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I've got a job that I've been asked to do which involves writing a program to determine where various people are to work on a given day.
For example the input may be:
4-6pm, Site A
1-2pm, Site B
9-11am & 2-4pm Site A
Essentially there can be many sites and people can work during multiple blocks. I get the feeling that this kind of problem has been solved long ago so rather than reinventing the wheel I was hoping somebody could point me in the direction of an elegant solution.
Edit: Reading similar questions I get the feeling that the problem may be NP complete. I don't need the most efficient solution only something that works and is reasonably ok.
Edit 2: To clarify, the output should be a timetable with people allocated such that gaps (instances where nobody is working) is as small as possible.
Looks like you are trying to solve a problem for which there are quite specialized software applications. If your problem is small enough, you could try to do a brute force approach like this:
Determine the granularity of your problem. E.g. if this is for a school time table, your granularity can be 1 hour.
Make instances for every time slot divided by the granularity. So for Site B there will be one instance (1-2pm), for size A there will be many instances (4-5pm, 5-6pm, 9-10am, 10-11am, ...).
Make instances for all the persons you want to assign
Then iteratively assign a person to a free time slot taking all of your constraints into account (like holiday, lunch break, only being able to do one thing at a time, maximum number of people per site, ...) until:
you have a valid solution (fine, print it out and exit the application)
you enter a situation where you still need to place persons but there is no valid time slot anymore. Then backtrack (undo the last action and try to find an alternative for it).
If your problem is not too big you could find a solution in reasonable time.
However, if the problem starts to get big, look for more specialized software. Things to look for are "constraint based optimization" and "constraint programming".
E.g. the ECLIPSe tool is an open-source constraint programming environment. You can find some examples on http://eclipseclp.org/examples/index.html. One nice example you can find there is the SEND+MORE=MONEY problem. In this problem you have the following equation:
S E N D
+ M O R E
-----------
= M O N E Y
Replace every letter by a digit so that the sum is correct.
This also illustrates that although you can solve this brute-force, there are more intelligent ways to solve this (see http://eclipseclp.org/examples/sendmore.pl.txt).
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.
I guess that there is probably no accurate algorithm for that, but maybe someone knows a good approximation or hints for developing it.
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".
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.
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).
Take a look at the 2 open source frameworks used by some of the finalists:
JBoss OptaPlanner (Java, open source)
Unitime (Java, open source) - more for universities
One of my half-term assignments was an genetic-algorithm school table generation.
Whole table is one "organism". There were some changes and caveats to the generic genetic algorithms approach:
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.
"Exchange" mutations were much more common than "Modify" mutations. Changes were only between parts of the gene that made sense - no substituting a teacher with a classroom.
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 calculation (on some 1GHz CPU) would take some 1h to stabilize near 10^+300, generating schedules that looked quite nice, for said 10x10x10 test case.
The problem is easily paralellizable by providing networking facility that would exchange best specimens between computers running the computation.
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.
This problem is tougher than it seems.
As others have alluded to, this is a NP-complete problem, but let's analyse what that means.
Basically, it means you have to look at all possible combinations.
But "look at" doesn't tell you much what you need to do.
Generating all possible combinations is easy. It might produce a huge amount of data, but you shouldn't have much problems understanding the concepts of this part of the problem.
The second problem is the one of judging whether a given possible combination is good, bad, or better than the previous "good" solution.
For this you need more than just "is it a possible solution".
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".
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"):
Sort activities, most difficult first. Not critical step, but speeds up the algorithm maybe 10 times or more.
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).
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.
Genetic algorithms are often used for such scheduling.
Found this example (Making Class Schedule Using Genetic Algorithm) which matches your requirement pretty well.
Here are a few links I found:
School timetable - Lists some problems involved
A Hybrid Genetic Algorithm for School Timetabling
Scheduling Utilities and Tools
This paper describes the school timetable problem and their approach to the algorithm pretty well: "The Development of SYLLABUS—An Interactive, Constraint-Based Scheduler for Schools and Colleges."[PDF]
The author informs me the SYLLABUS software is still being used/developed here: http://www.scientia.com/uk/
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.
