I don't know what is meant by "decision version of integer programming".
I know ILP, but this terminology has me confused.
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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 $k$.