We propose a flexible and general algorithmic framework for solving bi-objective mixed integer linear programming problems. The Pareto frontier of these problems can have a complex structure, as it can include isolated points, open, half-open and closed line segments. Therefore, its exact detection is an achievable though hard computational task. Operating in the criterion space, our algorithm features a simple structure that relies on an ordered exploration of the feasible region, making it relatively easy to implement. Our framework is oracle-based and iteratively solves two classes of subproblems: mixed integer and bi-objective linear programming problems. We introduce a family of cuts, called extreme-inequalities, to exclude dominated regions in the image space and improve the performance of the algorithm. Under specific assumptions, we show that the exact Pareto frontier can be detected in a finite number of iterations. Experimental results on a test-bed of instances and a comparison with the Triangle Splitting Method, a renowned criterion space method for bi-objective mixed integer problems, is presented, showing the notably good performance of our approach in terms of accuracy of the Pareto frontier detected and in terms of efficiency for medium size instances.