A local-search optimization pass is often used to "polish" the end answer for better results.
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.
Good luck !
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.
You can takle it with genetic algorithms, yes. But you shouldn't :). It can be too slow and parameter tuning can be too timeconsuming etc.
There are successful other approaches. All implemented in open source projects:
Constraint based approach
Implemented in UniTime (not really for schools)
You could also go further and use Integer programming. Successfully done at Udine university and also at University Bayreuth (I was involved there) using the commercial software (ILOG CPLEX)
Rule based approach with heuristisc - See Drools planner
Different heuristics - FET and my own
See here for a timetabling software list
I think you should use genetic algorithm because:
It is best suited for large problem instances.
It yields reduced time complexity on the price of inaccurate answer(Not the ultimate best)
You can specify constraints & preferences easily by adjusting fitness punishments for not met ones.
You can specify time limit for program execution.
The quality of solution depends on how much time you intend to spend solving the program..
Genetic Algorithms Definition
Genetic Algorithms Tutorial
Class scheduling project with GA
Also take a look at :a similar question and another one
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.
I don't know any one will agree with this code but i developed this code with the help of my own algorithm and is working for me in ruby.Hope it will help them who are searching for it
in the following code the periodflag ,dayflag subjectflag and the teacherflag are the hash with the corresponding id and the flag value which is Boolean.
Any issue contact me.......(-_-)
periodflag.each do |k2,v2|
if(TimetableDefinition.find(k2).period.to_i != 0)
subjectflag.each do |k3,v3|
if (v3 == 0)
if(getflag_period(periodflag,k2))
#teachers=EmployeesSubject.where(subject_name: #subjects.find(k3).name, division_id: division.id).pluck(:employee_id)
#teacherlists=Employee.find(#teachers)
teacherflag=Hash[teacher_flag(#teacherlists,teacherflag,flag).to_a.shuffle]
teacherflag.each do |k4,v4|
if(v4 == 0)
if(getflag_subject(subjectflag,k3))
subjectperiod=TimetableAssign.where("timetable_definition_id = ? AND subject_id = ?",k2,k3)
if subjectperiod.blank?
issubjectpresent=TimetableAssign.where("section_id = ? AND subject_id = ?",section.id,k3)
if issubjectpresent.blank?
isteacherpresent=TimetableAssign.where("section_id = ? AND employee_id = ?",section.id,k4)
if isteacherpresent.blank?
#finaltt=TimetableAssign.new
#finaltt.timetable_struct_id=#timetable_struct.id
#finaltt.employee_id=k4
#finaltt.section_id=section.id
#finaltt.standard_id=standard.id
#finaltt.division_id=division.id
#finaltt.subject_id=k3
#finaltt.timetable_definition_id=k2
#finaltt.timetable_day_id=k1
set_school_id(#finaltt,current_user)
if(#finaltt.save)
setflag_sub(subjectflag,k3,1)
setflag_period(periodflag,k2,1)
setflag_teacher(teacherflag,k4,1)
end
end
else
#subjectdetail=TimetableAssign.find_by_section_id_and_subject_id(#section.id,k3)
#finaltt=TimetableAssign.new
#finaltt.timetable_struct_id=#subjectdetail.timetable_struct_id
#finaltt.employee_id=#subjectdetail.employee_id
#finaltt.section_id=section.id
#finaltt.standard_id=standard.id
#finaltt.division_id=division.id
#finaltt.subject_id=#subjectdetail.subject_id
#finaltt.timetable_definition_id=k2
#finaltt.timetable_day_id=k1
set_school_id(#finaltt,current_user)
if(#finaltt.save)
setflag_sub(subjectflag,k3,1)
setflag_period(periodflag,k2,1)
setflag_teacher(teacherflag,k4,1)
end
end
end
end
end
end
end
end
end
end
end
I'd just like someone to verify whether the following problem is NP-complete or if there is actually a better/easier solution to it than simple brute-force combination checking.
We have a sort-of resource allocation problem in our software, and I'll explain it with an example.
Let's say we need 4 people to be at work during the day-shift. This number, and the fact that it is a "day-shift" is recorded in our database.
