Is there any mixed integer linear programming module capable of handling integer variables exactly?

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I have a mixed integer linear programming (MILP) problem with binary variables. I tried to treat binary variables as constrained real variables, but I could not provide a good solution to my problem. Are there any softwares which are able to give integer values to integer variables? Could you please recommend me some good modules?

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If you define variables as binary or integer and let MILP solvers (e.g., CBC/CPLEX/Gurobi) solve your model, the solution will have those variables as binary or integer. It cannot be any other way - these are hard constraints.

On the other hand, if your model specifies these variables as constrained real variables, then you have to set up the objective, and possibly other constraints in such a way that these variables still turn out to be binary/integer. Otherwise, they will just show up as real numbers in the solution. There are ways to do this, and is considered the crux of modeling optimization problem, and can only be done in certain situations.

Leaving this aside, this thread might help you get a list of solvers where your binary/integer constraints are honored.