In this paper, we propose a beamforming design that jointly considers two conflicting performance metrics, namely the sum rate and fairness, for a multiple-input single-output non-orthogonal multiple access system. Unlike the conventional rate-aware beamforming designs, the proposed approach has the flexibility to assign different weights to the objectives (i.e., sum rate and fairness) according to the network requirements and the channel conditions. In particular, the proposed design is first formulated as a multi-objective optimization problem, and subsequently mapped to a single objective optimization (SOO) problem by exploiting the weighted sum approach combined with a prior articulation method. As the resulting SOO problem is non-convex, we use the sequential convex approximation technique, which introduces multiple slack variables, to solve the overall problem. Simulation results are provided to demonstrate the performance and the effectiveness of the proposed approach along with detailed comparisons with conventional rate-aware-based beamforming designs.