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 characterize the optimal allocation policy as a two-threshold policy in which the optimal allocation quantities are spoon-shaped in terms of the current maximum demand shortfall. We further show that the thresholds and optimal allocation quantity for each community are nondecreasing in resource levels, realized demand from the current community, and weight of the efficiency objective. Based on these results, we propose a simple heuristic policy and numerically show that it performs well and generates fair allocations in a stochastic majorization order. The numerical results show that adding a small weight to the fairness objective significantly improves the system's fairness at a small efficiency cost. Moreover, the optimal initial capacity level is increasing (decreasing) in demand variance if the efficiency weight is small (large). Our theoretical analysis can be extended to the fill rate-based fairness metric.