In this thesis, we analyze how sequentially introducing decision variables into an integer program (IP) affects the value function and its level sets. We use a Gilmore-Gomory approach to find parametrized IP value functions over a restricted set of variables. We introduce the notion of maximal connected \color{black}subsets of level sets - volumes in which changes to the constraint right-hand side have no effect on the value function - and relate these structures to IP value functions and optimal solutions.