I am trying to make a program in which, over 5 days, a new seating arrangement is produced every day. There is a constraint that must be followed:
People are seated in groups of 4, and cannot be in the same group with another person more than once over the course of the 5 days. There are roughly 30-35 people in total.
I have done a bit of research already, and it seems that this is a Constraint Satisfaction Problem, but I was wondering if this kind of problem has a more specific name, or specific algorithm to be used with it.
In addition, is it possible to determine if this constraint is able to be met with a given sample size? I am new to the realm of graph theory, and do not know which algorithms are viable in this situation.
The link below is a good starting point for the Social Golpher Problem. It contains solutions for the simpler cases (including some with 4-tables) and gives pointers to some more advanced papers: http://www.mathpuzzle.com/MAA/54-Golf%20Tournaments/mathgames_08_14_07.html
I need to develop a Course Timetabling software which can allot timeslots and rooms efficiently. This is a curriculum based routine, not post-enrollment based. And efficiently means classes are assigned timeslots according to staff time preferences and also need to minimize 1st year-2nd year class overlap so that 2nd year students can retake the courses they've failed to pass.(and also for 3rd-4th yr pair).
Now, at first i thought that would be an easy problem, but now it seems different. Most of the papers i've looked on uses Genetic Algorithm/PSO/Simulated Annealing or these type of algorithm. And i'm still unable to interpret the problem to a GA problem.
what i'm confused about is why almost none of them suggests DFS or Graph-coloring algorithm?
Can someone explain the scenario if DFS/graph-coloring is used? Or why they aren't suggested or tried.
My experience with solving this problem for a complex department, is that the hard constraints (like no overlapping of courses that are taken by the same population, and hard constraints of the teachers) are rather easily solvable by exact methods. I modeled the problem with 0-1 integer linear programming, and solved it with a SAT-based tool called minisat+. Competitive commercial tools like cplex can also solve it.
So with today's tools there is no need to approximate as suggested above, even when the input is rather large.
Now, optimizing the solution is a different story. There can be many (weighted) objectives, and finding the solution that brings the objective to minimum is indeed very hard computationally (no tool that I tried can solve it within 24 hours), but they reach near optimum in a few hours (I know it is near optimum because I can compute the theoretical bound on the solution).
This document describes applying a GA approach to university time-tabling, so it should be directly applicable to your requirement: Using a GA to solve university time-tabling
I need an algorithm to solve a problem with the following conditions:
There is a set of "n" people and another set of "m" workshops, there are more people than workshops. Each person has chosen a subset of size "j" of the total workshops and has assigned values to each depending on how much they would like to assist that particular workshop. Now, every workshop only has a limited amount of vacancies.
Given these conditions the problem would be:
What is the best way to assign people to workshops, so that each person participates in the workshop which she considers most valuable (given the problem constraints, that is, if a person can´t participate in their first choice, then the algorithm should choose the second, third, fourth, and so on).
I think the problem is related to combinatorial optimization but I don't know much about algorithms. If anyone can tell me the name of one from which to start investigating, I'd be very grateful.
Thanks! And please excuse my english.
This is a matching problem with one-sided preferences (in the sense that people have preferences for the workshops, but not the other way around).
Here is an excellent paper that discusses this problem in more detail: https://mattmccutchen.net/lumc/index.html
An optimal solution to this problem isn't particularly clear. There are many different optimal (Pareto efficient) criterions. Unfortunately, the problem is NP-hard for many of them.
However, there are criterion with polynomial time algorithms. There is a nice list of these in the "Related work" section of the paper I linked.
I'm working on a simple application that will generate time table (daily planner) for schools. I've read up basics of algorithms, but confused as to where to start.
The problem:
Allocate teachers to classes taking into consideration a lot of constraints like:
1) Subject
2) Expertise of teacher
3) Not more than 2 classes continuously.. etc
It goes without saying that there should be no overlapping. Basically I need to assign N teachers to M classes with a fixed number of working hours everyday (8).
The inputs:
1) Total number of classes
2) Teachers along with their subject expertise
3) Subjects/Courses for each class
4) Number of lectures per class per day
5) Other flexible constraints like minimum/maximum working hours for a teacher per day, total working hours per teacher per week, etc
My questions:
1) Is it right to look at it as an assignment problem with multiple constraints?
2) Which algorithm should I use? (Hungarian algorithm?)
3) Should I start by getting the whole set of constraints at one go, and then generate the table, or should it be done in intermediate steps?
I'm a beginner to learning/implementing algorithms, so any help to point me in the right direction appreciated! Thanks.
