I have got a job scheduling problem. We are given start time, time to
complete the order, deadline.
It is given that start time + time to
complete <= deadline.
I have also been given the loss that will occur if I am not able to
complete the job before the deadline. I have to design an algorithm to minimize the loss.
I have tried changing the standard algorithm of dynamic programming for maximizing the profit in job scheduling but to no success.
What algorithm can I use to solve the question?
Dynamic Programming isn't the right approach based on what you're aiming to optimize. You can find the optimized schedule by using a greedy approach.
Here's a thorough guide with sample code for your desire language (C++), in the guide it assumes each jobs takes only 1 unit of time, which you can easily modify by using time_to_complete instead.
Your problem is similar to the knapsack one. Using a greedy approach is convenient if you aren't actually looking for the best possible solution, but just a "good enough" one.
The big pro of the greedy approach is that the cost is rather lower than other "more thorough" approaches but, if you need the best solution to your problem, I would say that backtracking is the way to go.
Since the deadline can be violated, the problems looks like a Total Weighted Tardiness Scheduling Problem. There are many flavors of it, but most problems under this umbrella are computationally hard, therefore Dynamic Programming (DP) would not be my first choice. In my experience, DP also poses difficulties during modeling and implementation. Same comment for mathematical programming "as-is". Some approaches that can be implemented more quickly are:
constraint programming: very small learning curve, and there are many libraries out there, included very good open source ones (most have C++ API). Bonus: constraint programming can demonstrate optimality.
ad hoc heuristics: (1) start with a constructive algorithm (like the greedy approach suggested by Ling Zhong and Flavio Giobergia), then (2) use some local search approach to improve if and finally (3) embed the approach into a metaheuristic scheme. This way you can build on top of the previous step, and learn a lot about the problem. Note: in general, heuristics cannot demonstrate optimality
special mention: local solver, a hybrid approach between the two above: it lets you model the problem using a formalism similar to constraint programming and then it solves it using heuristics. It is very easy to learn, it usually lets you get started quickly and, in my tests, it provides remarkably good results.
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I am developing an algorithm that has a large numerical computation part. My project supervisor recommended me to use Fortran because of this and so for the last weeks I've been work on it (so far so good). It would be a new version of an algorithm of his, which is basically a lot of numerical computing.
Mine, however, would have more "logic" to it. Without going into much detail, the brute force approach is done using just fortran because it's just 95% of reading from a file and doing the operations. However, the aim of the project is to provide an efficient algorithm to do this, I had been thinking about methods and wanted to start with a Greedy approach (something like Hill Climbing) and that got me thinking that for this part in particular, maybe it would be better to write the algorithm in C++ instead of Fortran.
So basically, how hard do you think it would be to develop the algorithm "logic" in C++ and then call Fortran whenever the bulk of the numerical computation has to be performed. Would it be worth it? Or should I just stick with one of the two languages?
Sorry if it is a very ignorant question but I can't get an idea of whether writing an algorithm such as Hill Climbing would be more difficult if done with Fortran instead of C++ and the benefits of Fortran in this case would be worth it.
Thanks for your time and have a nice day!
What is the best online solution to solve a linear programming problem?
I heard about several like Gurobi.
One thing I especially want is the possibility to get an approximate solution when the exact resolution takes too long.
The most comprehensive online optimization system is NEOS. It takes models in a variety of input formats and has a wide range of solvers.
Many solvers have settings to allow them to terminate early, even before optimality is reached, if you want an approximate and quick solution. But often your best bet in that case is to use a heuristic algorithm designed specifically for your problem.
I have an integer linear optimisation problem and I'm interested in feasible, good solutions. As far as I know, for example the Gnu Linear Programming Kit only returns the optimal solution (given it exists).
This takes endless time and is not exactly what I'm looking for: I would be happy with any good solution, not only the optimal one.
So a LP-Solver that e.g. stops after some time and returns the best solution he found so far, would do the job.
Is there any such software? It would be great if that software was open source or at least free as in beer.
Alternatively: Is there any other way that usually speeds up Integer LP problems?
Is this the right place to ask?
Many solvers provide a time limit parameter; if you set the time limit parameter, they will stop once the time limit is reached. If an integer feasible solution has been found, it will return the best feasible solution found to that point.
As you may know, integer programming is NP-hard, and there is a real art to finding optimal solutions as well as good feasible solutions quickly. To compare the different solvers, see Prof. Hans Mittelmann's Benchmarks for Optimization Software. The MILP benchmarks - particularly MIPLIB2010 and the Feasibility Benchmark should be most relevant.
