ∈
+
We solve the LP-relaxation of the following model:
SLIDE 78
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables: column generation based ILP
∑ ∑ ∑
∈ ∈ ∈
+ =
D d q C c c Mwd d P p dp dp y
z W y c WST z
) ~ ( ) (
min ) 2 ( D d y
d P p dp
∈ =
∑
∈
, 1
) (
∑ ∑
∈ ∈
∈ =
D d d P p dp cd p
q C c c lect y a ) ~ ( ), (
) (
) ~ ( ), (
) (
q C c c mwd z y b
c D d d P p dp cd p
∈ ≥ +
∑ ∑
∈ ∈
) ( , }, 1 , { d P p D d ydp ∈ ∈ ∈
) ~ ( , q C c zc ∈ Ζ ∈
+
We solve the LP-relaxation of the following model:
SLIDE 79
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for y variables: column generation based ILP
∑ ∑ ∑
∈ ∈ ∈
+ =
D d q C c c Mwd d P p dp dp y
z W y c WST z
) ~ ( ) (
min ) 2 ( D d y
d P p dp
∈ =
∑
∈
, 1
) (
∑ ∑
∈ ∈
∈ =
D d d P p dp cd p
q C c c lect y a ) ~ ( ), (
) (
) ~ ( ), (
) (
q C c c mwd z y b
c D d d P p dp cd p
∈ ≥ +
∑ ∑
∈ ∈
) ( , }, 1 , { d P p D d ydp ∈ ∈ ∈
) ~ ( , q C c zc ∈ Ζ ∈
+
We solve the LP-relaxation of the following model:
SLIDE 80
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for column generation based ILP
∑ ∑ ∑ ∑ ∑
∈ ∈ ∈ ∈ ∈
− − + −
) ~ ( ) ~ ( ) ( ~ ) (
min
q C c d q C c c c d H h q q hq CComp d H h ch c
u w W v ϕ δ ξ
q q d H h
q C c ch
~ ), ( , 1
) (
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( , 1
) (
q T t d H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
q q d H h
f hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q d H d H d H h
l f hq q C c ch ch ch
~ )}, ( ), ( { \ ) ( , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
δ ξ ξ ξ
q q d H h
l hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
) ~ ( ,
) (
q C c
c d H h ch
∈ ≥
∑
∈
ϕ ξ ) ( ), ~ ( }, 1 , { d H h q C c
ch
∈ ∈ ∈ ξ q q d H h
hq
~ ), ( }, 1 , { ∈ ∈ ∈ δ
) ~ ( }, 1 , { q C c
c
∈ ∈ ϕ
=1 if course c is given in time period h
SLIDE 81
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for column generation based ILP
∑ ∑ ∑ ∑ ∑
∈ ∈ ∈ ∈ ∈
− − + −
) ~ ( ) ~ ( ) ( ~ ) (
min
q C c d q C c c c d H h q q hq CComp d H h ch c
u w W v ϕ δ ξ
q q d H h
q C c ch
~ ), ( , 1
) (
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( , 1
) (
q T t d H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
q q d H h
f hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q d H d H d H h
l f hq q C c ch ch ch
~ )}, ( ), ( { \ ) ( , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
δ ξ ξ ξ
q q d H h
l hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
) ~ ( ,
) (
q C c
c d H h ch
∈ ≥
∑
∈
ϕ ξ ) ( ), ~ ( }, 1 , { d H h q C c
ch
∈ ∈ ∈ ξ q q d H h
hq
~ ), ( }, 1 , { ∈ ∈ ∈ δ
) ~ ( }, 1 , { q C c
c
∈ ∈ ϕ
=1 if there is an isolated lecture for curriculum q in time period h
SLIDE 82
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for column generation based ILP
∑ ∑ ∑ ∑ ∑
∈ ∈ ∈ ∈ ∈
− − + −
) ~ ( ) ~ ( ) ( ~ ) (
min
q C c d q C c c c d H h q q hq CComp d H h ch c
u w W v ϕ δ ξ
q q d H h
q C c ch
~ ), ( , 1
) (
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( , 1
) (
q T t d H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
q q d H h
f hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q d H d H d H h
l f hq q C c ch ch ch
~ )}, ( ), ( { \ ) ( , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
δ ξ ξ ξ
q q d H h
l hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
