I need to solve a job affectation problem and I would like to find preferably efficient algorithms to solve this problem.
Let's say there are some workers that can do several kind of tasks. We also have a pool of tasks which must be done every week. Each task takes some time. Each task must be taken by someone. Each worker must work between N an P hours a week.
This first part of the problem seems to be a good candidate for a constraint programming algorithm.
But here is the complication: because a worker can do different tasks they may also have preferences (or wishes). If one want to satisfy all wishes for everyone there is no solution to the problem (too many constraints).
So I need an algorithm to solve this problem. I don't want to reinvent the wheel if the perfect wheel already exists.
The algorithm must be fair (if one can define this word) so for example I should be able to add a constraint like "try to satisfy at least one wish per people". I'm not sure that this problem can be solved by Constraint Hierarchies methods described here: Constraint Herarchies. In fact I'm not sure that "fairness" and wishes can be expressed by valid constraints for this category of algorithms.
Is there a constraint programming expert to give me some advices ? Do I need to develop a new wheel with some heuristics instead of using efficient CP algorithms ?
Thanks !
Your problem is complex enough that a general solution will probably require formulating as a linear-integer problem. If on the other hand you are able to relax certain requirements, you may be able to use a simpler approach. For example, bipartite matching would allow you to schedule multiple workers to multiple jobs, and can even handle preferences, but would not be able to enforce general 'fairness' constraints. See e.g. this related SO question. Vertex colouring has efficient algorithms for enforcing job separation constraints.
Other posters have mentioned simplex and job shop scheduling. Simplex is an optimisation algorithm - it traverses a solution space seeking to maximise some objective function. Formulating the objective function can certainly be done, but is non-trivial. Classical job shop scheduling, like bipartite matching, can model some aspects of your problem, but not all. There are no precedence constraints, for example. There are extended versions that can handle some constraints, for example placing time bounds on tasks.
If you're interested in existing implementations, the Python networkx library has an implementation of this matching algorithm. An example of an open source timetabling program that might be of interest is Tablix.
I've done timetabling, which can be considered a form of constraint programming. You have hard (inviolable) constraints and soft constraints (such as interval preferences).
Linear integer programming usually becomes useless after more than 30 variables, and this can also be said about simplex.
It was trough domain-specific optimizations of heuristic algorithms that a solution was found.
The heuristic algorithms used were simmulated annealing, genetic algorithms, metaheuristic algorithms and similar, but in the end the best result were provided by an "intelligent" domain customized greedy search algorithm.
Basically, you might get some decent results with one of the heuristics here, but the main problem is being able to discern when a problem is overconstrained.
A great open-source tool for research is the HeuristicLab.
I agree with what have been proposed here. However, MIP (Mixed Integer Programming problems) of very large size (far beyond 30 variables !) are practically solved nowadays thanks to commercial codes (Xpress, Cplex, Gurobi) or open-source (Coin-Or/Cbc). Furthermore, fancy modeling languages such as OPL Studio, GAMS, AMPL, Flop ... allow to write easily mathematic models instead of using APIs.
You can take advantage of NEOS server (http://neos.mcs.anl.gov/neos/solvers/index.html) to try very esaily different MIPs available. You send your model in AMPL format . Although AMPL comes as a free limited version, NEOS can handle unlimited instances.
Modeling languages exist also for CP (COMET / OPL Studio) and Local Search (COMET).
Feel free to get in touch with me through my web site www.rostudel.com ('contact' page)
David
This sounds like job shop scheduling.
Related
I've implemented the algorithms marked as the correct answer in this question: What to use for flow free-like game random level creation?
However, using that method will create boards that may have multiple solutions. I was wondering if there is any simple restrictions or modification that can be made to the algorithm to make sure that there is only one possible solution?
