Modeling for the Equitable and Effective Distribution of Food Donations under Capacity Constraints

Food insecurity is an increasing threat to health and quality of life. In the United States, local food banks serve the population at risk for hunger to reduce food insecurity in their service area. We present and analyze several mathematical models to facilitate the equitable and effective distribution of donated food by a large local food bank among the population at risk for hunger. Demand typically exceeds the donated food supply, and is proportional to the poverty population within the service area. The food bank is required to distribute food donations in an equitable manner such that each person in poverty receives the same amount of food in each period. This objective conflicts with the goal of effectively distributing donated food by minimizing the amount of undistributed food.;We first develop deterministic network-flow models to minimize the amount of undistributed food while maintaining a user-specified upper bound on the deviation from perfect equity and derive closed-form optimal solutions. We then extend this model to obtain optimal policies for the allocation of additional receiving capacity to counties in the service area. These deterministic models show that locations with low capacity-to-demand ratios, bottlenecks, constrain the amount of food distributed owing to the equity requirement. Therefore, counties' capacities, which in practice are uncertain, strongly influence the optimal solution.;To address stochastic capacities, we develop a single-period model under which food distribution decisions are made before capacities at the receiving locations are known. After the capacities of the counties are observed, shipment decisions made at the beginning of the period can be corrected at additional cost. We prove that this model has a newsvendor-type closed-form optimal solution, which we use to develop a Myopic Heuristic for the multi-period problem. In the multi-period problem, supply is received at the beginning of each time period and shipments are made before observing the capacities for that time period. After capacities are observed, shipment decisions can be corrected by either shipping extra food from the branch to the counties or sending surplus food to waste from the counties. Any unshipped supply in the food bank is transferred to the following period as starting inventory. We use the structural properties of this problem to develop upper and lower bounds on the optimal shipment amounts and develop several heuristics that provide improvements on the Myopic Heuristic. An extensive numerical study demonstrates the promising performance of the heuristics.;Lastly, we develop a robust optimization model that allows the capacity parameters to vary within a range. We obtain conservative yet realistic solutions which focus on the capacity deviations at the bottleneck locations from their nominal values to achieve maximal influence on the objective. We also extend the deterministic food distribution model to obtain a different robust model that considers deviations in equity bounds while limiting the total level of inequity in the system. We develop algorithms to solve both of the robust models optimally and illustrate our results using historical data from our collaborating food bank.