This paper deals with a particular case of Bi-Objective Optimization called Bi-Objective Discrete Optimization (BODO), where the feasible set is discrete, and the two objectives take only positive values. Since the feasible set of a BODO problem is discrete and usually finite, it can theoretically be enumerated to identify the Pareto set, which consists of all Pareto-optimal solutions representing different trade-offs between two objectives. However, in general, this problem is algorithmically unsolvable. From a practical point of view, the Central Decision Maker (CDM) may be interested in a reduced Pareto set reflecting the own preference of the CDM, which a computationally tractable algorithm can obtain. In a recent paper [14], we introduced the concept of generalized Nash Fairness (ρ-N F) solution for Bi-Objective Minimization Problems, where their two positive objectives are to be simultaneously minimized. The ρ-N F solutions are the Pareto-optimal solutions achieving some proportional fairness between two objectives. The positive parameter ρ provided by the CDM reflects the relative importance of the first objective compared to the second one. We then focused on finding two particular ρ-N F solutions, called extreme ρ-N F solutions, having the smallest values of each objective. In this paper, our main contribution is a stronger result than in [14]: we generalize the concept of ρ-N F solution for BODO problems where each objective can be either maximized or minimized. We then show that the ρ-N F solution set is a subset of the Pareto set, and this inclusion can be strict. We also propose several algorithms for determining the ρ-N F solution set, including a recursive Newton-like algorithm. Finally, an illustrative example of BODO is given.