Creating unique Numberlink/Flow Free is very difficult. If you look at my algorithm proposal in the mentioned thread, you'll find an algorithm that lets you create puzzles with the necessary condition that solutions must not have a 2x2 square of the same color. The discussion at http://forum.ukpuzzles.org/viewtopic.php?f=3&t=41, however, shows that this is insufficient, since there are also many non-trivial non-unique puzzles.
From my looking into this problem, it seems the only way to solve this problem is to have a separate algorithm for testing uniqueness, and discarding bad instances. One solver that's made precisely for uniqueness testing algorithm is Imo's solver.
Another option is to use multiple different solvers and check that they come up with the same solution.
I think you should implement the solver, which finds all the solutions for some level. The simplest way is backtracking.
When you have many levels, take one by one and look for solutions. As soon as you find the second solution for some level, throw that level away.
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'm looking to find/create a routing algorithm that can be used to manage multiple vans performing deliveries as well as the loads of each of those vans.
Here's a rough specification of what I'm looking for..
The routes should be calculated in a fast and efficient manner
100+ vans / 1000+ packages / 1000+ dropoff points could be processed in one go
Each van could be a different size and have different weight restrictions
Each package could be a different size and weight
The packages should be organised onto the vans in a fair and economical manner, taking into account the routes, weight and size restrictions
The routes the vans should take should be economical and as short as possible (or a configurable balance between the two)
Vans could be limited to certain roads (low bridges, width, height and weight restrictions)
Some packages may be given timeslots for delivery
Has anyone seen this sort of thing before, and if so, any ideas as to what algorithm could be used to do this, or an example of how it could be done? I've seen a few university papers but they're quite old (probably fairly inefficient now) and don't handle the package management - they just presume all the vans and packages are the same size.
Any thoughts would be appreciated!
Rich
My impression is that this kind of problem routinely comes up in Operations Research, and the standard approach is to use a mixed integer programming solver. Here's an example of encoding a cargo shipping scheduling problem using MIP
Apparently 15 years of recent research in MIP made modern solvers 30,000 times faster than original ones.
If you want to make a solution from scratch, you could start by figuring out what your objective and constraints are, and then use some ideas from integer programming, like approximate branch-and-bound search.
pgRouting has a new function implementing a genetic algorithm for the Dial-a-Ride Problem: http://www.pgrouting.org/docs/1.x/darp.html
It's an extension of PostgreSQL/PostGIS and you can build an application with this. It also has functions for shortest path search, etc.
Any algorithm this specific is going to be proprietary and you will probably need to buy something. However, this sounds suspiciously like a problem that could be solved with a genetic algorithm implimentation. Here is some research I found:
http://www.ijimt.org/papers/38-M415.pdf
http://www.springerlink.com/content/w3165x33n24v8610/ (A book that looks like its focused on your problem)
http://www.computer.org/portal/web/csdl/doi/10.1109/ICCIT.2008.407
Just because an algorithm is old, doesn't mean its not efficient.
I'm writing a scheduling program with a difficult programming problem. There are several events, each with multiple meeting times. I need to find an arrangement of meeting times such that each schedule contains any given event exactly once, using one of each event's multiple meeting times.
Obviously I could use brute force, but that's rarely the best solution. I'm guessing this is a relatively basic computer science problem, which I'll learn about once I am able to start taking computer science classes. In the meantime, I'd prefer any links where I could read up on this, or even just a name I could Google.
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
There are several ways to do this
One approach is to do constraint programming. It is a special case of the dynamic programming suggested by feanor. It is helful to use a specialized library that can do the bounding and branching for you. (Google for "gecode" or "comet-online" to find libraries)
If you are mathematically inclined then you can also use integer programming to solve the problem. The basic idea here is to translate your problem in to a set of linear inequalities. (Google for "integer programming scheduling" to find many real life examples and google for "Abacus COIN-OR" for a useful library)
My guess is that constraint programming is the easiest approach, but integer programming is useful if you want to include real variables in you problem at some point.
