Since the advent of data clustering, the original formulation of the clustering problem has been enriched to incorporate a number of twists to widen its range of application. In particular, recent heuristic approaches have proposed to incorporate restrictions on the size of the clusters, while striving to minimize a measure of dissimilarity within them. Such size constraints effectively constitute a way to exploit prior knowledge, readily available in many scenarios, which can lead to an improved performance in the clustering obtained. In this paper, we build upon a modification of the celebrated k-means method resorting to a similar alternating optimization procedure, endowed with additive partition weights controlling the size of the partitions formed, adjusted by means of the Levenberg-Marquardt algorithm. We propose several further variations on this modification, in which different kinds of additional information are present. We report experimental results on various standardized datasets, demonstrating that our approaches outperform existing heuristics for size-constrained clustering. The running-time complexity of our proposal is assessed experimentally by means of a power-law regression analysis.