) ~ ( ,
) (
q C c
c d H h ch
∈ ≥
∑
∈
ϕ ξ ) ( ), ~ ( }, 1 , { d H h q C c
ch
∈ ∈ ∈ ξ q q d H h
hq
~ ), ( }, 1 , { ∈ ∈ ∈ δ
) ~ ( }, 1 , { q C c
c
∈ ∈ ϕ
=1 if course c uses
- ne or more time
periods of day d
SLIDE 83
7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for column generation based ILP
∑ ∑ ∑ ∑ ∑
∈ ∈ ∈ ∈ ∈
− − + −
) ~ ( ) ~ ( ) ( ~ ) (
min
q C c d q C c c c d H h q q hq CComp d H h ch c
u w W v ϕ δ ξ
q q d H h
q C c ch
~ ), ( , 1
) (
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( , 1
) (
q T t d H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
q q d H h
f hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q d H d H d H h
l f hq q C c ch ch ch
~ )}, ( ), ( { \ ) ( , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
δ ξ ξ ξ
q q d H h
l hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
) ~ ( ,
) (
q C c
c d H h ch
∈ ≥
∑
∈
ϕ ξ ) ( ), ~ ( }, 1 , { d H h q C c
ch
∈ ∈ ∈ ξ q q d H h
hq
~ ), ( }, 1 , { ∈ ∈ ∈ δ
) ~ ( }, 1 , { q C c
c
∈ ∈ ϕ
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7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for column generation based ILP
∑ ∑ ∑ ∑ ∑
∈ ∈ ∈ ∈ ∈
− − + −
) ~ ( ) ~ ( ) ( ~ ) (
min
q C c d q C c c c d H h q q hq CComp d H h ch c
u w W v ϕ δ ξ
q q d H h
q C c ch
~ ), ( , 1
) (
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( , 1
) (
q T t d H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
q q d H h
f hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q d H d H d H h
l f hq q C c ch ch ch
~ )}, ( ), ( { \ ) ( , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
δ ξ ξ ξ
q q d H h
l hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
) ~ ( ,
) (
q C c
c d H h ch
∈ ≥
∑
∈
ϕ ξ ) ( ), ~ ( }, 1 , { d H h q C c
ch
∈ ∈ ∈ ξ q q d H h
hq
~ ), ( }, 1 , { ∈ ∈ ∈ δ
) ~ ( }, 1 , { q C c
c
∈ ∈ ϕ
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7-11 Jan 2013, Aussois
2 Weekly Schedule Types ILP: subproblem for column generation based ILP
∑ ∑ ∑ ∑ ∑
∈ ∈ ∈ ∈ ∈
− − + −
) ~ ( ) ~ ( ) ( ~ ) (
min
q C c d q C c c c d H h q q hq CComp d H h ch c
u w W v ϕ δ ξ
q q d H h
q C c ch
~ ), ( , 1
) (
∈ ∈ ≤
∑
∈
ξ
) ~ ( ), ( , 1
) (
q T t d H h
t C c ch
∈ ∈ ≤
∑
∈
ξ
q q d H h
f hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ −
∑
∈ +
δ ξ ξ
q q d H d H d H h
l f hq q C c ch ch ch
~ )}, ( ), ( { \ ) ( , ) (
) ( 1 1
∈ ∈ ≤ − + −
∑
∈ + −
δ ξ ξ ξ
q q d H h
l hq q C c ch ch
~ ), ( , ) (
) ( 1
∈ ∈ ≤ + −
∑
∈ −
δ ξ ξ
) ~ ( ,
) (
q C c
c d H h ch
∈ ≥
∑
∈
ϕ ξ ) ( ), ~ ( }, 1 , { d H h q C c
ch
∈ ∈ ∈ ξ q q d H h
hq
~ ), ( }, 1 , { ∈ ∈ ∈ δ
) ~ ( }, 1 , { q C c
c
∈ ∈ ϕ
SLIDE 86
7-11 Jan 2013, Aussois
Computational Experiments
Set of instances:
- DDS1,…,DDS7 real-life cases from Italian universities
- test1,…,test4 proposed by Di Gaspero and Schaerf [2003]
- comp01,…,comp21 proposed for the ITC2007 (real-life cases from the
University of Udine)
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Computational Experiments
Set of instances:
- DDS1,…,DDS7 real-life cases from Italian universities
- test1,…,test4 proposed by Di Gaspero and Schaerf [2003]
- comp01,…,comp21 proposed for the ITC2007 (real-life cases from the
University of Udine) Computational experiments on all instances except from those whose optimal solution is zero.
