For medical image segmentation, most fully convolutional networks (FCNs) need strong supervision through a large sample of high-quality dense segmentations, which is taxing in terms of costs, time and logistics involved. This burden of annotation can be alleviated by exploiting weak inexpensive annotations such as bounding boxes and anatomical landmarks. However, it is very difficult to \textit{a priori} estimate the optimal balance between the number of annotations needed for each supervision type that leads to maximum performance with the least annotation cost. To optimize this cost-performance trade off, we present a budget-based cost-minimization framework in a mixed-supervision setting via dense segmentations, bounding boxes, and landmarks. We propose a linear programming (LP) formulation combined with uncertainty and similarity based ranking strategy to judiciously select samples to be annotated next for optimal performance. In the results section, we show that our proposed method achieves comparable performance to state-of-the-art approaches with significantly reduced cost of annotations.