Resource Allocation with Sigmoidal Demands: Mobile Healthcare Units and Service Adoption

Problem definition: Achieving broad access to health services (a target within the sustainable development goals) requires reaching rural populations. Mobile healthcare units (MHUs) visit remote sites to offer health services to these populations. However, limited exposure, health literacy, and trust can lead to sigmoidal (S-shaped) adoption dynamics, presenting a difficult obstacle in allocating limited MHU resources. It is tempting to allocate resources in line with current demand, as seen in practice. However, to maximize access in the long term, this may be far from optimal, and insights into allocation decisions are limited. Academic/practical relevance: We present a formal model of the long-term allocation of MHU resources as the optimization of a sum of sigmoidal functions. We develop insights into optimal allocation decisions and propose pragmatic methods for estimating our model’s parameters from data available in practice. We demonstrate the potential of our approach by applying our methods to family planning MHUs in Uganda. Methodology: Nonlinear optimization of sigmoidal functions and machine learning, especially gradient boosting, are used. Results: Although the problem is NP-hard, we provide closed form solutions to particular cases of the model that elucidate insights into the optimal allocation. Operationalizable heuristic allocations, grounded in these insights, outperform allocations based on current demand. Our estimation approach, designed for interpretability, achieves better predictions than standard methods in the application. Managerial implications: Incorporating the future evolution of demand, driven by community interaction and saturation effects, is key to maximizing access with limited resources. Instead of proportionally assigning more visits to sites with high current demand, a group of sites should be prioritized. Optimal allocation among prioritized sites aims at equalizing demand at the end of the planning horizon. Therefore, more visits should generally be allocated to sites where the cumulative demand potential is higher and counterintuitively, often those where demand is currently lower.