here thousands of variables or involved.and i implement this problem on GAMS software and i got some answer so I want to know that this solution is correct or not but how to check this?Any idea plz help as soon as possible.
I think it is best to read basic material on this topic:
1. http://onlinelibrary.wiley.com/doi/10.1002/mcda.4020030204/full
2. http://www.co.it.pt/~judice/Articles/AlvesDempeJudice2010.pdf
There are of course basic text books that will let you know - in principle - which properties that allow you to conclude that solutions exist. Ehrgott has written a good one,
More:
3. http://link.springer.com/book/10.1007%2F978-3-540-88908-3
4.Multiobjective Optimization: Principles and Case Studies by Yann Collette, Patrick Siarry
There are two questions raised: whether a given software provides a correct solution, and whether a given problem has a solution or not. The former is less easy to know, without knowing which problem is being solved, and what software is being used. The second is easier - perhaps - provided that the term "solution" is well-defined. There certainly are good books to read on the subject of characterizing solutions to bi- and multi-objective optimization problems, and criteria that let you know that a "solution" does exist. Which ones to you have, and do you need pointers to books?
Since integer programming is np-hard, the solution you got is not optimal (unless the problem was very small which does not seem to be the case) and so you just have a heuristic solution and want to know if it is feasible or not.
I would just replace the solution values (of all the variables) in the objective function and see if they are in fact a solution (if they satisfy the equation). I am assuming it is an exact problem (objective function f=0).
If its is a maximization or minimization problem, or if you want to make sure the solution is any good, maybe you can judge the quality of the solution experimentally from its comparison with randomly generated values for the variables. I don't know if the literature points to procedures like this, but if your solution clearly stands from randomly obtained "solutions" GAMS is probably doing things right. Check also what model GAMS is using, to be more certain that you are using the right one.
Kind regards.
Fernando.
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