  In this paper, we present an exact algorithm for optimizing two linear
fractional over the efficient set of a multi-objective integer quadratic
problem. This type of problems arises when two decision-makers, such as firms,
each have a preference function to optimize over the efficient set of a
multi-objective problem. The algorithm employs a branch-and-cut approach, which
involves: (1) exploring the solution space using a branch-and-bound strategy in
the decision space, and (2) eliminating inefficient solutions using a cutting
plane technique with efficient cuts constructed from the non-increasing
directions of objective functions. Additionally, integral tests are
incorporated to further ensure the efficiency of the obtained solutions.We
present a comprehensive example, accompanied by a step-by-step resolution, to
demonstrate the functioning of the algorithm.
