Linear fractional optimization for integer indefinite quadratic fractional problem

In this article, we propose a new exact method to solve an Integer Indefinite Quadratic Fractional Programming problem $ (\texttt{IIQFP}) $, in which the objective function is a product of two positive fractional functions subject to linear constraints and integer variables. This method is based on transforming the original problem $ (\texttt{IIQFP}) $ into an equivalent one that uses linear tools rather than non-linear ones. To do so, we define a bi-objective integer linear fractional programming problem $ (\texttt{BOILFP}) $ by separating the indefinite quadratic fractional objective function of the original problem. The corresponding efficient set of this latter problem is obtained by solving a sequence of linear problems, using a method based on a branch-and-bound algorithm, efficiency tests, efficient cuts and upper and lower bounds relative to ideal and nadir point. We define an equivalent problem in which the indefinite quadratic fractional objective function is optimized over the efficient solution set of the bi-objective problem. The integer optimal solution of the $ (\texttt{IIQFP}) $ is reached at one of the efficient solutions of $ (\texttt{BOILFP}) $, corresponding to the optimal value for the indefinite quadratic fractional objective function. An illustrative example is provided to explain the proposed method, followed by the convergence of the algorithm and a table of computational results.