In this paper, a bi-objective integer programming problem is analysed using the characteristic equation that was developed to solve a single-objective pure integer program. This equation can also provides other ranked solutions i.e. 2nd, 3rd,... best solutions. These solutions are potential non-dominated points for a bi-objective integer program, which is being investigated in this paper. A “C” code is developed to solve the characteristic equation, a tool which is not available in the IBM ILOG CPLEX library. Two versions of this algorithm are developed to identify the non-dominated points for the bi-objective integer programming problem. The second version improves on the first by reducing the number of search steps. Computational experiments are carried out with respect to the two algorithms developed in this paper and comparisons have also been carried out with one of the recently developed method, the balanced box method. These computational experiments indicate that the second version of the algorithm developed in this paper performed significantly better than the first version and out performed the balanced box method with respect to both CPU time and the number of iterations.