However, we don't require just anyone to fill those spots, there's some requirements that needs to be filled in order to fit the bill.
Of those 4, let's say 2 of them has to be a nurse, and 1 of them has to be doctors.
One of the doctors also has to work as part of a particular team.
So we have this set of information:
Day-shift: 4
1 doctor
1 doctor, need to work in team A
1 nurse
The above is not the problem. The problem comes when we start picking people to work the day-shift and trying to figure out if the people we've picked so far can actually fill the criteria.
For instance, let's say we pick James, John, Ursula and Mary to work, where James and Ursula are doctors, John and Mary are nurses.
Ursula also works in team A.
Now, depending on the order we try to fit the bill, we might end up deducing that we have the right people, or not, unless we start trying different combinations.
For instance, if go down the list and pick Ursula first, we could match her with the "1 doctor" criteria. Then we get to James, and we notice that since he doesn't work in team A, the other criteria about "1 doctor, need to work in team A", can't be filled with him. Since the other two people are nurses, they won't fit that criteria either.
So we backtrack and try James first, and he too can fit the first criteria, and then Ursula can fit the criteria that needs that team.
So the problem looks to us as we need to try different combinations until we've either tried them all, in which case we have some criteria that aren't filled yet, even if the total number of heads working is the same as the total number of heads needed, or we've found a combination that works.
Is this the only solution, can anyone think of a better one?
Edit: Some clarification.
Comments to this question mentions that with this few people, we should go with brute-force, and I agree, that's probably what we could do, and we might even do that, in the same lane that some sort optimizations look at the size of the data and picks different sort algorithms with less initial overhead if the data size is small.
The problem though is that this is part of a roster planning system, in which you might have quite a few number of people involved, both as "We need X people on the day shift" as well as "We have this pool of Y people that will be doing it", as well as potential for a large "We have this list of Z criteria for those X people that will have to somehow match up with these Y people", and then you add to the fact that we will have a number of days to do the same calculation for, in real-time, as the leader adjusts the roster, and then the need for a speedy solution has come up.
Basically, the leader will see a live sum information on-screen that says how many people are still missing, both on the day-shift as a whole, as well as how many people is fitting the various criteria, and how many people we actually ned in addition to the ones we have. This display will have to update semi-live while the leader adjusts the roster with "What if James takes the day-shift instead of Ursula, and Ursula takes the night-shift".
But huge thanks to the people that has answered this so far, the constraint satisfaction problem sounds like the way we need to go, but we'll definitely look hard at all the links and algorithm names here.
This is why I love StackOverflow :)
What you have there is a constraint satisfaction problem; their relationship to NP is interesting, because they're typically NP but often not NP-complete, i.e. they're tractable to polynomial-time solutions.
As ebo noted in comments, your situation sounds like it can be formulated as an exact cover problem, which you can apply Knuth's Algorithm X to. If you take this tack, please let us know how it works out for you.
It does look like you have a constraint satisfaction problem.
In your case I would particularly look at constraint propagation techniques first -- you may be able to reduce the problem to a manageable size that way.
What happens if no one fits the criteria?
What you are describing is the 'Roommate Problem' it is lightly described in this thesis.
Bear with me, I'm searching for better links.
EDIT
Here's another fairly dense thesis.
As for me I would most likely trying to find reduction to bipartite graph matching problem. Also to prove that problem is NP usually is much more complicated than staying you cannot find polynomial solution.
I am not sure your problem is NP, it does not smell that way, but what I would do if I was you would be to order the requirements for the positions such that you try to fill the most specific first since fewer people will be available fill these positions, so you are less likely to have to backtrack a lot. There is no reason why you should not combine this with algorithm X, an algorithm of pure Knuth-ness.
I'll leave the theory to others, since my mathematical savvy is not so great, but you may find a tool like Cassowary/Cassowary.net or NSolver useful to represent your problem declaratively as a constraint satisfaction problem and then solve the constraints.
In such tools, the simplex method combined with constraint propagation is frequently employed to deterministically reduce the solution space and then find an optimal solution given a cost function. For larger solution spaces (which don't seem to apply in the size of problem you specify), occasionally genetic algorithms are employed.
