The classical three-dimensional bin packing problem (3D-BPP) orthogonally packs a set of rectangular items with varying dimensions into the minimum number of three-dimensional rectangular bins. While ensuring the minimum number of bins used, the safety of the logistic operations is addressed with the complementary loadbalancing objective for which concepts such as orientation and stability are used in the literature though not at the same time. In this study, we extend the load-balanced 3D-BPP by combining both orientation and stability, and introducing a new concept called family unity which encourages packing a family of products (e.g., from the same order and with the same destination) together. Although item related concerns are very common in practice, there are no multi-objective studies in the bin packing literature that includes family unity concept. Therefore, this is the first study that proposes a multi-objective mixed integer programming model for the extended problem to determine the optimal packing plan that minimizes the number of bins used and the deviation of balance from the ideal barycenter while maximizing family unity ratio via a weighted objective function. A numerical example is provided to analyze the performance of the proposed model. Furthermore, a real-life container loading problem is solved and outputs of the study implies the practical advantages of including family unity and load-balance considerations in solving 3D-BPP problems.