In addition to selecting a good solver, there are many things that can be done to improve solve times including tuning the parameters of the solver and model reformulation. Many people in research and industry - including myself - spend our careers working on improving the solve times of MIP models, both in general and for specific models.
If you are an academic user, note that the top commercial systems like CPLEX and Gurobi are free for academic use. See the respective websites for details.
Finally, you may want to look at OR-Exchange, a sister site to Stack Overflow that focuses on the field of operations research.
(Disclaimer: I currently work for Gurobi Optimization and formerly worked for ILOG, which provided CPLEX).
If you would like to get a feasibel integer solution fast and if you don't need the optimal solution, you can try
Increase the relative or absolute Gap. Usually solvers have small gaps of say 0.0001% for relative gap. This means that the solver will continue searching for MIP solutions until it the MIP solution is not farther than 0.0001% away from the optimal solution. Increase this gab to e.g. 1%., So you get good solution, but the solver will not spent a long time in proving optimality.
Try different values for solver parameters concerning MIP heuristics.
CPLEX and GUROBI have parameters to control, MIP emphasis. This means that the solver will put more emphasis on looking for feasible solutions or on proving optimality. Set emphasis to feasible MIP solutions.
Most solvers like CPLEX, Gurobi, MOPS or GLPK have settings for gap and heuristics. MIP emphasis can be set - as far as I know - only in CPLEX and Gurobi.
A usual approach for solving ILP is branch-and-bound. This utilized the solution of many sub-LP (without-I). The finally optimal result is the best of all sub-LP. As at least one solution is found you could stop anytime and would have a best-so-far.
One package that could do it, is the free lpsolve. Look there at set_timeout for giving a time limit, and when it is ILP the solve function can return in SUPOPTIMAL the best_so_far value.
As far as I know CPLEX can. It can return the solution pool which contains primal feasible solutions in the search, and if you specify the search focus on feasibility rather on optimality, more faesible solutions can be generated. At the end you can just export the pool. You can use the pool to do a hot start so it's pretty up to you. CPlex is free now at least in my country as you can sign up as a researcher.
Could you take into account Microsoft Solver Foundation? The only restriction is technology stack that you prefer and here you should use, as you guess, Microsoft technologies: C#, vb.net, etc. Here is example how to use it with Excel: http://channel9.msdn.com/posts/Modeling-with-Solver-Foundation-30 .
Regarding to your question it is possible to have not a fully optimized solutions if you set efficiency (for example 85% or 0.85). In outcome you can see all possible solutions for such restriction.
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I'm looking to add some genetic algorithms to an Operations research project I have been involved in. Currently we have a program that aids in optimizing some scheduling and we want to add in some heuristics in the form of genetic algorithms. Are there any good libraries for generic genetic programming/algorithms in c++? Or would you recommend I just code my own?
I should add that while I am not new to c++ I am fairly new to doing this sort of mathematical optimization work in c++ as the group I worked with previously had tended to use a proprietary optimization package.
We have a fitness function that is fairly computationally intensive to evaluate and we have a cluster to run this on so parallelized code is highly desirable.
So is c++ a good language for this? If not please recommend some other ones as I am willing to learn another language if it makes life easier.
thanks!
I would recommend rolling your own. 90% of the work in a GP is coding the genotype, how it gets operated on, and the fitness calculation. These are parts that change for every different problem/project. The actual evolutionary algorithm part is usually quite simple.
There are several GP libraries out there ( http://en.wikipedia.org/wiki/Symbolic_Regression#Implementations ). I would use these as examples and references though.
C++ is a good choice for GP because they tend to be very computationally intensive. Usually, the fitness function is the bottleneck, so it's worthwhile to at least make this part compiled/optimized.
I use GAUL
it's a C library with all you want.
( pthread/fork/openmp/mpi )
( various crossover / mutation function )
( non GA optimisation: Hill-Climbing, N-M Simplex, Simulated annealling, Tabu, ... )
Why build your own library when there is such powerful tools ???
I haven't used this personally yet, but the Age Layered Population Structure (ALPS) method has been used to generate human competitive results and has been shown to outperform several popular methods in finding optimal solutions in rough fitness landscapes. Additionally, the link contains source code in C++ FTW.