If I remember correctly, NSolver also includes in sample code a simplification of an actual Nurse-rostering problem that Dr. Chun worked on in Hong Kong. And there's a paper on the work he did.
It sounds to me like you have a couple of separable problems that would be a lot easier to solve:
-- select one doctor from team A
-- select another doctor from any team
-- select two nurses
So you have three independent problems.
A clarification though, do you have to have two doctors (one from the specified team) and two nurses, or one doctor from the specified team, two nurses, and one other that can be either doctor or nurse?
Some Questions:
Is the goal to satisfy the constraints exactly, or only approximately (but as much as possible)?
Can a person be a member of several teams?
What are all possible constraints? (For example, could we need a doctor which is a member of several teams?)
If you want to satisfy the constraints exactly, then I would order the constraints decreasingly by strictness, that is, the ones which are most hardest to achieve (e.g. doctor AND team A in your example above) should be checked first!
If you want to satisfy the constraints approximately, then its a different story... you would have to specify some kind of weighting/importance-function which determines what we rather would have, when we can't match exactly, and have several possibilities to choose from.
If you have several or many constraints, take a look at Drools Planner (open source, java).
Brute force, branch and bound and similar techniques take to long. Deterministic algorithms such as fill the largest shifts first are very suboptimal. Meta-heuristics are a very good way to deal with this.
Take a specific look at the real-world nurse rostering example of Drools Planner. It's easy to add many constraints, such as "young nurses don't want to work the Saturday night" or "some nurses don't want to work to many days in a row".
I'm developing an application that optimally assigns shifts to nurses in a hospital. I believe this is a linear programming problem with discrete variables, and therefore probably NP-hard:
For each day, each nurse (ca. 15-20) is assigned a shift
There is a small number (ca. 6) of different shifts
There is a considerable number of constraints and optimization criteria, either concerning a day, or concerning an emplyoee, e.g.:
There must be a minimum number of people assigned to each shift every day
Some shifts overlap so that it's OK to have one less person in early shift if there's someone doing intermediate shift
Some people prefer early shift, some prefer late shift, but a minimum of shift changes is required to still get the higher shift-work pay.
It's not allowed for one person to work late shift one day and early shift the next day (due to minimum resting time regulations)
Meeting assigned working week lengths (different for different people)
...
So basically there is a large number (aout 20*30 = 600) variables that each can take a small number of discrete values.
Currently, my plan is to use a modified Min-conflicts algorithm
start with random assignments
have a fitness function for each person and each day
select the person or day with the worst fitness value
select at random one of the assignments for that day/person and set it to the value that results in the optimal fitness value
repeat until either a maximum number of iteration is reached or no improvement can be found for the selected day/person
Any better ideas? I am somewhat worried that it will get stuck in a local optimum. Should I use some form of simulated annealing? Or consider not only changes in one variable at a time, but specifically switches of shifts between two people (the main component in the current manual algorithm)? I want to avoid tailoring the algorithm to the current constraints since those might change.
Edit: it's not necessary to find a strictly optimal solution; the roster is currently done manual, and I'm pretty sure the result is considerably sub-optimal most of the time - shouldn't be hard to beat that. Short-term adjustments and manual overrides will also definitely be necessary, but I don't believe this will be a problem; Marking past and manual assignments as "fixed" should actually simplify the task by reducing the solution space.
This is a difficult problem to solve well. There has been many academic papers on this subject particularly in the Operations Research field - see for example nurse rostering papers 2007-2008 or just google "nurse rostering operations research". The complexity also depends on aspects such as: how many days to solve; what type of "requests" can the nurse's make; is the roster "cyclic"; is it a long term plan or does it need to handle short term rostering "repair" such as sickness and swaps etc etc.
The algorithm you describe is a heuristic approach.
You may find you can tweak it to work well for one particular instance of the problem but as soon as "something" is changed it may not work so well (e.g. local optima, poor convergence).
However, such an approach may be adequate depending your particular business needs - e.g. how important is it to get the optimal solution, is the problem outline you describe expected to stay the same, what is the potential savings (money and resources), how important is the nurse's perception of the quality of their rosters, what is the budget for this work etc.
