In many low and middle income countries, including Zambia, stockouts of life-saving medicines threaten the advancement of the Sustainable Development Goals (SDGs); it is therefore vital to reduce stockouts through the use of improved inventory control policies. The associated medicine distribution problem is challenging because it involves seasonality and uncertainty in both demand and lead-times, heterogeneous delivery locations, and lost demand. Besides service level and inventory costs, equity across delivery locations must also be considered This empirical study is based on an independently-validated simulation model constructed from extensive field data, and addresses the lack of rigorous recommendations of inventory policies in this context. It compares the current base-stock and other policies proposed in the practitioner's literature with an optimization-based policy adapted from research on industrial settings. Although the optimization-based policy may need more implementation efforts, it generally outperforms the other evaluated policies. The authors' results also suggest that the prevalent proportional inventory rationing rules may lead to substantial service level discrepancies between facilities. Finally, the performance metrics of service level and distribution equity can be at odds, prompting non-trivial design trade-offs and considerations. This work motivated the development of a digital distribution information system involving smartphones with barcode scanners deployed in 60 health centers, posts and district hospitals in Zambia until 2018.