You've picked a doozy of a problem to start with. Scheduling optimization like this is NP complete. There are a ton of papers on how to deal with problems like this the class of problem is known as constrain satisfaction. You can perform an exhaustive search which is easiest but is also very time consuming, if you have more than a handful of classes it won't work. You might take a look at solver foundation which is a suite of tool for .net for solving these sorts of things. Scott Hanselman did a podcast about it
here http://www.hanselminutes.com/default.aspx?showID=209 and you can find more about it here
http://code.msdn.microsoft.com/solverfoundation. If you fancy doing it yourself try looking at GSAT or otherwise some of these evolutionary algorithms look interesting http://www.springer.com/engineering/book/978-3-540-48582-7.
This question keeps coming up at least once a week here and the answers (including mine) are always the same. I think we should create a specific tag on scheduling algorithms if one doesn't exist.
I'll reiterate that the most suitable solution technique for scheduling problems like this are in the area of constraint programming. While other optimization techniques are OK for small problems, formulation often is painful for some constraints. Consider all of the integer variables you have to create for some simple constraints. Because the problem often requires a bunch of feasible schedules (or to determine infeasibility) rather than an optimal solution, CP is the preferred approach since that's what it's designed to do. Most other approaches require a user to "force" an optimality condition on the problem where one doesn't really exist.
This is a problem I've had on my mind for a long time. Being the son of a teacher and a programmer, it occurred to me early on... but I still haven't found a solution for it.
So this is the problem. One needs to create a time schedule for a school, using some constraints. These are generally divided in two categories:
Sanity Checks
A teacher cannot teach two classes at the same time
A student cannot follow two lessons at the same time
Some teachers must have at least one day off during the week
All the days of the week should be covered by the time table
Subject X must have exactly so-and-so hours each week
...
Preferences
Each teacher's schedule should be as compact as possible (i.e. the teacher should work all hours for the day in a row with no pauses if possible)
Teachers that have days off should be able to express a preference on which day
Teachers that work part-time should be able to express a preference whether to work in the beginning or the end of the school day.
...
Now, after a few years of not finding a solution (and learning a thing or two in the meanwhile...), I realized that this smells like a NP-hard problem.
Is it proven as NP-hard?
Does anyone have an idea on how to crack this thing?
Looking at this question made me think about this problem, and whether genetic algorithms would be usable in this case. However it would be pretty hard to mutate possibilities while maintaining the sanity check rules. Also it's not clear to me how to distinguish incompatible requirements.
A small addendum to better specify the problem. This is applied to Italian school style classrooms where all students are associated in different classes (for example: year 1 section A) and the teachers move between classes. All students of the same class have the same schedule, and have no choice over which lessons to attend.
I am one of the developer that works on the scheduler part of a student information system.
During our original approach of the scheduling problem, we researched genetic algorithms to solve constraint satisfaction problems, and even though we were successful initially, we realized that there was a less complicated solution to the problem (after attending a school scheduling workshop)
Our current implementation works great, and uses brute force with smart heuristics to get a valid schedule in a short amount of time. The master schedule (assignment of the classes to the teachers) is first built, taking in consideration all the constraints that each teacher has while minimizing the possibility of conflicts for the students (based of their course requests). The students are then scheduled in the classes using the same method.
Doing this allows you to have the machine build a master schedule first, and then have a human tweak it if needed.
The scheduler current implementation is written in perl, but other options we visited early on were Prolog and CLIPS (expert system)
I think you might be missing some constraints.
One would prefer where possible to have teachers scheduled to classes for which they are certified.
One would suspect that the classes that are requested, and the expected headcount in each would be significant.
I think the place to start would be to list all of your constraints, figure out a data structure to represent them.
Then create some sort of engine to that builds a trial solution, then evaluates it for fitness according to the constraints.
You could then throw the fun genetic algorithms part at it, and see if you can get the fitness to increase over time as the genes mix.
Start with a small set of constraints, and increase them as you see success (if you see success)
There might be a way to take the constraints and shoehorn them together with something like a linear programming algorithm.
I agree. It sounds like a fun challenge
This is a mapping problem:
you need to map to every hour in a week and every teacher an activity (teach to a certain class or free hour ).
Split the problem:
Create the list of teachers, classes and preferences then let the user populate some of the preferences on a map to have as a starting point.
Randomly take one element from the list and put it at a random free position on the map
if it doesn't cross any sanity checks until the list is empty. If at any certain iteration you can't place an element on the map without crossing a sanity check shift two positions on the map and try again.
When the map is filled, try shifting positions on the map to optimize the result.
In steps 2 and 3 show each iteration to the user: items left in the list, positions on the map and the next computed move and let the user intervene.
I did not try this, but this would be my initial approach.
