To combine different types of uncertain information from different sources under different frameworks, we need transformations between different frameworks. For the transformation of a fuzzy membership function (FMF) into a basic belief assignment (BBA), several approaches have been proposed. Among these approaches, the uncertainty optimization based transformations can provide BBAs without predefining focal elements. However, these two transformations, which respectively use the uncertainty maximization and minimization criteria, emphasize the extreme cases of uncertainty. We expect to obtain a BBA, which is the trade-off between the two BBAs obtained by solving the uncertainty maximization and minimization, to avoid extreme attitudinal bias. In this paper, we propose two transformations of an FMF into a BBA by using a user-specified weighting factor to obtain such a trade-off (or balanced) BBA. Some examples and related analyses are provided to show the rationality and effectiveness of the proposed transformations.