I have had similar problems. I used to have a complicated problem and defining a solution in terms of a fixed length vector was not desirable. Even a variable length vector does not look attractive. Most of the libraries focus on cases where the cost function is cheap to calculate which did not match my problem. Lack of parallelism is their another pitfall. Expecting the user to allocate memory for being used by the library is adding insult into injury. My cases were even more complicated because most of the libraries check the nonlinear conditions before evaluation. While, I needed to check the nonlinear condition during or after the evaluation based on the result of the evaluation. It is also undesirable when I needed to evaluate the solution to calculate its cost and then I had to recalculate the solution to present it. In most of the cases, I had to write the cost function two times. Once for GA and once for presentation.
Having all of these problems, I eventually, designed my own openGA library which is now mature.
This library is based on C++ and distributed with free Mozilla Public License 2.0. It guarantees that using this library does not limit your project and it can be used for commercial or none commercial purposes for free without asking for any permission. Not all libraries are transparent in this sense.
It supports three modes of single objective, multiple objective (NSGA-III) and Interactive Genetic Algorithm (IGA).
The solution is not mandated to be a vector. It can be any structure with any customized design containing any optional values with variable length. This feature makes this library suitable for Genetic Programming (GP) applications.
C++11 is used. Template feature allows flexibility of the solution structure design.
The standard library is enough to use this library. There is no dependency beyond that. The entire library is also a single header file for ease of use.
The library supports parallelism by default unless you turn it off. If you have an N-core CPU, the number of threads are set to N by default. You can change the settings. You can also set if the solution evaluations are distributed between threads equally or they are assigned to any thread which has finished its job and is currently idle.
The solution evaluation is separated from calculation of the final cost. It means that your evaluation function can simulate the system and keep a lot of information. Your cost function is called later and reports the cost based on the evaluation. While your evaluation results are kept to be used later by the user. You do not need to re-calculate it again.
You can reject a solution at any time during the evaluation. No waste of time. In fact, the evaluation and constraint check are integrated.
The GA assist feature help you to produce the C++ code base from the information you provide.
If these features match what you need, I recommend having a look at the user manual and the examples of openGA.
The number of the readers and citation of the related publication as well as its github favorite marks is increasing and its usage is keep growing.
I suggest you have a look into the matlab optimization toolkit - it comes with GAs out of the box, you only haver to code the fitness function (and a function to generate inital population eventually) and I believe matlab has some C++ interoperability so you could code you functions in C++. I am using it for my experiments and a very nice feature is that you get all sorts of charts out of the box as well.
Said so - if your aim is to learn about genetic algorithms you're better off coding it, but if you just want to run experiments matlab and C++ (or even just matlab) is a good option.
I am newbie for integer linear programming.
I plan to use a integer linear programming solver to solve my combinatorial optimization problem.
I am more familiar with C++/object oriented programming on an IDE.
Now I am using NetBeans with Cygwin to write my applications most of time.
May I ask if there is an easy use ILP solver for me?
Or it depends on the problem I want to solve ? I am trying to do some resources mapping optimization. Please let me know if any further information is required.
Thank you very much, Cassie.
If what you want is linear mixed integer programming, then I would point to Coin-OR (and specifically to the module CBC). It's Free software (as speech)
You can either use it with a specific language, or use C++.
Use C++ if you data requires lots of preprocessing, or if you want to put your hands into the solver (choosing pivot points, column generation, adding cuts and so on...).
Use the integrated language if you want to use the solver as a black box (you're just interested in the result and the problem is easy or classic enough to be solved without tweaking).
But in the tags you mention genetic algorithms and graphs algorithms. Maybe you should start by better defing your problem...
For graphs I like a lot Boost::Graph
I have used lp_solve ( http://lpsolve.sourceforge.net/5.5/ ) on a couple of occasions with success. It is mature, feature rich and is extremely well documented with lots of good advice if your linear programming skills are rusty. The integer linear programming is not a just an add on but is strongly emphasized with this package.
Just noticed that you say you are a 'newbie' at this. Well, then I strongly recommend this package since the documentation is full of examples and gentle tutorials. Other packages I have tried tend to assume a lot of the user.
For large problems, you might look at AMPL, which is an optimization interpreter with many backend solvers available. It runs as a separate process; C++ would be used to write out the input data.
Then you could try various state-of-the-art solvers.
Look into GLPK. Comes with a few examples, and works with a subset of AMPL, although IMHO works best when you stick to C/C++ for model setup. Copes with pretty big models too.
Linear Programming from Wikipedia covers a few different algorithms that you could do some digging into to see which may work best for you. Does that help or were you wanting something more specific?