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Eliz Cisneroz
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 classes case) for given class-subject-teacher associations. We can assume that we have sets of classes, lesson subjects and teachers associated with each other at the input and that timetable should fit between 8AM and 4PM.
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.
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:
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".
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:
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.
The International Timetabling Competition 2007 had a lesson scheduling track and exam scheduling track. Many researchers participated in that competition. Lots of heuristics and metaheuristics were tried, but in the end the local search metaheuristics (such as Tabu Search and Simulated Annealing) clearly beat other algorithms (such as genetic algorithms).
Rules were made for "illegal tables": two classes in the same classroom, one teacher teaching two groups at the same time etc. These mutations were deemed lethal immediately and a new "organism" was sprouted in place of the "deceased" immediately. The initial one was generated by a series of random tries to get a legal (if senseless) one. Lethal mutation wasn't counted towards count of mutations in iteration.
Small bonuses were assigned for bundling certain 2 hours together, for assigning same generic classroom in sequence for the same group, for keeping teacher's work hours and class' load continuous. Moderate bonuses were assigned for giving correct classrooms for given subject, keeping class hours within bonds (morning or afternoon), and such. Big bonuses were for assigning correct number of given subject, given workload for a teacher etc.
Teachers could create their workload schedules of "want to work then", "okay to work then", "doesn't like to work then", "can't work then", with proper weights assigned. Whole 24h were legal work hours except night time was very undesired.
The weight function... oh yeah. The weight function was huge, monstrous product (as in multiplication) of weights assigned to selected features and properties. It was extremely steep, one property easily able to change it by an order of magnitude up or down - and there were hundreds or thousands of properties in one organism. This resulted in absolutely HUGE numbers as the weights, and as a direct result, need to use a bignum library (gmp) to perform the calculations. For a small testcase of some 10 groups, 10 teachers and 10 classrooms, the initial set started with note of 10^-200something and finished with 10^+300something. It was totally inefficient when it was more flat. Also, the values grew a lot wider distance with bigger "schools".
Computation time wise, there was little difference between a small population (100) over a long time and a big population (10k+) over less generations. The computation over the same time produced about the same quality.
The resulting program never saw daylight outside getting me a good grade for the semester. It showed some promise but I never got enough motivation to add any GUI and make it usable to general public.
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.
For instance, is the same teacher working 5 days a week for X weeks straight? Even if that is a working solution, it might not be a better solution than alternating between two people so that each teacher does one week each. Oh, you didn't think about that? Remember, this is people you're dealing with, not just a resource allocation problem.
Even if one teacher could work full-time for 16 weeks straight, that might be a sub-optimal solution compared to a solution where you try to alternate between teachers, and this kind of balancing is very hard to build into software.
To summarize, producing a good solution to this problem will be worth a lot, to many many people. Hence, it's not an easy problem to break down and solve. Be prepared to stake out some goals that aren't 100% and calling them "good enough".
I work on a widely-used scheduling engine which does exactly this. Yes, it is NP-Complete; the best approaches seek to approximate an optimal solution. And, of course there are a lot of different ways to say which one is the "best" solution - is it more important that your teachers are happy with their schedules, or that students get into all their classes, for instance?
The absolute most important question you need to resolve early on is what makes one way of scheduling this system better than another? That is, if I have a schedule with Mrs Jones teaching Math at 8 and Mr Smith teaching Math at 9, is that better or worse than one with both of them teaching Math at 10? Is it better or worse than one with Mrs Jones teaching at 8 and Mr Jones teaching at 2? Why?
The main advice I'd give here is to divide the problem up as much as possible - maybe course by course, maybe teacher by teacher, maybe room by room - and work on solving the sub-problem first. There you should end up with multiple solutions to choose from, and need to pick one as the most likely optimal. Then, work on making the "earlier" sub-problems take into account the needs of later sub-problems in scoring their potential solutions. Then, maybe work on how to get yourself out of painted-into-the-corner situations (assuming you can't anticipate those situations in earlier sub-problems) when you get to a "no valid solutions" state.
Note that typically we are dealing with highly resource-constrained systems in school scheduling. Schools don't go through the year with a lot of empty rooms or teachers sitting in the lounge 75% of the day. Approaches which work best in solution-rich environments aren't necessarily applicable in school scheduling.
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.
I have designed commercial algorithms for both class timetabling and examination timetabling. For the first I used integer programming; for the second a heuristic based on maximizing an objective function by choosing slot swaps, very similar to the original manual process that had been evolved. They main things in getting such solutions accepted are the ability to represent all the real-world constraints; and for human timetablers to not be able to see ways to improve the solution. In the end the algorithmic part was quite straightforward and easy to implement compared with the preparation of the databases, the user interface, ability to report on statistics like room utilization, user education and so on.
This problem is MASSIVE where I work - imagine 1800 subjects/modules, and 350 000 students, each doing 5 to 10 modules, and you want to build an exam in 10 weeks, where papers are 1 hour to 3 days long... one plus point - all exams are online, but bad again, cannot exceed the system's load of max 5k concurrent. So yes we are doing this now in cloud on scaling servers.The "solution" we used was simply to order modules on how many other modules they "clash" with descending (where a student does both), and to "backpack" them, allowing for these long papers to actually overlap, else it simply cannot be done.So when things get too large, I found this "heuristic" to be practical... at least.
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