Your problem description isn't entirely clear, but if all you're trying to do is find a schedule which has no overlapping events, then this is a straightforward bipartite matching problem.
You have two sets of nodes: events and times. Draw an edge from each event to each possible meeting time. You can then efficiently construct the matching (the largest possible set of edges between the nodes) using augmented paths. This works because you can always convert a bipartite graph into an equivalent flow graph.
An example of code that does this is BIM. Standard graphing libraries such as GOBLIN and NetworkX also have bipartite matching implementations.
This sounds like this could be a good candidate for a dynamic programming solution, specifically something similar to the interval scheduling problem.
There are some visuals here for the interval scheduling problem specifically, which may make the concept clearer. Here is a good tutorial on dynamic programming overall.
I'm working on an interactive job scheduling application. Given a set of resources with corresponding capacity/availabilty profiles, a set of jobs to be executed on these resources and a set of constraints that determine job sequence and earliest/latest start/end times for jobs I want to enable the user to manually move jobs around. Essentially I want the user to be able to "grab" a node of the job network and drag that forwards/backwards in time without violating any of the constraints.
The image shows a simple example configuration. The triangular job at the end denotes the latest finish time for all jobs, the connecting lines between jobs impose an order on the jobs and the gray/green bars denote resource availabilty and load.
You can drag any of the jobs to compress the schedule. Note that jobs will change in length due to different capacity profiles.
I have implemented an ad-hock algorithm that kinda works. However there are still cases where it'll fail and violate some constraints. However, since job-shop-scheduling is a well researched field with lots of algorithms and heuristics for finding an optimal (or rather good) solution to the general NP-hard problem - I'm thinking solutions ought to exist for my easier subset. I have looked into constraint programming topics and even physics based solutions (rigid bodies connected via static joints) but so far couldn't find anything suitable. Any pointers/hints/tips/search key words for me?
I highly recommend you take a look at Mozart Oz, if your problem
deals only with integers. Oz has excellent support for finite domain
constraint specification, inference, and optimization. In your case
typically you would do the following:
Specify your constraints in a declarative manner. In this, you would
specify all the variables and their domains (say V1: 1#100, means
V1 variable can take values in the range of 1--100). Some variables
might have values directly, say V1: 99. In addition you would specify
all the constraints on the variables.
Ask the system for a solution: either any solution which satisfies
the constraints or an optimal solution. Then you would display this
solution on the UI.
Lets say the user changes the value of a variable, may be the start
time of a task. Now you can go to step 1 to post the problem to the
Oz solver. This time, solving the problem most probably will not take
as much time as earlier, as all the variables are already instantiated.
It may be the case that the user chose a inconsistent value. In that
case, the solver returns null. Then, you can take the UI to the earlier
solution.
If Oz suits your need and you like the language, then you may want to
write a constraint solver as a server which listens on a socket. This way,
you can keep the constraint solver separate from the rest of your code,
including the UI.
Hope this helps.
I would vote in favor of constraint programming for several reasons:
1) CP will quickly tell you if there is no schedule that satifies your constraints
2) It would appear that you want to give you users a feasible solution to start with but
allow them to manipulate jobs in order to improve the solution. CP is good at this too.
3) An MILP approach is usually complex and hard to formulate and you have to artificially create an objective function.
4) CP is not that difficult to learn especially for experienced programmers - it really comes more from the computer science community than the operations researchers (like me).
Good luck.
You could probably alter the Waltz constraint propagation algorithm to deal with changing constraints to quickly find out if a given solution is valid. I don't have a reference to hand, but this might point you in the right direction:
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TYF-41C30BN-5&_user=809099&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1102030809&_rerunOrigin=google&_acct=C000043939&_version=1&_urlVersion=0&_userid=809099&md5=696143716f0d363581a1805b34ae32d9
Have you considered using an Integer Linear Programming engine (like lp_solve)? It's quite a good fit for scheduling applications.