On the Fairness and Efficiency in Nonprofit Operations : Dynamic Resource Allocations

We study a sequential resource allocation problem balancing fairness and efficiency for nonprofit operations. (Un)fairness is measured by the expected maximum demand shortfall among all communities, and (in)efficiency is measured by the expected remaining resources after allocation. We prove that the optimal allocation policy for each community is a switching policy, and characterized by a spoon-shaped function in terms of the maximum demand shortfall in previous communities. We also prove that the optimal allocation quantities are increasing in terms of the remaining resource level and realized demand in the current community with bounded sensitivities. On the basis of these results, we propose a simple heuristic policy and numerically show that it performs quite well. Numerical results show that a little weight on fairness in the objective could improve the system's fairness significantly with a small efficiency cost. We also conduct numerical experiments to provide further explanations of the trade-off between fairness and efficiency and insights into the initial resource level decision. Our analysis can be fully extended to fill-rate based fairness metric to characterize the optimal allocation policy