Mathematical models are presented and analyzed to facilitate a food bank's equitable and effective distribution of donated food among the population at risk for hunger. Typically exceeding the donated supply, demand is proportional to the poverty population within the food bank's service area. The food bank seeks to ensure a perfectly equitable distribution of food, i.e., each county in the service area should receive a food allocation that is exactly proportional to the county's demand such that no county is at a disadvantage compared to any other county. This objective often conflicts with the goal of maximizing effectiveness by minimizing the amount of undistributed food. Deterministic network-flow models are developed to minimize the amount of undistributed food while maintaining a user-specified upper bound on the absolute deviation of each county from a perfectly equitable distribution. An extension of this model identifies optimal policies for the allocation of additional receiving capacity to counties in the service area. A numerical study using data from a large North Carolina food bank illustrates the uses of the models.A probabilistic sensitivity analysis reveals the effect on the models' optimal solutions arising from uncertainty in the receiving capacities of the counties in the service area.
The United Nations Sustainable Development Goals provide a road map for countries to achieve peace and prosperity. In this study, we address two of these sustainable development goals: achieving food security and reducing inequalities. Food banks are nonprofit organizations that collect and distribute food donations to food‐insecure populations in their service regions. Food banks consider three criteria while distributing the donated food: equity, effectiveness, and efficiency. The equity criterion aims to distribute food in proportion to the food‐insecure households in a food bank's service area. The effectiveness criterion aims to minimize undistributed food, whereas the efficiency criterion minimizes the total cost of transportation. Models that assume predetermined weights on these criteria may produce inaccurate results as the preference of food banks over these criteria may vary over time, and as a function of supply and demand. In collaboration with our food bank partner in North Carolina, we develop a single‐period, weighted multi‐criteria optimization model that provides the decision‐maker the flexibility to capture their preferences over the three criteria of equity, effectiveness, and efficiency, and explore the resulting trade‐offs. We then introduce a novel algorithm that elicits the inherent preference of a food bank by analyzing its actions within a single‐period. The algorithm does not require direct interaction with the decision‐maker. The non‐interactive nature of this algorithm is especially significant for humanitarian organizations such as food banks which lack the resources to interact with modelers on a regular basis. We perform extensive numerical experiments to validate the efficiency of our algorithm. We illustrate results using historical data from our food bank partner and discuss managerial insights. We explore the implications of different decision‐maker preferences for the criteria on distribution policies.
The optimal objective function value of the aggregated problem provides a lower bound on the optimal objective of the disaggregated (original) problem, i.e., 𝑍 * ≥ 𝑍 ̃ * (Litvinchev & Tsurkov, 2003). To prove that 𝑍 * = 𝑍 ̃ * , we will show that the models SM-A and SM are equivalent, i.e.,each feasible solution 𝛹 ̃≡ {𝑋 ̃ * , 𝑌 ̃ * , 𝑊 ̃ * } to SM-A with objective 𝑍 ̃, there exists a corresponding feasible solution 𝛹 ≡ {𝑋 𝑗 * , 𝑌 𝑗 * , 𝑊 𝑗 * ; 𝑗 ∈ 𝐽} to SM with objective 𝑍 = 𝑍 ̃ and for each feasible solution 𝛹 to SM with objective 𝑍, there is a corresponding feasible solution 𝛹 ̃ to SM with objective 𝑍 ̃= 𝑍 (Cormen, Leiserson, Rivest and Stein, 2003). Throughout this proof, the "disaggregated solution" will refer to the solution obtained through ( 23) -( 25) and the "aggregated solution" to the solution obtained through 𝑋 ̃= ∑ 𝑋 𝑙 𝑛 𝑙=1 , 𝑌 ̃= ∑ 𝑌 𝑙 𝑛 𝑙=1
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