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7-11 Jan 2013, Aussois
Computational Experiments
Set of instances:
- DDS1,…,DDS7 real-life cases from Italian universities
- test1,…,test4 proposed by Di Gaspero and Schaerf [2003]
- comp01,…,comp21 proposed for the ITC2007 (real-life cases from the
University of Udine) Computational experiments on all instances except from those whose optimal solution is zero. Tests on Intel Xeon E5310 (Dual Core, 1.6 GHz), 8 GB ram, Cplex 11.2.1 (one thread only) Time limit of 60 seconds for subproblem for x variables Total time limit of 40 CPU time units (22,800 seconds)
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Comparison between the proposed ILP formulations
Lower bound Computing time
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7-11 Jan 2013, Aussois
Computational Experiments
Lower bound derived from the solution of the subproblem on the x variables Computing time for the subproblem on the x variables Number of connected components for which we solve the IP model Number of connected components for which we solve the column generation model Number of times the subproblem for column generation (master) is solved Lower bound derived from the solution of the subproblem (slave) Computing time for the solving the subproblem (slave) Computing time for solving the column generation (master) Computing time for solving the subproblem on the y variables Obtained lower bound Total computing time Average percentage gap
x
lb
x
T IP LP iter
y
lb
s
T
m
T
y
T lb T
gap %
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7-11 Jan 2013, Aussois
Basic Formulation
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Basic Formulation
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Basic Formulation
The proposed method improves on the best-known LB in 9 out of 13 instances comp01-comp14. It proves optimality of the best-known solutions for DDS4 and comp20.
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Extended Formulation
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Extended Formulation
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The proposed method obtains the best-known or better LBs in 14 out of 26 instances, improving over the best-known LBs in 4 cases. In 7 out of 16 instances for which the optimal solution is known it is able to get a LB with the same value as the optimal solution. It proves optimality for the first time for test1.
Extended Formulation
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7-11 Jan 2013, Aussois
Comparison with state-of-the-art methods for the Extended Formulation
1T 10T 40T 1T 10T 40T 1T 10T 40T 1T 10T 40T 60.45 46.85 35.13 40.38 31.25 28.69 37.79 29.25 27.18 54.34 28.86 22.05 3 6 5 11 7
Lach and Luebbecke [2012] Burke et al. [2010] Hao and Benlic [2011] Proposed method
CPU %Gap #Best LB #Best LB Only
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7-11 Jan 2013, Aussois
Comparison with Asin Acha and Nieuwenhuis
T %Gap Best LB T %Gap Best LB 56,098 31.10 7 3,726 17.57 11 Asin Acha and Nieuwenhuis [2012] Proposed method
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7-11 Jan 2013, Aussois
Conclusion and Future Research
- The proposed method for computing lower bounds is based on splitting
the objective function into two parts and on formulating the two parts as ILPs.
- The computational experiments show that the proposed method was
able to improve some best known lower bounds and to prove optimality for some instances.
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7-11 Jan 2013, Aussois
Conclusion and Future Research
- The proposed method for computing lower bounds is based on splitting
the objective function into two parts and on formulating the two parts as ILPs.
- The computational experiments show that the proposed method was
able to improve some best known lower bounds and to prove optimality for some instances.
- Further research can be conducted to determine the optimal solution of
more instances, by applying branch and cut and price methods.
- This can be obtained by speeding up the computation through for