Inefficiency and inequity are two challenges that plague humanitarian operations and health delivery in resource-limited regions. Increasing capacity in humanitarian and health delivery supply chains is one option that has the potential to improve equity while maintaining efficiency. For example, the nonprofit organization Riders for Health has worked to increase capacity by providing reliable transportation to health workers in rural parts of sub-Saharan Africa; with more motorcycle hours at their disposal, health workers can perform more outreach to outlying communities. We develop a model using a family of fairness function to quantify the efficiency and equity of health delivery as capacity is increased via development programs. We present optimal resource allocations under utilitarian, proportionally fair, and egalitarian objectives and extend the model to include dual modes of transport and diminishing returns of subsequent outreach visits. Finally, we demonstrate how to apply our model at a regional level to provide support for humanitarian decision makers such as Riders for Health. We use data from the baseline phase of our evaluation trial of Riders for Health in Zambia to quantify efficiency and equity for one real-world scenario.