What is the decision version of integer programming?
It's the same was for all NP problems; the optimisation problem is


  Find a valid solution $s$ that minimises¹ $f(s)$!


and the corresponding decision problem is


  Is there a valid solution $s$ with $f(s) \leq k$?


You see that the former immediately solves the latter, and you can solve the former by using the latter with binary search over the set of feasible $k$.




Of course, you can have "maximise" here and "$\geq k$" in the decision version.