I've tackled similar planning/scheduling problems in the past and the AI technique that is often best suited for this class of problem is "Constraint Based Reasoning".
It's basically a brute force method like Laurenty suggested, but the approach involves applying constraints in an efficient order to cause the infeasible solutions to fail fast - to minimise the computation required.
Googling "Constraint Based Reasoning" brings up a lot of resources on the technique and its application to scheduling problems.
Answering my own question:
The FET project mentioned by gnud uses this algorithm:
Some words about the algorithm: FET
uses a heuristical algorithm, placing
the activities in turn, starting with
the most difficult ones. If it cannot
find a solution it points you to the
potential impossible activities, so
you can correct errors. The algorithm
swaps activities recursively if that
is possible in order to make space for
a new activity, or, in extreme cases,
backtracks and switches order of
evaluation. The important code is in
src/engine/generate.cpp. Please e-mail
me for details or join the mailing
list. The algorithm mimics the
operation of a human timetabler, I
think.
Link
Following up the "Constraint Based Reasoning" lead by Stringent Software on Wikipedia lead me to these pages which have an interesting paragraph:
Solving a constraint satisfaction
problem on a finite domain is an
NP-complete problem in general.
Research has shown a number of
polynomial-time subcases, mostly
obtained by restricting either the
allowed domains or constraints or the
way constraints can be placed over the
variables. Research has also
established relationship of the
constraint satisfaction problem with
problems in other areas such as finite
model theory and databases.
This reminds me of this blog post about scheduling a conference, with a video explanation here.
How I would do it:
Have the population include two things:
Who teaches what class (I expect the teachers to teach one subject).
What a class takes on a specific time slot.
This way we can't have conflicts (a teacher in 2 places, or a class having two subjects at the same time).
The fitness function would include:
How many time slots each teacher gives per week.
How many time slots a teacher has on the same day (They can't have a full day of teaching, this too must be balanced).
How many time slots of the same subject a class has on the same day (They can't have a full day of math!).
Maybe take the standard deviation for all of them since they should be balanced.
Looking at this question made me think
about this problem, and whether
genetic algorithms would be usable in
this case. However it would be pretty
hard to mutate possibilities while
maintaining the sanity check rules.
Also it's not clear to me how to
distinguish incompatible requirements.
Genetic Algorithms are very well suited to problems such as this. Once you come up with a decent representation of the chromosome (in this case, probably a vector representing all of the available class slots) you're most of the way there.
Don't worry about maintaining sanity checks during the mutation phase. Mutation is random. Sanity and preference checks both belong in the selection phase. A failed sanity check would drastically lower the fitness of an individual, while a failed preference would only mildly lower the fitness.
Incompatible requirements are a different problem altogether. If they're completely incompatible, you'll get a population that doesn't converge on anything useful.
Good luck. Being the son of a father with this sort of problem is what took me to the research group that I ended up in ...
When I was a kid my father scheduled match officials in a local sports league, this had a similarly long list of constraints and I tried to write something to help. When I got to University I even used it as my final year project eventually settling on a Monte Carlo implementation (using a Simulated Annealing model).
The basic idea is that if it's not NP, it's pretty close, so rather than assuming there is a solution, I would set out to find the best within a given timeframe. I would weight all the constraints with costs for breaking them: sanity checks would have huge costs, the preferences would have lower costs (but increasing for more breaks, so breaking it once would be less than half the cost of breaking it twice).
The basic idea is that I started with a 'random' solution and costed it; then made changes by swapping a small number of assignments, re-valued it and then, probalistically accepted or declined the change.
After thousands of iterations you inch closer to an acceptable solution.
Believe me, though, that this class of problems has research groups churning out PhDs so you're in very good company.
You might also find some interest in the Linear Programming arena, e.g. simplex and so on.
Yes, I think this is NP complete - or at least to find the optimal solution is NP complete.
I worked on a similar problem in college when i told a friend's father (who was a teacher) that I could solve his scheduling problems for him if he did not find a suitable program for it (this was back in 1990 or so)
I had no idea what I got myself into. Luckily for us all I had to do was find one solution that fit the constraints. But in my testing I was always worried about determining IF there was a solution at all. He never had too many constraints and the program used different heuristics and back tracking. It was a lot of fun.
I think Bill Gates also worked on a system like this in high school or college for his high school. Not sure though.
Good luck. All my notes are gone and I never got around to implementing a solution that I could market. It was a specialty project that I re-coded as I learned new languages (Basic, Scheme, C, VB, C++)
Have fun with it
i see that this problem can be solved by Prolog program by connecting it to a database
and the program can make the schedule given a set of constraints
read abt "Constraint satisfaction Problem prolog"