Umm, did you know that some ILP-solvers do quite a good job? Try AIMMS, Mathematica or the GNU programming kit! 600 Variables is of course a lot more than the Lenstra theorem will solve easily, but sometimes these ILP solvers have a good handle and in AIMMS, you can modify the branching strategy a little. Plus, there's a really fast 100%-approximation for ILPs.
I solved a shift assignment problem for a large manufacturing plant recently. First we tried generating purely random schedules and returning any one which passed the is_schedule_valid test - the fallback algorithm. This was, of course, slow and indeterminate.
Next we tried genetic algorithms (as you suggested), but couldn't find a good fitness function that closed on any viable solution (because the smallest change can make the entire schedule RIGHT or WRONG - no points for almost).
Finally we chose the following method (which worked great!):
Randomize the input set (i.e. jobs, shift, staff, etc.).
Create a valid tuple and add it to your tentative schedule.
If not valid tuple can be created, rollback (and increment) the last tuple added.
Pass the partial schedule to a function that tests could_schedule_be_valid, that is, could this schedule be valid if the remaining tuples were filled in a possible way
If !could_schedule_be_valid, simply rollback (and increment) the tuple added in (2).
If schedule_is_complete, return schedule
Goto (2)
You incrementally build a partial shift this way. The benefit is that some tests for valid schedule can easily be done in Step 2 (pre-tests), and others must remain in Step 5 (post-tests).
Good luck. We wasted days trying the first two algorithms, but got the recommended algorithm generating valid schedules instantly in under 5 hours of development.
Also, we supported pre-fixing and post-fixing of assignments that the algorithm would respect. You simply don't randomize those slots in Step 1. You'll find that the solutions doesn't have to be anywhere near optimal. Our solution is O(N*M) at a minimum but executes in PHP(!) in less than half a second for an entire manufacturing plant. The beauty is in ruling out bad schedules quickly using a good could_schedule_be_valid test.
The people that are used to doing it manually don't care if it takes an hour - they just know they don't have to do it manually any more.
Mike,
Don't know if you ever got a good answer to this, but I'm pretty sure that constraint programming is the ticket. While a GA might give you an answer, CP is designed to give you many answers or tell you if there is no feasible solution. A search on "constraint programming" and scheduling should bring up lots of info. It's a relatively new area and CP methods work well on many types of problems where traditional optimization methods bog down.
Dynamic programming a la Bell? Kinda sounds like there's a place for it: overlapping subproblems, optimal substructures.
One thing you can do is to try to look for symmetries in the problem. E.g. can you treat all nurses as equivalent for the purposes of the problem? If so, then you only need to consider nurses in some arbitrary order -- you can avoid considering solutions such that any nurse i is scheduled before any nurse j where i > j. (You did say that individual nurses have preferred shift times, which contradicts this example, although perhaps that's a less important goal?)
I think you should use genetic algorithm because:
It is best suited for large problem instances.
It yields reduced time complexity on the price of inaccurate answer(Not the ultimate best)
You can specify constraints & preferences easily by adjusting fitness punishments for not met ones.
You can specify time limit for program execution.
The quality of solution depends on how much time you intend to spend solving the program..
Genetic Algorithms Definition
Genetic Algorithms Tutorial
Class scheduling project with GA
Also take a look at :a similar question and another one
Using CSP programming I made programms for automatic shitfs rostering. eg:
2-shifts system - tested for 100+ nurses, 30 days time horizon, 10+
rules
3-shifts system - tested for 80+ nurses, 30 days time horizon, 10+ rules
3-shifts system, 4-teams - tested for 365 days horizon, 10+ rules,
and a couple of similiar systems. All of them were tested on my home PC (1.8GHz, dual-core). Execution times always were acceptable ie. for 3/ it took around 5 min and 300MB RAM.
Most hard part of this problem was selecting proper solver and proper solving strategy.
Metaheuristics did very well at the International Nurse Rostering Competition 2010.
For an implementation, see this video with a continuous nurse rostering (java).
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"