Optimization phase, when LC = 0. ⢠Prior is set of valid schedules. ⢠begins here at Cool = 10.7. ⢠continue optimizing until tired. Two phases in the computation ...
What is this problem? Dutch secondary school system A working algorithm
What is this problem?
• • •
Not Maximum Entropy Not Bayesian So, what are we doing here?
Large combinatorial problem
• • • • •
9000 lessons per week 95 teachers, 50% work part time 55 rooms, 2 locations 1200 students many other constraints and wishes
Dutch school system
• • • •
Lower level classes years 1 - 3, students 12 - 15 years old all classes have the same curriculum scheduling classes - teachers - rooms relatively easy
Upper levels
• • • • • •
Years 4 - 6, students 16 - 18 years old Common subjects ( Dutch, English, Science ) Branch subjects ( math & physics, biology & life science, economy, social science ) Optional subjects ( Spanish, drama, math++ ) Need for scheduling per individual student Hard problem
Students - subjects Name M/F Dutch
English
Math Bio Drama Math++
Frank M
1
1
0
1
0
0
Kevin m
1
1
1
0
1
1
Frieda f
1
1
1
0
1
0
1
1
0
1
0
0
Mark
m
Hierarchy in students
• • • •
Year: all studs of given year in given level, typically 200 students Cluster: associated with subject - students Class: 30 students, or less Group: all students with exactly the same subjects, typically 1 - 30 students
The partitioning of students in classes is the most computing intensive part
Subjects - hours - teachers Hours Classes
Dutch
4
5
Abe
Abe
Abe
Eve
Eve
English
3
5
Jan
Kay
Ivy
Jan
Kay
Math
3
3
John John John
Bio
2
2
Mary Mary
Drama
2
1
Lucia
Math++
2
1
John
Constraints
• • • • •
Lessons table (9000 lessons per week) Block hours, Combi hours Part time teaching Dutch labour laws Classes must fit in assigned rooms
Preferences
• • • • • • •
minimize number of idle hours minimize movements between locations avoid late hours in the day balance gender in classes balance class sizes same lessons always in same rooms et-cetera ...
A working algorithm
• • • • •
binary problem: student/teacher/room is occupied or not space is huge: 1.6 1010 search by MCMC algorithm of John Skilling there are many valid schedules choose one - based on preferences
MCMC
• • • • • •
20 members each member a trial schedule weighted sum of misfits: likelihood L evolve by stepping: compute ΔL accept step or reject step Decrease temperature of the system
MCMC
•
example steps:
• • • •
insert/delete 1 lesson for 1 class swap 2 lessons/hours/teachers in 1 class move student group to another class diffuse between members
MCMC at work
Two likelihoods
• •
Constraints likelihood: LC
•
any valid schedule MUST have LC = 0
Preference likelihood: LP
• •
minimization of LP ideally LP = 0, but this is unattainable in practice
Two phases in the computation
• •
Convergence phase, when LC > 0
•
Prior is availability of teachers*rooms
Optimization phase, when LC = 0
• • •
Prior is set of valid schedules begins here at Cool = 10.7 continue optimizing until tired
How many valid schedules are there?
•
Careful Combinatorics: about 103770 possibilities at Cool = 0
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