Algorithm for creating a school timetable

Question

I\'ve been wondering if there are known solutions for algorithm of creating a school timetable. Basically, it\'s about optimizing \"hour-dispersion\" (both in teachers and

Answer 1

This problem is NP-Complete!
In a nutshell one needs to explore all possible combinations to find the list of acceptable solutions. Because of the variations in the circumstances in which the problem appears at various schools (for example: Are there constraints with regards to classrooms?, Are some of the classes split in sub-groups some of the time?, Is this a weekly schedule? etc.) there isn't a well known problem class which corresponds to all the scheduling problems. Maybe, the Knapsack problem has many elements of similarity with these problems at large.

A confirmation that this is both a hard problem and one for which people perennially seek a solution, is to check this (long) list of (mostly commercial) software scheduling tools

Because of the big number of variables involved, the biggest source of which are, typically, the faculty member's desires ;-)..., it is typically impractical to consider enumerating all possible combinations. Instead we need to choose an approach which visits a subset of the problem/solution spaces.
- Genetic Algorithms, cited in another answer is (or, IMHO, seems) well equipped to perform this kind of semi-guided search (The problem being to find a good evaluation function for the candidates to be kept for the next generation)
- Graph Rewriting approaches are also of use with this type of combinatorial optimization problems.

Rather than focusing on particular implementations of an automatic schedule generator program, I'd like to suggest a few strategies which can be applied, at the level of the definition of the problem.
The general rationale is that in most real world scheduling problems, some compromises will be required, not all constraints, expressed and implied: will be satisfied fully. Therefore we help ourselves by:

• Defining and ranking all known constraints
• Reducing the problem space, by manually, providing a set of additional constraints.
  This may seem counter-intuitive but for example by providing an initial, partially filled schedule (say roughly 30% of the time-slots), in a way that fully satisfies all constraints, and by considering this partial schedule immutable, we significantly reduce the time/space needed to produce candidate solutions.
  Another way additional constraints help is for example "artificially" adding a constraint which prevent teaching some subjects on some days of the week (if this is a weekly schedule...); this type of constraints results in reducing the problem/solution spaces, without, typically, excluding a significant number of good candidates.
• Ensuring that some of the constraints of the problem can be quickly computed. This is often associated with the choice of data model used to represent the problem; the idea is to be able to quickly opt-for (or prune-out) some of the options.
• Redefining the problem and allowing some of the constraints to be broken, a few times, (typically towards the end nodes of the graph). The idea here is to either remove some of constraints for filling-in the last few slots in the schedule, or to have the automatic schedule generator program stop shy of completing the whole schedule, instead providing us with a list of a dozen or so plausible candidates. A human is often in a better position to complete the puzzle, as indicated, possibly breaking a few of the contraints, using information which is not typically shared with the automated logic (eg "No mathematics in the afternoon" rule can be broken on occasion for the "advanced math and physics" class; or "It is better to break one of Mr Jones requirements than one of Ms Smith ... ;-) )

In proof-reading this answer , I realize it is quite shy of providing a definite response, but it none the less full of practical suggestions. I hope this help, with what is, after all, a "hard problem".

Answer 2

Here are a few links I found:

School timetable - Lists some problems involved

A Hybrid Genetic Algorithm for School Timetabling

Scheduling Utilities and Tools

Answer 3

UPDATE: from comments ... should have heuristics too!

I'd go with Prolog ... then use Ruby or Perl or something to cleanup your solution into a prettier form.

teaches(Jill,math).
teaches(Joe,history).

involves(MA101,math).
involves(SS104,history).

myHeuristic(D,A,B) :- [test_case]->D='<';D='>'.
createSchedule :- findall(Class,involves(Class,Subject),Classes),
                  predsort(myHeuristic,Classes,ClassesNew),
                  createSchedule(ClassesNew,[]).
createSchedule(Classes,Scheduled) :- [the actual recursive algorithm].

I am (still) in the process of doing something similar to this problem but using the same path as I just mentioned. Prolog (as a functional language) really makes solving NP-Hard problems easier.

