In this paper, we consider a special case of Bi-Objective Optimization (BOO), called Bi-Objective Minimization (BOM), where two objective functions to be minimized take only positive values. As well as for BOO, most methods proposed in the literature for solving BOM focus on computing the Pareto-optimal solutions that represent different trade-offs between two objectives. However, it may be difficult for a central decision-maker to determine the preferred solutions due to a huge number of solutions in the Pareto set. We propose a novel criterion for selecting the preferred Pareto-optimal solutions by introducing the concept of rho-Nash Fairness (rho-NF) solutions inspired from the definition of proportional fairness. The rho-NF solutions are the Pareto-optimal solutions achieving some proportional Nash equilibrium between the two objectives. The positive parameter rho is introduced to reflect the relative importance of the first objective to the second one. For this work, we will discuss some existential and algorithmic questions about the rho-NF solutions by first showing their existence for BOM. Furthermore, the set of rho-NF solutions can be a strict subset of the Pareto set. As there are possibly many rho-NF solutions, we focus on extreme rho-NF solutions achieving the smallest values for one of the objectives. Then, we propose two Newton-based iterative algorithms for finding extreme rho-NF solutions. Especially, this algorithm can be modified for finding all rho-NF solutions. Finally, we present computational results on some instances of the Bi-Objective Shortest Path Problem.