Answer 4

My timetabling algorithm, implemented in FET (Free Timetabling Software, http://lalescu.ro/liviu/fet/ , a successful application):

The algorithm is heuristic. I named it "recursive swapping".

Input: a set of activities A_1...A_n and the constraints.

Output: a set of times TA_1...TA_n (the time slot of each activity. Rooms are excluded here, for simplicity). The algorithm must put each activity at a time slot, respecting constraints. Each TA_i is between 0 (T_1) and max_time_slots-1 (T_m).

Constraints:

C1) Basic: a list of pairs of activities which cannot be simultaneous (for instance, A_1 and A_2, because they have the same teacher or the same students);

C2) Lots of other constraints (excluded here, for simplicity).

The timetabling algorithm (which I named "recursive swapping"):

1. Sort activities, most difficult first. Not critical step, but speeds up the algorithm maybe 10 times or more.

2. Try to place each activity (A_i) in an allowed time slot, following the above order, one at a time. Search for an available slot (T_j) for A_i, in which this activity can be placed respecting the constraints. If more slots are available, choose a random one. If none is available, do recursive swapping:

   a. For each time slot T_j, consider what happens if you put A_i into T_j. There will be a list of other activities which don't agree with this move (for instance, activity A_k is on the same slot T_j and has the same teacher or same students as A_i). Keep a list of conflicting activities for each time slot T_j.

   b. Choose a slot (T_j) with lowest number of conflicting activities. Say the list of activities in this slot contains 3 activities: A_p, A_q, A_r.

   c. Place A_i at T_j and make A_p, A_q, A_r unallocated.

   d. Recursively try to place A_p, A_q, A_r (if the level of recursion is not too large, say 14, and if the total number of recursive calls counted since step 2) on A_i began is not too large, say 2*n), as in step 2).

   e. If successfully placed A_p, A_q, A_r, return with success, otherwise try other time slots (go to step 2 b) and choose the next best time slot).

   f. If all (or a reasonable number of) time slots were tried unsuccessfully, return without success.

   g. If we are at level 0, and we had no success in placing A_i, place it like in steps 2 b) and 2 c), but without recursion. We have now 3 - 1 = 2 more activities to place. Go to step 2) (some methods to avoid cycling are used here).

Answer 5

Generally, constraint programming is a good approach to this type of scheduling problem. A search on "constraint programming" and scheduling or "constraint based scheduling" both within stack overflow and on Google will generate some good references. It's not impossible - it's just a little hard to think about when using traditional optimization methods like linear or integer optimization. One output would be - does a schedule exist that satisfies all the requirements? That, in itself, is obviously helpful.

Good luck !

Answer 6

It's a mess. a royal mess. To add to the answers, already very complete, I want to point out my family experience. My mother was a teacher and used to be involved in the process.

Turns out that having a computer to do so is not only difficult to code per-se, it is also difficult because there are conditions that are difficult to specify to a pre-baked computer program. Examples:

• a teacher teaches both at your school and at another institute. Clearly, if he ends the lesson there at 10.30, he cannot start at your premises at 10.30, because he needs some time to commute between the institutes.
• two teachers are married. In general, it's considered good practice not to have two married teachers on the same class. These two teachers must therefore have two different classes
• two teachers are married, and their child attends the same school. Again, you have to prevent the two teachers to teach in the specific class where their child is.
• the school has separate facilities, like one day the class is in one institute, and another day the class is in another.
• the school has shared laboratories, but these laboratories are available only on certain weekdays (for security reasons, for example, where additional personnel is required).
• some teachers have preferences for the free day: some prefer on Monday, some on Friday, some on Wednesday. Some prefer to come early in the morning, some prefer to come later.
• you should not have situations where you have a lesson of say, history at the first hour, then three hours of math, then another hour of history. It does not make sense for the students, nor for the teacher.
• you should spread the arguments evenly. It does not make sense to have the first days in the week only math, and then the rest of the week only literature.
• you should give some teachers two consecutive hours to do evaluation tests.

As you can see, the problem is not NP-complete, it's NP-insane.

So what they do is that they have a large table with small plastic insets, and they move the insets around until a satisfying result is obtained. They never start from scratch: they normally start from the previous year timetable and make adjustments.